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Alternative utility functions: review, analysis and comparison

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Abstract

The paper reviews the development of von Neumann and Morgenstern (vNM) utility theory. Kahneman and Tversky’s (KT’s) prospect theory is introduced. The vNM utility function is compared and contrasted with KT’s value function. We prove the uniqueness of two popular utility functions. First, we show that all power utility functions possess constant RRA. And, we show that all exponential utility functions have constant ARA. The paper concludes by discussing applications, strengths and weaknesses of various utility functions.

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Fig. 1
Fig. 2
Fig. 3

Source: Harry Markowitz (1952), Fig. 5

Fig. 4
Fig. 5
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Fig. 9

Adapted from: Bodie et al. (2011), Figure 7.8, p. 210

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Notes

  1. Friedman and Savage’s (1948) article is not representative of the co-authors’ outstanding contributions to economics. Friedman and Savage (1948) and Friedman (1957, 1963) gave economics the permanent income hypothesis, consumption function, monetary economics and other important findings. Savage is also a top-notch economist. One co-author to this paper has a Ph.D. in finance and was a Federal Reserve Economist for 2 years, and he believes Milton Friedman was one of the best economists that ever walked the earth.

  2. Markowitz’s (1952) present value is at the origin in Fig. 3. Kahneman and Tversky (KT 1979) defined the status quo to be their reference point, and it is located at the origin of their value function. These three researchers all assert that their reference point should be at the origin. Note that below KT’s Eq. (2) in their 1979 paper KT explicitly acknowledge Markowitz’s contribution to their prospect theory.

  3. In this paper the values of \( \alpha_{1} = 1 \) and \( \alpha_{2} = 2 \) are control variables that are determined exogenously by the decision-maker rather than within the model. As a result, \( \alpha_{1} \) and \( \alpha_{2} \) do not have the meaningful shadow prices that are sometimes associated with Lagrangian multipliers.

  4. This paper uses single period vNM utility theory. An analysis of CRRA that goes beyond the scope of this paper could use Epstein–Zin recursive utility (Epstein and Zinn 1989).

  5. Under ordinal utility theory the negative exponential utility function is equivalent to the exponential utility function. See Norstad (2011) and Merton (1990). However, adding one to the equation would make these two utility functions differ under cardinal utility theory. In spite of the differences in their cardinal utility, the two functions behave similarly.

  6. Non-expected utility is an alternative to expected utility analysis. See Cho and Francis (1994) for an introduction.

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Acknowledgments

Funding was provided by Baruch College Fund.

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Authors and Affiliations

  1. Zicklin School of Business, Bernard Baruch College, City University of New York (CUNY), One Bernard Baruch Way, New York, NY, 10010-5585, USA

    Arie Harel, Jack Clark Francis & Giora Harpaz

  2. 129 Old North Stamford Road, Stamford, CT, 06905, USA

    Jack Clark Francis

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  1. Arie Harel
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  2. Jack Clark Francis
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Correspondence to Jack Clark Francis.

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Harel, A., Francis, J.C. & Harpaz, G. Alternative utility functions: review, analysis and comparison. Rev Quant Finan Acc 51, 785–811 (2018). https://doi.org/10.1007/s11156-017-0688-z

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