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The arguments of utility: Preference reversals in expected utility of income models


There is a debate in the literature about the arguments of utility in expected utility theory. Some implicitly assume utility is defined on final wealth whereas others argue it may be defined on initial wealth and income separately. I argue that making income and wealth separate arguments of utility has important implications that may not be widely recognized. A framework is presented that allows the unified treatment of expected utility models and anomalies. I show that expected utility of income models can predict framing induced preference reversals, a willingness to pay-willingness to accept gap for lotteries, and choice-value preference reversals. The main contribution is a theorem. It is proved that for all utility functions where initial wealth and income enter separately, either there will be preference reversals or preferences can be represented by a utility function defined on final wealth alone.

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Fig. 1


  1. 1.

    Other authors have also presented expected utility models that are explicitly not defined on wealth alone. An early example is Markowitz (1952) and a more recent one Sugden (2003).

  2. 2.

    Another response to Rabin’s paradox is the argument that other theories are vulnerable to analogous calibrations, so calibration may be a problem for all decision theories, not just expected utility of final wealth. Cox and Sadiraj (2006) present a concavity calibration proposition for small and large stake risk aversion that applies to expected utility of income models and some non-expected utility theories. Rubinstein (2006) considers time preferences and presents a calibration showing how constant discounting and seemingly plausible inter-day discounting predict implausible degrees of discounting over longer periods.

  3. 3.

    This analogy was introduced by Tversky and Kahneman (1981).

  4. 4.

    In Section 2.2 the state-by-state approach is compared to an approach similar to that taken by Koszegi and Rabin (2007).

  5. 5.

    The potential for anomalies does not go unnoticed by Cox and Sadiraj. For instance in their footnote 8 they write that the expected utility of initial wealth and income model they introduce “does not rule out certain types of anomalies (see Rubinstein 2006 for an illustration). Detailed analysis of possible “money pump” preference cycles and other violations of full rationality are beyond the scope of the present paper, which is concerned with the implications of concavity calibration for decision theories.”

  6. 6.

    This example is a slight modification of one Kahneman and Tversky (1982) use. It shows how the same decision problem can be framed in different ways and how different frames can lead people to choose different options.

  7. 7.

    It would be a simple extension to include an act representing background risk.

  8. 8.

    Other aspects of Koszegi and Rabin’s model (such as separating standard consumption utility and loss gain utility and making the reference point a person’s recent rational expectations about outcomes) are not used in this paper.

  9. 9.

    That choice-value preference reversals can occur in an expected utility model is not a new result. For instance, Sugden (2003) shows how similar results occur in an expected utility model where utility is defined on satisfaction and changes in satisfaction.

  10. 10.

    See Wakker (2008) for a discussion of the characteristics of such functions.


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I am grateful to Thomas Epper, Jacob Goeree, Konrad Mierendorff, Chris Starmer, Jingjing Zhang, and participants at the FUR XIV International Conference at Newcastle University, as well as the editor and one anonymous referee, for comments. I would like to thank the Swiss National Science Foundation (grant SNSF 138162) and the European Research Council (grant ESEI-249433) for financial support.

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Correspondence to Luke Lindsay.

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Lindsay, L. The arguments of utility: Preference reversals in expected utility of income models. J Risk Uncertain 46, 175–189 (2013).

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  • Expected utility theory
  • Risk aversion
  • Preference reversals

JEL Classification

  • C90
  • D81