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Normalized measures of concavity and Ross’s strongly more risk averse order

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Abstract

The Arrow-Pratt (A-P) definitions of absolute and relative risk aversion dominate the discussion of risk aversion and defining “more risk averse”. Ross (Econometrica 49:621–663, 1981) notes, however, that being A-P more risk averse is not sufficient for addressing many important comparative static questions. Consequently he introduces “a new and stronger measure for comparing two agents’ attitudes towards risk…”. Ross does not provide a corresponding measure of risk aversion. This paper uses a normalized measure of concavity to characterize the Ross definition of strongly more risk averse on bounded intervals. Other properties and uses of these normalized measures of concavity are also presented.

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Notes

  1. Pratt (1990) points out several weaknesses of the Ross strongly more risk averse order.

  2. The correct upper and lower bounds of outcome distributions are often specific to a particular decision problem. For many utility functions, utility is defined for every level of wealth, but the restriction to an interval is imposed to give Ross comparability over that interval. Thus, for the same utility function and different decision problems, the interval [a, b] can differ in location and size.

  3. For a utility function such as the negative exponential function which displays A-P constant absolute risk aversion, any finite values for a and b are allowed. For constant relative risk aversion, however, the logarithmic and power functions require that a > 0 be satisfied.

  4. Friedman and Savage (1948) use the second derivative of u(x) to define the utility premium.

  5. Eeckhoudt and Schlesinger (2009) discuss the utility premium in the context of savings and other decisions and attribute the utility premium concept to Friedman and Savage (1948). The utility premium is also studied in detail in Stone (1970) and used in Menezes et al. (1980), Eeckhoudt and Schlesinger (2006), Menegatti (2007) and Crainich and Eeckhoudt (2008), among others.

  6. In addition to being arbitrary to a scale factor, -u″(x) is a less desirable measure of risk aversion because of this unit issue as well.

  7. Normalization factors exist that allow similar results in the competitive firm and insurance demand decision models.

  8. Using the rate of substitution, instead of the risk premium, to quantify an individual’s reaction to risk increases turns out to be even more significant when the risk increases are of a higher degree, as shown in Liu and Meyer (2013a).

  9. In their study of the relationship between changes in risk aversion and changes in prudence (− u‴(x)/u″(x)), Eeckhoudt and Schlesinger (1994) also work with these families of utility functions. They point out that, using the parameterization of the current paper, prudence remains the same but the A-P risk aversion increases as αu increases. Note also that these families of utility functions include the so-called “linex” family, where w(x) takes the form of any CARA utility function, as a special case (Denuit et al. 2013).

  10. For a summary of the issues in this line of research, see Gollier (2001).

References

  • Arrow, K. J. (1974). Essays in the theory of risk-bearing. Chicago: Markham.

    Google Scholar 

  • Crainich, D., & Eeckhoudt, L. (2008). On the intensity of downside risk aversion. Journal of Risk and Uncertainty, 36, 267–276.

    Article  Google Scholar 

  • Denuit, M., & Eeckhoudt, L. (2010). Stronger measures of higher-order risk attitudes. Journal of Economic Theory, 145, 2027–2036.

    Article  Google Scholar 

  • Denuit, M., Eeckhoudt, L., & Schlesinger, H. (2013). When Ross meets Bell: The linex utility function. Journal of Mathematical Economics, 49, 177–182.

    Article  Google Scholar 

  • Eeckhoudt, L., & Schlesinger, H. (1994). Increases in prudence and increases in risk aversion. Economics Letters, 45, 51–53.

    Article  Google Scholar 

  • Eeckhoudt, L., & Schlesinger, H. (2006). Putting risk in its proper place. American Economic Review, 96, 280–289.

    Article  Google Scholar 

  • Eeckhoudt, L., & Schlesinger, H. (2009). On the utility premium of Friedman and Savage. Economics Letters, 105, 46–48.

    Article  Google Scholar 

  • Friedman, M., & Savage, L. J. (1948). The utility analysis of choices involving risk. Journal of Political Economy, 56, 279–304.

    Article  Google Scholar 

  • Gollier, C. (2001). The economics of risk and time. Cambridge: The MIT Press.

    Google Scholar 

  • Liu, L., & Meyer, J. (2013a). Substituting one risk increase for another: a method for measuring risk aversion. Journal of Economic Theory, forthcoming.

  • Liu, L., & Meyer, J. (2013b). Larger and riskier. Working paper.

  • Machina, M. J., & Neilson, W. S. (1987). The Ross characterization of risk aversion: Strengthening and extension. Econometrica, 55, 1139–1149.

    Article  Google Scholar 

  • Menegatti, M. (2007). A new interpretation for the precautionary saving motive: A note. Journal of Economics, 92, 275–280.

    Article  Google Scholar 

  • Menezes, C., Geiss, C., & Tressler, J. (1980). Increasing downside risk. American Economic Review, 70, 921–932.

    Google Scholar 

  • Pratt, J. (1964). Risk aversion in the small and in the large. Econometrica, 32, 122–136.

    Article  Google Scholar 

  • Pratt, J. (1990). The logic of partial-risk aversion: Paradox lost. Journal of Risk and Uncertainty, 3, 105–113.

    Article  Google Scholar 

  • Ross, S. A. (1981). Some stronger measures of risk aversion in the small and in the large with applications. Econometrica, 49, 621–663.

    Article  Google Scholar 

  • Rothschild, M., & Stiglitz, J. (1970). Increasing risk I: A definition. Journal of Economic Theory, 2, 225–243.

    Article  Google Scholar 

  • Stone, B. K. (1970). Risk, return and equilibrium. Cambridge: The MIT Press.

    Google Scholar 

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Authors and Affiliations

  1. Private Enterprise Research Center, Texas A&M University, College Station, TX, 77843, USA

    Liqun Liu

  2. Department of Economics, Michigan State University, East Lansing, MI, 48824, USA

    Jack Meyer

Authors
  1. Liqun Liu
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  2. Jack Meyer
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Correspondence to Liqun Liu.

Additional information

Support from the Private Enterprise Research Center at Texas A&M University is gratefully acknowledged. We thank Louis Eeckhoudt and Stephen Ross for helpful comments and suggestions on an early draft. Two reviewers also provided helpful comments. An early version of this paper was presented at Risk and Choice: A Conference in Honor of Louis Eeckhoudt, Toulouse, France, July 13, 2012. We thank conference participants, especially Ed Schlee, for valuable comments.

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Liu, L., Meyer, J. Normalized measures of concavity and Ross’s strongly more risk averse order. J Risk Uncertain 47, 185–198 (2013). https://doi.org/10.1007/s11166-013-9173-9

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