Two-period SOE Exercise

In this exercise we will set up and solve a simple 2 period, deterministic small open economy modelwith investment in physical capital and government expenditure.

Consider a small open economy populated by a large number of households with utility function

U = u(c1) + βu(c2)

where β is the subjective discount factor. Suppose the economy faces constant world interest rate r∗

at which households can borrow abroad. Output is produced according to the production function

y = Af(k)

where A is total factor productivity and k is physical capital. In period 1 the economy has no debt(b1 = 0) and a given level of initial capital k1. The government uses exogenous public consumptiong1, g2 in period 1 and 2 respectively. Capital depreciates at rate δ ∈ [0, 1].

1. Write down the maximization problem of the economy

2. De�ne an equilibrium of the economy

3. Characterize the equilibrium (that is, derive the �rst order conditions)

Assume β(1 + r) = 1 and u(c) = log c (you can also assume f(k) = kα if you want).

4. Solve for the equilibrium (that is, express consumption, capital and investment in both periodsas functions of parameters/ given variables only). What are the equilibrum trade balance,current account and debt levels?

You do not need to solve this all the way out, just far enough to be able to answer the followingquestions.

5. What is the e�ect on the equilibrium, including the international accounts, of a temporaryincrease in public spending (an increase in g1)? What about an increase in g2?

6. How does the current account adjust to a temporary productivity shock (a change in A1)?

7. If time: What is the sign of the response to a change in A2?

1