Problem set 8.
Question 1
Consider an economy where agents live for two periods and face an idiosyncratic
income process: yt ∈ {y 1 , y 2 , .., y N } for t = 1, 2, and 0 < y i < y i+1 , the income
PN
process is iid and the probability that income y i is realized is πi , with i=1 πi = 1.
Assume that there are many agents in the economy, and hence the law of large
numbers holds and the fraction of agents with income y i in either period is πi .
Agents maximize the following utility:
N
X

πi u(ci1 ) + β
i=1
N
X


πj u(ci,j
2 )
j=1
where ci1 is consumption in the first period when income in period 1 is y1 = y i ,
and ci,j
2 is consumption in the second period, when income in period 1 and 2 are
respectively y1 = y i and y2 = y j .
1. AUTARKY – Find the autarky solution, and express U A , the expected lifetime
value.
2. SELF INSURANCE – Now assume that agents can only save (not borrow), at a
rate R =
1
β.
State the agents problem, and describe the optimal consumption plan.
3. COMPLETE MARKETS – The agents have access to unrestricted insurance and
credit markets. Consider the following planner’s problem:
max
N
X
{ci1 },{ci,j
2 } i=1

πi (y1i − ci1 ) + β
N
X
s.t.
N
X


πj (y2j − ci,j
2 )
j=1

πi u(ci1 ) + β
N
X


πj u(ci,j
2 ) ≥ U0
j=1
i=1
Show that there is a U0 s.t. this problem leads to the same solution as the usual
agent’s utility maximization problem with the budget constraint
N
X

πi (y1i − ci1 ) + β
i=1
N
X
j=1
How does U0 compare to UA ?
Question 2
1


πj (y2j − ci,j
2 ) ≥0
A consumer faces the following problem:
max ∞ E0
{ct }t =0
s.t.
∞
X
β t U (ct )
t=0
ct + at+1 = (1 + r)at + yt
and
at+1 ≥ −φ
The income process yt ∈ Y = {y 1 , y 2 , ..., y N } (with 0 < y i < y i+1 ) follows a
Markov chain π(y 0 |y) = P rob(yt+1 = y 0 |yt = y). The consumer also faces a natural
borrowing constraint, i.e. φ =
y1
r ,
where r is the constant real interest rate. This
borrowing constraint states that the agent is allowed to borrow up to an amount
that she will be able to repay with probability one.
1. Can you think of a joint condition on the utility function u and on the transition
matrix π such that the natural borrowing constraint is never binding for the agent
(i.e. the solution of the problem above is always interior for all t)?
2. Suppose that the agent does not face any borrowing constraint. What joint condition on u and on the income process (i.e., on Y and on π) do we need to impose
to insure that at > 0 at every t?
Question 3
Assume that the consumer has CRRA period utility over consumption given
by
c1−γ
1−γ
u(c) =
with γ > 0, discounts the future at rate β ∈ (0, 1) and faces a constant interest rate
R. Assume that consumption is conditionally log-normal with mean Et ln(ct+1 ) =
µt and variance vt .
1. Show that the optimal consumption path follows:
Et ∆ ln(ct+1 ) =
1
1
ln βR + γvt .
γ
2
2. Based on the equation above, does the agent display precautionary saving behavior?
3. Suppose we tested the Permanent Income Hypothesis (PIH) by running the
regression
∆ ln(ct+1 ) = α0 + α1 yt + εt+1 .
2
where yt is past income and we found that α1 is significantly different from zero.
Based on your analysis above, could you say that this result means necessarily a
rejection of the PIH?
Question 4
Assume that labor income is the sum of two orthogonal components, a permanent component ytp which follows a martingale, and a transitory component ut
that is independently distributed over time:
yt = ytp + ut
p
+ vt
ytp = yt−1
Note that vt is the innovation to the permanent component, independently distributed over time. Assume that E(vt ) = E(ut ) = 0 and that the two shocks are
orthogonal, ut ⊥vt .
Use this specification of the income process to express the change in consumption as a function of the permanent innovation, vt and the transitory innovation,
ut . Use the equation for the change in consumption that we derived in the lecture
notes:
j
∞ r X
1
∆ct =
(Et yt+j − Et−1 yt+j ).
1 + r j=0 1 + r
Question 5*
Carefully explain why an efficient allocation in complete markets must satisfy:
∂Ui
∂ci (s)
∂Ui
∂ci (s0 )
=
∂Uj
∂cj (s)
∂Uj
∂cj (s0 )
Question 6*
Assume that there are no aggregate shocks and we are in an endowment economy,
P
i.e. i ω i (s) = ω for every s. Show that in a Pareto optimal allocation agents fully
insure themselves, i.e. ci (s) = ci (s0 ) for every s, s0 .
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