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CHAPTER 6

COST–BENEFIT AND COST–EFFECTIVENESS ANALYSES AND ASSESSMENTS

6.1 INTRODUCTION

The broad goals of cost–benefit analysis and cost–benefit assessment are to provide procedures for the estimation and evaluation of the benefits and the costs associated with alternative courses of action, including their analysis and assessment. In many cases it will not be possible to obtain a completely economic evaluation of the benefits of proposed courses of action. In this case the word benefit is replaced by the term effectiveness, and we determine a cost–effectiveness analysis and assessment rather than a strictly economic analysis and assessment of the net benefits associated with alternative courses of action.

We may view economics as a descriptive or as a normative science. The often retrospective study of the decisions people and organizations make with respect to the employment of scarce resources for production and consumption is a descriptive study. The study of the decisions that people and organizations should make with respect to resource allocations is a normative study. Our studies of the behavior of firms and consumers in Chapters 2 to 4 are basically studies of the normative behavior of individual firms and consumers. Often descriptive behavior will be approximately the same as normative behavior, at least in a substantive or “as if” fashion. Of course, normative implies value judgments. Here “normative” would have to imply “assuming that firms and consumers wish to maximize their profit and utility, respectively.” Some related comments may be found in Section 1.8.

In this chapter we restate the fundamental assumptions and requirements for rational (unaided) economic behavior. As we shall see, these requirements are somewhat difficult, especially with respect to their information demands, for consumers and firms to meet, in an unaided, descriptive, or positive sense. They are especially difficult to meet when there are groups, either of individuals or of firms, involved.

In Chapter 5 we introduced concepts from welfare or normative economics. In these situations, there is invariably more than one person or firm present. Thus questions of equity in the distribution of allocations will necessarily arise, unless some very restrictive assumptions are imposed. These assumptions will ensure that all of the requirements for the existence of perfect competition are met. When there is more than one person’s judgment involved, questions generally arise concerning how to combine the judgments of the individuals involved. There are a large number of ways to accomplish this combination, and five are particularly important here. We illustrate these for the case where there are two alternatives under consideration.

A brief discussion of each of these criteria is in order. Obviously unanimity is very desirable, but it will often not exist. Majority rule may lead to significant group intransitivity, as we have seen in Chapter 5, even when the preference structure of each individual in the group is transitive. For example, if there are three people with preferences among three alternatives a > b > c, b > c > a, and c > a > b >, then two out of the three people express the preferences a > b, b > c, and c > a and there is a major intransitivity. This can be, in principle, completely resolved by using a cardinal preference scale such that a scalar social welfare function results. There are considerable pragmatic difficulties, however, in constructing a cardinal scalar social welfare function.

Pareto superiority is only a slight weakening of the unanimity conditions in that individuals are now allowed to be indifferent or to judge alternatives as incomparable. It is doubtful that Pareto superiority will exist among many alternatives in realistic situations. Thus we will be left with a Pareto-efficient frontier with no way of determining a single Pareto-superior alternative. The concept of potential Pareto superiority is then of major importance. It is the potential Pareto-superiority criterion that forms the basic criterion for cost–benefit analysis. It is not especially different from the social welfare superiority criterion for cases where a strictly economic analysis can be performed. A key concept contained in the potential Pareto-superiority criterion, which is not explicitly contained in the scalar social welfare criterion, is that of gainers potentially compensating losers. This implies a precise measurement of costs and benefits, which will often be very difficult to accomplish. It will require a cardinal preference scale.

In Section 6.2 we introduce some necessary preliminary concepts. Included among these are various subjects from engineering economics, including net present worth (NPW), rate of return, depreciation, inflation, opportunity costs, portfolios of projects, and discount rates. Then we turn our attention to a more formal definition of cost–benefit analysis. Following this, we discuss the role of consumer and producer surplus, shadow pricing, and valuation of unmarketed goods in cost–benefit analysis. Finally, we present a brief discussion of cost–effectiveness analysis. The generic principles and concepts of cost–benefit and cost–effectiveness analyses are equally applicable in the private and public sectors. Our discussions focus on the usage of these important concepts in both sectors.

6.2 THE TIME VALUE OF MONEY

Generally, it is not fully meaningful to simply add monetary values when these exist at different points in time. Few of us would be just as happy with $10,000 received five years from now as we are to have the $10,000 now. There are several reasons for this. Normally we value present consumption more than we do future consumption. Also, opportunities may be lost due to not having capital at an earlier point in time. Consequently, we are generally willing to pay a premium for present consumption, or the opportunity for consumption, which is a measure of the worth of money over time. This premium is known as the discount rate or interest rate. This interest rate will vary with the supply and demand for money and with the risk that is associated with the venture. As we have noted, we will not specifically deal with risk and uncertainty considerations here, despite their great importance. The interest rate will also have to include the inflation rate.

6.2.1 Present and Future Worth

When we deposit an amount P0 into a savings account at year 0 and the yearly interest earned on the investment is i times the principal, then we expect to earn iP0 in interest in one year, and to have available at the end of the first year the principal plus interest, or F1 = P0 + iP0 = P0(1 + i). If this amount is invested for the next year, we will have F2 = P1(1 + i) = F1(1 + i) = P0(1 + i)2 at the end of the second year. At the end of N years we will have accumulated

(6.1)

In a similar way, the present value P0 of a future amount FN is

(6.2)

Often amounts are invested over a number of years. As we have noted, it would not be generally meaningful to simply add monies that occur at different points in time. But it is generally meaningful to add the present values of amounts An that occur at different points in time. The relation

(6.3)

expresses the worth, really the discounted worth, at time 0 of an amount A, at time n, when the interest rate is i. We simply add these amounts for all n under consideration and obtain

There is no need for the interest to remain constant from year to year. Should the interest rate from year k to year k + 1 be ik, then we have

(6.4)

(6.5)

This expression represents the present worth of amounts An invested for n years, where the interest rate varies from year to year.

In some cases the annual amounts of a benefit or a cost are constant. Then, if the discount rate is constant over the period under consideration, relatively simple closed-form expressions for present worth result. Many ways can be used to obtain these. One simple way is to note that the present worth, at time 0, of an amount 1 invested forever on an annual basis beginning at year 1 is

This follows in a simple manner from

where a = (1 + i)−1. The present value at time N of an amount i invested annually from time N + 1 is also i−1, and the present value at time 0 of this is i−1 (1 + i)−N. We subtract this from the expression i−1 and obtain the expression for the present worth, at time 0, of an amount 1 invested annually for N years. We have therefore determined that the present (discounted) worth P0 of a constant amount A invested at each of the N periods from n = 1 to N is given by

(6.6)

We may easily calculate the future worth, at the Nth period, of annual payments from this relation as

(6.7)

We now drop the subscripts on P and F for convenience. Relations such as these are especially useful for calculating the amount of annuity payments that an initial principal investment will purchase, or for calculating constant-amount mortgage payments. We must, however, be sure that we understand the precise meaning of the calculation, as we will soon see.

Other factors such as inflation and depreciation (or appreciation) need to be considered when evaluating the present or future worth of a project. For the most part these can be considered just as if they were interest. For example, the worth at year n + 1 of an investment at year n of Pn that is subject to interest i, inflation r, and depreciation d is given by

(6.8)

Here in is the interest in year n, rn the inflation in year n, and dn the depreciation in year n. If we assume that this is augmented by an amount An+1 in year n + 1, then the worth of the investment at the start of the next period is

(6.9)

or

(6.10)

We can solve this difference equation with arbitrary values of the parameters in, rn, dn, and An+1 to yield the future worth of any given investment path and annual investment over time.

In the simplest case the initial investment Pn,n is zero and the annual investment and rates are constant. Then it is a simple matter for us to show that the result of an investment over N years is given by the future amount at year N:

(6.11)

Here

If the percentage rates are all quite small, little error results from using the approximation obtained by dropping the products of small terms. We have

such that we simply use an effective interest rate that is the true interest rate less inflation and depreciation.

6.2.2 Economic Appraisal Methods

There are several existing methods that we can use to yield a single number that is reflective of the economic value of a project. The crudest of these do not consider the time value of money at all. The payback period, for example, is the time required from the start of a project before total cash flow becomes positive. Generally, projects involve initial outflows of capital followed by returns on the investment. As can be easily shown, the payback period is a rather naive criterion to use in evaluating projects.

Another common method of project evaluation is called the internal rate of return (IRR). It is that interest rate that will result in economic benefits from the project being equal to economic costs, assuming that all cash flows can be invested at the IRR. The assumption that it is possible to invest cash flows from the project at the IRR will often be incorrect. Thus the number that results from the IRR calculation may give an unfortunate impression of the actual return on investment (ROI). It tends to favor short-term projects that yield benefits quickly, as contrasted with projects that yield long-term benefits only, because of this reinvestment at the constant IRR assumption.

The relation from which the IRR may be calculated is easily obtained. If we assume that a project, of duration N years, has benefits Bn in year n and costs CN, then the present value of the benefits and costs are, assuming that the interest rate is constant,

(6.12)

(6.13)

The NPW of the project is given by

(6.14)

To obtain the IRR we set the NPW equal to zero and solve the resulting Nth-order algebraic equation for the IRR.

Three other criteria result from these relations. We have just defined the NPW criterion as the relation from which the IRR is obtained. We note, however, that the interest rate used in the NPW calculation is the actual interest rate, whereas the interest rate obtained from the IRR criterion is a fictitious rate that assumes reinvestment possibilities at the IRR. The benefit–cost ratio (BCR) is just the ratio of benefits to cost and is given by

The ROI is just the ratio of the net present value to the net present costs and is given by

Sometimes the ROI criterion is called the net benefit to cost ratio (NBCR) or the NPW to cost ratio.

It is of considerable interest to contrast and compare these criteria so as to enable a determination of the criterion, or criteria, most appropriate in particular circumstances. We will first consider the selection of a single project from several. An immediate problem that arises is that NPW and IRR may lead to conflicting results. The BCR criterion may result in a different ranking also. Unless costs are constrained, there is no real reason to use a BCR criterion, however.

Two projects may have the NPW versus interest rate curves shown in Fig. 6.1. Here we see that if the actual interest rate i is less than interest rate , the rate at which the NPW of the two investments are the same, we prefer investment A over investment B if we use the NPW criterion. If the interest rate i is greater than , we prefer investment B. However, if the actual interest rate is greater than IRRB, we would prefer not to invest at all, if this is possible, since the NPW of each investment is negative.

The IRR of project A is IRRA and that of project B is IRRB. Since IRRB is greater than IRRA, we would prefer project B to project A by the simplest IRR criterion. Some analysts would calculate a differential IRR by assuming one project as a base project and calculate a differential IRR (ΔIRR) for switching from the first project to the second. If, for some reason, we are able to select project A as the base project, then the ΔIRR curve is obtained from ΔNPW = NPWB − NPWA, as shown in Fig. 6.2. The ΔIRR of the differential investment is i. If i is greater than the actual interest rate i, or possibly a minimum attractive rate of return (MARR), then we should select project B. This is opposite to the conclusion obtained before. So perhaps we should have computed ΔNPW = NPWA − NPWB, but there is no logic to suggest this. Let us examine these two criteria in greater detail such that we obtain a greater appreciation for what they actually measure.

6.2.3 Comparison of Net Present Worth and Internal Rate of Return Criteria

To set the stage for a comparison of the NPW and IRR as investment rules, it is of value to discuss desirable properties for an investment rule.

The following are four essential properties or assumptions that any rational investment rule or criterion should satisfy:

Property 1 is trivially desirable, since we cannot obtain a valid measure of the worth of an investment unless we consider all the cash flows that constitute that investment. The OCC is the interest rate that can be obtained on an investment, and we will obtain an unrealistic picture of the worth of an investment unless we consider this and discount the investment at this or some other specified (social) discount rate. Wealth or benefit maximization is clearly the scalar performance criterion we should use, as this, broadly defined, represents economic effectiveness. Finally, it is desirable that we evaluate a project independently of all others, perhaps by comparing it to an absolute standard. This assures us that economic evaluation will be relatively simple to accomplish. For n projects we need only accomplish n evaluations, as contrasted with a much larger number that would result if the evaluations were not independent. Also, there can exist transitivity difficulties in making selective pairwise alternative comparisons. Thus we prefer to not use evaluation criteria that call for pairwise comparisons or, at least, to approach pairwise comparisons with the greatest of caution.

Some of the many problems with the IRR criterion are the following:

As a consequence of violating rational rule 2, due to either or both problems A and B, use of the IRR as a decision criterion will often lead to choice of an inferior investment. To illustrate a simple example of this, consider two investment alternatives that yield the NPW versus discount rate curves shown in Fig. 6.3. The IRR criterion will select project 2 as being better than project 1, whereas in reality project 1 is better at the OCC shown in Fig. 6.3. The reason why the IRR criterion errs here, and in other cases as well, is that the unstated implicit assumption is made that all cash flows can be reinvested at a rate equal to the IRR. This implicit assumption defies common sense and logic. Thus use of the IRR as a criterion does not assure selection of the project that maximizes wealth.

One way in which some people propose to avoid this fundamental flaw with the IRR criterion is to use the IRR criterion on the cash flow differences between two investment alternatives. But the difficulties with the IRR do not end by using these differential calculations. Calculations using the incremental rate of return using the algorithms often posed may give precisely the wrong answer. The following steps are often recommended to accomplish the differential IRR calculations:

The typical steps in the incremental rate of return analysis for more than two alternatives are the following:

In all of this analysis, it is implicitly assumed that all positive differential increments of investments—all investments with ΔIRRs that exceed the MARR—are desirable. In every situation, the best project will invariably depend on the assumed discount rate. There cannot, in general, be any “best project” that is independent of the OCC. Thus our “desirable property 2” is indeed desirable. Even if these complexities and difficulties were not present, there are still other problems with the IRR criterion:

The conclusion that the IRR is only of any value when one has a very “simple” investment, such as an initial cost followed by a constant steady stream of benefits, appears correct and unavoidable. But let us continue with our discussion of the virtues of this IRR criterion. Another problem is the following:

There have been a number of efforts to salvage the IRR concept by developing a taxonomy of investments and then developing IRR procedures for each resulting category. It is perhaps easiest to distinguish between what we might regard as simple conventional and nonsimple nonconventional investments. A simple conventional investment is one in which there is an initial flow of capital into the project in years 0, 1,..., M and an outflow of cash from the project in later years M + 1, M + . . ., N. A nonsimple nonconventional investment is one that is not simple or conventional—an investment has one or more positive cash inflows into the investment after a positive cash outflow from the project. For example, the cash flow from Example 6.1 represents a conventional investment, whereas that in Example 6.2 represents a nonconventional investment. A conventional investment is one whose cumulative net worth from 0 to a point in time T necessarily improves after some point in time where the cash flows first become (and remain) positive. In this sense, a conventional investment is necessarily a pure investment. A nonconventional investment may be a pure investment, but it may also be a mixed investment in the sense that the net worth from 0 to a point in time T may well not be monotonically increasing in T for sufficiently large T; it may become negative for sufficiently large T. A mixed project investment is one that has unrecovered investment balances such that the firm “loans” money to the project and overrecovered investment balances such that the firm “borrows” money from the investment. At a valid IRR, a pure investment is one such that the firm never “borrows” from the project. It is to obtain this valid IRR that we corrected the conventional IRR calculations in Example 6.2, which represents a nonsimple mixed investment strategy. When investments are nonsimple and mixed, we need to use two interest rates: an IRR, perhaps more properly called return from the project on invested capital, and a return on capital loaned to the firm from the project that can only earn at the OCC.

It is easy to examine the cash flow from a project to determine whether we are dealing with a simple or nonsimple investment. If a sequence of positive returns to the firm follows a sequence of negative returns (or investments), we have a simple investment. A simple investment is necessarily pure, and there can exist only one positive IRR.

If we have a nonsimple investment, we can calculate the project investment balances as a function of time using sequences of an IRR that is not yet known and the OCC discount rate. The value of IRR that would cause this balance to go to zero would be determined. If this computed IRR is between the values of interest that produce positive NPW at that point in time, we can generally assume that the project is returning money to the firm at that point in time. After this, the project is loaning money to the firm, and the OCC should be used to compute balances at the next period. A general statement of an appropriate algorithm is available, and contained in the work by Bussey (1978), but we will not present it here.

We have presented a number of discussions here to indicate that the IRR is the interest rate earned only on the unrecovered balances of an investment such that a precise zero balance of costs and benefits occurs at the assumed end of the project. As we will indicate by means of an example, the IRR is not just a rate of interest on the positive investments into the project. When there are positive recovered investment balances, or intermediate cash flows to the firm from the projects, as typically there will be when we compare two project alternatives, we must use a correct interpretation of the ΔIRR calculations if we are to obtain consistent results. Generally this is complex. Even when we have initial simple pure investments, we can only determine whether or not a project is acceptable in terms of whether or not IRR > MARR. We cannot ever justify using IRR as a figure of merit to use in ranking projects. To do this we must use the corrected ΔIRR concept, which generally results in a nonsimple mixed investment project and a tedious evaluation effort.

In this section, we have examined some salient attributes of the IRR criterion. Of four desired conditions for a rational economic decision criterion, the IRR criterion is shown to satisfy only one of them, that is, we can consider all cash flows. The NPW criterion can, however, accommodate all four criteria. In particular, it obeys value additivity; it correctly discounts at the OCC or social discount rate; and finally, it results in maximization of profit or worth. Furthermore, it is compatible with most forms of microeconomic and welfare economics used today as a basis for planning and decision making in industry and government. Thus it is the criterion of choice to use in comparing individual projects and, as we will see, for portfolios of projects.

Our arguments thus far in this section show that the IRR can often be expected to be a very unreliable criterion, despite the fact that it is, in practice, often used. There is nothing in our argument thus far, however, that assures us that the NPW criterion is the criterion of choice. It is possible to pose a set of very reasonable axioms, or properties of an investment decision, that only the NPW criterion satisfies. We will now do this.

We wish to specify an investment criterion for investments over the interval (0, N). We will require that the investment criterion be

Next we state five axioms or desirable properties that should be possessed by a preference ordering of cash flow vectors.

Axiom 1. Continuity.

If x is preferred to y [x > y], then for sufficiently small ε > 0, cash flow vector x − ε is also preferred to y [x − ε > y]. This continuity axiom is very reasonable in that it assures us that the preference criterion and associated preferences will not be schizophrenic for small arbitrary changes in the return of the investments.

Axiom 2. Dominance.

If cash flow x is at least as large as y at every period in (0, N) and strictly greater in at least one period, then x > y. This dominance criterion is equivalent to greed—more is preferred to less—or consumer satiation. All it says is that if xi ≥ yi for all i and xi > yi for at least one i, then x ≥ y and x > y.

Axiom 3. Time Value of Money.

If two investments x and y are identical except that an incremental cash flow obtained by investment x at some period n does not result until period n + 1 for investment y, then x is preferred to y. This investment axiom is equivalent to impatience. We would prefer to have an incremental amount of money now than at some future time.

Axiom 4. Consistency at the Margin.

We prefer cash flow x to y if and only if the differential cash flow x − y = [x0 − y0, x1 − y1,...,xN − yN]T is preferred to a cash flow of 0 = [0, 0,...,0]T.

Axiom 5. Consistency in Time.

If we shift all investment returns by some number of periods, the preference order is unchanged.

Surely, it would be difficult to argue that any of these axioms are unreasonable. If only Axioms 1–4 are satisfied, then it is possible to show2 the following:

Theorem A.

The only preferences that satisfy Axioms 1–4 are those given by the NPW criterion in which interest or discount rates, which may vary from period to period, are positive.

This theorem just says that we should value investment x according to an NPW criterion, or decision rule, in which we obtain the present value of the investment component at the nth period xn by using the standard discounting relation

(6.15)

and then add together these present values over all periods to obtain

(6.16)

or

(6.17)

Here ij − 1 is the interest received in the time interval from period j − 1 to present j.

When we also impose Axiom 5, we require further that the interest or discount rate be constant throughout the investment time interval (0, N). Thus we have the following:

Theorem B.

The only preferences that satisfy Axioms 1–5 are those given by the NPW criterion in which the interest or discount rates are constant and positive.

If we accept these axioms as reasonable, then we can only accept the NPW criterion, or one that would yield the same preference ordering, as truly reasonable. Since the IRR criterion does not generally yield the same preferences as the NPW criterion, it must be rejected as an inferior criterion.

6.2.4 Benefit–Cost Ratio and Portfolio Analysis

In this section we discuss calculation of the BCR. Also, we indicate why and where it is such an important criterion. To do this we need to introduce some concepts from portfolio analysis that concern decisions with a constrained budget. Generally BCRs are always defined by discounted benefits and discounted costs. We compute the BCR from Equations 6.12 and 6.13:

(6.18)

As is easily seen, we do not have to discount the benefits and costs to the present time to determine the BCR. Any time will do. Nor do we need to do this to obtain the ROI as BCR/(1 + BCR). It is convenient, however, to discount benefits and costs to the present time, for then we can easily obtain the NPW from PB0 − PC0.

It is easily seen that ranking projects by BCR will generally lead to results different from those obtained by the use of NPW.

6.2.5 The Discount Rate

In our discussions thus far in this chapter, we have assumed that there is a market rate of interest, which we have called the discount rate or opportunity cost of capital. However, a perfect capital market will generally not exist, and there is no single interest rate. Several related approaches to determining a discount rate are possible: marginal interest rates, marginal time preference rates (MTPRs) for individuals, corporate discount rates, MARRs, government borrowing rates, and the social opportunity costs of capital.

Corporate discount rates include a premium for risk, a markup for corporate taxes on incomes above a certain level, and a profit to be returned to shareholders. For example, if government bonds return 10% and the corporation expects a 4% premium for risk to be reasonable, then the corporation will have to obtain almost 28% ROI to provide a 14% return to stockholders. This is the case for “large” businesses. Small business corporations can elect to pass all earnings directly to shareholders, who must pay ordinary income taxes on these earnings rather than capital gains taxes, which are generally lower than taxes on ordinary income.

Market interest rates for government and corporate bonds vary, for example, with the perceived risk that the lender of capital takes on an investment. We can compute a market value of outputs, perhaps as a function of industry, and the inputs to the project, all measured in dollars and discounted in time. Then if a return of $112 results from an input investment of $100, we would say that the market interest rate, which would be the realized MARR, is 12%. This would need to be essentially doubled by corporations because of taxes they must pay. In the last paragraph of this section we describe a normative approach to determining a discount rate. Here we describe a descriptive approach.

The individual MTPR or personal discount rate is the rate at which an individual is willing to trade off present personal consumption for future personal consumption. Related to this is the concept of a personal social discount rate. The personal discount rate represents personal deferred consumption preferences. The social discount rate of an individual represents that individual’s preferences in terms of societal behavior. Many modifications to rates such as these can be made. One might believe that individuals will set too high a personal social discount rate to enjoy consumption today at the expense of future generations. Thus one might argue, as does the Pigouvian3 discount rate, for a social discount rate that is lower than the personal social discount rate of present-day individuals.

Doing this will, of course, enhance the NPW of projects with very long-term payoffs. But it seems reasonable to infer a minimum social discount rate on the market rate for the private or public organization potentially undertaking the project. The reasoning behind this is relatively straightforward. Projects must be paid for. If future costs and benefits are discounted at too low a rate, an unfortunately erroneous inference of ability and willingness to pay for the project in the future is obtained. But development projects must be paid for in one way or another. If too low a discount rate is assumed, then the project will not be really paid for when higher existing market rates extract their toll. So the burden of paying for the project will fall on groups that are perhaps not even identified in a cost–benefit analysis.

It would seem better for arguments to be made that important attributes of projects have been omitted and that the benefits of a project are greater than those identified by an analysis than to distort the analysis through use of an unreal discount rate. Thus the argument that the social discount rate should represent the opportunity costs of the project—the returns that could be obtained from these funds if used on some other project—is a very potent one. In other words, resources must be used in the most productive way. To not acknowledge this is to forget, in effect, that there are always resource constraints on the use of capital. The task should be to define productive in an appropriate way and not to artificially lower discount rates such that projects are justified that cannot pay their own way and which thereby cast the burden of payment in an undefined and unexplored manner, perhaps even on the unborn generations the artificially low discount rate is supposed to project. This seems to be wishful thinking in a most maladroit form.

Most of the discussion in this section applies to cost–benefit analysis and discount rates in the public sector. In the private sector, appropriate definition and use of a discount rate seems much less ambiguous and much more straightforward than in the public sector. In fact, it is the effect on the private sector of capital allocations in the public sector that acts to complicate resource allocation concerns in the private sector. At least three related questions arise:

The answer to these questions is clearly project dependent. We might expect to get considerably different answers, for example, for national defense projects and energy projects. Although these questions are both interesting and important, we will not answer them here.

So, where does this leave us with respect to the selection of a discount rate? One of at least three judgment situations typically exists relative to the choice of a discount rate. First, the rate may be fixed by corporate or government policy, and there is then little that can be done except to use the established rates. Second, the decision maker may have strong personal feelings about the appropriate rate. This is especially the case in private sector efforts. One role of the economic systems analyst and assessor is to work with clients in solving problems. This can include consultations with the decision maker concerning the appropriate rate to use, but, ultimately, the decision maker’s wishes must prevail.

In the third situation in which the economic systems analyst and assessor is free to work with the client in the specification of a discount rate, there will generally be no substitute for a sensitivity analysis that will allow determination of the critical discount rate at which decisions switch. Often these critical discount rates are such that arguments about the actual discount rate to use in a problem are futile, in that any reasonable rate will produce the same best decision. A reexamination of the graphs of NPW versus interest rates that have been produced in this section cannot help but reinforce this conclusion. An important use of sensitivity analysis is to allow pertinent parameters to vary over the imprecise values identified by the client.

6.3 IDENTIFICATION OF COSTS AND BENEFITS

Identification and quantification of the benefits and costs of possible alternative courses of action, or projects, or decisions, is usually a difficult task. It is generally not as difficult, perhaps, as formulation of the issue and identification of the alternatives themselves, but it is still not an easy task.

Here we use the word benefits to mean the possible effects of a project, both positive benefits and negative benefits, or disbenefits. We must first identify benefits and then we should quantify them by assigning a value to them. Many benefits (and disbenefits) will be intangible and will occur to differing groups or individuals in differing amounts. Problems with intangibles may be especially difficult in the public sector, where associated agencies are designed primarily to deliver services or public goods, rather than products for individual consumption. A major goal, however, of a private sector organization is profit maximization, and it is relatively easy to measure profit as a benefit. The benefits of a public service, such as a school, or a public good, such as a subway system, are much more difficult to define and identify because they are intangible or indivisible (or both). The very political environment of many public sector efforts further complicates measurements—and this means that factors other than variables associated with efficiency, economy, and equity need to be measured.

One valuation philosophy that we might adopt is based on two premises:

The fundamental conclusion of microeconomics, under perfect competition conditions, ensures us that these two premises should, under ideal conditions, serve as good guideposts for cost–benefit analysis. As we have seen, when faced with a given price for a good, the rational individual seeking to maximize satisfaction will purchase a number of units of that good so that, at the margin, the individual’s willingness to pay for that good is precisely equal to its price.

There is much that is contained in these two premises, and in this fundamental conclusion of the microeconomic theory of the consumer, that needs to be more fully explained. Willingness to pay seems like a reasonable concept, but it is a meaningful measure of value only if it is based on a sound knowledge of all the true benefits of a project. Premise A also implies that the consumer knows best what is good for them, or consumer sovereignty, and that the existing distribution of income is necessarily the most equitable. Postulate B assumes additivity and the equal weighting of individual values. Under these conditions, society is better off if one individual gains a value of 2 + ε from a project while another individual gains nothing than it is if each individual gains one unit of value.

Thus, there are information-related measurement difficulties associated with these premises, and there are potential equity problems as well. Measurement problems arise from the difficulty of assigning comparable values that reflect the value of a complex project. Generally, decomposition of the benefit of a project into attributes and assigning number values to these attributes will result in a more reliable indication of the value of a project than unaided wholistic judgment. A number of studies in behavioral psychology have shown this. We will look at ways to accomplish this decomposition later.

The implementation of any project will alter the supply of “inputs,” which are consumed by the production or service process, and the supply of “outputs,” the products or services that result from the project. The identification and quantification of the benefits and costs of projects involve exploring the difference between inputs and outputs with and without the “project.”

This notion is a relatively simple and comfortable one when we are evaluating operational-level projects such as the possible introduction of word processing equipment into an office to replace electric typewriters. However, strategic-level concerns, especially in the face of changing environments, involve changed situations that might prevail if a project is not implemented. Many projects are considered for implementation because of contingencies that will result, if they materialize, in increased costs and/or reduced benefits if one or more new projects are not implemented. In situations such as this, the “no-project” or do-nothing alternative must be described as evolving over time and it must include benefit reductions, or cost increases. Useful descriptions of situations such as these often require complex judgments.

The development of alternative future scenarios is generally of value in situations such as these. It would take us somewhat away from our primary objectives to describe scenario construction in detail; however, the essential steps are easily stated. Definitive discussions of scenario construction are found in Porter et al. (1980). These involve the fundamental steps of the systems process. The formulation step involves identification of the needs, constraints, and alterables that may influence environmental change. Potential impacts from these alterables are determined in the analysis step, and these may be evaluated in an interpretation effort if desired. The result of this is a set of possible future scenarios that may or may not occur. The probabilities of each scenario occurring may be obtained. Proposed projects, including the do-nothing alternative, are then embedded into these scenarios and evaluated.

Usually a financial accounting of the various alternative projects results in the information that is needed to identify and quantify economic benefits and costs, perhaps in the format of a balance sheet as illustrated in Fig. 6.4. Adjustments to the financial accounting data are often needed. Sometimes we will find that the financial accounting will neglect some benefits and costs, particularly those of a secondary and/or intangible nature, that are necessary for a useful analysis of the economic benefits and costs of a project. Also, it will often be necessary to adjust or revalue financial data to reflect the fact that market prices either do not exist or do not reflect true economic value even if they do exist. We will now identify some types of economic information that may be missing in financial data; then we will discuss the important topic of “shadow” pricing to obtain true economic value from market prices.

Among information that reflects economic benefits and costs and that may be omitted, or need exclusion, from financial data are the following:

To be sure, there are many cases in which market prices exist but where they are not reflective of the economic value of a product or service. A perfectly competitive economy, as we have discussed throughout this book, is one such that the price of everything precisely represents the value the last unit of that product or service contributes to production and consumption. Under conditions of perfect competition, there will generally exist an economic equilibrium. In this economic equilibrium, the “best” use of all productive units, yielding economic efficiency, will be achieved. There will be no alternate use of the resources of labor, capital, and land that would result in more efficient production and in greater total satisfaction in consumption. But markets are imperfect, and so prices will not, in all cases, perfectly reflect value. This inadequacy of market prices as a true value measurement is a strong reason why we should make a separate determination of value, and not just equate value to price, in all instances except those of private individual investments, where market price is what the individual is concerned with.

In those cases where the production of goods or services increases with the price of the good or service remaining constant, the social benefit of increased production is just the fixed price times the change in production quantity. However, when there is a substantial change in price, we must use an alternate approach. We need to determine a quantity called the compensating variation, which, we will show, is essentially equivalent to the concepts of consumer surplus and producer surplus. Our earlier discussions concerning consumer and producer surplus in Chapter 4 are of interest here.

The effects on the welfare of individuals that result from changes in the prices of products and services are measured by the consumer surplus. People obtain similar effects from changes in the wages of the factor services that they supply. These changes are denoted as producer surplus. A producer surplus also results to producers who are able to sell goods at a price higher than the smallest price they are willing to accept for these products and services.

We have indicated that willingness to pay is equivalent to the approximate area under a demand curve. A useful way to interpret a demand curve follows from the willingness to pay concept. Let D(q) be the demand curve for a specified time interval. Then if the price is constant at p0 the consumer will purchase q0 over that interval; if the price is p1 the consumer will purchase q1 for that interval; and if the price is p2 the consumer will purchase q2. The demand curve does not mean that, strictly speaking, if the price is initially at p0 the consumer will instantly purchase q0; it says that over the time horizon for which the demand curve was constructed, the consumer will purchase q0 if the price is constant at po. What this might also mean is that if the price drops to p1 during the time interval for which the demand curve is valid, the consumer will purchase an additional q1 − q0 times the fraction of the total demand horizon time that the price is p1. If it drops again to p2 the consumer might purchase an additional q2 − q1 times the fraction of the total demand horizon time that the price is p1. While this interpretation needs to be verified against whatever construct was used to derive it, and modified to give it precision for the true dynamic case, it is easy to see that it is this interpretation on which the consumer surplus concepts are based. We have commented before on the dynamic interpretation of demand curves in Chapter 1.

To derive a measure of the value of a price decrease, it is convenient to pose the question in the following way: What is the maximum amount of money one would be willing to pay to buy as much as they want of the product or service at the decreased price rather than at the initial price? The answer should be that amount of money that results in the same level of utility at the lower price as at the higher price. This amount of money is called the compensating variation. To determine a consumer’s compensating variation, we must either know the consumer utility function or measure the compensating variation, perhaps through measurement of the utility function.

The following are two approximations that are generally valid and will eliminate the need to measure the compensating variation:

Some discussions in Chapter 4 contain additional commentary relative to consumer surplus and these approximations. If the price changes are small, then, as we have previously noted, we do not really need the consumer surplus concept. In advanced economies, almost every product or service purchased, with housing as one possible exception, is a small proportion of total consumer expenditures. Thus consumer surplus is a close measure of the value of a price change. This measure is the compensating variation.

What we are noting here is that when implementation of a project results in lowered prices to consumers, this effect must be considered in determining the value of the project, as consumers would be willing to pay more for some of the quantity of the product or service consumed than they pay with the project. We have shown that the compensating variation for an individual is the value measure to the consumer that results from the project. Also, we have indicated that this compensating variation is in practical circumstances equivalent to the consumer surplus. If the social welfare gain is set equal to the sum of the individual welfare gains, then the aggregation of the compensating variation over all people gives a measure of social welfare. A government agency may wish to assign a higher value for consumer surplus accruing to poor people than it does to rich people. Also, the government might wish to encourage consumption rather than savings and may wish to assign a higher weight to consumer surplus components that result in consumption as contrasted with consumer surplus components that result in investments. Squire and van der Tak (1975) expanded considerably on the concept of “weight” in cost–benefit analysis, and their work should be consulted for further details.

Shadow prices may also be used as a measure of project value. As we have seen, the dual variables of linear programming, the Lagrange multipliers, may be interpreted as shadow prices. These represent the amount by which a cost function will change with changes in the marginal unit of a product or service that is consumed. Thus when we maximize social welfare, the Lagrange multiplier takes on particular meaning. Lagrange multipliers, which are shadow prices, are the social values of goods that are created by a project. The need for these shadow prices arises when market prices do not reflect social value. When a value other than market price is used in a cost–benefit analysis, that value is a shadow price. The justification for shadow pricing is that decisions must be made, and decisions imply that valuations have been made. If market values are not available, or appropriate, then other values must be used. These values should reflect social values if they are to improve decisions in a social context.

We have already discussed some approaches to determining “shadow prices” in this section. It is useful to view shadow prices as the dual variables in a linear program. We now consider a simple economy. There are two types of products: final consumption products X and raw materials Y. Society has somehow valued the final goods by associating prices with them, pi, i = 1, 2,..., N. A linear technology, through which the raw materials are transformed into final consumption products,

or

is assumed to exist. Here Xi is the number of units produced of product i and Yji the amount of raw material j used in the production of product i. Aij are nonnegative production process parameters. The raw material available for use is constrained. The production process must also satisfy the raw material constraints

Our goal here is to maximize the social value of production. This is

Thus we wish to maximize

subject to the equality constraint

where Yi represents the vector of raw materials going into product Xi. We also have the inequality constraint equation

In this section, we have discussed approaches toward the identification and quantification of costs and benefits. The fundamental concept here is willingness to pay. Much of our discussion concerned approaches that could be used to identify the value of costs and benefits when prices did not exist or when they were biased. It is not easy to get people to “willingly” reveal their willingness to pay. The clever use of questionnaires and public participation efforts are sometimes useful toward these ends, in addition to experimental and empirical results.

6.4 THE IDENTIFICATION AND QUANTIFICATION OF EFFECTIVENESS

In Section 6.3 we discussed various approaches toward the identification and quantification of economic costs and benefits associated with alternative action options or projects. Often there will be a variety of reasons why people will be uncomfortable with providing a strict economic measure for benefits. The word effectiveness is often used for benefit when a strictly economic valuation is not intended. When effectiveness is substituted for benefit we obtain a cost–effectiveness analysis.

In cost–effectiveness analysis, we desire to rank projects in terms of economic costs and effectiveness. The reason for this is that there are noncommensurate attributes of a project. Certainly we would wish to eliminate dominated or inferior projects—projects that are more expensive and less effective than other projects—from consideration for selection. Beyond this, a cost--effectiveness analysis does not specify which of the several nondominated projects is “best.” This can be accomplished if one is willing to trade off cost for effectiveness, so as to obtain a scalar performance index. It can be done by considering cost as one of the attributes in the effectiveness evaluation approach we will now briefly describe. This is but one of the many generally similar approaches in the subject area of decision analysis, an important field, but one that we are unable to explore in any depth here.

The effectiveness of an alternative is the degree to which that alternative is perceived by the decision maker as satisfying identified objectives. The effectiveness assessment approach described here provides an explicit procedure for the translation of qualitative impressions of values, or effectiveness indices, into a quantitative evaluation of alternatives when the impacts of the alternatives are described by multiple attributes. This is accomplished by identifying and organizing the attributes or proposed alternatives into a tree-type hierarchy, attribute tree, or worth structure that is used, together with measures of effectiveness, to compare alternatives as a basis of choice making. Effectiveness assessment is most appropriate when a single approach is desired that will enable us to specify and interpret the effectiveness of the impacts of proposed policies on individuals or groups, and to rank or prioritize programs in terms of effectiveness such as to enable the selection of that policy which has maximum effectiveness for a particular group.

The typical final results or product of the effectiveness evaluation approach we describe is an explicit evaluation of the worth or value of the outcomes or impacts associated with specific proposed policies in terms of the attributes or objectives that have led to the policy proposals or projects. The procedure that we describe also results in an easily communicable picture of the effectiveness, value, or importance that individuals or groups place on different attributes or objectives associated with the impacts of the proposed projects. Typically, also, there results a significant amount of learning by decision makers concerning their own preference structure and the consistency of their evaluations and associated decisions with their preferences. An increased understanding of the decision situation through a careful definition of alternatives, outcomes, and their relationship, and of decision-maker preferences for possible outcomes is another result of the effort. Generally an effectiveness assessment study involves the following major steps:

Effectiveness assessment can be very useful for the interpretation and evaluation of the results of an analysis effort. To use the approach we need a set of attribute or objective measures deemed to be of importance for issue resolution, information on the relative importance of attributes or objectives, and sufficient knowledge about project alternatives and their outcomes to be able to assign effectiveness scores to the attribute measures that characterize the impacts of each outcome.

6.5 SUMMARY

As we have noted, cost–benefit analysis is a method used by systems analysts and assessors to aid decision makers in the evaluation and comparison of proposed alternative plans or projects. Objectives must be identified and alternatives generated and defined carefully prior to initiation of formal analysis efforts. Then the costs as well as the benefits of proposed projects are identified. These costs and benefits are next quantified and expressed in common economic units whenever this is possible. Discounting is used to compare costs and/or benefits at different points in time. Present worth is the discount criterion of choice. Overall performance measures, such as the total costs and benefits, are computed for each alternative. In addition to this quantitative analysis, an account is made of qualitative impacts due to intangibles such as social, esthetic, and environmental effects. Equity considerations are considered to determine the distribution of costs and benefits across various societal groups. The cost–benefit method is based on the principle that a proposed economic condition is superior to the present state if the total benefits of a proposed project exceed the total costs, so that if there are provisions whereby gainers compensate losers, everyone is better off. Distribution or equity issues are addressed in the qualitative part of the analysis. Results are presented to the decision makers, who may use them to select one or more of the proposed project alternatives.

Among the results of a cost–benefit analysis are the following:

The following major activities are generally accomplished in a cost–benefit analysis:

Figure 6.7 illustrates typical steps of a cost–benefit analysis. The following are appropriate conditions for the use of a cost–benefit approach:

Cost–effectiveness analysis is, as we have seen, a very similar approach to analyze, assess, and compare the costs and benefits of various alternative projects. In principle, it is possible to adjust alternative plans of different generic types such that the same effectiveness index results from each project. Then the least costly project should be selected. In practice, this is difficult to do for many “discrete” projects in which alternatives have parameters that are continuously adjustable. Methods such as multiattribute utility theory can be used to evaluate effectiveness. In conjunction with cost analysis we are able to use the resulting effectiveness indices to assist in making trade-offs between the quantitative and qualitative attributes of the alternative projects.

BIBLIOGRAPHY AND REFERENCES

Our first discussions in this chapter concerned the time value of money. A classic paper in this area, which first discussed some of the fundamental limitations with the IRR criterion and presented a Lagrange multiplier solution to the capital budgeting under constraint problem, is

Lorie J, Savage LJ. Three problems in capital rationing. J Business 1955;28(4):229–239.

The mathematical optimization approach to capital budgeting is discussed in

Peterson PP, Fabozzi FJ. Capital budgeting: theory and practice. Hoboken, NJ: Wiley; 2002.

Weingartner HM. Mathematical programming and the analysis of capital budgeting problems. Chicago: Markham Publishing; 1967.

The axiomatic development that justifies the net present worth concept for project evaluation may be found in

Williams AC, Nassar JJ. Financial measurement of capital investment. Manag Sci 1966;12(12):85l–863.

There are many discussions of discount rates in the literature; especially recommended are

Baumol WJ. On the social rate of discount. Am Econ Rev 1968;58:788–802.

Marglin SA. The social rate of discount and the optimal rate of investment. Q J Econ 1963;77:95–111.

Contemporary works that present discussions of cost–benefit analysis and references to many cost–benefit studies that have been performed include

Brent RJ. Applied cost benefit analysis. 2nd ed. Northampton, MA: Edward Elgar; 2006.

Mishan EJ, Quah EH. Cost–benefit analysis. 5th ed. New York: Routledge; 2007.

Prest AR, Turvey R. Cost benefit analysis: a survey. Econ J 1965;75:683–735.

Sassone PG, Schaffer WA. Cost–benefit analysis—a handbook. New York: Academic; 1978.

Squire L, van der Tak HG. Economic analysis of projects. Baltimore: John Hopkins University; 1975.

Sugden R, Williams A. The principles of practical cost–benefit analysis. London: Oxford University; 1978.

Most of the aforementioned works concern cost–benefit analysis in the public sector. Works that discuss engineering project investments and decisions in the private sector and which are especially recommended include

Bussey LE. The economic analysis of industrial projects. Englewood Cliffs, NJ: Prentice-Hall; 1978.

Copeland TE, Weston IF. Financial theory and corporate policy. Reading, MA: Addison-Wesley; 2006.

Rose LM. Engineering investment decisions. Amsterdam: Elsevier; 1976.

The above-mentioned three texts discuss aspects of resource allocation under uncertainty considerations, an important topic that we have been unable to discuss here.

Two recent works that integrate public and private sector considerations are

Miller C, Sage AP. Application of a methodology for evaluation, prioritization and resource allocation to energy conservation program planning. Comput Electr Eng 1981;8(1):49–67.

Miller C, Sage AP. A methodology for the evaluation of research and development of projects and associated resource allocation. Comput Electr Eng 1981;8(2):123–152.

Especially valuable are two general works in systems science and engineering that provide much discussion concerning cost–benefit and cost–effectiveness analyses as well as related topics:

de Neufville Richard, Stafford JH. Systems analysis for engineers and managers. New York: McGraw-Hill; 1971.

Porter AL, Cunningham SW. Tech mining: exploiting new technologies for competitive advantage. Hoboken, NJ: Wiley; 2004.

Porter AL, Rossini FA, Carpenter SR, Roper AT. A guidebook for technology assessment and impact analysis. New York: North-Holland; 1980.

1Due to Lorie and Savage (1955).

2See Williams and Nassar (1966).

3Named after Cecil Pigou, a noted nineteenth-century economist.