On Risk Preferences and Curvilinear Utility Curves

Kuznar’s (CA 42:432–40) proposed relationship between social status (wealth), risk preference, and utility bears further scrutiny. The author proposes “that there will be a general curvilinear relationship between wealth and risk sensitivity in which the poor and the wealthy will prefer risky prospects whereas those with more moderate wealth will be risk-averse” (p. 437). He attempts to clarify this claim with his figure 3, which expresses a relationship between status/wealth and utility. Here I reproduce that figure with some expositional additions (fig. 1). Kuznar’s hypothesized relationship actually contradicts the claims/predictions that he apparently derives from it (captured in the above quotation). In fact, figure 1 shows that an individual at a particular wealth level may be risk-averse, risk-prone, or risk-neutral depending on the size of the stakes in the gamble. Consider a focal poor herder facing a choice between two options: (1) a sure gain of three cows and (2) a 50% chance at six cows and a 50% chance of zero cows. From figure 1 we see that U3 gives the utility received from gaining three cows (option 1) and U6 the utility received from gaining six cows. To figure out which choice our focal herder makes, assuming that his decisions are governed by expected utility maximization and the proposed relationship, we must calculate and compare the expected utilities from the two options. The expected utility of option 1 is simply U3, and the expected utility of option 2 is U6—(0.5)(U6) (0.5)(U0); U0 p 0. For a risk-neutral in1 2 dividual, N6 gives the utility from receiving six cows, and N6 gives the expected utility of option 2. Noting 1 2 that N6 p 2U3 (because of the linear relationship between them), we see that a risk-neutral individual will be indifferent between the two options. From figure 1, we can also observe that U6 1 N6, which implies that U N6—meaning that the utility of option 2 is greater 1 1 1 6 2 2 than that of option 1 ( U6 1 U3). Thus, our herder prefers 1 2 the risky option (option 2). To generalize the analysis we could simply replace three cows with the variable w (6 cows p 2w). This part is consistent with Kuznar’s analysis.