# Continuum limit of a weighted mean of heterogeneous discount factors

I try to find the limit of the following expression of a weighted mean of a series of discount rate $\rho_{i}$

Here is the expression ;

$$\rho_{\eta}\left(t\right)=\frac{\sum_{i}^{N}\rho_{i}\left(\omega_{i}e^{-\rho_{i}t}\right)^{\frac{1}{\eta}}}{\sum_{i}^{N}\left(\omega_{i}e^{-\rho_{i}t}\right)^{\frac{1}{\eta}}}$$

where $\omega$ is the weight for each $\rho$ and $t$ stands for time and $\eta$ is intertemporal elasticity of substitiution

and let's say for $\omega_{i}\rightarrow\rho^{k-1}e^{-\frac{\rho}{\theta}}$ which is so-called Gamma distribution, where $k$ and $\omega$ are shape and scale parameters.

Conceptually, I have understood that continuum limit is the case where $N\rightarrow\infty$ and we should approximate the term by a continuum, which is probably an integral but I can not figure out how to find the continuum limit of this expression. Any suggestion or hint ?

PS. The question comes from the following paper (page 14, equation 37)

• Shouldn't this depend on the properties of $\rho_i$ and $\omega_i$? – HRSE Mar 8 '16 at 10:27
• @HRSE Thanks, I edited question by adding distribution for $\omega_{i}$ but as $\rho_{i}$ is a personal discount for agent $i$ I think it does not matter so much for limit. – optimal control Mar 8 '16 at 11:02