Elasticity of intertemporal substitution
Elasticity of intertemporal substitution (or intertemporal elasticity of substitution, EIS, IES) is a measure of responsiveness of the growth rate of consumption to the real interest rate. If the real interest 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. 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 gross real interest rate formula_2 will be given by the following condition: A quantity of money formula_4 invested today costs formula_5 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 formula_6, where formula_7 is the net interest rate (if it yielded more, then the agent could make himself better off by saving  more). Solving for the gross interest rate, we see that In logs, we have Since formula_10 for small formula_7 (logs are very close to percentage changes) we have 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  formula_15 be given by Since this utility function belongs to the family of CRRA utility functions we have formula_17 Thus, This can be rewritten as Hence, applying the above derived formula Continuous time. Let total lifetime utility be given by formula_21 where formula_22 is shorthand for formula_23, formula_24 is the utility of consumption in (instant) time t, and formula_25 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 formula_27 If the utility function formula_28  is of the CRRA type: formula_29 (with special case of formula_30 being formula_31) then the intertemporal elasticity of substitution is given by formula_32. 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, 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.