# 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).

• If you're willing to accept \$5, and no less, to give up a lottery ticket, it means you're willing to pay \$5, and no more, to buy it back. – Michael Jun 10 '20 at 20:04

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 $$P$$.

• Good answer. Is there a more general class of utility functions for which the two coincide? Maybe constant absolute risk aversion utility functions? – brunosalcedo Jun 11 '20 at 0:57
• That was really helpful, thank you. So to compute P you just equate the utility of the initial wealth W with the expected utility after "paying to take the gamble" and solve for P, correct? – rmorel Jun 11 '20 at 1:16
• @brunosalcedo indeed, I tested a more generic gamble with a simple exponential utility function defined as $u(W) = (1 - e^{-W/a})/a$ with a > 0. The values indeed coincide in this case (with inverted signal of course). So maybe, for constant risk aversion utility functions that might always hold true. – rmorel Jun 11 '20 at 2:35
• @brunosalcedo: Yes, CARA utility functions eliminate the wealth effect and give $P=C$ here. – VARulle Jun 12 '20 at 7:21
• On further reflection, this is all down to Jensen's Inequality (en.wikipedia.org/wiki/Jensen%27s_inequality) plus a concave utility function. – Tiercelet Jun 12 '20 at 19:20

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}