ec1545 pset2 International Macroeconomics Harvard

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Economics 1545: Problem Set 2
Professor Ken Rogoff
Due: Wednesday, October 14, 2015 at the start of class
I. Dornbusch Model of Exchange Rate Overshooting
Consider a small open economy, such that it takes the world interest rate as given. The home nominal
interest rate i depends on the (fixed) foreign interest rate i∗ and the expected rate of depreciation of the
exchange rate:
it+1 = i
∗ + et+1 − et
a. In a sentence or two explain in what way the above “uncovered interest rate parity” equation assumes
that there is no risk, or alternatively, that investors are risk neutral.
Continuing with the setup, real money demand depends on the nominal interest rate i and real output y:
mt − pt = −ηit+1 + φyt
Moreover, it is assumed that the money market and the bond market are always in equilibrium, so the above
two equations always hold at any point in time.
Aggregate demand depends on the real exchange rate:
ydt = y + δ(et + p
∗ − pt − q), δ > 0
where the real exchange rate q is defined by
qt ≡ et + p∗ − pt
and q is the long run equilibrium real exchange rate. Assume q is fixed. All variables are in logs.
If the domestic price level p were free to move instantaneously, then the goods market would always clear.
But a critical assumption here is that the price of goods is sticky, so that in the short run there can be
excess demand or supply in the goods market. We will make the Keynesian assumption that absent market
clearing, the actual level of output is demand determined and equal to ydt .
Instead of adjusting instantaneously, prices are assumed to adjust gradually both in response to excess
demand ydt − y and to a forward looking inflation adjustment term p˜t+1 − p˜t:
pt+1 − pt = ψ
(
ydt − y
)
+ (p˜t+1 − p˜t)
where p˜ is the price level that would clear the goods market given the actual exchange rate e:
p˜t ≡ et + p∗ − q
As a side note, an alternate way to incorporate forward looking price adjustment into the model would be to
assume prices adjust to current excess demand and to the level of inflation that would pertain if prices were
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completely flexible. This approach gives similar qualitative answers, but it is more cumbersome analytically.
b. Normalizing the various constant terms to zero p∗ = i∗ = y = 0, show that equilibrium in the above
dynamic exchange rate and price model is governed by the following two dynamic (difference) equations:
∆qt+1 ≡ qt+1 − qt = −ψδ(qt − q)
∆et+1 ≡ et+1 − et = et
η
− (1− φδ)qt
η
−
(
φδq +mt
η
)
c. Consider a situation in which the money supply is initially assumed fixed forever at m and then jumps
unexpectedly to m′, a one-time permanent increase in the money supply. Recall that if the economy is
initially in steady state then prior to the money supply increase p = m and e = m+ q. In addition, the long
run neutrality of money implies that p′ − p = e′ − e = m′ −m where prime superscripts denote the new
long run steady state. Under what conditions does the (log) nominal exchange rate initially overshoot its
long run equilibrium in the sense that it jumps more than m′ −m in response to the initial shock to money
demand. You can provide a graphical answer.
d. Suppose instead of there being a once and for all increase in the money supply, there is a once and for all
unanticipated increase in the equilibrium real exchange rate from q to q′. Does the exchange rate overshoot
its long run equilibrium in this case?
e. What is the half life of shocks to the real exchange rate in this model if the only source of shocks are one
time permanent changes in the money supply? That is, suppose an unexpected monetary shock causes the
real exchange rate to move around. After such a shock, and assuming no more shocks, how many periods
does it take for half the shock to the real exchange rate to damp out?
f. Suppose that the government executes a surprise increase in the rate of money growth from zero to
mt+1−mt = g. Can you show graphically what happens to the real exchange rate in this case? Is there any
way for the monetary authorities to change the level of the money supply, while at the same time announcing
a new rate of money supply growth, so that output is unaffected? Does your answer suggest that there might
be ways for high inflation economies to reduce inflation painlessly? Does your analytical answer make sense
in practical terms (i.e. is it a credible policy)?
Hint: This question is difficult. But in this case, it is easier to use e−m instead of the nominal exchange rate
on your axes. You may also wish to consult Section 9.2 of Obstfeld and Rogoff for information on solving
the Dornbusch model when the money supply is not constant.
II. Debt Overhang and Debt Forgiveness
Consider a small open economy that inherits a very large amount of debt, D, which is scheduled to be repaid
in the second period. The country is inhabited by a representative agent with utility function
U = logC1 + β logC2
where Ci is consumption in period i. The country is endowed with first period (per capita) income Y1.
Second-period output is given by
Y2 = K
α
2
Victoria
Highlight
where K ≥ 0 is first-period investment (so C1 = Y1 − K). The country’s creditors cannot touch its first
period income or capital investment, but they can seize a fraction η of second-period income Y2, up to the
point where they are repaid in full. Thus debt repayments are given by
R = min [ηY2, D]
You may think of the initial level of debt as being infinite in the sense that even if the country were to set
K = Y1 (so that C1 = 0), it would still be the case that D > ηY2.
a. Characterize the country’s optimal choice of investment, KD, and the implied level of repayments,
η
(
KD
)α
.
b. Now assume that entering period 1, creditors decide to partly forgive the country’s debt, writing down its
face value to η
(
KD
)α
, the amount they expect to be repaid if they do nothing. Does this cost the creditors
anything? Does the country benefit? (Be careful to recognize that the “tax” on investment becomes zero
once K > KD.)
c. Assume that the creditors have no interest in the welfare of the country. Can they improve on b. by
forgiving a larger or smaller amount? You may simply answer this question using a graph.
d. What is the best that creditors can do? No closed form solution exists, so it is sufficient to set up the
optimization problem without solving it.
III. Debt Buybacks
A small country has foreign debt D coming due at the end of the (one and only) period. Its output Y is
uncertain according to following simple probability distribution: Y = z with probability 0.5, and Y = Z
with probability 0.5. Finally, ηz < D < ηZ, where η is the percent of the country’s output that can be
seized by creditors in the event of default.
a. What is the market value of the country’s debt V (D) = E [min (ηY,D)]?
b. Will the country’s debt sell at discount? That is, is price p = V (D)/D < 1? If so, at what price will it
sell?
c. Suppose that in addition to Y , the country also has a small amount of liquid assets (“cash”) C, and that
it uses the cash to buy back a portion of its debt at market prices, prior to the realization of Y . Denote the
post buyback debt level as D −X, where X is the amount of debt purchased in the buyback. Explain why,
if creditor participation in the buyback is totally voluntary, we must have
V (D −X)
D −X =
C
X
where V (D −X)/(D −X) is the post-buyback price. (Note that C/X is the average amount creditors who
sell get.)
d. Generally, why is the buyback a bad deal for the country here, assuming that creditors have no ability
to seize the country’s cash assets in the event of default?
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IV. Moral Hazard and Lending
Suppose a country is populated by entrepreneurs with utility function U = C2, who live in a small country
that can borrow abroad at world interest rate r.Each entrepreneur has first-period endowment Y1, which can
be used to invest in a project that yields output Z with probability pi(I), and yields nothing with probability
1−pi(I) (where pi′(I) > 0 and pi′′(I) < 0). First-period endowment Y1, however, is insufficient to achieve the
efficient level of investment I¯, defined implicitly by pi′(I¯)Z = 1 + r. Thus entrepreneurs would like to borrow
D = I¯ −Y1, but they are constrained by the fact that foreign creditors can observe only whether the project
actually succeeds or fails, and cannot observe investment I directly. Potential creditors worry that once the
entrepreneur has been given funds, she will sneak them into secret foreign bank accounts rather than invest.
Under this setup, including the information constraints, equilibrium investment is governed by the following
two equations:
pi(I)P (Z) = (1 + r)(I − Y1) (1)
pi′(I)[Z − P (Z)] = 1 + r (2)
where I − Y1 = D gives the amount the entrepreneur borrows, and P (Z) is the payment to the creditor
if the project succeeds (obviously, P (Z) < Z). Equations (1) and (2) govern the determination of I and P (Z).
a. Illustrate the effect of a rise in first-period endowment Y1 on investment. Why does this model suggest
that the marginal product of capital is likely to be higher in poor countries than in rich countries? It is
sufficient to illustrate your answer graphically if you can explain what the curves are and what the underlying
mechanism is.
b. Why might this model suggest that in a world where wealth is distributed unequally across countries,
average world output might be lower than in a world where wealth is distributed more equally? (Remember
that in this model there are decreasing expected returns to investment.)
c. Suppose a country’s government inherits a foreign debt D that must be paid off by taxes in the second
period on successful entrepreneurs. What effect will this have on investment? If you cannot answer this
analytically, answer graphically or intuitively.
V. Empirical Exercise
Two important measures of duress around country defaults are the ratio of external debt to government
revenues, and the ratio of total public debt to revenues. You are asked to look at the recent restructuring
in Greece (2012) to put this episode in historical perspective. You are to update Table 8.1 in This Time is
Different by Reinhart and Rogoff, adding this episode.
Table 8.1 (note: not Figure 8.1) can be found at http://www.carmenreinhart.com/this-time-is-different. To
find data on Greece, please utilize public sources. It is acceptable to mix numbers from closely related years:
if you cannot find the 2012 numbers, you may mix numbers from the interval 2012-2014.
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