### August 2014

```Question 1 (100 total points)
I. (70 points) Consider the following consumption-savings problem:
max
E0
∞
{ct }t=0
∞
X
β t u(xt , ct )
t=0
subject to:
xt+1 = Rxt − ct + yt+1
where R > 0 , xt and ct are scalars, yt+1 is i.i.d. with distribution F and mean y¯,
and x0 is given. Note the timing convention that ct is chosen before yt+1 is realized.
(a) Write down the Bellman equation for the problem.
i. Under what conditions will there be a unique solution to the Bellman equation?
ii. Under what conditions will the value function be increasing?
iii. Under what conditions will the value function be differentiable?
(b) Find the first order condition and envelope condition, and derive the Euler
equation.
(c) Suppose that preferences are given by:
1
u(x, c) = − (c − c¯)2 ,
2
for some constant c¯ > 0, and suppose that income yt+1 is normally distributed
N (¯
y , σ 2 ). Find explicit expressions for the value function V (x) and the optimal
decision rule c(x).
(d) Now suppose that the consumption-savings problem depicts the decision problem
of a representative agent, where R is the gross rate of return on a risk-free
bond (in zero net supply) and yt is the endowment process. Define a recursive
competitive equilibrium in this environment.
(e) Continuing with the special case of preferences as above, find the equilibrium
gross interest rate R and the price of a claim to the endowment process.
II. (30 points) Consider a cash-in-advance model in which there are two types of goods:
c1 requires money Mt to purchase, while c2 can be purchased on credit. The two
goods are technologically equivalent, as the endowment et can be converted one-forone into either of them, so et = c1t + c2t . Suppose that et follows a Markov process
with transition density Q(e0 |e). A representative agent in this economy thus solves:
max
{c1t ,c2t ,Mt }
E0
∞
X
β t u(c1t , c2t )
t=0
subject to the technological constraint, the budget constraint:
Pt c1t + Pt c2t = Pt et + Mt − Mt+1 ,
Pt c1t ≤ Mt .
(a) Write down the Bellman equation for the representative household and find the
optimality conditions.
(b) Consider a steady state equilibrium in which the endowment is constant et = e,
the money supply grows at a constant rate: Mt+1 = µMt , and real balances
Mt /Pt are constant. What is the minimal level of µ that will support a steady
state monetary equilibrium?
Question 2 (100 total points)
Part A. (50 points) Consider the following endowment economy. There are two periods
t = 1, 2 and two agents i = 1, 2. Let yti denote household i’s endowment in period t. Endowments in both periods are known with certainty. Agents have utility functions U i (ci1 , ci2 ),
where cit is agent i’s consumption in period t. Assume each U i is strictly increasing.
a) Define a sequential markets equilibrium.
b) Define an Arrow-Debreu equilibrium.
c) Show that a sequential markets equilibrium allocation is an Arrow-Debreu equilibrium
allocation.
d) Show that an Arrow-Debreu equilibrium allocation is Pareto optimal.
Part B. (50 points) Consider the following economy. There are identical households
with utility function
∞
X
β t log(ct )
t=0
where 0 < β < 1 and ct is consumption. Output yt is produced (exogenously) on a farm,
and shares of this farm are traded as a perfectly divisible security each period in competitive
markets. Households therefore face the following budget constraint in each period:
ct + pt at+1 ≤ (pt + yt )at ,
where pt is the share price in period t and at is the number of shares the household brings
into period t. Let a
¯ denote the total number of shares outstanding.
a) Define an equilibrium for this model economy.
b) Derive a closed form solution for the equilibrium stock price pt (i.e., an equation that
gives the equilibrium pt as a function of the model primitives β and (ys )∞
s=0 ).
```