# Elasticity of intertemporal sustitution with composite CRRA function

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:

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

• you can still take derivative of such function by chain rule
– 1muflon1
Nov 16, 2020 at 12:59
• Can you explain that in an answer? Nov 16, 2020 at 13:15

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.

• 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. Nov 16, 2020 at 17:27
• @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
Nov 16, 2020 at 17:30