<p>Gains and losses presuppose a reference point, which is not a feature in standard expected utility theory. In this theory, the only argument in the utility over wealth is <span class="math-container">$w$</span>, the absolute level of wealth. A common form of utility function is the <em>constant relative risk aversion</em> (CRRA) form: 
<span class="math-container">\begin{equation}
u(w)=\frac{w^{1-\rho}}{1-\rho},
\end{equation}</span>
where <span class="math-container">$\rho$</span> is the <a href="https://en.wikipedia.org/wiki/Risk_aversion#Relative_risk_aversion" rel="noreferrer">coefficient of relative risk aversion</a>. Note that this incorporates the commonly seen <span class="math-container">$\sqrt w$</span> and <span class="math-container">$\ln w$</span> forms when <span class="math-container">$\rho$</span> is <span class="math-container">$\frac12$</span> and <span class="math-container">$1$</span>, respectively. As is obvious, there is no term that helps differentiate between gains and losses in this formulation. The graph for a standard EU function with risk aversion is the same as any strictly increasing and strictly concave function. </p>

<p>Prospect theory is motivated by empirical regularities found in psychology experiments. A typical utility form is 
<span class="math-container">\begin{equation}
u(x,r)=
\begin{cases}
(x-r)^\alpha &amp;\text{if }x\ge r\\
-\lambda(r-x)^\beta &amp;\text{if }x&lt;r
\end{cases}
\end{equation}</span>
where <span class="math-container">$x$</span> is the payoff from some choice alternative (or "prospect" if the choice involves risks), <span class="math-container">$r$</span> is the reference point (which is <span class="math-container">$0$</span> in your example), <span class="math-container">$\alpha,\beta\in(0,1)$</span> are parameters that govern the degree of risk preference, and <span class="math-container">$\lambda$</span> is a parameter that separately captures <a href="https://en.wikipedia.org/wiki/Loss_aversion" rel="noreferrer">loss aversion</a>. Tversky and Kahneman (1992) estimate that <span class="math-container">$\alpha=\beta=0.88$</span> and <span class="math-container">$\lambda=2.25$</span>. </p>

CC BY-SA 4.0
2019-07-22T17:50:53.363Z
Thank you for your response! So, in an expected utility theory framework, can $u(w)$ ever take a negative input $w<0$? Or is this not possible because of the intuition that, unless we somehow incorporate debt, an individual cannot have negative wealth?
leecarvallo
CC BY-SA 4.0
2019-07-22T22:12:40.490Z
@leecarvallo: Standard EU framework only work with nonnegative values of $w$.
Herr K.
CC BY-SA 4.0
2019-07-23T09:04:44.953Z
@HerrK. Is there anything in the vNM axioms that excludes the function $u(w) = \sqrt{w + 100}$? This would work with some negative values.
Giskard
CC BY-SA 4.0
2019-07-23T16:27:27.393Z
@Giskard: I guess presuming some initial positive wealth level is one way of dealing with losses within the EU framework, but the *overall wealth level* would still have to be nonnegative. For example, the function you give would be ill-equipped to evaluate a bet where you either win or lose \$200 with equal probability.
Herr K.
CC BY-SA 4.0
2019-07-23T19:31:01.860Z
Yes, but my point is that there does not seem to be anything in vNM that explicitly forbids negative values. You could define a function $u(w) = \sqrt{w}$ if $w \geq 0$ and $u(w) = -w^2$ if $w < 0$. This is defined for all $w \in \mathbb{R}$. Does this somehow go against the axioms?
Giskard
CC BY-SA 4.0
2019-07-23T20:17:44.420Z
@Giskard: You're right; the vNM axioms do not preclude negative values in the argument of Bernoulli utility function. But when it comes to modeling risk attitudes, the mainstream textbooks do typically maintain the assumption of nonnegative wealth. Perhaps I was not being very precise in referring to this as the "standard EU framework"; the "standard model of risk preference" would probably be a better description.
Herr K.
CC BY-SA 4.0
2019-07-24T20:05:39.627Z
To expand on @HerrK., standard usage doesn't allow negative arguments because utility theory assumes you have utility over *actual outcomes* rather than *outcomes relative to reference* as in prospect theory (also, if you want to get formal, U should be over baskets of goods, not money), and few real-world outcomes are measured using negative values. Can't have -2 apples.
NickCHK
CC BY-SA 4.0
2019-07-24T20:57:30.513Z
@NickCHK: Thanks for the addition. Please note that, in the context of discussing risk preferences, it is typical to assume that utility is defined over *monetary* outcomes rather than consumption bundles.
Herr K.
CC BY-SA 4.0
2019-07-22T17:42:28.520Z
2019-07-22T17:53:10.843Z
2019-07-22T17:53:10.843Z
Herr K.
<p>I've been reading about prospect theory lately and have read often that prospect theory predicts people will be risk averse in gains and risk seeking in losses. This statement is typically supplemented with the phrase ", contrary to the standard expected utility theory model". </p>

<p>However, I'm not sure how expected utility theory deals with losses. Is it that EU theory assumes people who are risk averse in gains will be risk averse in losses as well? If so, how do you model this mathematically?</p>

<p>If our risk-averse EU maximizer has <span class="math-container">$u(x)=\sqrt{x}$</span> for <span class="math-container">$x&gt;=0$</span>, would the equivalent function in losses be <span class="math-container">$u(x) = -x^2$</span> for <span class="math-container">$x&lt;0$</span>? If so, wouldn't that mean that they are overweighting the loss component relative to the gain component - which is not an element of expected utility theory.</p>

<p>Graphically, a prospect theory value function is typically drawn as <a href="https://i.sstatic.net/VczV9.png" rel="noreferrer"><img src="https://i.sstatic.net/VczV9.png" alt="enter image description here"></a>
. </p>

<p>How would you draw the equivalent graph for an expected utility theory value function?</p>

<microeconomics><expected-utility><theory>
How does expected utility theory treat losses?
leecarvallo
# How does expected utility theory treat losses?

(Asked by: username_0 on 2019-07-22)

I've been reading about prospect theory lately and have read often that prospect theory predicts people will be risk averse in gains and risk seeking in losses. This statement is typically supplemented with the phrase ", contrary to the standard expected utility theory model".

However, I'm not sure how expected utility theory deals with losses. Is it that EU theory assumes people who are risk averse in gains will be risk averse in losses as well? If so, how do you model this mathematically?

If our risk-averse EU maximizer has $u(x)=\sqrt{x}$ for $x>=0$, would the equivalent function in losses be $u(x) = -x^2$ for $x<0$? If so, wouldn't that mean that they are overweighting the loss component relative to the gain component - which is not an element of expected utility theory.

Graphically, a prospect theory value function is typically drawn as [![enter image description here](https://i.sstatic.net/VczV9.png)](https://i.sstatic.net/VczV9.png)
.

How would you draw the equivalent graph for an expected utility theory value function?

**Tags:** microeconomics, expected-utility, theory

---

## 1 answers

### Answer by username_1 on 2019-07-22. Score: 6

Gains and losses presuppose a reference point, which is not a feature in standard expected utility theory. In this theory, the only argument in the utility over wealth is $w$, the absolute level of wealth. A common form of utility function is the *constant relative risk aversion* (CRRA) form:
\begin{equation}
u(w)=\frac{w^{1-\rho}}{1-\rho},
\end{equation}
where $\rho$ is the [coefficient of relative risk aversion](https://en.wikipedia.org/wiki/Risk_aversion#Relative_risk_aversion). Note that this incorporates the commonly seen $\sqrt w$ and $\ln w$ forms when $\rho$ is $\frac12$ and $1$, respectively. As is obvious, there is no term that helps differentiate between gains and losses in this formulation. The graph for a standard EU function with risk aversion is the same as any strictly increasing and strictly concave function.

Prospect theory is motivated by empirical regularities found in psychology experiments. A typical utility form is
\begin{equation}
u(x,r)=
\begin{cases}
(x-r)^\alpha &\text{if }x\ge r\\
-\lambda(r-x)^\beta &\text{if }x
where $x$ is the payoff from some choice alternative (or "prospect" if the choice involves risks), $r$ is the reference point (which is $0$ in your example), $\alpha,\beta\in(0,1)$ are parameters that govern the degree of risk preference, and $\lambda$ is a parameter that separately captures [loss aversion](https://en.wikipedia.org/wiki/Loss_aversion). Tversky and Kahneman (1992) estimate that $\alpha=\beta=0.88$ and $\lambda=2.25$.

> **Comments:**
> **username_0**: Thank you for your response! So, in an expected utility theory framework, can $u(w)$ ever take a negative input $w<0$? Or is this not possible because of the intuition that, unless we somehow incorporate debt, an individual cannot have negative wealth?  
> **username_1**: @leecarvallo: Standard EU framework only work with nonnegative values of $w$.  
> **username_2**: @HerrK. Is there anything in the vNM axioms that excludes the function $u(w) = \sqrt{w + 100}$? This would work with some negative values.  
> **username_1**: @Giskard: I guess presuming some initial positive wealth level is one way of dealing with losses within the EU framework, but the *overall wealth level* would still have to be nonnegative. For example, the function you give would be ill-equipped to evaluate a bet where you either win or lose \$200 with equal probability.  
> **username_2**: Yes, but my point is that there does not seem to be anything in vNM that explicitly forbids negative values. You could define a function $u(w) = \sqrt{w}$ if $w \geq 0$ and $u(w) = -w^2$ if $w < 0$. This is defined for all $w \in \mathbb{R}$. Does this somehow go against the axioms?  
> **username_1**: @Giskard: You're right; the vNM axioms do not preclude negative values in the argument of Bernoulli utility function. But when it comes to modeling risk attitudes, the mainstream textbooks do typically maintain the assumption of nonnegative wealth. Perhaps I was not being very precise in referring to this as the "standard EU framework"; the "standard model of risk preference" would probably be a better description.  
> **username_3**: To expand on @HerrK., standard usage doesn't allow negative arguments because utility theory assumes you have utility over *actual outcomes* rather than *outcomes relative to reference* as in prospect theory (also, if you want to get formal, U should be over baskets of goods, not money), and few real-world outcomes are measured using negative values. Can't have -2 apples.  
> **username_1**: @NickCHK: Thanks for the addition. Please note that, in the context of discussing risk preferences, it is typical to assume that utility is defined over *monetary* outcomes rather than consumption bundles.  

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