metadata: authors: 0: author: Hevia, Constantino
1: author: Nicolini, Juan Pablo
count: World
docna: 0: docna: Optimal devaluations
docty: Policy Research Working Paper
theme: Micro, Small and Medium Enterprise support
majtheme: Financial and private sector development
subtopic: Macro-Fiscal Policy,Economic Adjustment and Lending,Public Sector Economics,Currencies and Exchange Rates,International Trade and Trade Rules,Macroeconomic Management,Public Finance Decentralization and
            Poverty Reduction
teratopic: Private Sector Development,Macroeconomics and Economic Growth,Finance and Financial Sector Development
abstracts: cdata!: According to the conventional wisdom,
            when an economy enters a recession and nominal prices adjust
            slowly, the monetary authority should devalue the domestic
            currency to make the recession less severe. The reason is
            that a devaluation of the currency lowers the relative price
            of non-tradable goods, and this reduces the necessary
            adjustment in output relative to the case in which the
            exchange rate remains constant. This paper uses a simple
            small open economy model with sticky prices to characterize
            optimal fiscal and monetary policy in response to
            productivity and terms of trade shocks. Contrary to the
            conventional wisdom, in this framework optimal exchange rate
            policy cannot be characterized just by the cyclical
            properties of output. The source of the shock matters: while
            recessions induced by a drop in the price of exportable
            goods call for a devaluation of the currency, those induced
            by a drop in productivity in the non-tradable sector require
            a revaluation.
repnme: repnme: Optimal devaluations
display_title: Optimal devaluations
listing_relative_url: /research/2009/05/10543491/optimal-devaluations
topicv3: Government Expenditure m1326649 1112,Trade m1326924 698,Exchange Rate Policy m1327143 1124,Monetary Policy m1327609 444,Trade Balance m1327679 1136
publishtoextweb_dt: 2010-07-01T00:00:00Z
historic_topic: Private Sector Development,Macroeconomics and Economic Growth,Finance and Financial Sector Development
no_of_pages: 35
model_extractions: WPS4926 Policy ReseaRch WoRking PaPeR 4926

# Optimal Devaluations

_Constantino Hevia Juan Pablo Nicolini_ The World Bank Development Research Group Macroeconomics and Growth Team May 2009
Policy ReseaRch WoRking PaPeR 4926

# **Abstract**

According to the conventional wisdom, when an economy enters a recession and nominal prices adjust slowly, the monetary authority should devalue the domestic currency to make the recession less severe. The reason is that a devaluation of the currency lowers the relative price of non-tradable goods, and this reduces the necessary adjustment in output relative to the case in which the exchange rate remains constant. This paper uses a simple small open economy model with sticky prices to characterize optimal fiscal and monetary policy in response to productivity and terms of trade shocks. Contrary to the conventional wisdom, in this framework optimal exchange rate policy cannot be characterized just by the cyclical properties of output. The source of the shock matters: while recessions induced by a drop in the price of exportable goods call for a devaluation of the currency, those induced by a drop in productivity in the non-tradable sector require a revaluation.

This paper—a product of the Growth and the Macroeconomics Team, Development Research Group—is part of a larger effort in the department to understand the role of fiscal and monetary policy in developing countries. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The author may be contacted at chevia@worldbank.org.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
# Optimal Devaluations

Constantino Hevia Juan Pablo Nicolini<sup>�</sup> The World Bank Universidad Di Tella

> �This paper started after a lunch with Eduardo Levy Yeyati and Ernesto Schargrodsky, and grew out of several interesting discussions and conversations with Pedro Teles. We also would like to thank Andres Neumeyer, Rodolfo Manuelli, Fernando Alvarez, Pat Kehoe and Raphael Bergoeing for comments. Finally, we want to thank Charles Engel for a very clarifying discussion. But all errors are our own. The ...ndings, interpretations, and conclusions expressed in this paper are entirely those of the author. They do not necessarily represent the view of the World Bank, its Executive Directors, or the countries they represent.

1
# INTRODUCTION

In most economies that can be described as small and open, the Central Bank targets the value of the nominal exchange rate by intervening in the foreign currency market. Sometimes, the nominal exchange is ...xed at a given value for a period of time; most often, the interventions respond to economic conditions in a discretionary way.

According to the conventional wisdom, when the economy enters a recession it is convenient to devalue the domestic currency due to the existence of price stickiness. It is argued that a devaluation lowers the relative price of non-tradable goods–that would not fall otherwise because nominal prices are sticky–and, therefore, allows for a smaller adjustment in quantities, making the recession less severe<sup>1</sup> . This conventional wisdom has been supported by the implications of static reduced form models in which the way ...rms that cannot adjust prices behave, given that constraint, is not formally modeled. The policy implication derived from these models is attractive in that it can be characterized in a simple way: the nominal exchange rate must be counter-cyclical.

In this paper we characterize the optimal exchange rate policy in a fully speci...ed dynamic general equilibrium model of a small open economy in which the price frictions are explicitly modeled and ...rms have rational expectations. The model identi...es the distortion that price stickiness introduces and the way this distortion interacts with other ones, like income taxes and monopoly power. We can formally analyze how these di¤erent distortions should be traded o¤ to maximize welfare, as it is traditionally done in the second-best literature. The purpose of the paper is to explore if a standard general equilibrium dynamic Ramsey model of a small-open economy supports the conventional wisdom.

We study a labor only representative agent economy with monopolistic competition–so ...rms have power to set prices–a cash-in-advance constraint, a ...nal non-tradable good, and

> 1This was the logic behind several devaluations, like the ones in Brazil (1999) Argentina (2001) and, to some extent, Spain, UK and other western European countries in the early 90 ́s.

2
two tradable intermediate inputs, one that can be exported and the other that must be imported–so we can analyze the optimal policy response following both real exchange rate and terms of trade shocks. Our model extends the closed economy model in Correia, Nicolini and Teles (2008) to allow for international trade in goods and international capital mobility.

We show that the conventional wisdom (that is, that a devaluation must follow a recession) does not hold even in the simple model we analyze: it depends on the source of the shock generating the ‡uctuation and on the modeling details. For example, if the recession is due to a decline in the international price of exportable goods, the nominal exchange rate should be devalued. On the other hand, if the recession is due to a negative productivity shock in the non-tradable sector, the nominal exchange rate should be revalued. In addition, modeling details such as the distinction between cash and credit goods or which is the sector where the price friction is present also matter. For example, as we impose a cash-in-advance constraint solely on consumers, it is never optimal to move the nominal exchange rate after government spending shock, but an active exchange rate policy is called for after those same shocks if the government also faces a cash-in-advance constraint.

We derive general principles for the conduct of optimal monetary policy which hold in other contexts<sup>2</sup> : optimal policy ought to fully stabilize the “right”price index and reproduce the ‡exible prices allocation. However, the way that principle translate into actual policy instruments depends on both the source of the shock and on details of the economy such as where exactly the rigidity is, which are the traded goods and which are the cash goods.

The policy instruments that we consider are labor income taxes, dividend taxes, a tax on the return to foreign assets, and monetary policy. We also allow the government to issue state contingent bonds. With these instruments, the government has a complete set of instruments, as de...ned in Chari and Kehoe (1999). As it is standard in Ramsey analysis, we abstract from time inconsistency and assume full commitment. Thus, whichever role the

> 2See for instance Obstfeld and Rogo¤ (2000) and Woodford (2003).

3
exchange rate can have in fostering good (or bad) reputation will be absent in this analysis. Following the seminal work of Obstfeld and Rogo¤ (1995, 1996) there has been a growing interest in studying optimal policy in open economies with frictions in the setting of prices. A branch of the literature, like Obstfeld and Rogo¤ (2000), Engel (2001) and many others<sup>3</sup> focused on the two-countries case. This literature emphasizes the relationship between the strategic interactions in two-countries models and optimal exchange rate policy, and in most cases it focused on the ‡exible versus ...xed exchange rate regimes debate. In addition, it ignores - with the exception of Adao, Correia and Teles (2005) - the interaction between exchange rate and monetary policy with distortionary ...scal policy, an issue that we formally address.

The case of the small open economy we focus has been the subject of several papers. Gali and Monacelli (2005) and de Paoli (2004) study a similar problem but do not consider the ...scal instruments as we do. Their approach only allows solving for linear approximations very useful to compare alternative rules, but less so to characterize properties of the optimal policy as we do. Cunha (2001) analyzes a model similar to ours but without a complete set of instruments and he can only solve some examples. Closer to ours is the work of Faia and Monacelli (2004a) and (2004b), who use a framework similar to ours, but without jointly considering ...scal and monetary instruments.

The main contribution of this paper is the focus: rather than studying the welfare properties of families of policy rules in general, we want to answer the following question: should a government of a small open economy devalue during a recession? The key di¤erence with the literature is the joint analysis of monetary and ...scal policy with distortionary taxation. It turns out that with a complete set of instruments, the analysis of the solution is rather simple since all known results on the Ramsey literature can be applied. Therefore, we can

> 3An incomplete list also includes Corsetti and Pesenti (2001) and (2005), Devereux and Engel (2003), Benigno and Benigno (2003), Duarte and Obstfeld (2004), Ferrero (2005) and Adao, Correia and Teles (2005).

4
provide a very sharp answer to the question we focus on.

The paper is organized as follows. Section 2 spells out the model. Section 3 solves for the optimal allocation and characterizes the optimal monetary and exchange rate policy as a function of the shocks to the economy. Section 4 discusses extensions and concludes.

# 1. THE MODEL

We model a dynamic small open economy with uncertainty. Time is discrete and the state of the economy at period t = 0; 1; ::: is denoted by st, which belongs to a ...nite set of events, with s0 given. Let s<sup>t</sup> = (s0; s1; :::; st) be the history of events up to period t, and let � (s<sup>t</sup> ) be the probability of s<sup>t</sup> conditional on s0.

Households: There is a representative household who derives utility from a composite good and leisure according to where 0 < �< 1; U (�) is increasing in each argument and concave, ` (s<sup>t</sup> ) denotes leisure, and c (s<sup>t</sup> ) is a composite ...nal goods de...ned as The variable c (i; s<sup>t</sup> ) denotes consumption of a ...nal good of variety i 2 [0; 1], and �> 1 is the elasticity of substitution across varieties.<sup>4</sup> We assume that all varieties are non-traded, so they must be produced domestically.

Markets are complete. We let B (s<sup>t</sup> ; st+1) and B<sup>�</sup> (s<sup>t</sup> ; st+1) be one-period bonds denominated in domestic and foreign currency respectively. These bonds are issued at s<sup>t</sup> and pay

> 4Existence of an equilibrium requires �> 1.

5
one unit of the corresponding currency at t + 1 on the event st+1 and zero otherwise. The price in domestic and foreign currency of these claims are Q (s<sup>t+1</sup> js<sup>t</sup> ) and Q<sup>�</sup> (s<sup>t+1</sup> js<sup>t</sup> ) respectively. For k > 0, let Q �s<sup>t+k</sup> js<sup>t�</sup> = Q (s<sup>t+1</sup> js<sup>t</sup> ) Q (s<sup>t+2</sup> js<sup>t+1</sup> ) :::Q �s<sup>t+k</sup> js<sup>t+k�1�</sup> be the price of one unit of currency at s<sup>t+k</sup> in units of currency as s<sup>t</sup> . A similar de...nition holds for Q<sup>��</sup> s<sup>t+k</sup> js<sup>t�</sup> : As long as a no-arbitrage condition between domestic and foreign bonds is satis...ed (see condition (18) below), we can assume, without loss of generality, that households do not hold foreign bonds.<sup>5</sup> Therefore, for all periods, households face the budget constraint where W (s<sup>t</sup> ) (1 � �<sup>n</sup> (s<sup>t�1</sup> )) is the net of taxes nominal wage rate, M<sup>h</sup> (s<sup>t</sup> ) is the household’s demand for currency, and Pt (i; s<sup>t</sup> ) is the price of the ...nal good of variety i. We assume that initial nominal wealth is zero: M�1 + B0 = 0.<sup>6</sup> We model a money demand by imposing a cash-in-advance constraint on a subset of goods. The timing protocol follows Lucas (1984) and we assume that goods in the set � � [0; 1] are cash goods, so we impose the following cash-in-advance constraint By making � a proper subset of the unit interval, we allow for cash and credit goods in the model.

> 5Note that markets are complete with the domestic securities, so the foreign currency denominated securities are redundant. It is convenient to introduce these bonds for two reasons, ...rst, to let all intertemporal trade with the rest of the world be carried out by issuing foreign securities, and second, to derive the interest parity conditions.

> 6We assume this in order to avoid the well known problem of hyperin‡ation incentives in period 0 (Lucas and Stokey 1983).

6
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~~(—~~ ) ~~7——~~ fo ~~rd~~ cy) ~~(—)~~

*

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the compact notation described above.

The introduction of taxes on foreign securities is useful for two reasons: ...rst, it roughly captures the idea of taxing international capital ‡ows, and second, they are used to implement the Ramsey allocation. In the last section we discuss the consequences of eliminating these taxes, and how they can be substituted by consumption taxes. In addition, since taxation of dividends is equivalent to lump-sum taxes, any optimal policy will set dividend taxes to its highest feasible value �<sup>d</sup> (s<sup>t</sup> ) = 1 for all s<sup>t</sup> .<sup>8</sup> Tradeable Sector: The non-tradeable varieties are produced using two tradeable inputs. One of them can be produced domestically and the other must be imported.<sup>9</sup> Output of the home - exportable - input is given by the linear production function where A (st) is a technology shock and n (s<sup>t</sup> ) is labor. Competitive pricing requires where Px (s<sup>t</sup> ) is the domestic currency price of the home input.

As this is a small open economy, we assume that international trade is carried out in foreign currency. Absence of arbitrage opportunities implies the purchasing power parity conditions

> 8The results also hold if � d (st) = 0 or if it is constrained to be a positive number lower than one. The problem is much simpler with our assumption, otherwise the governemnt will try to manipulate other instruments, like monetary policy or exchange rates in order to tax pro...ts.

> 9This allows consideration for terms of trade shocks.

9
() Cyc ( ) ~~———5~~ ~~GA)~~ () ( ~~=a(-b)~~
We assume that each variety is produced by a monopolist.<sup>11</sup> We also assume that there is price stickiness. In particular, ...rms i 2 [0; �] are constrained to set prices at period t conditional on the information available up to period t � 1. Equivalently, we can think of those ...rms, called sticky ...rms, as setting prices one period in advance.<sup>12</sup> The other ...rms, called ‡exible ...rms, are allowed to set prices at t conditional on s<sup>t</sup> .

The demand faced by ...rm i is y<sup>d</sup> (i; s<sup>t</sup> ) = c (i; s<sup>t</sup> ) + g (i; s<sup>t</sup> ), and di¤ers according to the cash-credit characteristic of the good. Equations (6), (7) and (9) imply where Note that both the marginal cost and the elasticity of demand that each monopolist face is the same across varieties. Thus, from the point of view of preferences and technologies, the model exhibits total symmetry. However, the frictions in transactions and in the setting of prices do introduce heterogeneity across ...rms. Indeed, there are four types of ...nal goods ...rms: ‡exible ...rms with and without cash-in-advance constraint, and sticky ...rms with and without cash-in-advance constraints.

Flexible ...rms face the static optimization problem of maximizing nominal dividends,<sup>13</sup>

> 11This assumption is essential to have ...rms that can set prices in advance.

> 12The results also hold for other forms of price stickyness, as Calvo staggered pricing for instance. For details, see Correia, Nicolini and Teles (2002).

> 13Strictly speaking, full taxation of dividends imply that the pricing and production decisions of the ...rms are indeterminate. It is convenient, instead, to think of each ...rm maximizing after-tax dividends for �<sup>d</sup> j<sup>< 1,</sup> and then considering the limit economy as limj!1 �<sup>d</sup> j<sup>= 1forallst.</sup>

11
( ~~ra *~~ <u>UT]</u> ) ~~§>_ apo~~ 0 — € W)EC = ON 0) ( -) ~~7~~ _ ~~)———~~ ( ) ~~——____~~ () ()( ) ~~-——~~ ( ) 1 a!
re ed ee ne ~~so~~ () , ) Br) 8 ()x() ~~——~~ ( ) (Ct ~~)~~
Market Clearing: Feasibility in the ...nal goods, labor and currency market require De...nition: An allocation a and a price system P are contingent sequences a = fc (i), `, x (i), m (i), y (i)g and P = fQ, R, Px, Pm, P (i), W g for i 2 [0; 1].

De...nition: Given a government policy !, an allocation a and a price system P are an equilibrium if (i) households solve their utility maximization problem; (ii) the price of the home input satis...es (10); (iii) ...nal goods producers act optimally: (12) hold for all i 2 [0; 1] and they follow the pricing rules (14) if i 2 (�; 1] and (15) if i 2 [0; �]; (iv) the market clearing conditions (19), (20) and (21) are satis...ed; (v) the economy-wide feasibility constraint (17) holds; (vi) the no-arbitrage conditions (11) and (18) hold; and vii) net nominal interest rates are non-negative, R (s<sup>t</sup> ) � 1 for all s<sup>t</sup> . (By Walras’Law, the government budget constraint is also satis...ed.)

# 2. THE RAMSEY PROBLEM

As it is standard in the literature, we assume the government can commit to a particular policy chosen at period 0. We de...ne an allocation rule a (!) as the mapping from the set of government policies into equilibrium allocations. Speci...cally, a (!) is the set of allocation that satisfy conditions i) to vii) in de...nition 2 given !, and we call these allocations implementable. The Ramsey problem is to choose the implementable allocation that maximizes the household’s utility (1). Let the solution, called the Ramsey allocation, be denoted by a<sup>R</sup> .

14
a,T) aD’) () () ( ~~)ea(4)~~ 8 ()x() ~~——~~ ( ) ( - () ~~)-—>>—~~ gf 0) Q)()C- (RO)
Cy) ~~—{~~ / (0) — = () **(=** 7()} <!-- Start of picture text --> /<br><!-- End of picture text --> L a,T) a,T) a,T) a,T) a,T) a,T)
<!-- Start of picture text --> a,T)<br><!-- End of picture text --> a,T) Qa r Qa r a,T) € 2 () ( ) BrC)t (C0) OC) AL OC) C)C)0C> OI - « [0 ) (= ( ~~)=5(4)~~ ] ¢ (— () ~~)-—3—~~ Bar € GOA
Lemma 1: The relaxed Ramsey allocation satis...es c (i; s<sup>t</sup> ) = c (s<sup>t</sup> ) for all i and s<sup>t</sup> . In e addition, if a is implementable, then g (i; s<sup>t</sup> ) = g (st) ; x (i; s<sup>t</sup> ) = x (s<sup>t</sup> ) and m (i; s<sup>t</sup> ) = m (s<sup>t</sup> ) for all i and s<sup>t</sup> : Furthermore, prices satisfy Pt (i; s<sup>t</sup> ) = Pt (s<sup>t</sup> ) for all i and s<sup>t</sup> ; and the Friedman rule is optimal, that is R (s<sup>t</sup> ) = 1 for all s<sup>t</sup> : Proof: Taking the necessary ...rst order conditions of the Lagrangian with respect to m (i; s<sup>t</sup> ) and c (i; s<sup>t</sup> ) ; we obtain The last two equations and the de...nition of c (s<sup>t</sup> ) imply c (i; s<sup>t</sup> ) = c (s<sup>t</sup> ) for all i and s<sup>t</sup> . It follows from the de...nition of Pt (s<sup>t</sup> ) and conditions (6) and (7) that Pt (s<sup>t</sup> ) = Pt (i; s<sup>t</sup> ) for all i and s<sup>t</sup> . Then (8) and the previous results imply R (s<sup>t</sup> ) = 1 for all s<sup>t</sup> : Further, g (i; s<sup>t</sup> ) = g (st) for all i and s<sup>t</sup> follows from (9), and equations (22) and (12) imply m (i; s<sup>t</sup> ) = m (s<sup>t</sup> ) and x (i; s<sup>t</sup> ) = x (s<sup>t</sup> ) for all i and s<sup>t</sup> .� Corollary: If �> 0; all prices at period t depend on s<sup>t�1</sup> . That is, the price of the ...nal goods are perfectly forecastable one period in advance.

The intuition is as follows: the set J<sup>e</sup> does not restrict the behavior of the ...nal goods ...rms. Since any pair of ...nal goods c (i; s<sup>t</sup> ) and c (j; s<sup>t</sup> ) for i 6= j are imperfect substitutes in the homogeneous consumption aggregate c (s<sup>t</sup> ), it is optimal to set a symmetric allocation in which c (i; s<sup>t</sup> ) = c (s<sup>t</sup> ) for all i. If implementable, a symmetric allocation requires an equal price across varieties. The prices that matter are those faced by the households and the government, and incorporate the ...nancial cost of holding currency in order to buy cash-goods. Therefore, this requires an identical nominal price P (i; s<sup>t</sup> ) across varieties and 18
a,T) (0) () X= (0) 0) Ab 0) O- (C—O) Dr Dr (() ()4 ~~—+~~ 7) UT (() ()a 64
Feasibility in the ...nal goods market becomes Equations (25), (26) and (27) together with constraints (23) and (24) characterize the relaxed Ramsey allocation, where the trade balance becomes Speci...cally, equations (25) and (26) can be used to solve for consumption and leisure as a function of the multipliers � and �. Then we can use (12) and (27) to solve for m (s<sup>t</sup> ) and x (s<sup>t</sup> ) as a function of � and �. The present value constraints (23) and (24) can be used to ...nd the values for � and �.

Equations (25), (26), (27) and (28) imply that the Ramsey allocation only depends on the realization of the stochastic processes dated at period t, as government purchases g (st), productivity shocks A (st) and Z (st), and the terms of trade shock d<sup>�</sup> (st), and not on the history of realizations s<sup>t</sup> . In addition, government expenditure shocks do not a¤ect c (s<sup>t</sup> ) nor ` (s<sup>t</sup> ) since both can be solved as a function of the shocks A (st), Z (st) and d<sup>�</sup> (st), and of the two multipliers � and �, which do not depend on the particular realization of g (st). For example, if government expenditures increase, equation (27) shows that in order to keep private consumption constant, the usage of the tradeable inputs m (s<sup>t</sup> ) and x (s<sup>t</sup> ) increase. Then, (28) shows that the trade balance decreases. In other words, the availability of international credit allows the planner to insure all government expenditure shocks through borrowing and lending.

In what follows we assume that utility is separable between consumption and leisure and 20
that is well behaved, so U<sup>e</sup> (c; `) = H (c; �) � V (1 � `; �) is increasing and concave.<sup>20</sup> We will analyze these cases ...rst, and we discuss other cases in the last section.

Response to Shocks: Here we study how the di¤erent shocks a¤ect the optimal allocation. Suppose that we want to analyze a shock to the ...nal good technology Z (st). Strictly speaking, given s<sup>t�1</sup> ; the analysis compares two states st and s<sup>0</sup> t<sup>suchthatZ (st)>Z (s0</sup> t<sup>) :</sup> However, given the stationarity of the Ramsey allocation in our set-up, this is equivalent to analyze the experiment Z (st) > Z (st�1). Thus, both interpretations follow from our analysis. In addition, since government expenditure shocks were analyzed above, here we focus on productivity shocks Z (st) and A (st) ; and terms of trade shocks d<sup>�</sup> (st).

1) Consider a negative shock to the ...nal goods technology Z (st). Leisure remains constant, consumption decreases. The change in the trade balance and the change in usage of inputs m (st) and x (st) are indeterminate.

Proof: Conditions (25) and (26) imply that c (st) decreases and ` (st) remains constant. From (27) it follows that x (st) could either increase or decrease. Hence from (12) and (28), m (st) and TB (st) could increase or decrease.� The constancy of labor supply is a consequence of the separable utility function and the fact that the country has access to international capital markets. This can be seen in equation (26), where it is evident that the allocation of labor depends solely on the shocks to the home input sector and the multipliers, which capture the wealth of the country in present value terms. The usage of inputs x (st) and m (st) could increase or decrease depending on the curvature (i.e. the willingness to substitute intertemporally) of the utility function.

2) Consider a negative shock to the terms of trade d<sup>�</sup> (st) : Leisure remains constant and consumption decreases. The usage of the foreign input m (st) decreases. The change in x (s<sup>t</sup> ) and TB (st) are indeterminate.

> 20In this case, charaterizing the reaction of the optimal allocation after a shock is equivalent to a ...rst best and the intuitions are straighforward.

21
Proof: (25) and (26) imply that c (st) decreases and ` (st) does not change. From (27) it follows that m (st) decreases. Then (12) imply that x (st) could increase or decrease. Finally, from (28), TB (st) could increase or decrease.� Intuitively, a negative shock to the terms of trade has, in addition to a negative income e¤ect, a substitution e¤ect. The country is poorer since the price of the exportable input is lower. The negative income e¤ect implies a reduction in consumption. Labor remains constant for the same reason as in 1) above. The income e¤ect implies a lower usage of the inputs x (st) and m (st) to produce less ...nal goods. But the reduction in the relative price of x (st) relative to m (st) implies a substitution toward the home input. Both e¤ects imply an reduction in m (st) ; but the net e¤ect in x (st) and the trade balance, are indeterminate.

3) Consider a negative shock to the home input technology A (st) : c (st) ; m (st) and x (st) do not change. Leisure increases and the trade balance decreases.

Proof: Equations (12), (25) and (27) imply that neither c (st) ; m (st) ; nor x (st) change. Condition (26) implies that ` (st) increases, and hence, from (28), TB (s<sup>t</sup> ) also decreases.� The shock to the home-input technology determines when it is good a time to export and when it is not. Periods with low values of A (st) are bad periods to export, hence leisure increases and the trade balance decreases.

The table below summarizes these results Table I. Response of the allocation to negative shocks Note that for any shock, either consumption or labor remains constant. This is because of the joint e¤ect of two assumptions: separability of labor and consumption in preferences and 22
() (7) Ly r P () ~~Soy~~ yn ~~"~~ () ~~a (Fe~~ ) © "C= ())
the household’s equilibrium condition (4) after using (10), (11) and (31).

The equilibrium policies and prices are obtained as follows. Given the initial price P0 (s<sup>�1</sup> ) = P0<sup>s,(29)determinesrecursivelythepriceofthe...nalgood.Thecash-in-advanceconstraint</sup> (30) determines the money supply at period t that implements the price Pt (s<sup>t�1</sup> ). Condition (31) determines the foreign-input price Pm (s<sup>t</sup> ) and (32) pins down the labor tax rate �<sup>n</sup> (s<sup>t</sup> ). The exchange rate and home-input price follow from (11), and (10) determines the nominal wage rate W (s<sup>t</sup> ) : The prices Q (s<sup>t+1</sup> js<sup>t</sup> ) follow from (5), and the taxes on foreign securities �<sup>�</sup> (s<sup>t+1</sup> ) are pinned down from (18). (The decentralization of asset holdings is described in the appendix.) Policy Response to Shocks: In all cases, the behavior of labor taxes is indeterminate, except for a government purchase shock, where they stay constant. Moreover, note that none of the instruments or prices change with government expenditure shocks. This is a consequence of the previously mentioned result that expenditure shocks are fully insured through international capital markets. We study the optimal response of policy after the other three shocks.

1) Negative shock to the ...nal goods technology Z (st). Consumption decreases, and labor remains constant. Since Pt (s<sup>t�1</sup> ) is given, it follows from (30) that a lower consumption is implemented through a reduction in M (s<sup>t</sup> ) : In addition, the input prices, the nominal wage rate and the exchange rate all decrease. The decline in Pm (s<sup>t</sup> ) ; Px (s<sup>t</sup> ) and E (s<sup>t</sup> ) follow from (31) and (11), and the decrease in W (s<sup>t</sup> ) follows from (10).

2) Negative shock to the terms of trade d<sup>�</sup> (st) : Consumption decreases, labor remains constant and m (st) decreases. The decentralization of a terms of trade shock di¤ers depending on whether the decline in d<sup>�</sup> (st) is driven by a lower Px<sup>�(st)orahigherP �</sup> m<sup>(st).Inany</sup> case, from (30), the lower consumption level is implemented through a decline in M (s<sup>t</sup> ).

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Condition (31) implies that Pm (s<sup>t</sup> ) increases but d<sup>�</sup> (st) Pm (s<sup>t</sup> ) does not change.<sup>21</sup> Since Px (s<sup>t</sup> ) = d<sup>�</sup> (st) Pm (s<sup>t</sup> ), the home-input price does not change, and therefore, (10) implies that W (s<sup>t</sup> ) also remains constant. If the negative shock to d<sup>�</sup> (st) is driven by a decline in Px<sup>�(st) ;(11)impliesthatE (st)increases.Ifthedeclineind�(st)isdrivensolelybyan</sup> increase in Pm<sup>�(st) ; from (11) and (31), the nominal exchange rate does not change. (Due to</sup> this result, the next table considers changes in Px<sup>�(st)andP �</sup> m<sup>(st)separately.)</sup> 3) Negative shock to the home-input technology A (st). Consumption remains constant and labor decreases. (30) implies that M (st) does not change. (31) implies that Pm (s<sup>t</sup> ), and from the (11) conditions, Px (s<sup>t</sup> ) and E (s<sup>t</sup> ) do not change. Finally, condition (10) implies that W (s<sup>t</sup> ) declines.

The following table summarizes the previous results Table II. Response of the optimal policies and prices to negative shocks Table II summarizes the main message of the paper: it is not possible to characterize optimal policy by the cyclical properties of the economy even in this very simple model. The table considers several negative exogenous shocks hitting the economy. Some shocks call for a depreciation of the exchange rate, others call for an appreciation of the exchange rate, and others call for a constant exchange rate.

> 21The last result is special to the Cobb-Douglas speci...cation.

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# 4. EXTENSIONS AND CONCLUSIONS

In this section we discuss extensions to the model we just analyzed and show that they reinforce the results obtained. We assumed that labor enters only in the production of the tradeable input and that consumption and labor were separable in preferences. These two assumptions imply that either leisure or consumption would remain constant after any of the shocks analyzed and made the discussion of the decentralization straightforward and simple. The model can be extended by relaxing any of these assumptions. For instance, if we introduce labor as an additional input in the production of the continuum of ...nal goods, a negative shock to the home-input technology A (st) can be shown to imply a devaluation of the exchange rate and an increase in the money supply.<sup>22</sup> These results contrasts with those in Table II, where the nominal exchange rate and the money supply do not change. We draw two conclusions from the discussion. First, that modelling details are important for the qualitative predictions of the model, and second, that the main message that it is not possible to characterize optimal policy in terms of simple rules based on the cyclical properties of the economy is a robust result.

Another key assumption in the model is the existence of taxes on foreign securities �<sup>�</sup> (s<sup>t</sup> ) : This instrument gives the planner the ability to manipulate the intertemporal prices faced by the domestic residents without violating the interest parity condition (18). The same allocation can be implemented if, instead of having the taxes �<sup>�</sup> (s<sup>t</sup> ), the government has access to a uniform consumption tax over varieties i 2 [0; 1] : If, however, the government does not have access to consumption taxes, the constraint (18) binds. The allocation will be di¤erent, and welfare will be lower. Further, even though the consumption aggregate c (s<sup>t</sup> ) is homogeneous of degree one in the individual varieties, a condition known to imply that the Friedman rule is optimal with a cash-in-advance constraint on a subset of varieties (Chari,

> 22This version of the model is analyzed in Hevia and Nicolini (2004).

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Christiano and Kehoe, 1991), it will not be so in this case, since the government does not have a complete set of instruments. Intuitively, there is a trade-o¤ between distorting the relative consumption of cash and credit varieties, and the constraint imposed by (18), which ties down the intertemporal prices faced by households. By manipulating the nominal interest rate the government a¤ects the intertemporal prices, but distorts the relative consumption of varieties.<sup>23</sup> The same happens if the tax rates cannot be made state contingent or if the government cannot issue state contingent debt.

We imposed frictions in the setting of ...nal good prices. An alternative, widely used in the literature, is to assume frictions in the setting of wages. In addition, we imposed a one period rigidity, while most of the literature assumes staggered price setting. If we relax both assumptions, the Ramsey allocation is still independent of both the set of cash goods and the set of ...rms with sticky wages, and the same as in our model. However, the decentralization would be very di¤erent, since monetary and exchange rate policy would aim at stabilizing nominal wages, rather than prices. So, the optimal response of the nominal exchange rate— to implement the same Ramsey allocation— would be di¤erent in general.

Finally, if the conditions imposed on the utility function in Section 3 are not satis...ed, some of the conclusions regarding the optimal response of the allocation may be di¤erent. This is a standard feature of optimal taxation Ramsey problems and means that some of the results described in Table I may be di¤erent from what we obtained. However, given a particular movement of the optimal allocation, the way to decentralize it is the same as before.

This discussion reinforces the message of the paper: optimal policy is about dealing with distortions in response to shocks. The optimal exchange rate policy critically depends on which is the set of instruments available, on how -given those instruments- the optimal allocation responds to the shocks and on the mapping from policies to allocations.

> 23These results are discussed in Hevia and Nicolini (2004).

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There is a general policy principle that can be derived from our analysis, which is consistent with previous work: price stability is optimal, since this price stability is the one that avoids the distortion between ‡exible and sticky varieties. But to make this principle operational, one needs to know, ...rst, where the price stickiness is, and second, which is the transmission mechanism from the exchange rate to those sticky prices. The qualitative relationship between movements in aggregate output and optimal movements on the exchange rate critically depends on those modeling details.

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Given ea and P0 (s<sup>�1</sup> ) = P0<sup>s; (A5) determines recursively the price level Pt(st�1) that justi...es</sup> the Friedman rule R (s<sup>t</sup> ) = 1. Given the sequence of prices, (A2) determines the required money supply, and the equilibrium prices Q (s<sup>t+1</sup> js<sup>t</sup> ) follow from (5). The pricing equation (14) together with the condition Pt (i; s<sup>t</sup> ) = Pt (s<sup>t�1</sup> ) can be used to pin down the foreign-input price Pm (s<sup>t</sup> ) : Given Pm (s<sup>t</sup> ) ; (11) determines the nominal exchange rate E (s<sup>t</sup> ) and the home input price Px (s<sup>t</sup> ) : The wage rate W (s<sup>t</sup> ) follows from (10), and (4) pins down the labor tax rate �<sup>n</sup> (s<sup>t</sup> ). Finally, the stock of domestic debt B (s<sup>t</sup> ) is obtained from (A3).

By construction, ‡exible and sticky ...rms ...nd it optimal to set a common price Pt (s<sup>t�1</sup> ) : Since all the conditions of an equilibrium are satis...ed, the relaxed Ramsey allocation is implementable with a unique policy !<sup>R</sup> : Thus, ea is also the Ramsey allocation.

Case II. Flexible economy, i.e. � = 0: In this case, as in Lucas and Stokey (1983), monetary policy fE; M g and domestic bonds policy fBg are indeterminate. Given the e allocation a; the real wage W (s<sup>t</sup> ) =Pt (s<sup>t</sup> ) is determined from (10) and (14). Then, (4) pins down the labor tax rate �<sup>n</sup> . To see the indeterminacy of monetary policy, note that by symmetry, all ...nal goods ...rms set a common price Pt (s<sup>t</sup> ) given by (14). This price, in general, depends on the current shock st. Speci...cally, given the allocation, any sequence of prices satisfying (A1) can be an equilibrium path for the price level. In particular, the equilibrium price sequence Pt (s<sup>t�1</sup> ) obtained in case I can be implemented as an equilibrium in a ‡exible prices economy. Then, (14) and (11) determines the nominal exchange rate E (s<sup>t</sup> ), and given the prices and allocation, (A3) pins down the bonds holdings B: The rest of the equilibrium prices are obtained as in case I.

Case III. Fully sticky prices economy, i.e. � = 1. This case also features an indeterminacy, but of a very di¤erent nature. Here, all policy instruments except labor taxes and nominal exchange rates are uniquely determined. First, given P0<sup>sand the allocation ea, the sequence of</sup> prices Pt (s<sup>t�1</sup> ) is obtained from (A5). The money supply M (s<sup>t</sup> ) and the equilibrium prices Q (s<sup>t+1</sup> js<sup>t</sup> ) are given by (A2) and (5). To see the indeterminacy, note, ...rst, that in period 0 any combination of �<sup>n</sup> 0<sup>and W0satisfying (4) can be implemented as an equilibrium.Second,</sup> using R (s<sup>t</sup> ) = 1; (4), (5), (10) and (11) into (15) for all t � 1, and noting that Pt (s<sup>t�1</sup> ) is independent of st, we obtain where Any labor tax rate �<sup>n</sup> (s<sup>t</sup> ) satisfying (A6) can be decentralized. In particular, any pair of labor tax policies �<sup>n</sup> 1<sup>(st)and�n</sup> 2<sup>(st)satisfying(A6)decentralizestheRamseyallocation.</sup> According to (5), however, they induce di¤erent nominal wages W1 (s<sup>t</sup> ) and W2 (s<sup>t</sup> ), which, by virtue of (10) and (11) induce di¤erent nominal prices fE1 (s<sup>t</sup> ) ; Px1 (s<sup>t</sup> ) ; Pm1 (s<sup>t</sup> )g and fE2 (s<sup>t</sup> ) ; Px2 (s<sup>t</sup> ) ; Pm2 (s<sup>t</sup> )g.

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In all three cases I, II and III, there are prices and policy instruments !<sup>R</sup> such that relaxed Ramsey allocation ea is an equilibrium allocation. Therefore, ea is indeed the Ramsey allocation a<sup>R</sup> :� 31