Chapter 9: Money and the Open Economy(Monetary Theory and Policy, 3rd ed.)�

Carl E. Walsh

March 2009

1 Introduction

The analysis in earlier chapters was conducted within the context of a closedeconomy. Many useful insights into monetary phenomena can be obtained whilestill abstracting from the linkages that tie di¤erent economies together, butclearly many issues do require an open-economy framework if they are to beadequately addressed. New channels through which monetary factors can in-�uence the economy arise in open economies. Exchange-rate movements, forexample, play an important role in the transmission process that links mone-tary disturbances to output and in�ation movements. Open economies face thepossibility of economic disturbances that originate in other countries, and thisraises questions of monetary policy design that are absent in a closed-economyenvironment: Should policy respond to exchange-rate movements? Should mon-etary policy be used to stabilize exchange rates? Should national monetarypolicies be coordinated?In this chapter, we begin in section 2 by studying a two-country model based

on Obstfeld and Rogo¤ (1995, 1996). The two-country model has the advan-tage of capturing some of the important linkages between economies while stillmaintaining a degree of simplicity and tractability. Because an open economy islinked to other economies, policy actions in one economy have the potential toa¤ect other economies. Spillovers can occur. And policy actions in one countrywill depend on the response of monetary policy in the other. Often, because ofthese spillovers, countries attempt to coordinate their policy actions. The roleof policy coordination is examined in section 3.Section 4 considers the case of a small open economy. In the open-economy

literature, a small open economy denotes an economy that is too small to a¤ectworld prices, interest rates, or economic activity. Since many countries reallyare small relative to the world economy, the small-open-economy model providesa framework that is relevant for studying many policy issues.

� c The MIT Press. Draft chapter; do not circulate without approval.

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The analyses of policy coordination and the small open economy are con-ducted using models in which behavioral relationships are speci�ed directlyrather than derived from underlying assumptions about the behavior of individ-uals and �rms. As a result, the frameworks are of limited use for conductingnormative analysis as they are unable to make predictions about the welfare ofthe agents in the model. This is one reason for beginning the discussion of theopen economy with the Obstfeld-Rogo¤ model; it is based explicitly on the as-sumption of optimizing agents and therefore o¤ers a natural metric�in the formof the utility of the representative agent�for addressing normative policy ques-tions. This chapter ends by returning, in section 5, to a class of models based onoptimizing agents and nominal rigidities. These models are the open-economycounterparts of the new Keynesian models of Chapter 6.

2 The Obstfeld-Rogo¤Two-Country Model

Obstfeld and Rogo¤ (1995, 1996) examine the linkages between two economieswithin a framework that combines three fundamental building blocks. The �rstis an emphasis on intertemporal decisions by individual agents; foreign tradeand asset exchange open up avenues for transferring resources over time that arenot available in a closed economy. A temporary positive productivity shock thatraises current output relative to future output induces individuals to increaseconsumption both now and in the future as they try to smooth the path ofconsumption. Since domestic consumption rises less than domestic output, theeconomy increases its net exports, thereby accumulating claims against futureforeign output. These claims can be used to maintain higher consumption in thefuture after the temporary productivity increase has ended. The trade balancetherefore plays an important role in facilitating the intertemporal transfer ofresources.Monopolistic competition in the goods market is the second building block

of the Obstfeld-Rogo¤ model. This by itself has no implications for the e¤ectsof monetary disturbances, but it does set the stage for the third aspect of theirmodel�sticky prices. Since we have already discussed these basic building blocks,we will focus on the new aspects introduced by open-economy considerations.Detailed derivations of the various components of the model are provided inthe appendix to this chapter. It will simplify the exposition to deal with anonstochastic model in order to highlight the new considerations that arise inthe open-economy context.Each of the two countries is populated by a continuum of agents, indexed

by z 2 [0; 1], who are monopolistic producers of di¤erentiated goods. Agentsz 2 [0; n] reside in the home country, while agents z 2 (n; 1] reside in the foreigncountry. Thus, n provides an index of the relative sizes of the two countries.If the countries are of equal size, n = 1

2 . Foreign variables are denoted by asuperscript asterisk (�).

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The present discounted value of lifetime utility of a domestic resident j is

U j =1Xt=0

�t

"logCjt + b log

M jt

Pt� k

2yt(j)

2

#, (1)

where Cjt is agent j0s period-t consumption of the composite consumption good,

de�ned by

Cjt =

�Z 1

0

cjt (z)q

� 1q

, 0 < q < 1, (2)

and consumption by agent j of good z is cj(z), z 2 [0; 1]. The aggregate domesticprice de�ator P is de�ned as

Pt =

�Z 1

0

pt(z)q

q�1

� q�1q

. (3)

This price index P depends on the prices of all goods consumed by domesticresidents (the limits of integration run from 0 to 1). It incorporates pricesof both domestically produced goods fp(z) for z 2 [0; n]g and foreign-producedgoods fp(z) for z 2 (n; 1]g. Thus, P corresponds to a consumer price indexconcept of the price level, not a GDP price de�ator that would include only theprices of domestically produced goods.Utility also depends on the agent�s holdings of real money balances. Agents

are assumed to hold only their domestic currency, so M jt =Pt appears in the

utility function (1). Since agent j is the producer of good j, the e¤ort of pro-ducing output yt(j) generates disutility. A similar utility function is assumedfor residents of the foreign country:

U�j =

1Xt=0

�t

"logC�jt + b log

M�jt

P �t� k

2y�t (j)

2

#,

where C�j and P � are de�ned analogously to Cj and P .Agent j will pick consumption, money holdings, holdings of internationally

traded bonds, and output of good j to maximize utility subject to the budgetconstraint

PtCjt +M

jt + PtTt + PtB

jt � pt(j)yt(j) +Rt�1PtB

jt�1 +M

jt�1.

The gross real rate of interest is denoted R, and T represents real taxes minustransfers. Bonds purchased at time t� 1, Bjt�1, yield a gross real return Rt�1.As in our analysis of chapter 2, the role of T will be to allow for variations inthe nominal supply of money, with PtTt = (Mt �Mt�1). Dividing the budgetconstraint by Pt, one obtains

Cjt +M jt

Pt+ Tt +B

jt �

�pt(j)

Pt

�yt(j) +Rt�1B

jt�1 +

�1

1 + �t

�M jt�1

Pt�1, (4)

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where �t is the in�ation rate from t� 1 to t. To complete the description of theagent�s decision problem, we need to specify the demand for the good the agentproduces. This speci�cation is provided in the appendix (section 7.1), where itis shown that the following necessary �rst order conditions can be derived fromthe individual consumer/producer�s decision problem:

Cjt+1 = �RtCjt (5)

kyjt = q

�1

Cjt

� yjtCwt

!q�1(6)

M jt

Pt= bCjt

�1 + itit

�, (7)

together with the budget constraint (4) and the transversality condition

limi!1

iYs=0

R�1t+s�1

Bjt+i +

M jt+i

Pt+i

!= 0.

In these expressions, it is the nominal rate of interest, de�ned as Rt(1+�t+1)�1.In (6), Cwt � nCt + (1 � n)C�t is world consumption, where Ct =

R n0Cjt dj

and C�t =R 1nCjt dj equal total home and foreign consumption. Equation (5)

is a standard Euler condition for the optimal consumption path. Equation (6)states that the ratio of the marginal disutility of work to the marginal utilityof consumption must equal the marginal product of labor.1 Equation (7) is thefamiliar condition for the demand for real balances of the domestic currency,requiring that the ratio of the marginal utility of money to the marginal util-ity of consumption equal it=(1 + it). Similar expressions hold for the foreignconsumer/producer.We have yet to introduce the exchange rate and the link between prices for

similar goods in the two countries. Let St denote the nominal exchange rate,de�ned as the price of foreign currency in terms of domestic currency. A risein St means that the price of foreign currency has risen in terms of domesticcurrency; consequently, a unit of domestic currency buys fewer units of foreigncurrency. So a rise in St corresponds to a fall in the value of the domesticcurrency.While St is the exchange rate between the two currencies, the exchange

rate between goods produced domestically and goods produced in the foreigneconomy will play an important role. The law of one price requires that good zsell for the same price in both the home and foreign countries when expressed

1See (107) in the appendix.

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in a common currency.2 This requires

p(z) = Sp�(z).

It follows from the de�nitions of the home and foreign price levels that

Pt = StP�t . (8)

Any equilibrium must satisfy the �rst order conditions for the agent�s de-cision problem, the law-of-one-price condition, and the following additionalmarket-clearing conditions:

Goods market clearing: Cwt = n

�pt(h)

Pt

�yt(h)+(1�n)

�p�t (f)

P �t

�y�t (f) � Y wt ,

where p(h) and y(h) are the price and output of the representative home good(and similarly for p�(f) and y�(f)), and

Bond market clearing: nBt + (1� n)B�t = 0.

From the structure of the model, it should be clear that one-time propor-tional changes in the nominal home money supply, all domestic prices, and thenominal exchange rate leave the equilibrium for all real variables una¤ected�the model displays monetary neutrality. An increase in M accompanied by aproportional decline in the value of home money in terms of goods (i.e., a pro-portional rise in all p(j)) and a decline in the value of M in terms of M� (i.e.,a proportional rise in S) leaves equilibrium consumption and output in bothcountries, together with prices in the foreign country, unchanged.If we consider the model�s steady state, the budget constraint (4) becomes

C =p(h)

Py(h) + (R� 1)B, (9)

where B is the steady-state real stock of bonds held by the home country. Forthe foreign country,

C� =p(f)

Py(f)� (R� 1)

�n

1� n

�B. (10)

These two equations imply that real consumption equals real income (the realvalue of output plus income from net asset holdings) in the steady state.

2While the law of one price is intuitively appealing and provides a convenient means oflinking the prices p(j) and p�(j) to the nominal exchange rate, it may be a poor empiricalapproximation. In a study of prices in di¤erent U.S. cities, Parsley and Wei (1996) �nd ratesof price convergence to be faster than in cross-country comparisons, and they conclude thattradable-goods prices converge quickly. Even so, the half-life of a price di¤erence among U.S.cities for tradables is estimated to be on the order of 12-15 months.

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2.1 The Linear Approximation

It will be helpful to develop a linear approximation to the basic Obstfeld-Rogo¤model in terms of percentage deviations around the steady state. This servesto make the linkages between the two economies clear, and provides a base ofcomparison when, in the following section, we employ a more traditional openeconomy model that is not directly derived from the assumption of optimizingagents. Using lower case letters to denote percentage deviations around thesteady state, the equilibrium conditions can be expressed as

pt = npt(h) + (1� n) [st + p�t (f)] (11)

p�t = n [pt(h)� st] + (1� n)p�t (f) (12)

yt =1

1� q [pt � pt(h)] + cwt (13)

y�t =1

1� q [p�t � p�t (f)] + cwt (14)

nct + (1� n)c�t = cwt (15)

ct+1 = ct + rt (16)

c�t+1 = c�t + rt (17)

(2� q)yt = (1� q)cwt � ct (18)

(2� q)y�t = (1� q)cwt � c�t (19)

mt � pt = ct � �(rt + �t+1) (20)

m�t � p�t = c�t � �(rt + ��t+1), (21)

where � = �=(���) and � is 1 plus the steady-state rate of in�ation (assumedto be equal in both economies). Equations (11) and (12) express the domesticand foreign price levels as weighted averages of the prices of home- and foreign-produced goods expressed in a common currency. The weights depend on therelative sizes of the two countries as measured by n. Equations (13) and (14)are derived from (99) of the appendix and give the demand for each country�soutput as a function of world consumption and relative price. Increases in worldconsumption (cw) increase the demand for the output of both countries, whiledemand also depends on a relative price variable. Home country demand, forexample, falls as the price of home production p(h) rises relative to the homeprice level. Equation (15) de�nes world consumption as the weighted averageof consumption in the two countries.Equations (16)-(21) are from the individual agent�s �rst order conditions (5),

(6), and (7). The �rst two of these equations are simply the Euler condition forthe optimal intertemporal allocation of consumption; the change in consumptionis equal to the real rate of return. Equations (18) and (19) are implied by optimalproduction decisions. Finally, (20) and (21) give the real demand for home andforeign money as functions of consumption and nominal interest rates. While

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both countries face the same real interest rate rt, nominal interest rates maydi¤er if in�ation rates di¤er between the two countries.The equilibrium path of home and foreign production (yt; y�t ), home, domes-

tic, and world consumption (ct; c�t ; cwt ), prices and the nominal exchange rate

(pt(h); pt; p�t (f); p�t ; st), and the real interest rate (rt) must be consistent with

these equilibrium conditions.3 Note that subtracting (12) from (11) implies

st = pt � p�t , (22)

while the addition of n times (13) and (1 � n) times (14) yields the goodsmarket-clearing relationship equating world production to world consumption:nyt + (1� n)y�t = cwt .

2.2 Equilibrium with Flexible Prices

The linear version of the two-country model serves to highlight the channelsthat link open economies. Using this framework, we �rst discuss the role ofmoney when prices are perfectly �exible. As in the closed-economy case, thereal equilibrium is independent of monetary phenomena when prices can move�exibly to o¤set changes in the nominal supply of money.4 Prices and thenominal exchange rate will depend on the behavior of the money supplies in thetwo countries, and the adjustment of the nominal exchange rate becomes part ofthe equilibrating mechanism that insulates real output and consumption frommonetary e¤ects.The assumption of a common capital market, implying that consumers in

both countries face the same real interest rate, means from the Euler conditions(16) and (17) that ct+1�c�t+1 = ct�c�t ; any di¤erence in relative consumption ispermanent. And world consumption cw is the relevant scale variable for demandfacing both home and domestic producers.

2.2.1 Real-Monetary Dichotomy

With prices and the nominal exchange rate free to adjust immediately in theface of changes in either the home or foreign money supply, the model displaysthe classical dichotomy discussed in section 5.3.1 under which the equilibriumvalues of all real variables can be determined independently of the money supplyand money demand factors. To see this, de�ne the two relative price variables�t � pt(h)� pt and ��t � p�t (f)� p�t . Equations (11) and (12) imply that

n�t + (1� n)��t = 0,3Equations (20)-(21) di¤er somewhat from Obstfeld and Rogo¤�s speci�cation due to dif-

ferences in the methods used to obtain linear approximations. See Obstfeld and Rogo¤ (1996,chapter 10).

4Recall from the discussion in chapter 2 that the dynamic adjustment outside the steadystate is independent of money when utility is log separable, as assumed in (1). This resultwould also characterize this open economy model if it were modi�ed to incorporate stochasticuncertainty due to productivity and money growth rate disturbances.

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while (13) and (14) can be rewritten as

yt = ��t1� q + c

wt

y�t = ���t1� q + c

wt .

These equations, together with (15)-(19), su¢ ce to determine the real equilib-rium. The money demand equations (20) and (21) determine the price paths,while (22) determines the equilibrium nominal exchange rate given these pricepaths. Thus, an important implication of this model is that monetary policy (de-�ned as changes in nominal money supplies) has no short-run e¤ects on the realinterest rate, output, or consumption in either country. Rather, only nominalinterest rates, prices, and the nominal exchange rate are a¤ected by variations inthe nominal money stock. One-time changes in m produce proportional changesin p, p(h), and s.Changes in the growth rate of m a¤ect in�ation and nominal interest rates.

Equation (20) shows that in�ation a¤ects the real demand for money, so di¤erentrates of in�ation are associated with di¤erent levels of real money balances.Changes in nominal money growth rates produce changes in the in�ation rateand nominal interest rates, thereby a¤ecting the opportunity cost of holdingmoney and, in equilibrium, the real stock of money. The price level and thenominal exchange rate jump to ensure that the real supply of money is equal tothe new real demand for money.Equations (21) can be subtracted from (20), yielding

mt �m�t � (pt � p�t ) = (ct � c�t )� �(�t+1 � ��t+1),

which, using (22), implies5

mt �m�t � st = (ct � c�t )� �(st+1 � st). (23)

Solving this forward for the nominal exchange rate, the no-bubbles solution is

st =1

1 + �

1Xi=0

��

1 + �

�i �(mt+i �m�

t+i)� (ct+i � c�t+i)�. (24)

Since (16) and (17) imply that ct+i � c�t+i = ct � c�t , the expression for thenominal exchange rate can be rewritten as

st = �(ct � c�t ) +1

1 + �

1Xi=0

��

1 + �

�i �mt+i �m�

t+i

�.

The current nominal exchange rate depends on the current and future path of thenominal money supplies in the two countries and on consumption di¤erentials.

5This uses the fact that �t+1 � ��t+1 = (pt+1 � p�t+1)� (pt � p�t ) = st+1 � st.

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The exchange rate measures the price of one money in terms of the other, and, as(24) shows, this depends on the relative supplies of the two monies. An increasein one country�s money supply relative to the other�s depreciates that country�sexchange rate. From the standard steady-state condition that �Rss = 1 andthe de�nition of � as �=(��� �), the discount factor in (24), �=(1 + �), is equalto �=�� = 1=Rss �� = 1=(1 + iss). Future nominal money supply di¤erentialsare discounted by the steady-state nominal rate of interest. Because agentsare forward-looking in their decision making, it is only the present discountedvalue of the relative money supplies that matters. In other words, the nominalexchange rate depends on a measure of the permanent money supply di¤erential.Letting xt+i � (mt+i �m�

t+i)� (ct+i � c�t+i), the equilibrium condition for thenominal exchange rate can be written as

st =1

1 + �

1Xi=0

��

1 + �

�ixt+i =

1

1 + �xt +

�

1 + �

1Xi=0

��

1 + �

�ixt+1+i

=1

1 + �xt +

�

1 + �st+1:

Rearranging and using (24) yields

st+1 � st = �1�(xt � st)

= �1�

"(mt �m�

t )�1

1 + �

1Xi=0

��

1 + �

�i(mt+1+i �m�

t+1+i)

#.

Analogously to Friedman�s permanent income concept, the term

1

1 + �

1Xi=0

��

1 + �

�i(mt+1+i �m�

t+1+i)

can be interpreted as the permanent money supply di¤erential. If the currentvalue of m�m� is high relative to the permanent value of this di¤erential, thenominal exchange rate will fall (the home currency will appreciate). If st re�ectsthe permanent money supply di¤erential at time t, and mt is temporarily highrelative to m�

t , then the permanent di¤erential will be lower beginning in periodt+ 1. As a result, the home currency appreciates.An explicit solution for the nominal exchange rate can be obtained if speci�c

processes for the nominal money supplies are assumed. To take a very simplecase, suppose m and m� each follow constant, deterministic growth paths givenby

mt = m0 + �t

andm�t = m�

0 + ��t.

Strictly speaking, (24) applies only to deviations around the steady state andnot to money-supply processes that include deterministic trends. However, it is

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very common to specify (20) and (21), which were used to derive (24), in termsof the log levels of the variables, perhaps adding a constant to represent steady-state levels. The advantage of interpreting (24) as holding for the log levels ofthe variables is that we can then use it to analyze shifts in the trend growth pathsof the nominal money supplies, rather than just deviations around the trend. Itis important to keep in mind, however, that the underlying representative-agentmodel implies that the interest rate coe¢ cients in the money-demand equationsare functions of the steady-state rate of in�ation. We assume this is the samein both countries, implying that the � parameter was the same as well. Theassumption of common coe¢ cients in two country models is common, and wewill maintain it in the following examples. The limitations of doing so shouldbe kept in mind.Then (24) implies

st = �(ct � c�t ) +1

1 + �

1Xi=0

��

1 + �

�i[m0 �m�

0 + (�� ��)(t+ i)]

= s0 + (�� ��)t� (ct � c�t ) ,

where s0 = m0�m�0+ �(����).6 In this case, the nominal exchange rate has a

deterministic trend equal to the di¤erence in the trend of money growth rates inthe two economies (also equal to the in�ation rate di¤erentials since � = � and�� = ��). If domestic money growth exceeds foreign money growth (� > ��),s will rise over time to re�ect the falling value of the home currency relative tothe foreign currency.

2.2.2 Uncovered Interest Parity

Real rates of return in the two countries have been assumed to be equal, so theEuler conditions for the optimal consumption paths (16 and 17) imply the sameexpected consumption growth in each economy. It follows from the equality ofreal returns that nominal interest rates must satisfy it � �t+1 = rt = i�t � ��t+1,and this means, using (22), that

it � i�t = �t+1 � ��t+1 = st+1 � st.

The nominal interest rate di¤erential is equal to the actual change in the ex-change rate in a perfect-foresight equilibrium. This equality would not hold inthe presence of uncertainty, since then variables dated t + 1 would need to bereplaced with their expected values, conditional on the information available attime t. In this case,

Etst+1 � st = it � i�t , (25)

and nominal interest rate di¤erentials would re�ect expected exchange-ratechanges. If the home country has a higher nominal interest rate in equilib-rium, its currency must be expected to depreciate (s must be expected to rise)to equalize real returns across the two countries.

6This uses the fact thatP1i=0 ib

i = b=(1� b)2 for j b j< 1.

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This condition, known as uncovered nominal interest parity, links interestrates and exchange rate expectations in di¤erent economies if their �nancialmarkets are integrated. Under rational expectations, we can write the actualexchange rate at t+1 as equal to the expectation of the future exchange rate plusa forecast error 't uncorrelated with Etst+1: st+1 = Etst+1 + 't. Uncoveredinterest parity then implies

st+1 � st = it � i�t + 't+1.

The ex post observed change in the exchange rate between times t and t + 1is equal to the interest-rate di¤erential at time t plus a random, mean zeroforecast error. Since this forecast error will, under rational expectations, beuncorrelated with information, such as it and i�t , that is known at time t, wecan recast uncovered interest parity in the form of a regression equation:

st+1 � st = a+ b(it � i�t ) + 't+1, (26)

with the null hypothesis of uncovered interest parity implying that a = 0 andb = 1. Unfortunately, the evidence rejects this hypothesis.7 In fact, estimatedvalues of b are often negative.One interpretation of these rejections is that the error term in an equation

such as (26) is not simply due to forecast errors. Suppose, more realistically,that (25) does not hold exactly:

Etst+1 � st = it � i�t + vt,

where vt captures factors such as risk premia that would lead to divergencesbetween real returns in the two countries. In this case, the error term in theregression of st+1�st on it�i�t becomes vt+'t+1. If vt and it�i�t are correlated,ordinary least squares estimates of the parameter b in (26) will be biased andinconsistent.Correlation between v and i� i� might arise if monetary policies are imple-

mented in a manner that leads the nominal interest-rate di¤erential to respondto the current exchange rate. For example, suppose that the monetary authorityin each country tends to tighten policy whenever its currency depreciates. Thiscould occur if the monetary authorities are concerned with in�ation; deprecia-tion raises the domestic currency price of foreign goods and raises the domesticprice level. To keep the example simple for illustrative purposes, suppose thatas a result of such a policy, the nominal interest rate di¤erential is given by

it � i�t = �st + ut; � > 0,

where ut captures any other factors a¤ecting the interest rate di¤erential.8 As-sume u is an exogenous, white noise process. Substituting this into the uncov-

7For a summary of the evidence, see Froot and Thaler (1990). See also McCallum (1994a),Eichenbaum and Evans (1995), and Schlagenhauf and Wrase (1995).

8The rationale for such a policy is clearly not motivated within the context of a model withperfectly �exible prices in which monetary policy has no real e¤ects. The general point is toillustrate how empirical relationships such as (26) can depend on the conduct of policy.

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ered interest parity condition yields

Etst+1 = (1 + �)st + ut + vt, (27)

the solution to which is9

st = ��

1

1 + �

�(ut + vt) .

Since this solution implies that Etst+1 = �Et(ut+1 + vt+1)=(1 + �) = 0, theinterest parity condition is then given by

Etst+1 � st = ��

1

1 + �

�(ut + vt) = it � i�t + vt

or it � i�t = (ut � �vt)=(1 + �).What does this imply for tests of uncovered interest parity? From the solu-

tion for st, st+1� st = � (ut+1 � ut + vt+1 � vt) =(1+�). The probability limitof the interest-rate coe¢ cient in the regression of st+1� st on it� i�t is equal to

cov(st+1 � st; it � i�t )var(it � i�t )

=

1(1+�)2

��2u � ��2v

�1

(1+�)2 (�2u + �

2�2v)=

�2u � ��2v�2u + �

2�2v,

which will not generally equal 1, the standard null in tests of interest parity.If u � 0, the probability limit of the regression coe¢ cient is �1=�. That is,the regression estimate uncovers the policy parameter �. Not only would aregression of the change in the exchange rate on the interest di¤erential notyield the value of 1 predicted by the uncovered interest parity condition, butthe estimate would be negative.McCallum (1994a) develops more fully the argument that rejections of un-

covered interest parity may arise because standard tests compound the paritycondition with the manner in which monetary policy is conducted. While un-covered interest parity is implied by the model independently of the mannerin which policy is conducted, the outcomes of statistical tests may in fact bedependent on the behavior of monetary policy, since policy may in�uence thetime-series properties of the nominal-interest-rate di¤erential.As noted earlier, tests of interest parity often report negative regression

coe¢ cients on the interest rate di¤erential in (26). This �nding is also consistentwith the empirical evidence reported by Eichenbaum and Evans (1995). Theyestimate the impact of monetary shocks on nominal and real exchange ratesand interest-rate di¤erentials between the United States and France, Germany,Italy, Japan, and the United Kingdom. A contractionary U.S. monetary-policyshock leads to a persistent nominal and real appreciation of the dollar and

9From (27), the equilibrium exchange-rate process must satisfy Etst+1 = (1+�)st+ut+vtso that the state variables are ut and vt. Following McCallum (1983a), the minimal statesolution takes the form st = b0 + b1(ut + vt). This implies that Etst+1 = b0. So the interestparity condition becomes b0 = (1 + �) (b0 + b1(ut + vt)) + ut + vt, which will hold for allrealizations of u and v if b0 = 0 and b1 = �1=(1 + �).

12

a fall in i�t � it + st+1 � st, where i is the U.S. interest rate and i� is theforeign rate. Uncovered interest parity implies that this expression should haveexpected value equal to zero, yet it remains predictably low for several months.Rather than leading to an expected depreciation that o¤sets the rise in i, excessreturns on U.S. dollar securities remain high for several months following acontractionary U.S. monetary-policy shock.10

2.3 Sticky Prices

Just as was the case with the closed economy, �exible-price models of the openeconomy appear unable to replicate the size and persistence of monetary shockson real variables. And just as with closed-economy models, this can be remediedby the introduction of nominal rigidities. Obstfeld and Rogo¤ (1996, chapter10) provide an analysis of their basic two-country model under the assumptionthat prices are set one period in advance.11 The presence of nominal rigiditiesleads to real e¤ects of monetary disturbances through the channels discussedin chapter 5, but new channels through which monetary disturbances have reale¤ects are now also present.Suppose p(h), the domestic currency price of domestically produced goods, is

set one period in advance and �xed for one period. A similar assumption is madefor the foreign currency price of foreign-produced goods, p�(f). Although p(h)and p�(f) are preset, the aggregate price indices in each country will �uctuatewith the nominal exchange rate according to (11) and (12). Nominal deprecia-tion, for example, raises the domestic price index p by increasing the domesticcurrency price of foreign-produced goods. This introduces a new channel, oneabsent in a closed economy, through which monetary disturbances can have animmediate impact on the price level. Recall that in the closed economy, there isno distinction between the price of domestic output and the general price level.Nominal price rigidities implied the price level cannot adjust immediately tomonetary disturbances. Exchange-rate movements alter the domestic currencyprice of foreign goods, allowing the consumer price index to move in responseto such disturbances even in the presence of nominal rigidities.Now suppose that in period t the home country�s money supply rises unex-

pectedly relative to that of the foreign country.12 Under Obstfeld and Rogo¤�ssimplifying assumption that prices adjust completely after one period, botheconomies return to their steady state one period after the change in m. But

10Eichenbaum and Evans measure monetary policy shocks in a variety of ways (VAR innova-tions to nonborrowed reserves relative to total reserves, VAR innovations to the federal fundsrate, and Romer and Romer�s 1989 measures of policy shifts). However, the identi�cationscheme used in their VARs assumes that policy does not respond contemporaneously to thereal exchange rate. This means that the speci�c illustrative policy response to the exchangerate that led to (27) is ruled out by their framework.11They also consider the case in which nominal wages are preset.12An unexpected change is inconsistent with the assumption of perfect foresight implicit in

the nonstochastic version of the model derived earlier. However, the linear approximation willcontinue to hold under uncertainty if future variables are replaced with their mathematicalexpectation.

13

during the one period in which product prices are preset, real output and con-sumptions levels will be a¤ected.13 And these real e¤ects mean that the homecountry may run a current account surplus or de�cit in response to the changein m. This e¤ect on the current account alters the net asset positions of thetwo economies and can a¤ect the new steady-state equilibrium.Interpreting the model consisting of (11)-(21) as applying to deviations

around the initial steady state, the Euler conditions (16) and (17), imply thatct+1� c�t+1 = ct� c�t . Since the economies are in the new steady state after oneperiod (i.e., in t+1), ct+1�c�t+1 � C is the steady-state consumption di¤erentialbetween the two countries. But since we also have that ct�c�t = ct+1�c�t+1 = C,this relationship implies that relative consumption in the two economies imme-diately jumps in period t to the new steady-state value. Equation (23), whichexpresses relative money demands in the two economies, can then be writtenmt �m�

t � st = C � �(st+1 � st). Solving this equation forward for the nominalexchange rate (assuming no bubbles),

st =1

1 + �

1Xi=0

��

1 + �

�i �mt+i �m�

t+i � C�.

If the change in mt � m�t is a permanent one-time change, we can let �

m�m� without time subscripts denote this permanent change. The equilibriumexchange rate is then equal to

st =1

1 + �

1Xi=0

��

1 + �

�i(� C) = � C. (28)

Since �C is a constant, (28) implies that the exchange rate jumps immediatelyto its new steady state following a permanent change in relative nominal moneysupplies. If relative consumption levels do not adjust (i.e., if C = 0), then thepermanent change in s is just equal to the relative change in nominal moneysupplies . An increase inm relative tom� (i.e., > 0) produces a depreciationof the home country currency. If C 6= 0, then changes in relative consumptiona¤ect the relative demand for money from (20) and (21). For example, if C > 0,consumption, as well as money demand, in the home country is higher thaninitially. Equilibrium between home money supply and home money demandcan be restored with a smaller increase in the home price level. Since p(h) andp�(f) are �xed for one period, the increase in p necessary to maintain real moneydemand and real money supply equal is generated by a depreciation (a rise ins). The larger the rise in home consumption, the larger the rise in real moneydemand and the smaller the necessary rise in s. This is just what (28) says.While we have determined the impact of a change in m � m� on the ex-

change rate, given C, the real consumption di¤erential is itself endogenous.To determine C requires that we work through several steps. First, the lin-ear approximation to the current account relates the home country�s accu-mulation of net assets to the excess of its real income over its consumption:13 If we consider a situation in which the economies are initially in a steady state, the preset

values for p(h) and p�(f) will equal zero.

14

b = yt+[pt(h)� pt]� ct = yt� (1�n)st� ct, where pt(h)� pt = �(1�n)st fol-lows from (11) and the fact that pt(h) is �xed (and equal to zero) during periodt. Similarly, for the foreign economy, �nb=(1 � n) = y�t + nst � c�t . Together,these imply

b

1� n = (yt � y�t )� (ct � c�t )� st. (29)

From (13) and (14), yt � y�t = st=(1� q), so (29) becomes

b

1� n =�

q

1� q

�st � (ct � c�t ) =

�q

1� q

�st � C, (30)

where we have made use of the de�nition of C as the consumption di¤erential.The last step is to use the steady-state relationship between consumption,

income, and asset holdings given by (9) and (10) to eliminate b in (30) byexpressing it in terms of the exchange-rate and consumption di¤erences. In thesteady state, b is constant and current accounts are zero, so consumption equalsreal income inclusive of asset income. In terms of the linear approximation, (9)and (10) become

c = rb+ y + [p(h)� p] = rb+ y � (1� n) [s+ p�(f)� p(h)] (31)

and

c� = ��

n

1� n

�rb+ y� + n [s+ p�(f)� p(h)] . (32)

From the steady-state labor-leisure choice linking output and consumption givenin (18) and (19), (2� q)(y� y�) = �(c� c�), and from the link between relativeprices and demand from (13) and (14), y � y� = [s+ p�(f)� p(h)] =(1 � q).Using these relationships, we can now subtract (32) from (31), yielding

C =

�1

1� n

�rb+ (y � y�)� [s+ p�(f)� p(h)]

=

�1

1� n

�rb+ q (y � y�)

=

�1

1� n

�rb+

�q

q � 2

�C.

Finally, this yields

b =

�1� nr

��2

2� q

�C. (33)

Substituting (33) into (30), [2=(2� q)] C=r = qst=(1 � q) � C. Solving for sin terms of C,

st = C, (34)

where = (1 � q) [1 + 2=r(2� q)] =q > 0. But from (28) we have already seenthat st = � C, so C = � C. It follows that the consumption di¤erentialis C = =(1 + ). The equilibrium nominal exchange-rate adjustment to a

15

permanent change in the home country�s nominal money supply is then givenby

st =

�

1 +

� < .

With > 0, the domestic monetary expansion leads to a depreciation thatis less than proportional to the increase in m. This induces an expansion indomestic production and consumption. Consumption rises by less than income,so the home country runs a current account surplus and accumulates assetsthat represent claims against the future income of the foreign country. Thisallows the home country to maintain higher consumption forever. As we haveseen, consumption levels jump immediately to their new steady state with C ==(1 + ) > 0. To provide a rough magnitude for , if q = :9 and r = :05, � 4, so st = :8 while C = :2.The two-country model employed in this section has the advantage of being

based on the clearly speci�ed decision problems faced by agents in the model.As a consequence, the responses of consumption, output, interest rates, and theexchange rate are consistent with optimizing behavior. Unanticipated mone-tary disturbances can have a permanent impact on real consumption levels andwelfare when prices are preset. These e¤ects arise because the output e¤ectsof a monetary surprise alter each country�s current account, thereby alteringtheir relative asset positions. A monetary expansion in the home country, forinstance, produces a currency depreciation and a rise in the domestic price levelp. This, in turn, induces a temporary expansion in output in the home country(see 13). With consumption determined on the basis of permanent income, con-sumption rises less than output, leading the home country to run a trade surplusas the excess of output over domestic consumption is exported. As payment forthese exports, the home country receives claims against the future output ofthe foreign country. Home consumption does rise, even though the increase inoutput lasts only one period, as the home country�s permanent income has risenby the annuity value of its claim on future foreign output.A domestic monetary expansion leads to permanently higher real consump-

tion for domestic residents; welfare is increased. This observation suggests thateach country has an incentive to engage in a monetary expansion. However, ajoint proportionate expansion of each country�s money supply leaves m � m�

unchanged. There are then no exchange-rate e¤ects, and relative consumptionlevels do not change. After one period, when prices fully adjust, a proportionalchange in p(h) and p�(f) returns both economies to the initial equilibrium. Butsince output is ine¢ ciently low because monopolistic competition is present, theone-period rise in output does increase welfare in both countries. Both coun-tries have an incentive to expand their money supplies, either individually orin a coordinated fashion. But the analysis we have carried out has involvedchanges in money supplies that were unexpected. If they had been anticipated,the level at which price setters would set individual goods prices would have in-corporated expectations of money-supply changes. As we have seen in chapter5, fully anticipated changes in the nominal money supply will not have the real

16

e¤ects that unexpected changes do when there is some degree of nominal wageor price rigidity. As we will see in chapter 8, the incentive to create surprise ex-pansions can, in equilibrium, lead to steady in�ation without the welfare gainsan unanticipated expansion would bring.Unanticipated, permanent changes in the money supply can have permanent

e¤ects on the international distribution of wealth in the Obstfeld-Rogo¤ modelwhen there are nominal rigidities. Corsetti and Presenti (2001) have developed atwo-country model with similar micro foundations as the Obstfeld-Rogo¤modelbut in which preferences are speci�ed so that changes in national money suppliesdo not cause wealth redistributions. Corsetti and Presenti assume the elasticityof output supply with respect to relative prices and the elasticity of substitutionbetween the home-produced and foreign-produced goods are both equal to 1.Thus, an increase in the relative price of the foreign good (a decline in theterms of trade) lowers the purchasing power of domestic agents, but it alsoleads to a rise in the demand for domestic goods that increases nominal incomes.These two e¤ects cancel, leaving the current account and international lendingand borrowing una¤ected. By eliminating current account e¤ects, the Corseti-Presenti model allows for a tractable closed-form solution with 1-period nominalwage rigidity, and it allows them to determine the impact of policy changes onwelfare.A large and growing literature has studied open-economy models that are

explicitly based on optimizing behavior by �rms and households but which alsoincorporate nominal rigidities. Besides the work of Obstfeld and Rogo¤ (1995,1996) and Corsetti and Presenti (2001), examples include Betts and Devereux(2000), Obstfeld and Rogo¤ (2000), Benigno and Benigno (2001), Corsetti andPresenti (2002), and Kollmann (2001). Lane (2001) and Engle (2002) providesurveys of the �new open economy macroeconomics.�

3 Policy Coordination

An important issue facing economies linked by trade and capital �ows is the roleto be played by policy coordination. Monetary policy actions by one countrywill a¤ect other countries, leading to spillover e¤ects that open the possibilityof gains from policy coordination. As demonstrated in the previous section,the real e¤ects of an unanticipated change in the nominal money supply in thetwo-country model depend on how m�m� is a¤ected. A rise in m, holding m�

unchanged, will produce a home country depreciation, shifting world demandtoward the home country�s output. With preset prices and output demand de-termined, the exchange-rate movement represents an important channel throughwhich a monetary expansion a¤ects domestic output. If both monetary author-ities attempt to generate output expansions by increasing their money supplies,this exchange-rate channel will not operate, since the exchange rate depends onthe relative money supplies. Thus, the impact of an unanticipated change in mdepends critically on the behavior of m�.This dependence raises the issue of whether there are gains from coordinating

17

monetary policy. Hamada (1976) is closely identi�ed with the basic approachthat has been used to analyze policy coordination, and in this section, we de-velop a version of his framework. Canzoneri and Henderson (1989) provide anextensive discussion of monetary policy coordination issues; a survey is providedby Currie and Levine (1991).Suppose we consider a model with two economies. We will assume that

each economy�s policy authority can choose its in�ation rate and, because ofnominal rigidities, monetary policy can have real e¤ects in the short run. Inthis context, a complete speci�cation of policy behavior is more complicatedthan in a closed-economy setting; we need to specify how each national pol-icy authority interacts strategically with the other policy authority. We willexamine two possibilities. First, we consider coordinated policy, meaning thatin�ation rates in the two economies are chosen jointly to maximize a weightedsum of the objective functions of the two policy authorities. Second, we con-sider noncoordinated policy, with the policy authorities interacting in a Nashequilibrium. In this setting, each policy authority sets its own in�ation rate tomaximize its objective function, taking as given the in�ation rate in the othereconomy. These clearly are not the only possibilities. One economy may act asa Stackelberg leader, recognizing the impact its choice has on the in�ation rateset by the other economy. Reputational considerations along the lines we willstudy in chapter 10 can also be incorporated into the analysis (see Canzoneriand Henderson 1989).

3.1 The Basic Model

The two-country model is speci�ed as a linear system in log deviations arounda steady state and represents an extension to the open-economy environment ofthe sticky-wage, AS-IS model of chapter 6. The LM relationship is dispensedwith by assuming that the monetary policy authorities in the two countries setthe in�ation rate directly. An asterisk will denote the foreign economy, and �will be the real exchange rate, de�ned as the relative price of home and foreignoutput, expressed in terms of the home currency; a rise in � represents a realdepreciation for the home economy. If s is the nominal exchange rate and p(h)and p�(f) are the prices of home and foreign output, then � = s+ p�(f)� p(h).The model should be viewed as an approximation that is appropriate whennominal wages are set in advance so that unanticipated movements in in�ationa¤ect real output. In addition to aggregate supply and demand relationships foreach economy, an interest parity condition links the real interest-rate di¤erentialto anticipated changes in the real exchange rate:

yt = �b1�t + b2 (�t � Et�1�t) + et (35)

y�t = b1�t + b2 (��t � Et�1��t ) + e�t (36)

yt = a1�t � a2rt + a3y�t + ut (37)

y�t = �a1�t � a2r�t + a3yt + u�t (38)

18

�t = r�t � rt + Et�t+1. (39)

Equations (35) and (36) relate output to in�ation surprises and the real ex-change rate. A real exchange-rate depreciation reduces home aggregate supplyby raising the price of imported materials and by raising consumer prices rel-ative to producer prices. This latter e¤ect increases the real wage in terms ofproducer prices. Equations (37) and (38) make demand in each country anincreasing function of output in the other to re�ect spillover e¤ects that ariseas an increase in output in one country raises demand for the goods producedby the other. A rise in �t (a real domestic depreciation) makes domesticallyproduced goods less expensive relative to foreign goods and shifts demand awayfrom foreign output and toward home output.To simplify the analysis, the in�ation rate is treated as the choice variable

of the policy maker. An alternative approach to treating in�ation as the policyvariable would be to specify money demand relationships for each country andthen take the nominal money supply as the policy instrument. This wouldcomplicate the analysis without o¤ering any new insights.A third approach would be to replace rt with it � Et�t+1, where it is the

nominal interest rate, and treat it as the policy instrument. An advantageof this approach is that it more closely re�ects the way most central banksactually implement policy. Because a number of new issues arise under nominalinterest rate policies (see section ?? and chapter 8), we will simply interpretpolicy as choosing the rate of in�ation in order to focus, in this section, onthe role of policy coordination. Finally, a further simpli�cation is re�ected inthe assumption that the parameters (the a0is and b

0is) are the same in the two

countries.Demand (ut; u�t ) and supply (et; e

�t ) shocks are included to introduce a role

for stabilization policy. These disturbances are assumed to be mean zero, seriallyuncorrelated processes, but we allow them to be correlated so that it will bepossible to distinguish between common shocks that a¤ect both economies andasymmetric shocks that originate in a single economy.Equation (39) is an uncovered interest rate parity condition. Rewritten in

the form rt = r�t +Et�t+1��t, it implies that the home country real interest ratewill exceed the foreign real rate if the home country is expected to experience areal depreciation.Evaluating outcomes under coordinated and noncoordinated policies requires

some assumption about the objective functions of the policy makers. In modelsbuilt more explicitly on the behavior of optimizing agents, alternative policiescould be ranked according to their implications for the utility of the agents inthe economies. Here we simply follow a common approach in which polices areevaluated on the basis of loss functions that depend on output variability andin�ation variability:

Vt =1Xi=0

�i��y2t+i + �

2t+i

�(40)

19

V �t =1Xi=0

�ih��y�t+i

�2+���t+i

�2i. (41)

The parameter � is a discount factor between 0 and 1. The weight attached tooutput �uctuations relative to in�ation �uctuations is �. While these objectivefunctions are ad hoc, they capture the idea that policy makers prefer to mini-mize output �uctuations around the steady state and �uctuations of in�ation.14

Objective functions of this basic form have played a major role in the analysisof policy. They re�ect the assumption that steady-state output will be inde-pendent of monetary policy, so policy should focus on minimizing �uctuationsaround the steady state, not on the level of output.The model can be solved to yield expressions for equilibrium output in each

economy and for the real exchange rate. To obtain the real exchange rate,�rst subtract foreign aggregate demand (38) from domestic aggregate demand(37), using the interest parity condition (39) to eliminate rt � r�t . This processyields an expression for yt � y�t . Next, subtract foreign aggregate supply (36)from domestic aggregate supply (35) to yield a second expression for yt � y�t .Equating these two expressions and solving for the equilibrium real exchangerate leads to the following:

�t =1

Bfb2(1 + a3) [(�t � Et�1�t)� (��t � Et�1��t )]

+ (1 + a3) (et � e�t )� (ut � u�t ) + a2Et�t+1, (42)

where B � 2a1 + a2 + 2b1(1 + a3) > 0. An unanticipated rise in domesticin�ation relative to unanticipated foreign in�ation or in et relative to e�t will in-crease domestic output supply relative to foreign output. Equilibrium requiresa decline in the relative price of domestic output; the real exchange rate rises(depreciates), shifting demand toward domestic output. If the domestic aggre-gate demand shock exceeds the foreign shock, ut � u�t > 0, the relative price ofdomestic output must rise (� must fall) to shift demand toward foreign output.A rise in the expected future exchange rate also leads to a rise in the currentequilibrium �. If � were to increase by the same amount as the rise in Et�t+1,the interest di¤erential rt�r�t would be left unchanged, but the higher � would,from (35) and (36), lower domestic supply relative to foreign supply. So � risesby less than the increase inEt�t+1 to maintain goods market equilibrium.

15

Notice that (42) can be written as �t = AEt�t+1+vt, where 0 < A < 1 and vtis white noise, since the disturbances are assumed to be serially uncorrelated andthe same will be true of the in�ation forecast errors under rational expectations.It follows that Et�t+1 = 0 in any no-bubbles solution. The expected future realexchange rate would be nonzero if either the aggregate demand or aggregatesupply shocks were serially correlated.14The steady-state values of y and y� are zero by de�nition. The assumption that the policy

loss functions depend on the variance of output around its steady-state level, and not on somehigher output target, is critical for the determination of average in�ation. Chapter 8 dealsextensively with the time-inconsistency issues that arise when policy makers target a level ofoutput that exceeds the economy�s equilibrium level.15The coe¢ cient on Et�t+1, a2=B is less than 1 in absolute value.

20

Now that we have obtained an expression for the equilibrium real exchangerate, this can be substituted into the aggregate supply relationships (35) and(36) to yield

yt = b2A1 (�t � Et�1�t) + b2A2 (��t � Et�1��t )�a2A3Et�t+1 +A1et +A2e�t +A3(ut � u�t ) (43)

y�t = b2A2 (�t � Et�1�t) + b2A1 (��t � Et�1��t )+a2A3Et�t+1 +A2et +A1e

�t �A3(ut � u�t ). (44)

The Ai parameters are given by

A1 =2a1 + a2 + b1(1 + a3)

B> 0

A2 =b1(1 + a3)

B> 0

A3 =b1B> 0.

Equations (43) and (44) reveal the spillover e¤ects through which the in-�ation choice of one economy a¤ects the other economy when b2A2 6= 0. Anincrease in in�ation in the home economy (assuming it is unanticipated) leadsto a real depreciation. This occurs since unanticipated in�ation leads to a homeoutput expansion (see 35). Equilibrium requires a rise in demand for homecountry production. In the closed economy, this occurs through a fall in thereal interest rate. In the open economy, an additional channel of adjustmentarises from the role of the real exchange rate. Given that Et�t+1 = 0, the in-terest parity condition (39) becomes �t = r�t � rt, so, for given r�t , the fall inrt requires a rise in �t (a real depreciation), which also serves to raise homedemand.The rise in �t represents a real appreciation for the foreign economy, and

this raises consumer-price wages relative to producer-price wages and increasesaggregate output in the foreign economy (see 36). As a result, an expansion inthe home country produces an economic expansion in the foreign country. Butas (42) shows, a surprise in�ation by both countries leaves the real exchange rateuna¤ected. It is this link that opens the possibility that outcomes will dependon the extent to which the two countries coordinate their policies.

3.2 Equilibrium with Coordination

To focus on the implications of policy coordination, we will restrict attentionto the case of a common aggregate supply shock, common in the sense that ita¤ects both countries. That is, suppose et = e�t � "t, where "t is the commondisturbance. For the rest of this section, we assume u � u� � 0, so that "represents the only disturbance.

21

In solving for equilibrium outcomes under alternative policy interactions, theobjective functions (40) and (41) simplify to a sequence of one-period problems(the problem is a static one with no link between periods). Assuming that thepolicy authority is able to set the in�ation rate after observing the supply shock"t, the decision problem under a coordinated policy is

min�;��

�1

2

��y2t + �

2t

�+1

2

h� (y�t )

2+ (��t )

2i�

subject to (43) and (44).16 The �rst order conditions are

0 = �b2A1yt + �t + �b2A2y�t

=�1 + �b22A

21 + �b

22A

22

��t + 2�b

22A1A2�

�t + �b2"t

0 = �b2A2yt + �b2A1y�t + �

�t

=�1 + �b22A

21 + �b

22A

22

���t + 2�b

22A1A2�t + �b2"t,

where we have used the fact that A1+A2 = 1 and the result that the �rst orderconditions imply Et�1�t = Et�1�

�t = 0.17 Solving these two equations yields

the equilibrium in�ation rates under coordination:

�c;t = ��c;t = ��

�b21 + �b22

�"t � ��c"t. (45)

Both countries maintain equal in�ation rates. In response to an adverse supplyshock (" < 0), in�ation in both countries rises to o¤set partially the decline inoutput. Substituting (45) into the expressions for output and the equilibriumreal exchange rate,

yc;t = y�c;t =

�1

1 + �b22

�"t < "t

and�t = 0.

The policy response acts to o¤set partially the output e¤ects of the supply shock.The larger the weight placed on output in the loss function (�), the larger thein�ation response and the more output is stabilized. Because both economiesrespond symmetrically, the real exchange rate is left una¤ected.18

16 In de�ning the objective function under coordinated policy, we have assumed that eachcountry�s utility receives equal weight.17Writing out the �rst order condition for �t in full, 0 = �t+�b22

�A21 +A

22

�(�t � Et�1�t)+

2�b22A1A2 (��t � Et�1��t )+2�b2"t. Taking expectations conditional on time t�1 information

(i.e., prior to the realization of "t), we obtain Et�1�t = 0.18This would not be the case in response to an asymmetric supply shock. See problem 4.

22

3.3 Equilibrium without Coordination

When policy is not coordinated, some assumption must be made about thenature of the strategic interaction between the two separate policy authorities.One natural case to consider corresponds to a Nash equilibrium; the policyauthorities choose in�ation to minimize loss, taking as given the in�ation ratein the other economy. An alternative case arises when one country behavesas a Stackelberg leader, taking into account how the other policy authority willrespond to the leader�s choice of in�ation. We will analyze the Nash case, leavingthe Stackelberg case to be studied as a problem at the end of the chapter.The home policy authority picks in�ation to minimize �y2t + �2t , taking �

�t

as given. The �rst order condition is

0 = �b2A1yt + �t

=�1 + �b22A

21

��t + �b

22A1A2�

�t + �b2A1"t

so that the home country�s reaction function is

�t = ���b22A1A21 + �b22A

21

���t �

��b2A11 + b22A

21

�"t. (46a)

A rise in the foreign country�s in�ation rate is expansionary for the domesticeconomy (see 43). The domestic policy authority lowers domestic in�ation topartially stabilize domestic output. A parallel treatment of the foreign countrypolicy authority�s decision problem leads to the reaction function

��t = ���b22A1A21 + �b22A

21

��t �

��b2A1

1 + �b22A21

�"t. (47)

Jointing solving these two reaction functions for the Nash equilibrium in�ationrates yields

�N;t = ��N;t = ��

�b2A11 + �b22A1

�"t � ��N"t. (48)

How does stabilization policy with noncoordinated policies compare with thecoordinated policy response given in (45)? Since A1 < 1,

j�N j < j�cj .

Policy responds less to the aggregate supply shock in the absence of coordina-tion, and as a result, output �uctuates more:

yN;t = y�N;t =

�1

1 + �b22A1

�"t >

�1

1 + �b22

�"t.

Because output and in�ation responses are symmetric in the Nash equilibrium,the real exchange rate does not respond to "t.

23

Why does policy respond less in the absence of coordination? For eachindividual policy maker, the perceived marginal output gain from more in�ationwhen there is an adverse realization of " re�ects the two channels through whichin�ation a¤ects output. First, surprise in�ation directly increases real outputbecause of the assumption of nominal rigidities. This direct e¤ect is given bythe term b2 (�t � Et�1�t) in (35). Second, for given foreign in�ation, a rise inhome in�ation leads to a real depreciation (see 42) and, again from (35), the risein �t acts to lower output, reducing the net impact of in�ation on output. With�� treated as given, the exchange-rate channel implies that a larger in�ationincrease is necessary to o¤set the output e¤ects of an adverse supply shock.Since in�ation is costly, the optimal policy response involves a smaller in�ationresponse and less output stabilization. With a coordinated policy, the decisionproblem faced by the policy authority recognizes that a symmetric increasein in�ation in both countries leaves the real exchange rate una¤ected. Within�ation perceived to have a larger marginal impact on output, the optimalresponse is to stabilize more.The loss functions of the two countries can be evaluated under the alternative

policy regimes (coordination and noncoordination). Because the two countrieshave been speci�ed symmetrically, the value of the loss function will be thesame for each. For the domestic economy, the expected loss when policies arecoordinated is equal to

Lc =1

2

�1

1 + �b22

���2".

When policies are determined in a Nash noncooperative equilibrium,

LN =1

2

�1 + �b22A

21

(1 + �b22A1)2

���2".

Because 0 < A1 < 1, it follows that Lc < LN ; coordination achieves a betteroutcome than occurs in the Nash equilibrium.This example appears to imply that coordination will always dominate non-

coordination. It is important to recall that we considered the case in which theonly source of disturbance was a common aggregate-supply shock. The case ofasymmetric shocks is addressed in problem 4. But even when there are onlycommon shocks, coordination need not always be superior. Rogo¤ (1985a) pro-vides a counterexample. His argument is based on a model in which optimalpolicy is time inconsistent, a topic we will cover in chapter 9, but we can brie�ydescribe the intuition behind Rogo¤�s results. A coordinated monetary expan-sion leads to a larger short-run real-output expansion because it avoids changesin the real exchange rate. But this fact increases the incentive to engineer a sur-prise monetary expansion if the policy makers believe the natural rate of outputis too low. Wage and price setters will anticipate this tactic, together with theassociated higher in�ation. Equilibrium involves higher in�ation, but becauseit has been anticipated, output (which depends on in�ation surprises) does notincrease. Consequently, coordination leads to better stabilization but higher

24

average in�ation. If the costs of the latter are high enough, noncoordinationcan dominate coordination.The discussion of policy coordination serves to illustrate several important

aspects of open-economy monetary economics. First, the real exchange rate isthe relative price of output in the two countries, so it plays an important role inequilibrating relative demand and supply in the two countries. Second, foreignshocks matter for the domestic economy; both aggregate-supply and aggregate-demand shocks originating in the foreign economy a¤ect output in the domesticeconomy. As (43) and (44) show, however, the model implies that commondemand shocks that leave u � u� una¤ected have no e¤ect on output levels orthe real exchange rate. Since these shocks do a¤ect demand in each country, acommon demand shock raises real interest rates in each country. Third, policycoordination can matter.While the two-country model of this section is useful, it has several omis-

sions that may limit the insights that can be gained from its use. First, theaggregate-demand and aggregate-supply relationships are not derived explicitlywithin an optimizing framework. As we saw in chapter 5 and in the Obstfeld-Rogo¤ model, expectations of future income will play a role when consumptionis determined by forward-looking, rational economic agents. Second, there is norole for current-account imbalances to a¤ect equilibrium through their e¤ects onforeign asset holdings. Third, no distinction has been drawn between the priceof domestic output and the price index relevant for domestic residents. The lossfunction for the policy maker may depend on consumer price in�ation. Fourth,the in�ation rate was treated as the instrument of policy, directly controllableby the central bank. This is an obvious simpli�cation, one that abstracts fromthe linkages (and slippages) between the actual instruments of policy and therealized rate of in�ation. Although such simpli�cations are useful for address-ing many policy issues, chapters 9-11 will examine these linkages in more detail.Finally, the model, like the Obstfeld-Rogo¤ example, assumed one-period nomi-nal contracts. Such a formulation fails to capture the persistence that generallycharacterizes actual in�ation and the lags between changes in policy and theresulting changes in output and in�ation.

4 The Small Open Economy

A two-country model provides a useful framework for examining policy interac-tions in an environment in which developments in one economy a¤ect the other.For many economies, however, domestic developments have little or no impacton other economies. Decisions about policy can, in this case, treat foreigninterest rates, output levels, and in�ation as exogenous because the domesticeconomy is small relative to the rest of the world. The small open economyis a useful construct for analyzing issues when developments in the country ofinterest are unlikely to in�uence other economies.In the small-open-economy case, the model of the previous section simpli�es

25

to becomeyt = �b1�t + b2 (pt � Et�1pt) + et (49)

yt = a1�t � a2rt + ut (50)

�t = r�t � rt + Et�t+1. (51)

The real exchange rate � is equal to s + p� � p, where s is the nominalexchange rate and p� and p are the prices of foreign and domestic output, allexpressed in log terms. The aggregate supply relationship has been writtenin terms of the unanticipated price level rather than unanticipated in�ation.19

The dependence of output on price surprises arises from the presence of nominalwage and price rigidities. With foreign income and consumption exogenous, theimpact of world consumption on the domestic economy can be viewed as one ofthe factors giving rise to the disturbance term ut.Consumer prices in the domestic economy are de�ned as

qt = hpt + (1� h)(st + p�t ), (52)

where h is the share of domestic output in the consumer price index, while theFisher relationship links the real rate of interest appearing in (50) and (51) withthe nominal interest rate,

rt = it � Etpt+1 + pt. (53)

Uncovered interest parity links nominal interest rates. Since i� will be exogenousfrom the perspective of the small open economy, we can write (51) as

it = Etst+1 � st + i�t , (54)

where i� = r�+Etp�t+1�p�t . Finally, real money demand is assumed to be given

bymt � qt = yt � cit + vt. (55)

Notice that the basic structure of the model, like the closed-economy modelsof chapter 6 based on wage and/or price rigidity, displays the classical dichotomybetween the real and monetary sectors if wages are �exible. That is, if wagesadjust completely to equate labor demand and labor supply, the price-surpriseterm in (49) disappears.20 In this case, (49)-(51) constitute a three-equationsystem for real output, the real interest rate, and the real exchange rate. Us-ing the interest parity condition to eliminate rt from the aggregate demandrelationship, and setting the resulting expression for output equal to aggregate

19Since pt � Et�1pt = pt � pt�1 � (Et�1pt � pt�1) = �t � Et�1�t, the two formulationsare equivalent.20Recall from chapter 6 that the assumption behind an aggregate-supply function such as

(49) is that nominal wages are set in advance on the basis of expectations of the price level,while actual employment is determined by �rms on the basis of realized real wages (andtherefore on the actual price level).

26

supply, yields the following equation for the equilibrium real exchange rate inthe absence of nominal rigidities:

(a1 + a2 + b1)�t = a2�r�t + Et�t+1

�+ et � ut.

This can be solved forward for �t:

�t =1Xi=0

diEt

�a2r

�t+i + et+i � ut+ia1 + a2 + b1

�

= d1Xi=0

diEtr�t+i +

et � uta1 + a2 + b1

where d � a2=(a1 + a2 + b1) < 1 and the second equals sign follows from theassumption that e and u are serially uncorrelated processes. The real exchangerate responds to excess supply for domestic output; if et � ut > 0, a real de-preciation increases aggregate demand and lowers aggregate supply to restoregoods market equilibrium.The monetary sector consists of (52)-(55), plus the de�nition of the nom-

inal exchange rate as st = �t � p�t + pt. When wages and prices are �exible,these determine the two price levels p (the price of domestic output) and q (theconsumer price index). From the Fisher equation, the money demand equation,and the de�nition of qt,

mt � pt = yt + (1� h)�t � c (rt + Etpt+1 � pt) + vt.

Because the real values are exogenous with respect to the monetary sector whenthere are no nominal rigidities, this equation can be solved for the equilibriumvalue of pt:

pt =

�1

1 + c

� 1Xi=0

�c

1 + c

�iEt (mt+i � zt+i � vt+i) ,

where zt+i � yt+i+ (1� h)�t+i� crt+i: The equilibrium pt depends not just onthe current money supply, but also on the expected future path of m. Since (52)implies pt = qt � (1� h)�t, the equilibrium behavior of the domestic consumerprice index qt follows from the solutions for pt and �t.When nominal wages are set in advance, the classical dichotomy no longer

holds. With pt � Et�1pt a¤ecting the real wage, employment, and output, anydisturbance in the monetary sector that was unanticipated will a¤ect output,the real interest rate, and the real exchange rate. Since the model does notincorporate any mechanism to generate real persistence, these e¤ects last onlyfor one period.With nominal wage rigidity, monetary policy a¤ects real aggregate demand

through both interest-rate and exchange-rate channels. As can be seen from(50), these two variables appear in the combination a1�t�a2rt. For this reason,the interest rate and exchange rate are often combined to create a monetary

27

conditions index ; in the context of the present model, this index would be equalto rt � a1�t=a2. Variations in the real interest rate and real exchange rate thatleave this linear combination unchanged would be neutral in their impact onaggregate demand, since the reduction in domestic aggregate demand caused bya higher real interest rate would be o¤set by a depreciation in the real exchangerate.

4.1 Flexible Exchange Rates

Suppose that nominal wages are set in advance, but the nominal exchange rateis free to adjust �exibly in the face of economic disturbances. In addition,assume that monetary policy is implemented through control of the nominalmoney supply. In this case, the model consisting of (49)-(54) can be reducedto two equations involving the price level, the nominal exchange rate, and thenominal money supply (see the appendix for details). Equilibrium will dependon expectations of the period t + 1 exchange rate, and the response of theeconomy to current policy actions may depend on how these expectations area¤ected.To determine how the exchange rate and the price level respond to mone-

tary shocks, we will assume a speci�c a process for the nominal money supply.To allow for a distinction between transitory and permanent monetary shocks,assume

mt = �+mt�1 + 't � 't�1, 0 � � 1, (56)

where ' is a serially uncorrelated white noise process. If = 0, mt follows arandom walk with drift �; innovations ' have a permanent impact on the levelof m. If = 1, the money supply is white noise around a deterministic trend.If 0 < < 1, a fraction (1� ) of the innovation has a permanent e¤ect on thelevel of the money supply.To analyze the impact of foreign price shocks on the home country, let

p�t = �� + p�t�1 + �t, (57)

where � is a random white noise disturbance. This allows for a constant averageforeign in�ation rate of �� with permanent shifts in the price path due to therealizations of �.Using the method of undetermined coe¢ cients, the appendix shows that the

following solutions for pt and st are consistent with (49)-(54), and with rationalexpectations:

pt = k0 +mt�1 +B2[1 + c(1� )]

K't � 't�1

+[(A2 �B2)ut �A2et �B2vt]

K(58)

st = d0 +mt�1 � p�t�1 � �t �B1[1 + c(1� )]

K't

� 't�1 +[(B1 �A1)ut +A1et +B1vt]

K, (59)

28

0 1 2 3 4 5 6 7 8 9 10

Periods

γ = 0.0

γ = 0.5

γ = 1.0

Figure 1: Nominal Exchange Rate Response to a Monetary Shock

where A1 = h� a1 � a2, A2 = 1 + c� A1 > 0, B1 = �(a1 + a2 + b1 + b2) < 0,B2 = a1+ a2+ b1 > 0, and K = � [(1 + c)B1 + b2A1]. The constant k0 is givenby

k0 = (1 + c)�+

�c� a2(1� h� b1)

a1 + b1

�r�,

and d0 = k0 � ��.Of particular note is the way a �exible exchange rate insulates the domestic

economy from the foreign price shock �. Neither p�t�1 nor �t a¤ects the domesticprice level under a �exible exchange-rate system (see 58). Instead, (59) showshow they move the nominal exchange rate to maintain the domestic currencyprice of foreign goods, s+ p�, unchanged. This insulates the real exchange rateand domestic output from �uctuations in the foreign price level.With B2[1 + c(1 � )]=K > 0 and �B1[1 + c(1 � )]=K > 0, a positive

monetary shock increases the equilibrium price level and the nominal exchangerate. That is, the domestic currency depreciates in response to a positive moneyshock. The e¤ect is o¤set partially the following period if > 0. The shapeof the exchange rate response to a monetary shock is shown in �gure 6.1 fordi¤erent values of .

The < 1 cases in �gure 6.1 illustrate Dornbusch�s overshooting result(Dornbusch 1976). To the extent that the rise in m is permanent (i.e., < 1),the price level and the nominal exchange rate eventually rise proportionately.

29

With one-period nominal rigidities, this occurs in period 2. A rise in the nomi-nal money supply that increases the real supply of money reduces the nominalinterest rate to restore money market equilibrium. From the interest parity con-dition, the domestic nominal rate can fall only if the exchange rate is expectedto fall. Yet the exchange rate will be higher than its initial value in period 2, soto generate an expectation of a fall, s must rise more than proportionately tothe permanent rise in m. It is then expected to fall from period 1 to period 2;the nominal rate overshoots its new long-run value.The Dornbusch overshooting result stands in contrast to Obstfeld and Ro-

go¤�s conclusion, derived in section 8.2.3, that a permanent change in the nom-inal money supply does not lead to overshooting. Instead, the nominal ex-change rate jumps immediately to its new long-run level. This di¤erence resultsfrom the ad hoc nature of aggregate demand in the model of this section. Inthe Obstfeld-Rogo¤ model, consumption is derived from the decision problemof the representative agent, with the Euler condition for consumption linkingconsumption choices over time. The desire to smooth consumption impliesthat consumption immediately jumps to its new equilibrium level. As a result,exchange-rate overshooting is eliminated in the basic Obstfeld-Rogo¤ model.One implication of the overshooting hypothesis is that exchange-rate move-

ments should follow a predictable or forecastable pattern in response to mon-etary shocks. A positive monetary shock leads to an immediate depreciationfollowed by an appreciation. The path of adjustment will depend on the extentof nominal rigidities in the economy, since these in�uence the speed with whichthe economy adjusts in response to shocks. Such a predictable pattern is notclearly evident in the data. In fact, nominal exchange rates display close torandom-walk behavior over short time periods (Meese and Rogo¤ 1983). In aVAR-based study of exchange-rate responses to U.S. monetary shocks, Eichen-baum and Evans (1995) do not �nd evidence of overshooting, but they do �ndsustained and predictable exchange-rate movements following monetary-policyshocks. A monetary contraction produces a small initial appreciation, with thee¤ect growing so that the dollar appreciates for some time. However, in a studybased on more direct measurement of policy changes, Bonser-Neal, Roley, andSellon (1998) �nd general support for the overshooting hypothesis. They mea-sure policy changes by using data on changes in the Federal Reserve�s target forthe federal funds rate, rather than the actual funds rate, and restrict attentionto time periods during which the funds rate was the Fed�s policy instrument.

4.2 Fixed Exchange Rates

Under a system of �xed exchange rates, the monetary authority is committedto using its policy instrument to maintain a constant nominal exchange rate.This commitment requires that the monetary authority stand ready to buy orsell domestic currency for foreign exchange to maintain the �xed exchange rate.When it is necessary to sell foreign exchange, such a policy will be unsustainableif the domestic central bank�s reserves of foreign exchange are expected to go

30

to zero. Such expectations can produce speculative attacks on the currency.21

In our analysis here, we will deal only with the case of a sustainable �xed rate.And to draw the sharpest contrast with the �exible-exchange-rate regime, wewill assume that the exchange rate is pegged. In practice, most �xed-exchange-rate regimes allow rates to �uctuate within narrow bands.22

Normalizing the �xed rate at st = 0 for all t, the real exchange rate equalsp�t � pt. Assuming the foreign price level follows (57), the basic model becomes

yt = �b1(p�t � pt) + b2(pt � Et�1pt) + et

yt = a1(p�t � pt)� a2rt + ut

rt = r� + �� � (Etpt+1 � pt) .

The nominal interest rate has been eliminated, since the interest-parity condi-tion and the �xed-exchange-rate assumption imply that i = r�+��. These threeequations can be solved for the price level, output, and the domestic real interestrate. The money demand condition plays no role because m must endogenouslyadjust to maintain the �xed exchange rate.Solving for pt,

pt =(a1 + b1)p

�t � a2 (r� + ��) + a2Etpt+1 + b2Et�1pt + ut � et

a1 + a2 + b1 + b2.

Using the method of undetermined coe¢ cients, one obtains

pt = p�t �a2r

�

a1 + b1+

ut � et � b2�ta1 + a2 + b1 + b2

. (60)

Comparing (60) to (58) reveals some of the major di¤erences between the�xed and �exible exchange-rate systems. Under �xed exchange rates, the aver-age domestic rate of in�ation must equal the foreign in�ation rate: E (pt+1 � pt) =E�p�t+1 � p�t

�= ��. The foreign price level and foreign price shocks (�) a¤ect

domestic prices and output under the �xed-rate system. But domestic distur-bances to money demand or supply (' and v) have no price level or outpute¤ects. This situation is in contrast to the case under �exible exchange ratesand is one reason why high-in�ation economies often attempt to �x their ex-change rates with low in�ation countries. But when world in�ation is high,a country can maintain lower domestic in�ation only by allowing its nominalexchange rate to adjust.The e¤ects on real output of aggregate demand and supply disturbances

also depend on the nature of the exchange-rate system. Under �exible ex-change rates, a positive aggregate-demand shock increases prices and real out-put. Goods-market equilibrium requires a rise in the real interest rate and a realappreciation. By serving to equilibrate the goods market and partially o¤set therise in aggregate demand following a positive realization of u, the exchange-rate

21See Krugman (1979) and Garber and Svensson (1995).22Exchange-rate behavior under a target zone system was �rst analyzed by Krugman (1991).

31

movement helps stabilize aggregate output. As a result, the e¤ect of u on y issmaller under �exible exchange rates than under �xed exchange rates.23

The choice of exchange-rate regime in�uences the manner in which economicdisturbances a¤ect the small open economy. While the model examined heredoes not provide an internal welfare criterion (such as the utility of the represen-tative agent in the economy), such models have often been supplemented withloss functions depending on output or in�ation volatility (such as we used insection 3 to study policy coordination), which are then used to rank alternativeexchange-rate regimes. Based on such measures, the choice of an exchange-rate regime should depend on the relative importance of various disturbances.If volatility of foreign prices is of major concern, a �exible exchange rate willserve to insulate the domestic economy from real exchange-rate �uctuationsthat would otherwise a¤ect domestic output and prices. If domestic monetaryinstability is a source of economic �uctuations, a �xed-exchange-rate systemprovides an automatic monetary response to o¤set such disturbances.The role of economic disturbances in the choice of a policy regime is an im-

portant topic of study in monetary policy analysis. It �gures most prominentlyin discussion of the choice between using an interest rate or a monetary aggre-gate as the instrument of monetary policy. This topic forms the major focus ofchapter 9.

5 Open-EconomyModels with Optimizing Agentsand Nominal Rigidities

Chapter 7 developed a dynamic, general equilibrium model based on optimiz-ing households and �rms but in which prices were sticky. This model could besummarized in terms of an expectational IS curve, relating aggregate output toexpected future output and the real interest rate, and an in�ation adjustment re-lationship, in which current in�ation was a function of expected future in�ationand real marginal cost. Real marginal cost was a function of output relative tothe �exible-price equilibrium level of output. The model was closed by assumingthat the monetary authority determined the nominal rate of interest. In recentyears, a number of authors have extended the basic new Keynesian frameworkto the open-economy context. Examples include McCallum and Nelson (2000b),Clarida, Galí, and Gertler (2001, 2002), Gertler, Gilchrist, Natalucci (2001), andMonacelli (2005). See also Galí and Monacelli (2005) and Galí (2008, Chapter7). Lubik, T. and F. Schorfheide (2007), Adolfson, Laséen, Lindé, and Vallani(2007, 2008), and Adolfson, Laséen, Lindé, and Svensson (2008) provide ex-amples of estimated DSGE model for policy analysis based on the frameworkdiscussed in this section. Lubik and Schorfheide (2005) have jointed estimateda two-country open economy DSGE model with nominal rigidities using US andEuro Area data. These models di¤er from the models discussed in section 2.323This is consistent with the estimates for Japan reported in Hutchison and Walsh (1992).

Obstfeld (1985) discusses the insulation properties of exchange rate systems.

32

in that an in�ation-adjustment model based on Calvo (1983) replaces the as-sumption of 1-period nominal rigidity. In this section, we develop an exampleof an open economy new Keynesian model.

5.1 A Model of the Small Open Economy

Suppose there are two countries. The home country is denoted by the super-script h, while the superscript f denotes the foreign country. The countriesshare the same preferences and technologies. Both produce traded consump-tion goods which are imperfect substitutes in utility. The foreign country is astand-in for the rest of the world, and the home country is small relative to theforeign country.

5.1.1 Households

Households consume a CES composite of home and foreign goods, de�ned as

Ct =

�(1� ) 1a

�Cht� a�1

a + 1a

�Cft

� a�1a

� aa�1

(61)

for a > 1. As in chapter 6, we assume Ch (and Cf ) are Dixit-Stiglitz aggregatesof di¤erentiated goods produced by domestic (and foreign) �rms. The domestichousehold�s relative demand for Ch and Cf will depend on their relative prices.Let Pht (P

ft ) be the average price of domestically (foreign) produced consump-

tion goods. The problem of minimizing the cost Pht Cht + P ft C

ft of achieving a

given level of Ct yields the �rst order conditions

Pht = �t

�(1� ) 1a

�Cht� a�1

a + 1a

�Cft

� a�1a

� 1a�1

(1� ) 1a�Cht�� 1

a

and

P ft = �t

�(1� ) 1a

�Cht� a�1

a + 1a

�Cft

� a�1a

� 1a�1

1a

�Cft

�� 1a

,

where � is the Lagrangian multiplier on the constraint. Using (61), these �rstorder conditions can be written as

Pht = �tC1at (1� )

1a

�Cht�� 1

a ) Cht = (1� )�Pht�t

��aCt

and

P ft = �tC1at

1a

�Cft

�� 1a

.) Cft =

P ft�t

!�aCt.

These equations imply in turn that the home country�s relative demand forhome and foreign produced goods depends on their relative price:

Cht

Cft=

�1�

� Pht

P ft

!�a. (62)

33

Substituting the solutions for Cht and Cft into the de�nition of the aggregate

consumption bundle yields

Ct =

24(1� )�Pht�t

�1�a+

P ft�t

!1�a35 aa�1

Ct.

Dividing both sides by Ct and solving for �t yields

�t =

�(1� )

�Pht�1�a

+ �P ft

�1�a� 11�a

� P ct . (63)

as the aggregate (consumer) price index.Household utility depends on its consumption of the composite good and on

its labor supply. Assume that

U(Ct; Nt) =C1��t

1� � �N1+�t

1 + �. (64)

Intertemporal optimization implies the standard Euler condition,

C��t = �EtRt

�P ctP ct+1

�C��t+1, (65)

where Rt is the (gross) nominal rate of interest, while optimal labor-leisurechoice requires that the marginal rate of substitution between leisure and con-sumption equal the real wage. This last condition takes the form

N�t

C��t=Wt

P ct, (66)

where Wt is the nominal wage.Assume that the law of one price holds. This implies that

P ft = StP�t , (67)

where P �t is the foreign currency price of foreign-produced goods and St is thenominal exchange rate (price of foreign currency in terms of domestic currency).For simplicity, assume that all foreign goods sell for the price P ft . This speci-�cation assumes complete exchange rate pass-through; given P �t , a 1% changein the exchange rate produces a 1% change in the domestic currency price offoreign produced goods P ft . Given that we have assumed the law of one price,the price levels in the domestic and foreign countries are linked by

P ct = StPc�t .

For the rest of the world, we ignore the distinction between the CPI and theprice of domestic production, so P c�t = P �t .

24

24That is, the rest of the world is large relative to the home country, so changes in theprice of home produced goods has little impact on the consumer price index for the rest ofthe world.

34

It will be useful to de�ne the terms of trade and the real exchange rate. Theterms of trade is equal to the relative price of foreign and domestic goods:

�t �P ftPht

=StP

�t

Pht. (68)

The real exchange rate is the price of foreign produced goods (in terms of do-mestic currency) relative to the home country�s consumer price index:

Qt =StP

�t

P ct=

�PhtP ct

��t. (69)

As was the case with the new Keynesian model of chapter 6, the focus will beon percentage deviations around the steady state. Let lowercase letters denotepercentage deviation around the steady state of the corresponding uppercaseletter. Then (68) and (69) can be expressed as

�t = st + p�t � pht

andqt = st + p

�t � pct .

Using the de�nition of the terms of trade in (63), we obtain

pct = (1� )pht + pft = pht + �t, (70)

Equations (61)-(66) can be written as

ct = (1� )cht + cft (71)

cft = �a�t + cht (72)

ct = Etct+1 ��1

�

��it � Et�ct+1 � �

�(73)

where � = ��1 � 1 and�nt + �ct = wt � pht � �t, (74)

where �ct = pct�pct�1. Combining (71) and (72), ct = cht� a�t. De�ning in�ationin the prices of domestically produced goods as �ht = pht � pht�1, in�ation in theconsumer price index equals

�ct = pct � pct�1 = �ht + (�t � �t�1). (75)

Note that the real interest rate is de�ned in terms of consumer price in�ation �ct .In turn, this measure of in�ation depends on the rate of in�ation of domesticallyproduced goods �ht and the rate of change in the terms of trade.Clarida, Galí, and Gertler (2001) add a stochastic wage markup �wt to (74)

to represent deviations from the marginal rate of substitution between leisureand consumption:

�nt + �ct + �wt = wt � pht � �t. (76)

They motivate this markup as arising from the monopoly power of labor sup-pliers who set wages as a markup over the marginal rate of substitution. Themarkup is then assumed to be subject to exogenous stochastic variation.

35

5.1.2 International risk sharing

We assume agents in both economies have access to a complete set of inter-nationally traded securities. The time t home currency price of a bond thatpays o¤ one unit of the domestic currency at time t + 1 is R�1t , and the Eulercondition given by (65) can be written as

�Et

�P ctP ct+1

��Ct+1Ct

���=1

Rt. (77)

Since residents in the rest of the world also have access to these same �nancialsecurities, intertemporal optimization implies

C���t

StP �t= �Et

�Rt

St+1P �t+1

�C���t+1 .

The left side of this expression shows the marginal utility cost at time t of using1=St units foreign currency (worth 1=P �t each in terms of consumption goods) toobtain one unit of the domestic currency. This is invested in the domestic bond,yielding at time t + 1 a gross return Rt. This can be converted into Rt=St+1units of foreign currency, worth Rt=

�St+1P

�t+1

�in terms of foreign consumption,

yielding expected utility of Et�Rt=St+1P

�t+1

�C���t+1 . Thus, the right side is the

marginal bene�t in terms of utility from this �nancial transaction. Since thispayout occurs at time t+ 1, it must be discounted at the rate � to compare tothe cost of purchasing the �nancial asset. Along the optimal path, the marginalcosts and bene�ts are equal. Rearranging, we have that

�Et

�P �tP �t+1

��StSt+1

��C�t+1C�t

���=1

Rt.

Using the law of one price and the de�nition of the real exchange rate, thiscondition can be written as

�Et

�P ctP ct+1

��QtQt+1

��C�t+1C�t

���=1

Rt.

Hence,

�Et

�P ctP ct+1

��Ct+1Ct

���=

�1

Rt

�= �Et

�P ctP ct+1

��QtQt+1

��C�t+1C�t

���,

which implies

Ct = �Q1�t C

�t , (78)

where � is a constant of proportionality. For convenience, adopt � = 1 as anormalization. This is consistent with a symmetric initial condition with zeronet foreign asset holdings. In terms of a �rst order linear approximation aroundthe steady state,

ct = c�t +

�1

�

�qt = c�t +

�1� �

��t. (79)

36

This last equation has employed the fact that

qt = st + p�t � pct = �t + p

ht � pct = (1� )�t. (80)

5.1.3 Uncovered interest parity

If R�t is the foreign interest rate, then

�Et

�P �tP �t+1

��C�t+1C�t

���=

1

R�t.

Linearizing this condition yields

��Etc

�t+1 � c�t

�= r�t � Et��t+1

and doing the same for (77) implies

� (Etct+1 � ct) = rt � Et�ct+1.

Using (79), we have that�rt � Et�ct+1

���r�t � Et��t+1

�= � (Etct+1 � ct)� �

�Etc

�t+1 � c�t

�= Etqt+1 � qt.

Subtracting the in�ation terms from each side and using the de�nition of thereal exchange rate yields the condition for uncovered interest parity:

rt = r�t + Et (st+1 � st) .

The domestic nominal interest rate is equal to the foreign (rest of the world)nominal interest rate plus the expected rate of depreciation in the domesticcurrency.We can also express the uncovered interest rate parity condition in real

terms by subtracting in�ation in domestic produced goods prices and using thede�nition of the terms of trade:

rt � Et�ht+1 = ��t + (Et�t+1 � �t) (81)

where �� � r�t � Et��t+1.

5.1.4 Domestic Firms

The analysis of domestic �rms parallels the approach followed in chapter 7, andfurther details can be found there.In each period, there is a �xed probability 1 � ! that the �rm can adjust

its price. When it can adjust, it does so to maximize the expected discountedvalue of pro�ts. Each domestic �rm faces the identical production function,

Y ht = e"tNt,

37

and a constant elasticity demand curve for its output. The �rm�s real marginalcost is equal to

MCt =Wt=P

ht

e"t,

where Wt=Pht is the real product wage and e

"t is the marginal product of la-bor. In terms of percentage deviations around the steady state, this expressionbecomes

mct = wt � pht � "t. (82)

Following the derivation of chapter 6, the in�ation rate for the price indexof domestically produced goods is

�ht = �Et�ht+1 + �mct, (83)

where � = (1� !) (1� �!) =!. Combining (83) with (75),

�ct = �Et�ct+1 + �mct � � (Et�t+1 � �t) + (�t � �t�1). (84)

In�ation, as measured by the consumer price index, depends on expected fu-ture in�ation and real marginal cost, the same two factors that appeared in theclosed-economy in�ation adjustment equation. The di¤erence in the open econ-omy is that, given real marginal cost, in�ation in the consumer price index canbe a¤ected by current and expected future changes in the terms of trade. A risein the terms of trade (�t � �t�1 > 0) implies an increase in the relative price offoreign goods. Because foreign goods prices are included in the consumer priceindex (see 70), a rise in their relative prices increases consumer price in�ation.An expected future rise in the terms of trade reduces current consumer pricein�ation since, for a given Et�ct+1, this rise in the terms of trade must imply afall in expected future domestic goods in�ation and therefore a fall in currentdomestic goods in�ation.

5.1.5 The Foreign Country

To keep the analysis simple, we have assumed the foreign country is large relativeto the home country. This is taken to mean that it is unnecessary to distinguishbetween consumer price in�ation and domestic in�ation in the foreign country,and that domestic output and consumption are equal. Trade does take place,however, and so goods produced in the home country are sold to domesticresidents and to foreigners. Let ch�t be the foreign country�s consumption of thedomestically produced good (as a percentage deviation from the steady state).The foreign country�s demand for the home country�s output depends on theterms of trade. Assuming that foreign households have the same preferences asthose of the home country (so the demand elasticity is the same),

Ch�

t = �at Yft ,

orch

�

t = a�t + yft , (85)

38

where yft is foreign income (in terms of the percent deviation from the steadystate). The Euler condition for foreign country households implies

y�t = Ety�t+1 �

�1

�

��ift � Et��t+1

�,

or��t � i�t � Et��t+1 = �

�Ety

�t+1 � y�t

�. (86)

5.1.6 Equilibrium Conditions

We can now bring together the conditions for each market that need to besatis�ed in equilibrium.

The goods market Equilibrium requires that domestic production equal theconsumption of the domestically produced good. Since the domestic good isconsumed by both domestic residents and by residents of the rest of the world,equilibrium requires that

Yt = (1� )�PhtP ct

��aCt +

�PhtStP �t

��aY �t .

Using the earlier result that Ct = Q1�t C

�t = Q

1�t Y

�t to eliminate Y

�t and employ-

ing the de�nition of the real exchange rate as Qt = StP�t =P

ct , the goods market

equilibrium condition can be written as

Yt =

�PhtP ct

��a h(1� ) + Qa�

1�

t

iCt.

Taking a �rst order linear approximation of this equation around the steadystate yields25

yt = ct +� w�

��t, (87)

where w = a� + (a� � 1) (1� ).Since c�t = y�t , and from (79) and (80), ct = c�t +

�1�

�qt = c�t +

�1� �

��t, (87)

can be written as

yt = ct +� w�

��t = c�t +

�1� �

+ w

�

��t

= y�t +

�1

�

��t, (88)

where � � �= [1� (1� w)].25This makes use of the fact that in the steady state, Q = 1, so Y =

�Ph=P c

��aC, and of

the de�nition of the terms of trade, which implies pht � pct = � �t.

39

We can use these results in the Euler condition (73) to obtain the smallopen economy version of the expectational IS relations. Starting with the Eulercondition, use (87) to replace consumption with yt �

� w�

��t, yielding

yt = Etyt+1 ��1

�

��it � Et�ct+1 � �

��� w�

�(Et�t+1 � �t)

Because Et�ct+1 = Et�ht+1 + Et��t+1 (see 75),

yt = Etyt+1 ��1

�

��it � Et�ht+1 � �

�+

�(1� w) (Et�t+1 � �t) .

Equation (88) implies (Et�t+1 � �t) = � �Et (yt+1 � yt)�

�Ety

�t+1 � y�t

��,

and using this expression to eliminate the expected change in the terms of tradefrom the Euler condition, we obtain

yt = Etyt+1 ��1

�

��it � Et�ht+1 � ��

�(89)

where�� � �� � (1� w)

�Ety

�t+1 � y�t

�.

Equation (89) is the small open economy equivalent to the closed economyexpectational IS curve. There are two primary di¤erences between the openand the closed economy versions of this relationship. First, the elasticity ofdemand with respect to the real interest rate is no longer equal to the elasticityof intertemporal substitution, 1=�. Instead, it equals 1=� = [1� (1� w)] =�.From the de�nition of w,

� =�

1� 2 [1� a�]

which depends on the �openness� of the economy through (the larger is,the smaller the share of home produced goods in the consumption bundle ofdomestic residents) and a, the price elasticity of demand for home producedand foreign produced goods.

5.1.7 In�ation adjustment

To determine the rate of in�ation of domestically produced goods, (82) and (83)imply

�ht = �Et�ht+1 + �

�wt � pht � "t

�.

The real consumption wage is wt � pct , and this is related to the real productwage wt� pht by the terms of trade: pct = pht + �t. Since households equate thereal consumption wage to the marginal rate of substitution between leisure andconsumption, (66) implies that

�nt + �ct = wt � pct = wt � pht � �t.

40

Hence, real marginal cost is wt � pht � "t = �nt + �ct + �t � "t. We can nowuse (87) and (88) to eliminate consumption and the terms of trade to obtain anexpression for real marginal cost solely in terms of domestic output and foreignvariables:

�ht = �Et�ht+1 + � [(� + � ) yt � (� � � )y�t � (1 + �)"t]

or

�ht = �Et�ht+1 + � (� + � ) (yt � ~yt) , (90)

where ~yt � [(� � � )y�t � (1 + �)"t] = (� + � ).Notice that because �� and ~yt depend only on exogenous shocks or foreign

variables, (89) and (90) constitute a two equation model for domestic outputand in�ation once a speci�cation for monetary policy in terms of the nominalrate of interest has been added.Clarida, Galí, and Gertler (2002) relax the assumption that the foreign coun-

try is large and examine the role of policy coordination. To carry out this ex-amination, assume that both countries are subject to nominal price rigidities.In this case, (88) is modi�ed to becomes

~xt = ~xft +

�1

�

�~�t,

where ~�t, ~xt and ~xft are de�ned as gaps relative to the outcome when prices

are �exible in both countries. Using this equation to derive real marginal cost,Clarida, Galí, and Gertler �nd that

�ht = �Et�ht+1 + �~xt + �~x

ft

for domestic goods in�ation in the home country, with in�ation in the foreigncountry satisfying a similar equation. The spillover e¤ect of the output gapon in�ation in the other country gives rise, in general, to gains from policycoordination.

5.2 The Relationship to the Closed Economy NK Model

The small open economy model consisting of (89) and (90) is identical in formto the closed economy new Keynesian model of Chapter 7. Output satis�esan Euler condition with the real interest rate de�ned in terms of domesticallyproduced goods and in�ation of domestically produced goods is forward lookingand depends on an output gap measure. The parameters in these relationshipsdi¤er from those in the close economy model as they depend on factors relatedto the openness of the economy.To further draw the parallel with the closed economy model, de�ne the

output gap asxt � yt � ~yt. (91)

41

Then we can rewrite (89) and (90) as

xt = Etxt+1 ��1

�

��it � Et�ht+1 � ~�t

�(92)

�ht = �Et�ht+1 + �(� + � )xt. (93)

where~�t = �� + � Et (~yt+1 � ~yt) . (94)

We now have reduced the small, open economy model to a form that exactlyparallels that of the closed economy if we can show that the output gap xt =yt � ~yt is the gap between output and the �ex-price equilibrium price. That is,we need to show that with �exible prices, ~yt is the equilibrium output.The �exible-price equilibrium output is de�ned as the output level consistent

with real marginal cost equally its constant steady-state value. Since

mct = (� + � )yt + (� � � )y�t � (1 + �)"t,

it follows that the �ex-price output is

yflext =(� � �)y�t + (1 + �)"t

� + � ,

which is the de�nition assigned earlier to ~yt. Hence, xt is the �ex-price outputgap, just as it was in the closed economy new Keynesian model.Equations (93) and (92) are, as Clarida, R., J. Galí, and M. Gertler (2001,

2002) have emphasized, isomorphic to the closed economy new Keynesian model.26

The parallel is even strong if the central bank�s objective can be represented asminimizing a quadratic from of the output gap as de�ned in (91) and the in-�ation rate of domestically produced goods �ht . In this case, the central bank�spolicy involves minimizing

Et

1Xi=0

�ih��ht+i

�2+ �x2t+i

i(95)

subject to (93) and (92). This is exactly equivalent to the closed economy policyproblem studied in chapter 7 and so all the conclusions about policy reachedthere would apply without modi�cation to the small open economy. The criticalrequirement is that the in�ation rate appearing in the central bank�s objectivefunction must be �h and not the in�ation rate in consumer prices �c. If �c isthe relevant objective of the central bank (and most in�ation targeting centralbanks in small open economies de�ne their targets in terms of consumer pricein�ation), then (93) and (92) are not su¢ cient to determine either optimalpolicy or the economy�s equilibrium. Thus, an important issue for the analysisof monetary policy in the context of an open economy is determining the priceindex that is the appropriate objective of policy.26McCallum and Nelson (2000) have also developed a model that can be reduced to a version

isomotphic to the closed economy new Keynesian model. See problem 9.

42

5.2.1 Optimal policy in the small open economy

As demonstrated in the previous section, the analysis of optimal monetary policyin the small open economy would be equivalent to that in the closed economy ifthe central bank�s objectives were given by (95). In the absence of cost shocksto the in�ation equation for domestic goods prices, optimal policy would involvestabilizing the domestic price index.The basic intuition for such a policy cares over from the case of the closed

economy �if tax subsidies have dealt with the distortions associated with mo-nopolistic competition, then the role of monetary policy should be to eliminatethe distortions created by sticky prices. The central bank can eliminate thesedistortions by stabilizing prices, and since the sticky prices are the prices of do-mestically produced goods, the optimal policy would stabilize domestic prices.Unfortunately, as Galí and Monacelli (2005) discuss, this intuition is not

correct in general. Even if the distortion arising from the mark up is o¤set by�scal subsidies, there remains an additional factor present in the open economycontext that leads to a distortion and implies the �exible-price equilibrium is notfully e¢ cient.. Because foreign produced and domestically produced goods areimperfect substitutes, the central bank faces an incentive to a¤ect the terms oftrade. This is welfare improving and would cause the optimal policy to deviatefrom price stability.27

5.3 Imperfect Pass-Through

The isomporphism between the open and closed versions of the basic new Key-nesian model is a consequence of continuing to incorporate only a single nominalrigidity. Prices were assumed to be sticky, but nominal wages were assumed to be�exible. In addition, only the prices of domestically produced goods were takento be sticky. The home country prices of foreign produced goods were taken tobe perfectly �exible; they moved one-for-one with the nominal exchange rate;see (67).Adolfson (2001), Corsetti and Presenti (2002), and Monacelli (2005) provide

examples of models that allow for incomplete pass-through. When pass-throughis incomplete, the law of one price no longer holds. The law of one price allowsthe domestic currency price of foreign goods, pft , to be expressed as st + p�t ,where st is the nominal exchange rate and p�t is the foreign currency price offoreign goods (all expressed as percentage deviations from their steady-statevalues). The terms of trade are then equal to st + p�t � pht . With incompletepass-though, however, pft and st + p

�t can di¤er.

Following Monacelli (2005), the real exchange rate qt can be written as28

qt = st + p�t � pct = t + (1� )�t,

27See Corsetti and Presenti (2001). Benigno and Benigno (2003) consider this issue in atwo-country model.28This uses (70), which de�nes the consumer price index as (1� )pht + p

ft and the de�nition

of the terms of trade, �t = pft � pht .

43

where t = st+p�t �p

ft measures the deviation from the law of one price. In the

model of the previous section, the law of one price held, so t was identicallyequal to zero. Suppose pass-through is incomplete because of nominal rigidityin the price of imports, with only a fraction of importers adjusting their priceeach period, as in a standard Calvo-type model of price adjustment. Then thevariable t represents the marginal cost of importers, and the rate of in�ationin the average domestic currency price of foreign imports takes the form

�ft = �Et�ft+1 + �

f t,

where �ft = pft � pft�1 and the parameter �

f depends on the fraction of importprices that adjust each period.Letting zot denote the �ex-price equilibrium value of a variable zt, Monacelli

(2005) shows that with imperfect pass through, the output gap can be writtenas

xt =

�1 + w

�

�(�t � �ot ) +

�1 + (�a� 1)

�

�( t � ot ) ,

where w = (�a� 1) (2� ) as before. The real marginal cost of domestic �rmscan then be expressed as

mct �mcot =�

�

1 + w+ �

�xt +

�1� 1 + (�a� 1)

1 + w

�( t � ot ) .

In�ation in domestic producer prices is equal to

�ht = �Et�ht+1 + � (mct �mcot ) .

Thus, deviations from the law of one price, as measured by t � ot , directlya¤ect in�ation through their impact on marginal cost.Imperfect pass through represents a second nominal rigidity when combined

with the assumption of sticky prices. Not surprisingly therefore, it introducespolicy trade-o¤s much as the addition of sticky wages did in the basic newKeynesian model of chapter 7. Both the output gap and deviations from thelaw of one price a¤ect real marginal cost and, as a result, in�ation. Stabilizingin�ation in the face of a movement in t � ot requires that the output gap beallowed to �uctuate; stabilizing the output gap in the face of a movement in t � ot requires that in�ation �uctuate. With two sources of nominal rigidity�sticky prices and imperfect pass through �the central bank cannot undo thee¤ects of both distortions with it single policy instrument.

6 Summary

This chapter has reviewed various models that are useful for studying aspectsof open-economy monetary economics. A two-country model whose equilibriumconditions were consistent with optimizing agents was presented. This model,based on the work of Obstfeld and Rogo¤, preserved the classical dichotomy

44

between real and monetary factors when prices and wages were assumed to beperfectly �exible. In this case, the price level and the nominal exchange ratecould be expressed simply in terms of the current and expected future paths ofthe nominal money supplies in the two countries.Two primary lessons are that new channels by which monetary factors a¤ect

the economy are present in an open economy, and the choice of exchange-rateregime has important implications for the role of monetary policy. With stickynominal wages, monetary factors have important short-run e¤ects on the realexchange rate. Exchange-rate movements alter the relative price of the do-mestic good and the foreign good, a¤ecting aggregate demand and supply. Inaddition, consumer prices, because they are indices of domestic currency pricesof domestically produced and foreign-produced goods, respond to exchange-ratemovements.29 The impact of exchange-rate movements on consumer price in-�ation suggests that monetary policy may have more rapid e¤ects on in�ationin more open economies.Unlike the analysis of the closed economy, the Obstfeld-Rogo¤model implied

that monetary-induced output movements would have persistent real e¤ects byaltering the distribution of wealth between economies. In general, standardopen-economy frameworks used for policy analysis assume that the real e¤ectsof monetary policy arise only in the short run and are due to nominal rigidities.Over time, as wages and prices adjust, real output, real interest rates, and thereal exchange rate return to equilibrium levels that are independent of monetarypolicy. This long-run neutrality means that these models, like their closed-economy counterparts, imply the long-run e¤ects of monetary policy fall onprices, in�ation, nominal interest rates, and the nominal exchange rate. In theshort run, however, monetary policy can have important e¤ects on the mannerin which real output and the real exchange rate �uctuate around their longer-runequilibrium values.The �nal section of the chapter reviewed some extensions of new Keyne-

sian models to open-economy settings. In these models, households and �rmsare optimizing agents, but nominal rigidities lead to a staggered adjustment ofprices over time. In some cases, these models result in reduced-form equationsfor the output gap and in�ation that are identical in form to the closed-economyequivalents. However, as in closed economy models, policy trade o¤s are signif-icantly a¤ected with the addition of multiple sources of nominal rigidity. Themodel of imperfect pass through provided one empirically relevant example ofa nominal rigidity that is absence in the closed economy. As with models ofthe closed economy, recent empirical work with DSGE open economy modelssuch as Adolfson, Laséen, Lindé, and Vallani (2007, 2008) incorporate multiplesources of nominal (and real) frictions into the basic model structure reviewedin this chapter.

29Duguay (1994) provides evidence on these channels in the case of Canada.

45

7 Appendix

7.1 The Obstfeld-Rogo¤Model

This appendix provides a derivation of some of the components of the Obstfeld-Rogo¤ (1995, 1996) model.

7.1.1 Individual Product Demand

The demand functions faced by individual producers are obtained as the solutionto the following problem:

max

�Z 1

0

c(z)qdz

� 1q

subject toZ 1

0

p(z)c(z)dz = Z

for a given total expenditure Z. Letting � denote the Lagrangian multiplierassociated with the budget constraint, the �rst order conditions imply, for all z,

c(z)q�1�Z 1

0

c(z)qdz

� 1q�1

= �p(z).

For any two goods z and z0, therefore, [c(z)=c(z0)]q�1 = p(z)=p(z0), or

c(z) = c(z0)

�p(z0)

p(z)

� 11�q

.

If this expression is substituted into the budget constraint, we obtainZ 1

0

p(z)c(z0)

�p(z0)

p(z)

� 11�q

dz = c(z0)p(z0)1

1�q

�Z 1

0

p(z)q

q�1 dz

�= Z. (96)

Using the de�nition of P given in (3), both sides of (96) can be divided by P toyield

c(z0)p(z0)1

1�q

hR 10p(z)

qq�1 dz

ihR 10p(z)

qq�1 dz

i q�1q

=Z

P.

This can be simpli�ed to

c(z0)

�p(z0)

P

� 11�q

=Z

Por c(z0) =

�p(z0)

P

� 1q�1

C, (97)

where C = ZP is total real consumption of the composite good. Equation (97) im-

plies that the demand for good z by agent j is equal to cj(z) = [p(z)=P ]1=(q�1) Cj ,so the world demand for product z will be equal to

ydt (z) � n

�pt(z)

Pt

� 1q�1

Ct + (1� n)�p�t (z)

P �t

� 1q�1

C�t

=

�pt(z)

Pt

� 1q�1

Cwt , (98)

46

where Cw = nC + (1 � n)C� is world real consumption. Notice that we haveused the law of one price here, since it implies that the relative price for good z isthe same for home and foreign consumers: p(z)=P = Ep�(z)=EP � = p�(z)=P �.Finally, note that (98) implies

pt(z) = Pt

�ydt (z)

Cwt

�q�1. (99)

7.1.2 The Individual�s Decision Problem

Each individual begins period t with existing asset holdings Bjt�1 andMjt�1 and

chooses how much of good j to produce (subject to the world demand functionfor good j), how much to consume, and what levels of real bonds and money tohold. These choices are made to maximize utility given by (1) and subject tothe following budget constraint:

Cjt +Bjt +

M jt

Pt� pt(j)yt(j)

Pt+Rt�1B

jt�1 +

M jt�1Pt

+ � t

where � t is the real net transfer from the government and Rt is the real grossrate of return. From (99), agent�s j0s real income from producing y(j) will beequal to y(j)q (Cwt )

1�q, so the budget constraint can be written as

Cjt +Bjt +

M jt

Pt= yt(j)

q (Cwt )1�q

+Rt�1Bjt�1 +

M jt�1Pt

+ � t. (100)

The value function for the individual�s decision problem is

V�Bjt�1;M

jt�1

�= max

(logCjt + b log

M jt

Pt� k

2yt(j)

2 + �V�Bjt ;M

jt

�),

where the maximization is subject to (100). Letting � denote the Lagrangianmultiplier associate with the budget constraint, �rst order conditions are

1

Cjt� �t = 0 (101)

b

M jt

+ �V2

�Bjt ;M

jt

�� �tPt= 0 (102)

�kyt(j) + �tqyt(j)q�1 (Cwt )1�q

= 0 (103)

�V1

�Bjt ;M

jt

�� �t = 0 (104)

V1

�Bjt�1;M

jt�1

�= �tRt�1 (105)

V2

�Bjt�1;M

jt�1

�=�tPt. (106)

47

We also have the transversality condition limi!1Qik=0Rt+s�1

�Bjt+i +M

jt+i=Pt+i

�=

0.These �rst order conditions lead to the standard Euler condition for con-

sumption with log utility:Cjt+1 = �RtC

jt ,

which is obtained using (101), (104), and (105). Equations (103) and (101)imply that the optimal production level the individual chooses satis�es

yt(j)2�q =

q

k

(Cwt )1�q

Cjt. (107)

Equation (102) yields an expression for the real demand for money,

M jt

Pt= bCjt

�1 + itit

�,

where (1 + it) = Rt+1Pt+1=Pt is the gross nominal rate of interest from periodt to t+ 1. This expression should look familiar from chapter 2.

7.2 The Small-Open-Economy Model

This appendix employs the method of undetermined coe¢ cients to obtain theequilibrium exchange rate and price level processes consistent with (49)-(57).The equations of the model are repeated here, where the real exchange rate �thas been replaced by st+p�t�pt, rt by r��(st + p�t � pt)+Et

�st+1 + p

�t+1 � pt+1

�,

it by i�t + Etst+1 � st, and qt by pt + (1� h)(st + p�t � pt):

yt = �b1 (st + p�t � pt) + b2 (pt � Et�1pt) + et (108)

yt = a1 (st + p�t � pt)� a2

�r� � (st + p�t � pt) + Et

�st+1 + p

�t+1 � pt+1

��+ ut(109)

mt � [pt + (1� h)(st + p�t � pt)] = yt � c (i�t + Etst+1 � st) + vt (110)

mt = �+mt�1 + 't � 't�1, 0 � � 1 (111)

p�t = �� + p�t�1 + �t. (112)

Substituting the aggregate demand relationship (109), the money supplyprocess (111), and the foreign price process (112) into the money demand equa-tion (110) yields, after some rearrangement,

A1pt +A2st = C0 + �+mt�1 + 't � 't�1� (1� h+ a1)

�p�t�1 + �t

�� a2Etpt+1

+(a2 + c) Etst+1 � ut � vt, (113)

48

where A1 = h � a1 � a2, A2 = 1 � h + a1 + a2 + c > 0 and C0 = (c + a2)r� �

(1�h+a1�a2� c)��. In deriving (113), two additional results have been used:from (112), Etp�t+1 = 2�

� + p�t�1 + �t, and i�t = r� + Etp

�t+1 � p�t = r� + ��.

Using the aggregate supply and demand relationships (108) and (109),

B1pt +B2st = �b2Et�1pt + a2 (Etst+1 � Etpt+1)� (a1 + b1)��

+a2 (r� + ��) + et � ut � (a1 + b1)(p�t�1 + �t), (114)

where B1 = � (a1 + a2 + b1 + b2) < 0, and B2 = a1 + a2 + b1 > 0.The state variables at time t are mt�1, p�t and the various random distur-

bances. To rule out possible bubble solutions, we follow McCallum (1983a) andhypothesize minimum state variable solutions of the form:30

pt = k0 +mt�1 + k1p�t�1 + k2't + k3't�1 + k4ut + k5et + k6vt + k7�t

st = d0 +mt�1 + d1p�t�1 + d2't + d3't�1 + d4ut + d5et + d6vt + d7�t.

These implyEt�1pt = k0 +mt�1 + k1p

�t�1 + k3't�1

Etpt+1 = k0 +mt + k1p�t + k3't

= k0 + �+mt�1 + (1 + k3)'t � 't�1 + k1��� + p�t�1 + �t

�and

Etst+1 = d0 + �+mt�1 + (1 + d3)'t � 't�1 + d1��� + p�t�1 + �t

�.

These expressions for pt and st, together with those for the various expec-tations of p and s, can be substituted into (113) and (114). These then yield apair of equations that must be satis�ed by each pair (ki; di). For example, thecoe¢ cients on p�t�1 in (113) and (114) must satisfy

A1k1 +A2d1 = � (1� h+ a1)� a2k1 + (a2 + c) d1

andB1k1 +B2d1 = a2(d1 � k1)� b2k1 � (a1 + b1).

Using the de�nitions of Ai and Bi to cancel terms, the second equation impliesthat d1 = k1 � 1. Substituting this back into the �rst equation yields k1 = 0.Therefore, the solution pair is (k1; d1) = (0; 1). Repeating this process yieldsthe values for (ki; di) reported in (58) and (59).

30We have set the coe¢ cient on mt�1 equal to 1 in these trial solutions. It is easy to verifythat this assumption is in fact correct.

49

8 Problems

1. Suppose mt = m0+ mt�1 and m�t = m�

0+ �m�

t�1. Use (24) to show howthe behavior of the nominal exchange rate under �exible prices depends onthe degree of serial correlation exhibited by the home and foreign moneysupplies.

2. In the model of section 3 used to study policy coordination, aggregate-demand shocks were set equal to zero in order to focus on a commonaggregate-supply shock. Suppose instead that the aggregate-supply shocksare zero, and the demand shocks are given by u � x+ � and u� � x+ ��,so that x represents a common demand shock and � and �� are uncor-related country-speci�c demand shocks. Derive policy outcomes undercoordinated and (Nash) noncoordinated policy setting. Is there a role forpolicy coordination in the face of demand shocks? Explain.

3. Continuing with the same model as in the previous question, how are realinterest rates a¤ected by a common aggregate-demand shock?

4. Policy coordination with asymmetric supply shocks: Continuing with thesame model as in the previous two questions, assume that there are nodemand shocks but that the supply shocks e and e� are uncorrelated. De-rive policy outcomes under coordinated and uncoordinated policy setting.Does coordination or noncoordination lead to a greater in�ation responseto supply shocks? Explain.

5. Assume that the home country policy maker acts as a Stackelberg leaderand recognizes that foreign in�ation will be given by (47). How does thischange in the nature of the strategic interaction a¤ect the home country�sresponse to disturbances?

6. In a small open economy with perfectly �exible nominal wages, the textshowed that the real exchange rate and domestic price level were given by

�t =

1Xi=0

diEt

�a2r

�t+i + et+i � ut+ia1 + a2 + b1

�and

pt =

�1

1 + c

� 1Xi=0

�c

1 + c

�iEt (mt+i � zt+i � vt+i) ,

where zt+i � yt+i + (1� h)�t+i � crt+i: Assume that r� = 0 for all t andthat e, u, and z + v all follow �rst order autoregressive processes (e.g.,et = �eet�1 + xe;t for xe white noise). Let the nominal money supply begiven by

mt = g1et�1 + g2ut�1 + g3(zt�1 + vt�1).

Find equilibrium expressions for the real exchange rate, the nominal ex-change rate, and the consumer price index. What values of the parameters

50

g1, g2, and g3 minimize �uctuations in st? In qt? In �t? Are there anycon�icts between stabilizing the exchange rate (real or nominal) and sta-bilizing the consumer price index?

7. Equation (42) for the equilibrium real exchange rate in the two-countrymodel of section 3.1 takes the form �t = AEt�t+1 + vt. Suppose vt = vt�1 + t, where t is a mean-zero, white noise process. Suppose thesolution for �t is of the form �t = bvt. Find the value of b. How does itdepend on ?

8. Section 5.1 demonstrated how a simple open-economy model with nominalprice stickiness could be expressed in a form that paralleled the closed-economy new Keynesian model of chapter 6. Would this same conclusionresult in a model of sticky wages with �exible prices? What if both wagesand prices are sticky?

9. McCallum and Nelson (2000) McCallum and Nelson (2000) have proposeda new Keynesian open economy model in which imported goods are onlyused as inputs into the production of the domestic good and householdsconsumer only the domestically produced good. If et is the nominal ex-change rate, and st is the real exchange rate, The model can be summarizeby the following equations:

ct = Etct+1 � b1 [Rt � Et�t+1 � rt]

imt = yt � �stept = y�t + �

�st

st = et � pt + p�tRt = R�t + Etet+1 � et,

yt = (1� �)(nt + "t) + �imt

�t = �Et�t+1 + �xt

where imt denotes imports, ept denotes exports, and all variables areexpressed relative to their �exible-price equivalents. Foreign variables aredenoted by �. The linearized production function is

yt = (1� �)(nt + "t) + �imt

and the goods market equilibrium condition takes the form

yt = !1ct + !2gt + !3ept.

Show that this open economy model can be reduced to two equationscorresponding to the IS relationship and the Phillips curve that, whencombined with a spec�ciation of monetary policy, could be solved for theequilibrium output gap and in�ation rate. How does the interest elasticityof the output gap depend on the openness of the economy?

51

10. Assume the utility function of the representative household in a small openeconomy is

U = E0

1Xt=0

�t

(ln(Ct)�

N1+�

t

1 + �+

am1� m

�Mt

Pt

�1� m),

where C is total consumption, N is labor supply, and M=P is real moneyholdings. Ct is de�ned by (61), and utility is maximized subject to thesequence of constraints given by

PtCt+Mt+et1

1 + i�tB�t+

1

1 + itBt �WtNt+Mt�1+etB

�t�1+Bt�1+�t�� t.

Let Pht (Pft ) be the average price of domestically (foreign) produced con-

sumption goods.

(a) Derive the �rst order conditions for the household�s problem.

(b) Show that the choice of domestic-produced consumption goods rel-ative to foreign-produced consumption good basket depends on theterms of trade. Why is it not a function of the real exchange rate?

(c) Derive an expression of the price index Pt.

11. Using the �rst order conditions derived in question 10a, derive the Eulerequations and obtain the uncovered interest rate parity condition. Canyou provide economic intuition to explain this equation.

12. For the economy of questions 10 and 11, show how to derive the Fisherequation by introducing an extra asset into this economy and using theappropriate Euler equations.

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