Money-Metric Utility in Applied Welfare Analysis: A Saddlepoint Rehabilitation

Transcription

1 Money-Metric Utility in Applied Welfare Analysis: A Saddlepoint Rehabilitation M. Ali Khan and Edward E. Schlee Abstract The well-known fact that money-metric utility functions cannot be concave in consumption has led some scholars to argue against its use in applied welfare analysis. We prove two welfare results: that any competitive equilibrium allocation maximizes the sum of money-metric utilities at the competitive equilibrium price; and that the derivative of the sum of money-metric utilities equals seven other local measures of welfare in a general equilibrium setting, including one proposed by Radner (1993). The tool we use to prove these results is this: any solution to the consumer s problem for the general non-concave money metric representation must satisfy the famous saddlepoint inequalities from concave programming even if preferences are not convex. This fact allows us to calculate consumer demands as the solution to an unconstrained optimization problem. (192 words) Key words and phrases: Money metric, saddlepoint inequalities, first fundamental theorem, cost-benefit analysis JEL Classification Numbers: D110, C61, D610. This preliminary draft is not to be cited without the authors permission. A first draft of this work was completed during Khan s visit to the Department of Economics at Arizona State University, March 1-6, He is grateful to the department for its hospitality. A second draft was initiated during the second author s visit to the Ryerson University Economics department as Distinguished Lecturer. He thanks the department for its generous invitation and both authors are grateful to lecture participants for questions, in particular to Tsogbadral Galaabaatar. Department of Economics, The Johns Hopkins University. akhan@jhu.edu Department of Economics, Arizona State University. edward.schlee@asu.edu 1

2 1 Introduction In his influential text on choice and competitive markets, Kreps (2013) asks two questions: How many continuous utility functions give the same expenditure function? and Is an alleged expenditure function really an expenditure function. 1 While fully granting the interest in the question of the recoverability of continuous utility function, there is this irony that McKenzie (1957), in dispensing with the utility index in classical demand theory, perhaps un-intendedly, offered the expenditure function as a utility function in and of itself; see Khan and Schlee (2016) for a blow-by-blow account. To be sure, it took Samuelson (1974) to substitute a commodity bundle for the utility of the bundle as an argument in the expenditure function, to see a real-valued function of n + 1 variables as a real-valued function of 2n variables, and to change the nomenclature of an expenditure or a minimum income function to a money-metric utility. It is this change that recovers the inherent duality of the object and to convert it into a powerful engine of analysis. As is by now well-known, Samuelson (1974) used a money-metric to offer the fifth of his six definitions of complementary commodities, and thereby connect to basic and classical themes in consumer theory. His is a comprehensive application of a moneymetric. Khan and Schlee (2017) offered a more limited and sharper application. They invoked the result of Debreu (1976) on least concave representation of preferences to give an alternative proof of the Blackorby-Davidson theorem that money-metric utility is concave if and only if preferences are either homothetic (in the case of the consumption set being the nonnegative orthant of an Euclidean space) or quasihomothetic (in the case when the consumption set is bounded away from zero for each commodity) provided preferences are representable by an increasing, continuous, concave function. The nonconcavity property of the money-metric is elusive in that there is no understanding as to the ranges over which the money-metric is not concave. Indeed, Samuelson s conjecture that the money-metric is locally concave in the neighborhood of the bundle chosen at the base price still awaits a proof or a refutation. 2 Despite Samuelson s use of the money-metric as a vehicle for the for the formulation and analysis of complementarity in the classical theory of the consumer, he was skeptical of his use in welfare economics. He wrote: Whatever the use of the money-metric utility concept.., a warning must be given against its misuse. Since money can be added across people, those obsessed by Pareto optimality in welfare economics as against interpersonal equity may feel tempted to add money metric utilities across people and think that there is ethical warrant for maximizing the resulting sum. That would be an illogical perversion, and any such temptation should be resisted 1 As any reader schooled in the modern microeconomic foundations of the subject would undoubtedly notice, we have cribbed from David Kreps titles. The two questions title Section 10.5 and 10.7 of Kreps (2013). 2 See Footnotes 4 and 5 in Khan and Schlee (2017). 2

3 (p. 1266). The criticism of Blackorby and Donaldson (1988) was sharper. Invoking their theorem, and in particular their examples of preferences represented by money-metric utilities that are not concave no matter what reference price vector is chosen, they argued for the vitiation of the claim that money metrics are useful in applied welfare analysis because they represent preferences exactly. 3 They cited Gorman s result that a social ordering Ω is quasi-concave in individual allocations if the associated Bergson-Samuelson social welfare function W of individual utility representations is quasi-concave, and each of the individual utility levels are themselves concave. Quasiconcavity of Ω ensures that social judgements provide goods for everyone rather than giving them exclusively to the few. Samuelson called this requirement the foundation for the economics of the good society. Use of money metrics in a quasiconcave function W may result in a social-evaluation function corresponding to Ω that is not quasiconcave. 4 To be sure, it has long been understood that the simple sum of money metrics... makes the social ordering of allocations of goods and services across households depend on the reference price vector, and even for a single consumer, the closer the commodity bundle to the one that would be bought at the reference price vector, the more confident the predictions from the theory. 5 Put most succinctly, results obtained with the use of money-metrics need the qualification of their validity for changes in the small. The motivation for this paper is two-fold and can be succinctly stated. First, it is to articulate the observation that this non-concavity of the money-metric is perhaps not as dire as is feared by virtue of the fact that the classical consumer s constrained maximization problem can be converted into an (unconstrained) saddlepoint problem without any concavity assumptions when preferences are represented by the money-metric. After a brief introduction to the basic notation in Section 2, this saddlepoint result is presented in Section 3. It can then be used in the service of applied welfare analysis. It allows an alternative proof of the first fundamental theorem of welfare analysis. Given that this result does not rely on any convexity assumptions on preferences and technologies, this is hardly a surprise, but the proof deserves to be more-widely known; it is presented and connected to other antecedent proofs in Section 4. Furthermore, it allows an extension of Radner s (1978, 1993) result on cost-benefit in the small, and a useful comment to 3 See also Slesnick (1998, p. 1241), Blundell, Preston, and Walker (1994) and Hammond (1994). 4 See Blackorby and Donaldson (1988, p. 21). The authors adduce the example of an additive social welfare function W whose quasi-concavity necessitates, by the Debreu-Koopmans theorem, that all but one of the individual utility representations be concave, and the quasi-concavity of the remaining representation is only necessary but not sufficient. 5 For these observations, see respectively Blackorby and Donaldson (1988, p. 21) and Samuelson (1974, p. 1273). The former write, The social welfare function can be changed, as reference prices change, to undo this effect, but choosing an appropriate social welfare function in each case may be a delicate task. 3

4 the equivalence of welfare measures associated associated with the names of Dupuit, Slutsky, Divisia and Debreu, in addition to Radner, as exhibited in Schlee (2013a) and Schlee (2013b). As such, our rehabilitation is simply to render applied welfare analysis at the same conceptual level as other conventional measures. The concluding section 6 consolidates the various strands, and indicates directions for further analysis to establish the money-metric as an essential object for Walrasian equilibrium theory, and its classical consumer underpinnings. References Blackorby, C., and D. Donaldson (1988): Money metric utility: A harmless normalization?, Journal of Economic Theory, 46(1), Blundell, R., I. Preston, and I. Walker (1994): Introduction to applied welfare analysis, The Measurement of Household Welfare, pp Debreu, G. (1976): Least concave utility functions, Journal of Mathematical Economics, 3(2), Hammond, P. J. (1994): Money metric measures of individual and social welfare allowing for environmental externalities, in Models and Measurement of Welfare and Inequality, pp Springer. Khan, M. A., and E. E. Schlee (2016): On Lionel McKenzie s 1957 intrusion into 20th-century demand theory, Canadian Journal of Economics/Revue canadienne d conomique, 49(2), (2017): The nonconcavity of money-metric utility: a new formulation and proof, Economics Letters, 154, Kreps, D. M. (2013): Microeconomic Foundations I: Choice and Competitive Markets. Princeton University Press, Princeton, New Jersey. Radner, R. (1993): A note on the theory of cost-benefit analysis in the small, in Capital, Investment and Development, ed. by K. Basu, M. Majumdar, and T. Mitra, pp Blackwell. Samuelson, P. A. (1974): Complementarity: An essay on the 40th anniversary of the Hicks-Allen revolution in demand theory, Journal of Economic Literature, 12(4), Schlee, E. E. (2013a): Radner s cost-benefit analysis in the small: an equivalence result, Economics Letters, 120,

5 (2013b): Surplus maximization and optimality, The American Economic Review, 103, Slesnick, D. T. (1998): Empirical approaches to the measurement of welfare, Journal of Economic Literature, 36(4),