Cost is central to effective decision making in all organizations: public, private, and quasi-public (organizations that are neither fully public nor completely private). It is the cement that holds together the different elements of an organization into a viable operating structure. For government in particular, it has become the principal guiding force that underlies a majority of decisions it makes, especially those related to the provision of public goods and services. To put it simply, without some knowledge of costs it will be impossible for a government to know what kinds of goods and services it can provide, the level of their provisions, and the length of time for which they will be provided. Even for those goods and services that arc considered meritorious, such as education and health care, cost remains the single-most overriding concern.
This chapter introduces several key concepts that are useful in a cost study, in particular how costs are defined, measured, analyzed, and evaluated. In presenting this introduction, the chapter identifies and distinguishes between different categories of costs; illustrates the contrast between cost, expense, and expenditure; explains the rationale for using different types of prices in government; discusses the relationship between cost and inflation, and suggests a number of cost principles.
The term cost ordinarily means the value of economic resources used in the production or delivery of a good or service. The use of the word value has a special meaning in cost calculation: it indicates the actual dollar amount spent on materials, labor, and other factors of production. Since a government does not produce goods for direct public consumption (with some exceptions, such as electricity, water, and a few others), the costs a government incurs in providing goods and services include the costs of acquisition, maintenance, and delivery.
Costs are generally expressed in cost units. Л cost unit is a unit of a good or service in relation to which costs arc ascertained. It is the basic umt of measurement used in all cost calculations. Examples of cost units in government will be kilowatt hours of clcctricity produced, gallons of water supplied, tons of garbage collected, and so on. But costs can also be expressed in terms other than cost units, such as costs incurred by an agency, a department, a function, a program, or an activity. These individual units of a government to which costs are frequently charged arc called cost centers. Theoretically a cost center could be anything as long as it is possible to attribute a cost to a responsible body unit.
When costs are charged to a cost center, it entails only those costs that belong to it and not to any other entity. On occasions, one may find two or more cost centers sharing a service responsibility and, therefore, the costs associated with it. These types of costs arc generally known as joint costs. A good example of a joint cost in government will be a public safety program that is jointly offered by a city police department and a county sheriffs office where both organizations share the responsibility as well as the cost of running the program. Cost sharing for joint provision is an old concept, but is becoming increasingly popular, especially for communities that are financially strapped.
However, there is no one particular type or form of cost that can uniformly apply to all government activities. Different types and variations of costs exist, depending on the purpose they serve or expected to serve. This section presents several different types of costs that are used in varying degrees in a government. They include current versus historical costs, replacement versus sunk costs, fixed versus variable costs, direct versus indirect costs, controllable versus uncontrollable costs, one-time versus life-cycle costs, and marginal versus average cost.
Current versus Historical Costs. Current costs are the economic cost or value of a good or service consumed at the moment. The term current implies that the cost one incurred for a good or service in the past has no bearing on consumption in the present. To give an example, consider an agency that is using a certain item that it purchased for $ 115 a year ago which will cost $ 120, if purchased today. The current cost of the item will then be S120, even though the agency paid $115 for it.
Historical costs, on the other hand, arc the exact opposite of current costs. They arc the costs incurred or the amount paid at the time of purchase. Thus, if the agency paid $115 for the item a year ago, the historical cost will be S115, regardless of how much it costs today.
Replacement versus Sunk Costs. Л replacement cost is the cost one would pay to replace an existing asset (defined as anything of value acquired), including the cost due to any loss of time when the asset remained unproductive or could not have been used. For instance, let us say that it cost a government $7,500 to replace an old desktop computer it bought some years ago with a state-of-the-art system. Assume that it cost the government an additional $ 1,500 for the time it lost when the system could not have been fully used. The replacement cost for the computer will, therefore, be 7,500+1,500 = $9,000. Replacement costs generally do not include the costs of repair and maintenance since they are considered part of normal wear and tear.
In contrast, a sunk cost can be defined as the cost of a good acquired in the past that has no resale or salvage value. Thus, if the computer has no resale value (assuming it has become obsolete), but it cost the government $7,500 when first purchased, the sunk cost will be the entire amount of $7,500. The reason it is called a sunk cost because it is a cost that has occurred in the past and past costs, once incurred, cannot be changed.
Fixed versus Variable Costs. Л cost is considered fixed if it does not change, regardless of the quantity of goods produced (or services provided); that is, it is independent of any output quantity and is equal to some constant dollar amount. This is true for all fixed costs, but only in the short trun. In the long run, all costs are variable meaning that they do not remain fixed over a long period of lime. Examples of fixed costs will include rent, interest payments on debt, and depreciation on structure and equipment.
A variable cost, on the other hand, changes in direct proportion to changes in the quantity produced. It is fixed per unit of output, but varies in total as output level changes. To give an example, consider a case where the demand for a certain item used by a government increased by 10 percent from 50 units last year to 55 units this year. Assume that the item costs $45 a piece. The total variable cost, which constitutes, say 80 percent of all costs, will also increase by 10 percent from $1,800 (50x45x0.80 = $ 1,800) to $ 1,980; that is, (1,800x0.10) + 1,800 = $ 1,980. Wages, materials, and all kinds of office supplies are typical examples of variable costs.
There are situations, however, where a cost is neither entirely fixed nor completely variable. These types of costs are called mixed costs. Mixed costs, also known as semi-variable costs, are a common occurrence in government. For instance, a public utility's billing rate may contain a flat (fixed) charge per month that is independent of consumption plus a rate per unit in excess of the fixed monthly rate.
Direct versus Indirect Costs. A cost is said to be direct if it can be unequivocally traced to a specific task or objective. Since direct costs can be unequivocally traced to a specific task or objective, they do not involve any cost sharing between cost centers, which makes it possible to calculate them with relative case compared with other types of costs. Examples of direct costs will include wages, salaries, fringe benefits (such as insurance premiums, retirement contribution, vacation pay, etc.), and other non-personnel costs (such as materials and supplies).
In contrast, an indirect cost cannot be unequivocally attributed to a specific task or objective, although the cost may be necessary for the operation of an organization. Indirect costs, which are also known as overheads, generally include costs associated with indirect labor, supplies, overtime, depreciation, and repair and maintenance.
Controllable versus Uncontrollable Costs. A cost is considered controllable if there exists a significant degree of influence on its incurrence; that is, for a cost to be controllable, an organization must have sufficient discretion over it. Wages, salaries, fringe benefits, and various nonpersonncl costs are good examples of controllable costs.
Uncontrollable costs, on the other hand, are costs over which an organization does not have any discretion. Interest payments on debt, contract obligations, and entitlements arc among the best-known examples of uncontrollable costs. They arc called uncontrollable because when an organization, in particular a government, makes a commitment to provide certain goods and services, guaranteed by legislative and other actions, it bccomes incumbent on the part of the government to make sure that they arc fully provided. In other words, it cannot avoid the costs associated with the goods and services for the duration of commitment without violating any legal or contractual obligations.
One-time versus Life-Cycle Costs. One-time costs can be defined as costs that do not involve any additional commitment of resources once a good or scrvice has been provided. In other words, there are no subsequent obligations, including repair and maintenance, except for the initial cost of procurement. Seasonal contracts, with no direct or indirect benefits, arc good examples of one-time cost.
In contrast, a life-cycle cost includes all costs that arc associated with the ownership of an asset, including acquisition, operation, and maintenance over its useful life. Costs associated with buildings, equipment, and scrvice vehicles are typical examples of life-cycle costs.
Marginal versus Average Costs. A marginal cost is the cost of producing an additional unit of a good or scrvice. It is called marginal because it is the cost of producing the last unit. To give an example, say it costs a government S250 to collect 10 tons of garbage from a residential neighborhood and $285 to collect 11 tons, the marginal cost of garbage collection for the 11 th ton will be $35, which is the difference between the cost of collecting the 10th and the 11th ton of garbage.
On the other hand, an average cost is obtained by dividing the total cost by the quantity produced. For instance, if it costs a government $300,000 a year to collcct 50,000 tons of garbage, the average cost of garbage collection for the year will be $6 a ton.
When several different types of costs are involved in cost calculations, it is quite likely that there will be some overlap or that some of the costs will appear in more than one category. For instance, a cost can be direct as well as fixed, or it can be current and uncontrollable. It is, therefore, not uncommon to have more than one type of cost in a given problem. In fact, most real-world problems involve more than one cost category, which can lead to occasional computational problems, such as counting a single cost several times over (multiple counting). But as long as one is able to keep track of the differences between various costs and their sources of origin, dealing with
multiple categories should not pose any problem in cost calculations.
COST, EXPENSE, AND EXPENDITURE
It is often neccssary to make a distinction between three very important terms that frequently appear in cost calculations: cost, expense, and expenditure. As noted earlier, a cost is the value of economic resources used in the production or delivery of a good or service. An expense, on the other hand, is an expired cost resulting from a productiv e use of an asset. An asset is defined as anything of value acquired. Since a majority of public services are provided for direct and immediate consumption by the public, they do not have any acquired value and, consequently, are not regarded as assets.
On the other hand, when assets are expended in the course of providing a good or servicc, such as preventing a crime or providing health care for the elderly, they benefit the public a government serves. As such, they serve a productive purpose that is nccessary for the welfare of a society. Consequently, they are considered an expense. If such consumptions do not result in any benefit, direct or indirect, they are considered a waste or loss. Finally, an expenditure can be defined as an unadjusted cost; that Is, a cost that does not reflcct any changes in the consumption of an asset in the course of providing a good or service.
Reconciliation between Cost, Expense, and Expenditure
There is a simple explanation why it is necessary to distnguish between costs, expense, and expenditure. When a government acquires an asset or purchases goods and services in one period, it does not always consume them in the same period, or it may consume them at different rates over time. This creates a lag in consumption, but also affects the way in which costs and expenditures are calculated.1 From an accounting point of view, the lag indicates the amount of adjustment that is necessary for an expenditure to eventually become a cost. If there is no time lag, i .e., if the goods purchased in one period are consumed in the same period, then expenditures and costs will be the same. But when they are not consumed in the same period, an adjustment is neccssary to convert them to costs in order to maintain consistency in cost calculations.
I,et us look at a simple example to illustrate how one makes these adjustments in cost calculations. Suppose the public works department of a local government spent $225,000 last year on street cleaning, including $5,000 on materials and supplies, $70,000 on a new vehicle, and the rest on wages and salaries. This is the total amount of expenditure the government incurred last year to provide the service. Since most vehicles have a life expectancy of several years, we can assume that the vehicle purchased will be used next year, the year after, and so on, although it may lose some of its use value from wear and tear. By the same token, we can assume that not all of the materials and supplies will be fully consumed in the year in which they were purchased and that a portion of these materials and supplies will be available for consumption next year or the year after. Thus, to determine the cost of street cleaning to the government, we will need to make some adjustments to the total expenditure by taking into account these changes in consumption.
The following describes a simple two-step procedure for converting expenditures into costs. First, subtract the cost of items, such as the cost of the vehicle as well as the cost of materials and supplies, from total expenditure for the year when they were purchased. The rationale for this is that since these items were not fully consumed in the year in which they were purchased, a portion of them would be available for consumption in the future. Thus, subtraction allows for the adjustments to be made for the portion of the resources that were not consumed during the accounting period. Second, add the depreciation cost of the vehicle as well as the cost of the portion of the materials and supplies consumed last year to total expenditure. The latter is necessary to account for the amount of resources that have already been expended. In other words, it represents the expense of the vehicle (expired cost) plus the materials and supplies consumed last year. The cost of street cleaning, therefore, is the residue, i.e., the balance after all the adjustments have been made to the total expenditure. This is shown in Table 1.1.
According to the table, the total cost of street cleaning after all the adjustments have been made amounts to $163,000, which is $62,000 less than the total expenditure for the year under consideration. Thus, we can formally define the cost of a good
Table 1.1
Cost and Expenditure Calculation
$225,000.00 70,000.00
5,000.00 10,000.00
3,000.00 $163,000.00
Street Cleaning: FY 20XX
Total Expenditure:
Minus: Vehicle purchase
Materials and supplies (M&S)
Plus: Depreciation on vehicle
Materials and supplies expended
Total Cost: or service as the total expenditure for a given year minus the value of the materials and supplies purchased in that year plus the value of the materials and supplies used up during the year. However, one must be careful in using this simple approach since it can complicate the conversion process when too many cost items are involved.
In the example presented above, we made an attempt to adjust the cost of street cleaning by depreciating the value of the vehicle for the year during which it was used. Without this adjustment, we would have had a different cost figure than the one we obtained. Depreciation, therefore, plays a critical role in all cost calculations, especially where assets are involved with several years of useful life. Depreciation occurs because of wear and tear as well as technological obsolescence. When an asset depreciates, it loses some or all of its use value because it is wearing out and, as such, fails to perform to its fullest capacity. On the other hand, an asset may be slow in wearing out, but may still lose its usefulness and in the process its value because of changes in technology. In other words, with advancements in technology it may become obsolete. Thus, both wear and tear and technological obsolcsccnce must be taken into account when calculating the depreciation of an asset.
There are different ways of depreciating an asset, but the method we used in the strect-clcaning example is the one that is frequently used in depreciation calculation. It is called the straight-line method. According to this method, depreciation is calculated by dividing the cost of an asset by its useful life. Assume that the vehicle has a useful life of seven years and no salvage value. Therefore, dividing the purchase price by 7 would produce a depreciation of S 10,000. This is the amount we subtracted in Table 1.1 from total expenditure of the vehicle for wear and tear for that year.
An important characteristic of the straight-line method is that the depreciation of an asset is uniformly distributed over its useful life, which means that the depreciation will be the same for each of the seven years of the useful life of the vehicle, regardless
of the nature of wear and tear. We can formally express this as d,= d2=.....= d^ where
d stands for the depreciation of an asset and the subscripts 1,2,3,.....,T represent the
number of years the asset will remain in use. Thus, the depreciation for any period t will be
where dj is the amount of depreciation in year t, V, is the initial value or purchase
price of the asset, and T is the useful life in years.
We made an assumption early in the example that there was no salvage value at the end of the useful life of the vehicle, but what if there is a salvage value? In that case, we will need to modify our Equation 1.1 by subtracting the salvage value from the initial value of the asset to take into account the contribution this value makes toward the depreciable value of the asset (defined as the difference between the cost to acquire an asset and its expected salvage value). This is shown in Equation 1.2 below:
d-
[1.2]
I мi
where VM, represents the salvage value, and the rest of the terms are the same as before.
Let us assume that the vehicle has a salvage value of $14,000. The depreciable value of the vehicle will then be $56,000. Using the straight-line method, the depreciation for any year will be $8,000 (or 14.286 percent) of the depreciable value, that is, [(70,000-14,000)/7] = $8,000 or [(8,000/56,000) x 100] = 14.286. This is the amount (percentage) by which we would have depreciated the vehiclc to obtain the cost of street cleaning, instead of the $10,000 we used, had we taken the salvage value into consideration.
Other Methods of Depreciation
Besides the straight-line method, there are other, more sophisticated, methods one can use for depreciation calculation. Two of these methods that deserve special attention here are the declining balance depreciation and the sum-of-thc-years'-digits depreciation Both methods belong to a category of depreciation, known as accelerated depreciation In an accclerated depreciation, depreciation changes (increases) with time but not at a constant rate as in the straight-line approach.
Declining Balance Depreciation. In a declining balancc depreciation, depreciation is treated as a fixed percentage of the book value of an asset. The book value of an asset is the difference between the original value of an asset and the accumulated depreciation over time. For instance, if the original value of an asset acquired at the beginning of a fiscal year is $1,500 and the accumulated depreciation at the end of the fiscal year is $250, the book value of the asset (at the end of the year) will be $1,250.
We can formally illustrate this with the help of a simple expression. Let Pf be a fixed percentage of the book value of an asset at the end of the first year of its useful life. The depreciation for the first year can then be expressed as d, = V,Pf
(1.3]
where d, is ihe depreciation in year 1, V, is the initial value of the asset, and Pf is a fixed percentage of the book value.
Suppose we now assign a value, say of 28.57 percent to Pf, which is twice the rate we used for the straight-line method. Note that we arrived at this rate by using a standard approach used in depreciation calculation, where the useful life of an asset, T, is divided into 2 to produce a limiting value of 'Г. Therefore, for an asset with a useful life of seven years, the fixed percentage rate or limiting value will be 2/T = 2/7 = 0.2857. 'ITiis means that the depreciation of the vehicle at the end of the first year in our street-cleaning example will be d, = V,Pf = 70,000(0.2857) = $ 19,999. However, the adjusted book value (ABV) for the vehicle at the end of the first year will be $50,001, which can be obtained from the following expression:
ABV,= V,- V,Pf [1.4]
= V,(1-Pf) = 70,000(1-0.2857) = $50,001
Similarly, the depreciation and its corresponding ABV for the second year will be $ 14,285.29 and $35,715.71, respectively. That is,
d2 = V,(l-Pf)Pf [1.5]
= 70,000(1-0.2857)(0.2857) = $14,285.29
ABV2 = ABV,- d2 [1.6]
= V,(l-Pf)-V1(l-Pf)Pf = V,(1-Pf)' = 70,000(1-0.2857)2 = $35,715.71
Note that we could have also obtained the same result by dircctly substituting the values of ABV, and dj into the expression for ABV for the second year, that is, ABV2 = ABV,- d, = 50,001.00-14,285.29 = $35,715.71. For the Tlh year, the values, respectively, will be
dT-V,(l-Pf)T1Pf
(1.7]
П-8]
,T
ABVT- V,(l-Pf)
As a general rule, the fixed percentage Pf is limited to 2/T, where T is the useful life of an asset in years. Any value less than 2/T may be used, but is not usually recommended becausc it may produce a rate of depreciation that will be slower than the rate produced by the straight-line approach. When Pf is set as equal to 2/T, the declining balance is often referred to as the double declining balance depreciation.
Sum-of-the-Years'-Digits Depreciation. The sum-of-thc-years'-digits depreciation, on the other hand, is a much slower version of an accelerated depreciation. I"he difference between this and the declining balance depreciation is that it depreciates against the depreciable value, rather than against the full value of an asset. The result produces a rate of depreciation that is much slower than the declining balance depreciation, but still higher than the straight-line depreciation.
The way it is calculated is that each year a fraction, called fractional depreciation, is multiplied by a depreciable value to obtain that year's depreciation. That is.
[1.10]
d
(1.9]
where dsYD is the sum-of-thc-years'-digits depreciation, df is the fractional depreciation, and vd is the depreciable value. The fractional depreciation, in turn, is obtained by dividing the remaining year in the useful life of an asset by the sum-of-the-ycars'-digits, as shown below:
df---
' SYD
where ф is the fractional depreciation, Tr is the year remaining in the useful life of an asset, and SYD is the sum-of-the-ycars'-digits.
In general, the numerator for fractional depreciation, which may vary from year to year, represents the number of years remaining in the useful life of an asset. The denominator, on the other hand, is the sum of the digits representing the useful life of the asset in years. Note that the denominator can be obained by using a simple expression
syd-Ш^ 2
where SYD is the sum-of-the-years'-digits, and T is the useful life in years.
We can use the same street cleaning example to illustrate this procedure. Since the vehicle in our example has a useful life of 7 years, the denominator of the fraction, i.e., the sum of the digits, will be (T(T+l)]/2 = [7(7+l)]/2 = 28.00. From this, we can write the fractional depreciation for each year as follows: Year 1: 7/28 = 0.250, Year 2:6/28 = 0.214; Year 3: 5/28 = 0.179; Year 4: 4/28 = 0.143; Year 5: 3/28 = 0.107; Year 6: 2/28 = 0.071; and Year 7: 1/28 = 0.036. Next, we multiply the initial value of the vehicle by its corresponding fractional depreciation to produce the depreciation for that year. This is shown in Table 1.2.
To see if we have calculated it correctly, we can add the depreciation for every single year of the useful life of the asset. If the sum of these depreciations equals the initial cost of the asset, the result should be considered correct. The results presented in the table clearly show that the sum of the depreciations for all seven years adds up to $70,000, which is the inital value of the vehicle, meaning that our calculation is correct. If, on the other hand, we had taken the salvage value of $14,000 into consideration, then the sum-of-thc-years'-digits depreciation would have been $56,000, obtained by multiplying the depreciable value of $56,000 for each of the seven years
[1.11]
Table 1.2
Sum-of-the-Years'-Digits Depreciation
Year | Purchase Price ($)* | SYD Fraction | Depreciation ($) |
1 | 70,000.00 | 0.250 | 17,500.00 |
2 | 70,000.00 | 0.214 | 14,980.00 |
3 | 70,000.00 | 0.179 | 12,530.00 |
4 | 70,000.00 | 0.143 | 10,010.00 |
5 | 70,000.00 | 0.107 | 7,490.00 |
6 | 70,000.00 | 0.071 | 4,970.00 |
7 | 70,000.00 | 0.036 | 2,520.00 |
Total: | 1.000 | 70,000.00 |
♦Assumes no salvage value
by their corresponding S YD fractions, then adding the results, as before.
Other Methods of Depreciation. In addition to the ones discussed above, there are a couple of other methods that also deserve some attention here because of their popularity with utility companies. First is the well-known sinking fund depreciation. Л sinking fund is a fund in which money is deposited at regular intervals to pay for the principal and interests on a debt. According to this method, the depreciation for any year is the increase for that year, which is the deposit for the year plus any interest accrued for that year. The method produces a yearly depreciation schedule, which increases systematically with time and hence is viewed as the opposite of accelerated depreciation.
The second method, which is occasionally used in depreciation calculation in the private sector but has potential for use in government, especially for utilities, is the units of production depreciation. According to this method, the depreciation is based on the output of an asset, rather than on its useful life. To find the depreciation, the total output of an asset is first calculated and then depreciation is taken for each year based on the fraction of the total output produced in that year. If estimated correctly, the method can accurately reflect the costs associated with an asset.
A term that frequently appears alongside cost is price. A price is simply a measure of value. It is the premium we pay for the goods and services we consume. But determining price for public goods and services is far more complex than it is for private goods and services because public goods and services contain certain characteristics that make them considerably more different from the goods and services in the private sector.2 Important among the characteristics that contribute to this difference are:
1. Indivisihleness. The)- are produced in lump-sum, not in increments, i.e., in small amounts. In other words, they must be produced as a whole. Examples of this will include roads, bridges, highways, schools, hospitals, public safety, and so on.
2. Nonexclusion. Once produced, it is difficult to excludc individuals from not consuming these goods and services, although there are examples (such as water, sewer, electricity, etc., i.e., services that behave more or less the same way as private goods and services), where it is possible to exclude individuals if they arc not willing to pay a price for the goods and services they consume.
3. Free Riding. Since individuals cannot be excluded, they can consume these goods and services without having to pay for them. For instance, individuals residing in one community can easily commutc to other communities, work there, and
enjoy their public facilities, such as libraries, museums, parks, and other services
without necessarily paying for them.
These characteristics, which make public goods and services different from private goods and services, also make it difficult for a government to determine the exact quantity of these goods and scrviccs it must provide and the price it must charge the public for their consumption. As a result, there is a tendency in government to overproduce them, bccausc to produce less would mean a decrease in the total quantity available for consumption along with a possible reduction in public welfare. This is not an attractive alternative for any government or its elected officials since it will reduce the welfare of the voting public who has the ultimate say in determining who gets elected.
The reduction will similarly affect the bureaucrats because with reduced supply the agency budget, along with the range of services the agcncies provide, will also diminish. None of these prospects are in the best interest of the bureaucrats or the elected officials.3 From a purely economic point of view, however, over-production is costly, wasteful, and can easily lead to obsolescence.
Since a government cannot determine the price of many of the goods and services it provides, especially those that must be produced in lump sum, it collects taxes from the public to pay for them Therefore, taxation is a form of price the public must pay for the goods and services it consumes. However, even for those goods and services for which the public must pay a price, it is not always possible to determine the exact nature of this price. As a result, one does not find one but several different types of prices simultaneously in use in a government.
This section discusses several different pricing systems that one is likely to come across in a government: total cost pricing, marginal cost pricing, cost-plus pricing, transfer pricing, and IcvcJizcd pricing. While it may not be so obvious as to why so many different types of prices exist in a government, it is important to understand the principles that underlie them.
When a government collects money, in particular taxes, the question it must address is how much tax should it collect. Since a government does not operate primarily to make profit, in principle, then it must collect an amount of tax revenue that is equal to the total cost of providing goods and services In other words, the total revenue collected from taxes and other sources must be at least equal to the total cost of goods and services a government provides. This is known as total cost pricing. To give an example, suppose that it costs a government $8.25 million a year to provide public education for elementary school children. Suppose also that to pay for this service, the government must collect taxes all of which must come from, say, property taxes. Therefore, it is the price the public must be willing to pay to avail the service.
In reality, the revenue collected from a specific source does not always equal the cost of providing a good or service. To compensate for this discrepancy, a government may raise revenue from alternative sources or it may find it necessary to transfer revenue from those funds where there is a surplus. This practicc of moving resources from one fund to another to compensate for revenue gap is known as interfund transfer.
However, the burden of taxation does not fall evenly on all taxpayers. Some pay more taxes than others, depending on the tax structure, the liabilities an individual taxpayer carries, and the jurisdiction where the individual lives. For instance, if all of the S8.25 million needed to pay for elementary school education must come from property taxes alone, the burden of taxation on an individual taxpayer will vary depending on where the taxpayer resides and the value of his or her property. The tax paid by an individual taxpayer represents his or her share of the total price for education, even though the individual may not directly benefit from it.
One of the most widely used concepts in price behavior is the marginal cost pricing. Marginal cost pricing is based on the microeconomic principle that resources are best allocated if the prices of goods and services arc set equal to their marginal cost; that is, the cost of producing the last unit. For instance, if it costs a firm S150 to produce 25 units of a certain good, say X, and $ 158 to produce 26 units of the same good, the marginal cost, i.e., the cost of the 26th unit will be $8. The marginal cost of $8 will then serve as the price at which the good will be sold in the market. If this were a public good, then the marginal cost would measure the cost of resources to society for producing an additional unit of the good and the price would measure the value of the additional unit to the public, i.e, the consumer.4
Marginal cost pricing works well in situations where there is perfect competition. Perfect competition forces firms to be efficient in their resource use. A firm that does not use its resources efficiently will fail to compete with firms that are more efficient and will ultimately be thrown out of the market. Thus, at the level where price (P) is equal to marginal cost (MC), a firm is considered most efficient and the price reflects this efficiency. However, for marginal cost pricing to work the goods produced must be divisible so that they can be measured in discrete units, such as bushels of wheat produced or millions of burgers served.
Although, as a concept, marginal cost pricing makes pcrfcct sense when applied to business, it docs not work so well when applied to government, in particular where public goods and services are concerned. There are several explanations for this. First, most public goods, with the exception of those provided as proprietary or enterprise goods, arc not divisible. As a result, the marginal concept does not readily apply to them. Second, the market is neither perfect nor competitive with some exceptions, where a government has to compctc with the private sector for certain goods and services, such as electricity, parking garage, golf course, etc. This means that inefficiency in resource use can get easily translated into the price the consumers pay for the goods and services they consume. Third, and most important, since there is no competition for many of the goods and services a government provides (such as public safety), it is often regarded as a monopolist, although the behavior of a government monopolist is not quite the same as that of a monopolist in the private sector.
Theoretically for a monopoly firm to make profit it has to sell more goods to discourage others from entering the market. But to sell more, it must lower the price which is not difficult for a monopoly firm bccausc it can achieve economies of scale at a much faster rate than small, competitive firms. From the point of view of consumers, however, this makes sense because at a lower pricc the consumers will have the incentive to buy more goods until they are exhausted from the market.
However, for a government monopolist, the picture is somewhat different. If it lowers the price, even though it will increase consumer savings, the consumers may not necessarily be willing to buy more because there is no additional utility they can derive from further consumption of the goods once the basic requirements have been satisfied. In other words, one cannot consume more water, take more rides on public transportation, visit hospitals more frequently, or play more golf even if the price is more affordable. Therefore, by lowering price to increase consumption in order to maximize return may not work the same way for a government as it does for a monopolist in the private scctor.
Cost-plus pricing is primarily an extension of total cost pricing. It is obtained by adding a predetermined margin to the total cost of a good or scrvice. The justification for using cost-plus pricing in government is quite simple: firms and businesses and, to a certain extent, all governments need a cushion to provide safeguards against uncertainty, especially where fixed costs arc conccrned. Cost-plus pricing provides that cushion or safety margin needed to ensure that these costs can be eventually recovered.
For a monopoly firm that provides more than one type of good, cost-plus pricing is the surest way to recover the fixed costs of the monopoly (from different types of consumers). To recover these costs, one must first determine a safety margin, then add it to the total cost. There arc different ways of determining this margin. For instance, it can be determined by trade custom, by service usage, or it can be tailored according to the financial needs of an organization.
To give an example, consider a case where a government which operates a local health clinic determines that the total cost of running the clinic next year will be $8.5 million. Suppose the government discovers that it will need a safety margin of 6.25 percent to provide safeguards against uncertainly, without which it may experience a net loss of revenue. The cost-plus price for the operation will, therefore, be $9.031 million, obtained from the following expression:
= 8,500,000 + (0.0625)(8,500,000) = $9,031,250
where CPP is the cost-plus price, TC is the total cost of operation, and M, is the safety margin.
Although cost-plus pricing is the surest way to recover the fixed cost of an operation, there is a problem with this system that largely stems from the difficulty in setting the safety margin. Theoretically the margin should be determined in such a way that it does not drive the consumers away. This is possible only in situations where demand is less price-elastic. In other words, an increase in price will not have any serious dampening cffcct on consumption. But if there are enough substitutes in the market from which the consumers can choose (an option not available to most public goods and services), then higher safety margins will have a dampening effect on the firm or the organization raising the price. However, in a monopoly situation in which substitute products do not offer much competition to the firm's own product, this logic does not quite apply.
A pricing concept that applies to both public and private goods is the transfer price. Transfer pricing takes place where there is a transfer of services between the various units of an organization. In other words, it is the price at which the units within an organization exchange or trade their services. Since the services provided or exchanged arc mostly internal to the operation of an organization, it is also known as internal service(s) price.
The simplest way to determine this pnee is to set it arbitrarily since there are no actual monetary transactions involved in these transfers and administer it from the top. But there is a problem with this approach in that it could lead to inefficiency in service transfer because an arbitrarily determined price docs not reflect the true value, i.e., the real worth of a service. A logical alternative, therefore, would be to use the market price, that is, the pricc a service would have commanded if it were sold in the market. However, the use of the market pricc depends on a couple of conditions: (1) the transfer price must not be higher than the market price to maintain its competitiveness, (2) the unit producing the service should have full acccss to the market, i.e., it should have the option to sell it outside. For instance, if the market pricc is higher than the price at which it can sell it to other units internally, then it should have the option to sell the service outside. There are, however, circumstances where this rule may not apply, especially if the services are restrictive in nature, such as those related to national security or defense.
Another alternative to the problem, in particular when one fails to use the market price or when the market price fails to reflect the competitive price, is to let the units negotiate among themselves and settle on a price that will be satisfactory to all the parties conccrned. It is called a negotiated price. The rationale for using negotiation as a tool to establish a price is that it will stimulate competition and also encourage noncompetitive (inefficient) units to be more competitive.
Lcvelizcd pricing is used mostly for public utilities, such as electricity. It is unique in some sense that, unlike other pricing systems, the price the public utilities chargc is often regulated by regulatory commissions. Consequently, utility companies as well as others who arc responsible for these services arc careful about what they includc in setting these prices. Since investments in utilities can cost large sums of money, utility companies are allowed a return on their investments. In most instances, they arc allowed to treat the interest payment on investment as well as other costs, such as depreciation, as expense and includc them in the price the consumers would pay for these services. The pnee, therefore, becomes a cost to the consumers that they must absorb to compensate for the interest and related costs to the producers.
The calculation of this price is quite simple. It is obtained by dividing the net revenue from a utility service for a given year by the amount of scrvicc to be provided for that year, and is given by the expression
where PLis the levclized price per unit of utility, 7t, is the net revenue in year t, and Q, is the quantity of goods or services produced for time t. Note that л is calculated by taking the difference between estimated revenue for year t and the estimated expense for the same year, including interest expense, depreciation, etc.
To give an example, consider a case where a government expects to earn $ 1.525 million in net revenue from its electric utility fund next year, based on an estimated production of 35 million kilowatts of electricity. The lcvelizcd price per kilowatt of electricity will be a little over 4 cents, obtained by dividing n, by Qt.This is the amount of cost the consumers must absorb for each unit of electricity they would consumc.
In addition to the ones discussed above, there are other types of prices that one is likely to come across in a government. Important among them are: break-even price (a price at which total revenue equals total cost); strategic price (a price used for a specific good or service in the short run, which may or may not be consistent with the long-run pricc objective of an organization); penalty pricc (a price that an individual or a firm has to pay for delinquent behavior, such as penalty for pollution, payment for delinquent taxes, etc.); and pricc subsidy (subsidies a firm or business receives from a government to compensate for the difference between a prevailing price in the market and what they would have earned under normal conditions). All of these prices are used in varying degrees depending on the condtions that call for their use.
Inflation is an increase in the general price level. Inflation affects both the consumer and the producer in the same way by lowering the value of money they spend on goods and services. From a cost point of view, inflation is a factor in how costs are measured, evaluated, and compared. Costs reflect the changes in inflation. Therefore, to get an accurate measure of costs, unaffected by inflation, one must take out the effects of inflation from costs. The simplest way to remove the effects of inflation from costs is to deflate it by a price index. A pricc index compares the average pricc of a group of commodities in one period with the average price for the same group in another period. As a general rule, priccs are determined first for a base period and the prices of all subsequent periods are then measured in relation to the base-year price.
The price index that is most commonly used in cost adjustment is the consumer pricc index (CPI), which is based on the price of a group of basic items required by an average family. The Bureau of Labor Statistics, which calculates the CPI, uses a formula based on an index, called the Laspcyrcs Index. The index is based on three sets of information: the average quantity of an item consumed in the base period, the price in the base period, and the price in the current period. It can be expressed as
[1.14]
where L stands for Laspeyres Index, Xq is the average quantity of cach item consumed by individuals (usually the families in the wage-earner group) in the base period, P0 is the pricc in the base period, and P, is the price in the current period.
Using this index as a basis, one can deflate the costs for any number of years. The following steps are typically used for this purpose: (1) list the cost figures for each year in chronological order, (2) list the price index corresponding to the costs for each year, (3) divide each cost figure by its associated price index; and (4) multiply the quotient by 100 to express the deflated cost in percentage terms.
We can formally express this as
DC--—- x 100
Price Index
where DC is the deflated cost.
Table 1.3 presents the street-cleaning example again to demonstrate the eflfccts of inflation on the costs of providing the service. According to the table, it cost the city $ 132,000 in year 1 to provide the service, $137,000 in year 2, $ 143,000 in year 3, and so on. However, when these costs were adjusted for inflation, they turned out to be much lower (as expected). What this means is that had there been no inflation, the real costs and the initial costs of the service would have been the same. In other words, the difference between the current costs of an operation and its real costs is the cost of inflation.
In recent years, governments, in particular the federal government, have been using the GNP Implicit Price Deflator instead of the traditional consumer price index for deflating costs. The GNP Implicit Price Deflator is an index that takes into account not only the changes in prices over time but also the changes in the quality of goods and services individuals consume.
When a cost is incurred by an organization, it is usually the result of one of two things: either due to a commitment made by the organization or owing to a third-party effect. A majority of costs are due to the commitments an organization makes and, as
[1.15]
Table 1.3
Costs of Street Cleaning (Adjusted)
Year | Costs ($) | Price Index | Deflated Costs ($) |
1 | 132,000 | 110 | 120,000.00 |
2 | 137,000 | 115 | 119,130.43 |
3 | 143,000 | 122 | 117,213.11 |
4 | 149,000 | 130 | 114,615.38 |
5 | 154,000 | 140 | 110,000.00 |
6 | 163,000 | 152 | 107,236.84 |
such, they cannot be avoided in most instances. This is particularly true for governments where costs result from policy decisions often with long-term consequences. Therefore, it is important that these decisions are made on a sound understanding of the costs that underlie them The following provides some basic guidelines that can be of great help in this regard:
There should be a direct relationship between a cost and the purpose it serves. This is the first rule of cost commitment. There must be a clearly defined objective for an activity for which costs will be incurred. Setting clear-cut objectives can not only help an organization identify the costs it will incur for specific activities but also facilitate its resource allocation, improve organizational performance, and increase accountability in the long run.
Costs should be charged only when they are incurred. Although most organizations are careful about it, occasional glitches can occur in an accounting system where a cost unit could be charged before a cost has been actually incurred. This can happen especially where recurring programs or activities are involved. Regular monitoring of accounting and related activities can easily correct this problem.
Costs should be properly segregated to avoid the problems of multiple counting. Serious problems may arise if costs are not properly traccd to their sources of origin Particularly in government, where revenues and expenditures arc required by law to be segregated by funds, any mix-up in costs can affect the fund balance which, in turn, can affect future financial management decisions of an organization. With good record-keeping practices, this can be easily avoided.
Past costs should be avoided in cost calculations. It is an axiom in cost studies that only those costs that arc relevant to a specific activity should be included in cost calculations. Past costs, while they may be significant, are not considered relevant cxccpt when they are used for forecasting purposes since they have already been incurred. I Iowcvcr, if a cost has been incurred, it can be included in cost calculation, provided that it still has some realizable values left that can be retained or replaced.
All cost transactions should be recorded on a double-entry accounting system, where possible. The double-entry accounting system has become a standard practice for most organizations, including government. But for cost accounting, a lot of organizations still use a single-entry accounting system. In a single-entry system, cost ledgers are prepared differently from the financial ledgers (as in fund accounting), yet the two use the same basic data. This creates a discrepancy in cost analysis, but it can be corrected with a double-entry accounting system.
Abnormal cost data should be separately treated from normal data. This is more of a statistical than an accounting problem, but it can easily influence the outcome lor the latter. From time to time organizations will encounter costs that arc clearly outside the realm of normal data. In statistics, they are callcd outliers. Their presence in normal data can distort figures and provide misleading information to the decision makers. They should not be totally thrown out of cost calculations but, like most outliers, they should be treated separately.
Data used in cost studies should be verified periodically for validity and reliability. Like any financial data, cost data should be verified on a regular basis to see if they arc corrcct and follow proper cost accounting procedures. This is known as cost auditing. Cost auditing ensures that the data used in accounting are correct and that cost units, cost centers, and cost accounts have been appropriately charged.
Cost statements should be presented fairly and accurately. Finally, all financial organizations, including government, are required to provide fair and full disclosure of all their transactions and changes in financial position, which can also be extended to include cost statements. Failure to accurately measure the costs and present them in a consistent manner can have serious impacts on organizational performance both in current and in future years. Inaccuracy can enter into cost statements when, for instance, contingencies are added to cost calculations. If contingencies arc neccssary, they should be maintained in a separate account and not merged with accounts for regular and normal activities.
This chapter has presented an overview of a number of cost concepts that arc useful in government, such as current versus historical cost, replacement versus sunk cost, fixed versus variable cost, and so on. The chapter has also made an attempt to distinguish among several interrelated concepts that frequently appear alongside cost, such as expense, expenditure, depreciation, and inflation. Some of these terms, in particular cost, expense, and expenditure arc often used interchangeably in government, which can create unnecessary problems in cost calculations. Therefore, the distinction is necessary to understand the basic differences that exist among these conccpts.
The chapter has also presented some broad discussions on why it is difficult to determine the price of goods and services in the public sector, in particular why it is necessar>' to have different pricing systems for the same government. Included in these discussions were total-cost pricing, marginal cost pricing, cost-plus pricing, transfer pricing, and levclized pricing. The chapter concluded with a brtief presentation of some basic principles that are useful in all cost studies.
Notes
1.R.C. Kory and P. Rosenberg, "Costing Municipal Services," in J. Matzer (ed.) Practical Financial Management. Washington, DC: International City Management Association, 1984: 50-62.
2. PA. Samuclson, "A Pure Theory of Public Expenditure," in R. Staff and F. Tannian (eds.) Externalities: Theoretical Dimensions of Political Economy. New York, NY: Duncllen, 1984: 89-92.
3. W.A. Niskanen, Jr., Bureaucracy and Representative Government. Chicago, IL: Aldinc-Altherton, 1971.
4. R A. Musgrave and P.B. Musgrave, Public Finance in Theory and Practice. New York, NY: McGraw-Hill, 1989.
A fundamental principle that guides the cost behavior of an organization is that there exists a direct relationship between cost and output. Although variables other than output, such as the price of factors of production, the structure of an organization, and the letter's ability to efficiently utilize the input resources may have a bearing on cost, they are usually held constant in most cost analysis, especially in the short run. In the long run, all costs and related factors of production are variable. Since organizations at different levels of production operate differently, it is difficult to determine a priori the time penod for short run. However, the time period for long run can be determined by the relationship between inputs and the production process. In general, the more capital intensive an operation, i.e., the more capital an organization uses in relation to labor, the longer it takes for the production proccss to change.
This chapter discusses three basic elements of cost bchvior that are important to understanding the nature of costs and how they affect the functioning of an organization: time frame of costs, general properties of cost functions, and cost estimation. Of these, cost estimation is particularly important for determining the nature of current as well as future costs of an organization.
Although it may be difficult to determine the exact time period that separates a short-run from a long-run cost, the conccpts nevertheless have important implications for decision making in an organization. For instance, the short-run is an operating concept. As an operating concept, it deals with the day-to-day operations of an organization. Thus, when an organization wants to deal with its routine activities (such as collecting garbage, supplying water, or fixing traffic lights), it relies on its short-run cost functions. On the other hand, the long run, which includes many short runs, is a planning concept and, as a planning conccpt, it deals with non-routine activities of an organization. These activities usually take a longer time to complete and often have impacts that extend over a long period of time. Thus, when an organization plans to make major changes in its operation, such as improving traffic congestions on major highways or upgrading the delivery system for health care, it relies on its long-run cost functions.
Since in the long run, all factors of production are variable, it is possible to make adjustments in the factors more easily in the long run than in the short run. To a large measure, this is due to the fact that once a production process has been set in motion, it is not possible for an organization to easily changc the usage of its input factors. Some inputs such as plant and equipment may have already been committed to the process, thus making it difficult for the organization to change these factors without affecting the efficiency of operation.
Short-Run Costs
As noted before, the short run is a period during which some of the input factors, such as machines, tools, and equipment remain fixed. Since they remain fixed, they produce a cost to the organization that is independent of the level of output. Two factors determine the cost behavior in the short run: the price of variable inputs such as labor, and the production function that underlies a production process. A production function shows the relationship between input and output based on the state of technology. If technology remains constant in the short run, it is likely that the production function will also remain constant in the short run.
To understand the cost structure of an organization and sec how it affects the decision making in the short run, it is necessary to become familiar with some of the cost terms we introduced earlier, in particular the average cost (AC), fixed cost (FC), variable cost (VC), marginal cost (MC), and total cost (TC). In reality, all of these costs can be directly obtained from observations on total cost. For instance, if we can write total cost as the sum of fixed and variable costs using the standard expression 'ГС - FC+VC, then we can obtain the average cost by dividing the total cost by the quantity of output, Q, such that AC = TC/Q = (FC/Q) + (VC/Q). The last two terms represent the average fixed and average variable costs, respectively.
Furthermore, if we assume Q to be discrete, then we can obtain the marginal cost by taking the ratio of changes in total cost to changes in output; that is, MC = Д ТС/Д Q. On the other hand, if it is continuous, then MC = dTC/dQ, which is the first derivative of the cost Junction. Table 2.1 presents a cost schedule showing the relationship between fixed, variable, total, average, and marginal costs based on cost data obtained for a hypothetical water department of a local government.
According to the table, it costs the department $20 in fixed cost and $3.50 in variable cost to produce the first thousand gallons of water. While the fixed cost remains the same for any amount of water produced, the variable cost increases as the amount produced increases. For instance, it costs almost 5 limes as much to produce
Tabic 2.1
Cost Schedule for Water Department
Quantity (Gallons) | FC ($) | VC (S) | TC ($) | AC ($) | MC (S) |
1,000 | 20 | 3.50 | 23.50 | 23.50 | - |
2,000 | 20 | 5.00 | 25.00 | 12.50 | 1.50 |
3,000 | 20 | 6.00 | 26.00 | 8.67 | 1.00 |
4,000 | 20 | 7.00 | 27.00 | 6.75 | 1.00 |
5,000 | 20 | 8.75 | 28.75 | 5.75 | 1.75 |
6,000 | 20 | 11.96 | 31.96 | 5.33 | 3.21 |
7,000 | 20 | 17.29 | 37.29 | 5.33 | 5.33 |
8,000 | 20 | 26.69 | 46.69 | 5.84 | 9.40 |
9,000 | 20 | 45.00 | 65.00 | 7.22 | 18.31 |
10,000 | 20 | 73.20 | 93.20 | 9.32 | 28.20 |
7,000 gallons of water or 13 times as much to produce 9,000 gallons of water as it does to produce 1,000 gallons of water. Since the fixed cost remains constant, the only factor that affects the total cost of production is the variable cost.
Elasticity of Cost We now introduce a term that frequently appears in discussions on cost behavior, known as elasticity. Elasticity measures the responsiveness of a variable (called the dependent variable) as a result of a changc in an another variable (called the independent variable). The responsiveness is measured by a coefficient, known as elasticity coefficient, e, that is determined by two interrelated factors: its sign and magnitude. The sign reflects the direction of movement between two variables, and the magnitude indicates the degree of responsiveness of the dependent variable to a changc in the independent variable.
As a general rule, if |e| = 1, it is called unit elastic, meaning that a one percent changc in the independent variable will lead to a one percent changc in the dependent variable. If |e|>l, it is called elastic, meaning that a 1 percent changc in the independent variable will lead to a more than 1 percent change in the dependent variable. If, on the other hand, |e| < 1, it is called inelastic, meaning that a 1 perccnt change in the independent variable will lead to a less than 1 percent change in the dependent variable.
We can formally present the elasticity concept to measure how responsive a cost function is to a change in output. The following expression can be used to measure the elasticity of cost:
ATC/TC
e° " AQ/Q [2.1]
where ec is the elasticity of cost, АТС is the change in total cost, and AQ is the change in quantity.
We can simplify the expression in liquation 2.1 by rewriting it as
ATC/TC MC [2.21
AQ/Q AC
where TC is the total cost, MC is the marginal cost, AC is the average cost, and Q is the quantity.
According to Equation 2.2, elasticity is simply a ratio of two costs: marginal and average. It means that if one has information on a cost function, in particular average and marginal costs, along with data on output quantity, one should be able to determine the change in total cost for a unit change in output.
To illustrate the conccpt, let us look at the water production problem again. For instance, at Q = 6,000, the elasticity of cost is 0.6; that is, ec = MC/AC = 3.21/5.33 = 0.6. What this means is that for a one pcrccnt change in output, total cost will change by 0.6 percent, i.e., cost is inelastic at Q = 6,000. By the same token, cost is elastic at higher levels of output. For instance, at Q = 9,000, the elasticity is 2.54; that is, 18.31/7.22 = 2.54, meaning that it changcs by a more than proportionate change in output. In conventional economic terms, it means that it will cost the department more at a higher level of output, especially when it exceeds 8,000 gallons of water, than it will when the output level is low. In other words, when the output reaches a level where the production cost increases at a faster rate it becomes necessary to make adjustments in the production process to bring the costs down, which then becomes a long-run cost problem.
Point versus Arc Elasticity. The elasticity conccpt discussed above is known as point elasticity becausc it represents incremental movements from point to point along a (cost) curvc. But there are situations where we may need to measure elasticity over a large segment of the curve. In that ease, we need to use what is commonly known as arc elasticity. The arc elasticity for a total cost function can be written as
в' ■[ ТС'-ТС» и Q'Q° ] [2.3]
0 (TCj.TC^ (QsQJ/2
where e'c is the arc elasticity, and the subscripts 1 and 0 are the new and the initial costs and quantities, respectively. The purpose of dividing the denominator by 2 is that it measures the average distance or the distance halfway between the designated end points of the arc.
To give an example, take two output quantities from the previous problem, Q0= 5,000 and Q,= 8,000 and two total costs corresponding to these outputs, TC0 = $28.75 and TC,= $46.69, then substitute these values into Equation 2.3. The resulting coefficient will be the arc elasticity, as shown below:
(TCrTC0)/2 (Qj.Q^/2 (TC/TCqXQ^QQ)
"(TCj.TCoXQrQo) (46.69-28.75X8*5) "(46.69*28.75X8-5)
(17.94X13) " (75.44X3) -1.03
The result indicates that the total cost is elastic over the range of the curvc designated by the end points of the arc, (TC0, Q0) and (TC„ Q,). To put it simply, it represents the average elasticity for the range of output between Q0 and Q„ i.e., between 8,000 and 5,000 gallons of water.
Both point and arc elasticities are two very good examples of how elasticity, as a concept, can be used to see how cost behaves in response to a change in the behavior of an independent variable. Although our example has been limited to cost behavior in the short run, it can be used in any situation to measure changes in any function, including long-run costs as long as there is a logical justification for including the variables in elasticity calculation.
A note of caution is appropriate here. When elasticity is used for long-run cost functions, the results may be less than precise because so many other factors may intervene in the relationship between a dependent and an independent variable. Since elasticity measures a one-to-one relationship, it cannot account for any intervention that may affect the reponsiveness of the variable whose elasticity is being measured. Long-Run Costs
As we noted earlier, the short-run cost is an operating conccpt, which means that at any given point in time an organization incurs a set of fixed costs for a level of operation over which it has no control. Therefore, any effort on the part of the organization to increase its efficiency must comc from a reduction or adjustment in its variable costs. Contrary to this, an organization dealing with long-run costs docs not have any such problem. Since all factors of production are variable in the long run, it is possible for the organization to select the most efficient (least cost) combination of factors at which its operation will be optimal.
An important implication of this behavior is that in the long run it is possible for organizations to have greater economics of scalc. Economies of scale occur in an organization when the average cost of production decreases as a result of improvements in the production proccss. But economies of scalc cannot be maintained throughout the process Eventually, costs will rise. The eventual rise in costs is due to what is known as diseconomies of scale, which occur oncc the level of operation becomes too large making it difficult for the decision makers to efficiently coordinate all the activities of an organization.
All cost functions have one characteristic in common: they all contain two basic parameters, one represent fixed costs and the other variable costs. Fixed costs, as we know, are costs that do not change in the short run over a given range of output, whereas variable costs change, within a given range, in direct proportion to changcs in output. Note that the notion of range is important in cost functions because it sets the limits within which one's definition of cost remains valid.
Since fixed costs are constants, they can be assumed away in a cost function. That is, if one would remove the parameter representing the fixed cost from a cost function, it would leave the function with variable costs only, meaning that fixed costs do not have any effect on the function. In other words, the fixed cost will be zero. We can explain this with the help of a simple linear cost function, where we define total cost as a function of an output quantity, Q, such that TC = f(Q). The linear equation corresponding to this function can be written as
TC = a+bQ [2.4]
where a is the parameter representing fixed costs, and b is the variable cost per unit of output, Q.
If, according to the points suggested above, the fixed costs do not have any effect on total cost, TC, Equation 2.4 will becomc
[2-5]
TC = bQ
meaning that that there are no fixed costs, i.e., a=0. This also means that if we can draw a line corresponding to this equation on a two-dimensional plane, one representing total cost and the other quantity, it will pass through the points of origin of the two axes.
Linear versus Nonlinear Cost Functions
For most organizations, it is possible to identify at least three types of cost ftinct-ions: linear, quadratic, and cubic. A linear cost function shows a straight-line relationship between cost and a set of independent variables that are directly related to it. Equation 2.4 is a typical example of a linear cost function with one independent variable.
Both quadratic and cubic functions, on the other hand, show a nonlinear relationship that changes in relation to the behavior of one or more independent variables, such as the rate of change in output, changes in direct labor, and so on. For instance, a typical quadratic function describing the cost behavior of an organization has the form, given by the expression
TC = a + bQ - cQ
,2
[2-61
where TC is the total cost, Q is the quantity, and a, b and с are the parameters representing fixed and variable costs, respectively.
Likewise, a typical cubic function describing the cost behavior of an organization can be written as
TC = a + bQ - cQ2 + dQ
,з
[2.7]
where the terms of the expression arc the same as before.
Nonlinear cost functions, by their very nature, are more complex than a linear (cost) function. As such, the cost terms one derives from these functions, in particular
average and marginal costs, are also more complex. As a general rule, the average cost of a linear or quadratic cost function is linear, but is quadratic for a cubic function. On the other hand, the marginal cost is constant for a linear (cost) function, whereas it is linear for a quadratic function and quadratic for a cubic function. Table 2.2 presents some of these properties of a linear, quadratic, and cubic function.
An advantage of using these functions (both linear and nonlinear) is that once we know the average and marginal costs, we can directly estimate the elasticity of costs from these functions without having to go through the formalities of Equation 2.1. Table 2.2 shows the elasticities associated with these functions. It should be worth noting, however, that although most cost functions in the real world are nonlinear, there is a specific advantage of using a linear cost ftinction. It is easier to estimate costs empirically with a linear function than with a nonlinear one. Nonlinear functions, such as cubic functions, often require theoretical generalizations, which make it difficult to fit these curves to empirical data.
Cost functions and their properties serve two important objectives in all cost studies: (1) they help us to understand the patterns of cost behavior over time, and (2) they provide us with the information necessary to estimate the current as well as the future cost of operation for an organization. Cost estimation is an important element of cost behavior. Without good estimates of cost, it will be difficult for an organization to effectively plan its current as well as future activities. There is a wide range of methods with varying degrees of sophistication and complexity that can be used for cost estimation. This section discusses three such methods, sclcctcd primarily for their
Table 2.2
Properties of Cost Functions
Basic Form | Average Cost | Marginal Cost | Elasticity of Cost |
TC=a+bQ | a/Q+b | b | b/(a/Q+b) |
TC=a+bQ-cQ2 | a/Q+b-cQ | b-2cQ | (b-2cQ)/ (a/Q+b-cQ) |
TC=a+bQ -cQ2+dQJ | a/Q+b-cQ+dQ2 | b- 2cQ+3dQ2 | (b-2cQ+3dQ2)/ (a/Q+b-cQ+dQ2) |
simplicity and ease of use: the graphical method, the high-low method, and the regression method.
The graphical method is used to show the relationship between two variables (a dependent and an independent) by fitting a visual line into a set of data on a two dimensional plane. Usually the cost data arc presented on the vertical axis and the sendee data on the horizontal axis. For instance, if we can assume that there is a linear relationship between total cost, which we will call Y, and the level of service activity, which we will call X, such that Y = a+ЬХ, then using the graphical method we can estimate the parameters a and b of this relationship by visually fitting a line to the plotted observations, called coordinates.
We can use a simple example to illustrate this. Suppose a local government wants to estimate the costs of garbage collection, in particular the fixed and variable costs associated with the service. 1лЛ us say that the government has collected data on garbage collection (X) and its corresponding costs (Y) for the past 12 months. Our objective is to find the value of a and b, given the information we have on the two variables, X and Y. Table 2.3 presents these data.
To estimate the values of a and b, we begin first by plotting the data on a graph, as shown in Figure 2.1. Next, we visually construct a line that would pass through the data points in such a way that some of the observations will lie above the line and some below it, thus giving it the appearance of a good fit. As the figure shows, the line intercepts the cost axis roughly at SI ,500. This is the fixed cost, a, of providing the service. The variable cost, b, is given by the slope of the line. We can obtain this slope by taking the ratio of two changes, one in total cost (Д Y) and the other in the quantity of garbage collected (ДХ), as shown below:
b . A! [2.8]
AX
57,500 -40,000 2,000-1,200
17,500 800
- 21.875
The result produces a value of $21.88 (after rounding off). This is the variable cost
Table 2.3
Cost and Output Data for Garbage Collection
Month | Amount of Garbage Collected (Tons) | Total Cost (S) |
January | 1,500 | 50,000 |
February | 1,300 | 45,000 |
March | 1,200 | 42,500 |
April | 1,000 | 37,500 |
May | 1,200 | 40,000 |
June | 1,600 | 47,500 |
July | 2,000 | 55,000 |
August | 1,800 | 52,000 |
September | 1,700 | 50,000 |
October | 1,800 | 50,000 |
November | 2,000 | 57,500 |
December | 2,200 | 60,000 |
of collccting a ton of garbage for the government. The High-Low Method
Unlike the graphical method, where one tries to visually fit a line to a set of sample observations, the high-low method uses two observations, one high and one low, to estimate the slope, b, and the vertical intercept, a. The advantage of using this method is that it utilizes a minimal amount of information to estimate these parameters compared with any other method, including the graphical method.
To illustrate how the method is used in practicc, let us return to the garbage collection example. Assume that there is a linear relationship between total cost (Y) and the quantity of garbage collected (X). In fact, the high-low method always assumes a linear relationship between a dependent and an independent variable. As before, our objective is to find the values of a and b. We use the following procedure to determine these values: let YHand YLbe the total cost at the selected high and low points. Similarly, let XH and XL be the activity level at the respective points.
Sincc we assumed a linear relationship between the two variables, we can set up the problem in terms of two linear equations, given by
Y, Gost(SOOO)
Change in Y
1-Г~П-1--1-r
О 2 4 6 8 10 12 14 16 18 20 22 24
Figure 2.1 Changes in Cost Behavior
-X, Quantity (00)
Y„ = a+bX„ YL = a+bXL
12.91 [2.10]
where a is the intercept, and b is the slope. The equations indicate that at both levels, YH and Yl, the total cost consists of a fixed component, a, and a variable component equal to the rate, b, times the level of activity.
We begin with the slope, the variable component of the total cost fucntion since we need to have this information in order to compute the intercept. To obtain the slope,
we simply subtract Equation 2.10 from Equation 2.9, rearrange the terms, and solve for b. That is,
Yh-Yl- bXH- bX
и
н
L
60,000-37,500 2,200-1,000 22,500 1,200
18.75
The result produces a slope of 18.75 or $18.75. This is how much it costs the government to collect each ton of garbage, which is about three dollars less than what we obtained for the graphical method.
The calculation of the intercept term, a, is rather straightforward oncc we know the value of the slope parameter, b. We can use any of the two equations (2.9-2.10) for this purpose. Equations 2.13-2.14, which are obtained from Equations 2.9-2.10, show us below how this inetercept can be estimated from these equations:
[2.13]
[i|
YH = a + bX,
н
YL = a + bX,
[ii]
L
.\a-YL-bX
[2.14]
L
Suppose now that we want to use Equation 2.13. Therefore, substituting the respective values of the terms b, YH, and XH into Equation 2.13, we obtain the intercept, i.e., the estimate of a. That is,
а-у1гЬХн
.60,000.(18.75X2,200)
-60,000-41,250
-18,750
which produces a value of $18,750. This is the fixed cost of garbage collcction for the government.
Having obtained the values of a and b, it is important to find out if these estimates, in particular the intercept, arc correct. Sincc it was obtained from the estimate of the slope parameter, b, one needs to make sure that the intercept is estimated correctly. To accomplish this, we simply substitute the observed values of b, YL, and XL into the equation that was not considered (which is Equation 2.14) and see if it produces the same result. If it does, the estimate should be considered correct. That is,
a = Yl- bXL = 37,500-0 8.75)(1,000) = 37,500- 18,750 = 18,750
which yields exactly the same result as the one obtained from Equation 2.13, meaning that it has been estimated correctly. We can now put all of this information together and present the estimated equation
as
Y- 18,750 ♦ 18.75 X [2.15]
The result indicates that the fixed cost of garbage collection for our government, regardless of the amount collccted, is SI 8,750. The variable cost, on the other hand, is $18.75, which means that for each additional ton of garbage collected, the total cost will increase by a constant amount equal to the variable cost of $18.75.
What appears to be an obvious advantage for the high-low method can also be a disadvantage. Since the method does not utilize all the information in a data set, the estimated equation may not be an accurate representation of the exact relationship between the variables under study. In other words, it may not be the perfect line that can be fitted through the data points. This is where the regression method becomes useful in that it makes full use of all the available information and, as such, provides a much better measure of accuracy of the estimated equation than the high-low method.
The Regression Method
The regression method is by far the best of the three methods discussed here for cost estimation. Besides being able to fully utilize the available information in a data set, there are a number of other advantages that make the regression method considerably superior to either the graphical or the high-low method. For instance, it is computionally simple and docs not have cxccssivc data requirements. Another advantage of the method is that its parameters have ccrtain optimal properties that make it possible to apply various tests of significance. These tests are primarily used to determine the reliability of the estimated parameters as well as of the estimated equation. Furthermore, it is comprehensive in analytical details and frequently serves as the foundation for other more advanced methods used in cost studies.
This section focuses on a simple regression model involving two variables: one dependent and one independent, as shown below:
Y = a+bX [2.16]
where Y is the dependent and X is the independent variable, and a and b are the respective parameters, i.e., the intercept and the slope of the model. As before, we assume a linear relationship between the variables (although problems with nonlinear relationships can be just as effectively dealt with using the regression method).
Steps in a Regression Method. 'Hie basic idea behind the regression method is quite simple. As with the graphical and the high-low method, the objcctivc is to construct a line that would best fit a set of sample observations. However, unlike the first two methods, the regression method is based on a number of mathematical conditions that ensure that the line is the best that could be drawn for a data set. Important among these conditions are:
1. Specification of the model. The regression model must be specified in terms of the relationship between a dependent and one or more independent (explanatory) variables. For instance, for a two variable case, the model can be specified as Y - f(X), where Y is the dependent and X is the independent variable. For an n-variable case, it will be Y = f(X„X2,X3,......,XJ.
2. Expressing the precise mathematical relationship between the variables. The simplest way to achieve this is to assume a linear relationship between the variables in a model so that one can write it in the familiar form Y = a + bX, for
a two-variable case, and Y = a + bX, + bX2 +.....+ bX„, for an n-variable case.
Where the relationship is nonlinear, one can always linearize it by using measures, such as taking logarithms on both sides of the equation, unless the relationship is inherently nonlinear, in which case it must be solved without the benefit of transformation.
3. Estimating the model. One must be able to estimate the values of the parameters, along with ail the relevant statistics and set up the estimated equation, once the model has been estimated.
4. Testing for the significance of the estimated parameters and the model as a whole. The model must be tested for statistical significance of the estimated parameters to ensure that it is a good fit and that the estimated parameters did not occur by chance.
As a matter of convention, the signs of the parameters in a regression model (the intercept and the slope) arc always presented in positive terms although, in reality, the values of the parameters can be positive, negative, or zero. In general, a positive slope indicates a positive relationship between X and Y, meaning that a change in X in certain direction will lead to a change in Y in the same direction; that is, if X goes up, so will Y, and vice versa. A negative slope, on the other hand, indicates an inverse relationship between X and Y, meaning that a change in X in certain direction will lead to a change in the opposite direction. That is, if X goes up, Y will go down, and vice versa. The slope, however, cannot be zero because it will mean a total redundance of the independent variable, in which case it should not have been included in the first place.
The intercept term, on the other hand, can be positive, negative, or zero. When it is positive, it means Y is positive when X is zero. When it is zero, it means Y is zero when X is zero. In other words, there is no intercept; that is, the line passes through the points of origin, where X and Y intersect at 0. Finally, when the intercept is negative, it means that Y has a value that is less than zero when X is zero, in which case it depends on the researcher whether to keep or ignore the negative portions of the equation.
The Error Term. When a linear equation is expressed as Y = a+ЬХ, it implies that the relationship between X and Y is exact; that is, all the variation in Y is due entirely to the variation in X If this were true, then all the data points on a two-dimensional plane would fall on a straight line. However, when observations come from the real world and are plotted on a graph, as we saw in our garbage collection example, one will most certainly observe that they will not fall on a straight line. In other words, there will be some deviations of the observations from the line, however small they may be.
There are several explanations why these deviations occur:
1. Omission of variables. We may have excluded from our regression equation those explanatory variables that would have played a significant role in explaining the variations in the dependent variable.
2. Random behavior of human beings. The scatter points around the (regression) line may be due to the unpredictable elements in human behavior.
3. Imperfect specification of the mathematical form of the model. We may have linearized a nonlinear relationship or have left out some equations from the model that should have been included. This happens, especially when one does not have a full understanding of the factors affecting the model relationship.
4. Errors in aggregation When specific data are not available, one is occasionally forced to use aggregate data When this happens, it produces errors in aggregation, meaning that when data are aggregated they often add magnitudes referring to individuals whose behaviors arc dissimilar.
5. Errors in measurement. This refers to the deviations of observations from the line. These types of deviations occur often due to the methods employed in collecting and processing empirical data. This is a common problem in social science, where data are mostly generated from primary sources.
Of the five sources of errors mentioned here, the first four arc called the errors of omission, while the fifth is called the error of measurement. In order to correct for these errors, the convention is to introduce an error term into the standard regression equation (2.16), so that the new equation will now appear as
Y = a+bX+c [2.17]
where с is the error term. Therefore, the true relationship that connects the variables in this equation consists of two components, one represented by the line, a+ЬХ, and the other by the error term, e.
The Least Squares Criterion. The linear equation we have just introduced, Y = a+bX+c, holds true for situations where one is dealing with population data but, in reality, we seldom deal with population data. Instead, we collect a sample of observations on X and Y, specify the distribution of the error term, and try to obtain a satisfactory estimate of the true value of the parameters of the relationship. As noted earlier, we do so by fitting a regression line through the observations of a sample, assuming that it would be a good approximation of the true line. Therefore, the true relationship between X and Y can be expressed as Y = a+bX+e, and the true regression line as Y = a+bX.
On the other hand, the estimated relationship can be expressed as
Y- a.bX.e [2.18]
and the estimated regression line as
Y- fUbX [2.19]
л л
where Y is the estimate of Y given a specific value of X, a is the estimate of the true
A A
intercept, a, b is the estimate of the true slope, b, and e is the estimate of the true value of the error term, c. Note that the symbol, л, represents the estimated value of the parameters as well as of the regressed (dependent) variable in the equation.
The weakness of this procedure is that we can obtain an infinite number of regression lines by assigning different values to the parameters a and b that will fit the data. But our objective in regression analysis is to choose a line that will best approximate the true line. Therefore, we need to use some guidelines or criteria by which we can determine this line. The criterion we generally use for this purpose is the least squares criterion. According to this criterion, the line selected must be such that it will produce the smallest possible error for the observations from the line, i.e., it will be optimal. In other words, selcct the line for which the sum of the squares of the deviations of the observations from the line is the minimum.
Estimating the Parameters. The least squares criterion serves a very important purpose in regression analysis in that it allows us to estimate the parameters directly from it. The only requirement is that they must be estimated in such a way that they are efficient, i.e., they have the minimum variance (errors). To accomplish this, let us recall Equation 2.18 for the two variable case
Y- a.bX.e
which, with slight reorganization, can be written as
a -Y-Y
Our objective is to find the estimates of a and b in such a way that the sum of the squares of the deviations, 2(e)2, will be minimum. That is,
Minimize £ ё2.£ (Y-i-bX)2 [2.21 ]
with respect to a and b.
Therefore, to obtain the estimates of a and b at which the sum of the squared deviations will be minimum, we take the partial derivatives of E(e)2 with respect to a and b, and set them equal to 0. That is,
6a 6a
2£ (Y-a-bX)(-l)-0 Y*2na*2b£ X-0
for a. Similarly,
б£ё2 б£ (Y-a-bX)2 Q 6b 6b
2£ (Y-a-bXX-X)-O
-2£ YX.2a£X.2b]TX2-0
for б.
We can simplify these expressions by dividing them through -2, so that we have
£ Y-na-b£ X-0 [2.22]
£ YX-a£X-b£X2-0 [2.23]
With slight rearrangements, Equations 2.22 and 2.23 can be written as
[2.24]
£ Y-na-bj: X
£ YX-a£ X*b£X2 [2.25]
The new equations are called the normal equations. From these two equations, we can obtain the values of a and б by solving them simultaneously. The results of the solution are presented below:
[2.261
n n
^nSXY-SXSY [22?]
nEX2-(EX)2
Note that the values of a and б in Equations 2.26 and 2.27 correspond to the points where the first derivatives of the squared errors taken with respect to a and б (Equation 2.21) were zero, i.e., the sum of the squared deviations was minimum.
However, it is possible to show that these coefficients can also be obtained by the following equations. That is,
[2.28]
a-Y-bX
. E (X-XXY-Y)
where the equalions arc expressed in terms of the deviations from the means of the variables, X and Y.(,)
To illustrate their use, we can now apply the equations to the garbage collcction problem. To be consistent with our model, wc will call the total cost of garbage collection our dependent variable, Y, and the amount of garbage collccted for a given month our independent variable, X. Table 2.4 presents a set of summary measures that were computed for the data to help us estimate the parameters of the model and other relevant statistics.
Using the appropriate summary data from the table and substituting them into the respective terms in Equations 2.28 and 2.29, wc obtain the estimates of a and b, as shown below:
♦ E (X-XXY-Y)
E (x-x)2
27,508,333.33 " 1,549,166.68
-17.757
and
a-Y-bX
(,)To soc how these estimates are obtained, we can do the following: First, set the deviations of X and Y so that
у - Y-Y [21
[2.29]
Then, bring in the first of the two normal equations, Equation 2.21, and divide it through by N to obtain
[31
Y-a.bX
X-1,608.33 ; Y-48,916.67 2(X 2).32,590 ,000 .00 E(X-X)2-1,549,166.67 £(Y-Y)2-514 ,916,666.73 E(X-XXY-Y)-27,508 ,333 .33 E(Y-Y)2-26,455 ,087.80 E(Y-Y)2-488 ,449,950.40
oe = 2,645,508.80
-48,916 .67 -(17.757 XI ,608 .333 ) -20,357.11
As before, we can put all of this information together and present the estimated equation of the egression line as
(continued)
Equation (3) indicates that the means of both X and Y lie on the regression line. Next, we bring in Equation 2.17 and subtract Equation (3J from it, so that
a-20,357.11 b-17.76
Table 2.4
Summary Data for the Regression Method
Summary Data
Estimated Parameters
y-bx-e [7]
From this, we can obtain the function of the variable b (to be minimized) by squaring and
Y-20,357.11 ♦ 17.757 X
liquation 2.30 tells us is that, given the information we have on the problem, the estimated values of a and ft, respectively, are: $20,357.11 and $17,757, meaning that it will cost the government $17.76 in variable cost to collect each ton of garbage plus $20,357.11 in fixed costs, regardless of the amount of garbage collected. Interestingly, the results compare well with those we obtained earlier for a and b from the high-low method.
Significance Tests for the Estimated Parameters. So far our discussion has concentrated on the procedure for estimating the numerical values of the parameters of the regression equation. But wc do not know how good these estimates are. Therefore, to establish the goodness (reliability) of these estimates we need to conduct a statistical test, called the t test (used mostly for small samples involving 30 or fewer cases).
The purpose of this test is to determine whether the estimates of a and ft arc significantly different from zero. In other words, whether the sample from which they have been estimated could have come from a population whose true parameters are zero and, if so, what the chances are of that happening. What this means is that if the test results could show, with some degree of ccrtainty, that they did not come from the other population whose true parameters are zero, one should be able to consider the estimates to be statistically significant (acceptable), provided that everything else remains the same.
[2.30]
In general, the t values are computed by taking the ratio of the estimated parameters to their corresponding standard errors. That is,
(continued)
summing over N, such that
Finally, we take the derivative of this function (left side of Equation 7) with respect to b, and set it equal to 0 to obtain the estimate of b. That is,
db
Exy-b2xl-0 2xy-bExJ
[2.31] [2-32]
t(a).
S(a)
t(b>-
S(b)
where S is the standard error, and the rest of the terms are the same as before.
The standard errors are essentially the standard deviations of the estimated parameters and are given by the square roots of their variances, as shown below:
S(a)-v/VAR (a)
n£ (X-X)2
[2.33]
\
for S(a), and
S(b)-v/vAR (b)
E (x-x)2
(continued)
Now, substituting this (result) into Equation [3] would produce the estimate of a. That is,
i-Y-ЬХ I'0]
which means that once we have estimated the value of b, we can obtain the value of the intercept, a, without any difficulty.
[2.34]
for S(6), where о2 = Se2/(n-k), к is the number of parameters in the regression equation, and n is the sample size or the number of observations in a data set. The e2 is called the error variance, and is given by the expression (Y-Y)2; that is, e2 = (Y-(а+ВХ)]2= (Y-Y)2.
Now, to obtain the t values for our estimated parameters we simply substitute the values from Table 2.3 into the respective terms in Equations 2.31 -2.32, so that
/[(o2eXEX2)]/[ n£(X-X)2]
__20,357.11_
/[(2^645 ,508 .80X32,590,000 .00)7(12X1,549,166 .68)
20,357.11 20,357.11 ^4,637 ,823 .40 2,153.56
-9.453
is the observed t corresponding to a, and
/oVE (x-x)2
_17.757_
v/(2,645 ,508 .80)/(l ,549,166 .68)
17.757 17.757 "vШ* ^-307
-13.586
is the observed t corresponding to ft.
Tlie estimated equation in its complete form, including the t values, can, therefore, be written as
Y-20,357.11 ♦ 17.757 X (9.453) (13.586)
where the values in parentheses indicate the t values.
From this compact form, wc can test the significance of the estimated parameters a and б by comparing the observed t values against their theoretical (critical) values, which can be found in any standard t table. The rule for this comparison is simple: if the observed t is positive and greater than the critical value of t (for a two-tailed test) at a given level ofp, called the p value, it is considered statistically significant at that level. The opposite is true, when it is negative.
Since our observed t is greater than the critical value of t at the 0.05 level of p (with n-2 =12-2=10 degrees of freedom), i.e., t(2) = 9.453 > t(a) = 2.228, our estimated parameter, a, is statistically significant. Similarly, since t(6) = 13.586 > t(b)= 2.228, then for the same level of p and the degrees of freedom, the estimated parameter, 6, is also statistically significant. What this means is that there is only 5 percent chance that these results could have occurred by chance, i.e., due to error.
Confidence Intervals for Estimated Parameters. To say that our estimates are statistically significant does not mean that they are correct estimates of the true population parameters, a and b. It simply means that our estimates have come from a sample drawn from a population whose parameters are different from zero. Therefore, in order to determine how close the estimated parameters are to the true population parameters, we need to construct a confidcnce interval for these parameters. In other words, we need to establish, with a certain degree of confidence, two limiting values, one high (callcd the upper limit of the interval) and one low (called the lower limit of the interval), within which we can expect to find the true value of the population parameter, callcd 0.
The procedure for setting up a confidence interval is as follows:
c[e-S(e)te/2<o<e*S(6)te/2Hi-a) I2 36]
where С is called the confidencc statement, (1 -a) is the degree of confidcnce, 0 is the estimated value of the parameter 0, S(0) is the standard error of the estimate, and ta/2 is the critical t value (for two-tails) corresponding to a given level of a (probability of obtaining an observed statistic by error) and degrees of freedom.
Using the expression in Equation 2.36, we can construct the confidence intervals for our estimated parameters a and b, say at the 99 percent level of confidence. That
[2.35]
С [a-S(a)ta/2<a<a-S(a)te/2]-(l -а)
С[20,357.11-(2,153.56X3.17)<a<20,357.1Ь(2,153.56 Х3.17>0.99 С[20,357 .11 -6,826 .79<а<20,357 .11.6,826 .79J-0.99 С(13,530.32<а<27,183 .901-0.99
fora.
Similarly,
С [b-S(b)te/2<b<b»S(b)te/2]-( 1 - о)
С[17.76-(1.31(3.17)<6<17.76.(1.31 Х3.17)]-0.99
С[17.76-4.15<&<17.76.4.15]-0.99
С [13.61 <Ь<21.91 J-0.99
for б'.
'1Ъе results indicate that at the 99 percent level of confidence, the true parameters of the estimated equations, a and b, will lie between SI 3,530.32 and $27,183.90 for a, and between S13.61 and S21.91 for b. In other words, we arc 99 percent certain that the true parameters of the estimated equations will lie between these intervals. This is how sure we are about the reliability of our estimates. As a general rule, confidence intervals established at less than 90 percent level are not considered statistically significant (acceptable).
Goodness of Fit of the Regression Line. Once we have estimated the values of the parameters and set up the equation of the regression line, we need to determine how good the fit is to the sample observations, i.e., how closely the line fits the observed data points. In general, the closer the line to the observations, the belter the fit. However, to be able to precisely measure this fit we need to use a statistic, called the coefficient of determination, or simply R: The coefficient of determination measures the amount of variation in the dependent variable that is explained by the variations in the independent variables (assuming that there is more than one independent variable in the equation). In otlier words, it shows how well the observations fit around the regression line.
Since the total variation in a regression is equal to the variation explained by the regression line and the error term; that is, total variance = explained (regression) variance + error variance, the R2 can be computed by either of the following two expressions:
j.j £ 2 Explained Variance
Total Variance
e E (Y-Y)2
' £ (Y-Y)2 [2.37]
2 , Error Variance
- 1--
Total Variance
£ (Y-Y)2 f2'38!
[•) R2
Now, substituting the respective values from Table 2.3 into the above expressions, we can easily obtain the R2 for our garbage-collection problem, as shown below:
S(Y-Y)2 E(Y-Y)2
488 ,449 ,950 .40 514,916,666.73
- 0.9486 « 0.95
r2«
E (Y-Y)2
26,455 ,087.80 514 ,916 ,666 .73
- 1-0.0514
- 0.9486 к0.95
The results indicate that approximately 95 percent of the variation in total costs is explained by the equation of the regression line, i.e., by the variation in the quantity of garbage collcctcd, which is quite high considering that R2 ranges between 0 and I. The remaining 5 percent of the variation remains unaccountcd for by the regression line and is attributed to the error term, e. Overall, it appears that our equation was a good fit.
Although a high R2 reflects the existence of a strong relationship between a dependent and a set of independent variables, it does not say anything about the significance of the coefficient itself. To ensure that our observed R2 is significant, we use another statistical test, called the F test. The F test tells us whether it makes sense, from a statistical point of view, to use the regression line to describe the data.
The F statistic, the basis of this test, is computed by taking the ratio of the variance due to the regression line, i.e., the variance of the independent variable(s) to the variance due to the error term, adjusted for appropriate degrees of freedom. That is,
P Explained Variance /(k-1) Error Variance /(n-k)
. S(Y-Y)2/(k-l) [239]
£(Y-Y)2/(n-k)
where к is the number of parameters (or variables in a regression equation, including the dependent variable), and n is the sample size.
According to the equation for F statistic, if the explained variance is the same as the error variance, the ratio will be equal to one, meaning that the regression line is not a good fit. If it is less than one, it is much worse and should be considered unacceptable. However, as the explained variance increases in magnitude in relation to the unexplained, the F statistic also increases in value. As a general rule, the higher the explained variance in relation to the unexplained, the better the fit. Table 2.4 presents a table, callcd the ANOVA (or F) table, to show how the F statistic was computed based on the information contained in Table 2.3.
л
To determine if the estimated equation, Y = 20.357.11+17.757X, is statistically significant, wc need to compare the observed F, as we did for the observed ts, against the critical value of F. Since our observed F of 184 .434, with 1 degree of freedom for
Table 2.4
Analysis of Variance (ANOVA) Table
Sources | Degrees | Mean | ||
of | of | Square, | ||
Variation | Sum of Squares | Freedom | MS=B/C | F-value |
[A] | IB] | [C] | P] | |
k-1 | ||||
Due to | E(Y-Y)2 | = 2-1 | =488,449, | |
Regression | -488 ,449 ,950 .40 | =1 | 950.40 | MSR/MSE |
Z(Y-Y)2 | n-k | -(488,449, | ||
Due to | = 12-2 | =2,645, | 950.40)/ | |
Error | -26,455 ,087.80 | =10 | 508.78 | (2,645,508. |
78) | ||||
-184.434 | ||||
Due to | E(Y-Y)2 | n-1 | ||
Data | -514,905 ,038.20 | =12-1 | =46,809, | |
(Total) | =11 | 548.93 |
the numerator and 10 degrees for the denominator, is greater than the critical value of F of 10.04 at the 0.01 level of p (which can be obtained from any standard F table), we can say with 99 percent level of certainty that our estimated equation is statistically significant. That is, there is only 1 percent chance that it could have occurred by chance, i.e., due to error.
To further determine if our observed F has been computed correctly, we can add the explained and the error variances together to see if they equal the total variance. If they do, we can say that it has been calculated correctly, which appears to be the case here exccpt for the rounding-oflf errors.
Second-Order Tests. All regression models are based on a set of conditions, called model assumptions. Violations of these assumptions produce results that arc often unreliable. There arc a number of tests that can be conducted to see if the observed estimates have violated the assumptions that underlie a regression model. Important among these assumptions arc normality, homoskedasticity, serial independence, and no multicollinearity. All the assumptions, with the exception of no multicollinearity, are based on the distribution of the error term The tests used for this purpose arc called the second-order tests.
The Assumption of Normality. To start with, the normality assumption means that the error term, e, is normally distributed with a zero mean and constant variance, о 2C. That is,
e ~N(0,o2)
where N indicates the normality of the distribution.
The assumption of normality is necessary for two reasons: (1) to conduct the significance tests on the estimated parameters, and (2) to set up the confidence intervals. What this means is that when this assumption is violated, one cannot assess the statistical reliability of the tests of significance or the confidence intervals. For most statistical analysis, however, normality is assured as long as the sample size is large, usually 30 or more (based on a theorem in statistics, called the Central Limit Theorem).
The Assumption of Homoskedasticity. The homoskcdasticity assumption is based on the notion that the variance of the error term(s) is the same for all the values of the explanatory variables. That is,
[2.41]
Var(e).[E(ei)-E(e1)]2-E(ei)2-oc2
where the term К represents the expected value, ej is the error term for the /th observation, and the rest of the terms arc the same as before.
In general, when there is no homoskedasticity (a condition known as hctcro-skedasticity), it means that the estimated parameters do not have minimum variance (a violation of the condition for OLS). In other words, they are inefficient although they may be unbiased (defined as the difference between the cxpccted value of an estimator and the true parameter being zero). Inefficiency in this case means that it is not possible to conduct the tests of significance and construct confidence intervals on the estimated parameters. One way to correct the problem is to transform the original model in such a way that the transformed model will have a constant variance.
(2.40)
The Assumption of Serial Independence. Serial independence means that the error terms of one period arc not related to the error terms of the preceding period. In statistical terms, this means that the covariance of the terms is zero. That is,
[2.42]
Cov(e|ej).E[ei-E(e1)](ej-E(ej)].0 where ej*ej.
When the assumption of serial independence is violated, it indicates that there is a serial correlation, called autocorrelation, among the error terms. The presence of autocorrelation suggests that the values of the estimated parameters for any single sample are not correct. In fact, they may be underestimated, but not biased. Underestimation also means that any predictions based on these estimates will be inefficient in that their variances will be much larger compared with predictions based on estimates obtained from alternative methods.
However, the problem can be corrected in most instances with the help of a process known as first differencing, according to which the original equation is transformed by subtracting the value of each period for a variable from the value of its preceding period. This allows for removal of any effect that the error terms in the previous period may have on the error terms in the current period.
The Assumption of No Muhicollinearity. Finally, the assumption of no multcolli-ncarity means that the explanatory variables are not perfectly (linearly) correlated, i.e., the correlation between any two explanatory variables is not equal to 1. That is,
V,"1 [2.43]
where r stands for correlation, and and x, are the corresponding explanatory variables.
When multicollincarity is present in a relationship, it indicates that the estimates of the parameters are indeterminate. In other words, their standard errors becomc infinitely large, making the estimated coefficients statistically insignificant. The simplest way to correct the problem is either to increase the size of the sample or to systematically eliminate the redundant variables from the regression model, using methods such as step-wise regression, indirect least square, two-stage least square, instrumental variables, and principal component analysis. The end result will be a model that is efficient and will include only those variables that will explain most of the variation in the dependent variable.
Although no specific tests were done to see if any of the assumptions were violated in the currcnt example, a cursor)' glance at the results will indicate that autocorrelation was not a problem in this case nor was there any evidence of hetcroskedasticity that could have affected the efficiency of the model. Since wc were dealing with only one explanatory variable, the problem of multicollincarity was never a question.
Extension of the Simple Linear Model
So far we have used only one independent variable (quantity of garbage collcctcd) to estimate the fixed and the variable cost of the servicc for the government. In reality, we do not have one but several different independent variables, such as dircct labor, indirect labor, direct machine hours, indirect machinc hours, weather conditions, and so on that could have some effects on the dependent variable. Theoretically we could have m number of independent variables, instead of one, in which case wc need to expand our simple linear model (with one independent variable) to incorporate multiple independent variables.
The general cost function with multiple independent variables can thus be written
as
+ ЬтХш + e [2.44]
Y= a + b,X, + bjX2 + bjX3 +
Equation 2.44 is a typical expression for a regression model with m-independent variables in linear form.
It should be pointed out that when we are dealing with a multiple regression model, it is not possible to use the same computational procedure we used for the simple regression model. It will involve far too many computations that will be difficult to solve by hand. The alternative is to use matrix algebra or a statistical software that can deal with multidimensional^ in a regression problem more efficiently.
Before one can obtain the estimates of the parameters of the general linear model in Equation 2.44 using matrix algebra, it is necessary to set up the model in a format that is suitable for matrix formulation. Note that our model has m independent variables and one dependent variable; that is, m+1 variables, and n observations. Since there are n observations in the model, we can present it in terms of n equations, one for each observation. That is,
Y,= a, + b,Xn + b,X2l +
+ ЬшХш1 + e, [245]
Y2= a2 + b,X12 + b^ +
+ ЬЮХт2 + «2 I2 46]
1 12
Y„= an + blXln + + ..................... + Ь»хшп + Cn [2.47]
We can put these equations in matrix form so that we have
[2.48]
where
1 X11 X22 1 x12 x22
ml
'm2
x .
1 X,n *2n
X.
[2.49]
Note that Y is an nxl column vector of observations (for Y), X is an nxm matrix of observations (for Xs), b is an mxl column vcctor of parameters, and e is an nxl column vector of error terms. Note also that the column of Is in X matrix stands for the constant intercept in Equations 2.45-2.47.
Now, to obtain the estimates of b we need to minimize the sum of the squares of the residuals, 2(e)2, similar to Equation 2.21. That is,
D
i. l
Note that e' is the transpose (where rows becomc columns and columns becomc rows) of the vector of residuals, i.e., the estimated error terms, e, where e can be written as
b -
t - Y-tf
For convenience, we can simplify Equation 2.51 by rewriting and expanding it as
ё = Y - Y
or Y = Xb + I - t
Next, we take the vector of the estimated error terms e in Equation 2.52 and substitute it into the right-hand side of Equation 2.50 for minimization such that
Minimize 6'ё = (Y - X6)'(Y - Xb)
= Y'Y - Х'6'Y - XSY' + X'f>'Xb « Y'Y - 2X'f>'Y + Х'б'Хб [2.53]
A
[2.51]
Equation 2.53 is the function that we finally need to minimize. Since b'X'Y is a scalar (constant), it is equal to its transpose, 6XYV As before, to minimize the sum of the squared residuals we take the partial derivative of e'e with respect to I>, and set it equal to 0. That is,
-2X'Y + 2X'Xb-0
Finally, we divide both sides of the derivation above by -2, and rearrange the terms by changing the sides to obtain the estimates of our regression parameters. That is,
X'y.X'Xb fi-CXtty^X /Y)
where (X'X)"1 is called the inverse0* of (X'X), and I) is the vector of estimated parameters, i.e., the estimated (slope) coefficients for the regression model in Equation 2.44.
Forecasting Future Costs
[2.54]
Oncc we have estimated the parameters of a cost function for an organization and tested for their statistical significance, we can further extend the discussion by estimating the future costs of operation for the organization. This would, however, require that we have some information on the future values of the independent variablc(s) of the cost function. To give an example, let us go back to the garbage-collection problem again. We begin with the estimated equation:
Y - 20,357.11*17.757X
Assume now that we have some information on garbage collection for the month of January, next year. Let us say that it is 1,650 tons. Our objective is to find the corresponding cost for the month of January. Assuming that there has not been any changc in the cost components of our estimated equation, the total cost for the month of January next year will be
YJtnuey - 20,357 .11 ♦ 17.757X
= 20,357.11+17.757(1,650) = 20,357.11+29,299.05 = 49,656.16
or S49.656.16, which is about S350 lower than what it cost the same month the previous year.
In doing the forecast for our example, we assumed that we had some knowledge of the future valuc(s) of the independent variable, e.g., the amount of garbage to be collected in January, next year. When the future valuc(s) of the independent variables in a forecasting model are known a priori, it is called an unconditional forecast In contrast, when they are not known beforehand, which is frequently the case, it is callcd a conditional forecast. Given a choicc, most forecasters would prefer unconditional forecasts to conditional forecasts. But it is not always possible to know with certainty the future values of the independent variables so that one can incorporate tins information into the regression model to estimate the future values of the dependent (forecast) variable. As a result, conditional forecasts are more commonly used than unconditional forecasts.
Although they arc more frequently used, conditional forecasts have a major weakness in that they have a tendency to add to forecast errors. A forecast error is the difference between an actual and a forecast value. The error occurs due to the fact that when one tries to forecast the values of a forecast variable for a set of independent variables, one must first estimate the future values of the independent variables before they can be incorporated into the model to forecast the future values of the dependent variable. Since the forecast values, which themselves are estimates, depend on the esUmated values of the independent variables, the likelihood of a forecast error is higher in conditional forecasts than it is for unconditional forecasts.
To put it simply, error in forecasting is a common problem that all forecasters must learn to deal with. Since the problem is inherent in all forecasts, the best one can do is try to minimize the error so that one can establish a greater confidence on the forecasts. Several procedures have been suggested over the years to improve the situation. This includes measures, such as correct specification of the model, variable transformation, use of lagged as opposed to current data, and constant updating of the forecast values as more recent and up-to-date information becomes available.
Another, perhaps more attractive, alternative is to use time-series models, such as simple moving average, exponential smoothing, or Box-Jenkins,1 which are built on less restnetive assumptions than most regression models. Also, the data requirements for many of these models, with the exception of the Box-Jenkins, are not high, meaning that they are relatively easy to use, especially in situations where data are not readily available. (For an at length discussion on timc-scrics forecasting, see Gilchrist,1 Bails and Peppers,3 Pindyck and Rubinfeld,4 Montgomery and Johnson,5 and Box and Jenkins.6)
This chapter has provided a general discussion of cost behavior for an organization that also applies to government. Cost behavior deals with the way in which costs respond to changes in one or more independent variables. Several aspects of cost behavior were discussed in the chapter, including cost behavior in the short run and long run, properties of cost functions, and cost estimation. In the short run, costs of input factors such as rent, interest payment on debt, and depreciation on structure are fixed. Since these input factors have to be provided regardless of the quantity produced, the payments for these factors of production remain constant. As such, they arc called fixed costs of production. In the long run, all inputs are variable meaning that all costs are variable. As a result, costs in the long run are treated more as a planning horizon than as an operating cost for an organization.
On the other hand, the basic idea behind cost estimation is to estimate the relationship between costs and the variables affecting these costs. This chapter has presented three of the most widely used cost estimation techniques, namely the graphical method, the high-low method, and the regression method. Of these, the regression method is considered more rigorous and analytically more sophisticated than either the graphical or the high-low method. The reasons for this is that (1) it utilizes all the available information in a data set, (2) the estimated parameters are more precisc than those obtained by the other two methods, and (3) the estimated parameterd can be tested for their statistical significance. Perhaps the greatest advantage of the regression method is that once the parameters have been estimated, they can be tested for (statistical) siginificancc. Not only that, they can be used to forecast the costs of providing services in the future.
1. A. Khan, "Forecasting A Local Government Budget with Time-Scries Analysis." State and Local Government Review. Vol. 21, Fall 1989: 123-129.
2. W. Gilchrist, Statistical Forecasting. New York, NY: John Wiley and Sons, 1976.
3. D.G. Bails and L.C. Peppers, Business Forecasting: Forecasting Techniques and Application. Englcwood-Cliflfs, NJ: Prentice-Hall, 1982.
4. R.S. Pindyck and D.L. Rubinfeld, Econometric Models and Economic Forecasts. New York, NY: McGraw-Hill, 1991.
5. D.C. Montgomery and L.A. Johnson, Forecasting Time-Series Analysis. New York, NY: McGraw-Hill Book Company, 1976.
6. G.E.P. Box and G.M. Jenkins, Time Series Analysis: Forecasting and Control. San Francisco, CA: Holden-Day, 1976.
An important financial consideration for firms and businesses in the private sector is to be able to determine the level of operation at which they will have maximum profit on their investments or earn just enough income to break even. A government does not operate to maximize profit, but it is just as important for a government to know in advance the level of operation at which a project or activity will be economically and financially viable so that it will not overproduce or undergo a revenue loss. However, to determine the level at which an operation will be viable, one must be able to measure the costs and returns associated with it in precise monetary terms. But for a majority of public goods and services, with the exception of those considered as proprietary or enterprise goods, it is difficult to measure the costs and returns in precise monetary terms bccausc of the characteristics that separate them from private goods and services. Nonetheless, where it is possible to measure the costs and returns with some degree of precision, efforts should be made to do so to ensure the viability of a government activity.
This chaptcr presents four simple yet useful tools of cost analysis that have received considerable attention in the literature on cost studies in reccnt years: breakeven analysis, differential cost analysis, benefit-cost analysis, and cost-cffectivcncss analysis. Of these, break-even analysis has been extensively used in business and bcnefit-cost analysis in government. In fact, all four techniques can be used in varying degrees to determine the level of operation at which a given project or activity will be economically and financially viable.
Developed originally by Rautcnstrausch in the late 1930s, break-even analysis is considered as one of the most popular techniques used in cost analysis.1 Its purpose is to integrate the cost, revenue, and output of an activity in order to determine the effect it will have on alternative courses of action. The basic objective is to find, for a given cost, a level of operation at which there will be neither gain nor loss; that is, it will break even. The fundamental rule that underlies this objective is that at the lowest level of an activity cost exceeds revenue producing a loss, but as the level of activity increases revenue will increase at a faster rate than cost resulting in a situation where the two would become equal producing a break-even point. If the level of activity continues to increase, revenue will eventually cxcccd cost, producing a net gain until a point comes when it will begin to diminish again.
This limdamental law of operation that serves as the basis for break-even analysis is universal, but for the law to be of any practical significance one must be able to demonstrate how this relationship works in reality. This section presents examples of linear as well as nonlinear break-even analysis to provide support to this basic tenet and, in particular, to illustrate the relationship between cost, revenue, and output.
We begin our discussion with a simple assumption that there is a linear relationship between cost, revenue, and output level of an activity. The linearity assumption sets up a mathematical relationship between cost (in particular, fixed and variable), revenue, and output that can be expressed in terms of a straight line, provided that the following conditions hold: (I) the variable cost is constant and hcncc is linearly dependent on output; (2) the fixed cost is independent of output; (3) there are no other financial costs; (4) output can be increased without significantly affecting the cost structure; and (5) all output units are charged or sold at the same price.
These conditions imply a constancy of rate scale that does not vary for different levels of output. What this means is that the price a government charges for a unit of service it provides or the cost it incurs in providing that unit of servicc, docs not change regardless of the quantity provided. Since the price does not change, the total revenue from an operation can be represented by a linear curve emanating from the point of origin (as in a two-dimensional plane). Likewise, since the per unit cost does not change, the total cost of operation can also be represented by a linear curve emanating from the point of origin. Figure 3.1 shows this relationship between total revenue, total cost, and break-even quantity.
According to the figure, since both revenue and cost curves are straight lines, there is only one break-even point, which occurs at Q*. It is the point at which total revenue (TR) equals total cost (TC). The shaded areas above and below the total cost line represent net gain and loss, respectively.
Break-Even Quantity, Pricc, and Revenue. The mathematical relationships underlying the cost and revenue functions described above provide a basis for developing a set of simple algebraic expressions that can be used to deal with various break-even problems, such as break-even quantity, break-even price, break-even revenue, etc. We begin with the simplest one, the break-even quantity, as shown below:
Cost and Revenue ($)
r,oin /TDvrf*\
Q* Quantity (Q)
Figure 3.1 A Linear Break-Even Chart
TR - pQ [3.1]
TC = FC + (VC)(Q) [3.2]
where p is the price per unit of output, Q is the total quantity, FC is the fixed cost, and VC is the per unit variable cost associated with that output.
From these two simple equations, one can obtain the break-even quantity, Q*. by setting TR = TC, since at the break-even point total revenue must always equal total cost. We can formally express this relationship as
TR = TC TR-TC = 0 KpQ*))-[FC+VC(Q*)]-0 ((pQ*)]-[VC(Q*)] = FC Q*(p-VC) = FC
л Q*= FC/(p-VC) [3.3]
With slight algebraic manipulation, Equation 3.3 yields the break-even price, p*, corresponding to Q*. That is,
Q* = FC/(p*-VC) p*Q* - VC(Q*) = FC
p*Q*= FC + VC(Q*)
л p*= (FC/Q*) + VC [3.4]
Sincc it is derived from Q*, at the break-even price, p*. total revenue must be the same as total cost.
We can use a simple example to illustrate how these relationships work. Suppose the number of citations given by the police department of a large metropolitan government for traffic violations is 100,000 a year. Suppose also that the fixed cost associated with this activity is $2 million, with a variable cost of $25 per citation. The department chargcs $50 across the board for each citation, regardless of the nature of infraction Assume that the department is working at 100 percent level of its capacity. Our objective is to do two things: find the break-even quantity for the department and determine the net revenue from operation at the current rate of 100,000 citations a year.
Let us start with the break-even quantity for the department first. Since wc already know the price one will pay for an infraction, we can determine the break-even quantity by directly applying Equation 3.3 to the problem. That is,
Q* = FC/(p*-VC) = 2,000,000/(50-25) = 80,000
which is 80,000 citations a year.
This is the number (of citations) the department must produce to break even. At this rate, there should not be any gain or loss for the department, meaning that its total revenue must be equal to total cost. To see if this is true for the current example, we can substitute the value of Q* into Equations 3.1 and 3.2, then set them equal to each other, so that
TR = p*Q* = (50)(80,000) = $4,000,000.00 TC = FC + (VC)(Q*)
= 2,000,000+ (25)(80,000) = $4,000,000.00 л TR = TC = $4,000,000
The result produces a sum of $4 million for both TR and TC, which means that at the break-even point of 80,000 citations a year, total revenue is equal to total cost. In other words, at the break-even point, Q*. the net revenue is 0. That is,
к = TR-TC = 4,000.000 - 4,000,000 = 0
where п is the net revenue.
On the other hand, the net revenue at the current level of output is positive, not zero, since the department is producing 20,000 more citations than the break-even quantity, i.e., 100,000 instead of 80,000. The difference between the two output quantities, therefore, produces a net revenue of $500,000 for the department, as can be seen from the following calculations:
л = TR-TC [3.5]
= PQ-[FC-K VC)(Q)]
= [(50)(100,000)] - [2,000,000+(25)( 100,000)] = 5,000,000-4,500,000 = $500,000
Note that we could have also arrived at the same result by subtracting the net revenue at the break-even level of citations from the net revenue at the current level of citations. That is,
к = (pQ-{FC+(VC)(Q)}] -[P*QMFC+(VC)(Q*)}] [3 6] = [ {(50)( 100,000)} - {2,000,000+(25)( 100,000)} ] -[{(50)(80,000)}-{2,000,000+(25)(80,000)}] = 500,000-0 = $500,000
Sincc they producc the same result, it does not matter which expression is used.
Changes in Capacity. Earlier in the example we made an assumption that the department is working at 100 percent level of its capacity. In reality, very few organizations work at their fullest capacity level. For instance, if we assume that the department is working at 90 percent level of its capacity, instead of 100 pcrccnt, the net revenue from operation will be $250,000, instead of $500,000, which is $250,000 less than when we assumed that it was working at the fullest capacity level. That is,
л = [(0.90)(50X100,000)] - [2,000,000+(0.90)(25)( 100,000)) = 4,500,000 - 4,250,000 = $250,000
If, on the other hand, we assume that its capacity level increased to 105 percent from the current level of 100 percent, the net revenue for the department will also increase from $500,000 to $625,000, provided that no changes took place in its operating costs or the pricc it charges for an infraction. That is,
л = [(1 05)(50)( 100,000)] - [2,000,000 + (1.05)(25)( 100,000)] = 5,250,000 - 4,625,000 = $625,000
However, at the break-even level of output, where the net revenue is zero, the capacity level for the department is 80 percent (80,000/100,000) x 100 = 80). In other words, to break even the department only needs to work at the eighty percent level of its capacity, as shown below:
л = [(0.80)(50)( 100,000)] - [2,000,000 + (0.80)(25)( 100,000)] = 4,000,000 - 4,000,000 = 0
which also means that any drop in its capacity level below this percentage will produce a (net) negative return for the department.
Changes in Revenue, Output, and Price. Financial necessities often require that a government must find ways to increase its revenue to meet a specified goal or target. Theoretically a government can increase its revenue in one of three ways: by raising taxes, by increasing the pricc of goods and services it provides, and by increasing its output Sincc most governments are reluctant to raise revenue by increasing price or raising taxes, the alternative is to increase output for those goods and services where the potential exists for such an increase. For instance, a government can install more parking meters, issue more permits, give more citations, produce more electricity, collect refuse more frequently, add more hospital beds, and so on to increase the output quantity as long as it is possible to establish that such efforts will be economically viable, i.e., they will increase the net revenue of the government.
Operationally, then, how does one determine the level of output that will generate a desired level of revenue? To determine the level of output that will produce a desired level of revenue, the convention is to treat the target level as an increment of fixed cost since it can be considered as a constant that, like any fixed cost, docs not change in the short run. One can easily extend the expression in liquation 3.3 to obtain this output quantity, as shown below:
Q' = (FC+rc)/(p-VC) [3.7]
where Q' is the projected break-even quantity and the rest of the terms arc the same as before.
To illustrate the point, let us return to the traffic citation problem for a moment. Suppose the department now sets as its goal a net revenue of $ 1 million, instead of $500,000 it obtained from 100,000 citations given a year. Assuming no changcs in price or costs, the projected break-even quantity, Q', will thus be 120,000. That is,
Q' = (FC+n)/(p-VC)
= (2,000,000+1,000,000)/(50-25) = 120,000
This is the total number of citations the department must produce to generate a target net revenue of $1 million.
A question that becomes relevant here is what if the department does not have the physical capacity to raise its output from 100,000 citations a year to 120,000 but, at the same time, it needs to increase its net revenue to $1 million. This would obviously mean that the department must increase the price it presently charges for each citation in order to bring the revenue to the desired level, while keeping the level of output constant at 100,000 We can easily reorganize the expression in Equation 3.7 for projected output to obtain the new pricc, as shown below:
[3.8]
Q' = (FOn)/(p-VC) (p-VC) = (FC+tO/Q'
Л p' = [(FC+n)/Q'] + VC where p' is the new or projected price, and the rest of the terms are the same as before.
Therefore, applying the expression in Equation 3.8 to the current problem one will obtain a new pnee for the department, which is
[3.9]
p' = [(FC+ti)/QJ + VC = [(2,000,000 + 1,000,000)/100,000] + 25 = 30+25 = $55
This is the price the department must chargc for each citation to produce a net revenue of $1 million To sec whether or not it produces the target revenue, we can substitute the value of p' into the expression for net revenue, which is
л = p'Q - [FC+(VC)(Q)]
[(55)(100,000) ] - [2,000,000+(25)(100,000)]
= 5,500,000-4,500,000 = $1,000,000
The result yields a net revenue of $1 million, as expected. Nonlinear Break-Even Analysis
Although it is easy to deal with a linear break-even analysis, the real relationship between cost, revenue, and output may be a nonlinear one. The notion that per unit price or cost will always be a constant, regardless of the quantity of output consumed, is not a realistic assumption since in reality costs related to labor, maintenance, and utility change disproportionately, rather than in fixed, constant amounts. To give an example, recall the citation problem again. Assume now that we have a total revenue and a total cost function for citations given by a single officer, rather than the entire department. We can express the two functions in terms of two simple equations, one linear and one nonlinear (cubic), as given below:
[3.10]
[3.11]
TR= 150Q 'ГС = 0.002Q4.5Q2+100Q+75,000
where the pnee of an infraction is now set at $ 150 instead of $50, with a fixed cost of $75,000, and a variable cost that varies with different levels of output of the polynomial cost function
As before, our objective is to obtain a break-even quantity for the officer, given the information we have on the new price and the costs of operation for the department To make the problem a little more interesting, let us impose a restriction that at the new break-even level of output the net revenue will be maximum. In other words, the maximum revenue the officcr should be able to produce for the department, after adjusting for all costs, would be the amount at the break-even level of output. For most nonlinear break-even functions, there exists such a level of output that will produce this maximum, but to achieve this for the current problem we need to do two things. First, find the net revenue function for the operation. Second, take the derivative of this function with respect to Q, set it equal to 0, and determine the value corresponding to it. (The latter is known as the first order condition for optimization).
Thus, to obtain the break-even quantity for the oft'iccr we set up a new net revenue function by rearranging the terms of the equations (3.10-3.11), as can be seen from the following expressions:
л = TR-TC = 150Q-0.002QJ+1 5Q2- 100Q-7 5,000 = -0.002Q,+ 1.5Q2+50Q-75,000
Next, take the derivative of this function with rcspcct to Q and set it equal to 0 such that
dn/dQ = -0.006Q2+3Q+50 = 0
The result produces a quadratic function of the form
aQ2+bQ+c = 0 [3.12]
where a and b are two parameters (constants) associated with Q.
Equation 3.12 has two solutions, callcd roots, one high and one low, that are determined by a standard expression used for solving quadratic equations, given by the following equation:
When this equation is applied to the result obtained by taking the derivative of the net revenue function, it produces two output quantities, one negative and one positive, as shown below:
Q = [W(b2-4ac)]/2a = [-3±N/{(3)2-(4)(-0.006)(50)}J/[(2)(-0.006)] = [-3±v(10.2)]/(-0.012) = [-3±3.193]/(-0.012) л Q, = -16.083 Q2= 516.083
It is obvious that a negative output, given by Q,, docs not make any sense since output cannot be negative. So the second solution, Qj, is the one that should produce the maximum net revenue. This turns out to be $75,407.83, as can be seen from the following calculations:
7i = TR-TC = -0.002Q»+1.5Q2+50Q-75,000 = -0.002(516.083)» +1.5(516.083)2 +50(516.083)-75,000 = -274.908.81 +399,512.49+25.804.15-75,000 = -349,908.81 + 425,316.64 = S75,407.83
These results are shown in Figure 3.2. The figure also shows two break-even points, Q,* and Q2* (obtained graphically), at which the net revenue is zero. The quantities corresponding to these points are 250 for Q,* and 710 for Q2*. Note that the breakeven quantity at Q2* should have been 710.75, not 710 (as indicated). But one cannot have a decimal citation so we rounded it off to the nearest whole number (integer), which is 711.
To verify if the break-even points, i.e., the quantities obtained for the problem, are correct, we can substitute these quantities into the expression for net revenue. If the result produces a zero, it should be considered correct. For instance, at Q,* = 250, the net revenue is 0. That is,
О 100 200 300 400 500 600 700
Quantity (Q)
Figure 3.2 A Nonlinear Break-Even Chart
t: = TR-TC = -0.002QVl.5Q2+50Q-75,000 = -0.002(250),+1.5(250)'+50(250)-75,000 = -31,250+93,750+25,000-75,000 = -106,250+106,250 = 0
which is correct But at Q2* = 711, the net revenue is $-19.36. That is.
я = TR-TC = -0.002QJ+1 5Q2+50Q-75,000 = -0.002(711 )>+1.5(711 )2+50(711 )-75,000 = -718,850.86+758,281.50+35,550-75,000 = -793,850.86+793,831.50 = -19.36
which does not quite come out to be 0 due to rounding-ofT error, but it is the closest
value we could get.
The figure further reveals that any amount below Q,* or above Q2* will produce a negative return. Therefore, the objective of the department should be to stay within these two limits to remain economically viable.
Other Uses of Break-Even Analysis
The break-even analysis we have discussed up to this point deals with a single activity, but it can also be used for comparing multiple activities with varying levels of fixed and variable costs. Consider the example of a city that is planning to replace the incinerator for solid waste disposal. Projects, such as incinerators are highly capital intensive in that they require large initial costs, which often run into millions of dollars. Fortunately for most governments, the high initial cost is frequently offset by a low operating cost that is directly related to output. That is, at small levels of output the)' are costly, but as the level of output increases the cost goes down resulting mostly from economics of scale, i.e., decreasing average cost of production.
To illustrate the case, assume that the city in question is considering offers from two private firms, X and Y, both wanting to supply the city with a new incinerator. Assume further that we have two cost functions for the incinerator based on the information provided by the firms. For convenience, we present the functions in terms of two linear equations, one for X and the other for Y, as shown below:
[3.14] [3-15]
[3.16]
TCX= 20,500,000+25Q TCY= 25,000,000+10Q
According to the equations, project Y appears to be more capital intensive, but has a lower operating cost ($10 compared with $25 for project X). On the other hand, project X seems to be less capital intensive, but has a higher operating cost. Since both projects have some advantages and disadvantages, we need to have some guidelines or criteria by which we can make a comparative assessment of the projects. The criterion most frequently used in this case is the fixed-cost efficiency (defined as the relationship between output and the fixed cost of a project or activity). To determine this efficiency, the rule of thumb is to find an equilibrium quantity, Q", and use that as a basis for comparisons among projects. We can obtain this quantity for our two projects by simply setting the two equations equal to each other, then solving the expression for Q", as shown below:
TCX=TCY 20,500,000+25(2" = 25,000,000+10Q
15Q" = 4,500,000 л Q" = 300,000 tons/year
The result produces an equilibrium quantity of 300,000 tons a year. What this means is that at an output level of less than 300,000 tons a year, project X will be more efficient since it has a lower fixed cost. The explanation for this is quite simple: with low fixed costs, high variable cost project is preferred at low levels of output becausc the gain from low fixed costs initially offsets the high variable costs. As output level increases, fixed costs tend to decrease in relative importance for both projcct scalcs eventually reversing the benefits of the smaller scale. In other words, the gain in terms of low variable costs for the higher fixed cost projcct will more than offset the difference in fixed costs at Q greater than 300,000. To put it in another way, at any given level of output, the lower the fixed costs, the smaller the fraction of total revenue necessary to recover these costs.
Operating Leverage. We can extend the discussion by introducing into the above example an important break-even concept, called the operating leverage. Operating leverage is the measure of the extent to which fixed production facilities affect the operation of a projcct. In other words, it measures the sensitivity of an operation resulting from the use of fixed, rather than variable inputs. The degree of operating leverage for break-even analysis, for any amount of output, can be measured by a ratio, given by the return on sunk (irrecoverable) costs of an operation and its net revenue. We can formally express this ratio as
OL = [TR-VC(Q)]/n (3.17)
= [(PQ-VC(Q)]/K = Q(p-VC)/IQ(p-VC)-FC]
where OL is the operating leverage, Q(p-VC) is the sunk cost, and Q(p-VC)-FC is the net return.
Note that the expression p-VC in Equation 3.17 is called the contribution margin, which is obtained by taking the difference between pricc and variable costs. Contribution margin measures the contribution this difference makes toward recovering the fixed cost of an operation as well as any net gain from it. For instance, if a government charges $10 for a fishing permit and it costs the government $3 in variable costs, each permit issued recovers its variable cost plus $7. The $7 is the contribution margin since it contributes to the recover)' of the fixed cost plus any net gain from operation.
Let us now return to the incinerator example. Assume that it costs the government
S150 for cach ton of waste disposed. Thus, at an output level of 300,000 tons a year, project X will have an operating leverage of 2.21 with a contribution margin of $ 125, as can be seen from the following calculations:
OLx = 1300,000( 150-25)]/[300,000( 150-25)-20,500,000] = 37,500,000/17,000,000 = 2.21
On the other hand, at the same level of output, project Y will have an operating leverage of 2.47 and a contribution margin of SI 40, as shown below:
OLy = [300,000( 150-10)]/[300,000( 150-10)25,000,000] = 42,000,000/17,000,000 = 2.47
We can interpret these results the same way as one would interpret the elasticity of revenue resulting from a changc in output. As noted previously, the term elasticity means the degree to which a variable changes in response to a changc in another variable. The higher the degree of responsiveness, the greater the elasticity. Thus, in the current example, a 1 pcrccnt increase in output over the equilibrium quantity of 300,000 tons a year will result in 2.21 percent increase in revenue for project X. At the same level of output for project Y, which has a slightly higher operating leverage, a 1 percent increase in output will result in 2.47 percent increase in revenue. As a general rule, operating leverage is greater for capital-intcnsivc operations than it is for labor-intensive operations, meaning that capital-intensive operations will have a higher break-even point than labor-intensive operations. This is clcarly evident in the break-even quantity produced by the two projects, as shown below:
Q*x= FC/(P-VC) = 20,500,000/(150-25)= 164,000 tons/year Q\= FC/(P-VC) = 25,000,000/(150-10)= 178,571 tons/year
As the results indicate, project Y, which is more capital intensive of the two, has a higher break-even point than project X. In sum, how much cost advantage a capital intensive operation will enjoy depends on the level of output. At higher levels of output, capital-intensive operations have an advantage over labor-intensive operations, whereas at lower levels of output labor-intensive operations have the advantage.
Like break-even analysis, differential cost analysis deals with cost, revenue, and output of an activity. Its purpose is to use cost and other related information to determine the coursc of action that will producc the best possible return from that decision for an organization. For instance, when a government wants to initiate a new program or eliminate an existing one, or when it wants to reduce a service charge or buy a piecc of equipment, it must compare these decisions against the existing conditions to determine if the net return is higher in each instance to justify that course of action. Without this comparison, it will be difficult for a government to determine the relative worth of its decisions. As a general rule, a decision is considered worthwhile if it produces a return that is higher than the return produced under the existing conditions.
This section examines three empirical situations where differential cost analysis makes most sense: (1) to determine the effects of a change in cost on net return from an operation, (2) to determine the time it lakes to recover a cost, called payback period; and (3) to determine when to buy as opposed to lease, a frequently asked question in government procurement.
Earlier in the chapter, we defined net revenue as the difference between total revenue and total cost, but in differential cost analysis it is defined as the difference between two differentials: differential revenue and differential cost. Differential revenue is the difference between a current revenue and the revenue one will earn from an alternative. Similarly, differential cost can be defined as the difference between current cost and the cost under an alternative. Thus, in calculating net revenue, the emphasis is placed on the differentials rather than on the simple difference between cost and revenue.
The calculation of net return from these differentials is quite simple, provided that one has all the information one needs on differential cost and revenue. Equation 3.18 presents a simple expression for calculating this net return, as shown below:
n4 = [ДR x T]-[C0 + (ДС x T)] [3.18]
where л d is the net return from differentials, AR is the changc in revenue, C0 is the one-time cost (such as construction cost), Д С is the change in cost, excluding the onetime cost, and T is the time.
An Illustrative Example. We can use a simple example to illustrate this. Suppose that wc have a local government which owns and operates a parking garage with a capacity for parking 150 cars at any given time. The government charges $3.75 a day for parking, which runs from 7:00 in the morning to 7:00 in the evening. The garage remains closed after that time. The number of cars parked on average is 200 a day, including weekends. Assume that the demand for parking space has been steadily increasing in recent months. To deal with the problem, the city has dccided to upgrade the garage, although there arc other alternatives it could have pursued (such as selling it to a private provider, say for $ 1 million) The upgrading will involve increasing the capacity by another 150 more parking spaccs, thus raising the number of cars that can be parked on average to 400 a day.
Assume that it will cost the city $750,000 to upgrade the garage, which it will borrow as an intermediate-term loan from a local bank payable in three years. The city is expectcd to retire the principal on or before the end of this period, along with the interests it will accrue for the period of the loan Assume further that the city also plans to increase the parking fee by 75 ccnts, thereby raising the current price from $3.75 to S4.50 a day. Table 3.1 shows a three-year differential return, including the time it will take to even out the costs of upgrading.
According to the table, the city revenue from upgrade will increase by $383,250 a year, resulting in part from pricc increase and in part from the revenue generated from additional parking spaces. That is,
RU = Rap-REC [3.19]
= [(400)(365)(3.75)]+[(400)(365)(0.75)]-[(200)(365)(3.75)] = [(547,500+109,500)-273,750] = $383,250
where Ry is the revenue from upgrade, RAP is the revenue from alternative proposal, and Rec is the revenue from existing condition(s).
The operating cost will also increase by $77,750 a year, much of which will be due to (say) an increase in payroll, costs of additional repair and maintenance, and interest payments on debt. The table also shows a three-year net return from differentials, which stands at $166,500 (after all the adjustments have been made). I lowevcr, the unadjusted return for the city for the same period is much greater at $ 1,149,750, based on an annual return of $383,250; that is, 383,250 x 3 = $ 1,149,750.
Does this mean that the city will be better off by not selling the garage? The answer is "yes" since it will take the city about three years to break even, and another two years or so to recover the $1 million it will have earned if it had dccided to sell it, while retaining the ownership of the garage. That is, (657,000x2) - (99,250x2) = 1,314,000 -198,500 = $1,115,500 (assuming no changes in the cost of operation).
Tabic 3.1
Differential Cost Analysis
Existing Condition ($) | Alternative Proposal (S) | Difference ($) | |
A Revenue from: 1. Parking spacc 2. Price increase | 273,750 0 | 547,500 109,500 | 273,750 109,500 |
Total revenue/year: | 273,750 | 657,000 | 383,250 |
B. Cost due to: 1. Construction 2. Interest payment 3. Repair & maintenance 4. Payroll increase 5. Miscellaneous | 0 0 2,500 18,000 1,000 | 750,500 56,250 5,000 36,000 2,000 | 750,000 56,250 2,500 18,000 1,000 |
Total cost/ycar: | 21,500 | 99,250* | 77,750* |
C. Three-year differential analysis 1. Total differential revenue [383,250x3] 2. Total differential cost [750,000+(77,750x3)] | 1,149,750 983,250 | ||
Net return from differentials | 166,500 |
* Excludes the cost of construction
Payback Period
An important conclusion that emerges from the preceding example is that it will take the city about three years to recover the cost of the project. In other words, it is the amount of time the city will need for the earnings (cash inflows) from the garage to equal the cost (cash outflows) of its expansion. In cost analysis, this is known as payback period. Payback period simply means the time during which cash inflows from a project operation will equal its cash outflows. In other words, it is the time when the net cash flow (the difference between cash inflows and cash outflows) from
an operation will be zero.
The definition has an interesting implication in cost analysis in that it focuses primarily on recovery time, i.e., the time it takes to realize the full cost of a project, but not what happens after the costs have been recovered. Since it is mostly concerned with recovery time, payback is often used as a benchmark for project acceptance. The term benchmark in this case means the maximum number of years it will take an organization to recover the cost of a project. Thus, if it takes five years to recover the cost of a project, it is the time that should be used as the benchmark for that project. However, the time it takes to recover the cost of a project varies, depending on the size of the projcct and the flow of funds. For instance, a small project with low positive net flow can take a much longer time to realize the full cost of its investment than a large project with high positive net flow.
There are several advantages of using payback in cost analysis. One, it is simple to use sincc it requires only three sets of information: inflows, outflows, and the initial cost of investment. Two, it can help an organization determine how fast it can recover the cost of a project to avoid the problem of holding up cash for a long period of time. This is particularly important for organizations that arc strapped for cash. Three, and most important, it is flexible, meaning that it can be used as a supplementary' tool or in combination with other decision tools, such as discounted cash flows.
Let as look at a simple example to illustrate this. Suppose a state treasurer's officc wants to upgrade its computer system that is fast becoming obsolete. The system is primarily used for internal consumption, such as providing financial and accounting scrviccs to other departments of the government. Let us say that it will cost the government SI.2 million to upgrade the system, which the office expects to rccovcr from the revenue it will receive from the various user departments. Table 3.2 shows the expected cash flows (inflows and outflows) from the upgrading of the system.
As the table shows, it will take the government just about six years to recover the cost of the projcct, which is shown under the column, Net Flow. However, looking at the cash inflows for the project, it seems obvious that it will continue to produce returns beyond the payback period but, as we noted earlier, payback is not concerned with what happens once the cost has been recovered. This is a major weakness of payback since it fails to take into account the returns that most projects continue to generate beyond the period of cost recovery.
There are a couple of other weaknesses of the method that arc worth noting here. First, it takes a fragmented approach to projcct investment. What this means is that it does not consider the overall liquidity or cash flow position of an organization, but rather focuses on the cash flow resulting from a single project. Sccond, it frequently fails to consider the costs of funds ncccssary to support an investment even during the payback period. Cost of fund is the return on an alternative after all the adjustments have been made for costs. This is an important factor that should not be ignored in payback calculation because to ignore it could result in underestimating the payback.
Tabic 3.2
Expccted Outflows, Inflows, and Net Flows
Expected | Expected | Cumulative | ||
Year | Outflow ($) | Inflow ($) | Net Flow ($) | Net Flow ($) |
0 | (1,200,000) | - | (1,200,000) | (1,200,000) |
1 | - | 125,000 | (1,075,000) | (1,195,000) |
2 | - | 170,000 | (905,000) | (1,144,500) |
3 | - | 200,000 | (705,000) | (1,058,950) |
4 | - | 215,000 | (490,000) | (949,845) |
5 | - | 250,000 | (240,000) | (794,830) |
6 | - | 250,000 | 10,000 | (624,313) |
7 | - | 350,000 | 260,000 | (336,744) |
8 | - | 375,000 | 635,000 | 4,582 |
To give an example of how one would incorporate the cost of funds into payback calculation, let us go back to the table and look at the column, titled Cumulative Net Flow. According to the table, the net flow for the current year is $1.2 million since there is no cash inflow during this period. Let us assume that the government would like to have a return of 10 percent on this investment. Therefore, at a return rate of 10 percent on $ 1.2 million, the cost of funds corresponding to the net flow in the first year would be $ 120,000, which is obtained by multiplying the net flow by the rate of return; that is, 1,200,000 x 0.10 = $ 120,000.
Thus, to obtain the cumulative net flow for the first year we simply add this cost of fund to the net flow of $1.2 million and subtract the total from the inflow for the first year, which will producc a net flow of $1,195,000. That is,
CNF1, = (CNF10+CF0) - ENFlj (3.20)
= (1,200,000+( 1,200,000 x 0.10)] -125,000 = (1,200,000+120,000)- 125,000 = 1,320,000-125,000 = $1,195,000
where CNF1, is Ihe cumulative net flow at time 1, CNF^ is the cumulative net flow at time 0, CF0 is the cost of funds at time 0, and EIF1, is the expected inflow at time 1.
We do the same for the second year. That is,
CNF12 = (CNFl.+CF,) - ENF12 [3.21 ]
= [ 1,195,000 + (1,195,000 x 0.10)] - 170,000 = (1,195,000+119,500) - 170,000 = 1,314,500-170,000 = $1,144,500
and continue to repeat the process until the costs are fully recovered.
According to the table, this happens to be the eighth year. In other words, it will take eight years instead of six to recovcr the cost of the project. If we did not consider the cost of funds, we would have underestimated the payback period by two years.
Buying or leasing is a spccial case of differential cost analysis in which costs are compared not between an alternative and the existing conditions, but between two alternatives: buy or lease. Like any business organization, when a government decidcs to acquire an asset it has the option to buy or lease. Which option it uses depends on a number of factors, such as the cost of the asset, the length of time for which it will be used, the salvage value it will have at the end of its useful life, and so on. Although as a financial option leasing has been around for a long time, more governments arc using it today than ever before because of its viability as an alternative to buying
teasing can be defined as a process, whereby the owner of an asset, called lessor, enters into an agreement (lease) with the user of the asset, called lessee, to allow the latter to use the asset for a specified time at an agreed upon cost. Theoretically the lease cost must be less than the purchase price to justify the option to lease, but there are situations where buying may be a better alternative to leasing even if it costs a government more initially, provided that in the long run it will save the government more money.
There arc several reasons why leasing is considered a better alternative to buying. Important among them are:
1. It provides a viable alternative for acquiring assets for governments that have limited resources. Without the opportunity to lease, many of these governments will not be able to obtain the facilities and equipment they need
2. It can conserve or free up existing resources of a government that can be used
to meet other financial needs.
3. It offers a fast and flexible alternative to raising taxes or borrowing funds to pay pay for the facilities and equipment a government needs.
4. It provides 100 percent financing in most cases. In addition, costs such as delivery fees and installation charges are frequently included in lease payments, thereby reducing the cost burden for the government.
5. It avoids the risk of obsolescence. In an age of rapid technological changes, ownership docs not have a distinct advantage every time a new and improved product is introduced in the market.
6. It docs not affect a government's ability to borrow and docs not reduce its credit rating.
Although the advantages are obvious, a government considering leasing as an option must do some financial analysis to see if it has a definite cost advantage over buying. In reality, the actual analysis does not have to be all that complicated and can be accomplished in three easy steps: (1) determine the costs of buying as well as leasing; (2) calculate the difference between the two options; and (3) select the one that produces a lower cost to the government.
We can formally express this relationship between buying and leasing in terms of a simple mathematical expression, as shown below:
w<- (*VE o,-svT) -E (Lr°.) 13 22J
U> t-0
where Wc is the net cost of ownership, P0 is the purchase price at time 0, is the cost of leasing at time t, Ot is the operating cost at time t for the duration of time the asset will be in use, SVT is the salvage value at time 'Г, and t = 0, 1,2,......, T.
The first part of the equation represents the cost of ownership and the second part the cost of leasing. While the equation seems simple enough to be useful for everyday purpose, it has one major weakness in that it docs not take into consideration the time value of money, which is important if the item in question has a useful life that extends over several years.
Time Value of Money. Since leasing involves a commitment of resources that extends over several years into the future, it is necessary to introduce the notion of time value of money into the discussion. The time value of money simply means the value a given sum of money will have over time. Generally speaking, the value of money tends to decline as time progresses, indicating that with time money loses its value. Several factors explain for this apparent loss of value of money over time, such as inflation (it will cost us more to buy the same bundle of goods and services tomorrow than what it costs us today), our unwillingness to defer current consumption in favor of consumption in the future (we are better off consuming the rcsoucres today than postponing them for tomorrow bccausc they will cost more), uncertainty (we do not know what awaits us in the future), and so on.
Since money loses its value over time, it is necessary to make some adjustments in the cost of ownership to get an accurate picture of its real worth. The simplest way to accomplish this is to discount the cost of ownership (Wc) by an appropriate discount rate. The result would produce a discounted value of ownership, callcd the present value (PV) of ownership. By present value, we mean the value a future stream of costs and returns will have in today's terms. Why in today's terms? Because when a decision is made to acquire an asset, it involves a resource commitment at the time when the decision is made to acquire it, although the actual rctum(s) from it may not occur immediately or they may occur over a period of time.
To give an example, say that we invest an amount, X, today at a 5 percent rate of discount that will pay us exactly $100 a year from now, its value in today's terms will be $95.24. That is,
[3.23]
X(l+0.05)' = 100
X= 100/(1 -Ю.05)' = 100/1.05 = $95.24
In other words, this is how much $ 100 next year will be worth in today's terms, which is the same as saying that if we invest $95.24 today at a 5 percent rate of discount, it will produce $100 next year.
Similarly, if we invest an amount, X, today that will pay us exactly $100 two years from now, it will be worth $90.70 today. That is,
[3.24]
X(l+0.05)2 = 100
a
X= 100/(1-Ю.05У = $90.70
and so on.
To put these in present value terms, any amount invested today that will produce a return at some future points in time will be worth much less when expressed in today's terms.
We can now formally express the net cost of ownership in Equation 3.22 by rewriting it in present value terms, as shown below:
[3.25]
к™* - ip л:
UO (1.T)1 (l.r)T u> (1.Г)'
where NPVW is the net present value of ownership (defined as the difference between the present value of purchase or ownership and the present value of lease), r is the rate of discount, and the rest of the terms are the same as before. Note that the expression 0/(1+r)1 can also be written as 0,[l/(l+r)'], where l/(l+r) is called the discount factor PF).
Л discount factor is the value by which the costs and returns of a projcct are multiplied to produce a discounted stream of costs and returns. As a general rule, if the (net) present value of ownership is positive, leasing should be preferred since it will cost more to own than lease. On the other hand, if it is negative, purchasing should be preferred since it will cost more to lease than own.
We can use another example to illustrate this. Suppose a public works department needs a heavy duty truck for doing its normal repair and maintenance works. If the department has to buy the truck, it will cost it $65,000. If it has to lease, the rental cost will be $18,000, payable at the end of each period for three years (assuming the department will lease the truck for three years only). If purchased, there is a salvage value of $23,500 at the end of the asset's useful life. There is also an operating cost of $5,000, regardless of whether the vehicle is purchased or leased. Assume a discount rate of 9 percent. Table 3.3 shows the present value of ownership for the problem based on this discount rate.
According to the table, the net present value of ownership is $2,788.97, which is obtained by taking the difference between the sum of the present value of purchase and the present value of lease. That is,
[3.26]
NPVW = £(PV of purchase)-£(PV of lease)
= [(65,000.00+4,587.40+4,208.40+ 3,860.92)-16,647.99]
-[21,100.91+19,358.64 + 17,760.21] = 61,008.73 -58,219.76 = $2,788.97
where NPVW is the net present value of ownership. Now, to determine whether the department should buy or lease the vehicle we apply
Table 3.3
Present Value of Qwnership
Year | P($) | 0($) | SV($) | DF | PVof purchase ($) |
0 | 65000 | - | - | - | 65,000.00 |
1 | - | 5,000 | - | 0.917431 | 4,587.40 |
2 | - | 5,000 | - | 0.841680 | 4,208.40 |
3 | - | 5,000 | - | 0.771830 | 3,860.92 |
4 | - | - | 23,500 | 0.708425 | 16,647.99 |
Year | L($) | 0($) | Total cost ($) | DF | PVof lease ($) |
0
1 | 18,000 | 5,000 | 23,000 | 0.917431 | 21,100.91 |
2 | 18,000 | 5,000 | 23,000 | 0.841680 | 19,358.64 |
3 | 18,000 | 5,000 | 23,000 | 0.772183 | 17,760.21 |
4
the decision rule wc established earlier; that is, if the net present value of ownership is positive, it should lease; if not, it should purchase. Since the net present value of ownership is positive, the department should go with the option to lease.
We have taken a rather simplistic approach here to determine the cost of ownership. Although somewhat easy to understand, there is a weakness in this approach in that it tends to ignore intangible factors, such as convenience, location, and personal relationship between a lessee and a lessor that can affect the final cost. Another problem with this approach is that it is too broad and fails to include a host of other requirements that may also have an important determining influence on the final cost. For instance, a government may be required to pledge a collateral before acquiring an asset, or there may be restrictions on the use of the asset once acquired, or there may be special clauses requiring the government to buy a safety insurance to protect the asset against potential loss, theft, or damage. In each of these instances, there is a monetary implication that must be incorporated in the final calculation before making a decision whether to buy or lease.
BENEFIT-COST ANALYSIS
One of the earliest decision tools used in cost analysis with a long history of application in government is benefit-cost analysis (BCA) that dates back at least to the 17th century when Sir William Petty first introduced the conccpt, while studying the public health cost of combatting the plague in London, England, in 1667 2 Later on, it was popularized by Jules Dupit, a 19th century French economist, then by an Italian economist by the name of Vilfrcdo Pareto, and much later in the 1940s by two British economists, Nicholas Kaldor and John Hicks. In this country, it achieved a significant milestone with the passage of two major bills early in this ccntury, the River and Harbor Act of 1902 and the Flood Control Act of 1906. Today, hardly a bill passes or a decision is made in government without some reference to benefit-cost analysis of one form or another.
Considering its long history, benefit-cost analysis still remains one of the most popular and widely used tools of resource allocation available to a decision maker. When resources are scarce and the demand for goods and services exceeds the available resources, the rational choice facing a decision maker is to undertake those activities that will produce the greatest amount of return for the resources utilized This fundamental rule of efficiency that guides the allocation decision of a household or firm also guides a government in allocating its resources among competing needs and interests. Without this measure of efficiency, there will be very little basis for determining how best a government can utilize its resources or what it can do to improve its allocation decisions in the future
Steps in Benefit-Cost Analysis
Like most decision tools used in government for resource allocation, benefit-cost analysis can be carried out in a number of distinct phases or steps. These steps are: (1) define goals and objectives; (2) identify a projcct or a group of projects, (3) determine the costs and benefits associated with each project, (4) compare the projects using one or more decision rules; and (5) select the most appropriate project(s).
All benefit-cost analysis begins with a clear statement of goals and objectives For most organizations in the private sector, the objectives are frequently stated in monetary terms where the manager of a firm or business undertakes those measures that will maximize the flow of money income over time for its owners and stockholders. This model of private firms also applies to government, but only for the activities that are similar to those provided by the private sector.
As noted earlier, when a government provides goods and services that arc similar to those provided by the private sector, it can and often functions like a private firm using the conventional pricc mechanism and other behavioral rules of market operations. On the other hand, when it provides goods and services that cannot be treated in the same fashion as those provided by the private sector, like national security, it bases its decisions on much broader goals affecting the entire society than on some narrowly defined financial objectives. To put it in another way, for goods and services that cannot be sold to the public in a business like manner, governments base their decisions not on any particular financial criterion but on some underlying social objectives) that will benefit society at large. Social objcctivcs, therefore, remain at the center of most allocation decisions in government.
Since social objectives lie at the heart of most allocation decisions in government, the question is: how does one define and measure these objectives? There are two strands of arguments that have dominated the discussion in the economic literature on this subject for a long time.3 The first, rooted deeply in mainstream welfare economics, argues that bcncfit-cost analysis must be based on a set of normative considerations as to what the social objectives ought to be. The underlying notion here is that these objectives can be obtained from a consensus on value judgments of the individuals who make up the society. According to the second strand, a social objective can be defined as the one pursued by those responsible for making decisions that affect society in the aggregate and who are accountable to the public for those decisions. From the latter perspective, bcncfit-cost analysis is a process of appraising decision problems as viewed by a decision maker or a group of decision makers who hold the same view of society and how its welfare is measured. This is more consistent with financial decision-making in the private sector (and is the one used in the present discussion).
Assuming that one is able to define a set of goals and objectives in a clear and consistent manner, the next step in the process is to identify a project or a group of projects that will meet these objectives. Theoretically there is no limit to the number of projects an organization can consider at any one time, but in practice it depends on the goals, objectives, and priorities of the organization as well as the resources the projects will consume and the benefits they will produce. The central problem in any benefit-cost analysis is not how many projects are selected at any given time, but rather how one values the benefits and costs associated with them. In government, where one has difficulty in measuring social objcctivcs in precise monetary terms, the task of valuing the benefits and costs of a public project can be a real challenge for the decision makers From a practical point of view, one can always use the private sector experience as a starting point, where dollar values are commonly used for measuring benefits and costs.
Once the goals and objectives have been defined, the projects have been identified for evaluation consistent with those goals, and one has some information on their costs and benefits, the next step in the process is to apply one or more decision rules by which the projects can be compared and evaluated. There are three basic decision rules that are commonly used in benefit-cost analysis for this purpose: net benefit (N13), benefit-cost ratio (B:C), and net present value (NPV). Net benefit is the difference between total benefits and total costs. According to this simple rule, a project is admissible only if its net benefit is positive, that is, if its benefits exceed its costs. Wc can formally express this as: EB-EOO. If more than one projcct is involved, based on this rule, select the project with the highest net benefit.
Benefit-cost ratio, on the other hand, is the return one will earn for each dollar worth of investment. For instance, a benefit-cost ratio of 0.75 means a return of 75 cents for each dollar invested. According to this rule, a project is considered acceptable if it produces a benefit-cost ratio of greater than or equal to one. that is, B:C* 1. Obviously any project that produces a benefit-cost ratio of less than 1 is not worth considering since its costs will outweigh its benefits. When several projects are involved in an analysis, the rule says that only the projcct with the highest benefit-cost ratio should be accepted, provided that all of them have a benefit-cost ratio of at least one.
Finally, the net present value can be defined as the difference between a discounted stream of benefits and costs. As a general rule, for a projcct to be accepted its net present value must be positive. When more than one project is involved, the projcct with the highest positive net present value is the one usually accepted.
Let us look at a couple of examples to illustrate this. Our examples includc both single and multiple projects.
Benefit-Cost Analysis involving a Single Project. To begin with, consider a ease where a local government with limited resources wants to renovate a major facility that is primarily used for indoor activities, such as games and sports, exhibitions, etc. Let us say that the renovation will include (1) replacing the existing cooling system with a more energy efficient one, (2) increasing the seating capacity to the maximum limit, and (3) completing a number of minor repairs. The government expects that this will significantly increase its revenue, reduce repair and maintenance costs, and increase savings in energy costs. The decision facing the government is to determine if it is worthwhile to undertake the project. Since the problem deals with a single project, the only requirement for the decision maker(s) is to accept or reject the projcct. Table 3.4 shows these benefits and costs along with their associated net benefit and bencfit-cost ratio.
According to the table, it will cost the city approximately S5.5 million to do all the repairs the facility will need The total benefit from the renovation will be $6 million, thus producing a net benefit of $0.5 million and a benefit-cost ratio of 1.09. Since the net benefit is positive and the benefit-cost ratio is greater than 1, the project should be accepted. As a general rule, if B:C is greater than 1, net benefit will be positive, and vice versa.
Benefit-Cost Analysis involving Multiple Projects Let us look at another example, this time involving a government that is looking into five mutually exclusive projects for downtown redevelopment. By mutually exclusive, we mean the selection of a projcct that will automatically preclude the selection of any other. Sincc they are mutually exclusive, only one of the five projects can be accepted, but wc do not know
Table 3 .4
Benefit-Cost Analysis Involving A Single Project
Dollar Value ($)
3,500,000 1,500,000
500,000 5,500,000
Costs and Benefits
Direct Costs:
Replacing the A/C system Adding more scats Indirect Costs:
Repair and maintenance Total costs:
4,000,000
1,250,000 750,000 6,000,000
Direct Benefits:
Return from sales Indirect Benefits:
Savings in energy cost Savings in maintenance cost Total benefits:
SB- EC
= 6,000,000- 5,500,000 = $500,000
Benefit-Cost Ratio:
EB/EC
= 6,000,000/5,500,000 = 1.09
Net Benefit:
which one. As before, the decision facing the government is to select the project that would be acceptable to it based on the criteria suggested earlier. Table 3.5 shows the relevant information on costs and benefits, including both benefit-cost ratio and net benefit for each project.
A cursory glance at the table indicates that there is no one-to-one correspondence between net benefit and benefit-cost ratio, although in each instance the net benefit is positive and the benefit-cost ratio is greater than one. What this means is that a project with high net benefit does not necessarily have a high benefit-cost ratio. For instance, project C, which has the highest net benefit ($537,000) has a benefit-cost ratio of 1.43, as opposed to project B, which has the highest benefit-cost ratio (1 78), but a net benefit of only $339,300. Similarly, project A, which has the third highest benefit-cost ratio has a net benefit of just $62,500, which is barely above the project with the lowest benefit-cost ratio.
This raises an interesting dilemma for the decision makers: which project should
Table 3.5
Benefit-Cost Analysis Involving Multiple Projects
Project | Total Cost ($) | Total Benefit ($) | Net Benefit ($) |
A | 250,000 | 312,500 | 62,500 |
В | 435,000 | 774,300 | 339,300 |
С | 1,250,000 | 1,787,500 | 537,500 |
D | 725,000 | 761,250 | 36,250 |
E | 565,000 | 683,650 | 118,650 |
Project | B:C | Total Cost ($) | Cumulative Cost ($) |
В | 1.78 | 435,000 | 435,000 |
С | 1.43 | 1,250,000 | 1,685,000 |
A | 1.25 | 250,000 | 1,935,000 |
E | 1.21 | 565,000 | 2,500,000* |
D | 1.05 | 725,000 | 3,225,000 |
they select? The answer depends on the objective of the decision makers vis-a-vis the government. For instance, if the objective is to have maximum net benefit, then it should select the project that produces the highest net benefit, which in this case happens to be project C. On the other hand, if the objective is to have maximum return for each dollar worth of investment, then the decision should be to go with the project that produces the highest bcncfit-cost ratio. This will be project B.
Earlier in the example, we made an assumption that our projects were mutually exclusive. Suppose we relax this assumption by suggesting that the projects are not mutually exclusive and that the government would prefer to accept not one, but instead as many projects as the available resources will permit. Let us say that it will cost the government $3,225 million to undertake all five projects, but it does not have the resources to accept them all. Consider, for a moment, that our government has a constraint of $2.5 million, which means that given the amount of resources available to the government it can accept only four of the five projects: В, C, A, and E, in that order. If we assume a constraint of $2 million instead, only three of these projects, B, C, and A, could be considered, and so on. As noted before, the order of a project is important in benefit-cost analysis. Where a project stands in a rank order, determines whether or not it will be acccpted, although the order is likely to be different for different decision rules.
Time plays an important role in all benefit-cost analysis, especially for projects that have a life span of several years. As noted earlier, when time is brought explicitly into benefit-cost calculation, it changes the valuation of a project becausc of the need to discount it for benefits and costs that occur over a period of time. The critical element in this calculation is the rate one must use to discount these benefits and costs since setting this rate high or low will have a direct impact on the decision whether to acccpt or reject a project. For instance, a high discount rate will make a project less attractive to future consumption bccause, from the point of view of the consumers, they will be better off, i.e., derive more satisfaction, by consuming the resources now than postponing the consumption for sometime in the future. This is bccausc at a higher rate of discount money is worth more today than it is at some future points in time.
This leads us to the difficult choice of selecting a discount rate that will better reflect an individual's ability and his or her willingness to make sacrificc in favor of future consumption. The conventional wisdom in this regard is to use the inflation rate as а рюху for discount rate. Unfortunately, the use of inflation rate as a proxy provides a less than adequate measure becausc it fails to take into account the real earning capacity of funds invested in public projects. But nobody quite knows what the real earning capacity of these funds are and how to utilize that information to determine a rate that can be effectively used to discount a stream of benefits and costs. Several methods have been suggested over the years on how to deal with this problem that merit some attention here. Important among them arc: opportunity cost of capital, marginal rate of time preference, weighted average cost of capital, social rate of time preference, and borrowing rate of capital.
Opportunity Cost of Capital. Opportunity cost simply means the cost of an alternative or opportunity forgone. When the conccpt is applied to capital, it means the return a given capital would have earned from an investment had it been undertaken instead of the current investment. Suppose a government decides to undertake a project at a substantial cost of money that would benefit the community as a whole it serves. To finance the project, the government must impose a new tax that the community will have to be bear, but will eventually be repealed oncc the project takes off. Assume, for a moment, that if instead of paying taxes to support the project, the taxpayers would have invested the money in money market funds it would have earned them a return of 8.5 percent a year. This is the opportunity cost of capital, i.e., the opportunity the taxpayers (would) have forgone.
The opportunity cost of capital is based on a simple assumption that money, in particular savings, if not withdrawn through taxes, would have gone into private investment; that is, it will not remain idle. But this is hardly the case. Individuals do not spend all their savings on investments; some of it are held as cash or placed in demand deposits to meet short-term needs. Alternatively, they may decide to spend all of it on additional consumptions, such as going on vacations or buying luxury goods. This trade-off between investment and consumption makes it further difficult to determine the rate that will reflect the true opportunity cost of capital.
Marginal Rate of Time Preference. The assumption underlying the opportunity cost of capital is that money- used for public projects comcs entirely from private savings which, if not taxed away by government, would have gone into private investment. Assume now that money comes not from private investment, but from private consumption, the appropriate discount rate will then be the rate at which the utility or satisfaction one derives from the consumption of a dollar at the margin this year is equal to the utility one will derive from a marginal dollar at any future year. In common-sense terms, this means that the value, say of S100 consumed today is equal to (100X1.05)' = $ 105.00 next year, or (100)( 1,05)2 = $110.25 the year after and that this relationship will hold for all consccutive years.
As long as this relationship holds, people will be indifferent between currcnt and future consumption. Thus, when funding for a project comes from private consumption, the appropriate discount rate will be the marginal rate of time preference; that is, the rate that reflects the value an individual placcs on the consumption of a marginal dollar over time. This will be 5 percent in the currcnt example.
Weighted Average Cost of Capital. In reality, public investments include funds that come from both private investment and private consumption. Sincc funds come from private investment as well as from private consumption, a better alternative will be to take an average of the both * The result will be an average cost of capital, called the weighted average cost of capital. We can formally write it as
[3.27J
r' = sra+crb
where r' is the weighted average cost of capital, s is the proportion of resources that comcs from private investment, ra is the opportunity cost of capital, с is the proportion of resources that comes from private consumption, and rb is the marginal rate of time preference. Note that s and с are the weights corresponding to ra and rb, respectively.
Since the total amount of resources available to a government can come cither from private investment or consumption, or both, s+c must be always equal to one. To give an example, assume that we have information on all the terms in Equation 3.27 such that s = 0.8, ra = 0.085, с = 0.2, and rb = 0.065. The weighted average cost of capital will, therefore, be
r' = sra+crb = sra+(l-s)rb
= (0.8)(0.085)+(0.2)(0.065) = 0.068 + 0.013 = 0.081
or 8.1 percent.
Although useful as a concept, the weighted average cost of capital has a major drawback. The existence of market imperfections, such as financial regulations and taxes can greatly diminish the attractiveness of both the opportunity cost of capital and the marginal rate of time preference. This is because most individuals do not find financial regulations or taxes attractive, especially if it has to come at the expense of their investment or consumption decisions. Thus, any average based on either of the two alternatives as a means for discounting costs or benefits will not necessarily be accurate. A better alternative, especially from the perspective of society (since most public projects produce results that affect the society as a whole) will be to use a rate known as the social rate of time preference.
Social Rate of Time Preference. Social rate of time preference measures the valuation a society' places on current consumption in favor of consumption in the future. The assumption here is that there exists for the society a collective rate of time preference that relates the value of net benefits received at some future points in time to the value of an equivalent net benefit available today. This collective rate of time preference is generally regarded as the market rate of interest represented by long-term government securities. The rationale for using government securities in this ease is that they are riskless and, as such, safe to invest. Economists argue that the collective rate of time preference, as measured by the market rate of interest on riskless securities, is equal to the marginal rate of return on private capital which represents the opportunity cost of capital.5 If this equality does not hold, society will want to increase or decrease its savings until the collective rate of time preference just equals the marginal rate of return on invested capital (after allowing for risks).
Borrowing Rate of Capital. While they may appear attractive from a conceptual point of view, the methods discussed above are not always easy to operational ize. As a result, most governments prefer to use what is commonly known as the borrowing rate of capital. Governments, like individuals as well as private firms and businesses, frequently borrow money to finance their long-term projects and development activities. For a government to be able to borrow, it must compete with the private sector for capital in the capital market. This, in turn, increases the aggregate demand lor capital But how much capital is available at any given time and the extent to which the market is able to meet the demand will eventually determine the rate at which a government will be able to borrow.
Some economists argue that when a government competes with the private sector for capital, it crowds out the capital market.6 This is true only if it can be argued that the return from private investments is much greater than the return from public investments. Even if it were so, the comparison will not be easy to make because of the inherent differences that exist between private and public investments, and the intcrgcnerational effects these investments produce for society. Furthermore, public borrowing is often necessary to provide for economic and social infrastructure, such as roads, bridges, highways, utilities, education, and public safety, which arc essential for creating conditions for private investment Therefore, the argument that public borrowing comes at the expense of private investment does not necessarily bear out.
An Illustrative Example. To illustrate the role the time value of money plays in benefit-cost analysis, let us look at an example where a government is looking into two mutually exclusive projects, A and B, cach with a life span of five years. Let us say that the initial cost of construction for project A is SO.9 million and for project В it is $ 1 million. Both projects produce benefits that occur at different amounts over time. Assume a discount rate of 6 percent. Which of the two projects is worth undertaking for the government?
To dccidc which project our government should select, we can use all three measures, i.e., net benefit, benclit-cost ratio, and net present value but, given a choice, one should always use net present value because it takes the time value of money explicitly into consideration. We can formally express net present value as
T R т С
NPV - £
(3.28]
where NPV is the net present value, г is the rate of discount, B, is the benefit at time
t, C, is the cost at time t, and t = 0, 1,2,....., T.
Expanded fully, Equation 3.28 can be written as
♦ ♦
.2
(ЬГ)Т
NPV - (
(Ьг)° (Ьг)1 (Ьг)"
1 л 1
♦С
2
• [Вл-*В,-»В~-♦........»ВТ-1-[Сл-*с,-
(Ьг)° (Ьг)1 (Ьг)2 Т(Ьг)т °(Ы)° (Ьг)1
(ЬГ)2 '(Ьг)
where 1/(1 -Иг) is the discount factor (DF) one uses to multiply a stream of benefits and costs to produce a discounted stream of benefits and costs.
If there is a scrap or salvage value at the end of the useful life of a project, Equation 3.28 can be further written as
T R cv T p
NPV - [Г —--] - T--
uo(br)1 (br)T W) (Ьг)1
where S VT represents the salvage value at time T, and the rest of the terms as the same as before.
Since our problem involves a one-time cost and does not include any salvage value at the end of the useful lives of cither of the two projects, the net present value will become
npv . £ - C0 [3.30]
where C0 is the one-time cost. Note that a one-time cost means that there arc no subsequent costs a project would incur except for normal repair and maintenance which, in principle, should be treated as the operating cost for the government.
However, if costs occur over the entire length of a project's useful life, rather than one time, then they must be discounted the same way as one would discount a stream of benefits. A typical example of the latter would be when a government borrows funds to finance a project which it must pay in principal and interest at an agreed upon term until the amount borrowed is fully paid off. In this case, the costs the project would incur over time, i.e., the principal and interest plus any other costs, must be discounted by an appropriate rate to determine the present value of costs of the project (as in Equation 3.28).
Table 3.6 shows the benefits and costs for both projects, along with their
Tabic 3.6
Benefit-Cost Analysis with Time Value of Money
Project A
Year | Outlay ($) | Benefit ($) | DF | PVof Benefit ($) |
0 | 900,000 | - | - | - |
1 | - | 700,000 | 0.9434 | 660,380 |
2 | - | 450,000 | 0.8900 | 400,500 |
3 | - | 325,000 | 0.8396 | 272,870 |
4 | - | 275,000 | 0.7921 | 217,828 |
5 | - | 100,000 | 0.7473 | 74,730 |
Total: | 900,000 | 1,850,000 | - | 1,566,308 |
Project В | ||||
Year | Outlay ($) | Benefit ($) | DF | PVof Benefit ($) |
0 | 1,000,000 | - | - | - |
1 | - | 125,000 | 0.9434 | 117,925 |
2 | - | 200,000 | 0.8900 | 178.000 |
3 | - | 475,000 | 08396 | 398,810 |
4 | - | 750,000 | 0.7921 | 594,075 |
5 | - | 525,000 | 0.7473 | 392,333 |
Total: | 1,000,000 | 1,925,000 | - | 1,681,143 |
NPVA: $666,308.00 B:CA = 2.0556
NPVB : $681,143.00 B:CB = 1.9250
corresponding net present value and bcnefit-cost ratio. According to the table, if the objective of the government is to have maximum net present value, then it should select project В since it has a higher net present value. If, on the other hand, the objective is to get the most return for each dollar worth of investment, then projcct Л should be selected sincc it has the higher benefit-cost ratio of the two. Since a government does not undertake projects to maximize returns the same way a private firm or business does, net present value, rather than benefit-cost ratio, should be used when selecting a projcct.
Internal Rate of Return. Net present value works well if we know what the discount rate is or if we are able to obtain it from any of the several measures suggested here. But what if wc do not have this rate, or it is not necessary for us to know this rate a priori? In that case, the alternative is to use a measure, called the internal rate of return (IRR). An internal rate of return is the rate at which the sum of a discounted stream of benefits equals the sum of a discounted stream of costs. In other words, it is the rate at which the net present value is zero.
Wc can formally present the internal rate of return as
т В т С
where В and С are the streams of benefits and costs, respectively; i is the internal rate of return, i.e., the rate at which the sum of a discounted stream of benefits will equal the sum of a discounted stream of costs, and t is the time period that ranges from 0 to T.
Note that we have used i in Equation 3.31, instead of r. since we arc dealing with internal rate of return. In reality, when i is used, one can compare it with the opportunity cost of capital, r, to sec whether a project produces a positive net present value. As a general rule, if i>r, the net present value is greater than zero and the project should be accepted; otherwise, it should be rcjcctcd. In other words, if the internal rate of return is greater than the opportunity cost of capital, select the project, if not, reject it.
There are some advantages to using internal rate of return as a decision rule as opposed to net present value or benefit-cost ratio. For one, it can be empirically determined through trial-and-error. Since internal rate of return is defined as the rate at which the sum of a discounted stream of benefits equals the sum of a discounted stream of costs, a trial-and-error approach will produce the rate easily. Figure 3.3 shows how this rate can be determined. According to the figure, the internal rate of return at which the net present value becomes zero is approximately 17.5 percent. The figure also shows that there is an inverse relationship between net present value and discount rate; that is, the higher the discount rate, the lower the net present value,
NPV (SOOO)
and vicc versa.
In general, a high internal rate of return is associated with high early returns, meaning that benefits will occur at a faster rate during the early years of a project's life, then decline as time progresses. By the same token, a low internal rate of return would indicate a gradual increase in benefits, with higher returns taking place toward the later years, then declining again as time progresses. Thus, if the objective of the decision makers is to realize most of the returns in the early years of a project's useful life, then they should select a projcct with high internal rate of return This explains why firms and businesses have for a long time preferred internal rate of return to net present value because it is to the advantage of the owners or stockholders to know when the return will be maximum for their investment and when it will begin to diminish. (However, the trend has changed considerably in recent years as businesses are focusing more on maximizing revenue in the long run than maximizing profit in the short run).
Another reason why internal rate of return is considered a plausible alternative to both net present value and benefit-cost ratio is that since a government docs not necessarily invest in projects to maximize profit, logically then it should use a discount rate at which the net present value will be zero; that is, one in which there will be no monetary gain or loss for the government. But the problem with this argument is that if there is no net benefit from public investments then, in principle, it is not in the best interest of the society to undertake those investments. The decision makers will have a hard time justifying investments if they fail to produce a net gain for those who arc supposed to benefit from them. As common sense should tell us, a rational decision maker will always select those projects whose benefits outweigh their costs. This is where net present value has a clear advantage over internal rate of return and probably explains why it is more frequently used than any other alternative.
Projects with Different Life Spans
In our example of the two projects, A and B, both had exactly the same life span of five years each. In reality, projects seldom have identical life spans, meaning that their benefits and costs cannot be readily compared without adjusting for the differences in project lives. Without this adjustment, one may end up with projects that are less desirable even though they may have a higher net benefit or net present value. This can create a real problem for the decision makers if they have to go by the outcomes of these decisions. Since time difference in projects plays a critical role in benefit-cost analysis, there is a simple way to resolve the problem: take a number of short-lived projects and compare them with several larger ones in such a way that they all end up at an approximate common time.
To give an example, suppose that we have two mutually exclusive projects, X and Y, with a life span of 3 years for X, and 4 years for Y. To compare the projects, we take four X-type projects and three Y-type projects so that the sum total of their useful lives will be exactly the same, i.e., 12 years each. Assume that for each project we have costs that occur only one time in their life span. Assume also that we want to use net present value as the decision rule. Therefore, we can write the net present value for the two sets of projects as
.9
(Ьг)ш (Ьг)"
[3 32]
[3.33]
where NPV is the net present value, В is the benefit, С is the cost, r is the discount rate, and j, к, 1, and m represent the time streams corresponding to the projects in each group. Note that, in both cases, benefits lag costs by a year, which means that benefits do not begin to take place until the respective projects are completed.
If we ignore the assumption that our projects, instead of having one-time costs, have costs that are spread over their entire useful lives, Equations 3.32 and 3.33 will become
) d э /"• ( d ( p > r > г
npvx . (£ -A--E -SUE Л-Е ^TME —rZ Ai
И (ЬгУ )-« (ЬгУ к-* (Ьг) м (Ьг) 1-7 (Ьг)1 1.7 (ЬГ)1
►[Е
ю-10 (Ьг)т т-10 (Ьг)'
12
-Е
-]
[3.34]
Л Л ci Л вк «А ct " в. » с. npvy - [Е —*--Е —^МЕ —VE —ME —4-Е —л
>1 (ЬгУ и (1.гУ ы (Ьг) Ь5 (Ьг)к w (ЬгУ w (ЬгУ
[3.35]
where the terms of the equations arc the same as before.
Although it may appear simple, the actual process of selecting projects in order to organize them into groups can be quite difficult. Part of the problem is in finding appropriate projects that are similar so that they can be linked together in some meaningful ways. As a result, when one is considering projects with different life spans it is important that there are some commonalities among them that would justify their inclusion in a particular group. In other words, one must not arbitrarily lump different projects together simply because their combined life span will come out to be the same. To do so will not result in any better solution than when they are treated as individual projects.
When it becomes difficult to find projects that are similar and cannot be grouped together, one should try to use alternative methods to deal with the problem. Several methods have been suggested in recent years for this purpose. Two of these methods are discussed next.
Annual-Cost Method. The principle behind combining a group of projects with uneven life spans, although makes perfect sense, is time consuming because of the requirement that one must find a new duration for the common denominator for their useful lives. To avoid this problem, in particular for costs, most practitioners use an alternative approach known as the annual-cost method The essence of the method is
that it converts a projcct lasting T years with a present value, P VT, to an equivalent 1 -year cost (based usually at the end of the year). In other words, it transforms all systems on a 1 -year basis. The method can be formally expressed as
(br)T (lT)T-l
<V(Pvt>
13-36]
where CA is the annual (A) cost, PVT is the single present value of a time series T, and г is the rate of discount. Note that the second part of the expression is callcd the capital recovery factor, which is used to convcrt a single cost to an equivalent uniform end-of-year annual cost.
To give an example, consider a situtation where we have two projects, X and Y; X with a useful life of three years, and Y four years. Let us say that we have some information on the present value of both projects which, respectively, are: $512,725 for X and $615,975 for Y. Assume now a 7 percent rate of discount. The annual costs of the projects based on the above expression will, therefore, be
Ск.(РУт)
[3.37]
(Ьг)Т (Ьг)т-1
where CK is the capitalized (k) cost, P V is the present value of cost at time T, and г is the rate of dicsount Note that the second part of the expression represents the factor that converts a present value to capitalized cost, called the capitalized cost factor.
Сд.х о07j = (512,725)(0.3813) = $ 199,502.04 C„Y 007'4 = (615,975X0.2954) = $181,959.02
The results indicate that project Y is more economical even though it costs more. This can be seen from the cost ratio of the two projects, which is 1.0964, obtained by dividing the annual cost of X by the annual cost of Y.
Capitalized Cost. Another alternative to the problem is to use a method, called the capitalized cost Capitalized cost is the present value of annual costs. Like the annual-cost method, it also reduces the lives of projects to a common denominator. We can formally express the capitalized cost for a project as
To see how the method works, we can look at the same example we used for the annual cost method. Assuming that we have the same rate of discount as the annualized cost, the capitalized costs for the two projects will then be
= (512,725)(7.9013) = $4,051,194.00 = (615,975)(0.2954) = S3.354.599.90
r
As before, we can take the ratio of the two costs to see which project is more economical. It turns out to be project Y, as expected.
One of the essential requirements of all benefit-cost analysis is that one must be able to express the benefits and costs in precise monetary' terms. But expressing costs and benefits, in particular the latter in dollar terms is not always possible, especially where many of the benefits arc intangible in nature. The difficulty in quantifying benefits makes it more difficult to assign dollar values to them in a consistent manner. For instance, how would one assign dollar values to the benefits a neighborhood recreation facility produces for a community or to the lives of children a public safety program saves or to the improvements in public welfare a particular social policy generates?
When benefits arc difficult to measure in monetary terms, the alternative is to use a method known as cost-effectiveness analysis. In cost-effectiveness analysis, as in benefit-cost analysis, costs arc presented in monetary terms, while the benefits are presented by the amount of nonmonetary effects a project produces for a given cost, rather than by its dollar amount. The objective, however, is the same as in benefit-cost analysis in that it purports to select the best alternative from several possible choices. The only difference is that in cost-effectiveness analysis, the comparison is made in terms of their costs and their effectiveness without the requirement of expressing the latter in dollar terms.7
Like benefit-cost ratio, cost-effectiveness can be measured by a ratio, callcd the cost-effectiveness ratio. It is determined by comparing the monetary effects of a projcct or program with its nonmonetary effects, and can be expressed as
[3.38J
C:E - —— NME
where C:E is the cost-effectiveness ratio, С represents cost or monetary effects, and NME represents benefits or nonmonetary effects.
There is a similarity between cost-effectiveness ratio and the bcnefit-cost ratio in that both measure the effectiveness of funds invested in one or more projects in relation to the results they produce. But unlike the benefit-cost ratio, the decision rule for cost effectiveness does not revolve around a single measure, rather it depends on the goals and objectives a decision maker hopes to achieve from a given investment.
To give an example, consider a situation where the Federal Drug Administration (FDA) is looking into the cffccts of five different drugs ready to be introduced into the market for patients seriously afflictcd with a life-threatening disease, which we will call Xuben. Let us say that each drug was tested on 100 patients with varying levels of costs and recovery rates. Table 3.7 shows the results of these tests, along with their costs and effectiveness measures. According to the table, drug A has the lowest cost-cffectivencss ratio with a value of 17,857.14. What this means is that it will cost the government on average $17,857.14 for each successful treatment using drug A. In other words, it is the cost per recovery for drug A. Similarly, the cost per recovery for drug В is $40,882.35, and for drug С it is $33,538.46, and so on.
The result produces an interesting dilemma in that wc cannot quite tell which drug the administration should recommend by simply looking at the ratios since they arc not consistent either with costs or effectiveness (recovery rates). I lowever, if we had some knowledge of the decision makers' goals and objectives or the factors that are critical to their decision making, it will not be so difficult to resolve the problem. For instance, if we knew that costs were an important determining factor in making the choice, then drug A should be recommended since it has the lowest cost per recovery. In other words, it is the most cost-effective alternative for the money the government could successfully spend on a patient. On the other hand, if recovery rate is considered more important, thai drug D should be recommended since it has the highest recovery rate.
Present Value of Cost-Effectiveness
The cost-cffcctivencss ratios presenetd above are based on cost per recovery for a specific drug, but they may not be an adequate measure of effectiveness since we do not have any information on factors, such as life expectancy or the number of years a patient survives after receiving the initial treatment. Therefore, a much better alternative for the effectiveness measure will be to use the number of years a patient survives, rather than their recovery rates. From a time value perspective, this would further allow us to determine the present value of recovery by discounting the number of years a patient survives on average (for a given drug), similar to the present value calculation for benefit-cost analysis.
To illustrate the point suppose that we now have some information on the average number of years a patient survives after receiving the initial treatment from a drug. Assume that this average is roughly 12 years. We can use this average in the same way
Table 3.7
Cost-Effectiveness Analysis Involving Multiple Programs
Drug | Total Cost, С (S) | Effectiveness, NME (% of Recovery) | C:E ($/Recovery) |
A | 1,250,000 | 70 | 17,857.14 |
В | 3,475,000 | 85 | 40,882.35 |
С | 2,180,000 | 65 | 33,538.46 |
D | 4,325,000 | 90 | 48,055.56 |
E | 1,720,000 | 75 | 22,933.33 |
Drug | Years Survived | PV of Recovery (In Years) | PV of C:E (In S) |
A | 10 | 8.9286 | 1,999.934 |
В | 13 | 11.6072 | 3,522.155 |
С | 11 | 98215 | 3,414.800 |
D | 15 | 13.3929 | 3,588.136 |
E | 12 | 10.7143 | 2,140.442 |
as one would use a discount rate to calculatc the present value of a future stream of benefits and costs. The only difference is that our present value will now be regarded as the present value of recovery in life years. Therefore, to calculatc the present value of recovery for each drug, we simply incorporate the average number of years a patient survives (following an initial treatment) into the discount factor so that
£ —
PV
[3.39]
where PVS is the present value of recovery, l/( 1 +r) is the discount factor, and t is the number of years a patient survives, which ranges from 0 to T.
To determine the cost-effectiveness ratio for cach drug in terms of the effects of present value of recovery, we can rewrite the expression in Equation 3.38 by dividing the cost of treatment by the product of nonmonetary effects and the present value of recovery. Wc will call the resulting ratio, the present value of cost-effectiveness ratio, or simply the cost per present value of recovery. Equation 3.40 shows this ratio below:
(NME XPV,) [3.40]
where PVCE is the present value of cost-effectiveness ratio, С is the cost of treatment, NME is the nonmonetary effects, and PVS is the present value of recovery.
The bottom half of Table 3.7 shows these new ratios, together with the present value of recovery, and the average number of years a patient survives (for a given drug) after receiving the initial treatment. Note that the last column, the present value of cost-effectivencss ratio, is obtained by dividing the cost of treatment for each drug by the effectiveness rate times the present value of recovery. The result produces the cost per present value of recovery for cach drug; that is, l,2500/[(70)(8.9286] = $ 1,999.934 for drug A, 3,475,00/[(85)(l 1.6072)] = $3,522.155 for drug B, and so on.
The result does not seem to be inconsistent with what we observed earlier. For instance, if cost is an important determining factor in making decisions, then drug E, instead of A, should be rccommcndcd sincc it produces the most effcctivc result on a per patient basis. On the other hand, if the number of years a patient survives after the initial treatment is a crucial determining factor, then drug D should be selected since it will be far more effective in prolonging life than any of the other drugs used in the example.
The decision the FDA has to make on the basis of these simple rules may seem somewhat cut-and-dried, but one must realize that the purpose of using any measures, regardless of their methodological considerations, is to help the decision makers make a rational choicc when faced with difficult decision problems. In that regard, all measures, including cost-effectiveness analysis, must be treated as simple decision aids and nothing more. As anyone familiar with project evaluation should know, the ultimate decision will always rest with the decision makers, their familiarity with the problem, their goals and objectives vis-a-vis the government, and the constraints facing their decision.
For both benefit-cost and cost-effectiveness analysis, it is important to take an additional factor into consideration that has not received much attention in the public sector use of these methods: it is the role of uncertainty. Most decisions in today's world have some elements of uncertainly (as well as risk) that cannot be avoided The reason for this uncertainty is that not all components that go into the calculation of costs, benefits, or effectiveness are known with certainty. Since they are mostly estimates, they will not be the exact values of the costs and benefits that will occur over time. For instance, a project thai will cost $ 18 million to complete is an expected, but not an actual cost. Although, in theory, we expect Ihe difference between the two to be zero or negligible, in reality, it is seldom the case. Therefore, it must be supplemented with a measure (margin) of uncertainty. In other words, it must be added to the cost.
The problem the decision makers will encounter here is where to set this margin. One way to overcome the problem is to look at a number of similar projects and take an average of their margins. Another alternative would be to use a method, called the certainty equivalent method, where a decision maker's utility preference is directly incorporated into the investment process. A third alternative would be to use a method, called a risk-adjusted discount rate, where projects with higher variability in returns are discounted at a higher rate than projects with lower variability. In fact, there is a whole range of methods that can be used to deal with the uncertainty problem in benefit-cost analysis. However, the point that is being emphasized here is that without some recognition of uncertainly in bcnefil and cost calculations, it is quite likely that the decision makers will face a range of problems that they did not anticipate, such as unexpected cost overruns, delay in project completion, etc.
This chapter has presented a brief introduction to cost analysis in government in lerms of several commonly used techniques: break-even analysis, differential cost analysis, benefit-cost analysis, and cost-effectiveness analysis. Of these, break-even analysis is one of the oldest and perhaps the most widely used techniques in cost analysis. Its primary purpose is to bring together the cost, revenue, and output of an activity to determine the best course of action for a government. Like break-even analysis, differential cost analysis is used to determine an appropriate coursc of action, based on cost, rcvenume, and output level of an activity. Although they are not quite comparable, the advantage of using differential cost analysis as opposed to break-even analysis is that it gives the decision makers an opportunity to evaluate the viability of an alternative decision against the existing condition(s). The chapter, in particular, looked at three very special cases of differential cost analysis: the cffects of a change in cost and revenue on net return, recovery time for an investment or payback period, and knowing when to buy or lease. All three of these techniques arc useful as tools when comparing the merits of alternative decisions.
The two other techniques discussed in the chapter arc benefit-cost and cost-effectiveness analysis. Both of these decision tools, though in particular benefit-cost analysis, have been extensively used in government for all kinds of projeets and programs. As an evaluative method, benefit-cost analysis is not difficult to use, provided that one has information on benefits and costs that can be expressed in precise monetary terms. Where monetary measures are not available, it is possible to use surrogates or proxies, but defining surrogates that will accurately measure the benefits and costs of a project is a difficult exercise. The alternative is to use methods, such as cost-effectiveness analysis, which do not have some of these problems that are inherent in bcncfit-cost analysis.
Notes
1. W. Rautenstrausch, Economics of Business Enterprise. New York, NY: John Wiley and Sons, 1939.
2. M.S. Thompson, Benefit-Cost Analysis for Program Evaluation. Beverly Hills, СЛ: Sage Publications. Fourth printing, 1982: 1-2.
3. E.J. Mishan, Introduction to Normative Economics. New York, NY; Oxford University Press, 1981.
4. E.M. Gramlich, Benefit-Cost Analysis of Government Programs. Englcwood Cliffs, NJ: Prentice-Hall, 1981:98-100.
5. R.F. Mikcsell, The Rate of Discount for Evaluating Public Pojects. Washington, DC: American Enterprise Institute for Studies in Economic Policy, 1977.
6. R. Dornbusch and S. Fisher, Macroeconomics. New York, NY: McGraw-I lill Publishing Company, 1990: 149-157.
7. E.S. Quade, Analysis for Public Decisions. New York, NY: North Holland, 1982: 207-210.
Cosl accounting is the universal language of an organization. There is not a single organization anywhere that does not understand or is not concerned about costs. As organizations become complex, requiring more diverse and intelligent information to survive, the need for cost accounting becomes even more important. The primary objective of cost accounting is to provide cost and related information that is essential for an organization to determine how to run its internal operations; in particular, how to allocate its costs, how to control them, and what measures to take to improve its cost performance. In government, where efficiency and accountability have always been a concern, cost accounting can fill an important gap by producing this critical information to help overcome that concern
This chapter presents several topics that, according to conventional wisdom, constitute the essence of cost accounting. They are: job costing, process costing, variable costing, cost allocation, and cost control. Somewhat more specific than those discussed in the previous chapters, they provide the nuts and bolts of accounting for costs in an organization. The chapter begins with a brief background discussion on cost accounting and the system that underlies it before presenting these topics in details.
Cost accounting deals with the costs an organization incurs in canrying out its normal, everyday activities and the proccss of accounting for those costs. The process is complex, involving a number of distinct, fairly-detailed, and well-defined activities. These activities primarily include recording, analyzing, summarizing, evaluating, and interpreting costs and related information and communicating (reporting) the results to those who would directly benefit from them. In government, this would mostly include the chief administrator, elected officials, various agency heads, internal auditors, plus any other individual who is directly involved in the routine operations of a government. Although individuals outside of government may benefit from the information the process generates, it is primarily used for internal consumption.
Cost accounting is often discussed in conjunction with two other fields of account-ting, namely financial and managerial accounting. Although the three are discussed in the same vein, they arc not quite the same. What distinguishes them is the basis of use, i.e., who uses the information generated by the accounting process and the purpose it serves. For instance, financial accounting produces information that is primarily used for external consumption to meet the needs of outside individuals and organizations, such as the bond rating institutions, investors in government securities, concemcd citizens, and other external evaluators. Much of the information financial accounting produces is presented in the form of financial statements that contain detailed accounts of the financial position of a government. To ensure that these statements arc presented fairly and accurately, they are prepared according to a set of prescribed guidelines, known as generally accepted accounting principles (GAAP).
GAAP consists of a set of rules, conventions, and procedures that define the accepted accounting principles in operational terms. In government, the responsibility for establishing these rules falls on the Governmental Accounting Standards Board (GASB), an autonomous organization headed by a chairman, a full-time director, and several part-time members.1 The board reports directly to the central accounting organization, called the Financial Accounting Foundation (FAF). The foundation, in turn, sclccts its members and raises funds to support its activities and those of its pnvate sector counterpart, called the Financial Accounting Standards Board (FASB).
In contrast, managerial accounting produces information that is mostly used by agcncics, internal auditors, elcctcd officials, and others who are dircctly involved in the day-to-day operations of a government. Since the information produced by managerial accounting is largely used for internal consumption, it is not governed by the generally accepted accounting principles and, as such, is considered more flexible than financial accounting. It is also more eclectic, drawing heavily from other disciplines, such as planning, economics, mathematics, operations research, finance, and management.
In that regard, there is a similarity between cost and managerial accounting. Both rely heavily on other disciplines for substantive growth, both provide information that is used for internal consumption and, in both cases, their practices are not governed by GAAP. Because of these apparent similarities, the two fields are often treated as one, with one major difference: cost accounting is more concerned with the routine operations of an organization, whereas managerial accounting is more future oriented.
All costs boil down to three things: materials, labor, and overhead. A responsible manager must know how to keep track of these costs as they accumulate. In the private sector, managers keep track of their costs by assigning them to an individual consumer, a cost unit, or to a unit of a good produced. When costs are assigned to an individual consumer, a cost unit, or to a unit of a good produced, it is called unit costing. Unit costing is a useful method when applied to goods that are divisible and there is no frcc-nding problem such that one is able to trace the costs to an individual costing unit. Most private goods would fall into this category.
However, for a majority of public goods and services, with the exception of those provided by the proprietary funds, it is difficult to determine the exact quantity an individual consumes of these goods and services. As a result, public organizations often maintain cost information based on a specific job, service, or activity rather than on a per capita basis, although the latter may be implicit in most cost calculations. For instance, a county hospital may want to know the costs it would incur in providing a particular service, such as immunizing children against flu or providing outpatient care for the elderly. Similarly, a local school district may want to know the cost it would incur in running a computer literacy program for disadvantaged children or the costs a city recreation department would entail in organizing a summer youth festival.
In reality, one should not have much difficulty in assigning costs to a specific activity, especially if the costs are direct. When costs are indirect, i.e., when they cannot be directly attributed to a job or activity, it becomes difficult to assign them to specific activities, in which case one must use some rational basis for cost allocation. Proper assignment of costs is important for an organization to improve performance, ensure control, and increase efficiency in cost management. This process of assigning costs to a specific job, scrvice, or activity, and allocating them using established procedures or some rational basis to ensure control is known as cost accounting system. The five topics that arc to be discusscd next (job costing, process costing, variable costing, cost allocation, and cost control) are all integral to this system.
Job costing is an accounting system that traccs costs to a specific job, service, or activity. Tracing costs to a specific job makes it possible for an organization to determine the costs it will accumulate separately for materials, labor, and overhead. When materials and labor used in a job can be directly traced to it, they are called direct materials and direct labor. Indirect materials and labor that cannot be directly traced to a job are generally treated as overhead.
Job costing begins when an operating department receives instructions from the responsible authority to begin work on a job. During this stage materials arc issued, labor is expended, and overhead is incurred. Three things must take place to complete the process: a number of source documents must be prepared, the information must be entered into a journal, and the overhead rates must be determined.
Source documents are materials used to accumulate the costs for an individual job. The most frequently used source document is the job cost sheet, prepared primarily to accumulate and summarize cost information on direct materials, direct labor, and overhead. Л typical job cost sheet includes the following: job identification number, job description, beginning and completion dates, and costs of materials, labor, and overhead. The actual entries are made from materials requisition forms, labor cost sheets, and from other evidence of costs.
A materials requisition form provides a detailed account of the type and quantity of materials that arc to be drawn from a central location, such as a central warehouse, and the job to which the materials are to be chargcd. The labor cost sheet specifics the number of hours worked, the rate at which the labor was paid, and the total cost of the labor used. Both the requisition form and the labor cost sheet serve as a means for controlling the flow of materials and labor into the job completion process and, in particular, for making entries into the job cost sheet.
Table 4.1 presents a simple example of a job cost sheet for street repair for a local government. According to the table, it cost the government $220,000 in total repair, including $45,000 in direct materials, $125,000 in direct labor, and $50,000 in overhead. The table also provides information on the exact nature of the job performed, the beginning and completion dates, the job requisition number, and the department responsible for the job.
As noted earlier, job costing does not begin until the operating department receives notification that a service order has been issued for the job. Once the direct materials have been issued and the labor costs identified, the operating department makes entries into the job cost sheet charging the job noted on the sheet with the costs of materials and labor used. When the job is completed, the total costs of materials and labor plus the overhead are summarized and presented on the sheet.
As the information on materials, labor, and overhead costs bccomes available, they are entered into a journal, called the book of initial entry. The purpose of journalizing the entries is to show the flow of costs as they accumulate. Several basic transactions arc recorded in a journal, such as the purchase of materials, issuance of direct and indirect materials, accrual of direct and indirect labor, payment of labor, incurrence of overheads other than materials and labor, and so on.
To enter the transactions in a journal, the convention is to use a double-entry accounting system, called T-accounts. A T-account has two sides, a left side (called debit), and a right side (callcd crcdit). As a general rule, when liabilities are incurred they arc posted on the credit side and when assets are acquired they are posted on the debit side. Since costs arc considered a liability, any increase in costs is recorded on
Table 4.1
A Job Cost Sheet for Street Repair
Summary' Information
Job summary: Job number Date started Date completed Job description Job location
Cost summary:
Requisition number Responsible department
Actual cost:
Direct materials Direct labor
Overhead ($35,000/labor + $ 15,000/materials)
037-PW
August 15, 20XX August 31, 20XX
Repairing asphalt/Concrete, and leveling 19th Street and Flint
012-PW Public Works
$ 45,000 125,000 50,000
the credit side. By the same token, any decrease in costs is recorded on the debit side since it represents a decrease in liability. Convention dictates that when liabilities are incurred by a government, they should be recognized as such, and vouchers should be issued before paying off the liabilities in cash. Therefore, the journal entries must include a recognition of liability when vouchers are issued and a payment of liability when a cash payment is made.2
Since it is a double-entry accounting system, the two sides of a T-account must be equal, i.e., the entries must be balanced. Thus, if a government purchases a vehiclc, it will reflect an increase in its asset position and, as such, it should be debited. At the same time, it will also increase the liability of the government since it will have to pay for the vehicle and, therefore, it should be credited. If the vehicle is paid in full, the two sides should even out. On the other hand, if the payment is not made in full during the accounting period, the difference will appear as "balance" on the liability side of the ledger. The latter is known as the book of final entry.
To give an example of how this applies to job costing, let us go back to the street repair problem. Suppose the central warehouse of the government purchases materials for the job in the amount of $90,000, which include $65,000 in direct and $25,000 in indirect materials, and pays for them in cash. It is not unusual for a warehouse to have these materials in its inventory, in which case the journal entry would recognize them as a simple inventory account with no outstanding obligation or payment of liability to make.
Total cost:
Assume the first scenario that the government did not have the materials in its inventory and that the warehouse had to purchase them. The journal entries for the transactions can then be shown as
1. Purchase of Materials (Direct) 65,000 Purchase of Materials (Indirect) 25,000
Vouchers payable 90,000
To record purchase of materials
2. Vouchers Payable for Materials (Direct) 65,000 Vouchers Payable for Materials (Indirect) 25,000
Cash 90,000
45,000 15,000
To record payment of materials
When direct materials are issued for a job, they are recorded in an account, called Work in Process (WIP), which is really an inventory account. Indirect materials, on the other hand, are recorded in the overhead account because all costs associated with a job, except for direct materials and labor, are accumulated in this account. The journal entries for the WIP account for materials, based on the amount issued, can be presented as
WIP Materials (Direct) 45,000
Overhead Materials (Indirect) 15,000
Inventory of Materials (Direct) Inventory of Materials (Indirect)
To record issuance of materials
Similarly, the direct labor costs accumulated for a job are recorded in the WIP account as debit and the indirect labor costs are also debited to the overhead account. As before, to pay for the labor, vouchers must be issued (recognition of liability) and
paid out in cash (payment of liability). The entries for the labor cost transactions can thus be shown as
1. WIP Labor (Direct) Overhead Labor (Indirect) Vouchers payable
125,000 35,000
160,000
To record the costs of labor
2. Vouchers Payable for Labor (Direct) Vouchers Payable for Overhead Labor (Indirect) Cash
125,000
35,000
160,000
To record payment of labor
The process can be continued until every single transaction related to the job is exhausted. Once the journal entries arc completed, they are posted into a ledger (not shown here). A ledger is a group of accounts, where the data from a journal arc recorded and summarized by accounts. The summary information in the ledger is then transferred to the job cost sheet and serves as the database for it.
As noted before, overhead costs are indirect costs that cannot be attributed to a specific job. Some overheads, especially those related to utilities, are not always known until the end of a given period. Rather than hold the completed jobs in an inventory until all costs can be traced to it, it may be necessary to develop a system of allocating overhead costs to a job on a predetermined basis. The simplest way to calculate a predetermined overhead cost is to divide the estimated overhead for, say an entire year, by an appropriate base, such as direct material hours, direct labor hours, etc.
The following expression shows the procedure for calculating the predetermined overhead rate for direct labor hour:
[4.1]
where OR^ is the overhead rate for direct labor hour, TOC is the total overhead cost for the entire organization , DLH>i is the dircct labor hour attributable to job i, with i =
1,2,3,......, m. It is worth noting here that when actual data on overhead costs and
indirect labor are not available, one can use estimates of these costs, in which case the numerator of the expression in Equation 4.1 will represent the estimated total overhead costs and the denominator the estimated total dircct labor hours.
To give an example, assume that our government for the street repair problem has estimated that its total overhead costs and dircct labor hours for the year would be $ 15,850,000 and 317,000, respectively. Its predetermined overhead rate per dircct labor hour for the year, based on this information, will then be
15,850,000 317,000
OR
-50
DLH
which is $50.
Assume further that the street repair job requires a total of 700 hours of direct labor. Therefore, the total overhead cost for the repair, based on a labor cost of $50 an hour, will be $35,000. Since the selection of a predetermined rate requires that estimates be used for both the numerator (total overhead cost) and the denominator (direct labor hour), most practitioners prefer to use a rulc-of-thumb approach to determine this rate. It is obtained by taking the ratio of an agency budget to the total budget of a government.* For instance, if an agency has an annual budget of $2,382,500 that is approximately 5 percent of the total budget of the government, the overhead allocation for the agency will be SI 19,125, which is five percent of the total budget. The same principle can be applied to allocate the overhead costs for every agency in the government.
While the method suggested above is simple, there is a problem with it in that if the agency budget constitutes an unusually large (or small) fraction of the total budget, it may overestimate (or underestimate) the overhead allocation. Also, when money is frequently transferred between funds, a common practice in government, it may misrepresent the overhead allocation by the amount of the transfer. The alternative is to remove the etfects of interfund transfers before the method can be used.
The general position taken here, in particular for the predetermined rate, is that overheads should be allocated on the basis of the factors or activities that cause these costs. In other words, the more an activity causcs an overhead, the higher the cost that should be charged to it. fhe trick is to select a base, such as labor or machine hours that is commonly used by an organization so that its overhead application will be equitable among jobs. Since the focus is on the base, another term used for it is activity-based costing, or ABC in short.
Proccss costing is a system of accounting in which costs are assigned equally to all units of a good or service produced by an organization or its constituent units. Since costs are equally assigned to all units in a proccss costing, it is assumed that the units produced are homogeneous. As a result, process costing is useful in situations where there is a continuous flow of production, such as gas, water, and electricity, where product variation is almost nonexistent. But it can also be applied to situations where the units of production are discrete, rather than continuous, such as crime prevention, trash collection, treatment for diseases, and so on as long as one is able to measure them in distinct units and their essential characteristics remain the same throughout the proccss.
Since it is required that the units produced must be homogeneous, in process costing one accumulates costs by department rather than by job and assigns these costs equally to all units that pass through the department. Thus, in a proccss costing system, instead of using a job cost sheet, one uses a set of reports, called schedules. Three types of schedules are commonly used for this purpose: a schedule of equivalent pr<xluction, a schedule of cost analysis, and a schedule of cost summary
Schedule of Equivalent Production
When costs arc assigned equally to all units of a good or servicc as they pass through a department, convention dictates that only those units that have been completed should be taken into consideration. In proccss costing, this would includc completed as well as partially completed units. Together, these units (completed and partially completed) are called equivalent units. The rationale for including partially completed units in cost assignment is that even though they have not been fully completed, the department(s) involved in the proccss did incur some costs (matenals, labor, and overhead) for the portion of the job that has been completed. Therefore, to accurately measure the output produced during a given period, one must includc both completed and partially completed units.
Two methods are commonly used in calculating the equivalent units for a department: a weighted average method, and a first-in, first-out (FIFO) method. Ik>th methods are simple in concept and can be used together or interchangeably for the same problem.
Weighted Average Method. Acccording to this method, the equivalent units are calculated as the number of units completed during a given period plus the units that are incomplete times the percentage of their completion. In other words, equivalent units include both completed and partially completed units. To give an example, consider the police department of a large metropolitan government that has been investigating 125 cases of homicides this past year. Let us say that 45 of these eases have been solved completely by the department and 80 remain unsolved by the end of the year. Since 45 cases were fully completed, we can safely assume that these cases received 100 percent of materials, labor, and overhead needed to complete them
Of the cases that remain unsolved, assume that 75 percent were completed as to materials and 60 percent as to direct labor and overhead, called conversion, when the year ended. The equivalent units for the department, therefore, will be 60 for materials and 48 for conversion, obtained by the following expression:
[4.2]
EU - (UP)(PC)
where EU is the total equivalent units, UP is the units remaining in process, and PC is the percentage of completion.
To explain how the equivalent units for partially completed cases were obtained, we simply followed the equation (4.2) and multiplied the units remaining in process by the percentage completed. The result produced 60 units for materials and 48 units for conversion; that is, UP x PC = 80 x 0.75 = 60 and UP x PC = 80 x 0.60 = 48, respectively. However, in order to obtain the total equivalent units, it is necessary to modify Equation 4.2 so that the new equation will include the units that have been completed as well as those remaining in process, as shown below:
[4-3]
EU = UC + (UP x PC)
where EU is the equivalent units, UC is the units completed, and the rest of the terms are the same as before.
Now, applying the above expression in Equation 4.3 to the problem we can obtain the respective equivalent units. That is,
EU = UC + (UP x PC)
45 + (80x0.75)
= 45 + 60 = 105 units
for materials, and
EU = UC + (UP x PC)
= 45+ (80x0.60) = 45+48 = 93 units
for conversion.
Table 4.2 shows the total equivalent units for materials and conversion, together with information on the units to be accounted for by the department. Note that we made an implicit assumption in the example that all 125 eases of homicides occurred during the accounting period, which means that there were no carry overs from the previous year. As a result, no units appear under the columns for materials and conversion for the beginning inventory. 13ut if we changc the assumption that there were cam- overs from previous years, the result would be different. This is where the second method, FIFO, comes in
First-in, First-out Method. The calculation of equivalent units under the FIFO method is different from the weighted average method in that it takes into consideration the beginning inventory, i.e., the carry overs from previous periods. This is particularly important in situations, such as homicides, where the results (outputs) of a proccss do not always get completed within a given period and where there are frequent сагту overs from previous periods. In other words, there is a continuous flow of output in the production process such that the units remaining in process at the end of one period bccomc the beginning inventory in the next period.
Thas, in calculating the equivalent units for an organization one needs to take into consideration the units in the beginning inventory. In order to do so, we need to reformulate the expression in Equation 4.3 for equivalent units by taking the beginning inventory into consideration, as shown below:
EU = BI + UC + (UP x PC) [4.4J
where 131 rcsprcscnts the beginning inventory and the rest of the terms are the same as before.
Tabic 4.2 also shows the equivalent units for the department based on the FIFO method (lower half). According to the table, of the 45 units that were fully completed, 15 were in the beginning inventory. That means all 15 units, like the remaining 30 in this group, received full materials, labor, and overhead support needed for their completion. As a result, these 15 units also appear under the materials and conversion columns for the beginning inventory.
As for the units in ending inventory, assume that 60 perccnt were completed with regard to materials and 25 percent with regard to conversion, thereby producing a total
Tabic 4.2
Schedule of Equivalent Production Year Ended: August 31, 20XX
Stages of Completion | Units to be Completed | Equivalent Units | |
Materials | Conversion | ||
I. Without beginning inventory | |||
Beginning inventory | 45 | 45 | 45 |
Units started and completed | |||
Ending inventor)' | |||
Work in progress | 80 | - | - |
Materials at 75% | - | 60 | - |
Conversion at 60% | - | - | 48 |
Total units: | 125 | 105 | 93 |
II. With beginning inventory | |||
Beginning inventory | 15 | 15 | 15 |
Units started and completed | 30 | 30 | 30 |
Ending inventory | |||
Work in progress | 80 | - | - |
Materials at 60% | - | 48 | - |
Conversion at 25% | - | - | 20 |
Total units: | 125 | 93 | 65 |
of 93 and 65 equivalent units for the respective categories, as can be seen from the following expressions:
EU = BI + UC + (UPxPC) = 15 + 30+ (80x0.6) = 15 + 30 + 48 = 93 units
for materials, and
EU = BI + UC + (UP x PC)
= 15+ 30+ (80x0.25) = 15 + 30 + 20 = 65 units.
for conversion.
Note that these results also appear under the materials and the conversion columns in Table 4.2. It should be pointed out that there is a fundamental difference in application when these methods, in particular the FIFO, are used in government as opposed to when they are used in the private sector. For instance, when the FIFO is used in the private sector, the units in beginning inventory are usually completed first and then transferred out before beginning to work on the current inventory, hence the name first-in, first-out. But in government, the units in beginning inventory do not necessarily have to be completed before dealing with the units that began in the current period. They can be addressed simultaneously or in any sequence but, from an accounting point of view, they must always be recognized in calculating the total equivalent units for a given period.
Once we know the number of equivalent units in an inventory, we can computc the cost of total equivalent units as well as the cost per equivalent unit for a department. The procedures for calculating these costs are quite simple: the former is obtained by simply adding all the costs in an inventory, whereas the latter is obtained by dividing the total cost by the number of equivalent units. The procedures apply equally to both methods, the weighted average and the FIFO.
Table 4.3 presents a schedule of cost analysis for the department with and without the beginning inventory. As the table shows, the total cost for the department without the beginning inventory is $6.89 million, which is obtained by adding the costs of materials (SI,564,250) and the costs of conversion ($5,325,750). When these costs arc divided by the corresponding equivalent units (from Table 4.2), we obtain the cost per equivalent unit of $14,897.62 for materials and $57,266.13 for conversion. However, when the costs for beginning inventory are included in the schedule, it changes the picture somewhat. Assume that the total costs to be accounted for remain the same at $6.89 million, the cost per equivalent unit (with the beginning inventory included) will now increase to $16,819.89 for materials and $81,934.62 for conversion, as expected.
To verily' if our calculation is correct, we can multiply the cost per equivalent unit by the corresponding equivalent units for each cost category, then add them together to see if it produces a result that is equal to the total cost of operation for the department. Since it produces a total cost of $6.89 million, cxccpt for the rounding-off error, the result appears to be corrcct, as shown below:
Table 4.3
Schedule of Cost Analysis Year Ended August 31, 20XX
Cost Description | Cost ($) | ||
Materials | Conversion | Total | |
I. Without beginning | |||
inventory | |||
Costs for: | - | - | - |
Beginning inventor)' | 1,564,250 | 5,325,750 | 6,890,000 |
Current period | 105 | 93 | - |
Equivalent units | 14,897.62 | 57,266.13 | 72,163.75 |
Cost/Equivalent units | |||
II. With beginning | |||
inventory | |||
Costs for: | |||
Beginning inventor)' | 340,054.35 | 1,401,513.19 | 1,741,567.54 |
Currrent period | 1,224,195.66 | 3,924,236.82 | 5,148,432.48 |
Total: | 1,564,250.01 | 5,325,750.01 | 6,890,000.02 |
Equivalent units | 93 | 65 | - |
Cost/Equivalent units | 16,819.89 | 81,934.62 | 98,754.51 |
TC = (CPEUmalx EU) + (CPEU^ x EU) (4.5]
= (16,819.89 x 93) + (81,934.62 x 65) = 1,564,249.77 + 5,325,750.30 = $6,890,000.07 = $6,890,000.00.
where TC is the toal cost, CPEU is the cost per equivalent unit, and EU is the equivalent unit.
Schedule of Cost Summary
The final phase in process costing involves the construction of a schedule of cost summary, showing the costs for both beginning and ending inventories. Information contained in this schedule comes from the schedule of equivalent production and the unit cost analysis. Table 4.4 presents this schedule for materials as well as conversion costs.
According to the table, the cost of completed units without the beginning inventory is $3,247,368.75, which is obtained by multiplying the cost per unit by the number of units fully completed; that is, $72,163.75 x 45 = S3,247,368.75. Similarly, the costs of units in process arc obtained by multiplying the cost per equivalent unit by the equivalent units for materials and conversion. The result produces $893,857.20 in materials costs and $2,748,774.20 in conversion costs, with a total cost of $3,642,631.40. Together with the costs of completed units, they produce a total of $6.89 million, which is the total cost of operation for the department (after adjusting for the rounding-off error).
Similarly, the costs of equivalent units, with the beginning inventory included, also turn out to be $6.89 million. A breakdown of these costs will include $ 1,481,317.60
Tabic 4.4
Schedule of Cost Summary Year Ended: August 31,20XX
Cost Description | Costs of | Costs of Units in |
Completed Units ($) | Process ($) | |
I. Without beginning inventory | ||
Beginning inventory | - | - |
Units started and completed | ||
[45x$72,163.75] | 3,247,368.75 | - |
Ending inventory (WIP) | ||
Materials: 60x$ 14,897.62 | - | 893,857.20 |
Conversion: 48x57,266.13 | - | 2,748,774.20 |
Total: | 3,247,368.75 | 3,642,631.40 |
II. With beginning inventory | ||
Beginning inventory | ||
[15x$98,7 54.52] | 1,481,317.60 | - |
Units started and completed | ||
[30x$98,754.52] | 2,962,635.60 | - |
Ending inventory (WIP) | ||
Materials: 48x$ 16,819.89 | - | 807,354.72 |
Conversion:20x$81,934.62 | - | 1,638,692.40 |
Total: | 4,443,953.20 | 2,446,047.12 |
for completed units in beginning inventory, $2,962,635.60 for completed units that began in the current period, and $2,446,047.12 for partially completed units in ending inventory, including $807,354.72 in materials costs and $1,638,692.40 in conversion costs.
It is important to note that as long as there arc units remaining in a work in proccss account, they will automatically become a part of the beginning inventory for next period. Consequently, one must transfer the costs of completed units out of work in process inventory to show the balance at the end of the period. Take the cost for the department with beginning inventory as a case in point. The total cost of operation for the year was $6.89 million, of which the costs of completed units, including those that appear in the beginning inventory, were $4,443,953.20; that is, 1,481,317.60 + 2,962,635.60 = $4,443,953.20. The difference between $6.89 million and $4,443,953.20, which is approximately $2,446,047.12, then becomes the costs associated with the beginning inventory for next year. This should be added to next year's materials and conversion costs for all units (completed as well as partially completed) to produce the total cost to be accounted for that period and the proccss will continue.
In dealing with the problem in the current example, we made an assumption that we have a single production unit (the policc department) that is responsible for homicide investigations. The drawback of this assumption is that since we were dealing with a single production department, wc needed only one work in process inventory. In reality, there is usually more than one production department where the output from one department would pass to the next and eventually to the final inventory account. When more than one department is involved, the procss requires that the accounting system maintain as many work in process inventory accounts as there are departments in the proccss. But this will require more work since a separate set of schedules will have to be prepared for each department and the corresponding work in process will have to be reconciled in the final account.
Early in our discussion of job costing, we treated all costs as product costs, regardless of whether they were fixed or variable. In other words, wc treated the cost of a unit of production in terms of direct materials, dircct labor, and overhead. No attempt was made to separate fixed costs from variable costs. This proccss of cost assignment, where no distinction is made between fixed and variable costs is called absorption or full costing.
Since fixed and variable costs are treated as one in absorption costing, it is not useful for certain types of financial reporting, such as income statements which often require that some information be provided on contribution margin. As noted previously, a contribution margin is the difference between total revenue and total variable costs. The margin represents the rate at which the output of an operation contributes to an organization's fixed costs and net income or revenue. The method that clearly recognizes this in preparing a financial statement is the variable costing, also known as dircct costing.
Variable costing is mostly used for internal reporting since it provides additional information by disaggregating costs into two distinct categories, fixed and variable, whereas absorption costing is primarily used for external reporting, where disaggregation is not always required.
Variable versus Absorption Costing
Let us look at a simple example to illustrate the difference between the two costing methods (variable and absorption) and see if the difference between them can be reconciled. We begin with two simple income statements for an clcctric utility fund for a local government. Income statements in government are used primarily for proprietary funds, such as water, sewer, electricity, parking meters, and the like because of their similarity with private sector operations. Table 4.5 presents the two statements, one for absorption costing and the other for variable costing.
According to the table, the net income at the end of the year under absorption costing is obtained by subtracting the operating and the administrative costs of the government from its operating revenue. The operating revenue, in turn, is obtained by multiplying the total units (kilowatt hours) of electricity sold (consumed) by the price of electricity. Note that, for most organizations, operating costs arc usually separated from administrative costs to keep proper track of cost assignments. Assume that the government produced 100 million units of electricity last year, of which 80 million units were consumed by the residents at a price of $0.12 a unit. Accordingly, the operating revenue from the sale of electricity for the government would be 80,000,000 x 0.12 = $9,600,000.00, as shown in the table.
The operating costs, on the other hand, were obtained by multiplying the total units (of electricity produced) by the cost per unit of electricity. Assume now that the cost of producing one unit of electricity was $0.10, which included $0.04 in direct materials, $0.02 in direct labor, $0.01 in variable overhead, and $0.03 in fixed overhead. The total operating cost for the service, therefore, would be 80,000,000 x 0.10 = $8,000,000.00, with a gross margin (the difference between operating revenue and operating cost) of $1.6 million; that is, 9,600,000.00 - 8,000,000.00 = $1,600,000.00.
Note that under absorption costing, the net incomc is obtained by subtracting the administrative costs from the gross margin, where the latter is obtained by taking the difference between operating revenue and operating cost(s). Note also that under absorption costing, both fixed and variable costs assigned to the units sold are included in the operating costs, while the administrative costs arc treated separately. In contrast, the net income under variable costing is obtained by subtracting the fixed costs of operation from the contribution margin. The contribution margin, on the other
Table 4.5
Simplified Income Statements Year Ended: August 31, 20XX
Absorption Costing (S)
Cost Flow
Operating revenue (-) Operating costs = Gross margin (-) Administrative costs = Net Income
9,600,000 8,000,000 1,600,000 1,000,000 600,000
Variable Costing ($)
Operating revenue 9,600,000
(-) Variable (operating) costs 5,600,000
(-) Variable (administrative) costs 400,000
= Contribution margin 3,600,000
(-) Fixed overhead costs 3,600,000
(-) Fixed administrative Costs 600,000
- Net Income_0
Reconciliation
Difference in net income: $600,000.00
Reconciliation: [Variable costing income + (Inventory balance * Fixed overhead cost)] = 0 + (20,000,000 x 0.03) = 0 + 600,000 = $600,000.00 = Net income for variable costing
hand, is obtained by taking the difference between operating revenue and variable costs. The latter includes both variable operating and variable administrative costs.
Two things arc worth noting here. First, the overhead component of the cost is split between fixed overhead and variable overhead, where the latter is subsumed under variable operating costs. The result produces a variable cost that is considerably lower than the operating cost under absorption costing, e.g., $5.6 million as opposed to $8.0 million. In reality, the problem can be avoided by constructing a detailed income statement. Second, in calculating fixed overhead costs, consideration is given to the total units of goods and services produced, rather than the total units sold. The reason for this is that since these units were already produced and fixed overhead costs were incurred in producing them, they must be recognized in preparing the income statement.
Cost Flow
Another interesting thing to observe in the currcnt example is that net incomc under absorption costing for the government was $0.6 million, which became 0 under variable costing. In other words, under absorption costing our government had a net income of SO.6 million, which disappeared under variable costing. There is a simple explanation for this: the difference occurred due to the discrepancy between the total output produced and the actual quantity sold or consumed. As a general rule, if the amount of goods and services produced is higher than the amount sold, net incomc under absorption costing will be higher than net income under variable costing. The opposite is true when the amount produced is lower than the amount sold. If there is no discrepancy, i.e., if the amount produced is exactly the same as the amount sold the net income will be the same for both methods. This explains why the net income was higher under absorption costing in our example.
One of the simplest ways to explain the difference we have just observed in net incomc between variable and absorption costing is how one expenses the fixed overhead costs under the two methods. For instance, under variable costing the fixed overhead costs are expensed to the total units produced, i.e., charged to the period when they were incurred In contrast, under absorption costing they arc expensed to the total units sold, i.e., charged to the units when they were sold. The former is called period costing, while the latter is known as product costing. It is this difference between the two costing methods (period and product) that produces the difference in net income.
The following shows how the discrepancy occurred as a result of the differences in expensing the fixed overhead costs:
Fixed Overhead Expensed (Direct costing)
(100,000,000 x $0.03] $3,000,000.00
Fixed Overhead Expensed (Absorption costing)
(80,000,000 x $0.03] $2,400,000.00
Discrepancy: $600,000.00
According to the information presented above, if the fixed overhead costs were not expensed for the total amount of electricity produced under variable costing, the net income for both methods would have been identical. In other words, if the government had been able to sell 100 perccnt of the electricity it produced during the accounting period, there would not have been any discrepancy in net income.
Once there is a discrepancy in net income, it may be necessary to find ways in which the difference could be reconciled. Wc suggest here a simple procedure for reconciliation that will even out the difference in income flow under both methods. For instance, when the number of units sold is less than the total units produced, add the fixed overhead costs for the difference to the variable costing income. When it is more (assuming that there is a carry over from previous periods), do the opposite. Thus, if we multiply the 20 million units of electricity that remain unsold at the end of the year in our example by the fixed overhead costs of $0.3 and add this to the net incomc under variable costing, it will be the same as the net income under absorption costing, as can be seen from the folowing calculations:
variable costing (net) income + (inventory balance * fixed overhead costs) = 0+ (20,000,000x0.03) = 0 + 600,000 = $600,000.00,
which is exactly what we had for the net income under absorption costing. COST ALLOCATION
Cost allocation, also callcd indirect costing, deals with the process of allocating costs, in particular indirect costs, to one or more cost objectives. Cost objectives are objects, such as a service unit, a department, or an agency for which it is possible to measure costs separately. When costs are measured separately, it enables one to keep track of the origin and destination of costs which, in turn, makes it possible for an organization to identity the points of inefficiency in service operation.
All cost allocations are based on a number of key considerations that serve as guidelines for allocating costs to various cost objectives. Important among these considerations arc:
1. Costs should be allocated according to the degree of association or benefit received. This means that only those service departments that benefit from a given service should be included in cost allocation.
2. The inefticicncics in resource use of the servicc department should not be passed on to the user and other departments since the inefficiencies do not producc
any benefit for these departments.
3. Most important, the cost charged by a scrvicc department to a user department must be independent of the activities of all other departments. In other words, the benefit received by a user department for which a cost is charged must be unique to that department
Cost allocation is a simple exercise if cost flows in one direction from a single scrvicc department to one or more user departments. But when several service departments are involved in a process, i.e., when cost flows in multiple directions, allocation can become quite complex. This section presents three simple, but frequently discussed methods that can deal with different levels of complexity in cost allocation: direct method, step method, and reciprocal method.
Dircct Method
In a direct method, costs are allocated directly from a serv ice department to the user departments. In general, when n-uscr departments receive services from a service department, the costs arc allocated in direct proportion to the amount of services they receive. The proportionality factor, which serves as a basis for cost allocation, is callcd the cost allocation base. The cost allocation base for a service department reflects the amount of scrviccs it provides to other departments. In most instances, the base is selected in such a way that is relatively easy to measure. Examples of cost allocation base will be number of labor hours, machine hours, number of employees, and so on.
To give an example, consider a case where a certain department of a government, called X, provides (internal) services to four departments within the government: D„ D2, D3, and D'4. Let us say that it cost the department $ 1.5 million last year to provide these services. The total amount of services provided by the department was 100,000 units (cost allocation base) at a cost of $15 a unit. This included the cost of both machine (capital) and labor used by the department. The distribution of services for the four departments, say was 20,000 units for I)„ 30,000 units for D2, 40,000 units for Dj, and 10,000 units for D\.
We can use a simple expression to determine the cost allocation for the departments based on this proportionality factor, as shown below:
TCX.£ Cfj-C£ fj [4.6]
i-1 i>l
where TCX is the total cost of scrvicc for department X, с is the cost per unit, and ^ is
the fraction of scrvicc received by the ith department from X, with i = 1,2, 3,.......,
m. Note that we treated с as a constant on the assumption that the services provided by the department were identical.
Thus, substituting the above information into Equation 4.6 for the service received by each of the four departments, we obtain the total cost of the service provided by department X, as shown below:
TCx- c£ f, 1-1
- (15 X20,000 ,>( 15 X30,000 15 X40,000 3) 415X10,000 4)
- 300,000 |*450 ,000 2»600 ,000 3*150 ,000 4
- $1,500,000
where the subscripts 1, 2, 3, and 4 represent the departments D„ D2, D„ and I)*4, respectively.
According to the results presented above, of the $1.5 million in total cost incurred by the service department, $300,000 are allocated to D, for the services it received, $450,000 to D2 for its share of the services, $600,000 to D3, and 150,000 to D'4. These are the respective shares of the costs each department must bear for the serv ices they received from X.
While it is simple to use, the direct method has a major weakness in that it does not take into consideration the reciprocal relationship between departments. By reciprocal, we mean a manner of service provision where a department not only receives but also provides services to other departments. This is where the step method has an advantage over the direct method in that it takes this relationship explicitly into consideration.
According to this method, a department is selected first and a pro rata share of its cost is allocated to the rest of the departments. Next, a sccond department is selected and its costs, including those allocated from the first department, are then allocated to all the remaining departments, except for the one allocated first. The proccss is continued until all service costs have been fully allocated. What this means is that the allocations arc made in such a way that the departments whose costs have been previously assigned do not absorb any costs of the departments being allocated. This makes it possible to avoid any problem (such as double counting) that may result from reciprocal allocation.
However, the use of the method requires that some order be established in allocating departments. Two approaches have been traditionally used for this purpose. One, sclect a department that serves the largest percentage (fraction) of departments. Two, select the departments at random. The latter is recommended only in situations where the percentage or fraction of departments served is evenly distributed.
To illustrate the method, we can look at the service allocation problem again. Assume now that of the four departments that receive services from X, three arc service departments (Dlt Dj, and Dj), and the fourth is a user department, D'4, meaning that it does not provide any service to the other departments. In other words, the three service departments provide services to each other as well as to the fourth department, while the fourth department uses services provided by the three departments but does not provide any service in return.
Table 4.6 shows the services provided by the three departments to each other as well as to the user department. For instance, department Dj provides 20,000 units of service to D2, 30,000 units to D,, and 15,000 units to D'4. Similarly, Department D2 provides 25,000 units of scrviccs to D,, 45,000 units to D, and 30 units to D'4, and so on. The zeroes in the principal diagonal indicate that the services by a department to itself are ignored. The table also shows the costs prior to allocation that are to be allocated among the departments.
Wc begin with the order of the department first. Let us say that we use the first approach. Thus, based on the largest amount of scrviccs provided by a service department, wc select the order D„ D2, and D, such that (20,000+30,000) / 65,000 = 0.77 for D, > (25,000+45,000) / 100,00 = 0.60 for D2> (20,000/45,000) = 0.44 for D,. This means that D! provides more services than D2, which provides more services than D,. Accordingly, all three departments utilizing D,'s services (D2, D,and D'4) shared its costs of $450,000. The distribution of these costs to the three departments, respectively, are: $138,460 for D2, $207,690 for D„ and $103,850 for D'4.
Notice next that only two of the three departments utilizing D2's scrviccs shared its costs of $688,460, which included the original cost of $550,000 plus the $138,460 that came from D,. Thus, Dj was excluded from D2's cost allocation to avoid the problem of reciprocity, mentioned earlier.
Similarly, both D, and I)2 were excluded in D3's allocation of $1,370,770, which included the original $75,000 plus the costs that came from D, and D2. To see if our allocation was correct, we can add the service costs assigned to D'4 by all three departments. If the sum of these costs equals the sum of the costs prior to allocation, the allocation should be considered correct, which appears to be the case here.
As opposed to the step method, the reciprocal method uses a more direct approach
Tabic 4.6
Step and Reciprocal Methods Year Ended: August 31, 20XX
Department | Units of Service (000) | Cost Prior to Allocation ($000) | ||
D, | ||||
D, 0 25 20 450 D2 20 0 0 550 D, 30 45 0 750 D\ 15 30 25 700 Total: 65 100 45 2,450 | ||||
Step | Cost Allocation by Department ($000) | |||
D, | n2 | D, | ||
Cost prior to allocation 450 550 750 700 D, (20/65, 30/60,15/65) (450) 138.46 207.69 103.85 D2(45/75,30/75) 0 (688.46) 413.08 275.38 D,(25/25) 0 0 (1,370.77) 1,370.77 Total: 0 0 0 2,450.00 | ||||
n__• ___I | Cost Allocation by Department ($000) | |||
Reciprocal | D, | Dy | D'4 | |
Cost prior to allocation D,(.31, .46, .23) D2(.25, .45, .30) D,(.44,0,.56) Total: | 450 550 750 (1,558.221) 483.048 716.782 258.262 (1,033.048) 464.717 849.859 0 (1,931.499) 0 0 0 | 700 358.39 309.91 1,081.64 2,449.94 | ||
to cost allocation. Somewhat analytically more involved than the previous two methods, the reciprocal method considers all the scrvicc departments together by presenting the problem in terms of a system of simultaneous equations. The purpose of those equations is to find solutions for the cost of individual service departments. Once the solutions to the equations have been found, they can be used for allocating each department independently to all user departments.
To illustrate the use of the method, let us go back to the problem again. We start with three scrvice departments, D„ D2, and D„ as before. To be consistent with our earlier discussion, we begin with D„ although it does not make any difference which department we start with sincc the order is not important under the reciprocal method. Our objective is to set up an equation for D, that will include its original costs of $450,000 plus a share of the costs in D2, and Dv According to the table (4 .6), I), should absorb 25 percent of D2's cost (25,000/100,000), and 44 percent of D,'s (20,000/45,000). Therefore, the equation for D, will be
D, = 450,000 + 0.250D2 + 0.440D, [4.7]
Similarly, the equations for D2, and Dj will be
D, = 750,000 + 0.460D, + 0.450D2 [4.9]
We can now solve these equations simultaneously to find the costs for each scrvice department by substituting Equations 4.8 and 4.9 into Equation 4.7, first for D„ as shown below:
D, = 450,000 + 0.250(550,000 + .310D,) + 0.440(750,000 + 0.460D, + 0.450D2) = 450,000 + 137,500 + 0.0775D, + 330,000 + 0.202D, -Ю. 198D2 = 917,500 + 0.280D,+0.198D2
Substituting the right-hand side of the extended equation (4.7) for D2, we get
Dj = 917,500 + 0.280D, + 0.198(550,000 + 0.310D,) = 917,500 + 0.280D, + 108,900 + 0.061D, = 1,026,400+ 0.34 ID,
With slight reorganization, D, can be reduccd to
D, - 0.34ID, = 1,026,400 0.659D, = 1,026,400 л D, = $1,558,220.70»$1,558,221
Now, substituting the above result into Equation 4.8 we obtain the solution for D2, as shown below:
D, = 550,000+0.31(1,558,220.70) — 550,000 + 483,048.42 = $1,033,048.42 -$1,033,048
Similarly, substituting the values of D, and D2 into Equation 4.9 we get the following value of D3:
D, = 750,000 + .46(1,558,220.70) + 0.45(1,033,048.42) = 750,000 + 716,781.52 + 464,871.78 = $1,931,653.30 =$1,931,653
The results obtained for D,, D2, and D3 thus represent the cost amounts for each of the service departments. In other words, these are the amounts that are to be allocated to respective departments in proportion to the services they received from the service departments.
Table 4.6 further shows these allocations (bottom section). As the table shows, of the $1,558,221 obtained for D„ $483,048 are allocated to D2, $716,782 arc allocated to D,, and $358,39 to D'4. The process is repeated for D2 and D, to show similar allocations of service costs. Since the sum of the costs allocated to D'4 equals the total cost of service prior to allocation, except for the rounding-off errors, we can assume the results to be correct.
Reciprocal Allocation by Matrix Inversion. The procedure wc used to solve the system of equations above is called the substitution method, which works fine as long as the number of equations to be solved remains small. When the number increases in size, i.e., when there are many more service departments with complex (reciprocal) relationships involved, it becomes difficult to use the method because of the amount of substitutions that will be required. The alternative is to use matrix algebra, in particular matrix inversion.
Matrix inversion is a process in which a square matrix (one in which the number of rows is equal to the number of columns), when inverted and subsequently multiplied by the original matrix, produces an identity matrix (one in which the principal diagonals are all Is and the off-diagonal elements are all Os). To see how the method applies to reciprocal allocation, we begin with the equations for the three service departments and present them in terms of a system of simultaneous equations, as shown below:
D, = 450,000 + 0.25Dj + 0.44D, [4.10)
D, = 550,000 + 0.31D, + O.OODj 14.11 j
D,= 750,000 + 0.46D, + 0.45D2 [4.12]
With slight reorganization, the equations can be written as
1 00D, - 0.25D, - 0.44D, = 450,000 [4.13]
-0.31 Di -н 1.00D2 + 0.00D, = 550,000 [4.14]
-0.46D, - 0.45D2 + 1.001), = 750,000 [4.15]
[4.16]
Using the standard matrix notations, the equations can be further written as
AD = С
where A is the matrix of coefficients, D is a column vector of service departments, and С is a column vector of costs corresponding to each of these departments.
Expanding the terms of the expression in Equation 4.16, we can write it in full as
[4.17]
1.00 -0.25 -0.44 -0.31 1.00 0.00 -0.46 -0.45 1.00
"450 ,000" | |||
D2 | - | 550 ,000 | |
_750,000_ |
Now, to obtain the inverse of the matrix A we do the following: first, find a matrix of cofactors (submatrices with plus and minus signs), Лс, which arc obtained by taking the determinants (single values or scalars found only in square matrices) of the submatrices by removing a row and a column corresponding to an element of the matrix, cach time a determinant is calculated for one of the submatrices. Thus, for a 3x3 matrix, there will be nine determinants corresponding to nine submatrices. Equation 4.18 shows this cofactor matrix below:
[4.18J
"1.0000 0.31000 0.5995 0.4480 0.7976 0.5650 0.4400 0.1364 0.9225
Second, take the transpose of this matrix by changing the rows into columns, called an adjoint matrix, (A*)\ That is,
1.0000 0.4480 0.4400 0.3100 0.7976 0.1364 0.5995 0.5650 0.9225
[4.19]
(А<У -
Third, and finally, divide the adjoint matrix by the determinant of the original matrix, |A|, to obtain the inverse matrix, A"'. For a 3x3 matrix, the determinant can be obtained by the expression4
|A| = A,,A2JA,J + AI2A2JA„+ A,JA21Aj2- AjjAJJA,, - AuA2JA,2- A12A21AJJ
[4.20]
where the terms of the expression represent the coefficients of the original matrix, A.
Substituting the respective values into the expression in Equation 4.20, we obtain a determinant of 0.6587 for A. Therefore, dividing the adjoint matrix, (AC)T, by
0.6587 (the determinant of A) produces the inverse matrix for the problem, as shown below:
"1.5181 0.6803 0.6680 0.4706 1.2109 0.2071 0.9101 0.8578 1.4005
(AC)T
IA|
[4.21]
Next, to obtain the value of the column vector C, i.e., the cost of cach service department, we multiply the inverse matrix by C. That is,
"1.5181 0.6803 0.6680" 0.4706 1.2109 0.2071 0.9101 0.8578 1.4005
450,000 550,000 750 ,000
[4.22]
А С .
1,558,310 1,033 ,090 1,931,710
D.
D.
The result shows the costs associated with each sen-ice department which, respectively, are $1,558.310 for D„ $1,033,090 for D2, and $1,931,710 for D,. As expected, the results came out to be almost the same as those obtained by the substitution method (Table 4.6), except for the rounding-off errors.
The last and probably the most critical element in a cost accounting system is cost control. Cost control is the proccss of ensuring that the costs of operation for an organization do not exceed their target costs and that there are no cost overruns. While occasional cost overruns are not unlikely because of events and circumstances over which an organization may not have enough control, it is important that all organizations, large or small, adopt measures that woud ensure cost control as part of their normal operation. There is a diverse array of these measures, ranging from
cybernetics (the study of all forms of control) to decision techniques (such as those discussed in the next few chapters) that one can use for cost control. This section presents a brief discussion of four such measures that have received considerable attention in the accounting literature on cost control: variance analysis, ratio analysis, internal control, and planning and budgeting.
Variance Analysis
As the name implies, variance analysis deals with the difference, i.e., variance between actual and standard costs. Standard costs are predetermined costs based on what production or deliver)' costs should be under efficient conditions. Efficient condition generally means that all the factors that go into the production or delivery of a service for an organization are working at their fullest (optimum) capacity. It is a theoretical construct and, as such, serves as a reference point or standard for comparing observations on costs and performance.
It is important to recognize that work processes are random, even though the production of goods and services may follow identical, routine operations. What this means is that the materials and labor consumed in the production of a good or delivery of a service may not be exactly the same each time the good is produced or service delivered. Therefore, standards are simply averages around which one would expcct the actual costs to vary. If variances are a normal occurrcncc (which they are for most organizations), then performance should be acccptable as long as the variance is kept to a minimum or as long as it is within some intervals around the standard (as in an interval estimation, discusscd previously).
Two things are necessary to perform a succcssful variance analysis. One, it must be tied to a specific activity so that one is able to collect relevant data on it. Two, the data collected must be reliable; that is, they must be consistent so that the same results can be obtained from repeated applications. The latter is necessary to ensure that any deviation a variance analysis produces is not due to unreliability of data. Theoretically one could calculate as many different types of variances as data would permit but, for the purposes of illustration here, we will focus on five: price variance, usage variance, efficiency variance, overhead variance, and budget variance.
Price Variance. Price variance is the difference between actual and market price. Two types of price variances are commonly used in variance analysis, one related to materials used in the production or delivery of a good or service, called the materials price variance, and the other related to the price an organization charges for the goods and services it provides, callcd the service pricc variance. Cost accounting is primarily concerned with the former, i.e., the materials price variance.
Materials pricc variance is obtained by comparing the actual cost of purchase of materials with their market cost. The assumption here is that the market pricc of materials reflects a competitive price absent any distortions in the marketplace. If such distortions are present in a market place, the convention is to use the average price for materials and use that as a surrogate or proxy for the market price. To give an example, consider a federal agency that purchased 500 units of a certain item, say screw drivers, at a cost of $7,625. '1Ъе price the agency paid for cach unit, on average, is $15.25, while the market price is $12.50 per unit. At this price, the market cost of the units purchased will be $6,250. Therefore, the materials pricc variance for the agency is $1,375, obtained by taking the difference between the actual and the market price. That is,
MPV = MP - AP [4.23]
= 6,250 -7,625 =-SI,375
where MPV is the materials pricc variance, AP is the actual pricc paid for materials, and MP is the market price of materials. Since the difference is negative, this is the amount the agency has overpaid.
Usage Variance. Usage variance is the difference between the actual and the standard usage of materials. When the usage variance is multiplied by the price of the materials used, it produces the cost of overutilization (or undcrutilization) of materials. To give an example, suppose the public works department of a local government recently purchased 100 units of an item for a job that actually requires 80 units. The department used 95 units to complete the job, thereby producing a usage variance of 95-80 = 15. Note that the five units that remain unused become part of the inventory to be used at a later date (assuming that it will not become obsolete or lose its use-value).
Assume further that the department spent $45 for each unit, with a total cost of $4,500 for the 100 units it purchased. Since the department spent 15 units more than what is considered standard for the job, the cost of overutilization to the department would be $45.00 x 15 = $675.00. That is,
CO = (UR x c) - (UU x c) [4.24]
= (80 x 45) - (95 x 45) = 3,600 - 4,275 = -$675
where CO is the cost of overutilization, UR is the units required, UU is the units used, and с is the cost per unit.
Efficiency Variance. This is really an extension of the usage variance involving the usage of labor time. It is called the efficiency variance because it compares the actual labor time with the planned or standard labor time, whatever that standard is. As in usage variance, multiplying the observed variance by the price of labor, i.e., the wage rate, will produce the cost of ovcrutilization (or underutilization) of labor. To give an example, consider a local government that takes an average of 25 minutes to dispose of one ton of solid waste. Suppose the government spent 2,000 hours of labor time last year to dispose of25,000 tons of wastes which, based on the available standard time, would have taken 1,875 hours. The result produces an efficiency variance of 2,000-1,875= 125 hours.
Assume now that the government paid SI4.50 an hour for labor time, with a total labor cost of $29,000. Therefore, the total loss to the government due to over-utilization of labor is $ 1,812.50. That is,
[4.25]
LGO = TCL - SCL
= (2,000 x 14.50)-(1,875 x 14.50) = 29,000.00-27,187.50 = $1,812.50
where LGO is the loss to the government due to overutilization, TCL is the total cost of labor, and SCL is the standard cost of labor.
Overhead Variance. Overhead variance is the difference between actual and standard overhead costs. The computation of this variance is somewhat more involved than any other variance in that it includes both fixed and variable overhead costs. Take, for instance, a government that spent $25,500 in variable overhead costs and $74,500 in fixed overhead costs on a particular service, with a total overhead cost of $ 100,000. Let us say that it cost the government 750 hours in indirect labor to provide the scrvicc. The standard cost of labor is $18.50 an hour. The overhead variance, based on the above information, therefore, is $11,625.00. That is,
[4.26]
OV = AO С - SOC
= 100,000.00-[(750 x 18.50) + 74,500)] = 100,000.00-88,375.00 = $11,625.00
where OV is the overhead variance, AOC is the actual overhead cost, and SOC is the standard overhead cost.
This is the amount the government has overpaid in overhead costs for the scrvicc. Note that we left the fixed overhead cost unchanged because it is likely that it will be the same under both conditions. In the event that it is different, one can always adjust it by following the same procedure as the one we used for variable overhead.
Budget Variance. Budget variance is the difference between budgeted and actual cost of an operation. Budget variance is a common occurrence in most organizations, especially in government where budgeting is a recurring exercise. Several factors contribute to this variance, such as poor cost estimates, fluctuations in the economy, cost overruns due to unforeseen events, and so on. To give an example, suppose that it cost a government $1.925 million to run a social program last year that was intially budgeted for $1,682 million. The budget variance for the government, obtained by-taking the difference between the two, therefore, is -$0,243 million.
Although budget variance in cost accounting generally focuses on the cost side of an operation, it can also deal with the revenue or incomc side, in which case it is called an income variance. Income variance occurs when there is a deviation between budgeted (expected) income and actual income. For instance, consider a government that had expcctcd to cam $8,375 million in incomc from a scrvicc facility last year, but instead earned $7,986 million. The resultant variance in incomc for the government would be SO.389 million.
Decision Involving Variance Analysis. All variance analysis serves two important purposes. One, they help us determine if a problem has occurred in the course of providing a good or service. Two, they also help us trace the cost effect the problem will have on an organization. Therefore, a properly conducted variance analysis can not only identify where in the process a problem has occurrcd but can also save money for an organization by determining the cost effect the problem will have on its operations.
Doing a variance analysis involves costs. However, most of these costs entail the costs of labor and capital one would incur in collecting, refining, and analyzing cost and related data for these analyses. Although data requirements for individual analysis are usually low, they can be quite substantial as the number of analysis to be done increases. This raises an interesting question: at what point should an organization decide to do a variance analysis? Since cost is the primary factor in this case, the answer will depend on the cost of doing an anlysis as opposed to not doing an analysis. As a general rule, if the expected cost of doing an analysis is less than the cost of not doing an analysis, it should be worth undertaking.
We can formally express this relationship as
[4.27]
pC + (l-p)(C+V)<(l-p)D where p is the probability that a serious problem has occurred, С is the cost of doing an analysis, V is the cost of correcting a problem (if it is discovered from variance information that a serious problem has occurred that needs to be fixed), D is the cost of not correcting the problem, and (1 -p) is the probability that a serious problem has not occurred. Note that (1 -p) is the complementary probability of p; therefore, the sum of p and (1 -p) must be equal to 1.
Suppose now that we have the following information for an agency that wants to do a variance analysis: С = $750, V = $2,500, and D = $5,000. Let us assume that we do not have any information on p. Since we do not have any information on p, we can make a conservative assumption that it is 0.50, i.e., there is only a fifty percent chance that no problem has occurred. This is an acceptablc assumption in statistical analysis if it can be argued that the agency concerned is basically doing a good job. Therefore, substituting these values into the expression in Equation 4.27, we can determine if the expected costs of doing the analysis will be lower than not doing the analysis. That is,
pC + (l-p)(C+V)<(l-p)D or (0.5)(750) + (1-0.50)(750+2,500) < (1-0.50)(5,000) or 375 + 1,625 <2,500 or $2,000<$2,500
where $2,000 is the cost of doing the analysis, and $2,500 is the cost of not doing the analysis. Since the expected cost of doing the analysis is less than the cost of not doing the analysis, the analysis is justified.
Like variance analysis, ratio analysis can be used to determine the discrepancy between actual and standard performance in order to determine the cost effect this deviation will have on an organization. A ratio is a simple mathematical relationship between two quanlites As in variance analysis, there is a wide range of ratios that one can use but, for convenience, this section discusscs only three: efficiency ratio, capacity ratio, and activity ratio.
Efficiency Ratio. Efficiency ratios arc used for both labor and materials usage. An efficiency ratio for labor usage shows the relationship between actual labor hours and standard labor hours, and is given by the expression
£R Standard labor hours х1(Ю [4 щ
where ER stands for efficiency ratio.
To give an example, consider a government that has used 3,500 hours of labor for an activity, say Z, where the standard for the activity is 3,000 hours. Therefore, the efficiency ratio for the activity, based on this standard, is 0.8571, or (3,000/3,500) x 100 = 85.71 percent. As a general rule, the ratio must be greater than or equal to 1, or 100 percent, to be efficient. Since the observed ratio in our example is about 14 percent less than the acceptable ratio of efficiency (100.00-85.71 = 14.29), we can say that there has been an inefficency in labor use by the government by that amount.
Assume now that the standard cost of labor for the activity is $ 10.25 an hour, then based on the total labor hours used the government has spent $5,125 more than it should have; that is, (3,500-3,000) x 10.25 = $5,125.
Capacity Ratio. Capacity ratio, on the other hand, shows the relationship between the time spent on an activity against the estimated or budgeted time. It can be formally expressed as
Actual time ...
CR--xlOO Г4 991
Budgeted time 1'
where CR is the capacity ratio.
Unlike the efficiency ratio, to be efficient the capacity ratio must be less than or equal to 1, or 100 percent. This is becausc we arc using actual time in the numerator, which means that we expect the actual time not to exceed the budgeted time. To give an example, let us go back to the problem we have just discussed. Assume that the budgeted time for the activity is 3,300 hours. Budgeted times are usually based on past experiences of an organization and arc often used in efficiency calculation when standard times arc not available, or when an organization wants to set its own targets.
Given this information, it seems that the government has used 200 hours of more labor than budgeted, thereby producing a capacity ratio of 106.06 percent; that is, (3,500/3,300) x 100 = 106.06. In other words, it has used 6.06 percent more labor to complete the activity than allocated Translated in dollar terms, it implies that the government has actually spent $2,050 more in labor cost than it should have, if it had strictly gone by the budgeted time; that is, (3,500-3,300) x 10.25 = $2,050.
Activity Ratio. When an organization plans to undertake an activity, it is quite likely that its budgeted time or cost will not correspond to standard time or cost. This is due to the fact that organizations vary in their capacity when utilizing resources, in particular labor. The activity ratio shows this relationship by comparing the standard time (or cost) with the budgeted time (or cost). It is given by the expression
Standard time
AR- xlOO [4.30]
Budgeted time
where AR is the activity ratio. Since it docs not directly deal with actual time or cost, the activity ratio merely sets the limit for the budgeted time (or cost), which is used for calculating capacity and other ratios.
To give an example, let us go back to the same problem and use the same information we have used for calculating the capacity ratio. The result produces an activity ratio of0.9091, or 90.91 percent; that is, (3,000/3,300) x 100 = 90.91. What this means is that the government's allocation of time for the activity is 9.09 percent (100.00-90.91) above the standard time, which does not speak well for the efficiency of the government. In cost terms, it further means that the government's estimate of the budgeted cost exceeds the standard cost by S3,075; that is, (3,300-3,000) x 10.25 = $3,075.
Both variance and ratio analysis have been extensively used, especially in business. As can be seen from the preceding examples, they are simple to construct yet extremely useful in helping an organization control its costs, in particular those related to materials, labor, and overhead. Perhaps the greatest advantage of using these measures is their low data requirements. Both variance and ratio analysis require limited amount of information, meaning that the costs of doing these analyses are relatively low as long as the number of analysis to be done remains limited.
Internal control deals with a set of measures and guidelines that arc important for an organization to ensure that its resources are used for intended purposes. These measures are essentially the same for all organizations, although their application may vary. Some of these measures are simple requiring very little effort and resources, while others are exhaustive requiring detailed and careful analysis throughout the proccss We present here five such measures that are frequently used in cost control: responsibility assignment, expenditure control, cash control, performance auditing, and information management.
Responsibility Assignment. Responsibility assignment deals with the allocation of responsibilities to individuals in order to carry out the diverse tasks of an organization. Assigning responsibilities allows an organization to keep track of its activities consistent with its goals and objectives. This, in turn, makes it possible to hold individuals, m particular those who have been entrusted with such responsibilities, accountable when deviations occur. In a typical organization, the responsibilities follow through a chain of authority with appropriate separation of tasks. For instance, the accounting and finance functions of a government are vested in the finance department, usually headed by a director. Under the director, one would typically find a treasurer, a financial planner, and an accountant or controller plus a number of support personnel, cach with a set of clearly-defined responsibilities.
As a general rule, the tasks individuals perform in an organization must be non-overlapping to avoid the problem(s) of duplication. For instance, the responsibility of a treasurer in a government is to manage its cash and other crcdit activities, including cash-flow analysis, credit rating, and portfolio management. Similarly, the responsibility of a financial planner includes monitoring financial condition, raising funds needed for capital projects, and forecasting costs and technological changes, among other things. Finally, the responsibility of an accountant or a controller is to maintain records of financial transactions in a government and control its financial activities. The latter includes responsibilities, such as identifying deviations from planned and efficient performance, managing inventories, including fixed assets, payroll and matters related to revenue as well as information system and computer operations. The separation of respoasibilitics in this manner not only reduccs redundancy in operation but also cuts costs and makes coordination among various activities much easier.
Expenditure Control. Expenditure control is the first and foremost financial measure an organization uses to bring its expenditure within the limits of its resource capabilities. Expenditure control, especially in government, takes place in two ways: through regulating timing of obligations and expenditures, and publishing a number of financial and accounting reports on a regular basis.5 Regulating timing ensures that funds arc obligated for intended purposes and that their misuse is kept to a minimum. Most organizations use variance analysis, similar to the ones discussed earlier, to show the amount of funds an agency has received and the actual amount it has spent on a specific activity within a given period.
Similarly, publishing financial and accounting reports is essential to show what kinds of expenditures an organization has incurred and how they compare with those intended These reports, which should be published on a quarterly, if not on a weekly or monthly basis, can also serve as principal source documents for the annual financial report that mast financial organizations, including government, are required to publish at the end of each fiscal year.
Cash Control Cash control is the means by which an organization ensures control of its cash, i.e., its cash rcccipts and cash disbursements Cash control is extremely important in government, especially at the local level, where a government receives the bulk of its revenue during certain times of the year, while its expenditure takes place consistently throughout the year. This creates two operational problems: a cash surplus, when more revenue is received than expenditure incurred; and a cash shortage, when more expenditure is incurred than revenue received. Cash control helps a government maintain a balance between the two by ensuring that it has enough cash to carry' out its normal, day-to-day operations, while making sure that any surplus that is generated in the process docs not remain idle. When surpluses arc accumulated, they are usually invested in securities earning additonal revenue for the government.
The simplest way to control cash is to do a cash flow analysis. As discussed previously, a cash flow analysis is a fairly detailed and time-consuming cxcrcisc. It provides three vital information that is useful for financial planning of a government: cash inflows, cash outflows, and a balance that results from their difference, called the net flow. Inflows include the flow of funds from all sources of revenue, while outflows include the expenditures that have been incurred or will be incurred in the future. The analysis serves an important purpose by identifying in advance the cash position an organization will be in and how to deal with problems that it may encounter in the future, such as cash shortage.6
Table 4.7 presents the cash flows for a local government based on information on monthly receipts and disbursements over a three-month period. According to the table, the net flow for the government at the end of the first period is -S147,000, which then becomes the beginning balancc for the next period. In general, if the ending balancc. i.e., the balance at the end of a period is positive, it adds to the total revenue for the next period and the converse is true, when it is negative. The proccss can be repeated for as many periods as one wants to. In reality, however, a cash flow analysis for more than six or eight periods may not be useful because of the likelihood of error that increases with time in predicting future flows.
Performance Auditing. Auditing is a proccss of investigating the accuracy of the information a government provides on its financial, accounting, and other activities for a given fiscal year. There are three specific objectives an auditing process serves for an organization: (1) determine the extent to which it has been fair and accuralc in presenting its financial position; (2) determine whether or not it has complied with the staled laws, regulations, and mandates that influence its financial transactions; and (3) determine the extent to which it has been able to achicve its stated goals and objectives. The first two objectives together are called financial auditing, while the third is known as performance auditing.
Financial auditing is based on a set of standards by which one can measure and evaluate the financial position of a government, while performance auditing is much broader in scope. There are no set standards that can accuratcly guide and measure the performance of an organization. However, by focusing on goals and objectives, performance auditing can address the broader question of cfficicncy and effectiveness, in particular how an organization carries out its operations and the manner in which it utilizes its resources in achieving those goals and objectives.7 In other words, by placing emphasis on goals and objectives and their achievement, performance auditing attempts to measure an organization's bottom-line performance.
Since performance auditing is primarily conccmed with cfficicncy, it tends to pay more attention to the internal operations of an organization. This is where cost
Table 4.7
A Sample Cash Flow Analysis
Cash Flow | Monthly Flow ($000) | ||
January | February | March | |
Beginning Balance | (147) | 573 | |
Cash Receipts (Inflows): | |||
Tax revenue: | |||
Property tax | 27,238 | 28,365 | 28,956 |
(Local) Sales tax | 6,285 | 5,987 | 5,823 |
(Local) Income tax | 7,347 | 7,549 | 7,450 |
Nontax revenue: | |||
Charges and fees | 2,455 | 2.138 | 2,107 |
Fines and forcleitures | 1,570 | 1,296 | 1,325 |
Intergovcmtal transfer | 5,216 | 6,295 | 6,794 |
Miscellaneous | 1,079 | 1,512 | 1,928 |
Total inflows: | 51,190 | 53,142 | 54,383 |
Cash Disbursemnets (Outflows): | |||
Salaries and wages | 31,514 | 32,925 | 32,975 |
Materials and supplies | 3,295 | 2,764 | 2,516 |
Debt scrvice | 5,760 | 5,760 | 5,760 |
Benefit payments | 2,127 | 2.479 | 2,863 |
Capital outlay | 7,295 | 7,387 | 7,956 |
Miscellaneous | 1,346 | 1,095 | 1,215 |
(-) Total outflows: | 51,337 | 52,410 | 53,285 |
Changes in Cash Balance | |||
= Net cash flow | (147) | 720 | 1,098 |
+ Beginning balance | - | (147) | 573 |
= End balancc | (147) | 573 | 1,671 |
accounting becomes useful. As noted earlier, cost accounting produces cost and related information that is essential for an organization to run its day-to-day operations. Therefore, by producing accurate, objective, and reliable information, cost accounting can significantly ease the tasks of performance auditing.
Information Management. All organizations need to have a good information system in place to maintain quality internal control. Without good, reliable, and quality information and their management, most organizations will find it difficult to make effective decisions. Information management deals with the management of data that have been processed and analyzed to meet the specific needs of an organization.
There is a difference between information management and a term interchangeably used with it, called data management. Data management refers to the mangement of raw, unprocessed facts and figures that are gathered by an organization as part of its normal operation. As organizations increase in size and complexity, their information requirement also increases, thereby placing additional demands on information management. With an abundance of information and computing facilities these days, more organizations are focusing on what is known as intelligent use of information.8 By intelligent use of information, wc mean cutting through redundant and useless information to obtain those that have a dircct bearing on the functioning of an organization To make an intelligent use of information, one requires a clear understanding of the critical needs of an organization and developing an information base to meet those needs.
There is a direct relationship between cost accounting, planning, and control. Planning deals with the specification of goals, objectives, and missions of an organization and the development of plans to achieve them. An important component of a planning process is the development and execution of a budget. A budget is a plan for carrying out the routine and nonroutine activities of an organization and a commitment of resources to accomplish them. As a plan, it helps an organization coordinate these activities to ensure that they arc achieved in an orderly and timely manner.
Two types of budgets are generally prepared by most financial organizations: an operating budget and a capital budget. An operating budget, which is usually prepared on an annual basis, deals with the day-to-day operations of an organization. As such, it is concerned with materials, labor, overhead, expenditure, revenue, cash, and other routine activities. In contrast, a capital budget deals with the long-term activities of an organization, such as long-term assets, liabilities, capital structure, liquidity, and so on. The two, however, are interlinked by a process where the operating budget includes a capital component, callcd capital outlay, that deals exclusively with the capital expenditure for a budget year plus any other expenditure that is carried over from previous years. What makes this integration possible is the fact that capital budgets, although prepared for several years into the future, arc divided into annual segments so that capital outlay for any given year can be easily incorporated into the operating budget for that year.
Besides being a planning instrument, budgets are extremely useful as a control device. For instance, governments are often required either by law or by some established procedure to prepare their budgets according to a formal structure, callcd the budget format. The format, which frequently varies between governments, provides specific guidelines regarding how a budget should be prepared, including settmg goals, objectives, programs, and activities.
An important characteristic of all budgets, in particular an operating budget, is that once it has been approved and funds have been allocated for its execution it must be followed through. This may appear somewhat rigid to those who may not be familiar with public budgeting, but there is a rationale for this apparent inflexibility. In the private sector, competitive market forces serve as a vehicle to ensure that the resources of an organization are efficiently managed and utilized. In other words, when a private firm or business becomes inefficient, it will not make enough profit to stay in business, and will eventually be forced out of the market. Since no such forces exist in government, nonmarket mcchanisms, such as laws, regulations, manadates, and various compliance requirements arc necessary to achieve the same objective.
This chapter has presented a brief description of five of the most important activities in cost accounting: job costing, process costing, variable costing, cost allocation, and cost control. In job costing, costs are traced to a specific job, service, or activity By tracing costs to a specific job, an organization can estimate the costs associated with each job accurately. In contrast, in process costing costs arc assigned equally to all units of a job or service produced by an agency or department. The units in process costing are homogeneous, which makes it possible to assign the costs equally to these units. In variable costing, on the other hand, only variable costs are taken into consideration, rather than both fixed and variable costs. When the latter is considered in a system, it is known as absorption costing.
Cost allocation, the fourth element in this process, deals with the allocation of indirect costs to a department, an agency, or to any other responsibility center in an organization. In doing so, it not only assigns costs that are indirect to a department but also examines how costs that jomtly benefit several departments can be assigned. Finally, cost control deals with a set of procedues and measures that are useful in controlling the costs of an organization. A wide range of measures are currently available that can be used for this purpose. This chapter has focused on four such measures that are relatively easy to use, namely variance analysis, ratio analysis, (cash) internal control, and planning and budgeting.
1. R W. Ingram, et al. Accounting and Financial Reporting for Governmental and Nonprofit Organizations: Basic Concpts. New York, NY: McGraw-Hill, 1991.
2. E.S. Lynn and J.W. Norvillc. Introduction to Fund Accounting. Reston, VA: Reston Publishing Company, 1984.
3. J.T. Kelly. Costing Government Services: A Guide to Decision Making. Washington, DC: Government Finance Officers Association, 1984.
4. J.M. Gere and W. Weaver, Jr. Matrix Algebra for Engineers. New York, NY: D. Van Noslrand Company, 1965.
5. T.D. Lynch. Public Budgeting in America. Englcwood-Clifts, NJ: Prentice Hall, 1995.
6. A. Khan. "Cash Management: Basic Principles and Guidelines," in J. Rabin, et al., (eds ] Budgeting: Formulation and Execution. Athens, GA: Carl Vinson Institute of Government, The University of Georgia, 1996: 313-322.
7. R E. Brown, et al. Auditing Performance in Government. New York, NY: John Wiley and Sons, 1982.
8. C.J. Austin. Information Systems for Hospital Administration. Ann Arbor, MI: Health Administration Press, University of Michigan, 1979.
1 For a detailed discussion on matrix inversion, see Chapter 4.
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