On risk aversion, classical demand theory, and KM preferences

Abstract

Building on Kihlstrom and Mirman (Journal of Economic Theory, 8(3), 361–388, 1974)’s formulation of risk aversion in the case of multidimensional utility functions, we study the effect of risk aversion on optimal behavior in a general consumer’s maximization problem under uncertainty. We completely characterize the relationship between changes in risk aversion and classical demand theory. We show that the effect of risk aversion on optimal behavior depends on the income and substitution effects. Moreover, the effect of risk aversion is determined not by the riskiness of the risky good, but rather the riskiness of the utility gamble associated with each decision.

Appendix: Proofs

We combine the proofs of Propositions 1, 2, and 3. We first state the following Lemma which combines Lemma 2 in Kraus (1979) and conditions stated in Katz (1981).

Kraus-Katz Lemma

Let $x_{\tilde {\theta }}^{\ast } =\arg \max _{x} \mathbb {E}_{\tilde {\theta }} \varphi ( Z(x,\tilde {\theta }))$ such that $\partial ^{2} \varphi (Z(x,\theta ))/\partial x^{2} \vert _{x=x^{\ast }_{\tilde {\theta }}}<0$. If $\partial Z(x,\theta )/\partial \theta \vert _{x=x^{\ast }_{\tilde {\theta }}}>0$ and $\partial ^{2} Z(x,\theta ) / \partial x \partial \theta \vert _{x=x^{\ast }_{\tilde {\theta }}} < 0 (> 0)$, then $x^{\ast }_{\tilde {\theta }}$ increases (decreases) along with an increase in risk aversion. Alternatively, if $\partial Z(x,\theta )/\partial \theta \vert _{x=x^{\ast }_{\tilde {\theta }}}<0$ and $\partial ^{2} Z(x,\theta ) / \partial x \partial \theta \vert _{x=x^{\ast }_{\tilde {\theta }}} > 0 (< 0)$, then $x^{\ast }_{\tilde {\theta }}$ increases (decreases) along with an increase in risk aversion.

In the consumer’s problem, Z(x, θ) ≡ U(x, (I − P _x x) / P _y ), θ ∈ {I, P _x , P _y }. Suppose first that income is random as in Proposition 1, i.e., $\tilde {\theta } \equiv \tilde {I}$. Then, $\frac {\partial U( x,(I-P_{x}x)/P_{y})}{\partial I} > 0$, and $\frac {\partial ^{2} U( x,(I-P_{x}x)/P_{y})}{\partial x\partial I} = I{}E_{I} \equiv U_{12} \cdot \frac {x}{P_{y}} - U_{22} \cdot \frac {P_{x}x}{P_{y}^{2}}$ is of the same sign as the income effect related to a change in income. By Kraus-Katz Lemma, $x_{\tilde {I}}^{\ast }$ decreases along with an increase in risk aversion when the sure good is normal, i.e., $I{}E_{I} \vert _{x=x^{\ast }_{\tilde {I}}} > 0$. Suppose next that the price of the sure good is random as in Proposition 2, i.e., $\tilde {\theta } \equiv \tilde {P}_{x}$. Then, $\frac {\partial U( x,(I-P_{x}x)/P_{y})}{\partial P_{x}} < 0$, and $\frac {\partial ^{2} U( x,(I-P_{x}x)/P_{y})}{\partial x \partial P_{x}} = I{}E_{P_{x}} + S E_{P_{x}}$, where $I{}E_{P_{x}} \equiv -\left [ U_{12} \cdot \frac {x}{P_{y}}-U_{22} \cdot \frac {P_{x}x}{P_{y}^{2}} \right ]$ and $S E_{P_{x}} \equiv -\frac {U_{2}}{P_{y}} < 0$ are of the same sign as the income and substitution effects, respectively, related to a change in the price of the sure good. By Kraus-Katz Lemma, $x_{\tilde {P}_{x}}^{\ast }$ decreases along with an increase in risk aversion when the sure good is normal, i.e., $I{}E_{P_{x}} \vert _{x=x^{\ast }_{\tilde {P}_{x}}} < 0$. The effect of risk aversion for an inferior sure good depends on the relative strength of the income and substitution effects. Suppose finally that the price of the risky good is random as in Proposition 3, i.e., $\tilde {\theta } \equiv \tilde {P}_{y}$. Then, $\frac {\partial U( x,(I-P_{x}x)/P_{y})}{\partial P_{y}} < 0$, and $\frac {\partial ^{2} U( x,(I-P_{x}x)/P_{y})}{\partial x \partial P_{y}} = I{}E_{P_{y}} + S E_{P_{y}}$ where $I{}E_{P_{y}} \equiv -\left [ U_{12}\cdot \frac {x}{P_{y}}-U_{22}\cdot \frac {P_{x}}{P_{y}}\right ] \frac { I-P_{x}x}{P_{y}^{2}}$ and $S E_{P_{y}} \equiv \frac {U_{2}P_{x}}{P_{y}^{2}}>0$ are of the same sign as the income and substitution effects, respectively, related to a change in the price of the risky good. By Kraus-Katz Lemma, $x_{\tilde {P}_{y}}^{\ast }$ increases along with an increase in risk aversion when the good is inferior, i.e., $I{}E_{P_{y}} \vert _{x=x^{\ast }_{\tilde {P}_{y}}} < 0$. The effect of risk aversion for a normal sure good depends on the relative strength of the income and substitution effects.