<p>Suppose <span class="math-container">$c_t$</span> is some composite function of interest rate <span class="math-container">$r$</span>, e.g. <span class="math-container">$c_t(G(r_t))$</span>.</p>
<p>First the intertemporal elasticity of substitution is actually (IES) given by <span class="math-container">$\frac{\partial \ln(c_{t+1}/c_{t})}{\partial r}$</span> (or also <span class="math-container">$\frac{\partial \ln(c_{t+1}/c_{t})}{\partial \ln( u'(c_{t+1})/u'(c_t))}$</span>).</p>
<p>You can take the derivative above using chain rule for composite functions which says that <span class="math-container">$dF(G(x))/dx = \frac{dF}{dG}\frac{dG}{dx}$</span>.</p>
<p>So in the above case we would get:</p>
<p><span class="math-container">$$\frac{\partial \ln(c_{t+1}(G(r))/c_{t}(G(r)))}{\partial r} = \frac{\partial \ln(c_{t+1}(G(r))/c_{t}(G(r)))}{\partial G(r)} \cdot \frac{\partial G(r)}{\partial r} $$</span></p>
<p>You can directly apply this to the problem you mention in your question.</p>

CC BY-SA 4.0
2020-11-16T17:27:40.960Z
Perhaps I expresed myself not clearly enough. MY problem is the fact that the function G() enters substracting consumption in the traditional CRRA function not as a composite one.
mmendina
CC BY-SA 4.0
2020-11-16T17:30:09.173Z
@MartinMendina I don’t understand what you mean. Why would the fact that it is subtracted from consumption function matter at all? The above formula would work even with negative utility or consumption regardless if that makes economic sense.
1muflon1
CC BY-SA 4.0
2020-11-16T14:02:02.837Z
2020-11-16T14:02:02.837Z
1muflon1
<p>In the usual CRRA <span class="math-container">$\frac{c^{1-\sigma}-1}{1-\sigma}$</span> function we have that the intertemporal elasticity of sustitution  <span class="math-container">$\partial\frac{c_{t+1}}{{c_{t}}{\partial r}}$</span> is <span class="math-container">$\frac{1}{\sigma}$</span>.</p>
<p>But how can i calculate the IE of S when the CRRA function is the composite:</p>
<p><span class="math-container">$\frac{(c-G())^{1-\sigma}-1}{1-\sigma}$</span></p>
<p>Thanks in advance</p>

CC BY-SA 4.0
2020-11-16T12:59:33.773Z
you can still take derivative of such function by chain rule
1muflon1
CC BY-SA 4.0
2020-11-16T13:15:27.823Z
Can you explain that in an answer?
mmendina
<utility><consumer-theory><elasticity><dsge>
Elasticity of intertemporal sustitution with composite CRRA function
mmendina
# Elasticity of intertemporal sustitution with composite CRRA function

(Asked by: username_0 on 2020-11-16)

In the usual CRRA $\frac{c^{1-\sigma}-1}{1-\sigma}$ function we have that the intertemporal elasticity of sustitution $\partial\frac{c\_{t+1}}{{c\_{t}}{\partial r}}$ is $\frac{1}{\sigma}$.

But how can i calculate the IE of S when the CRRA function is the composite:

$\frac{(c-G())^{1-\sigma}-1}{1-\sigma}$

Thanks in advance

**Tags:** utility, consumer-theory, elasticity, dsge
> **Comments:**
> **username_1**: you can still take derivative of such function by chain rule  
> **username_0**: Can you explain that in an answer?  

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## 1 answers

### Answer by username_1 on 2020-11-16. Score: 1

Suppose $c\_t$ is some composite function of interest rate $r$, e.g. $c\_t(G(r\_t))$.

First the intertemporal elasticity of substitution is actually (IES) given by $\frac{\partial \ln(c\_{t+1}/c\_{t})}{\partial r}$ (or also $\frac{\partial \ln(c\_{t+1}/c\_{t})}{\partial \ln( u'(c\_{t+1})/u'(c\_t))}$).

You can take the derivative above using chain rule for composite functions which says that $dF(G(x))/dx = \frac{dF}{dG}\frac{dG}{dx}$.

So in the above case we would get:

$$\frac{\partial \ln(c\_{t+1}(G(r))/c\_{t}(G(r)))}{\partial r} = \frac{\partial \ln(c\_{t+1}(G(r))/c\_{t}(G(r)))}{\partial G(r)} \cdot \frac{\partial G(r)}{\partial r} $$

You can directly apply this to the problem you mention in your question.

> **Comments:**
> **username_0**: Perhaps I expresed myself not clearly enough. MY problem is the fact that the function G() enters substracting consumption in the traditional CRRA function not as a composite one.  
> **username_1**: @MartinMendina I don’t understand what you mean. Why would the fact that it is subtracted from consumption function matter at all? The above formula would work even with negative utility or consumption regardless if that makes economic sense.  

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