Economia V; Instructor: Marco Del Negro

Problem set 6

The setup of the problem is as follows:

• the economy has two periods, 1 and 2.
• there is only one agent in the economy, and her initial resources at the beginning of period 1 are equal to Y1=2.
• Assume that the agent owns the firm directly. The production function is Cobb-Douglas, and N (labor) is fixed to 1, $\alpha$ is equal to .5, and A is equal to 2:
  
  $$F(K,N)=2 K^{\frac{1}{2}}$$
  
  

• capital depreciates completely at the end of the period ($\delta=1$)
• capital mobility is "quasi-perfect": the agent can borrow from "Merrill Lynch" at the world interest rate r^w, which is 0%, only to finance investment (K2), but cannot borrow from Merrill Lynch in order to finance consumption (in other words, the agent is liquidity constrained, in the sense that $C1 \leq Y1$). Of course, the agent can always choose to lend to Merrill Lynch at the world interest rate, if she wants to.
• the utility function is logarithmic, that is:
  
  $$U(C1,C2)=\ln{C1}+\beta\ln{C2}$$
  
  
  with $\beta=\frac{1}{1+r^w}$.

[hint: solve the problem as if the constraint $C1 \leq Y1$ did not exist. Find the solution for C1 and then check whether $C1 \leq Y1$. If this is the case, then you are all set]
a) Knowing that Merrill Lynch is happy to finance any level of investment, compute how much capital K2 is invested in the country in period 1. [hint: capital K2 will have to be such that MPK=1+r^w]
c) Write the budget constraint of the agent is the first and in the second period (call D the amount of debt borrowed on international financial markets -Merrill Lynch- in period 1).
d) Solve the agent's problem and find the optimal consumption in periods 1 and 2 (C1 and C2). [hint: from b) you already know what K2 is. Therefore C1 and C2 can be found simply applying the formulas from permanent income.]
e) Solve the problem with Y1=.5, and show that it is still the case that in equilibrium MPK=1+r^w.