ec1545 pset3 International Macroeconomics Harvard

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Economics 1545: Problem Set 3
Professor Ken Rogoff
Due: Monday, November 9, 2015 at the start of class
I. Speculative Attacks on Exchange Rate
Suppose demand for money in a small open economy is characterized by:
mt − et = −ηe˙t + yt
where mt is the log of the money supply, et is the log of the exchange rate, and yt is the log of output. For
notation, e˙t =
de
dt , and we assume that the second derivative of et with respect to time is zero (i.e. changes
in the exchange rate are linear). In this economy, yt grows at rate θ.
The central bank controls the money supply, which is backed by a portfolio of domestic bonds and reserves.
Through this balance sheet relationship, we have the size of the money stock (where the following variables
are in levels, not logs):
Mt = Bt + EtRt
where Bt is the level of domestic bonds and Rt is the level of reserves, and Mt and Et represent the money
stock and exchange rate (both in levels).
Assume that the central bank is required to issue domestic bonds at rate µ (and this takes priority over
everything else), such that:
B˙t
Bt
= b˙t = µ
Moreover, assume that µ > θ.
a. The central bank starts by trying to enforce a fixed exchange rate, such that Et = E (i.e. et = e¯). At
what rate is the central bank changing reserves (i.e. what is r˙t)?
b. If a country is growing very quickly, such that θ is big, will the country be accumulating or losing reserves?
If a country is issuing high debt, such that µ is big, will the country be accumulating or losing reserves?
c. Write the equation governing the shadow exchange rate when the central bank runs out of reserves.
d. Assume that µ is very large, and graph the shadow exchange rate. Why does a collapse occur when
e˜T = e¯?
e. Graph the central bank’s reserve holdings throughout the scenario described above.
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f. Thus far we have assumed a passive central bank that gets attacked. Assume the central bank actually
has a utility function that cares about two things: the level of its reserves, and whether it actually enforces
its mandate. Assume that the central bank gets flow utility from holding reserves (discounted at rate ρ) and
gets flow disutility of Γ for breaking its mandate. Thus, assuming it gets attacked at time T (and loses all
its reserves at that point), its utility function at t = 0 looks like this:
U =
∫ T
0
e−ρtRtdt−
∫ ∞
T
e−ρtΓdt
Given this formulation, find the total utility for the passive central bank. You may assume that the bank
starts with reserves R¯, and for notational simplicity you may assume that Rt declines at rate λ (i.e. r˙t =
λ < 0), such that Rt = R¯e
λt.
g. Assuming the speculators follow the plan above, show that the central bank always has a profitable
deviation from the passive policy.
h. Now we will compute the equilibrium of this game. Assume that the central bank cannot change policy
on an instantaneous basis (perhaps the governors of the central bank only meet once a week). Thus, the
central bank instead faces the choice of changing its policy at one of the discrete intervals: {..., T −∆, T, T +
∆, T + 2∆, ...} where ∆ > 0. The bank’s analysts compute the optimal policy, and they decide that, based
on the central bank’s utility function, the peg should be actively abandoned at T −∆ instead of the previous
policy of passively abandoning at T . What will the shadow exchange rate be at T − ∆? Assuming the
country starts with extremely large reserves, how does it compare to the shadow exchange rate in part c at
time T? Finally, what is the intuition for the difference?
i. Now return to the speculators: what will they do at time T −∆, after they forecast the central bank’s
response? Specifically describe what happens to reserves at time T −∆. Will they go to zero?
j. How will the central bank respond in turn? Assuming that speculators plan to attack at T −∆, give a
simple argument for abandoning the peg at T − 2∆. Will this work? Algebraic answers are not required.
k. Assuming that speculators plan to attack at T −∆, give a simple argument for abandoning the peg at
T instead. Provide the utility condition that will drive this, and draw the graph of reserves over time if the
central bank deviates to abandoning at T .
l. Is either of the two strategies above (abandoning at T − 2∆, abandoning at T ) a Nash equilibrium?
Explain why or why not, and describe informally what kind of equilibria this setup could support.
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II. Liquidity Traps
Consider a closed economy in which there is no investment. The representative agent has exogenous income
stream Yt and has utility given by:
U =
∞∑
s=t
βs−t log(Cs)
This economy is governed by the cash-in-advance constraint. Moreover, we will assume (for the moment)
that prices are flexible. Since the only two sources of final demand are consumer demand, Ct, and government
spending, Gt, goods market clearing requires that:
Yt = Ct +Gt
Government spending does not enter private utility. Moreover, both the government and private sector have
cash-in-advance constraints. Just as PtCt ≤ MPRt where MPRt is private holdings of money, we also have
PtGt ≤MGt , where MGt is government holdings of cash. Money market equilibrium of course implies that:
MPRt +M
G
t ≤Mt
where Mt is the total money supply at time t.
The Euler equation derived from the representative agent’s utility maximization problem is:
u′(Ct) = β(1 + rt+1)u′(Ct+1) ∀ t
where rt+1 is the interest rate on real bonds between period t and period t+ 1.
a. Write the equation that relates the nominal interest rate it+1 (on a one-period bond that pays off a
non-indexed cash amount in period t+ 1) in terms of Yt, Gt, Pt, Yt+1, Gt+1, Pt+1.
b. Assume that Gs is fixed for the current and future periods (Gs = G¯ ∀ s), assume that M is fixed for all
periods after the present period Ms = M¯ ∀ s > t, and assume that income Ys = Y is constant ∀ s > t but
initial income Yt > Y .
1. Rearrange the equation from a. to depict a liquidity trap in this economy.
2. Aside from being unable to lower interest rates, what variable(s) can the central bank not change in
this liquidity trap? (Hint: remember that prices are flexible, while output is exogenous.)
c. Assume the economy is in the liquidity trap discussed above.
1. Can a permanent rise in G¯ for all periods (including the current one) solve the problem?
2. Can a temporary rise only in the current period solve the problem?
3. Finally, what happens if the government plans to raise current period G only (future government
spending will drop back to G¯), but no one believes the government and everyone thinks the rise will
be permanent for political economy reasons?
d. Until now, we have assumed that money cannot pay a negative interest rate. Let us now suppose that
currency is electronic and there is no paper currency so there is no zero bound on the nominal interest rate,
because electronic currency can have a negative interest rate. How would this affect your answers to part
b.? Are negative prices possible?
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