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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,
is equal to .5, and A is equal to 2:
- capital depreciates completely at the end of the period (
)
- capital mobility is "quasi-perfect": the agent can borrow from
"Merrill Lynch" at the world interest rate rw, 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
). 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:
with
.
[hint: solve the problem as if the constraint
did not exist.
Find the solution for C1 and then check whether
.
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+rw]
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+rw.
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Marco Del Negro
2000-02-22