# Utility functions that exhibit nonconstant elasticity of marginal utility

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.

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 $$\gamma c$$.