Exercise session 4
Course: Mathematical methods in Economics
Lecturer: Dmytro Vikhrov
Date: March 12, 2013.
Problem 1 (wealth transfer over time)
Suppose that a consumer lives for two periods with instantaneous utility in each period
equal to U(ct). The consumer can transfer her wealth across periods by buying bonds b in t = 1
and selling them at t = 2. The interest rate paid on bonds is R and the discount factor is β.
Consumer’s flow of income in respective periods is y1 and y2.
1. Explain the meaning of discount factor β and bonds b.
2. Carefully setup the maximization problem.
3. Derive the Euler equation for consumption. Explain how the consumer decides to shift
consumption over time.
4. For U(ct) = log(ct) derive the optimal consumption sequence, {c∗
1, c∗
2} and bond holdings
b∗. Interpret your findings.
5. Define the competitive equilibrium and find the market clearing R∗. Compute ∂R∗
∂β , ∂R∗
∂y1
,
∂R∗
∂y2
.
Problem 2 (pay-as-you-go social security system)
Consider the two-period model from Problem 1. Assume that workers get income y while
young and no income while old. The government taxes the young generation at per-unit tax τ
and distributes the proceeds amongst the old generation. Each population cohort grows in size,
Lt+1 = (1 + g)Lt.
1. Draw a graph to depict the intuition of the overlapping generations.
2. Setup the government budget constraint. In what cases can the government run a deficit
of the PAY-GO pension system. What tools does the government have to reduce the
deficit?
3. Setup the consumer maximization problem.
4. Derive the optimal consumption sequence {c∗
1, c∗
2}.
5. Derive ∂c∗
∂g and ∂c∗
∂τ and interpret your findings.
6. In what cases does the consumer wish to opt out of the mandatory PAY-GO system?
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Problem 3 (funded social security system)
Assume the two period setup as in the previous problem. At t = 1, The consume has a
choice to invest in bonds b that yield an interest rate R at t = 2 or participate in the government
funded scheme. In this scheme the consumer is taxed at the per-unit rate τ and the proceeds
are invested at the rate r. Setup the consumer maximization problem and derive conditions for
the consumer to participate in the government funded scheme.
Problem 4 (infinite period setup)
Recall the setup of problem 1, however now assume that the consumer is infinitely lived.
Setup the maximization problem and derive the Euler equation for consumption.
Problem 5 (dynamic programming)
Setup the maximization problem of an infinitely lived agent in the recursive form.
1. Carefully define the state and control variables.
2. Write down conditions under which the policy function y = π(x) maximizes the value
function.
3. Find the Euler equation for consumption.
4. Using the Envelope theorem, establish that V (x) = ∂U(x,y∗)
∂x . State the transversality
conditions.
Problem 6 (DP example 1)
Consumer’s maximization problem is given by:
max
{kt, ct}
∞
t=0
βt
log(ct)
s.t.: kt+1 = kα
t − ct, k(0) > 0.
1. Setup the Bellman equation and take the first - order conditions.
2. Somebody tells you that the policy function is of the form π(x) = Axα. Find the parameter
A.
Problem 7 (DP example 2)
Consider the following DP problem:
max
{kt, ct}
∞
t=0
βt
log(ct)
s.t.: ct = (1 + r)at − at+1 + w, a(0) > 0.
1. Setup the Bellman equation and take the first - order conditions.
2. Guess and verify that the solution is a policy function π(x) = A + Bln(x).
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