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. It determines how individuals choose to allocate their consumption between the present and the future and hence consumption and saving behavior. The elasticity of intertemporal substitution is equal to the reciprocal of the elasticity of marginal utility with respect to consumption.

Mathematical definition

The definition depends on whether one is working in discrete or continuous time. We will see that for a particular functional form of utility, the two approaches yield the same answer. It is simpler to abstract away from specific financial settings and analyze how the change in consumption is affected by a change in marginal utility rather than in interest rates (which in most models will be equivalent given an optimizing agent). The below functional forms assume that utility from consumption is time additively separable.

Continuous time

Let total lifetime utility be given by

$U=\int _{0}^{T}e^{-\rho t}u(c_{t})dt$

where $c_{t}$ is shorthand for $c(t)$, $u(c(t))$ is the utility of consumption in (instant) time t, and $\rho$ 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,

$RRA=-{\frac {d(u'(c_{t}))}{d(c_{t})}}{\frac {c_{t}}{u'(c_{t})}}=-u''(c_{t}){\frac {c_{t}}{u'(c_{t})}}$

then the elasticity of intertemporal substitution is defined as

$EIS=-{\frac {\partial ({\dot {c}}_{t}/c_{t})}{\partial ({\dot {u}}'(c_{t})/u'(c_{t}))}}=-{\frac {\partial ({\dot {c}}_{t}/c_{t})}{\partial (u''(c_{t}){\dot {c}}_{t}/u'(c_{t}))}}={\frac {\partial ({\dot {c}}_{t}/c_{t})}{\partial (RRA\cdot ({\dot {c}}_{t}/c_{t}))}}={\frac {1}{RRA}}=-{\frac {u'(c_{t})}{u''(c_{t})\cdot c_{t}}}$

If the utility function $u(c)$ is of the CRRA type:

$u(c)={\frac {c^{1-\theta }-1}{1-\theta }}$ (with special case of $\theta =1$ being $u(c)=ln(c)$)

then the intertemporal elasticity of substitution is given by ${\frac {1}{\theta }}$. 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.

Discrete time

We total lifetime utility is given by

$U=\sum _{t=0}^{T}\beta ^{t}u(c_{t})$

Then the elasticity of intertemporal substitution is defined as

$EIS=-{\frac {\partial \ln(c_{t+1}/c_{t})}{\partial \ln(u'(c_{t+1})/u'(c_{t}))}}=-{\frac {\partial (c_{t+1}/c_{t})}{\partial (u'(c_{t+1})/u'(c_{t}))}}{\frac {u'(c_{t+1})/u'(c_{t})}{(c_{t+1}/c_{t})}}$

If we hold $c_{t+1}$ constant than this simplifies to the same expression as in the continuous time case

$-{\frac {u'(c_{t})}{u''(c_{t})\cdot c_{t}}}$

For the CRRA utility mentioned above, the IES becomes

$-{\frac {\partial \ln(c_{t+1}/c_{t})}{\partial \ln(u'(c_{t+1})/u'(c_{t}))}}=-{\frac {\partial \ln(c_{t+1}/c_{t})}{\partial \ln((c_{t+1}/c_{t})^{-\theta })}}={\frac {\partial \ln(c_{t+1}/c_{t})}{\partial \theta \ln(c_{t+1}/c_{t})}}={\frac {1}{\theta }}$.

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.