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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.

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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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