Sciences Po – Graduate Macroeconomics 2
Problem set 1
Question 1
Formulate the problem of the central planner, who is maximizing the expected
present discounted value from consumption of the representative household. The
representative household lives forever, discounts utility at rate β ∈ (0, 1), and has
an instantaneous utility function of U (C) = ln(C), and the population is normalized to 1. The production of the economy in period t is Yt = Zt F (Kt , 1) = Zt Ktα ,
where α ∈ (0, 1), Kt is the available capital stock, and Zt is a stochastic productivity measure. The depreciation rate of capital is 1, i.e. capital fully depreciates in
each period. Use the Lagrangian method to derive the optimal consumption plan.
Hint: Follow these steps:
1. Formulate the planner’s problem.
2. Combine the FOCs to get the Euler equation.
3. Denote by st the saving rate in period t: Ct = (1 − st )Yt , and use the complementary slackness constraint (i.e. that the resource constraint is satisfied with
equality) to express capital, Kt+1 as a function of st and Yt .
4. Use these to get a simplified version of the Euler equation, which depends
on st , st+1 , Kt , Zt , Zt+1 , α, β.
5. Show that the optimal saving rule is st = αβ.
Question 2
Consider the following problem:
max
∞
{ct }t=0
∞
X
β t ln ct
t=0
subject to:
P∞
t=0 ct
≤ k0
ct ≥ 0 for ∀t ≥ 0
k0 is given
Use the Lagrangian method to derive the optimal consumption plan for the cake
eating problem. The name of the problem comes from the first constraint: there is
a certain amount of capital, that is not productive and can be consumed over time.
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Hint: Follow these steps:
1. Set up the Lagrangian.
2. Combine the FOCs to get the Euler equation.
3. Use the complementary slackness constraint (i.e. that the resource constraint
is satisfied with equality).
4. How much of the cake is left at time T ?
5. Is the transversality condition satisfied?
6. What if the resource constraint is not satisfied?
Question 3
Consider the main maximization problem of Lecture 1. How did we show that the
solution to the centralized and decentralized problem is the same? Why does the
zero profit result hold at the firm’s optimum? Why is this an important result?
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