<ol>
<li>Yes.</li>
<li>In general not.</li>
</ol>

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

CC BY-SA 4.0
2020-06-11T00:57:22.387Z
Good answer. Is there a more general class of utility functions for which the two coincide? Maybe constant absolute risk aversion utility functions?
brunosalcedo
CC BY-SA 4.0
2020-06-11T01:16:29.757Z
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
CC BY-SA 4.0
2020-06-11T02:35:17.740Z
@brunosalcedo indeed, I tested a more generic gamble with a simple [exponential utility function](https://en.wikipedia.org/wiki/Exponential_utility) 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
CC BY-SA 4.0
2020-06-12T07:21:53.970Z
@brunosalcedo: Yes, CARA utility functions eliminate the wealth effect and give $P=C$ here.
VARulle
CC BY-SA 4.0
2020-06-12T19:20:37.433Z
On further reflection, this is all down to Jensen's Inequality (https://en.wikipedia.org/wiki/Jensen%27s_inequality) plus a concave utility function.
Tiercelet
CC BY-SA 4.0
2020-06-12T20:35:25.010Z
@rmorel (1st comment): Yes, that's correct.
VARulle
CC BY-SA 4.0
2020-06-11T00:13:37.137Z
2020-06-11T00:19:46.207Z
2020-06-11T00:19:46.207Z
VARulle
<h1>Introduction</h1>

<p>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).</p>

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

<h1>Questions</h1>

<ol>
<li><p>Can you apply the certainty equivalent concept to calculate the fair value that the individual would be willing <strong>to pay to enter the gamble</strong> g (instead of giving up on the gamble)?</p></li>
<li><p>In that case, is it the same value to give up on the gamble?</p></li>
</ol>

<h1>References</h1>

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

CC BY-SA 4.0
2020-06-10T20:04:16.377Z
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
<expected-utility><rational-expectations><measuring-utility>
Fair value that a risk averse individual would pay to enter a gamble
rmorel
# Fair value that a risk averse individual would pay to enter a gamble

(Asked by: username_0 on 2020-06-10)

# 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).

**Tags:** expected-utility, rational-expectations, measuring-utility

> **Comments:**
> **username_1**: 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.  

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## 1 answers

### Answer by username_2 on 2020-06-11. Score: 2 | Accepted answer

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 $P$.

> **Comments:**
> **username_3**: Good answer. Is there a more general class of utility functions for which the two coincide? Maybe constant absolute risk aversion utility functions?  
> **username_0**: 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?  
> **username_0**: @username_3 indeed, I tested a more generic gamble with a simple [exponential utility function](https://en.wikipedia.org/wiki/Exponential_utility) 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.  
> **username_2**: @username_3: Yes, CARA utility functions eliminate the wealth effect and give $P=C$ here.  
> **username_4**: On further reflection, this is all down to Jensen's Inequality (https://en.wikipedia.org/wiki/Jensen%27s_inequality) plus a concave utility function.  
> **username_2**: @username_0 (1st comment): Yes, that's correct.  

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