<p>Consider a function like <span class="math-container">$u(c) = e^{\gamma c}$</span>, where <span class="math-container">$\gamma &gt; 0$</span>.</p>
<p>Corresponding Marginal Utility is <span class="math-container">$\text{MU}=\dfrac{du}{dc}=\gamma e^{\gamma c}$</span>.</p>
<p>We get that the elasticity of marginal utility is equal to <span class="math-container">$\gamma c$</span>, which is increasing in <span class="math-container">$c$</span>.</p>
<p>If you're looking for a function which is concave in <span class="math-container">$c$</span>, then you can consider <span class="math-container">$u(c) = 1-e^{-\gamma c}$</span>, where <span class="math-container">$\gamma &gt; 0$</span>. This one also has the elasticity of marginal utility equal to <span class="math-container">$\gamma c$</span>.</p>

CC BY-SA 4.0
2024-02-18T06:18:19.220Z
2024-02-18T06:41:20.343Z
2024-02-18T06:41:20.343Z
Amit
<p>Consider there is a single good in the economy. The class of utility functions for which the elasticity of marginal utility <span class="math-container">$\eta$</span> is constant is given by
<span class="math-container">$$U(C)=\frac{C^{1-\eta}}{1-\eta}$$</span>
for <span class="math-container">$\eta&gt;0$</span> and <span class="math-container">$\eta\neq1$</span> and
<span class="math-container">$$U(C)=\ln C$$</span>
for <span class="math-container">$\eta=1$</span>.</p>
<p>Is there a well-known example of a utility function that exhibits increasing/decreasing elasticity of marginal utility?　</p>
<p><strong>edit1:</strong>
I want a class of utility functions such that whether they exhibit increasing or decreasing elasticity of marginal utility is governed by a single parameter. I tried something like this: Suppose the elasticity of marginal utility is given by <span class="math-container">$\eta(C)$</span>, then for some constant <span class="math-container">$c_1,c_2$</span>,
<span class="math-container">$$U(C)=c_1\int_1^C\exp\left(-\int_1^x\frac{\eta(y)}{y}dy\right)dx+c_2$$</span>
Then I tried to find a function <span class="math-container">$\eta(C)$</span> with a single parameter that governs whether it is increasing or decreasing and yield a nice expression for <span class="math-container">$U$</span>.</p>
<p><strong>edit2:</strong>
I ended up with the following expression,
<span class="math-container">$$U(C)=\int_0^{\frac{\eta}{\psi}C^\psi}t^{\frac{1}{\psi}-1}\exp(-t)dt=\gamma\left(\frac{1}{\psi},\frac{\eta}{\psi}C^\psi\right),$$</span>
whose elasticity of marginal utility is given by <span class="math-container">$\eta C^\psi$</span>. Here <span class="math-container">$\gamma$</span> denotes the lower incomplete gamma function. I am not sure if it is appropriate to work on this class, though.</p>
<p><strong>edit3:</strong>
I realized that the utility function above is valid only for <span class="math-container">$\psi&gt;0$</span>, otherwise the integral does not converge.</p>

<utility><elasticity>
Utility functions that exhibit nonconstant elasticity of marginal utility
kenji
# Utility functions that exhibit nonconstant elasticity of marginal utility

(Asked by: username_0 on 2024-02-18)

Consider there is a single good in the economy. The class of utility functions for which the elasticity of marginal utility $\eta$ is constant is given by
$$U(C)=\frac{C^{1-\eta}}{1-\eta}$$
for $\eta>0$ and $\eta\neq1$ and
$$U(C)=\ln C$$
for $\eta=1$.

Is there a well-known example of a utility function that exhibits increasing/decreasing elasticity of marginal utility?

**edit1:**
I want a class of utility functions such that whether they exhibit increasing or decreasing elasticity of marginal utility is governed by a single parameter. I tried something like this: Suppose the elasticity of marginal utility is given by $\eta(C)$, then for some constant $c\_1,c\_2$,
$$U(C)=c\_1\int\_1^C\exp\left(-\int\_1^x\frac{\eta(y)}{y}dy\right)dx+c\_2$$
Then I tried to find a function $\eta(C)$ with a single parameter that governs whether it is increasing or decreasing and yield a nice expression for $U$.

**edit2:**
I ended up with the following expression,
$$U(C)=\int\_0^{\frac{\eta}{\psi}C^\psi}t^{\frac{1}{\psi}-1}\exp(-t)dt=\gamma\left(\frac{1}{\psi},\frac{\eta}{\psi}C^\psi\right),$$
whose elasticity of marginal utility is given by $\eta C^\psi$. Here $\gamma$ denotes the lower incomplete gamma function. I am not sure if it is appropriate to work on this class, though.

**edit3:**
I realized that the utility function above is valid only for $\psi>0$, otherwise the integral does not converge.

**Tags:** utility, elasticity

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## 1 answers

### Answer by username_1 on 2024-02-18. Score: 1

Consider a function like $u(c) = e^{\gamma c}$, where $\gamma > 0$.

Corresponding Marginal Utility is $\text{MU}=\dfrac{du}{dc}=\gamma e^{\gamma c}$.

We get that the elasticity of marginal utility is equal to $\gamma c$, which is increasing in $c$.

If you're looking for a function which is concave in $c$, then you can consider $u(c) = 1-e^{-\gamma c}$, where $\gamma > 0$. This one also has the elasticity of marginal utility equal to $\gamma c$.


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