Fair value that a risk averse individual would pay to enter a gamble

Question

Introduction

Assuming an individual (or corporation) with risk aversion and a von Neumann-Morgenstern utility curve and given a gamble g with E(g) > 0. From what I researched, certainty equivalent is defined as the amount of wealth we could offer with certainty that would make him indifferent between accepting that wealth with certainty and facing a gamble g (Jehle and Reny, 2011, Chapter 2, pg. 112).

From that definition, the certainty equivalent is the amount of value that the individual would accept to give up on the gamble g.

Questions

1. Can you apply the certainty equivalent concept to calculate the fair value that the individual would be willing to pay to enter the gamble g (instead of giving up on the gamble)?

2. In that case, is it the same value to give up on the gamble?

References

Jehle, G. A. and Reny, P. J., 2011. Advanced Microeconomic Theory (3rd Edition). ISBN-10: 0273731912. Pearson; 3rd edition (April 30, 2011).

Answer 1

1. Yes.
2. In general not.

Let's say the individual has initial wealth $W$ and the gamble $g$ has payouts $0$ and $G$, each with probability $1/2$. As you say, the certainty equivalent $C$ of the gamble is the amount $C$ with $$u(W+C)=(u(W)+u(W+G))/2.$$ Now the same individual would be willing to pay at most $P$ to enter the gamble, where $$u(W)=(u(W-P)+u(W+G-P))/2.$$ So if $C$ is the certainty equivalent of $g$ at wealth level $W$, then $P$ is the certainty equivalent of $g$ at wealth level $W-P$. If $u$ is linear, then $P=C=G/2$ follows, but in general $C$ will differ from $P$.

Answer 2

The equivalence doesn't seem to work CRRA utility functions (unlike CARA functions, as mentioned in the comments above). Here's an example: Consider an agent with wealth $W$ and utility function $u(W)=ln(W)$. He faces a gamble $\{\frac{1}{2}\circ 0; \frac{1}{2} \circ W\}$. That is, the gamble doubles his wealth with prob 0.5, and gives nothing with prob 0.5.

Certainty Equivalent \begin{align} ln(W+C)&=\frac{ln(W) + ln(2W)}{2}\\ \implies W+C &= W\sqrt{2}\\ \implies C &= W(\sqrt{2}-1) \approx \textbf{0.414 W}. \end{align}

Payment for Gamble \begin{align} ln(W) &= \frac{ln(W-P)+ln(2W-P)}{2}\\ \implies W^2 &= 2W^2 - 3WP + P^2\\ \implies P &= \frac{3W\pm\sqrt{9W^2-4W^2}}{2}\\ \implies P &= W\big[\frac{3\pm\sqrt{5}}{2}\big] \approx 2.61 W \text{ or } \textbf{0.381 W}. \end{align}