Chapter 14THE THEORY OF COST-BENEFIT ANALYSISJEAN DREZE AND NICHOLAS STERN *London School of Economics1. Basic principles1.1. IntroductionCost-benefit analysis is very widely used and it is therefore important that itsmethods be properly understood. In this chapter we try to contribute to theunderstanding by giving a formal description of the subject and examining thetheoretical basis for some of the techniques which have become accepted tools ofdecision-making around the world.The purpose of cost-benefit analysis is to provide a consistent procedure forevaluating decisions in terms of their consequences. This might appear as anobvious and sensible way to proceed, but it is by no means the only one(examples of alternative procedures are majority voting, collective bargaining, theexercise of power, or the assertion of rights). So described, cost-benefit analysisclearly embraces an enormous field. To keep our subject-matter manageable, weconfine most of our attention in this chapter to its best-known and mostimportant application: the evaluation of public sector projects. Nevertheless, the*The chapter was written at different places in the period 1982-85 including the Indian StatisticalInstitute and the Development Economics Research Centre (DERC) at the University of Warwick.Financial support was received from the Economic and Social Research Council through the DERCand the programme "Taxation, Incentives and the Distribution of Income" at the London School ofEconomics.We have benefited greatly from the comments and suggestions of E. Ahmad, K.J. Arrow, A.B.Atkinson, A. Auerbach, C.L.G. Bell, C.J. Bliss, V.K. Chetty, D. Coady, A.S. Deaton, P.A. Diamond,W.E. Diewert, A.K. Dixit, J.H. Dreze, G.S. Fields, R. Guesnerie, P.J. Hammond, I. Heggie, C. Henry,G.J. Hughes, B.R. Ireton, P.O. Johansson, D. Lal, I.M.D. Little, M. Marchand, D.M.G. Newbery,K.W.S. Roberts, A. Sandmo, M.FG. Scott, J. Seade, A.K. Sen, M.A.M. Smith, S. Venu, and S.Wibaut. Bibliographical assistance was provided in the summer of 1982 by Graham Andrew to whomwe are very grateful. All errors are ours.Handbook of Public Economics, vol. II, edited by A.J. Auerbach and M. Feldstein 1987, Elsevier Science Publishers B. V. (North-Holland)

910J. Drze and N. Sterntheory we develop also offers clear guidelines for the evaluation of governmentdecisions in such varied fields as tax, trade or incomes policies, the provision ofpublic goods, the distribution of rationed commodities, or the licensing of privateinvestment.We shall concentrate on theory. Furthermore, we shall not attempt to present asurvey or summary of the vast literature on the theoretical aspects of the subject.Rather, we shall put forward a view of how cost-benefit analysis should proceed,give a fairly unified account of the most salient results of the theoreticalliterature, show how the framework encompasses a number of approaches to thedefinition and formulation of cost-benefit problems, and then discuss implications for a number of practical issues.Accordingly, the contents of the paper are as follows. In Section 1 weintroduce the basic concepts of cost-benefit analysis for project evaluation. Inparticular we show how and when shadow prices can be used to constructcost-benefit tests which evaluate projects in terms of their net effect on socialwelfare. For this to be the case, the shadow price of a commodity must be definedas the total impact on social welfare of a unit increase in the net supply of thatcommodity from the public sector. In order for this definition to be operational,it must be possible to predict all the repercussions of a project. We shall embodythis idea in the notion of a "policy" and emphasise the close relationship betweenshadow prices and the choice of policies. We attempt, in Section 2, to drawtogether some important results selected from the theoretical literature, byanalysing a single model and following the principles outlined in Section 1. InSection 3 we review a number of more practical issues at the centre of theliterature on applied project evaluation (the treatment of traded and non-tradedgoods; the discount rate; the shadow wage, and so on) in the light of the previousresults. Section 4 contains concluding comments.1.2. Project evaluation, cost-benefit analysis, and shadow pricesIn this subsection we introduce some basic concepts which will be usedthroughout. They are given formal structure, discussed and developed in the restof the paper.By a project, we shall understand a change in the net supplies of commoditiesfrom the public sector. The term "public sector" is interpreted here in asomewhat restricted sense, which will be clarified below; however, the theory wedevelop also provides precise guidelines for the evaluation of projects in theprivate sector. The analysis will be conducted from the point of view of aplanner, who has to assess projects and who has preferences over states of theeconomy, embodied in a well-defined objective function or "social welfare"

Ch. 14: Theory of Cost-Benefit Analysis911function. The interpretation, specification and necessity of the objective functionwill be discussed in detail below.The process of judging whether or not a project should be accepted is calledproject evaluation. Cost-benefit analysis is the examination of a decision in termsof its consequences or costs and benefits. The shadow price of a good measuresthe net impact on social welfare of a unit increase in the supply of that good bythe public sector.In the context of project evaluation a cost-benefit test is a simple decision rulewhich consists of accepting only those projects which make a positive profit atshadow prices. As we show below, our definition of shadow prices ensures thatshadow profits are precisely a (first-order) measure of the net effect of aproject on social welfare, so that cost-benefit tests succeed in identifying welfareimproving projects.In order to evaluate a project from the point of view of its consequences, it iscrucial to have a model which predicts the total effect on the state of the economyof undertaking a particular project. This total effect involves a comparison of theeconomy "with" the project and the economy "without" it. Formally, weembody the relation between a project and its consequences in the notion of a"policy", i.e. a rule which associates a state of the economy with each publicproduction plan. It is a recurring theme of this chapter that different policiescorrespond to different rules for shadow pricing. To the extent that severalpolicies are genuinely available, we argue that the policy and project should beselected with respect to the same criterion, the level of social welfare. We alsoexamine closely the special case where there is no real choice and only one policyis available to the planner.The two basic ingredients of the approach to cost-benefit analysis which isadopted in this chapter are therefore the ability to predict consequences (amodel) and the willingness to evaluate them (an objective function).A major purpose of using the techniques of cost-benefit analysis, and particularly shadow prices, is to allow decisions at the level of the enterprise in thepublic sector. In principle one could imagine a planner who is endowed withinformation on the working of the entire economy and well informed aboutpossible projects, who could calculate the level of social welfare associated withany possible course of action. Formally this is how most optimising modelsappear. We know, however, that it is generally impossible for one office or bureauto be fully acquainted with possibilities and difficulties at each enterprise andhousehold. Thus, we seek to leave many decisions at a level closer to theindividual enterprise but to provide information by which individual decisionsmay be co-ordinated. With this information each enterprise can take decisionswhilst exploiting its own detailed knowledge of its own circumstances. Thus, ourapproach does not assume full knowledge of production possibilities but is

912J. Drize and N. Sternsimply concerned with providing information to public enterprises for the appraisal of their own projects.1.3. The basic theory of shadow pricesWe develop in this subsection a model which formalises the concepts introducedabove. It will be given further structure in Sections 2 and 3.Our economy consists of "private agents" and a "planner". The theory as suchallows the identity of the planner to be interpreted in various ways, e.g. at oneextreme the planner could represent a powerful government agency controllingmany policy instruments, and at another it could designate a analyst solelyconcerned with the evaluation of a single project. For purposes of interpretation,we shall also speak of the planner as "the government", bearing in mind,however, that when governments are not "monolithic", all the agencies not underthe control of the planner should formally be included in the set of privateagents.Private agents behave systematically, in response to a vector s (sk) of signalssummarising all the relevant variables affecting their behaviour (prices, taxes,quantity constraints, etc.). Thus, given the vector s, called the environment, oneknows exactly how a private agent will respond, e.g. his net demands or suppliesand his level of utility or profit. In particular the vector E of aggregate netdemands for commodities from private agents is assumed to be a well-definedfunction of s. This is not restrictive provided the vector of signals is definedcomprehensively (e.g. it could include scale factors for constant returns to scaleindustries - see Section 2.3.5). Commodities are indexed by i, taking into account,if necessary, the time of their delivery and the state of the world. Problems raisedby time and uncertainty will be discussed separately in Section 3. The net supplyof the public sector, or public production plan, is represented by the vector z withcomponents z i (where z i 0 if the ith commodity is neither used nor producedin the public sector). The public sector is identified with the set of firms directlyunder the control of the planner; in particular the planner should have fullcontrol over both the production plan and profits of these firms - the notion isfurther discussed in Section 2.3.1.Two types of constraints restrict the set of environments which may realistically be considered by the planner as feasible. The scarcity constraints require thematching of net supplies and demands. In addition, side constraints describe anyfurther limitations on the selection of s by the planner - e.g. permissible tax ratesmay be restricted, or quotas which he cannot influence may apply. Formally,these constraints are respectively written asE(s) -z O(1.1)

Ch. 14: Theory of Cost-Benefit Analysis913ands ES,(1.2)where E(s) is the vector of net demands from the private sector, and S is theopportunity set of the planner.We write (1.1) with strict equality since otherwise the use of some of the netpublic supply would not be described. To write it as an inequality constraintwould involve the unnecessary assumption that free disposal is possible. Freedisposal is an aspect of production possibilities and it may or may not be aproperty of the public sector production set (denoted by Z). We shall not alwaysassume that the public production set is known to the planner (indeed, as wehave argued above, the use of cost-benefit techniques may aim partly at avoidingthe difficulties associated with centralising such knowledge). Rather, we shallconsider an arbitrary initial value of z, and represent a small project as aninfinitesimal perturbation dz of this public production plan. A small project isfeasible if (z dz) E Z. We shall not be concerned with assessing the feasibilityof projects, but rather with appraising the desirability of a priori specified, andpresumably feasible, projects. The concentration on small projects motivates ouruse of differential techniques; all the functions appearing in this paper areassumed to be once continuously differentiable [the reader who wishes to pursuenon-differentiability, corner solutions and the like should see Guesnerie (1979)].The normative element of the model consists of an objective function whichreflects the planner's preferences between different environments:V: s - V(s)(1.3)(recall that the behaviour of private agents can always be inferred from s). Asbefore we also speak of V(.) as the social welfare function. Together (1.1), (1.2)and (1.3) constitute the model of the planner.For future reference we should point out what is involved in writing theconstraints in the above manner. First, note that writing demands and preferences as functions of s in (1.1) and (1.3) (rather than of s and z) involves no lossof generality since s is defined to include all the variables relevant to thebehaviour of private agents. Secondly, it is not restrictive in (1.2) to regard S asindependent of z, since one can always substitute for z using (1.1). Thirdly, wecould not use this last procedure where there is a constraint linking s with theproduction plan of individual firms in the public sector. Examples where thismight arise are externalities from a specific public firm to consumers or privatefirms, or a firm-specific budget constraint (of the Boiteux type) applying to apublic firm. At this stage such problems are precluded by the notion of fullcontrol (by the planner) over the public sector; however, they will be explicitlyexamined in Section 2.

914J. Drize and N. SternBy a (feasible) policy, we shall understand a function, denoted by 0p( ), whichassociates with each public production plan z an environment s such that (s, z)satisfies (1.1) and (1.2). We assume that at least one feasible policy exists. This isnot very restrictive since it amounts to saying that any public production plan iscompatible with at least one environment (at least around the initial value of z).Once a policy (0 is specifi