~A4-

(al2) 8, = slexplce-sV/ Almas + pA f expl (e-s)/Alugas

+ aL G_— 9) — Awl = 0. = Aw] f exp[w(t-s) Ju ds.

According to (All) and (Al2), the flex-price exchange rate and
domestic output price depend both on future expected monetary disturbances
(represented by nm.) and future expected real disturbances (represented by

us). However the flex-price real exchange rate a. defined as é. - p>

t
is given by

(A13) a = -w i exp[w(t - s)]u.ds.

The real exchange rate thus depends exclusively on current and anticipated
future real shocks; it is not influenced by monetary factors. Expression
(Al3) reflects the real-monetary dichotomy that characterizes the flex-price

exchange-rate model.

2. Sticky Prices and the Barro-Grossman Adjustment Rule

To introduce sticky prices and the Barro-Grossman adjustment scheme,
assume that the domestic ouput price is a predetermined variable and replace

the goods-market equilibrium condition (A3) with the equation
Al so . _ ss pa
(Al4) p, = mIo(e, - p, + u,) - co(é, - BL] +B:

where B, again represents the flex-price equilibrium output price derived
in the previous section. Equilibrium ee and Pe in the present model
satisfy (A2) and (Al14), while é. and B, satisfy (A2) and (A3). It follows

that under the Barro-Grossman price dynamics, the exchange rate and output


-13-

The following result makes apparent the essential reason for the

two models' structural equivalence:

COROLLARY. Along the saddlepath of the Mussa model, the output price p

obeys a differential equation having the same form as the Barro-Grossman

equation (10). More precisely, along the saddlepath of the Mussa model,

as) p= 6g4- yy) +5

where

_ 8L = 0) + 206]
(9) 8 = GF bas) F208) *

PROOF. Letting q, once again denote the real exchange rate e, 7 Phe we

may use (3) to write excess demand in the Mussa model as
d M ~ M 2
(20) ¥, — ¥ = O04, - 4.) - aoa, - 4,)-
Equation (A28) of the appendix implies that on the saddlepath of the Mussa

model,

(21) ap - a, = 86a - a).

Equations (20) and (21) together imply that
d = M_~ = Mx ~M_sx
(22) ¥_ — ¥ OC + G00) (a, - 4.) = O(1 + bao) f(e, - €.) - @, - BDI.

Using equations (16a) and (16b), we have

Me
Mog ~a(p, - Py)
(23) & ~ &e > Ta Say F NOOT *

Combining (22) and (23) yields

Mo _ -[( =) +20] 9d _
(24) Pe Pe = a+ bao) + AOD Oe ~ +

Because, by (16b), ae = -06 (Py - P,) + By along the Mussa model's saddlepath,

(24) implies that


REFERENCES

Barro, R.J. and H.I. Grossman, Money, Employment and Inflation, (London:

Cambridge University Press, 1976).

Buiter, W.H. and M.H. Miller, "Monetary Policy and International Competitive-
ness: The Problems of Adjustment," Oxford Economic Papers 33 (July,
1981), 143-175.

» "Real Exchange Rate Overshooting and the Output Cost

of Bringing Down Inflation," European Economic Review 18 (May/June, 1982),

85-123.

Dornbusch, R., "Expectations and Exchange Rate Dynamics," Journal of Political

Economy 84 (December ,1976), 1161-1176.

Flood, R., "Comment on the Buiter and Miller Paper," European Economic Review

18 (May /June, 1982a), 125-127.

» "Sticky Prices and Inventory Adjustment," Board of Governors of

the Federal Reserve System, mimeo (February »1982b).

Frankel, J. A., "On the Mark: A Theory of Floating Exchange Rates Based

on Real Interest Differentials," American Economic Review 69

(September,1979), 610-622.

Gray, M. R. and S. J. Turnovsky, "The Stability of Exchange Rate Dynamics

under Myopic Perfect Foresight," International Economic Review 20

(Ocotober 1979), 643-660.

Green, J. and J. J. Laffont, "Disequilibrium Dynamics with Inventories and

Anticipatory Price-Setting," European Economic Review 16 (May,1981),

199-221.


-4-

are exogenous: m is the logarithm of the nominal money supply; a is the
share of home goods in the domestic consumer price index; y is the logarithm
of the flow of perishable home output; r* is the nominal interest rate on
foreign-currency bonds; and u is a shock to foreign demand for domestic
output. a is assumed to be fixed, and y and r* are assumed to be fixed and
equal to zero. The exogenous foreign-currency price of imports is likewise
assumed to be constant, with its natural logarithm normalized to zero.

The model assumes rational expectations; this amounts to perfect
foresight in the absence of unanticipated shocks. Thus, there is no
distinction between actual and anticipated rates of change of e and p.
Equation (1) is the money-demand schedule, which relates the demand for
price-index deflated nominal balances to the home interest rate and income.
Equation (2) reflects the assumption that home-currency and foreign-
currency bonds are perfect substitutes. Equation (3) is the aggregate-
demand schedule, which posits that demand for domestic output depends on
the terms of trade, tastes (as represented by u), the real rate of interest,
and income. Finally equations (4) and (5) require continuous clearing
of both the money and goods markets. Condition (5), which entails full
flexibility of domestic prices, will be relaxed in subsequent sections.

The flex-price equilibrium values of the exchange rate and domestic
output price are denoted by é and p. Following Sargent and Wallace [1973],
we close the model with the assumption that the economy must always lie on

6/

its unique conditionally stable saddlepath:i— As the appendix shows, the

6/ In models with a well-defined stationary long-run equilibrium, the
saddlepath is the unique path converging to long-run equilibrium and thus

is the unique path implying nonexplosive behavior for the economy. Saddlepoint
stability results from the self-fulfilling nature of rational expectations.

(FOOTNOTE CONTINUES)


-AS-

price obey the equations

cas) ép = [G - a) /Alep + (a/A)Py >

(ale) Be = mId(e, - pL) - ao(é - BL],

where ef ze - é. and Pt =P, - Py- Equations (A15) and (A16),
when combined, yield

can 9p - Bit TAG eae) 9: - Gat +2)

Together (Al5) and (Al7) describe an autonomous differential equation system
i ' '
ine and Pre

The characteristic roots of that system, ny and Ng» are given by

2
_ -[Amd + too - (1 - o)] [And + tao - (1 - a] 0) 4s
Tye Ny = TAC = Tao) +f meq ~ nao? + hay?

Provided that the condition 1 - tao > 0 is met, ny > 0 and Ny <0, as required
for saddlepath stability when one of the two endogenous variables is predetermined.
By imposing the requirement that the economy be on the stable saddlepath,

we obtain the rational-expectations solution

-(Pg - Bodo

(A18) e, = Ta =a - any) exp(nyt) + & >

(A19) Py = (Pg - Bo) exp(ngt) +B, -

According to (Al8) and (A19), ee and P, converge to their flex-price
values é. and B, at a rate given by Ing] At any point, the deviation
of actual e, or p, from its flex-price value is proportional to the initial

discrepancy between the predetermined initial output price Po and its

flex-price level Po given by (Al0). If there is no disequilibrium initially,


-A7-

The Laplace transform, applied to (A23), yields the equation

4 +(L-w+ 89) dq + @(8d + &L(u,)
(2+ 09) (2 = w) .

(425) L(a,) =

Equation (A25) is based on the normalization Uy = 0. After partial-

fraction expansion, (A25) becomes

4g = 89a do * S449 1
w+ 6p Tog! + peg? + Leg

L(a,) = ~¢

The Laplace transform theorem now imples that

dg = 4p do * S49
(a26) a, = ~ ore? exp(-6¢t) + ore? exp(wt) +0 A exp[W(t-s) usds
dg - &dg dg + oq

= -SF eG? exp(-O0ot) + (oa? +W ; exp (-us)u ds] exp (Wt)

f exp[w(t - s)Ju.ds .
t s

The saddlepath assumption requires that the coefficient of exp(wt)

in (A26) be zero. Thus, do and dg must satisfy the relation

(427) 4 -8q5 = -(@ + 89) [49 +0 4 exp (-ws)u ds].
Combining (Al13), (A26), and (A27), we find that
(A28) 4 = Gg - dq) exp(-Opt) + 4, -

The effect of Mussa's price-adjustment mechanism is to drive the real
exchange rate toward its flex-price level at a rate given by 0.

To solve for ep we write (A24) in terms of deviations from flex-price

equilibrium values,


~A2-

By solving these simultaneous equations we obtain

[AQ - w) - ale + OPg + Ww- &)L(m,) - ow Cu, )
(a4) Le.) = ME - G/Ie ~ w] ,

(C-a) = Awleg + [At - (1-a)Ipg + (w~ 2)L(m,) +L (1-0) - ARIEL (uy)
ATR = AA) = wl :

(85) L(p,) =

A partial-fraction expansion of (A4) leads to the representation

OPy - (a + aden 1 1;

(46) Le) =ranyoas ‘aan > Toe

reo af) ®

* tam ~olt- Gy 7 Ta

t
+ XT Sat L(ffexpl(t - s)/A] ~ explu(t ~ s)]}m, 4s)
~ FTAA Sap EGCAM) expl(e - 971 - w explact ~ 5)1)m,48)
(orn) t
- Way) > oT Lftexpl (t - s)/d] - exp{w(t - s)]}u,ds).

In deriving (A6), we have used the convolution property (iii) and the facts
that Llexp(t/A)] © [2 - (L/A)]TM*, Llexp(wt)] = = a).
The Laplace transform theorem allows us to infer from (A6) that

the exchange rate path has the form

(1 - AW - ae, + ap a(e, - p,)
(47) = 2 2 exptte/a) + = 2 exp tat)
-i 5 exp[(t - s)/A]m_ds - ow _ fexp[(t - s)/A] - exp[w(t - s)]}u_ds
Xo OP 8 1 - Aw 0 Pp P s

It is convenient to rewrite (A7) in the equivalent form


fully flexible. The resulting price adjustment scheme is
. d ~
(10) p=tmy -y) +5,

where p is the flex-price equilibrium output price discussed in the previous
section. This type of pricing rule is suggested by Barro and Grossman {1976]
in a closed-economy setting, although they do not assume rational expectations
regarding 5.27/ It is easy to see that the Barro-Grossman rule reduces to
Dornbusch's rule (8) when the exogenous variables are fixed except for
unanticipated jumps, for in that case, 3 = 0. Similarly, the rule reduces
to (9) when the only expected change in the exogenous variables is the
growth of money at a constant rate. However, the rule does not reduce to
Dornbusch's rule (8) in the case of the one-time expected money increase
analyzed by Gray and Turnovsky [1979] and by Wilson [1979]. If it is
announced at time t = 0 that the money stock will increase by an amount

Am at time t = T, then Py = (1/d) dm exp[(t - T)/A] > 0 for OS t <T.

That the Barro-Grossman rule is immune to the criticisms levelled

at rules (8) and (9) is clear. Perfectly anticipated shocks -- whether
real or nominal -- do not cause disequilibrium when price adjustment is
given by (10). Since the equilibrium flexible-price p does not jump

discontinuously at any point in response to a change announced "infinitely

"in advance, p will fully trace its movements under (10).

far
Mussa [1977, 1982] suggests an alternative rule, and demonstrates
that it, too, renders disequilibrium price adjustment unnecessary when

shocks are perfectly anticipated. The Mussa rule is given by

(4) p= 6G4-y) +3

12/ See Barro and Grossman, [1976, p. 178], equation (4.26).


3. STICKY DOMESTIC PRICES AND ALTERNATIVE PRICE-ADJUSTMENT RULES
When the price of domestic output is sticky, the goods~market clearing
condition (5) need not hold. Following Dornbusch [1976], we assume that the
domestic output price is a predetermined or nonjumping variable that adjusts
gradually to eliminate goods-market disequilibrium. Dornbusch, who assumes that
the exogenous variables are constant except for one-time unanticipated jumps,

postulates the following price-adjustment scheme:
. d
(8) pty - y)-

While specification (8) is entirely appropriate given the environment
Dornbusch assumes, it may become inappropriate once anticipated shocks or
trend movements in exogenous variables are introduced .£/
Gray and Turnovsky [1979] and Wilson [1979] use the Dornbusch model
with adjustment rule (8) to analyze a one-time anticipated increase in
the money supply. The solution these authors derive is qualitatively
sensible when the disturbance is expected to occur in the near future, but
problems arise when (8) is used to study money shocks that are anticipated
long before they occur. To see this, consider the economy's behavior as
the date of the future money increase recedes infinitely far into the
future. As the adjustment period preceding the intended policy act grows
longer, the disequilibrium caused by the announcement of that act should
disappear. In the limit of a perfectly anticipated money-stock increase --
one that is anticipated "infinitely far" in advance -- the domestic output

price and the exchange rate should rise gradually and in proportion toward

the long-run equilibrium levels associated with the post-disturbance stock

8/ Mussa [1982] has emphasized these problems.


-5-

rational-expectations solution paths for é and B are then given by

(6) é. = ; fexpl (t-s) /Alm,ds + is {lexp[ (t-s)/A] - exp[w(t-s) ]}usds
7) BR =F fexp[ (t-s) /Almjas + 722 fexpl (t-s)/Alu,és

- acne) — Aa) fexplu(t-s) ]uas :

where WTM = g/oa.—!/ Under rational expectations, equilibrium nominal prices
depend on the future expected paths of money and the autonomous component
of aggregate demand. Note that monetary factors affect é and By by equal
amounts, whereas demand shifts affect these variables differentially in
general. Thus, while real shocks must be accommodated by shifts in the

flex-price real exchange rate 4a,

- By» a, depends exclusively on
factors that shift aggregate demand y‘, and not on money. In the flex-price
model set out here, there is a complete dichotomy between real and monetary
phenomena. This classical dichotomy would disappear, for example, if real
balances mm - OP, -qa- ae, were an additional argument in the aggregate
demand function. To simplify the analysis, we abstract from real baiance

effects.

(FOOTNOTE CONTINUED FROM PREVIOUS PAGE)

Whether there is a stationary long-run equilibrium or a moving long-run
equilibrium, the saddlepath can be defined as the unique path of the
economy along which prices do not depend in part on pure speculative
"bubbles" unrelated to actual market conditions. The appendix discusses

the mathematical implementation of this definition in the present context.

7/ These solutions are derived by the method of Laplace transforms. The
Laplace transform technique is a convenient one for rational-expectations

models, although other solution techniques are available.


-A3-

co cs

{(1-Aura)eg + Pg - [(1/A)-w) fexp(-s/A)m ds - awfexp(-s/d)u.ds}
0 s 0 §

To nw exp(t/d)

[aleg - Py) + ow é exp (-ws)u ds]

+ Tove exp (wt)

cy

+ ; exp[ (ts) /A]m,ds + cet {expt (t-s)/A] - exp[w(t-s) ] }u,ds.

Equation (A8) expresses the path of the exchange rate as a function of its
own initial value (ep) and the initial value of the price of domestic output
(9) + These two initial conditions are uniquely determined by the saddlepath
assumption, which requires that the coefficients of the explosive "bubble"
terms, exp(t/A) and exp(wt), be zero (see Sargent and Wallace [1973]). The

requirement that the economy be on its saddlepath implies [by (A8)] that

~

(A9) =i A exp (-s/A)m ds +

~{r flex (-s/A) - exp (-ws) }usds,

>I

>I

(aio) 3, < ES exp(-s/Aymds + —M%— f exp (-s/A)u_ds
Po 0 P s 1- dw 0 P s

[@ = a) - rw) F

Tow 4 exp (-is)u ds ,
where a tilde denotes a saddlepath equilibrium value for the flex-price
model. Combining (A8), (A9), and (Al0), we obtain the rational-expectations

equilibrium value of the exchange rate

zig Hewpl(t-5)/A1 - expla(t-si Jud.

(All) é. = xf exp[ (t-s) /A]m ds +

A derivation similar to the foregoing shows that the flex-price equilibrium

domestic output price can be written as


~8-

1i/

rule to environments of secular inflation.—’ These authors modify (8)
by adding to the excess demand term the current rate of nominal money

growth. The resulting adjustment rule is
(9) penyt-y) +a.

Specification (9) is consistent with an assumption that all changes in the
monetary growth rate are unanticipated. But the rule implies unreasonable
asymptotic behavior when anticipated changes are analyzed. Consider, for
example, a perfectly anticipated increase in the monetary growth rate. With
higher trend money growth, long-run m - p is lower. Thus, a perfectly
anticipated money-growth increase should cause m - p to decline over time
but should not occasion disequilibrium. Because p is predetermined, however,
rule (9) implies that m - p can fall over time only if the announcement of
the future increase in m creates an excess demand for output that does

not vanish as the date of the increase becomes infinitely distant.

While the price-adjustment rules described so far are unsatisfactory
in many situations, a generalization which contains these rules as special
cases is adequate for the analysis of any type of disturbance. Assume
now that p is a function not only of current disequilibrium, but also of

the rate at which the price of domestic output would increase if that price were

il/ Like Dornbusch [1976], section V, Buiter and Miller analyze a model
in which output is demand-determined and therefore endogenous. In that
model, the disequilibrium term entering their price adjustment rule
depends on the difference between actual output and full-employment
potential output. Our analysis applies with only minor modifications to

variable-output models; see footnote 15 below.


-R2-

Henderson, D. W., "The Dynamic Effects of Exchange Market Intervention
Policy: Two Extreme Views and a Synthesis," in H. Frisch and

G. Schwodiauer, eds., The Economics of Flexible Exchange Rates,

supplement to Kredit und Kapital 6 (Berlin: Duncker and Humblot, 1980).

Kouri, P. J. K., "The Exchange Rate and the Balance of Payments in the
Short Run and in the Leng Run: A Monetary Approach," Scandinavian

Journal of Economics 2 (May,1976), 280-304.

Liviatan, N., "Anti-inflationary Monetary Policy and the Capital Import

Tax," Warwick Economic Research Paper, No. 171; University of Warwick, 1980.

Lucas, R. E. Jr., "Econometric Policy Evaluation: A Critique," in

K. Brunner and A. H. Meltzer, eds., The Phillips Curve and Labor

Markets (Amsterdam: North Holland, 1976).

McCallum, B. T., "Rational Expectations and Macroeconomic Stabilization

Policy: An Overview," Journal of Money, Credit, and Banking 12

(November, 1980), 716-746.

Mussa, M., "A Dynamic Theory of Foreign Exchange," in M. Artis and

A. Nobay, eds., Studies in Modern Economic Analysis: The Proceedings

of the Association of University Teachers of Economics (Oxford:

Basil Blackwell, 1977).

» "Sticky Prices and Disequilibrium Adjustment in a Rational

Model of the Inflationary Process," American Economic Review 71

(December, 1981), 1020-1027.


International Finance Discussion Papers
Number 223

May 1982
Revised, May 1983

EXCHANGE RATE DYNAMICS WITH SLUGGISH PRICES UNDER ALTERNATIVE PRICE-
ADJUSTMENT RULES

by

Maurice Obstfeld and Kenneth Rogoff

NOTE: International Finance Discussion Papers are preliminary materials
circulated to stimulate discussion and critical comment. References in
publications to International Finance Discussion Papers (other than an
acknowledgment by a writer that he has had access to unpublished
material) should be cleared with the author or authors.


-11-

4, THE STRUCTURAL EQUIVALENCE OF THE BARRO-GROSSMAN AND MUSSA
PRICE-ADJUSTMENT RULES

The following result clarifies the relationship between the Barro-

Grossman and Mussa pricing rules:

THEOREM. Provided that the condition

(13) 1 - Tao > 0

is satisfied, the Barro-Grossman price adjustment scheme (10) and the Mussa

price adjustment scheme (11) yield Structurally equivalent exchange rate

models.

PROOF. The appendix demonstrates [see equations (A18) and (A19)] that
when (13) holds, the rational-expectations equilibrium of the model described

by equations (1) through (4) and the Barro-Grossman rule (10) is given by

-(Py - Boa

BG _ ‘0 7 Po ~
(14a) e “T@> as Trg] exp (nyt) + é.
(14) mo = (Pg ~ Podexp(ngt) +B,»
where

2
_ -Umd + moo - (1 - 0)) © [Amo + t00 - (1 - J 1 5
(15) 12 PGE) t aa ~ nao)” + Gry <O>

and é and B, again denote the equilibrium solutions for the flex-price model of

section 2. When the Mussa rule (11) is substituted for (10), the rational-

expectations equilibrium is given by

-(Pp - Boda
M o 7” Po ~
(16a) & = Ta =) + N00) exp(-0gt) + é,
(16b) pu = (Pg - Bo) exp(-8Gt) + BL


~-10-

where p is defined as the domestic output price that would clear the goods
market given the actual (possibly disequilibrium) values of the endogenous

variables e, p, r, é, and p. More formally, P, is defined by the condition
12 =0O=e= -?p - -oap. - - é
(12) y=0 ole, Pet u) olr, op, (1 a)é.] .

The difference between p and p deserves emphasis. p is the output price
that would prevail in a hypothetical Walrasian general equilibrium with
fully flexible prices. p is the output price that would clear the goods
market given current levels of the sticky-price system's endogenous variables.

Alternative microeconomic rationales for the Mussa price-adjustment
rule are presented in McCallum [1980], Mussa [1981], and Flood [1982b].
Mussa's derivation, for example, assumes monopolistic firms for whom price
changes are costly. Flood's inventory-adjustment story assumes that firms
set their prices a period in advance of market transactions.—!

The Barro-Grossman rule (10) and the Mussa rule (11) appear quite
unrelated, and it is natural to ask how these different price-adjustment

schemes affect the dynamic behavior of the economy. We address this question

in the next section by comparing explicit solutions of models incorporating

the two rules.

13/  Flood's justification of the Mussa rule is based on work of Green
and Laffont [1981]. Rotemberg [1981] points out some limitations of

Mussa's [1981] rationale.


of money .2/ However, this equilibrium scenario is impossible with adjustment-
rule (8), because the price level cannot rise in the absence of excess
demand. Even perfectly anticipated money shocks must cause goods-market

disequilibrium, 20/

Similar difficulties surround the Frankel [1979], Liviatan [1980],

and Buiter-Miller [1981, 1982] extensions of Dornbusch's price-adjustment

+ Prices naturally exhibit this behavior in the flex-price model of the
previous section. When the money shock is expected to occur in finite
time and prices are flexible, the domestic output price and exchange rate,
after an initial equiproportionate jump when the announcement is made, rise
smoothly and in proportion so that the economy is at its long-run equilibrium
when the money supply increases. This type of adjustment to imperfectly
anticipated shocks is impossible in the sticky-price setting, but it is
still true that, after the initial announcement, price evolve smoothly.

In particular, the exchange rate cannot jump when the expected money
increase takes place, for an anticipated discrete jump would entail an
unexploited opportunity to earn an infinite instantaneous rate of return

on foreign bonds.

10/ To be precise, the problems with the Gray-Turnovsky and Wilson analyses
are not caused exclusively by the price-adjustment scheme these authors adopt.
An additional source of nonneutrality is their assumption that aggregate
demand is a function of the nominal, rather than the real, domestic interest
rate. It is easy to see that in a well-specified model, the nominal

interest rate would rise over time during the adjustment to a perfectly

anticipated money increase while the real interest rate would remain constant.


~A6-

a divergence between Poy and By can arise only as the result of previously

unanticipated information arriving at time t = 0.

3. Sticky Prices and the Mussa Adjustment Rule

The Mussa model results from replacing (Al4) by the price-adjustment

rule

A2 pb, = - - é - 7 >

(A20) By 8{o(e, PL + uy) 0 (6, BY] + Pps

where Py is defined by the condition

(A21) ole, -P + u,) - aoe, - BL) =0.

If we differentiate (A21) and substitute the result into (A20), we obtain
A22) § = 0f _ _ sos . w os
(A22) By. Biote, Pet u) 00 (,. 5] +e +a, (1/) c, BY)

(recall that © = $/oa). Using the definition of the real exchange rate,

Mm

>» we may rewrite (A22) as a single, second-order differential

q e

che 7 Pe
equation in qe

(A23) 4, - o(1 - 800)4, - w8q, = 08du, + ob +

Even though the level of q depends on the initial sticky nominal price of
domestic goods, the adjustment law for qq involves real factors only.

The model consisting of (A2) and (A22) will be solved in three steps.
First, (A23) will be solved to obtain the rational-expectations path of

the real exchange rate q: Second, equation (A2), rewritten as
(a24)-& = G/A)e, - G/A)a, - m./A,

will be used in conjunction with the solution for q to obtain the path
of ee and the initial values & and do: Third, the identity p, =e, - 4

will be used to derive the path of the domestic output price.


-3-

the money supply or in the equilibrium terms of trade, for example, cause
no disequilibrium in a well-specified model, but do induce movements of the
system. In this section, we focus on the second source of dynamics by
solving a standard exchange rate model under the temporary assumption that
domestic prices are fully flexible, or, alternatively, that all movements
in the exogenous variables affecting the economy are perfectly anticipated.
The equilibrium path of this flexible-price model provides a limiting
benchmark for the sticky-price models analyzed later. This benchmark is

a natural generalization of the fixed "long-run" equilibrium appearing in
sticky-price models whose exogenous variables are static except for one-
time unexpected jumps.

The exchange-rate model used here is of the extended small-country

variety, and is based on work of Dornbusch [1976] and Mussa [1977, 19821.2/

It is described by the following equations:

(1) nt - ap - (1 - aje = - Ar + py (money demand)

(2) r=ré¥+é (uncovered interest parity)
(3) y’ = o(e - p +u) - ofr - ap - (1 - a)é] + yy (aggregate demand)

(4) a! =m (money-market equilibrium)

(5)

(goods-market equilibrium)

Here, p is the logarithm of the money price of domestically-produced goods;
e is the logarithm of the exchange rate (defined as the domestic-currency
price of foreign currency); and r is the domestic nominal interest rate.

5 eae ;
Dots over variables indicate rates of change.2/ The remaining variables

4/ See the Dornbusch and Mussa papers for more detailed expositions.

5/ Unless otherwise stated, these rates of change are right-hand derivatives.


-14-

M_ Of -a) +299] 4 z
Pe + baa) + x66) Ye ~ + Pee

The foregoing corollary shows why the Mussa model is necessarily
saddlepath stable. Although prices in that model do adjust according
tc a Barro-Grossman-type rule, the model can never be unstable because,

by (19), the stability criterion
(25) 1 - ao > 0

is always satisfied [cf. equation (13)]. By assuming the Mussa rule, we
effectively limit the parameter 6 to values between 0 (8 = 0) and l/ao (6 = +=).
Even though the Mussa rule places an upper bound on the excess-demand coefficient
6 appearing in (18), there is no upper bound on the speed at which goods-

market disequilibrium is eliminated. As was pointed out above, the Mussa

model converges to the equilibrium flex-price model as 6 + + and 6 + 1/ao.

A consequence of these findings is that the speed of goods-market
adjustment under the Barro-Grossman rule (10) becomes infinite as 7 + l/ao.
Accordingly, that rule yields a saddlepoint-stable model for any speed of
goods-market adjustment. Contrary to appearances, the stability condition

15
(13) does not limit one to adjustment speeds which are not too great .—

15/ The isomorphism results presented here are readily extended to
variable-output versions of the exchange rate model, like that of Buiter

and Miller [1981, 1982] and those compared by Flood [1982a]. The stability
condition when the Barro-Grossman mechanism is used is slightly more complicated
in the variable-output case if money demand depends on actual output.

The stability criterion, in that case, is A(1 - y) + aoW- Amao > 0.

Otherwise, the isomorphism between variable-output models incorporating

the Barro-Grossman rule and those incorporating the Mussa rule can be

proved as in the text.


APPENDIX

This appendix calculates explicit solution paths for the flexible-price
exchange rate model and the two sticky-price models discussed in the text.
These rational-expectations models are solved by the method of Laplace
transforms, which is equivalent to the operator solution procedure described

by Sargent (1979 ).2/ The Laplace transform of a function f, is defined by
(Al) L(£_) = f exp(-2t)f£ dt,
t 0 t

and is a function of 2.2/ The key theorem invoked below is that a
continuous function is uniquely determined on (0, ~) by its Laplace transform
(see Sokolnikoff and Redheffer [1966]). It is easy to verify that: (i)

L(+) is linear; (11) L(f.) = AL(E.) - fos (iii) L(E,)L(8,) = L( Fees).

These three properties will be used repeatedly in what follows.

1. The Flex-Price Model

The flex-price model may be written in the form

(a2) é = fa - a)/rle, + (a/A)p, - m,/d

(43) op, = “we, + UP, + e, 7 wu

t t

where w = ¢/00. The Laplace transform of the system given by (A2) and

(A3) may be written in matrix notation as

Lim, )
@ = a) a t
nn one wen] [eo -
. L(m,)
w- G52 2-2-4 ]] ep [Pg = aE - whee.)

1/ The advantages of the Laplace transform method are that it is completely
algorithmic and that it produces solutions which are expressed in terms of the
state variables' initial positions.

2/ The transform is defined only for & such that the integral in (Al) converges.


-2-

disturbances or nonstationary long-run equilibria. Second, the Barro-Grossman
rule contains the rules of Dornbusch, Frankel, Liviatan, and Buiter and
Miller as special cases, and thus has an intuitive interpretation.

The paper's central result is that the Mussa and Barro-Grossman
rules, though apparently quite dissimilar, yield structurally equivalent ex-
change rate modeis.2/ Thus, despite the key role of disequilibrium price

dynamics, the choice between the two adjustment mechanisms is not necessarily

a critical one.

2. THE LIMITING FLEXIBLE-PRICE MODEL

The dynamics of a sticky-price exchange rate model with rational
expectations can be decomposed into two components. The first component
is caused by the system's adjustment to current disequilibrium. The second
component is caused by movement of the equilibrium that would obtain if all

prices were fully flexible! Perfectly predictable trend movements in

2/ The models explored below assume that agents have perfect foresight, but
the structural equivalence result would carry over to an explicitly stochastic
environment such as the one assumed by Mussa [1982]. In a stochastic setting,
structural equivalence is the same as econometric observational equivalence.
The "perfect foresight" assumption allows for the initial arrival of ‘previously
unanticipated information concerning the future paths of relevant exogenous var-
iables. After the initial moment, however, agents' expectations are fulfilled.
3/ Even if prices are flexible and exogenous variables are stationary,
dynamics may arise from the adjustments of nonmonetary asset stocks to
long-run desired levels, as in Kouri [1976]. The class of models studied

in this paper abstracts from the dynamics of asset accumulation. For an

analysis of asset accumulation in a sticky-price model, see Henderson [1980].


-R3-
Mussa, M., "A Model of Exchange Rate Dynamics," Journal of Political
Economy 90 (February, 1982), 74-104.

Rotemberg, J. J., "Fixed Costs of Price Adjustment and the Cost of

Inflation," sioan School of Management, M.I.T., mimeo (December ,1981).

Sargent, T.J., Macroeconomic Theor » (New York: Academic Press, 1979).
Sargent, T.J. and N. Wallace, "The Stability of Models of Money and

Perfect Foresight," Econometrica 41 (November, 1973), 1043-1048.

Sokolnikoff, I. S. and R. M. Redheffer, Mathematics of Physics and Modern

Engineering, 2nd edition, (New York: McGraw-Hill, 1966).

Wilson, C., "Anticipated Shocks and Exchange Rate Dynamics," Journal of

Political Economy 87 (June, 1979), 639-647.


EXCHANGE RATE DYNAMICS WITH SLUGGISH PRICES UNDER ALTERNATIVE PRICE-
ADJUSTMENT RULES

Maurice Obstfeld and Kenneth Rogof ¢l/

1. INTRODUCTION

Recent research on the asset-market approach to exchange rates has
incorporated short-run Keynesian price rigidities into models assuming
rational expectations. These sticky-price models generally exhibit classical
properties in the long run, but allow for temporary goods-market disequilibrium
in response to real and monetary shocks that are less than perfectly anticipated.
A critical element in these models is the mechanism determining how domestic
goods prices adjust over time in response both to current disequilibrium
and to expectations of future events.

In his seminal paper on exchange-rate determination with sticky
domestic prices, Dornbusch [1976] assumed that the nominal price of domestic
output is a predetermined variable that moves in response to current goods-
market disequilibrium only. Mussa [1977, 1982] has criticized the simple
Dornbusch adjustment rulé as being inadequate in situations where future
disturbances are anticipated or where the long-run equilibrium of the economy
moves over time. Frankel [1979], Liviatan [1980], and Buiter and Miller
(1981, 1982] introduce trend inflation into the Dornbusch model by linking
price adjustment to the underlying (constant) money growth rate in addition
to direct goods-market pressure. An alternative price-adjustment scheme
allowing for very general moving long-run equilibria is derived by Mussa [1981].

The paper compares the price-adjustment rule of Mussa [1981] to a
rule advanced by Barro and Grossman [1976] in a closed-economy context. The
interest of the Barro-Grossman rule is twofold. First, like Mussa's rule,

that of Barro and Grossman is appropriate in models with anticipated future

1/ Columbia University and the Board of Governors of the Federal Reserve System,
respectively. This paper represents the views of the author (authors) and should

not be interpreted as reflecting the views of the Board of Governors of the Federal

Reserve System or other members of its staff.

aT


-12-

[see the appendix, equations (A35) and (A36)]. The two sets of solutions can be made
numerically equal by choosing @ (the speed-of-adjustment parameter in the Mussa
model) so that

(17) 9 = [Amd + tao - (1 - a)] + (One + me - a=o1' +

Bs
y"
2oxC1 = Tra) 4(od)2(. = tao)? y

pa - TA0.

for when @ is so chosen, -@ = Ny [cf. (15)]. The two models are therefore
structurally equivalent,

It is important to note that the theorem holds only along the
saddlepath. The models are not structurally equivalent elsewhere.

When condition (13) is violated (as it necessarily is for m sufficiently
large), the model with the Barro-Grossman adjustment rule has two characteristic roots
with positive real part. Saddlepath stability requires that a negative root
be associated with each predetermined variable in the model; and because p
is predetermined, a rational-expectations equilibrium will not exist in
general when (13) fails and the model is unstable 24/ In contrast, the
Mussa rule necessarily yields saddlepath stability for any nonnegative
values of the system's parameters; and as the speed-of-adjustment parameter
6 + +0, the Mussa model converges smoothly to the flex-price model. It
seems worrisome, at first glance, that under the Barro-Grossman rule, the
economy becomes unstable and does not converge to the flex-price model as
m1 + +, We will argue shortly that when the price adjustment implied by
the Barro-Grossman mechanism is interpreted properly, (13) always holds

and this apparent convergence problem disappears.

147 the instability is due to the fact that when (13) does not hold, a
positive shock to aggregate demand (an increase in u) leads, ceteris

paribus, to falling prices.


-15-

5. CONCLUSION

This paper has studied the consequences of adopting alternative
sticky-price adjustment rules in exchange rate models characterized by
moving long-run equilibria. A price-adjustment scheme suggested by Barro
and Grossman [1976] in a different context was shown to be a natural general-
ization of less versatile price-adjustment schemes advanced by Dornbusch [1976],
by Frankel [1979], by Liviatan [1986], and by Buiter and Miller [1981, 1982].
It was also demonstrated that use of the Barro-Grossman rule results in an
exchange rate model that is structurally equivalent to one based on the
apparently quite different price-adjustment rule proposed by Mussa [1977,
1982]. The choice between the Barro-Grossman and Mussa rules is therefore
not critical for many theoretical and empirical applications 2°! Unlike the
simpler price-adjustment rules used in earlier studies, either yields
sensible results for any expected path of the exogenous variables driving

the system.

Columbia University, U.S.A.

Board of Governors of the Federal Reserve System, U.S.A.

16/ As Lucas [1976] argues, changes of policy regime could alter the
parameters appearing in the two rules. In some cases it may be easier
to model this possibility using the Mussa rule, which at present has a

somewhat better-developed microeconomic rationale.