Consider a two-period, small, open, endowment economy without investment and government expenditures, but with durable consumption goods. Purchases of durable consumption goods in period 1, denoted C1, continue to provide utility in period 2. The utility of households in period 2 depends on purchases of durable consumption goods in period 2 , denoted C2, and on the un-depreciated stock of durables purchased in period 1. Durable consumption goods are assumed to depreciate at the rate [0, 1]. Household preferences are described by the following utility function

U =ln(C1)+ln(C2 +(1)C1)

Assume that the world interest rate is given by r, the endowment in period one is denoted by Y1 and the endowment in period 2 is denoted by Y2. Finally assume that the initial asset position, B1, is zero.

1. (a) [1 points] State the household's budget constraints in periods 1 and 2.
2. (b) [2 points] Find the equilibrium values of consumption and net exports in both periods. Will households smooth consumption over time? Provide intuition.

Assume that the world interest rate is r = 0.1 per year, that the endowment in period one is Y1 = 1, and that the endowment in period 2 is Y2 = 1.1.

(c) [1 points] Assume now that = 1. Find the equilibrium values of consumption and net exports in periods 1 and 2.

1. (d) [1 points] Suppose the country experiences a boom in period 1. Specifically, output in period 1 increases from 1 to 2 (Y1 = 2). The output shock is temporary in the sense that output in period 2 is unchanged. Continue to assume that = 1, that is, that consumption is nondurable. Are net exports in period 1 countercyclical, that is, does the change in net exports have the opposite sign as the change in GDP? Why or why not. Find the change in net exports in period 1.
2. (e) [3 points] Continue to assume that Y1 increases in period 1 from 1 to 2. But now do not impose that = 1. How much durability does one need to ensure that the response of net exports in period 1 is countercyclical. Provide intuition for your answer.

Consider a two-period, small, open, endowment economy without investment and government expenditures, but with durable consumption goods. Purchases of durable consumption goods in period 1, denoted C1, continue to provide utility in period 2. The utility of households in period 2 depends on purchases of durable consumption goods in period 2 , denoted C2, and on the un-depreciated stock of durables purchased in period 1. Durable consumption goods are assumed to depreciate at the rate [0, 1]. Household preferences are described by the following utility function

U =ln(C1)+ln(C2 +(1)C1)

Assume that the world interest rate is given by r, the endowment in period one is denoted by Y1 and the endowment in period 2 is denoted by Y2. Finally assume that the initial asset position, B1, is zero.

1. (a) [1 points] State the household's budget constraints in periods 1 and 2.
2. (b) [2 points] Find the equilibrium values of consumption and net exports in both periods. Will households smooth consumption over time? Provide intuition.

Assume that the world interest rate is r = 0.1 per year, that the endowment in period one is Y1 = 1, and that the endowment in period 2 is Y2 = 1.1.

(c) [1 points] Assume now that = 1. Find the equilibrium values of consumption and net exports in periods 1 and 2.

1. (d) [1 points] Suppose the country experiences a boom in period 1. Specifically, output in period 1 increases from 1 to 2 (Y1 = 2). The output shock is temporary in the sense that output in period 2 is unchanged. Continue to assume that = 1, that is, that consumption is nondurable. Are net exports in period 1 countercyclical, that is, does the change in net exports have the opposite sign as the change in GDP? Why or why not. Find the change in net exports in period 1.
2. (e) [3 points] Continue to assume that Y1 increases in period 1 from 1 to 2. But now do not impose that = 1. How much durability does one need to ensure that the response of net exports in period 1 is countercyclical. Provide intuition for your answer.