ec1545 pset1International Macroeconomics Harvard

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Economics 1545: Problem Set 1
Professor Ken Rogoff
Due: Monday, September 21, 2015 at the start of class
I. Aggregation and Risk-Sharing with Complete Markets
Consider a stochastic two-period endowment economy in which each agent i, i ∈ {1, ..., N} has the following
utility function:
U =
(
(Ci1)
1−ρ
1− ρ
)
+ E
(
β(Ci2)
1−ρ
1− ρ
)
The agent faces the following budget constraints in the first and second period, respectively:∑
s
p(s)
1 + r
Bi2(s) = Y
i
1 − Ci1
Ci2(s) = Y
i
2 (s) +B
i
2(s)
where Y i2 (s) and C
i
2(s) respectively denote agent i’s output and consumption in period 2 and state s. B
i
2(s)
represents the agent’s holdings of a bond that pays off one unit in period 2 and state s. Finally, the price
of the state s bond in period 1 is p(s) and ρ is agent i’s level of absolute risk aversion. Let pi(s) denote the
probability of state s. Assume that the total population of the economy is N .
a. Write the first-order Euler condition for an individual’s consumption that relates Ci1, C
i
2(s), p(s), pi(s),
and 1 + r (the real interest rate on a riskless bond).
b. Write the first-order Euler condition for per capita consumption, defining
C1 ≡ 1
N
N∑
i=1
Ci1
C2(s) ≡ 1
N
N∑
i=1
Ci2(s)
Does your answer require all agents to have identical levels of consumption in each period, or just identical
growth rates of consumption?
c. The last section defines a country-level Euler equation. Now assume that there are two countries, both
with the same utility functions (although the home country has parameter ρ and the foreign country has
paramter ρ∗). Moreover, assume that the two countries have different output streams — one country has
first period income Y1 and second period income Y2(s) while the other country has first period income Y
∗
1
and second period income Y ∗2 (s). Finally, assume that there are complete state-contingent markets in this
global economy. Show, analytically, in what sense consumption risk is shared. It is sufficient to show your
result in terms of relating consumption growth rates across countries (log(C2/C1) and log(C
∗
2/C
∗
1 )).
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II. The Low Interest Rate Puzzle
Consider the complete markets model as in Problem 1 above, with the same utility function:
U =
(
C1−ρ1
1− ρ
)
+ E
(
βC1−ρ2
1− ρ
)
a. Show that the Euler condition can be written as
(1 + r)−1 = E
{
β
(
C2
C1
)−ρ}
On page 313 of Obstfeld and Rogoff, it is shown that if consumption is lognormally distributed, then the
above equation can be approximated as
r ≈ log(1 + r) = ρE
{
log
(
C2
C1
)}
− ρ
2
2
Var
{
log
(
C2
C1
)}
− log(β)
You do not need to derive this approximation, you may take it as given. Use this to answer the remaining
questions.
b. Explain why it is a puzzle that real interest rates are often 1% or less for long periods of time, assuming
ρ = 2 and β ≈ 1. Would the puzzle necessarily be as great if, instead of US data, one used consumption
data for an emerging market with much higher consumption volatility?
c. How high would the variance of consumption have to be to get an interest rate under 1%? Assume ρ = 2
and β ≈ 1.
d. Suppose instead of a simple lognormal distribution for consumption, consumption is driven by two shocks:
a lognormal shock (as above), and a rare disaster shock. (In most periods, the rare disaster is zero, but on
occasion there is a very large negative shock, per the Poisson distribution.) Can you intuit how this might
affect the equilibrium interest rate?
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III. Diamond-Dybvig Model of Bank Runs
Consider a simple closed economy model is which there are three dates, 0, 1, 2 and a large number of
consumers, each endowed at time 0 with one unit of a consumption good. Consumption can take place either
at date 1 or at date 2. Consumers only learn at the beginning of date 1 whether they are an “impatient”
consumer, who only gets utility from date 1 consumption, or a “patient” consumer who is indifferent between
period 1 and 2 consumption. λ1+λ of consumers turn out to be impatient and
1
1+λ turn out to be patient
(everyone knows these probabilities in advance). Impatient consumers turn out to have utility function:
u(c1) = log c1
where c1 is an individual’s consumption in period 1 (omitting the subscript i where it is obvious). Patient
consumers have utility function
u(c1, c2) = log(c1 + c2)
where c2 is consumption at date 2.
At time zero, the individuals have a choice between storing their good for later use (with no depreciation)
or investing their money in a bank. If they invest at bank, they must decide at time 1 whether to withdraw
their money (as permitted by their deposit contract) or leave it in. The bank can hold some fraction of the
deposits as cash (with no return, and thus gross return 1) and invest the rest in an illiquid project with
gross return R > 1, payable only at date 2. If a bank tries to liquidate its illiquid investments prematurely
at time 1 (perhaps to deal with a bank run), it gets only return ηR, where η < 1.
a. Discuss why the optimal bank contract solves the following:
max
c1,c2
λ
1 + λ
log c1 +
1
1 + λ
log c2
subject to
λc1 +
c2
R
≤ 1 + λ
(Hint: this is very closely related to maximizing an individual’s expected utility.) What assumption does
this make about the banking market? What is the interpretation of this second constraint?
b. Suppose first that the bank can observe consumers’ types at time 1, and will only let truly impatient
consumers withdraw at time 1. What is the optimal contract? (That is, what values of c1, c2 would the bank
offer to solve the problem in part a. above?) How does this compare to the agents’ consumption choices
without a bank?
c. Now suppose that banks cannot observe agents type. Why is it necessary for any contract to obey the
“incentive compatibility constraint” u(c1) ≤ u(c2)? A verbal answer will suffice.
d. Assume banks must deal with a “sequential-servicing constraint” which is a jargonistic way of saying that
they must serve customers according to the order they stand in line. Why is the arrangement above subject
to bank runs in the first period in which even patient consumers try to withdraw their money? Very briefly,
why would a government-financed deposit insurance scheme prevent bank runs? Would such a scheme need
to be able to draw on funds outside the banking system to work?
e. Is there any policy the bank could invoke to prevent runs without the help of the government?
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f. Up till now, we have been dealing with theoretical banks facing an infinite number of customers. Now
consider a bank that serves N customers (each with one dollar), and to keep the math simple, assume
λ
1+λ =
1
1+λ = 0.5. A regulator tells this bank that they have to make promises that they can hold 98% of
the time (based on the sample mix of patient and impatient customers) — the bank can only default in the
extreme 1% case that there are too many patient or too many impatient customers.
For instance, if N = 2, then from the binomial distribution, we know that the probability that both are
impatient is 0.52 = 0.25 and the probability that both are patient is 0.52 = 0.25 also, and so the bank would
have to hold both depositors’ money in the liquid asset and offer c1 = c2 = 1. But if N = 7, then the
probability that all 7 are impatient or are patient is 0.57 ≈ 0.008, and so the bank can hold at least one
dollar in each of the liquid and the illiquid asset without violating the regulator’s constraints.
Under this new rule, assume N is large (e.g. N > 50) but not infinite. Setup the optimization problem for
the best fixed contract the the bank can offer its clients as a function of N , but you do not need to solve.
Hint: utilize the normal approximation to the binomial distribution.
g.Intuitively, do customers do better or worse as N →∞?
h. Despite the careful calculations of the bank staff, why might the bank default more than 2% of the time
in equilibrium?
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