Independence of initial wealth for Constant Absolute Risk Aversion

Question

Suppose a consumer's preference over wealth gambles (lotteries) can be represented by a twice differentiable Von Neumann Morgenstern utility function. Show that the consumer's preference over gambles are independent of his initial wealth if and only if his utility function displays Constant Absolute Risk Aversion (CARA).

Answer 1

Suppose a fair gamble pays $G = \pm \epsilon$ where $\displaystyle P(G=\epsilon ) = P(G = -\epsilon) = \frac{1}{2}$.

From the classic work described in

Pratt, J.W. (1964) "Risk-Aversion in the Small and in the Large'" Econometrica 55,143-54

for small gambles, the absolute amount an agent is willing to pay to avoid a gamble of a given size is determined by the coefficient of absolute risk aversion.

For a rough argument, we have a risk averse agent with initial wealth $W_0$ and utility function $U$ willing to pay $\delta$ to avoid the gamble such that $\delta$ is determined by

$$U(W_0 - \delta) = E[U(W_0+G)]=\frac{1}{2}U(W_0 +\epsilon) + \frac{1}{2}U(W_0- \epsilon).$$

Using the Taylor expansion for U around $W_0$ we have

$$U(W_0) -U'(W_0)\delta + \frac{1}{2} U''(W_0)\delta^2 + \ldots \\ = \frac{1}{2} [U(W_0) +U'(W_0)\epsilon + \frac{1}{2} U''(W_0)\epsilon^2 + \ldots ] \\ +\frac{1}{2} [U(W_0) -U'(W_0)\epsilon + \frac{1}{2} U''(W_0)\epsilon^2 + \ldots ] \\ = U(W_0) + \frac{1}{2}U''(W_0) \epsilon^2 + \ldots $$

Solving for $\delta$ for small $\epsilon$ we get

$$\delta \approx \frac{\epsilon^2}{2}\left[- \frac{U''(W_0)}{U'(W_0)} \right].$$

The premium to avoid a gamble is independent of initial wealth if and only if there is a constant $\gamma$ such that

$$- \frac{U''(W)}{U'(W)} = \gamma .$$

Solving this differential equation we get

$$U(W) = C_1 - e^{C_2} \frac{e^{-\gamma W}}{\gamma}.$$

Without loss of generality (from the invariance properties of utility functions) we can set $C_1 = C_2 = 0$ to obtain the CARA utility function

$$U(W) = - \frac{e^{-\gamma W}}{\gamma}.$$