Elasticity of intertemporal substitution

Elasticity of intertemporal substitution (or intertemporal elasticity of substitution) is a measure of responsiveness of the growth rate of consumption to the real interest rate. If the real rate rises, current consumption may decrease due to increased return on savings; but current consumption may also increase as the household decides to consume more immediately, as it is feeling richer. The net effect on current consumption is the elasticity of intertemporal substitution.

The definition depends on whether one is working in discrete or continuous time. We will see that for CRRA utility, the two approaches yield the same answer. The below functional forms assume that utility from consumption is time additively separable.

Total lifetime utility is given by

In this setting, the real interest rate will be given by the following condition:

A quantity of money ${\displaystyle Q}$ invested today costs ${\displaystyle Qu'(c_{t})}$ units of utility, and so must yield exactly that number of units of utility in the future when saved at the prevailing gross interest rate ${\displaystyle R}$. (If it yielded more, then the agent could make himself better off by saving more.)

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