Question

Consider the following two-period, endowment open economy model. A representative consumer with utility represented by U(C1, C2) decides consumption C1 and C2, and bond holdings B1 and B2. Initial bonds are B0, and interest rates are r0 and r1. Endowments are Q1 and Q2. In equilibrium, B2 = 0 and normalize the prices to one in each period. The consumer's budget constraint in periods 1 and 2 is respectively:

C1 + B1 - B0 = r0B0 + Q1
C2 + B2 - B1 = r1B1 + Q2

(a) Write the consumer's first-order conditions assuming c1 > 0 and c2 > 0 in equilibrium.

Answer 1

The first-order conditions for a consumer's utility maximization in a two-period endowment open economy require setting the partial derivatives of the utility function equal to the respective price ratios adjusted for interest rates, under the constraint that bond holdings by period 2 are zero.

Step-by-step explanation:

The question asks about finding the first-order conditions for a consumer's utility maximization problem in a two-period endowment open economy. Given the utility function U(C1, C2), a representative consumer chooses consumption in both periods (C1 and C2) and bond holdings to maximize their utility subject to the given budget constraints.

To derive the first-order conditions, we calculate the partial derivatives of the utility function with respect to C1 and C2, and set them equal to the price ratios (adjusted for interest rates), since the prices are normalized to one.

The first-order condition for C1 will be:

U1(C1, C2) = (1 + r0)

This implies that the marginal utility of consumption in period 1 should be equal to the cost of forgoing that consumption, adjusted for the interest rate in period 1.

The first-order condition for C2 will be:

U2(C1, C2) = (1 + r1)

This implies that the marginal utility of consumption in period 2 should be equal to the cost of forgoing that consumption, adjusted for the interest rate in period 2. Since B2 = 0, it indicates that the consumer plans to hold no bonds at the end of period 2.