Exercise 10 Consider a closed economy that exists for two periods with a production function given by Yt 5 Kα

t L12α t ;

where αAð0; 1Þ, Kt is the capital endowment and Lt is the labor endowment for the economy for period t. Assume that the labor endowment is equal to the size of the population in this economy, which we will assume is equal to 1 for each period. Further assume that the economy receives a capital stock of K1 . 0 in period 1, and the capital depreciates in such a way as that capital stock in period 2 is given by K2 5 K1 1 I1, where I1 represents investment in period 1.

Chapter 4 • What Is the Optimum Current-Account Level? 99 Representative agent preferences in this economy are represented by the following utility function:

U1 5 lnðC1Þ 1 β lnðC2Þ:

a. Derive the production possibilities frontier in the economy, assuming that K3 5 0.

b. Illustrate by means of a graph the production possibilities frontier obtained in item (a).

How does the concavity of the curve change with variations in α?

c. How does the existence of a government that consumes positive quantities of product in period 2 affect the production possibilities frontier of the representative agent?

d. Compute the interest rates in autarky in the case where α 5 1.

e. Illustrate the equilibrium in autarky by means of a graph that relates savings and investment.