Cost-Effectiveness Analysis

The effectiveness of a program is measured by the extent to which, if implemented, some desired goal or objective will be achieved. Since a goal usually can be achieved in more than one way, the analytical task is to determine the most effective approach from among several alternatives. The preferred alternative either (1) produces a desired level of performance at the minimum cost or (2) achieves the maximum level of performance possible for a given level of cost. Although costs can ordinarily be expressed in monetary terms, levels of achievement are usually represented by non-monetary indexes, or measures of effectiveness. Such indices measure the direct and indirect effects of resource allocations.

Output Orientation

Techniques of cost-effectiveness analysis originated in the early 1970s and initially were used in situations where benefits could not be measured in units commensurable with costs. In these early applications, the level of effectiveness or output was usually taken as a given. Several alternative methods of achieving this level were then examined in order to identify the alternative with the lowest costs. These initial studies revealed many important aspects of decision making with respect to the allocation of scarce resources.

In contemporary applications of cost-effectiveness analysis, the emphasis is on program objectives and on the use of effectiveness measures to monitor progress toward agreed-upon objectives. The extended time horizon adopted in cost-effectiveness analysis leads to a fuller recognition of the need for life-cycle costing--that is, analysis of costs over the estimated duration of the program or project.

Cost-effectiveness analysis can be viewed as an application of the economic concept of marginal analysis. The analysis must always move from some base that represents existing capabilities and existing resource commitments. The objective is to determine what additional resources are required to achieve some specified additional performance capability. Thus, the focus is on incremental costs.

Effectiveness measures involve a basic scoring technique for determining the increments of output achieved relative to the investment of additional increments of cost. Effectiveness measures are often expressed in relative terms--for example, percentage increase in some measure of educational attainment, percentage reduction in the incidence of a disease, or percentage reduction in unemployment. These measures facilitate comparisons and the rank-ordering of alternatives in terms of the costs involved in achieving identified goals and objectives. However, since benefits are not converted to the same common denominator, the merit of any single project cannot be ascertained. Nor is it possible to compare which of two or more projects with different objectives will produce the better returns on investment. It is only possible to compare the relative efficacy of program alternatives with the same or similar goals and objectives.

Types of Analyses

Three supporting analyses are required under the cost-effectiveness approach:

(1) Cost-goal studies are concerned with the identification of feasible levels of achievement.

(2) Cost-effectiveness comparisons assist in the identification of the most effective program alternative.

(3) Cost-constraint assessments determine the cost of employing less than the most optimal program.

The objective of a cost-goal study is to develop a cost curve for each program alternative. This curve approximates the sensitivity of costs (inputs) to changes in the level of goal achievement (outputs). Costs may change in direct proportion to the level of achievement; that is, each additional increment of cost may produce the same increase in output. However, if output increases more rapidly than costs, then the program alternative is operating at a level of increasing return. This condition is represented by a positively sloped curve that rises at an accelerating rate, as illustrated by the initial segment of cost curve B in Exhibit 10. If costs increase more rapidly than output, the program alternative is operating in an area of diminishing returns (as in the upper segment of cost curve B).

Exhibit 10. Cost-Effectiveness Analysis in Graphic Form

Cost-effectiveness analysis requires a model that can relate incremental costs to increments in achievement. For some types of problems, practical models can be developed with relative ease. For other problems, cost curves can be approximated from historical data. As the input-output relationships associated with various program alternatives are better understood, the construction of cost curves and effectiveness scales should become increasingly more sophisticated.

Assuming that the costs associated with different achievement levels can be determined for each alternative, the problem remains of how to choose among these alternatives. In principle, the rule of choice should be to select the alternative that yields the greatest excess of positive effects (attainment of objectives) over negative impacts (resources used, costs, and negative spillover effects). In practice, however, this ideal criterion is seldom applied, as there is no practical way to subtract dollars spent from the non-monetary measures of effectiveness.

The best approach, therefore, may be a cost-effectiveness comparison of program alternatives, as illustrated in Exhibit 10. Alternative A achieves the first level of output (O1) at a relatively modest level of cost (C1A), whereas nearly twice the amount of resources (C1B) would be required to achieve the same level of effectiveness using alternative B. Both alternatives achieve the second level of output (O2) at the same level of cost (C2). Alternative B requires a lower level of resources (C3B) to achieve the third level of output (O3). And only alternative B achieves the fourth level of output (O4), since the program cost curve of alternative A is not projected to reach this level of effectiveness.

Which of these two program alternatives is more desirable? To answer that question, it is necessary to define the optimum envelope formed by these two cost curves. If resources in excess of C2 are available, then alternative B is clearly the better choice. However, if available resources are less than C2, alternative A provides greater effectiveness for the dollars expended.

In general, it may not be possible to choose between two alternatives simply on the basis of cost-effectiveness unless one alternative dominates at all levels of goal achievement. Usually, either a desired level of performance must be specified and then costs minimized for that effectiveness level, or a cost limit must be specified and achievement maximized for that level of resource allocation.

In practice, organizations may adopt programs that are do not the most effective technically available. Among the more obvious reasons for this are legal constraints, technical capacity, employee rights, union rules, and community attitudes. The purpose of a cost-constraint assessment is to examine the impact of these factors by comparing the cost of the program that might be adopted if no constraints were present with the cost of the constrained program.

This analysis, shown graphically in Exhibit 11, starts with the expressed goal O1 and two programs (P constrained and P not-constrained). P not-constrained represents the most effective program as determined by cost-effectiveness analysis. The constrained program, however, may be the only program available. The cost of the constraints to the agency is the difference between the program cost of P constrained and P not- constrained

Exhibit 11. Cost-Constraint Analysis in Graphic Form

Once this cost differential has been identified, decisions can be made as to the feasibility of eliminating the constraints. This assessment gives decision makers an estimate of how much would be saved by the relaxation of a given constraint. By the same token, the cost of the constraint suggests of the amount of resources that might be committed to overcoming it. In some cases, however, maintaining a constraint may be more important for social or political reasons than implementing a more effective program.

Optimum Envelope

Significant shifts in the configuration of the cost curves frequently occur in the formulation of program alternatives as additional levels of effectiveness are sought. Thus, program A may provide the most desirable ratio at one level of effectiveness (and cost), whereas at a higher level of effectiveness (and cost), some other program may provide the more desirable ratio.

The following case study illustrates this situation. Assume that some 3,000 workers in a given state become unemployed each year due to technical obsolescence, that is, the jobs for which they are trained and skilled are eliminated through the mechanization of industrial processes. The state seeks to establish an effective program to retaining all or a significant portion of these workers to new skills through an intensive one-year training course. To provide this training, it is necessary to develop regional training centers, build new facilities, hire new instructional personnel, and so forth. It is anticipated that the program will operate over a ten-year period. Through this program, it is anticipated that the workers will be employable at a desirable skill level ten years earlier than if they had to attain these skills on their own.

Two alternative programs are identified to meet these objectives. Program A is an equipment-intensive approach, involving extensive use of programmed learning techniques, tape libraries to upgrade basic skills, the use of computers to achieve self-paced learning, and so forth. This program only requires five instructors per training center and has a trainee-instructor ratio of 60 to 1. Program B is a teacher-oriented approach, involving team-teaching techniques. It requires 20 instructors per training center and has a 10 to 1 trainee-instructor ratio. The trainee capacity at training centers for program A is 300 and for program B, 200. The costs for each program are summarized in Exhibit 12.

Exhibit 12. Alternative Program Costs

Items of Cost Program A Program B
Development costs $13,000,000 $1,000,000
Investment per training center $1,500,000 $ 500,000
Operating costs per year per center $1,750,000 $3,000,000

Exhibit 13. Total Costs over Ten Years

Program A
Number of Centers 0 3 6 10
Trainees 0 900 1,800 3,000
Research & Development $13,000,000 $13,000,000 $13,000,000 $13,000,000
Investment -- $4,500,000 $9,000,000 $15,000,000
Operations -- $5,250,000 $10,500,000 $17,500,000
Totals $13,000,000 $22,750,000 $32,500,000 $45,000,000
Program B
Number of Centers 0 5 9 12 15
Trainees 0 1,000 1,800 2,400 3,000
Research & Development $1,000,000 $1,000,000 $1,000,000 $1,000,000 $1,000,000
Investment -- $2,500,000 $4,500,000 $6,000,000 $7,500,000
Operations -- $15,000,000 $27,000,000 $36,000,000 $45,000,000
Totals $1,000,000 $18,500,000 $32,500,000 $43,000,000 $53,500,000

It is now possible to examine how costs and benefits (program effectiveness) are related in the tests for preferredness. Since decision-makers do not know the level of training that can be supported given limited resources, it is necessary to develop a schedule of costs and benefits over the full range of workers to be trained each year (i.e., 0 to 3,000). Program A would require 10 training centers for 3,000 trainees and program B would require 15 centers. The development, investment, and ten years of operating costs are summarized in Exhibit 13 for various levels of coverage.

Based on these data, it is possible to identify the best program given either a fixed budget or a specified level of benefits. Program B is preferred for all budgets under $32.5 million because it would have a greater trainee capacity. Conversely, for all trainee loads less than 1,800, Program B is preferred because it will cost less than Program A. For budgets above $32.5 million or trainee loads above 1,800, Program B is preferred. For example, at a budget of $26 million, Program A has an annual capacity of 1,200 trainees, but Program B could accommodate 1,400 trainees each year for $25.5 million. However, if the objective is to handle an annual training load of 2,400, then Program A would cost $39 million, whereas Program B would cost $43 million. This brief example illustrates how fixed costs (i.e., R&D costs), investment costs (per center), and variable costs (operating costs) can impact the overall cost configuration in different ways.

Risk and Uncertainty

Financial planning and management often is concerned with future events that are inevitably characterized by uncertainty. It is important to recognize such uncertainty and to explicitly deal with it from the outset. Strategic decisions should involve an assessment of uncertainty and risk based on available estimates of alternative payoffs or gains. A risk is taken no matter what the decision. Even the decision to do nothing involves the risk of lost opportunity. An effective financial manager, whether in the public or private sector, must be aware of how opportunity, innovation, and risk are interrelated and must be willing to take risks appropriate to his or her level of responsibility.

Converting Uncertainty to Risk

One financial manager's uncertainty may be another's acceptable risk. What one manager may interpret as an uncertain situation to be avoided, another may see as an opportunity, albeit involving some risk. Although the two terms often are mistakenly used interchangeably, the distinction between uncertainty and risk is important in fiscal management.

Certainty can be defined as a state of knowledge in which the specific and invariable outcomes of each alternative course of action are known in advance. The key to certainty is the presence of only one state of nature (although under some circumstances, numerous strategies maybe applied to achieve that state). This condition enables the manager to predict the outcome of a decision with 100 percent probability.

Uncertainty can be defined as a state of knowledge in which one or more courses of action may result in a set of possible specific outcomes. The probabilities of these outcomes, however, are neither known or meaningful. As Archer has observed, uncertainty involves a range of conditions in which probability distributions vary from a condition of relative confidence, based on objective probabilities, to a condition of extreme uncertainty, with little or no information as to the probable relative frequency of particular events. [12]

If a program manager is willing to assign objective or subjective probabilities to the outcome of uncertain events, then such events may be said to involve risk. Risk is a state of knowledge in which each alter-native leads to one of a set of specific outcomes, each outcome occurring with a probability that is known to the decision maker. More succinctly, risk is reassurable uncertainty. Risk is measurable when decision expectations or outcomes can be based on statistical probabilities. The event of a Republican or Democratic victory in any given election is an uncertain outcome. The event of drawing a red card from a well-shuffled deck is an example of a risky outcome with a probability of 50 percent.

Uncertainty, Risk, and Probability Functions

In financial planning and management, risk and uncertainty must be confronted from two primary sources: (1) statistical uncertainty, and (2) uncertainty about the state of the real world in the future. The first type of uncertainty is usually less troublesome to handle. It arises from chance elements in the real world and would exist even if the second type of uncertainty were zero. Monte Carlo and related probability techniques can be used to deal with statistical uncertainty when it is encountered. [13]

Establishing a probability function can bring problems within more manageable bounds by reducing uncertainty to some level of risk that may be tolerable, depending on the risk threshold of the manager or organization. Probabilities can be established either a posteriori (by induction or empirical measurement) or a priori (by deduction or statistical inference).

The basic conditions necessary to establish a posteriori probability are: (1) the number of cases or observations must be sufficiently large to exhibit statistical stability; (2) the observations must be repeated in the appropriate population or universe; and (3) the observations must be made on a random basis. The inductive approach offers the maximum opportunity for applied decision theory, because the number and range of situations in which such objective probabilities can be applied are increasing significantly.

Under the deductive, or a priori approach, a probability statement is not intended to predict a particular outcome for a given event. Rather, it asserts that in a large number of situations with certain common characteristics, a particular outcome is likely to occur. In short, a statistical inference is made regarding the probable outcome of an uncertain event or series of events.

Uncertainty and Cost Sensitivity

The second type of uncertainty--uncertainty about the future state of the real world--is more troublesome for fiscal management. In such cases, the use of sophisticated statistical techniques may be little more than expensive window dressing. When the environment is uncertain, an expected value approach often must be applied. Expected value is deter-mined by multiplying the value products across all possible outcomes. In mathematical terms, expected value (EV) can be expressed as:

where P stands for probability, $ stands for the value of an outcome, and

Several techniques utilizing the concept of expected value have been developed to analyze uncertainty about the future state of events. These techniques include: (1) sensitivity analysis, (2) contingency analysis, and (3) a fortiori analysis. Each of these techniques is applicable in cost analysis under varying circumstances. The purpose here is not to present a "how-to" approach, but rather to identify the conceptual framework underlying these methods.

Sensitivity analysis is designed to measure (often quite crudely) the possible effects that variations in uncertain decision elements (for example, costs) may have on the alternatives under analysis. In most strategic decisions, a few key parameters exhibit considerable uncertainty. The analyst must determine a set of expected values for these parameters (as well as other parameters). Recognizing that these expected values may be, at best, "guesstimations," the analyst may use several values (optimistic, pessimistic, and most likely) in an attempt to ascertain how sensitive the results might be to variations in the uncertain parameters.

Exhibit 14 illustrates how sensitivity analysis can be used to determine the variations in rankings among several alternatives, based on anticipated costs. First, the analyst sets the expected values for all costs that are certain (for which some reliable basis exists for establishing an estimated cost). Three values for the uncertain costs are then determined. The optimistic cost represents an assessment of cost based on the assumption that everything will go right with the project--that all of the uncertainty is resolved favorably. The pessimistic cost represents the opposite assumption. The most likely cost figure falls somewhere in between these two extremes.

___________________________________________________________________________________

Exhibit 14. Illustration of Sensitivity Analysis

Cost Levels Alternative A Alternative B Alternative C
Expected Values of Certain Costs $90,000 $80,000 $100,000
Optimistic Expected Values of Uncertain Costs $10,000 $30,000 $20,000
Expected Values of All Costs $100,000 $110,000 $120,000
Rankings 1 2 3
Pessimistic Expected Values of Uncertain Costs $110,000 $115,000 $90,000
Expected Values of All Costs $200,000 $195,000 $190,000
Rankings 3 2 1
Most Likely Expected Values of Uncertain Costs $60,000 $40,000 $70,000
Expected Values of All Costs $150,000 $120,000 $170,000
Rankings 2 1 3
Composite Expected Values $155,000 $140,500 $166,000

Two points concerning uncertainty are illustrated in Exhibit 14. First, the range of uncertainty may vary from alternative to alternative (for alternative A, the uncertain range is $10,000 to $110,000; for alternative B, $30,000 to $115,000; and for alternative C, $20,000 to $90,000). Second, uncertain costs may not always be the critical factor in determining the "best" alternative. For example, although uncertain costs for alternative C vary over the narrowest range, this alternative still ranks third except under conditions of high, or pessimistic, uncertain costs.

Probability theory also can be applied in connection with sensitivity analysis. Assume, for example, that the probability of the most likely costs being realized is 50 percent; the most pessimistic costs, 30 per-cent; and the most optimistic costs, 20 percent. The composite expected values for all costs are shown at the bottom of Exhibit 15. Given these probability assumptions, alternative B is clearly the preferred alternative.

Contingency analysis is designed to examine the effects on alternative choices when a relevant change is postulated in the evaluation criteria. This approach can also be used to determine the effects of a major change in the general decision environment, or "ground rules," within which the problem situation exists. In short, contingency analysis is a "with and without" approach. In the field of public health, for ex-ample, alternative approaches to environmental health might be evaluated with and without a major new code enforcement program. In a more local context, a public service organization might evaluate various sites for the location of its headquarters under existing conditions of client distribution and access routes. Additional evaluations might then be made, assuming different client distributions and other route configurations.

A fortiori analysis (from the Latin, meaning "with stronger reason") is a method of deliberately "stacking the deck" in favor of one alternative to determine how it might stand up in comparison to other approaches. Suppose that, prior to analysis, the governing board strongly favors alternative C. In performing the analysis on C in comparison to the other feasible alternatives, a deliberate choice is made to resolve any major uncertainties in favor of C. The analyst would then determine how each of the other alternatives compared under these circumstances. If some alternative other than C looks good (that is, if C does not show up "with stronger reason" to be the best alternative), there may be a very strong case for dismissing the initial intuitive judgment in favor of C. This type of analysis can be carried out in a series of trials, with each alternative, in turn, being favored in terms of the major uncertainties.

These three techniques for dealing with uncertainty may be useful not only in a direct analytical sense; they may also contribute indirectly to the resolution of problem situations. Through sensitivity and contingency analyses, for example, it may be possible to gain a better understanding of the really critical uncertainties of a given problem. With this knowledge, a new alternative might be formulated that would provide a reasonably good hedge against a range of more significant uncertainties. This is often difficult to do. When it can be accomplished, however, it may offer one of the best ways to offset the uncertainties of a problem situation.

Uncertainty, Risk, and Expected Utility

The assumption that people actually behave rationally in the manner suggested by the mathematical notion of expected value often is contradicted by observable behavior in risky situation. People are willing to buy insurance, for example, even though they know that the insurance company makes a profit. People are willing to buy lottery tickets even though the chances of winning are minimal.

Consideration of the problem of insurance and the so-called "St. Petersburg paradox" led Daniel Bernoulli, an eighteenth-century mathematician to propose that these apparent contradictions could be resolved by assuming that people act so as to maximize their expected utility, rather than expected value. Thus, people buy insurance because the consequences against which they are insured are significant in view of the costs. People are willing to invest small amounts of money in lottery tickets, even though the probability outcome is highly uncertain, because the payoff is so high relative to their expected utility.

Extensive research has been performed in the area of risk and uncertainty because the behavior of decision makers often appears to violate commonly accepted axioms of rational behavior. Although no exact probabilities may exist for the success or failure of a particular event, has observed that an individual with "clear-cut, consistent preferences over a specified set of strategies. . . will act as if he has assigned probabilities to various outcomes." [14] The values for the probabilities will be unique for each individual and not unlike the values of utility that might be assigned to an individual through a study of his or her social preferences. The obverse of social preferences, of course, is risk aversion, a subject on which opinions vary. [15]

As most economists will now admit, utility theory alone cannot resolve the disputes over social preference and/or aversion to risk. There are numerous situations in which fiscal managers will have to obtain a more careful reading of the various utility functions or preferences of their clientele and the organization as a whole. As Stokey and Zeckhauser explain, strategic choice under uncertainty is a threefold process: [16]

(1) Alternatives must be assessed to determine what probabilities and payoffs are implied for individual members of the organization and its clientele.

(2) Attitudes toward risk of these individuals must be evaluated to determine the certainty equivalents of these probabilities and payoffs.

(3) Having estimated the equivalent benefits that each alternative offers to different members of the organization/clientele, the decision maker must select the preferred outcome.

Although this process may sound simple, it often is very complex in application. Some basic tools been developed to aid in unraveling these complexities. [17] These techniques can be brought into play, however, only after the manager has a fairly good understanding of organizational and/or clientele preferences. Once the groundwork for approximating utility has been laid, the fiscal manager will be better prepared to address uncertainties in a more systematic fashion.

A basic objective of fiscal management is to reduce uncertainty by bringing to light information that will clarify relationships among elements in the decision process. This reduction of uncertainty may cause the risk associated with a particular choice: (1) to remain unchanged; (2) to decrease (as in the case where a reduction in uncertainty permits the assessment of more definitive probabilities); or even (3) to increase (as happens when the additional information reveals risk factors that previously were unknown). Thus, although risk and uncertainty are inter-related, they must be treated independently in many situations.

Summary

In the allocation of limited fiscal resources, it may be assumed that most organizations consider both the payoffs and the pitfalls associated with various program requirements. These assessments, however, are often haphazard and uncoordinated, with little systematic effort to quantify benefits or to include all costs appropriate to the particular alternatives under consideration.

Strategic funds programming is a future-oriented approach that can be helpful in determining where discretionary funds to implement new programs and strategies may be available within the fiscal structure of an organization. The techniques used in programming strategic funds help to identify feasible options under various fiscal assumptions. The fiscal manager, however, must still make an assessment of risks and payoffs before the "best" option is selected.

Various techniques of financial ratio analysis--such as liquidity measures, leverage ratios, measures of profitability, and asset utilization ratios--have been widely used for many years to indicate the relative well-being of business organizations. These ratios, however, tend to be retrospective and static in nature and are of only limited application in public resource management.

In recent years, interactive computer software has become a significant analytical tool for fiscal planning, making possible on-line, real-time decision support systems. Traditional methods of financial analysis use hindsight to determine why things went wrong. Computer-assisted methods of fiscal planning provide a basis on which to anticipate (and accommodate) change before its full impact occurs. Most computer-based systems for fiscal planning can also be used to analyze risk and uncertainty.

Factors influencing future costs must be examined as part of the fiscal planning process. Monetary costs--research and development costs, investment costs, and the costs of operations, maintenance, and replacement--are commonly reflected in financial accounts. In fiscal planning, however, it often is necessary to look beyond these monetary costs to opportunity costs, associated costs, and social costs. A thorough cost analysis must also distinguish among (1) fixed and variable costs, (2) recurring costs, and (3) marginal or incremental costs. These costs should be examined over the life of the project or program under analysis. The need to adopt an extended time dimension in such cost assessments has led to the development of cost-benefit analysis.

Cost-benefit and cost-effectiveness analysis can be applied at two pivotal points in the evaluation of resource commitments. In the planning stage, cost-benefit analyses are based on anticipated costs and benefits. Such analyses are not necessarily empirically based. After a program or project has been implemented and shown to have a significant impact, cost-benefit and cost-effectiveness analyses can be used to assess whether the costs of the program are justified by the magnitude of net outcomes. Such after-the-fact analyses should be based on detailed studies of available empirical data.

Cost-benefit and cost-effectiveness models need not be adopted "whole cloth." A number of subroutines may be introduced into ongoing cost analysis procedures. Decision inputs can be developed to include considerations of time preference and marginal productivity of capital investment. The techniques of cost curve analysis can be applied to a variety of decision situations. The examination of expenditures in terms of program objectives and the evaluation of total benefits for alternative program expenditures can be important derivatives of cost-benefit techniques. The extended time horizon adopted in these analytical methods leads to a fuller recognition of the need for life-cycle costing and benefits analysis. The importance of incremental costing, sunk costs, and inheritable assets also is underlined by this extended perspective. Cost-goal and cost-constraint analyses add other important dimensions to the information available to the decision maker. As the complexity of the resource allocation problem becomes more evident, other subroutines may be adopted, depending on the availability of data and the capabilities of the analyst.

Uncertainty can be reduced and risk can be brought within tolerable limits through the generation of management information that clarifies critical relationships among elements in the decision process. Various methods have been formulated for converting uncertainty to risk --including the use of objective and subjective probabilities and the techniques of sensitivity analysis, contingency analysis, and a fortiori analysis. The concept of expected utility has been touched upon in this chapter in an effort to provide the reader with a broader understanding of the critical dimensions of strategic decisions.

Endnotes

[1] William H. Hausell, Jr., "Forward" to Long-Term Financial Planning: Creative Strategies for Local Government (Chicago: International City Management Association, 1987), p. 2

[2] Ibid., pp. 2-5

[3] See: Allen Schick, The Road to PPB: The Stages of Budget Reform in Public Budgeting, Program Planning, and Evaluation, Fremont J. Lyden and Ernest Miller, editors (Chicago: Markham, 1968), pp. 18-31.

[4] Jeffrey Chapman, "The Future of Local Government," Long-Term Financial Planning: Creative Strategies for Local Government (Chicago: International City Management Association, 1987), p. 5

[5] For a more extensive discussion of these issues, see Herbert B. Mayo, Basic Finance: An Introduction to Money and Financial Management (Philadelphia: W.A. Saunders Company, 1980).

[6] Alan J. Rowe, Richard O. Mason, and Karl E. Dickel, Strategic Management and Business Policy: A Methodological Approach (Reading, MA: Addison-Wesley, 1982), p. 102.

[7] U.S. Congress, House Subcommittee on Evaluation Standards, Report to the Interagency Committee on Water Resources, Proposed Practices for Economic Analysis of River Basin Projects (Washington, DC: U.S. Government Printing Office, May 1958), p. 9.

[8] For a further discussion, see: Ronald H. Coase, "The Problem of Social Cost," Journal of Law and Economics 3 (October 1960): 1-44.

[9] A.R. Prest and R. Turvey, "Cost Benefit Analysis: A Survey," The Economic Journal (1965), P. 583.

[10] Anatol Rapoport, "What Is Information" ETC: A Review of General Semantics 10 (Summer 1953), p. 252.

[11] Otto Eckstein, Water Resource Development (Cambridge, MA: Harvard University Press, 1958).

[12] Stephen H. Archer, "The Structure of Management Decision Theory," Academy of Management Journal 8 (December 1964), p. 283.

[13] For a discussion of Monte Carlo techniques, see: E.S. Quade, Analysis for Public Decisions (New York: American Elsevier, 1975).

[14] Sheen Kassouf, Normative Decision-Making (Englewood Cliffs, NJ: Prentice-Hall, 1970), p. 46

[15] For a broader discussion, see Jack Hirshleifer and David L. Shapiro, "The Treatment of Risk and Uncertainty," in Robert H. Haveman and Julius Margolis (eds.), Public Expenditures and Policy Analysis, 2nd ed. (Chicago, Ill.: Rand McNally, 1977), pp. 180-203.

[16] Edith Stokey and Richard Zeckhauser, A Primer for Policy Analysis (New York: Norton, 1978), p. 252.

[17] Howard Raiffa, Decision Analysis (Reading, MA.: Addison-Wesley, 1968). For an introductory discussion of Markov chains, see Stokey and Zeckhauser, A Primer for Policy Analysis, chap. 7.

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