Exercise 6 Consider a two-period intertemporal model, where the representative agent has a utility function given by U1 5 C12σ 1 1 2 σ 1 β C12σ 2 1 2 σ ;

where Ct in consumption for the period t; σ . 0 is the preference parameter, and βAð0; 1Þ is the intertemporal discount rate. In period tAf1; 2g, this individual receives an exogenous flow of income Yt. There is a financial market that remunerates savings at the rate of r. The interest rate charges on debt are also given as r.

a. Write and interpret the intertemporal budget constraint for the agent. Illustrate this constraint with a graph, along with the map of the indifference curves of the representative agent.

b. Derive the first-order condition for the consumer’s problem. Obtain the Euler equation and present its economic interpretation.

c. Obtain the intertemporal elasticity of substitution for the representative consumer. [Hint:

The mathematical expression for the elasticity of substitution is given by

η 5 ðdðc1=c2Þ=dTMSÞðTMS=ðc1=c2ÞÞ, where TMS is the marginal rate of substitution.]