ECOM130
Problem set 4
Answer 3 questions.
Weights: All questions carry the same weight.
1. Assume that
u(c) =
1
1− αc
1−α, α > 0.
Assume that Rt is independently and identically distributed and is such that ER
1−α
t <
1/β. Consider the problem
maxE
∞∑
t=0
βtu(ct), 0 < β < 1,
subject to At+1 ≤ Rt(At − ct), A0 > 0 given. Here, At are savings and Rt is a rate
of return. It is assumed that ct must be chosen before Rt is observed. Show that the
optimal policy function takes the form ct = λAt and derive an explicit formula for λ.
(Hint: consider a value function of the general form v(A) = BA1−α, for some constant
B.)
2. Consider the following problem. A worker’s per-period utility u depends on the
amount of market-produced goods consumed, c1t and also on the amount of home-
produced goods, c2t (for example, entertainment, leisure). In order to acquire market-
produced goods, the worker must allocate some amount of time, l1t, to market activ-
ities that pay a salary wt, measured in terms of consumption good. The worker takes
wages as given and beyond the worker’s control. There is no borrowing or lending. It
is known that the market wage evolves according to the law of motion wt+1 = h(wt).
The quantity of home-produced goods depends on the stock of ”expertise” that the
worker has at the beginning of the period, which we label at. This stock depreciates
at the rate δ and can be increased by allocating time to nonmarket activities. To
summarize the problem, the individual agent maximizes
∞∑
t=0
βtu(c1t, c2t), 0 < β < 1,
subject to
c1t ≤ wtl1t
c2t ≤ f(at)
1
at+1 ≤ (1− δ)at + l2t
l1t + l2t ≤ l
wt+1 = h(wt)
a0 given.
(a) Formulate the problem as a dynamic programming problem.
(b) What additional assumptions you have to make so that the solution to the Bell-
man equation is unique?
3. Consider a deterministic model with a stock of factories and houses, kMt ≥ 0 and
kHt ≥ 0. While it takes one period to complete the construction of houses, it takes
four periods to complete the construction of factories. Once the construction of a
factory has started, it has to be completed. A house of size xHt whose construction
started in period t becomes a part of the housing stock in period t+ 1. We denote by
sjt a factory of size s that in period t is j periods away from completion. (A factory of
size s whose construction started in period t is thus in period t four periods away from
completion and is denoted by s4t, while a factory that is almost complete in period
t and will become a part of the capital stock in period t + 1 is thus denoted by s1t.)
The stock of factories evolves as kM,t+1 = (1−δM )kMt+s1t, while the stock of houses
evolves as kH,t+1 = (1 − δH)kHt + xHt. The construction process of factories is such
that a fraction φj of the factory size must be built during the construction phase j.
Factories and houses are owned by a representative consumer with a life-time utility
function ∞∑
t=0
βtu(ct, kHt)
and a per-period resource constraint
ct = F (kMt)− (φ1s1t + φ2s2t + φ3s3t + φ4s4t)− xHt,
where ct ≥ is non-housing consumption and F (.) is a production function.
(a) Suppose that the consumer maximizes the life-time utility. Write the maximiza-
tion problem in a recursive form.
(b) Impose a minimum number of assumptions so that the Bellman equation has a
unique fixed point and a convergence to it is guaranteed from any initial guess.
(c) Impose a minimum number of assumptions so that you can apply the Benveniste-
Scheinkman theorem.
(d) Obtain the first-order and Benveniste-Scheinkman conditions.
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4. Consider the model with distortionary income taxes, a lump sum tax, and government
spending presented in the lecture on recursive competitive equilibrium. (a) Define the
RCE for this economy, including all the conditions and functions that are part of this
equilibrium. (b) Is the equilibrium efficient? What is the optimal tax policy? (c)
Explain how to characterise the equilibrium numerically. Are these methods reliable?
5. There are two agents i ∈ {1, 2} who live forever and get stochastic endowments (e1, e2)
every period. Denote the state of the world at time t as st ∈ {1, 2}. If st = 1, the
endowments are (e, 1− e), and if st = 2, (1− e, e), where e > 1/2. The realization of
st is i.i.d. over time. Denote the probability that agent 1 gets the high endowment
(e1 = e) as pi, so the probability that agent 2 gets the high endowment is 1 − pi (do
not assume that pi = 1/2). Each individual has preferences
E
∞∑
t=0
βtu(ct
with the standard assumptions. Assume that the market opens before the realization
of s0.
(a) Define the Arrow-Debreu equilibrium. Denote the price of Arrow-Debreu securities
following history st, at time 0, as qt(s
t). Obtain an expression for consumption allo-
cations, and the equilibrium price and quantities of securities bought/sold, assuming
that ∞∑
t=0
∑
st
qt(s
t) = 1.
(b) Define a recursive equilibrium. Denote the price of Arrow securities as Q(s′ | s).
Obtain an expression for consumption allocations, and the equilibrium prices and
quantities of securities bought/sold. Relate the Arrow security prices to that of Arrow-
Debreu securities. Is Q(.) contingent on s, today’s state?
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