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The nonconcavity of money-metric utility: A new formulation and proof

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  • Ali Khan, M.
  • Schlee, Edward E.

Abstract

We offer a new, succinct proof of the fact that the money metric utility is concave for any preference relation representable by a concave function if and only if the indirect utility is affine in wealth. Our proof exploits the existence of a least concave representation established in Debreu (1976), and brings into salience the observation that the money-metric utility to be itself a least-concave representation of the preferences if it is concave. This observation is apparently new.

Suggested Citation

  • Ali Khan, M. & Schlee, Edward E., 2017. "The nonconcavity of money-metric utility: A new formulation and proof," Economics Letters, Elsevier, vol. 154(C), pages 10-12.
    • Handle: RePEc:eee:ecolet:v:154:y:2017:i:c:p:10-12
    DOI: 10.1016/j.econlet.2017.02.007
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    References listed on IDEAS

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    1. Kihlstrom, Richard E. & Mirman, Leonard J., 1974. "Risk aversion with many commodities," Journal of Economic Theory, Elsevier, vol. 8(3), pages 361-388, July.
    2. M. Ali Khan & Edward E. Schlee, 2016. "On Lionel McKenzie's 1957 intrusion into 20th-century demand theory," Canadian Journal of Economics, Canadian Economics Association, vol. 49(2), pages 589-636, May.
    3. Lionel McKenzie, 1957. "Demand Theory Without a Utility Index," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 24(3), pages 185-189.
    4. David Gale, 1967. "On Optimal Development in a Multi-Sector Economy," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 34(1), pages 1-18.
    5. Blackorby, Charles & Donaldson, David, 1988. "Money metric utility: A harmless normalization?," Journal of Economic Theory, Elsevier, vol. 46(1), pages 120-129, October.
    6. Kannai, Yakar, 1977. "Concavifiability and constructions of concave utility functions," Journal of Mathematical Economics, Elsevier, vol. 4(1), pages 1-56, March.
    7. Ali Khan, M. & Piazza, Adriana, 2012. "On the Mitra–Wan forestry model: A unified analysis," Journal of Economic Theory, Elsevier, vol. 147(1), pages 230-260.
    8. Weymark, John A, 1985. "Money-Metric Utility Functions," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 26(1), pages 219-232, February.
    9. Kannai, Yakar, 1980. "The ALEP definition of complementarity and least concave utility functions," Journal of Economic Theory, Elsevier, vol. 22(1), pages 115-117, February.
    10. Debreu, Gerard, 1976. "Least concave utility functions," Journal of Mathematical Economics, Elsevier, vol. 3(2), pages 121-129, July.
    11. Samuelson, Paul A, 1974. "Complementarity-An Essay on the 40th Anniversary of the Hicks-Allen Revolution in Demand Theory," Journal of Economic Literature, American Economic Association, vol. 12(4), pages 1255-1289, December.
    Full references (including those not matched with items on IDEAS)

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    Cited by:

    1. Schlee, Edward E. & Ali Khan, M., 2023. "Money-metrics in local welfare analysis: Pareto improvements and equity considerations," Journal of Economic Theory, Elsevier, vol. 213(C).
    2. Piera Mazzoleni & Elisa Pagani & Federico Perali, 2019. "The curvature properties of social welfare functions," Working Papers 493, ECINEQ, Society for the Study of Economic Inequality.
    3. Piera Mazzoleni & Elisa Pagani & Federico Perali, 2023. "On the Curvature Properties of “Long” Social Welfare Functions," Mathematics, MDPI, vol. 11(7), pages 1-14, March.

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    More about this item

    Keywords

    Money metric; Expenditure function; Least-concave representation;
    All these keywords.

    JEL classification:

    • D11 - Microeconomics - - Household Behavior - - - Consumer Economics: Theory
    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • D61 - Microeconomics - - Welfare Economics - - - Allocative Efficiency; Cost-Benefit Analysis

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