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.
Contents
Mathematical definition
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.
Discrete time
Total lifetime utility is given by
In this setting, the real interest rate will be given by the following condition:
A quantity of money
Solving for the real interest rate, we see that
In logs, we have
Logs are very close to percentage changes, so we can interpret
The elasticity of intertemporal substitution is defined as the percent change in consumption growth per percent increase in the net interest rate:
By substituting in our log equation above, we can see that this definition is equivalent to the elasticity of consumption growth with respect to marginal utility growth:
Either definition is correct, however, assuming that the agent is optimizing and has time separable utility.
Example
Let utility of consumption in period
Since this utility function belongs to the family of CRRA utility functions we have
This can be rewritten as
Hence, applying the above derived formula
Continuous time
Let total lifetime utility be given by
where
then the elasticity of intertemporal substitution is defined as
If the utility function
then the intertemporal elasticity of substitution is given by
Ramsey Growth model
In the Ramsey growth model, the elasticity of intertemporal substitution determines the speed of adjustment to the steady state and the behavior of the saving rate during the transition. If the elasticity is high then large changes in consumption are not very costly to consumers and as a result if the real interest rate is high they will save a large portion of their income. If the elasticity is low the consumption smoothing motive is very strong and because of this consumers will save a little and consume a lot if the real interest rate is high.
Estimates
Empirical estimates of the elasticity vary. Part of the difficulty stems from the fact that microeconomic studies come to different conclusions than macroeconomic studies which use aggregate data. A meta-analysis of 169 published studies reports a mean elasticity of 0.5, but also substantial differences across countries.