<blockquote>
  <p>How can someone claim that this return is guaranteed, since we need to use <span class="math-container">$\mathbb{V}ar(\widetilde{W})$</span> to measure the utility gain of an individual? </p>
</blockquote>

<p>Maybe I misunderstand your point, but though <span class="math-container">$\widetilde{W}$</span> is a random variable, <span class="math-container">$\mathbb{E}(\widetilde{W})-\frac{\rho}{2}\mathbb{V}ar(\widetilde{W})$</span> and hence <span class="math-container">$CE(\widetilde{W})$</span> is a real number. So <span class="math-container">$CE(\widetilde{W})$</span> is a measure of a certain income which to the consumer has the same utility as <span class="math-container">$\widetilde{W}$</span>.</p>

CC BY-SA 4.0
2020-06-06T22:52:48.637Z
Why $CE(\widetilde{W})$ is a real number? What is the intuition behind this?
Hunger Learn
CC BY-SA 4.0
2020-06-07T09:25:01.357Z
The "*intuition*" is that $\mathbb{E}(\widetilde{W})$ is a number, $\frac{\rho}{2}$ is a number and $\mathbb{V}ar(\widetilde{W})$ is a number, so performing the mathematical operations $$\mathbb{E}(\widetilde{W})-\frac{\rho}{2}\mathbb{V}ar(\widetilde{W})$$ gives you a number, which you defined as $CE(\widetilde{W})$.
Giskard
CC BY-SA 4.0
2020-06-06T22:34:37.413Z
2020-06-06T22:34:37.413Z
Giskard
<p>The certainty equivalent is a guaranteed return that someone would accept now, rather than taking a chance on a higher, but uncertain, return in the future. Otherwise, some definitions say that the certainty equivalentis the mean income of a gamble. If we see the vast literature of microeconomic and game theory papers, then we shall see that they assume CARA Gaussian normal preferences (or negative expnential utility, i.e. <span class="math-container">$\mathbb{U}(\widetilde{W})=-e^{\rho\widetilde{W}}$</span>). Specifically the certainty equivalent in such case is <span class="math-container">$$CE(\widetilde{W})=\mathbb{E}(\widetilde{W})-\frac{\rho}{2}\mathbb{V}ar(\widetilde{W})$$</span>
How can someone claim that this return is guaranteed, since we need to use <span class="math-container">$\mathbb{V}ar(\widetilde{W})$</span> to measure the utility gain of an individual? </p>

CC BY-SA 4.0
2020-06-07T18:01:49.203Z
While $\widetilde W $ is a random variable, $\mathbb Var (\widetilde W )$ is not random.
Herr K.
<microeconomics><game-theory><mathematical-economics>
Certainty Equivalent
Hunger Learn
# Certainty Equivalent

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

The certainty equivalent is a guaranteed return that someone would accept now, rather than taking a chance on a higher, but uncertain, return in the future. Otherwise, some definitions say that the certainty equivalentis the mean income of a gamble. If we see the vast literature of microeconomic and game theory papers, then we shall see that they assume CARA Gaussian normal preferences (or negative expnential utility, i.e. $\mathbb{U}(\widetilde{W})=-e^{\rho\widetilde{W}}$). Specifically the certainty equivalent in such case is $$CE(\widetilde{W})=\mathbb{E}(\widetilde{W})-\frac{\rho}{2}\mathbb{V}ar(\widetilde{W})$$
How can someone claim that this return is guaranteed, since we need to use $\mathbb{V}ar(\widetilde{W})$ to measure the utility gain of an individual?

**Tags:** microeconomics, game-theory, mathematical-economics

> **Comments:**
> **username_1**: While $\widetilde W $ is a random variable, $\mathbb Var (\widetilde W )$ is not random.  

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## 1 answers

### Answer by username_2 on 2020-06-06. Score: 5 | Accepted answer

> How can someone claim that this return is guaranteed, since we need to use $\mathbb{V}ar(\widetilde{W})$ to measure the utility gain of an individual?

Maybe I misunderstand your point, but though $\widetilde{W}$ is a random variable, $\mathbb{E}(\widetilde{W})-\frac{\rho}{2}\mathbb{V}ar(\widetilde{W})$ and hence $CE(\widetilde{W})$ is a real number. So $CE(\widetilde{W})$ is a measure of a certain income which to the consumer has the same utility as $\widetilde{W}$.

> **Comments:**
> **username_0**: Why $CE(\widetilde{W})$ is a real number? What is the intuition behind this?  
> **username_2**: The "*intuition*" is that $\mathbb{E}(\widetilde{W})$ is a number, $\frac{\rho}{2}$ is a number and $\mathbb{V}ar(\widetilde{W})$ is a number, so performing the mathematical operations $$\mathbb{E}(\widetilde{W})-\frac{\rho}{2}\mathbb{V}ar(\widetilde{W})$$ gives you a number, which you defined as $CE(\widetilde{W})$.  

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