1
Problem set 4
This homework is due on April 19th (Wednesday) in class. For Q2 and Q3, you have to submit your
matlab codes to Yu.
1) (Uncertainty) Consider an economy in which there is a representative consumer who lives forever.
In each period t = 0, 1, ..., one of two random events occurs, st = θ1 or st = θ2 . A state is
represented by s. At t = 0, the initial state is s0 and a stationary Markov process given by a 2 × 2
matrix with elements πi,j = prob(st+1 = j|st = i) governs the probability of future states. Let
πt (st ) be the induced probability of a particular event history. The representative consumer has the
utility function,
X
β t πt (st )u(ct (st ), 1 − lt (st )).
st ∈S t
Here S is the all possible event histories and 0 < β < 1. Assume the nice and usual properties
about u. The consumer is endowed with one unit of labor in each state and k¯0 units of capital in
the initial state. Feasible consumption/investment plans satisfy the following feasibility:
ct (st ) + kt+1 (st ) − (1 − δ)kt (st−1 ) ≤ θt (kt (st−1 ))α (lt (st ))1−α , ∀st ∈ S t ,
where θt takes on one of two values as determined by the current event, where θ1 < θ2 .
a) Define an Arrow-Debreu (AD) equilibrium for this economy.
b) Define a sequential market (SM) equilibrium for this economy. Carefully state a proposition or
proporsitions that establish the essential equivalence between the objects in the AD equilibrium
and those in the SE equilibrium.
c) Specify the dynamic programming (DP) problem that such an allocation must satisfy by writing
down its Bellman’s equation. Suppose that you have a solution to this DP problem. Explain
carefully how you could calculate the values of the variables in the definition of an AD equilibrium
in any state. Explain carefully how you could calculate the values of the variables in the definition
of an SM equilibrium in any state.
d) Consider an economy with two types of consumers, rather than one, but otherwise identical to
that in parts (a)–(c). Define a SM equilibrium.
e) Does the equilibrium allocation in part (d) solve a DP problem? Carefully explain why or why
not.
2) (Life-cyle model) Solve for the life-cycle model where agents live up to 90 years (assume they
are born at 16) and have the wage profile (the data are posted on the website: epsilon − f ile.txt
for wages). Agents are assumed to have standard log utility over consumption (u(c) = log(c)). Set
a real interest rate to .0525, and a discount factor to .98. Make sure that your accuracy criteria is
2
.0001. Solve this problem by the three methods that we discussed in class. Plot age profiles for
consumption and assets.
3) (Life-cyle model) Now, we extend the life-cycle model we used in the previous problem. Solve
for the steady state of an economy where agents live up to 90 years (assume they are born at 16)
and have the wage profile (again, posted on the website). Each period there is an inflow of new
borns of measure 1. In other respects, the economy is a standard growth model. Let the agents
have standard per period preferences over consumption (log utility). Let the labor share of output be
.67, the depreciation rate .05, the wealth to output ratio 4, and the discount rate be .99. Make sure
that your accuracy criteria is .0001. Solve the problem of the agents by the ‘backward recursion’
(Method 2). Plot age profiles for consumption and assets.
4) (Baseline RBC model) Suppose the standard stochastic business cycle model with two inputs of
production capital and labor.
Yt = ezt Ktθ Nt1−θ .
The shock z follows AR(1) process: zt+1 = ρzt + , is i.i.d. random variable. Per period utility
function is log(c) + α(1 − n).
a) Write down a Competitive equilibrium in a recursive form.
b) Write down equilibrium conditions to characterize equilibrium. (at this stage, you don’t need to
characterize equilibrium).