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Foundations of International Macroeconomics^1
Workbook^2
Maurice Obstfeld, Kenneth Rogoff, and Gita Gopinath
Chapter 8 Solutions
- First think about the continuous-time case. At time t = 0 the market believes the governmentís promise that it will peg the exchange rate at its t = T level from time T onward. In that case, we can show that the exchange rate is indeterminate, so that the governmentís policy is not coherent (in that it does not tie down a unique market equilibrium). Let perfect-foresight equilibrium be described by the continuous-time Cagan model [eq. (70) on p. 559 of the book], mt = et − η e˙t.
Let eaT be an arbitrary time-T exchange rate and suppose the market firmly expects that rate to prevail. Then the preceding Cagan equation, coupled with the terminal condition eT = eaT , shows that the exchange rate path
et =^1 η
Z (^) T t^ exp[(t^ −^ s)/η]msds^ + exp[(t^ −^ T^ )/η]e
aT
will equilibrate markets for t ∈ [0, T ]. Next consider the monetary authorityís position at time T when con- fronted with this exchange rate path. The authority has no choice, in view of its vow of a constant exchange rate from T on, but to set the fundamental mT = eaT and to hold mt = eaT for all t > T. Why? Were the authority to act otherwise, the exchange rate would deviate from eaT at some point in the (^1) By Maurice Obstfeld (University of California, Berkeley) and Kenneth Rogoff (Prince- ton University). ∞cMIT Press, 1996. (^2) ∞cMIT Press, 1998. Version 1.1, February 27, 1998. For online updates and correc- tions, see http://www.princeton.edu/ObstfeldRogoffBook.html
interval [T, ∞), in violation of the governmentís initial pledge. In short, the authority must validate (ratify) any market expectation whatsoever. Any initial exchange rate can be an equilibrium because each is conditioned on a different expectation of what the (constant) money supply path will be from date T onward. The situation is subtly different in discrete time, in which case the model is mt = et − η (et+1 − et).
The basic reason discrete time makes a difference is that now, if the market firmly foresees a date T rate of eaT , there are two distinct ways the author- ity can fulfill its promise of a constant exchange rate from date T on (two alternatives which collapse to one in continuous time). First, the authority could still set mt = eaT for t = T and for t > T , as in the continuous-time settingñin which case the exchange rate again is not uniquely determined. Alternatively, if the authority can commit not to adjust mT fully to validate eaT , but instead only to set mt = eaT for t strictly greater than T only, we again get an exchange rate path constant at eaT starting on date T. In this second case, however, the exchange rate is uniquely determined. To fix ideas, suppose the authority can commit to a fixed value mT for date T. In that case, since the date T + 1 rate will be pegged at its date T level, the Cagan equation says that the exchange rate must satisfy
eT = mT
on date T , a well-defined solution. The distinction we make here actually does arise in practice. For example, the Maastricht Treaty on European Union (in effect) links to prior market values the ìirrevocably fixedî bilateral currency conversion factors that will apply to member currencies when European economic and monetary union (EMU) starts on 4 January 1999. (EMU starts officially on Friday, 1 January 1999, but the first business day for the new European System of Central
(c) With the new budget constraint, the first-order conditions become:
Bs+1: u^0 (Cs)g
μ (^) Ms Ps
∂ = (1 + r)βu^0 (Cs+1)g
√ Ms+ Ps+
! ,
Ms:
u^0 (Cs)
g
μ (^) Ms Ps
Ps
− Cs g
≥ (^) Ms Ps
¥ (^) g^0
μ (^) Ms Ps
Ps
= βu^0 (Cs+1)
" g
√ Ms+ Ps+
! 1 Ps+
.
One can then rewrite the money demand equation as
Cs
g^0
μ (^) Ms Ps
∂
g
μ (^) Ms Ps
∂ 2 =^
is+ 1 + is+^.
(d) The analysis here parallels that in the text.
- (a) The constraint follows by straightforward addition. Because domestic money is a nontraded asset, the present value of private and government spending must equal the present value of the economyís tradable resources, which in turn equals the sum of the present value of output and the economyís net foreign financial wealth.
(b) The answer does not change. The variable Bt in part a, the overall net foreign assets of the economy as a whole, equals the sum of domestic government and private-sector net assets, Bt = B tg + Bp t [recall eq. (7) from Chapter 3]. Equation (38) in Chapter 8 would change, however, in that (1 + r)Bp t rather than (1 +r)Bt would appear on its right-hand side. The last equation in footnote 26, p. 537 (the government budget constraint) would also differ, in that (1 + r)B tg would be added to its right-hand side.
(c) Let us take the setup of Chapter 4, but with the services of money being the nontraded good and with is+1/(1 + is+1) that goodís date s price in
terms of the tradable, consumption (recall section 8.3.3). When θ = 1 (the Cobb-Douglas case), footnote 22, Chapter 4, tells us that
Ω
μ C, MP
∂ = C
γ (^) (M/P ) 1 −γ γγ^ (1 − γ)^1 −γ
and that
P (^) sc =
√ is+ 1 + is+
! 1 −γ .
Equation (26) of Chapter 4, translated to apply to the current setting, is the Euler equation for real consumption,
Cγ s+
√ Ms+ Ps+
! 1 −γ
" (1 + r)P (^) sc P (^) sc+
#σ βσC sγ
μ (^) Ms Ps
∂ 1 −γ .
With θ = 1, we also have that
Ms Ps^ =
√ 1 − γ γ
! (P (^) sc )−^1 /(1−γ)^ Cs
[eq. (40) in Chapter 8]. Substituting this relation into the Euler equation preceding it, we derive
Cs+1 =
√ P (^) sc P s c+
!σ− 1 (1 + r)σβσCs,
which parallels eq. (34) in Chapter 4 for the case θ = 1. If you combine this equation with the intertemporal constraint derived in part a of this exercise, the result is
Ct =
(1 + r)Bt + P∞ s=t
≥ (^1) 1+r
¥s−t (Ys − Gs) P∞ s=t [(1 +^ r)σ−^1 βσ]
s−t ≥^ P (^) tc P (^) sc
¥σ− 1.
The definition of the consumption-based real interest rate (section 8.3.3) leads to the equivalent formula
Ct =
(1 + r)Bt + P∞ s=t
≥ (^1) 1+r
¥s−t (Ys − Gs) P∞ s=t
hQs v=t+1(1 +^ rc v )
iσ− 1 βσ(s−t)
in the limit as h → 0. Plugging this result into eq. (82) on p. 571 leads to
G(k) = k + ημG 0 (k) + ηv
2 2
G 00 (k)
as the differential equation any exchange rate solution must satisfy [cf. eq. (84), p. 572]. A general solution is of the form
G(k) = k + α + b 1 exp(λ 1 k) + b 2 exp(λ 2 k)
where the bís are arbitrary constants. Because internal consistency requires that k + α + b 1 exp(λ 1 k) + b 2 exp(λ 2 k)
= k + ημ [1 + λ 1 b 1 exp(λ 1 k) + λ 2 b 2 exp(λ 2 k)]
2 2
h (λ 1 )^2 b 1 exp(λ 1 k) + (λ 2 )^2 b 2 exp(λ 2 k)
i ,
α = ημ and λ 1 and λ 2 are the two roots of the quadratic equation
ηv^2 2
λ^2 + ημλ − 1 = 0.
The particular target zone solution still satisfies the ìsmooth pastingî con- ditions G^0 (k) = 0 at the top and bottom of the band. The argument is the same as in section 8.5.4, because at the edges of the zone Et {dkt+h} still changes discontinuously even when μ 6 = 0 (movements that would drive the exchange rate out of the band suddenly are prohibited).
- (a) Intuitively, spending a domestic currency unit on the least-cost con- sumer basket such that C = Ω(C 1 , ..., CN ) = 1 yields 1/P units of C. Alter- natively, spending the currency unit on any individual commodity j yields 1 /pj units of that good, each increasing C by Ωj ≡ ∂C/∂Cj. At an optimum these two uses of the currency unit must have the same impact on C, so that the equality 1 P =
pj
∂C
∂Cj
follows. Alternatively and more formally, consider the optimization problem that defines the CPI P , which is to minimize the expenditure P = PNj=1 pj Cj subject to Ω(C 1 , ..., CN ) = 1. One way to solve this problem is to set up the Lagrangian
L =
X^ N j=
pj Cj − λ [Ω(C 1 , ..., CN ) − 1].
The first-order optimality conditions are (for all j):
∂L ∂Cj^ =^ pj^ −^ λΩj^ = 0.
Since Ω(C 1 , ..., CN ) is linear homogeneous, we have
X^ N j=
Ωj Cj = Ω(C 1 , ..., CN ) = 1
at the optimum. Thus, multiplying the preceding first-order condition by Cj and summing over all j, we derive
P =
X^ N j=
pj Cj = λ
X^ N j=
Ωj Cj = λ.
This equality, however, allows us to write the first-order condition as
Ωj =
∂C
∂Cj^ =^
pj P.
(b) The difference between the ex post real return on a nominal Home- currency bond and that on a nominal Foreign-currency bond is
(1 + it+1)Pt Pt+1^ −^
(1 + i∗ t+1)P (^) t∗ P (^) t∗+1^ ,
where P and P ∗^ are the consumption-based price levels measured in Home and Foreign currency, respectively. Observe that these consumption-based price levels will be linked by purchasing power parity (absent trade barriers),
Plainly Siegelís paradox does not apply.
(d) The result follows immediately from part a, where it was shown that ∂C/∂Cj = pj /P.
(e) The condition that consumers equate their marginal rates of substitution to relative prices implies that
Cx,t Cy,t^ =
√ γ 1 − γ
! Etp∗ y,t px,t^ ,
so that we can express the spot exchange rate as
Et =
√ 1 − γ γ
! √ px,tCx,t p∗ y,tCy,t
! .
In a perfectly pooled risk-sharing equilibrium, Cx,t = C x∗,t = Xt/2 and Cy,t = C y∗,t = Yt/ 2. Moreover, using the (binding) cash-in-advance constraints for the two currencies, Mt = px,tXt and M t∗ = p∗ y,tYt, we can express the equation for the spot exchange rate as
Et =
√ 1 − γ γ
! √ Mt M t∗
! .
Using the result in part d:
Ft =
Et
( Et+1uj (Ct+1) pj,t+
)
Et
( uj (Ct+1) pj,t+
) (^) , j = x, y.
In combination with the cash-in-advance constraints and the preceding rela- tionship between the spot exchange rate and money supplies, we have, for j = x,
Ft =
√ 1 − γ γ
! (^) Et
( Cx,t+ M t∗+1^ ux(Ct+1)
)
Et
( Cx,t+ Mt+1^ ux(Ct+1)
If money and output shocks have statistically independent distributions, how- ever, we may factor out the terms in consumption above and write
Ft =
√ 1 − γ γ
! (^) Et
( 1 M t∗+
) Et {Cx,t+1ux(Ct+1)}
Et
( 1 Mt+
) Et {Cx,t+1ux(Ct+1)}
√ 1 − γ γ
! (^) Et
( 1 M t∗+
)
Et
( 1 Mt+
(f) The result in part a and the cash-in-advance constraint imply that
M = pxCx = P Cx^ ∂C ∂Cx
From part e we therefore obtain the ìrisk neutralî forward exchange rate, eq. (107) in Chapter 8, by again using the assumption that outputs and monies are independently distributed random variables:
Ft =
Et
(√ 1 − γ γ
! √ Mt+ M t∗+
! 1 Mt+
)
Et
( 1 Mt+
)
Et
( Et+1^1 Mt+
) Et
( Cx,t+1^ ∂Ct+ ∂Cx,t+
)
Et
( 1 Mt+
) Et
( Cx,t+1 ∂^ ∂CCt+ x,t+
)
Et
( Et+
√ Cx,t+ Mt+
! ∂Ct+ ∂Cx,t+
)
Et
(√ Cx,t+ Mt+
! ∂Ct+ ∂Cx,t+
)
Et {Et+1/Pt+1} Et { 1 /Pt+1}.
The reason for the nonzero covariance is that temporally adjacent overlapping multiperiod forward-rate errors share common innovations. They share in- novations because the maturity of the forward contract (two weeks) is longer than the sampling interval (one week). We can show through similar steps that for j > 1, Cov {et+2 − ft, 2 , et+j+2 − ft+j, 2 } = 0.
(b) A series sampled biweekly would not be serially correlated. This result follows easily from the argument of part a. In this case there are no overlap- ping multiperiod forecast errors.
(c) We outline how a General Method of Moments (GMM) estimator could be used to test the hypothesis that Et(et+2 − ft, 2 ) = 0. Define the two-period- ahead forward-rate forecast error as ≤t+2, 2 ≡ et+2 − ft, 2. The null hypothesis states that the forward rate is equal to the conditional expectation of the two-period-ahead spot rate, and is therefore the best (most efficient) unbi- ased predictor of the spot rate. This property implies that ≤t+2, 2 will be uncorrelated with any information available at time t. One possible way to test this implication is to run the following regression [which is similar to equation (110) in the text],
et+2 − et = a 0 + a 1 (ft, 2 − et) + ≤t+2, 2 , (1)
where the difference ft, 2 − et is the forward premium on date t. The forward- market ìefficiencyî test asks whether one can reject the joint null hypothesis that a 0 = 0 and a 1 = 1. Under the null hypothesis, ≤t+2, 2 equals the forward- rate forecast error and so the orthogonality conditions E{≤t+2, 2 } = 0 and E{(ft, 2 − et) ≤t+2, 2 } = 0 are satisfied for a 0 = 0 and a 1 = 1. The specification implies that the efficient GMM estimator of a 0 and a 1 is the same as the ordinary least squares (OLS) estimator based on all the available (weekly) observations of spot and two-week-ahead forward rates. However the usual OLS standard errors would not be appropriate. When we use the weekly series of two-week-ahead prediction errors, we face a problem of serial corre- lation in ≤t+2, 2 (as discussed in part a).
To describe the GMM estimator, define the coefficient column vector θ = [a 0 a 1 ]^0 , along with the vectors
xt ≡
^1 ft, 2 − et
and
wt(θ) ≡ xt ∑ ≤t+2, 2 (θ) =
^ ≤t+2,^2 (θ) (ft, 2 − et) ≤t+2, 2 (θ)
.
By eq. (1) above,
≤t+2, 2 (θ) ≡ et+2 − et − [a 0 + a 1 (ft, 2 − et)].
The efficient GMM estimator (EGMM) for a sample of size T is derived as
àθEGMM( Φà) = argmin (^) θ wØ(θ)^0 Φà−^1 wØ(θ),
where
w Ø(θ)= T^1
X^ T t=
wt(θ),
X^ ∞ j=−∞
Γj ,
Γj = E {wt(θ)wt−j (θ)^0 } ,
and Φà is a consistent estimate of Φ. Under standard regularity conditions, the estimate àθEGMM is asymptotically normal with mean θ. One can estimate the asymptotic covariance matrix of àθEGMM by
T
√ (^) XT
t=
xtx^0 t
!− 1 Φ^ à
√ (^) XT
t=
xtx^0 t
!− 1 .
Under the null hypothesis the coefficient vector is θ 0 = [0, 1]^0. One can per- form a Wald/Likelihood Ratio/Lagrange Multiplier test of the joint null hy- pothesis. (For details on GMM estimation and hypothesis testing, see R. Davidson and J. G. Mackinnon, Estimation and Inference in Econometrics, Oxford, 1993.)
