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 ;


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)

: http://www.nber.org/papers/w18999.pdf

  • 2
    $\begingroup$ Shouldn't this depend on the properties of $\rho_i$ and $\omega_i$? $\endgroup$ – HRSE Mar 8 '16 at 10:27
  • $\begingroup$ @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. $\endgroup$ – optimal control Mar 8 '16 at 11:02

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