Elasticity of intertemporal substitution (or intertemporal elasticity of substitution) is a measure of responsiveness of the growth rate

Economic growth

In economics, economic growth is defined as the increasing capacity of the economy to satisfy the wants of goods and services of the members of society. Economic growth is enabled by increases in productivity, which lowers the inputs for a given amount of output. Lowered costs increase demand...

 of consumption

Consumption (economics)

Consumption is a common concept in economics, and gives rise to derived concepts such as consumer debt. Generally, consumption is defined in part by comparison to production. But the precise definition can vary because different schools of economists define production quite differently...

 to the real interest rate

Real interest rate

The "real interest rate" is the rate of interest an investor expects to receive after allowing for inflation. It can be described more formally by the Fisher equation, which states that the real interest rate is approximately the nominal interest rate minus the inflation rate...

. If the real rate rises, future consumption may increase due to increased return on savings; but future consumption may also decline as the saver decides to save less given that he can get a higher return on what he does save. The net effect on future consumption is the elasticity of intertemporal substitution.

Mathematical definition

The definition depends on whether one is working in discrete or continuous time. We will see that for CRRA

Risk aversion

Risk aversion is a concept in psychology, economics, and finance, based on the behavior of humans while exposed to uncertainty....

 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 invested today costs 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 . (If it yielded more, then the agent could make himself better off by saving more.)

Solving for the real interest rate, we see that
In logs, we have
Logs are very close to percentage changes, so we can interpret as an net interest rate like 5%, whereas is the corresponding gross interest rate like 1.05.

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.

Continuous time

Let total lifetime utility be given by



where is shorthand for , is the utility of consumption in (instant) time t, and is the time discount rate. First define the measure of relative risk aversion (this is useful even if the model has no uncertainty or risk) as,
then the elasticity of intertemporal substitution is defined as



If the utility function is of the CRRA type:

(with special case of being )

then the intertemporal elasticity of substitution is given by . In general, a low value of theta (high intertemporal elasticity) means that consumption growth is very sensitive to changes in the real interest rate. For theta equal to 1, the growth rate of consumption responds one for one to changes in the real interest rate. A high theta implies an insensitive consumption growth.

Ramsey Growth model

In the Ramsey growth model

Ramsey growth model

The Ramsey–Cass–Koopmans model or the Ramsey growth model is a neo-classical model of economic growth based primarily on the work of the economist and mathematician Frank P. Ramsey, with significant extensions by David Cass and Tjalling Koopmans...

, the elasticity of intertemporal substitution determines the speed of adjustment to the steady state

Steady state

A system in a steady state has numerous properties that are unchanging in time. This implies that for any property p of the system, the partial derivative with respect to time is zero:...

 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

Consumption smoothing

Consumption smoothing is the economic concept used to express the desire of people for having a stable path of consumption.Since Milton Friedman's permanent income theory and Modigliani and Brumberg life-cycle model, the idea that agents prefer a stable path of consumption has been widely accepted...

 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.