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A user’s guide to economic utility functions

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  • Charles E. Phelps

    (University of Rochester)

Abstract

Economic modeling of behavior under uncertainty has almost exclusively used one of two approaches. First comes the mean–variance tradeoff, central to the financial economics literature. The alternative approach uses a specific function to assess expected utility. In this approach, almost all of the economics literature has used either the exponential utility (EU) function, with constant absolute risk aversion (CARA) or the power utility (PU) function with constant relative risk aversion (CRRA). Using a Taylor Series expansion, I show that higher-order terms (skewness and kurtosis) can significantly affect estimates of expected utility. I provide specific formulaic guidance allowing economists to assess when these terms become important. This guidance uses readily observable parameters such as mean, median and variance in a wide array of non-Gaussian statistical distributions. I next review two generalizations, one for CRRA utility, hyperbolic absolute risk aversion (HARA), and one for CARA utility, exponential power (EP). I then introduce the possibility of risk-seeking behavior, both in standard economic theory and in the “value” portion of prospect theory (PT), and provide a new Generalized Logistic Utility (GLU) that automatically incorporates such behavior in its functional form. Next, I introduce issues involved in modeling utility that includes a non-marketable component such as health, the environment, or altruism, and discuss how the choice of utility function alters these models. I then assess the choice between exact utility functions and Taylor Series approximations. I conclude by discussing methods to estimate utility function parameters and methods to choose among alternative estimates.

Suggested Citation

  • Charles E. Phelps, 2024. "A user’s guide to economic utility functions," Journal of Risk and Uncertainty, Springer, vol. 69(3), pages 235-280, December.
  • Handle: RePEc:kap:jrisku:v:69:y:2024:i:3:d:10.1007_s11166-024-09443-5
    DOI: 10.1007/s11166-024-09443-5
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    1. Lakdawalla, Darius N. & Phelps, Charles E., 2023. "The Generalized Risk-Adjusted Cost-Effectiveness (GRACE) Model for Measuring the Value of Gains in Health: An Exact Formulation," Journal of Benefit-Cost Analysis, Cambridge University Press, vol. 14(1), pages 44-67, March.
    2. Danyang Xie, 2000. "Power Risk Aversion Utility Functions," Annals of Economics and Finance, Society for AEF, vol. 1(2), pages 265-282, November.
    3. Tunçel, Tuba & Hammitt, James K., 2014. "A new meta-analysis on the WTP/WTA disparity," Journal of Environmental Economics and Management, Elsevier, vol. 68(1), pages 175-187.
    4. Marquis, M Susan & Holmer, Martin R, 1996. "Alternative Models of Choice under Uncertainty and Demand for Health Insurance," The Review of Economics and Statistics, MIT Press, vol. 78(3), pages 421-427, August.
    5. Robert M. Solow, 1956. "A Contribution to the Theory of Economic Growth," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 70(1), pages 65-94.
    6. Tversky, Amos & Kahneman, Daniel, 1992. "Advances in Prospect Theory: Cumulative Representation of Uncertainty," Journal of Risk and Uncertainty, Springer, vol. 5(4), pages 297-323, October.
    7. Philippe Weil, 1990. "Nonexpected Utility in Macroeconomics," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 105(1), pages 29-42.
    8. Levon Barseghyan & Francesca Molinari & Ted O'Donoghue & Joshua C. Teitelbaum, 2018. "Estimating Risk Preferences in the Field," Journal of Economic Literature, American Economic Association, vol. 56(2), pages 501-564, June.
    9. Levy, Moshe & Nir, Adi Rizansky, 2012. "The utility of health and wealth," Journal of Health Economics, Elsevier, vol. 31(2), pages 379-392.
    10. Viscusi, W. Kip & Moore, Michael J., 1989. "Rates of time preference and valuations of the duration of life," Journal of Public Economics, Elsevier, vol. 38(3), pages 297-317, April.
    11. Saku Aura & Peter Diamond & John Geanakoplos, 2002. "Savings and Portfolio Choice in a Two-Period Two-Asset Model," American Economic Review, American Economic Association, vol. 92(4), pages 1185-1191, September.
    12. Harry Markowitz, 1952. "The Utility of Wealth," Journal of Political Economy, University of Chicago Press, vol. 60(2), pages 151-151.
    13. Larry G. Epstein & Stanley E. Zin, 2013. "Substitution, risk aversion and the temporal behavior of consumption and asset returns: A theoretical framework," World Scientific Book Chapters, in: Leonard C MacLean & William T Ziemba (ed.), HANDBOOK OF THE FUNDAMENTALS OF FINANCIAL DECISION MAKING Part I, chapter 12, pages 207-239, World Scientific Publishing Co. Pte. Ltd..
    14. Peter P. Wakker, 2008. "Explaining the characteristics of the power (CRRA) utility family," Health Economics, John Wiley & Sons, Ltd., vol. 17(12), pages 1329-1344, December.
    15. Elamin H. Elbasha, 2005. "Risk aversion and uncertainty in cost‐effectiveness analysis: the expected‐utility, moment‐generating function approach," Health Economics, John Wiley & Sons, Ltd., vol. 14(5), pages 457-470, May.
    16. Evans, William N & Viscusi, W Kip, 1991. "Estimation of State-Dependent Utility Functions Using Survey Data," The Review of Economics and Statistics, MIT Press, vol. 73(1), pages 94-104, February.
    17. Merton, Robert C., 1971. "Optimum consumption and portfolio rules in a continuous-time model," Journal of Economic Theory, Elsevier, vol. 3(4), pages 373-413, December.
    18. Charles A. Holt & Susan K. Laury, 2002. "Risk Aversion and Incentive Effects," American Economic Review, American Economic Association, vol. 92(5), pages 1644-1655, December.
    19. Amy Finkelstein & Erzo F. P. Luttmer & Matthew J. Notowidigdo, 2013. "What Good Is Wealth Without Health? The Effect Of Health On The Marginal Utility Of Consumption," Journal of the European Economic Association, European Economic Association, vol. 11, pages 221-258, January.
    20. Viscusi, W Kip & Evans, William N, 1990. "Utility Functions That Depend on Health Status: Estimates and Economic Implications," American Economic Review, American Economic Association, vol. 80(3), pages 353-374, June.
    21. Gary S. Becker & Tomas J. Philipson & Rodrigo R. Soares, 2005. "The Quantity and Quality of Life and the Evolution of World Inequality," American Economic Review, American Economic Association, vol. 95(1), pages 277-291, March.
    22. E. S. Phelps & R. A. Pollak, 1968. "On Second-Best National Saving and Game-Equilibrium Growth," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 35(2), pages 185-199.
    23. Harry M. Markowitz, 2010. "Portfolio Theory: As I Still See It," Annual Review of Financial Economics, Annual Reviews, vol. 2(1), pages 1-23, December.
    24. Kimball, Miles S, 1990. "Precautionary Saving in the Small and in the Large," Econometrica, Econometric Society, vol. 58(1), pages 53-73, January.
    25. Edoh Y. Amiran & Daniel A. Hagen, 2003. "Willingness To Pay and Willingness To Accept: How Much Can They Differ? Comment," American Economic Review, American Economic Association, vol. 93(1), pages 458-463, March.
    26. Aurelie Meunier & Louise Longworth & Manuel Gomes & Sreeram Ramagopalan & Louis P. Garrison & Sanjay Popat, 2023. "Distributional Cost-Effectiveness Analysis of Treatments for Non-Small Cell Lung Cancer: An Illustration of an Aggregate Analysis and its Key Drivers," PharmacoEconomics, Springer, vol. 41(8), pages 1011-1025, August.
    27. W. Kip Viscusi & William N. Evans, 1998. "Estimation Of Revealed Probabilities And Utility Functions For Product Safety Decisions," The Review of Economics and Statistics, MIT Press, vol. 80(1), pages 28-33, February.
    28. Joseph S. Pliskin & Donald S. Shepard & Milton C. Weinstein, 1980. "Utility Functions for Life Years and Health Status," Operations Research, INFORMS, vol. 28(1), pages 206-224, February.
    29. Robert E. Hall & Charles I. Jones, 2007. "The Value of Life and the Rise in Health Spending," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 122(1), pages 39-72.
    30. Carroll, Christopher D & Kimball, Miles S, 1996. "On the Concavity of the Consumption Function," Econometrica, Econometric Society, vol. 64(4), pages 981-992, July.
    31. Sven Tengstam, 2014. "Disability And Marginal Utility Of Income: Evidence From Hypothetical Choices," Health Economics, John Wiley & Sons, Ltd., vol. 23(3), pages 268-282, March.
    32. Hakansson, Nils H, 1970. "Optimal Investment and Consumption Strategies Under Risk for a Class of Utility Functions," Econometrica, Econometric Society, vol. 38(5), pages 587-607, September.
    33. W. Kip Viscusi, 2019. "Utility functions for mild and severe health risks," Journal of Risk and Uncertainty, Springer, vol. 58(2), pages 143-166, June.
    34. Gary S. Becker & Robert J. Barro, 1988. "A Reformulation of the Economic Theory of Fertility," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 103(1), pages 1-25.
    35. Lillard, Lee A & Weiss, Yoram, 1997. "Uncertain Health and Survival: Effects on End-of-Life Consumption," Journal of Business & Economic Statistics, American Statistical Association, vol. 15(2), pages 254-268, April.
    36. James E. Smith & Ralph L. Keeney, 2005. "Your Money or Your Life: A Prescriptive Model for Health, Safety, and Consumption Decisions," Management Science, INFORMS, vol. 51(9), pages 1309-1325, September.
    37. Atkinson, Anthony B., 1970. "On the measurement of inequality," Journal of Economic Theory, Elsevier, vol. 2(3), pages 244-263, September.
    38. Hanemann, W Michael, 1991. "Willingness to Pay and Willingness to Accept: How Much Can They Differ?," American Economic Review, American Economic Association, vol. 81(3), pages 635-647, June.
    39. Kevin M. Murphy & Robert H. Topel, 2006. "The Value of Health and Longevity," Journal of Political Economy, University of Chicago Press, vol. 114(5), pages 871-904, October.
    40. Feldstein, Martin & Friedman, Bernard, 1977. "Tax subsidies, the rational demand for insurance and the health care crisis," Journal of Public Economics, Elsevier, vol. 7(2), pages 155-178, April.
    41. Shogren, Jason F. & Seung Y. Shin & Dermot J. Hayes & James B. Kliebenstein, 1994. "Resolving Differences in Willingness to Pay and Willingness to Accept," American Economic Review, American Economic Association, vol. 84(1), pages 255-270, March.
    42. Toda, Alexis Akira, 2021. "Necessity of hyperbolic absolute risk aversion for the concavity of consumption functions," Journal of Mathematical Economics, Elsevier, vol. 94(C).
    43. Angus Deaton, 2008. "Income, Health, and Well-Being around the World: Evidence from the Gallup World Poll," Journal of Economic Perspectives, American Economic Association, vol. 22(2), pages 53-72, Spring.
    44. Sloan, Frank A. & Kip Viscusi, W. & Chesson, Harrell W. & Conover, Christopher J. & Whetten-Goldstein, Kathryn, 1998. "Alternative approaches to valuing intangible health losses: the evidence for multiple sclerosis1," Journal of Health Economics, Elsevier, vol. 17(4), pages 475-497, August.
    45. Gilbert W. Bassett, Jr., 1987. "The St. Petersburg Paradox and Bounded Utility," History of Political Economy, Duke University Press, vol. 19(4), pages 517-523, Winter.
    46. Arthur E. Attema & Jona J. Frasch & Olivier L’Haridon, 2022. "Multivariate risk preferences in the quality‐adjusted life year model," Health Economics, John Wiley & Sons, Ltd., vol. 31(2), pages 382-398, February.
    47. Kenneth J. Arrow, 1974. "The Use of Unbounded Utility Functions in Expected-Utility Maximization: Response," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 88(1), pages 136-138.
    48. Christensen, Laurits R & Jorgenson, Dale W & Lau, Lawrence J, 1975. "Transcendental Logarithmic Utility Functions," American Economic Review, American Economic Association, vol. 65(3), pages 367-383, June.
    49. Eeckhoudt, Louis R & Hammitt, James K, 2001. "Background Risks and the Value of a Statistical Life," Journal of Risk and Uncertainty, Springer, vol. 23(3), pages 261-279, November.
    50. Juan Carlos CóRdoba & Marla Ripoll, 2017. "Risk Aversion and the Value of Life," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 84(4), pages 1472-1509.
    51. Willig, Robert D, 1976. "Consumer's Surplus without Apology," American Economic Review, American Economic Association, vol. 66(4), pages 589-597, September.
    52. Charles E. Phelps, 2023. "Optimal health insurance," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 90(1), pages 213-241, March.
    53. Maiwenn J. Al & Talitha L. Feenstra & Ben A. van Hout, 2005. "Optimal allocation of resources over health care programmes: dealing with decreasing marginal utility and uncertainty," Health Economics, John Wiley & Sons, Ltd., vol. 14(7), pages 655-667, July.
    54. Zeckhauser, Richard, 1970. "Medical insurance: A case study of the tradeoff between risk spreading and appropriate incentives," Journal of Economic Theory, Elsevier, vol. 2(1), pages 10-26, March.
    55. James K. Hammitt, 2013. "Positive versus Normative Justifications for Benefit-Cost Analysis: Implications for Interpretation and Policy," Review of Environmental Economics and Policy, Association of Environmental and Resource Economists, vol. 7(2), pages 199-218, July.
    56. Darius N. Lakdawalla & Charles E. Phelps, 2022. "A guide to extending and implementing generalized risk-adjusted cost-effectiveness (GRACE)," The European Journal of Health Economics, Springer;Deutsche Gesellschaft für Gesundheitsökonomie (DGGÖ), vol. 23(3), pages 433-451, April.
    57. Mulligan, Karen & Baid, Drishti & Doctor, Jason N. & Phelps, Charles E. & Lakdawalla, Darius N., 2024. "Risk preferences over health: Empirical estimates and implications for medical decision-making," Journal of Health Economics, Elsevier, vol. 94(C).
    58. Ted O'Donoghue & Jason Somerville, 2018. "Modeling Risk Aversion in Economics," Journal of Economic Perspectives, American Economic Association, vol. 32(2), pages 91-114, Spring.
    59. Lakdawalla, Darius N. & Phelps, Charles E., 2020. "Health technology assessment with risk aversion in health," Journal of Health Economics, Elsevier, vol. 72(C).
    60. Peter P. Wakker, 2008. "Explaining the characteristics of the power (CRRA) utility family," Health Economics, John Wiley & Sons, Ltd., vol. 17(12), pages 1329-1344.
    61. Merton, Robert C, 1969. "Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case," The Review of Economics and Statistics, MIT Press, vol. 51(3), pages 247-257, August.
    62. Milton Friedman & L. J. Savage, 1948. "The Utility Analysis of Choices Involving Risk," Journal of Political Economy, University of Chicago Press, vol. 56(4), pages 279-279.
    63. Acland, Daniel & Greenberg, David H., 2023. "The Elasticity of Marginal Utility of Income for Distributional Weighting and Social Discounting: A Meta-Analysis," Journal of Benefit-Cost Analysis, Cambridge University Press, vol. 14(2), pages 386-405, July.
    64. MOSSIN, Jan, 1968. "Optimal multiperiod portfolio policies," LIDAM Reprints CORE 19, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    65. William N. Evans & W. Kip Viscusi, 1993. "Income Effects and the Value of Health," Journal of Human Resources, University of Wisconsin Press, vol. 28(3), pages 497-518.
    66. Keeler, Emmett B. & Rolph, John E., 1988. "The demand for episodes of treatment in the health insurance experiment," Journal of Health Economics, Elsevier, vol. 7(4), pages 337-367, December.
    67. Brown, Thomas C. & Gregory, Robin, 1999. "Why the WTA-WTP disparity matters," Ecological Economics, Elsevier, vol. 28(3), pages 323-335, March.
    68. Eric French & John Bailey Jones, 2004. "On the distribution and dynamics of health care costs," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 19(6), pages 705-721.
    69. Cass, David & Stiglitz, Joseph E., 1970. "The structure of investor preferences and asset returns, and separability in portfolio allocation: A contribution to the pure theory of mutual funds," Journal of Economic Theory, Elsevier, vol. 2(2), pages 122-160, June.
    70. Deaton, Angus S & Muellbauer, John, 1980. "An Almost Ideal Demand System," American Economic Review, American Economic Association, vol. 70(3), pages 312-326, June.
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