Consider a two-period model of a small open economy with a single good each period. 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, and ln denotes the natural logarithm. In period 1, the household receives an endowment of Q1 = 5. In period 2, the household receives profits, denoted by ?2, from the firms it owns. Households and firms have access to financial markets where they can borrow or lend at the interest rate r1. (r1 is the interest rate on assets held between periods 1 and 2.).
Representative firm borrows D1f in period 1 to make investment I1 that enable the firm to produce goods in period 2. The production technology in period 2 is given by
Q2 = ?(I1),
where Q2 and I1 denote, respectively, output in period 2 and investment in period 1.
Assume that there exists free international capital mobility and that the world interest rate, r*, is 10% per period (i.e., r* = 0.1). Finally, assume that the economyâ€™s initial net foreign asset position is zero (B0* = 0).
a) Compute the firmâ€™s optimal level of period-1 investment and period-2 profit.
b) State the maximization problem of the representative household and solve for the optimal levels of consumption in periods 1 and 2.