Problem 1 (An Economy with Investment)
Consider a two-period model of a small open economy with a single good. Let
preferences of the representative household be described by the utility function
ln C1 + ln C2,
where C1 and C2 denote consumption in periods 1 and 2, respectively. In period
1, the household receives an endowment of Q1 = 10. In period 2, the household
receives profits, denoted by Π2, from the firm it owns. The firm invests in period
1 to be able to produce goods in period 2. The production technology in period 2
is given by
Q2 = √I1
where Q2 and I1 denote output in period 2 and investment in period 1, respec-
tively. The household and the firm have access to financial markets where they
can borrow or lend. Assume that there exists free international capital mobility
and that the world interest rate, r∗, is 10% each period (i.e., r0 = r1 = r∗ = 0.1,
where rt is the interest rate on assets held between periods t and t + 1). Finally,
assume that the economy’s initial net foreign asset position is zero (B0 = 0).
(a) Compute the firm’s optimal levels of investment in period 1 and the profits in
period 2.
(b) Solve for the optimal levels of consumption in periods 1 and 2.
(c) Find the country’s net foreign asset position at the end of period 1 and, for
each of the periods 1 and 2, the country’s savings, trade balance and current
account balance.
(d) Now consider an investment surge. Specifically, assume that as a result of a
technological improvement, the production technology becomes Q2 = 2√I1.
Find the period 1 equilibrium level of investment, the period 1 and period 2
equilibrium levels of savings, the trade balance, the current account balance,
and the country’s net foreign asset position at the end of period 1. Compare
your results with those obtained in parts (a)-(c) providing interpretation and
intuition.
Consider a two-period model of a small open economy with a single good. Let
preferences of the representative household be described by the utility function
ln C1 + ln C2,
where C1 and C2 denote consumption in periods 1 and 2, respectively. In period
1, the household receives an endowment of Q1 = 10. In period 2, the household
receives profits, denoted by Π2, from the firm it owns. The firm invests in period
1 to be able to produce goods in period 2. The production technology in period 2
is given by
Q2 = √I1
where Q2 and I1 denote output in period 2 and investment in period 1, respec-
tively. The household and the firm have access to financial markets where they
can borrow or lend. Assume that there exists free international capital mobility
and that the world interest rate, r∗, is 10% each period (i.e., r0 = r1 = r∗ = 0.1,
where rt is the interest rate on assets held between periods t and t + 1). Finally,
assume that the economy’s initial net foreign asset position is zero (B0 = 0).
(a) Compute the firm’s optimal levels of investment in period 1 and the profits in
period 2.
(b) Solve for the optimal levels of consumption in periods 1 and 2.
(c) Find the country’s net foreign asset position at the end of period 1 and, for
each of the periods 1 and 2, the country’s savings, trade balance and current
account balance.
(d) Now consider an investment surge. Specifically, assume that as a result of a
technological improvement, the production technology becomes Q2 = 2√I1.
Find the period 1 equilibrium level of investment, the period 1 and period 2
equilibrium levels of savings, the trade balance, the current account balance,
and the country’s net foreign asset position at the end of period 1. Compare
your results with those obtained in parts (a)-(c) providing interpretation and
intuition.
