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

<p>Here is the expression ;</p>

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

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

<p>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.</p>

<p>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 ? </p>

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

<p>: <a href="http://www.nber.org/papers/w18999.pdf" rel="nofollow">http://www.nber.org/papers/w18999.pdf</a></p>

CC BY-SA 3.0
2016-03-08T10:27:44.470Z
Shouldn't this depend on the properties of $\rho_i$ and $\omega_i$?
HRSE
CC BY-SA 3.0
2016-03-08T11:02:31.397Z
@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
<discount><heterogeneity>
Continuum limit of a weighted mean of heterogeneous discount factors
optimal control
# Continuum limit of a weighted mean of heterogeneous discount factors

(Asked by: username_0 on 2016-03-08)

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)

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

**Tags:** discount, heterogeneity

> **Comments:**
> **username_1**: Shouldn't this depend on the properties of $\rho_i$ and $\omega_i$?  
> **username_0**: @username_1 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.  

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## 0 answers

