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A simple approach to international monetary policy coordination
Journal of International Economics 57 (2002) 177–196 www.elsevier.com / locate / econbase
A simple approach to international monetary policy coordination Pierpaolo Benigno* Department of Economics, New York University, New York, USA and CEPR, London, UK Received 12 May 1999; received in revised form 15 January 2001; accepted 7 April 2001
Abstract This paper analyzes the strategic interaction between the monetary policymakers of two countries, in an intertemporal general equilibrium model with nominal rigidities and imperfect competition. It offers an excursus on non-cooperative towards cooperative solutions. In a non-cooperative equilibrium the monopolistic allocation prevails in both countries, because of the incentive to use strategically the terms of trade. In a cooperative solution where both policymakers internalize the externalities given by the terms of trade, the competitive allocation is reached. However, cooperation can be counterproductive. We then characterize a problem of delegation in which the set of choice is restricted to the Pareto efficient allocations and in which the participation constraints implied by the non-cooperative equilibrium are taken into account. 2002 Elsevier Science B.V. All rights reserved. Keywords: Open-economy macro; Monetary coordination JEL classification: E42; F41; F42
Our analysis of monetary policy [ . . . ] relies on social welfare functions defined directly over macroeconomic outcomes and policy instruments. This lack of explicit microeconomic underpinnings reflects the state of the art in
*Tel.: 11-212-998-8958; fax: 11-212-995-4186. E-mail addresses: [email protected] (P. Benigno), http: / / homepages.nyu.edu / |pb50 (P. Benigno). 0022-1996 / 02 / $ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S0022-1996( 01 )00132-5
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much of the literature, which relies on linear versions of the Mundell– Fleming model.1
1. Introduction The state of the art has changed: Corsetti and Pesenti (2001) have shown how to reconcile rigor in microfoundations and ‘tractability’ through a two-country general equilibrium model with nominal rigidities and monopolistic competition. In their paper, once a closed-form solution is obtained, it is possible to analyze the effect of unanticipated monetary policy shocks, even of large size, and to compute well-defined welfare functions. We utilize their simple framework in order to address the issue of international monetary policy coordination. We study the normative implication of different forms of strategic interaction, thus revisiting the basic findings of the ad hoc literature on international monetary policy coordination, with no ‘lack of explicit microeconomic underpinnings’.2 Given that wages are predetermined, monetary policy has an influence on real activity and welfare. In this model, there are two distortions: one is ‘structural’ and is given by the monopolistic-competition factor market, the other is ‘strategic’ and is given by the terms of trade. In a closed-economy model, monetary policy would get rid of all the distortions induced by monopolistic competition. Real wages would be equated to the marginal cost of labour in terms of utility — the competitive allocation. In an open-economy model, when the policymakers act in a non-cooperative way, the ‘strategic’ distortion given by the terms of trade is in conflict with the objective of eliminating the monopolistic distortion. Were both policymakers to push their economies at the competitive level, each policymaker would have an incentive to contract its money supply. In fact the reduction in utility that comes from the decrease in consumption is more than offset by the reduction in the disutility of producing goods, since the ‘burden’ of production is shifted to the other country through an improvement of the terms of trade. The strategic use of the terms of trade is at the root of the ‘contractionary bias’. This contraction is larger, the higher are the degrees of openness of the economy and the higher is the degree of substitutability in utility between the home and foreign goods. Instead, in the traditional literature (e.g. Canzoneri and Henderson, 1991), the ‘contractionary bias’ emerges because the improvement of terms of trade can lower inflation. 1
Persson and Tabellini (1995, p. 2003). The traditional approach to international policy coordination is built in Hamada (1985) and in the first chapters of Canzoneri and Henderson (1991). The main conclusion of their analyses is that if policymakers cooperate, they could all be better off. On the other hand, Rogoff (1985) points out how monetary policy cooperation can be counterproductive when policymakers lack credibility. Persson and Tabellini (1995) emphasize the role of institutions in enforcing the cooperative solution. 2
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We investigate if other forms of interaction can improve upon the noncooperative equilibrium. Firstly, we consider the outcome of a cooperative agreement, in a decentralized setting, where both policymakers consider as given the exchange rate and thus, in a world of preset prices and wages, the terms of trade. By internalizing the distortions induced by the terms of trade, policymakers can also eliminate the distortion given by monopolistic competition. As in Rogoff (1985), cooperation reduces the incentive to contract money supply and, moreover, can be counterproductive. In fact, the welfare of the non-cooperative solution can be Pareto improved only if the difference in the ‘economic size’ of the countries is not large. We then investigate a form of centralized cooperation in which society delegates the conduction of monetary policy to a central institution. As the degree of substitutability in utility between the goods increases, there is more room for cooperation. In fact, the incentive to a ‘strategic’ use of the terms of trade is stronger and the ‘contractionary bias’ is larger. It follows that the competitive allocation can be implemented also for a large difference in ‘economic size’ across countries. In this case, there is a rationale for delegating the conduction of monetary policy to a central institution that completely internalizes the distortions induced by the terms of trade and achieves the competitive allocation. The work is organized as follows. In the next two sections, the Corsetti and Pesenti model is briefly outlined. Section 4 discusses the non-cooperative Nash equilibrium and its sub-optimality properties. Section 5 discusses the cooperative solution in a decentralized setting. Section 6 presents the characteristic of the Pareto frontier, while Section 7 studies the problem of optimal delegation.
2. Building blocks of the model In this section we briefly outline the key building blocks of the model following Corsetti and Pesenti (2001). This model belongs to a class of micro-founded models of imperfect competition with nominal rigidities, that have recently become a modern paradigm for analyzing international policy issues.3 The assumption of nominal rigidities — in this paper sticky wages — allows monetary policy to affect real variables at least in the short run. A monopolistic-competition factor market rationalizes the existence of nominal rigidities. The model includes two countries, home and foreign, each specialized in the production of a single traded good. In each country there is a continuum of economic agents, with population size normalized to 1. Home agents are indexed by j, foreign agents by j*. The lifetime utility of home agent j is given by 3
A partial list includes Svensson and van Wijnbergen (1989), Obstfeld and Rogoff (1995, 1996, 1998), and Devereux and Engel (1999). Lane (2001) offers a survey of this literature.
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C s jd M s jd k O b F]]] 1 x ln ]] 2 ] , s jd G. 12r P 2 `
t 2t
Ut ( j) 5
t
12 r
t 5t
t
t
t
2
Here b is the discount rate, equal to s1 1 dd 21 where d is the rate of time preference; 1 /r is the elasticity of intertemporal substitution; C is a consumption index; , denotes the amount of labour supplied by the agent. Within countries, agents have symmetric preferences and constraints. Across countries, individual preferences are also symmetric, but all the variables should be considered as starred variables.4 Agents’ wealth is allocated among two assets: real money holdings, M /P and M* /P*, respectively for the home and foreign agents; and an internationally traded bond denominated in composite consumption units, denoted B in what follows. Only agents within a country hold the money issued in that country, while the international bond is in zero net-supply worldwide. Real money balances enter in the utility function and they provide liquidity services. The Cobb–Douglas consumption index for the home agent is defined as Ct ( j) ;sCH,t ( j)dgsCF,t ( j)d 12g 0 , g , 1 where CH ( j) and CF ( j) are domestic consumption of the home and foreign goods, respectively, by individual j. Preferences on the two goods are identical across countries:
* ( j*)dgsC *F,t ( j*)d 12g. C *t ( j*) ;sC H,t Here we define g as the ‘economic’ size of the home country, i.e. the share of the home good in the consumption basket; while 1 2 g is the ‘economic’ size of the foreign country. The consumption-based price indexes that correspond to the above specification of preferences are Pt P t*
1 ; ] sPH,tdgsEt P *F,td 12g gW 1 ; ] sPH,t /EtdgsP *F,td 12g, gW
(1)
where gW ; g g (1 2 g )( 12g ) . In Eq. (1), PH is the price of home good in domestic currency, P F* is the price of foreign good in local currency, and E is the nominal exchange rate (domestic currency per unit of foreign currency). Production of the domestic (foreign) good requires a continuum of differentiated labour inputs that are supplied by domestic (foreign) agents. Technology is 4
We could have allowed for differences in the preferences toward liquidity and leisure across countries, but this would not have changed the analysis that follows.
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described by a linear-homogeneous CES production function, which assumes the following specification f / f 21
1
1E 1E
,t ( j)
Yt 5
f 21 / f
0
2
dj
f / f 21
1
Y t* 5
, *t ( j*)
f 21 / f
0
dj*
2
f . 1,
where Y and Y* denote output per capita, respectively, in the home and foreign countries and the parameter f is the elasticity of input substitution.5 While national firms act competitively, each economic agent is a monopoly supplier of one type of labour input. The labour demand for each type of agent in each country is
S D
Wt ( j) ,t ( j) 5 ]] PH,t
2f
Yt ,
S
W *t ( j) , *t ( j) 5 ]] PF,t
D
2f
Y *t ,
where W( j) is the nominal wage rate. In a symmetric equilibrium nominal wages are equal to product prices, whether wages are sticky or flexible. In the short run, the economies of both countries are characterized by predetermined nominal wages which are assumed to be fixed only for one period (short run).
3. Solving the model In what follows, given generic variables X for the home country and X* for the foreign country, we define world variables as XW 5 (X)g (X*)12g, while relative variables are defined as XR 5 X /X*. The model is solved considering an initial steady-state equilibrium in which the net bond position of each country is zero. Given the assumption of Cobb–Douglas preferences in consumption, the equilibrium holdings of the traded bond in each country are zero at any time. It is then possible to study the impact of permanent unanticipated changes in domestic and foreign money by using a closed-form solution to the model. Given that wages are fixed one period in advance, we can restrict our analysis to three periods only. In the initial period, indexed by 0, the economy is at a steady state. At time 1 the economy is hit by unanticipated permanent shocks to the money supply of each country, while wages are consistent with the initial pre-shocks steady state. In the final period the economy adjusts to a new steady state. In this new steady state 5
We could have allowed for different elasticities of input substitution across countries without affecting the general conclusions of the analysis.
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Table 1 Solution of the model a Determinants of home welfare C¯ 5 a 1 Y¯ 5 a 2 ¯ /P¯ 5 a 3 M ¯ ¯ W /MW )1 / r C 5 C(M 0 ¯ ¯ R /MR )12g (M ¯ W /MW )1 / r Y 5 Y(M 0 0 ¯ ¯ ¯ ¯ M /P 5 (M /P )(MW /MW0 )
Long-run Long-run Long-run Short-run Short-run Short-run
Prices ¯ ¯ *F /P¯ H 5 a 4 EP ¯ P¯ H 5 a 5 M ] 1 1 r 5 b 21 (MW /MW0 )21 ¯ ¯ R /MR ) * EP F /PH 5 (EP¯ *F /P¯ H )(M 0 E 5 E¯ 5 a 6 MR
Long-run terms of trade Long-run home good price Short-run real interest rate Short-run terms of trade Nominal exchange rate
consumption output real balances consumption output real balances
a The index R refers to ratios of home to foreign variables. The index W refers to geometric averages of home and foreign variables with weights g and 1 2 g. The constants are defined below, where the subscript 0 indexes pre-shock levels, and G ; [g /(1 2 g )] 12g (gW )2 / ( 11 r ) while F ; [(f 2 1) /(fk )]. a 1 5 G (F )1 / ( 11 r ) ; a 2 5 G ( 12 r ) / 2 F 1 / 2 (F )( 12 r ) / 2( 11 r ) ; a 3 5 x [(1 1 d ) /d ]G r (FW ) r / ( 11 r ) ; a 4 5 [g /(1 2 g )] 2( 11 r ) / 2 ; a 5 5 (a 1 ) r a 2 (a 3 )21 F 21 ; a 6 5 [g /(1 2 g )] 2 r .
variables are identified with upper bars, while the variables in the transition period are plain variables. The relevant results for the welfare analysis are summarized in Tables 1 and 2. Given that the equilibrium assets holding is zero, money is neutral in the long Table 2 Solution of the model a Determinants of foreign welfare C¯ * 5 a *1 Y¯ * 5 a *2 ¯ * /P¯ * 5 a *3 M ¯ W /MW )1 / r C* 5 C¯ *(M 0 2g ¯ 1 /r ¯ ¯ Y* 5 Y *(MR /MR 0 ) (M W /MW0 ) ¯ * /P* 5 M ¯ * /P¯ *(M ¯ W /MW ) M 0
Long-run Long-run Long-run Short-run Short-run Short-run
Prices ¯ ¯ F* /P¯ H 5 a 4 EP ¯* P¯ *F 5 a 5* M ] 1 1 r 5 b 21 (MW /MW0 )21 ¯ ¯ *F /P¯ H )(M ¯ R /MR ) EP *F /PH 5 (EP 0 ] E 5 E¯ 5 a 6MR
Long-run terms of trade Long-run foreign good price Short-run real interest rate Short-run terms of trade Nominal exchange rate
consumption output real balances consumption output real balances
a The index R refers to ratios of home to foreign variables. The index W refers to geometric averages of home and foreign variables with weights g and 1 2 g. the constants are defined below, where the subscript 0 indexes pre-shock levels, and G * ; [(1 2 g ) /g ] g (gW )2 / ( 11 r ) while F ; [(f 2 1) /(fk )]. a *1 5 (G *)(F )1 / ( 11 r ) ; a *2 5 (G *)( 12 r ) / 2 (F )1 / 2 (F )( 12 r ) / 2( 11 r ) ; a *3 5 x [(1 1 d ) /d ](G *) r (F ) r / ( 11 r ) ; a 4 5 [g /(1 2 g )] 2( 11 r ) / 2 ; a 5* 5 (a 1* ) r a 2* (a 3* )21 (F )21 ; a 6 5 [g /(1 2 g )] 2 r .
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run. In the short run a global money expansion affects positively both consumption and output in each country. A relative money expansion depreciates the exchange rate worsening the terms of trade. The latter effect shifts demand to the good produced in the home country. The elasticity of changes in home output to changes in the terms of trade is given by the parameter (1 2 g ), while (2g ) is the same elasticity for the foreign output. As g increases, the ‘economic’ size of the home country increases. It becomes more a ‘closed’ economy and thus less affected by terms of trade changes. On the opposite, the ‘economic’ size of the foreign country decreases, becoming then a more ‘open’ economy and thus more influenced by terms of trade changes.6 A domestic money expansion has a positive effect on home and foreign consumption, and on domestic output. However the effect on foreign output is ambiguous. There are two contrasting channels: on one side, the increase in aggregate consumption induces an increase in foreign output; on the other side, the worsening of the terms of trade tends to reduce it. The former effect is stronger when r , 1 while the second channel prevails when r . 1. As stressed by Corsetti and Pesenti (2001), this result is related to the complementary or substitutability of home and foreign goods in utility. Two goods are complements in utility if the marginal utility of one good increases as the consumption of the other good increases. With our specification of preferences, two goods are complements when the elasticity of intertemporal substitution is larger than the elasticity of intratemporal substitution. In the opposite case, they are substitutes. In this model the intratemporal elasticity of substitution is unitary while the intertemporal elasticity is 1 /r. It is then the case that a domestic money expansion increases foreign output if the two goods are complements in utility, decreases otherwise.
4. Non-cooperative solution to monetary interdependence The utility of the consumer is a natural welfare criterion for the monetary policymaker of each country. However, following the example of recent literature on monetary policy evaluation, we consider the contribution of the liquidity service of money to the utility of the consumers to be small in size.7 Monetary policy operates through changes in the money supply that, in an unanticipated way, can affect the equilibrium allocation. Given that money is neutral in the long run, we can focus only on the impact of this policy experiment on the short-run utility flow. The welfare criteria W and W*, for the home and foreign country, respectively, are defined as 6
In this model the degrees of openness of the home and foreign countries are inversely related. This is not necessarily the case in a model that includes both traded and non-traded goods. 7 We are considering a case in which x is approaching zero. Rotemberg and Woodford (1998), Obstfeld and Rogoff (1998), and Devereux and Engel (1999) use the same assumption.
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C 12 r k W ; ]] 2 ] Y 2 , 12r 2
(C*)12 r k W* ; ]]] 2 ] (Y*)2 . 12r 2
This model can be solved in a closed form, thus allowing the analysis of large unanticipated shocks. However, there are natural constraints on the amplitude of these shocks. To counteract this, we introduce some definitions. Definition 1. In a competitive allocation the real wages are equated to the marginal cost of labour in terms of utility. Definition 2. In a monopolistic allocation the real wages are above the marginal cost of labour in terms of utility. Thus, we assume that the shocks are bounded by the requirement that real wages should be at least equal to the marginal cost of labour in terms of utility. Formally these conditions can be stated, for the home and foreign country, as ≠C C 2 r ] $ k Y, ≠Y
≠C* (C*)2 r ]] $ k Y*, ≠Y*
which can be rewritten as 1 $ k C r 21 Y 2 ,
1 $ k (C*) r 21 (Y*)2 .
We start analyzing the interaction between the home and foreign country in the Nash equilibrium, where each policymaker maximizes its own country’s welfare with respect to its instrument, taking as given the policy of the other policymaker. Proposition 3. In a Nash equilibrium both countries are operating at a monopolistic allocation. ] The maximization of the home welfare W with respect to M yields the following first-order condition 1 5 k C r 21 Y 2 D,
(2)
where D has been defined as
S
D
12g D ; 1 1 ]] r . 1. g ] While maximization of W* with respect to M* yields 1 5 k (C*) r 21 (Y*)2 D*, where D* has been defined as
(3)
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g D* ; 1 1 ]] r . 1. 12g
S
D
The distance between the Nash allocation and the competitive allocation is given by the ‘open-economy’ distortions D and D*. These distortions are induced by the strategic interaction in an open-economy context. As the ‘economic’ size of a country increases, the closer it will operate to the competitive allocation. In approximating the closed-economy framework, we retrieve the result of Blanchard and Kiyotaki (1987): monetary policy should eliminate all the monopolistic distortions. However, in an open-economy model there is another distorting channel to consider: the terms of trade. A monetary expansion in the home country increases consumption and worsens the terms of trade. These two effects lead to an increase in home output. However, the increase coming from the ‘terms of trade’ channel is specific to the ‘openeconomy’ context, while the ‘aggregate consumption’ effect is also a characteristic of a ‘closed-economy’ model. Were both policymakers to expand the economy until the competitive allocation, as in a closed economy, each policymaker would have an incentive to contract its money supply. In fact, this contraction reduces in the same proportion consumption and the ‘aggregate consumption’ component of output. However, output is further reduced by the improvement in the terms of trade. The reduction in the utility derived from consumption is more than compensated by the reduction in the disutility of producing effort, because the ‘burden’ of production is shifted to the other country through the improved terms of trade. The incentive to a strategic use of the terms of trade is magnified when output reacts more to the ‘terms of trade’ channel than to the ‘aggregate consumption’ channel.8 This is the case when the 8 It happens that starting from the long-run equilibrium it can be optimal not only to maintain the monopolistic allocation but also to reduce the money supply. If we compute the derivative of the home utility with respect to its own money supply at the long run equilibrium, we observe that there is a contraction if and only if
˜ D , 0, 12F where we have defined
f 21 F˜ 5 ]] 5 kF. f In a closed economy D 5 1. It follows that the inequality above is never satisfied, because f . 1. As the ‘open-economy distortions’ increase, the inequality can be satisfied. Note that in the case in which the inequality is satisfied as equality — which might be the case only in an open economy — it is possible to define the existence of a time-consistent equilibrium for any arbitrary initial condition on the money supply. Time-consistent or sustainable equilibria have also been defined in closed economy models superimposing an upper bound on the money supply, as in Ireland (1997). In our context we leave wages as a predetermined, not forward looking variable, and we analyse the optimal stabilization policy of unanticipated monetary policy disturbances, as in Canzoneri and Henderson (1991).
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‘economic’ size of the country is small or the degree of substitutability in utility between the goods is high. As a consequence, each country’s distance from the monopolistic allocation is a decreasing monotone function of the ‘economic’ size of that country and of the degree of substitutability between the goods. Here we characterize the optimal choice of the money supplies in the Nash equilibrium. The relative unanticipated money innovation can be derived by taking the ratio of (2) and (3), using the reduced forms of Tables 1 and 2. We obtain ]N MR 5 MR 0sDRd 21 / 2 . (4) In the short run, both the exchange rate and the terms of trade (see Tables 1 and 2) are proportional to (4): so they are functions of the ‘open-economy’ distortions; ] moreover MR N is a monotone function of the relative ‘economic’ size of the countries, i.e. the larger the home country the more the terms of trade will be worsened and the more the home country will operate near the competitive allocation. Again, the larger country has no reasons to strategically use the terms of trade and the smaller country, while working less, can benefit from the higher consumption induced by the expansion in the larger economy. After substituting (4) into (2) it is possible to obtain the equilibrium level of money in the global economy as ]N 2 r / (11 r ) MW 5 MW0sF˜ d sDWd 2r / ( 11r ). The need for a coordinated action in monetary policy clearly emerges in this context. At a Nash equilibrium, the home policymaker optimizes its welfare taking as given the policy of the other country. Any further expansion in the money innovation decreases its welfare, because of the deterioration of the terms of trade. Were the policy expansion coordinated with the foreign authority, it would be possible to obtain an improvement in welfare relative to the Nash equilibrium. Proposition 4. The Nash equilibrium is not a Pareto efficient allocation. Here we search for an allocation that constitutes a Pareto improvement. We consider an expansion in the money supply of both countries that leaves the exchange rate and the terms of trade unchanged at the Nash equilibrium. In order to evaluate whether this allocation increases the welfare of both countries, we compute the derivative of the indirect utility of the home and foreign countries, ] ] ] ] with respect to M and M*, leaving MR 5MR N . We then evaluate this derivative at the Nash equilibrium. For the home country, we obtain sign
F G ≠W ] ] ≠M
] ]N M R 5M R
D21 5 sign [1 2 k (C N ) r 21 (Y N )2 ] 5 ]] . 0, D
(5)
where we have used (2). A similar condition holds for the foreign country. In the Nash equilibrium both countries restrict their monetary policy too much: they do not internalize the positive externalities given by a coordinated expansion. This is
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in the spirit of the contractionary bias result of the traditional literature on international policy coordination, but in a micro-founded setting (e.g. Canzoneri and Henderson, 1991). The incentive to contract is inherent in the strategic role of the terms of trade that enable the policymaker to shift the burden of production to the other country. It is then natural that the Nash equilibrium can be improved by internalizing the externality given by the terms of trade. Consider an equiproportionate money shock. This has no effect on the terms of trade. The only effect is on consumption and output. However, at the Nash equilibrium both countries operate under monopolistic competition and output is suboptimally low; it follows that this kind of expansion has a first order effect on the global welfare. It is then possible to improve the welfare of the Nash equilibrium even for larger deviations. Proposition 5. Any allocation in which both countries are simultaneously at the monopolistic allocation is not Pareto efficient. In fact (5) is always positive if evaluated at any allocation where a country is at a monopolistic level of production. It is worth stressing that there is room for a possible improvement only when agents in both countries maintain their monopolistic power; in fact, any expansion is bounded by the requirement that agents’ real wages do not fall below their marginal cost in terms of utility. When at least one country reaches the competitive allocation, it might not be possible to find an improvement. The constraint on the real wage will then be binding. The analysis above suggests as a possible efficient allocation the solution in which the policymakers restrain themselves from the possibility of influencing the terms of trade.
5. Cooperative solution in a decentralized setting Here we consider a solution where both countries agree on fixed terms of trade. In this solution both countries act in the same way: there is no leader and the relative money innovation is simultaneously determined by their best responses. Proposition 6. A cooperative solution in which both monetary policymakers internalize the externalities given by the terms of trade coincides with the competitive allocation. We compute the derivative of the indirect utility of the home and foreign countries with respect to their money supplies, when they take as given the relative prices. The first order conditions are:
F G ≠W ] ] ≠M
] MR
5 1 2 k (C) r 21 (Y)2 5 0,
(6)
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F G ≠W* ]] ] ≠M*
] MR
5 1 2 k (C*) r 21 (Y*)2 5 0.
(7)
Both policymakers find it optimal to increase their production until real wages are equated to the marginal costs in terms of utility, i.e. the competitive allocation. By internalizing the externalities given by the terms of trade, they can also eliminate the distortion induced by monopolistic competition. The ratio of (6) and (7) determines the optimal relative money innovation under such a cooperative agreement ]C MR 5 MR 0 . (8) Relative money innovation is fixed at its initial level and, moreover, it is independent of the relative size of the economies and of the interdependence parameter. In fact, the ‘open-economy’ distortions have been internalized. The optimal global money expansion is ]C 2 r / (11 r ) , (9) MW 5 MW0sF˜ d which is also independent of the ‘open-economy’ distortions. Using (8) and (9) we can rewrite the Nash equilibrium in relation to the cooperative allocation as ]N ]C MR 5MR sDRd 21 / 2 , (10) ] N ] C 2r / ( 11r ) MW 5MW D W .
(11)
The terms of trade of the two solutions are equal only if g 5 1 / 2; otherwise, in the Nash equilibrium they are proportional to g. The larger is the home country, the more its terms of trade are depreciated. Instead the global money expansion is always less in the Nash equilibrium because of the contractionary bias. The result — that cooperation increases the global money expansion — is consistent with the traditional literature on international policy coordination: as Rogoff (1985) pointed out, cooperation may remove the disincentive to inflate. However, as in Rogoff (1985), an increased cooperation does not necessarily increase welfare in both countries. In our context, it can be shown relying on a graphical simulation that the comparison between the Nash allocation and the cooperative solution can be related to the size of the country (g ) and to the elasticity of substitution ( r ). For each r, it is possible to determine a region of values of g centered around 1 / 2, (g L , g U ), in which the cooperative solution Pareto dominates the Nash equilibrium. That the cooperative solution is better than the Nash equilibrium for countries not so dissimilar in size can be understood in the following way. Consider a home country small in ‘economic’ size. At a Nash equilibrium, it operates at a monopolistic allocation. The distance, D, from the competitive allocation is large because of the improvement in the terms of trade. On the other
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hand, it benefits from higher consumption because the foreign country operates near the competitive level. The improvement of the terms of trade does not impair its purchasing power: ‘it consumes without working’. It follows that for low values of g, the home consumer prefers the Nash allocation to the cooperative allocation. This reasoning applies similarly to the foreign country. As r increases, the interval (g L , g U ) becomes larger. There is more room for cooperation to improve upon the Nash equilibrium, given that in the latter equilibrium the distance from the competitive allocation is large for both countries. Proposition 7. The cooperative solution is not credible. Starting from the cooperative solution, we analyze if one policymaker has the possibility to increase the welfare given that the other policymaker reacts with the best response. For the home policymaker, this problem can be written as C 12 r 1 ]] max U 5 2 ] k Y 2, ] 1 2 r 2 M s.t. k (C*) r 21 (Y*)2 5 1, where the constraint can be simplified to ] ] M* 5 [k (a *1 ) r 21 (a *2 )2 ] 2 r / (11 r )M gR(12 r ) / (11 r ) .
(12)
The foreign policymaker takes the terms of trade as given. The best response maps ] from MR to the choice of its money supply. After plugging the best response of the foreign policymaker into the indirect ] utility of the home policymaker, we maximize the latter with respect to M and evaluate the derivative at the competitive solution. The resulting expression is sign
F G
≠U F 12 r ] ] 5 sign f 2sC d s1 2 gdg , 0. ≠M
In order to improve the welfare of the competitive solution, the home policymaker must contract money supply, thus improving the terms of trade. Starting from the competitive allocation, this latter effect leads to an increase in real wages. It follows that this deviation is feasible and that the competitive solution is not an equilibrium (a symmetric reasoning can be applied to the foreign country). The derivatives above are also meaningful for the analysis of the ‘fixed exchange rate as a leadership commitment’ a´ la Canzoneri and Gray (1985). In this kind of solution the follower takes as given the exchange rate following the reaction function (12), while the leader implicitly decides the exchange rate. In our context the follower reaches the competitive level while the leader maintains the monopolistic power.
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6. Pareto frontier In this section we focus on a centralized solution where a common monetary policymaker decides on the money stance in both countries. We assume that this common institution conducts monetary policy by maximizing a linear combination of the welfares of the countries. In doing so, we are implicitly tracing the characteristics of the Pareto efficient allocations. The central policymaker maxi] ] mizes with respect to MR and MW the following welfare function WW ; a W 1 (1 2 a )W*, where a [ [0, 1] represents the weight attributed to the home welfare. The maximum is constrained by the usual condition that the equilibrium real wages of both representative agents do not fall below the marginal cost in terms of utility. Proposition 8. The Pareto frontier has the following characterization: (i) if sa / 1 2 ad ,sg / 1 2 gd r the home country is at the competitive allocation, while the foreign country is at a monopolistic allocation; (ii) if sa / 1 2 ad 5sg / 1 2 gd r both countries are at the competitive allocation; (iii) sa / 1 2 ad .sg / 1 2 gd r the home country is at the monopolistic allocation, while the foreign country is at the competitive allocation.9 An interesting implication of this proposition is that it is possible to implement the cooperative solution of the previous section, in which both countries are at a competitive allocation, by using a particular weight in the welfare of the central institution. In similar fashion, the cooperative solution is Pareto efficient. This is the case if a solves the following equation r a g ]] 5 ]] . 12a 12g
S
D S
D
When the goods are neither substitutes nor complements in utility, the cooperative solution can be implemented by assigning to each country a weight equal to its economic size. When the two goods are substitutes in utility ( r . 1), the cooperative solution is implemented by magnifying the size of the bigger country, while if the two goods are complements ( r , 1) the distance in the ‘economic’ size is reduced by the choice of the weight a. An implication of Proposition 8 is that the country with more bargaining power maintains the monopolistic power while the other reaches the competitive level. So far, we have described the set of Pareto efficient allocations. However, it can be the case that in some allocations one country would receive less welfare than what it would get under the Nash equilibrium. This happens when one country has little bargaining power in the centralized solution. It is then natural to restrict the 9
The proof is available in an Appendix under the homepage http: / / homepages.nyu.edu / |pb50 / work.html.
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set of weights a in a way to at least satisfy the two participation constraints given by the Nash equilibrium. Fig. 1 plots the set of values for a, for each value of g and for some r, that constitutes an improvement upon the Nash equilibrium. The lowest dashed line represents the constraint implied by the Nash equilibrium for the home country, while the highest dashed-dotted line is the constraint of the foreign country. Between these two lines, both countries are better off than they were in the Nash equilibrium. For any given pair (g, r ), it is possible to find an interval on a such that for all a belonging to that interval the welfare of each country under the Pareto-efficient allocation is better than the welfare under the Nash equilibrium. As the degree of substitutability in utility increases, the region of improvement increases. In fact, as substitutability increases, the strategic distortion induced by the terms of trade is exacerbated. The distance between the Nash allocation and the competitive allocation is large. There is more room for improving upon the non-cooperative allocation.
7. Optimal delegation In the previous section we demonstrated how the Nash allocation can be improved by a centralized solution in which a weighted average of the welfare of the countries is maximized. This centralized solution coincides with a Pareto efficient allocation. However, for each structure of the economy — in this case for each pair of g and r — there is a continuum of Pareto efficient allocations that represents an improvement upon the Nash equilibrium. In this section we select the Pareto efficient allocation that solves optimally a problem of delegation. The starting point for the analysis of delegation is the definition of the welfare of the society. For this purpose we use a population-weighted average of the welfares of the single countries defined as 1 1 WP ; ] W 1 ] W*, 2 2 where each country’s welfare receives a weight equal to its relative population size. Given its preferences, society appoints a central institution that maximizes an objective function represented by a weighted average of the welfare of the single countries, with weights a and 1 2 a Wa 5 a W 1 (1 2 a )W*. Given the structure of the economy and the welfare of society, the problem of delegation coincides with the optimal choice of the weight a that maximizes the welfare of society, WP , while satisfying the two individual rationality constraints
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Fig. 1. Nash-equilibrium constraints.
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given by the Nash equilibrium. We are implicitly assuming that each country has a reservation value equal to the value that it would have achieved in the noncooperative allocation. The problem of delegation is ‘trivial’, in the sense that the preferences of the central policymaker reflect the preferences of society, when both the participation constraints are not binding and the goods are neither complements nor substitutes in the preferences. In fact, if r 5 1 society would like to appoint a central institution with preferences that mirror its own preferences. In this case a is set equal to 1 / 2. Fig. 2 presents the numerical solution for some values of r and for a grid of values of g. The dashed and dashed-dotted lines represent the constraints given by the Nash allocation, the dotted line the values of a that implement the cooperative solution, while the solid line represents the values of a that solve the problem of delegation. In the area above the dotted line country H is at a monopolistic allocation while country F is at a competitive allocation. The opposite happens in the area below. The problem of delegation has the following characterization. In the case the two goods are complements in utility ( r , 1), the country with smaller ‘economic’ size receives a weight bigger than its population size. Instead, when the two goods are substitutes in utility ( r . 1), the country with larger ‘economic’ size receives a weight bigger than its population size. Moreover, as the degree of substitutability increases, society tends to appoint a central institution that implements the cooperative allocation. With a higher degree of substitutability, there is more room for cooperation. In fact, the distance between the Nash allocation and the competitive allocation is large for both countries. It is then possible to eliminate both the monopolistic distortion and the terms of trade distortion without impairing the participation constraints given by the Nash allocation. Instead, with lower degrees of substitutability, the smaller country needs to receive some rent from participating to the cooperative agreement. These are cases in which, in the Nash equilibrium, the smaller country can ‘consume without working’. We conclude that there is then a rationale for delegating monetary policy to an institution that internalizes the terms of trade distortion and conducts the economy to the competitive allocation. This possibility arises when the incentive to use the terms of trade strategically is strong.
8. Conclusions In this paper we have shown how a model of imperfect competition with nominal rigidities can be useful and powerful in addressing normative analyses. We have focused on monetary shocks with the aim of reviving the literature on international policy coordination in a micro-founded model. We have found several similarities: the ‘contractionary bias’ and the inefficient outcome of the Nash
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Fig. 2. Optimal delegation problem.
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equilibrium; the possibility that cooperation can be counterproductive; the rationale for delegating monetary policy to a central institution. We have offered different explanations: all the conclusions can be read in terms of the interaction between the monopolistic distortions and the terms of trade externalities, with a specific attention given to the different ‘economic’ sizes of the countries and to the degree of substitutability of the goods. As the degree of substitutability increases, the incentive to a strategic use of the terms of trade increases and the optimal delegation implies a common institution that leads the economies to the competitive allocation. There are two directions toward which further studies of international monetary policy coordination should be addressed. Firstly, in the model presented here, monetary policy is ‘passive’. Other sources of fluctuations, such as productivity or demand shocks, should be added in order to study the optimal stabilization policy under different monetary regimes. Secondly, the model presented here is ‘static’. A stochastic dynamic general equilibrium model can be more powerful in addressing the strategic interaction between monetary policymakers. Along these lines, a better characterization of the behavior of the monetary policymaker in its response to the shocks affecting the economy can be analyzed.
Acknowledgements I owe all my gratitude to Giancarlo Corsetti and Paolo Pesenti, in a special way to Paolo who helped me with useful and long conversations. I thank two anonymous referees for useful and detailed comments. I thank Patrizia Baudino, Gianluca Benigno, Ben Bernanke, Gita Gopinath, Kenneth Rogoff and Michael Woodford for useful suggestions. The usual disclaimers apply. Special thanks to Ary, Dino, Gian, Virginia and ‘Brat’. I gratefully acknowledge financial support from the Istituto Bancario San Paolo of Turin and from the Alfred P. Sloan Doctoral Dissertation Fellowship.
References Blanchard, O., Kiyotaki, N., 1987. Monopolistic competition and the effect of aggregate demand. American Economic Review 77, 647–666. Canzoneri, M., Gray, D., 1985. Monetary policy games and consequences of non-cooperative behaviour. International Economic Review 26, 547–562. Canzoneri, M., Henderson, D., 1991. Monetary Policy in Interdependent Economies. MIT Press, Cambridge. Corsetti, G., Pesenti, P., 2001. Welfare and macroeconomic interdependence. Quarterly Journal of Economics, in press. Devereux, M., Engel, C., 1999. Fixed versus floating exchange rates: how price setting affects the optimal choice of exchange-rate regime. NBER, Working paper no. 6867.
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Hamada, K., 1985. The Political Economy of International Monetary Interdependence. MIT Press, Cambridge. Ireland, P.N., 1997. Sustainable monetary policies. Journal of Economic Dynamics and Control 22, 87–108. Lane, P., 2001. The new open economy macroeconomics: a survey. Journal of International Economics 54, 235–266. Obstfeld, M., Rogoff, K., 1995. Exchange rate dynamics redux. Journal of Political Economy 103, 624–660. Obstfeld, M., Rogoff, K., 1996. Foundations of International Macroeconomics. MIT Press, Cambridge. Obstfeld, M., Rogoff, K., 1998. Risk and exchange rates. NBER, Working paper no. 6694. Persson, T., Tabellini, G., 1995. Double-edged incentives institutions and policy coordination. In: Grossman, G.M., Rogoff, K. (Eds.). Handbook of International Economics, Vol. 3. North-Holland, Amsterdam, pp. 1973–2030. Rogoff, K., 1985. Can international monetary policy coordination be counterproductive? Journal of International Economics 18, 199–217. Rotemberg, J., Woodford, M., 1998. An optimization based econometric framework for the evaluation of monetary policy. In: Bernanke, B.S., Rotemberg, J. (Eds.), NBER Macroeconomic Annual, 1997. MIT Press, Cambridge, pp. 297–346. Svensson, L.E.O., van Wijnbergen, S., 1989. Excess capacity, monopolistic competition, and international transmission of monetary disturbances. Economic Journal 99, 785–805.
A simple approach to international monetary policy coordination Pierpaolo Benigno* Department of Economics, New York University, New York, USA and CEPR, London, UK Received 12 May 1999; received in revised form 15 January 2001; accepted 7 April 2001
Abstract This paper analyzes the strategic interaction between the monetary policymakers of two countries, in an intertemporal general equilibrium model with nominal rigidities and imperfect competition. It offers an excursus on non-cooperative towards cooperative solutions. In a non-cooperative equilibrium the monopolistic allocation prevails in both countries, because of the incentive to use strategically the terms of trade. In a cooperative solution where both policymakers internalize the externalities given by the terms of trade, the competitive allocation is reached. However, cooperation can be counterproductive. We then characterize a problem of delegation in which the set of choice is restricted to the Pareto efficient allocations and in which the participation constraints implied by the non-cooperative equilibrium are taken into account. 2002 Elsevier Science B.V. All rights reserved. Keywords: Open-economy macro; Monetary coordination JEL classification: E42; F41; F42
Our analysis of monetary policy [ . . . ] relies on social welfare functions defined directly over macroeconomic outcomes and policy instruments. This lack of explicit microeconomic underpinnings reflects the state of the art in
*Tel.: 11-212-998-8958; fax: 11-212-995-4186. E-mail addresses: [email protected] (P. Benigno), http: / / homepages.nyu.edu / |pb50 (P. Benigno). 0022-1996 / 02 / $ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S0022-1996( 01 )00132-5
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much of the literature, which relies on linear versions of the Mundell– Fleming model.1
1. Introduction The state of the art has changed: Corsetti and Pesenti (2001) have shown how to reconcile rigor in microfoundations and ‘tractability’ through a two-country general equilibrium model with nominal rigidities and monopolistic competition. In their paper, once a closed-form solution is obtained, it is possible to analyze the effect of unanticipated monetary policy shocks, even of large size, and to compute well-defined welfare functions. We utilize their simple framework in order to address the issue of international monetary policy coordination. We study the normative implication of different forms of strategic interaction, thus revisiting the basic findings of the ad hoc literature on international monetary policy coordination, with no ‘lack of explicit microeconomic underpinnings’.2 Given that wages are predetermined, monetary policy has an influence on real activity and welfare. In this model, there are two distortions: one is ‘structural’ and is given by the monopolistic-competition factor market, the other is ‘strategic’ and is given by the terms of trade. In a closed-economy model, monetary policy would get rid of all the distortions induced by monopolistic competition. Real wages would be equated to the marginal cost of labour in terms of utility — the competitive allocation. In an open-economy model, when the policymakers act in a non-cooperative way, the ‘strategic’ distortion given by the terms of trade is in conflict with the objective of eliminating the monopolistic distortion. Were both policymakers to push their economies at the competitive level, each policymaker would have an incentive to contract its money supply. In fact the reduction in utility that comes from the decrease in consumption is more than offset by the reduction in the disutility of producing goods, since the ‘burden’ of production is shifted to the other country through an improvement of the terms of trade. The strategic use of the terms of trade is at the root of the ‘contractionary bias’. This contraction is larger, the higher are the degrees of openness of the economy and the higher is the degree of substitutability in utility between the home and foreign goods. Instead, in the traditional literature (e.g. Canzoneri and Henderson, 1991), the ‘contractionary bias’ emerges because the improvement of terms of trade can lower inflation. 1
Persson and Tabellini (1995, p. 2003). The traditional approach to international policy coordination is built in Hamada (1985) and in the first chapters of Canzoneri and Henderson (1991). The main conclusion of their analyses is that if policymakers cooperate, they could all be better off. On the other hand, Rogoff (1985) points out how monetary policy cooperation can be counterproductive when policymakers lack credibility. Persson and Tabellini (1995) emphasize the role of institutions in enforcing the cooperative solution. 2
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We investigate if other forms of interaction can improve upon the noncooperative equilibrium. Firstly, we consider the outcome of a cooperative agreement, in a decentralized setting, where both policymakers consider as given the exchange rate and thus, in a world of preset prices and wages, the terms of trade. By internalizing the distortions induced by the terms of trade, policymakers can also eliminate the distortion given by monopolistic competition. As in Rogoff (1985), cooperation reduces the incentive to contract money supply and, moreover, can be counterproductive. In fact, the welfare of the non-cooperative solution can be Pareto improved only if the difference in the ‘economic size’ of the countries is not large. We then investigate a form of centralized cooperation in which society delegates the conduction of monetary policy to a central institution. As the degree of substitutability in utility between the goods increases, there is more room for cooperation. In fact, the incentive to a ‘strategic’ use of the terms of trade is stronger and the ‘contractionary bias’ is larger. It follows that the competitive allocation can be implemented also for a large difference in ‘economic size’ across countries. In this case, there is a rationale for delegating the conduction of monetary policy to a central institution that completely internalizes the distortions induced by the terms of trade and achieves the competitive allocation. The work is organized as follows. In the next two sections, the Corsetti and Pesenti model is briefly outlined. Section 4 discusses the non-cooperative Nash equilibrium and its sub-optimality properties. Section 5 discusses the cooperative solution in a decentralized setting. Section 6 presents the characteristic of the Pareto frontier, while Section 7 studies the problem of optimal delegation.
2. Building blocks of the model In this section we briefly outline the key building blocks of the model following Corsetti and Pesenti (2001). This model belongs to a class of micro-founded models of imperfect competition with nominal rigidities, that have recently become a modern paradigm for analyzing international policy issues.3 The assumption of nominal rigidities — in this paper sticky wages — allows monetary policy to affect real variables at least in the short run. A monopolistic-competition factor market rationalizes the existence of nominal rigidities. The model includes two countries, home and foreign, each specialized in the production of a single traded good. In each country there is a continuum of economic agents, with population size normalized to 1. Home agents are indexed by j, foreign agents by j*. The lifetime utility of home agent j is given by 3
A partial list includes Svensson and van Wijnbergen (1989), Obstfeld and Rogoff (1995, 1996, 1998), and Devereux and Engel (1999). Lane (2001) offers a survey of this literature.
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C s jd M s jd k O b F]]] 1 x ln ]] 2 ] , s jd G. 12r P 2 `
t 2t
Ut ( j) 5
t
12 r
t 5t
t
t
t
2
Here b is the discount rate, equal to s1 1 dd 21 where d is the rate of time preference; 1 /r is the elasticity of intertemporal substitution; C is a consumption index; , denotes the amount of labour supplied by the agent. Within countries, agents have symmetric preferences and constraints. Across countries, individual preferences are also symmetric, but all the variables should be considered as starred variables.4 Agents’ wealth is allocated among two assets: real money holdings, M /P and M* /P*, respectively for the home and foreign agents; and an internationally traded bond denominated in composite consumption units, denoted B in what follows. Only agents within a country hold the money issued in that country, while the international bond is in zero net-supply worldwide. Real money balances enter in the utility function and they provide liquidity services. The Cobb–Douglas consumption index for the home agent is defined as Ct ( j) ;sCH,t ( j)dgsCF,t ( j)d 12g 0 , g , 1 where CH ( j) and CF ( j) are domestic consumption of the home and foreign goods, respectively, by individual j. Preferences on the two goods are identical across countries:
* ( j*)dgsC *F,t ( j*)d 12g. C *t ( j*) ;sC H,t Here we define g as the ‘economic’ size of the home country, i.e. the share of the home good in the consumption basket; while 1 2 g is the ‘economic’ size of the foreign country. The consumption-based price indexes that correspond to the above specification of preferences are Pt P t*
1 ; ] sPH,tdgsEt P *F,td 12g gW 1 ; ] sPH,t /EtdgsP *F,td 12g, gW
(1)
where gW ; g g (1 2 g )( 12g ) . In Eq. (1), PH is the price of home good in domestic currency, P F* is the price of foreign good in local currency, and E is the nominal exchange rate (domestic currency per unit of foreign currency). Production of the domestic (foreign) good requires a continuum of differentiated labour inputs that are supplied by domestic (foreign) agents. Technology is 4
We could have allowed for differences in the preferences toward liquidity and leisure across countries, but this would not have changed the analysis that follows.
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described by a linear-homogeneous CES production function, which assumes the following specification f / f 21
1
1E 1E
,t ( j)
Yt 5
f 21 / f
0
2
dj
f / f 21
1
Y t* 5
, *t ( j*)
f 21 / f
0
dj*
2
f . 1,
where Y and Y* denote output per capita, respectively, in the home and foreign countries and the parameter f is the elasticity of input substitution.5 While national firms act competitively, each economic agent is a monopoly supplier of one type of labour input. The labour demand for each type of agent in each country is
S D
Wt ( j) ,t ( j) 5 ]] PH,t
2f
Yt ,
S
W *t ( j) , *t ( j) 5 ]] PF,t
D
2f
Y *t ,
where W( j) is the nominal wage rate. In a symmetric equilibrium nominal wages are equal to product prices, whether wages are sticky or flexible. In the short run, the economies of both countries are characterized by predetermined nominal wages which are assumed to be fixed only for one period (short run).
3. Solving the model In what follows, given generic variables X for the home country and X* for the foreign country, we define world variables as XW 5 (X)g (X*)12g, while relative variables are defined as XR 5 X /X*. The model is solved considering an initial steady-state equilibrium in which the net bond position of each country is zero. Given the assumption of Cobb–Douglas preferences in consumption, the equilibrium holdings of the traded bond in each country are zero at any time. It is then possible to study the impact of permanent unanticipated changes in domestic and foreign money by using a closed-form solution to the model. Given that wages are fixed one period in advance, we can restrict our analysis to three periods only. In the initial period, indexed by 0, the economy is at a steady state. At time 1 the economy is hit by unanticipated permanent shocks to the money supply of each country, while wages are consistent with the initial pre-shocks steady state. In the final period the economy adjusts to a new steady state. In this new steady state 5
We could have allowed for different elasticities of input substitution across countries without affecting the general conclusions of the analysis.
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Table 1 Solution of the model a Determinants of home welfare C¯ 5 a 1 Y¯ 5 a 2 ¯ /P¯ 5 a 3 M ¯ ¯ W /MW )1 / r C 5 C(M 0 ¯ ¯ R /MR )12g (M ¯ W /MW )1 / r Y 5 Y(M 0 0 ¯ ¯ ¯ ¯ M /P 5 (M /P )(MW /MW0 )
Long-run Long-run Long-run Short-run Short-run Short-run
Prices ¯ ¯ *F /P¯ H 5 a 4 EP ¯ P¯ H 5 a 5 M ] 1 1 r 5 b 21 (MW /MW0 )21 ¯ ¯ R /MR ) * EP F /PH 5 (EP¯ *F /P¯ H )(M 0 E 5 E¯ 5 a 6 MR
Long-run terms of trade Long-run home good price Short-run real interest rate Short-run terms of trade Nominal exchange rate
consumption output real balances consumption output real balances
a The index R refers to ratios of home to foreign variables. The index W refers to geometric averages of home and foreign variables with weights g and 1 2 g. The constants are defined below, where the subscript 0 indexes pre-shock levels, and G ; [g /(1 2 g )] 12g (gW )2 / ( 11 r ) while F ; [(f 2 1) /(fk )]. a 1 5 G (F )1 / ( 11 r ) ; a 2 5 G ( 12 r ) / 2 F 1 / 2 (F )( 12 r ) / 2( 11 r ) ; a 3 5 x [(1 1 d ) /d ]G r (FW ) r / ( 11 r ) ; a 4 5 [g /(1 2 g )] 2( 11 r ) / 2 ; a 5 5 (a 1 ) r a 2 (a 3 )21 F 21 ; a 6 5 [g /(1 2 g )] 2 r .
variables are identified with upper bars, while the variables in the transition period are plain variables. The relevant results for the welfare analysis are summarized in Tables 1 and 2. Given that the equilibrium assets holding is zero, money is neutral in the long Table 2 Solution of the model a Determinants of foreign welfare C¯ * 5 a *1 Y¯ * 5 a *2 ¯ * /P¯ * 5 a *3 M ¯ W /MW )1 / r C* 5 C¯ *(M 0 2g ¯ 1 /r ¯ ¯ Y* 5 Y *(MR /MR 0 ) (M W /MW0 ) ¯ * /P* 5 M ¯ * /P¯ *(M ¯ W /MW ) M 0
Long-run Long-run Long-run Short-run Short-run Short-run
Prices ¯ ¯ F* /P¯ H 5 a 4 EP ¯* P¯ *F 5 a 5* M ] 1 1 r 5 b 21 (MW /MW0 )21 ¯ ¯ *F /P¯ H )(M ¯ R /MR ) EP *F /PH 5 (EP 0 ] E 5 E¯ 5 a 6MR
Long-run terms of trade Long-run foreign good price Short-run real interest rate Short-run terms of trade Nominal exchange rate
consumption output real balances consumption output real balances
a The index R refers to ratios of home to foreign variables. The index W refers to geometric averages of home and foreign variables with weights g and 1 2 g. the constants are defined below, where the subscript 0 indexes pre-shock levels, and G * ; [(1 2 g ) /g ] g (gW )2 / ( 11 r ) while F ; [(f 2 1) /(fk )]. a *1 5 (G *)(F )1 / ( 11 r ) ; a *2 5 (G *)( 12 r ) / 2 (F )1 / 2 (F )( 12 r ) / 2( 11 r ) ; a *3 5 x [(1 1 d ) /d ](G *) r (F ) r / ( 11 r ) ; a 4 5 [g /(1 2 g )] 2( 11 r ) / 2 ; a 5* 5 (a 1* ) r a 2* (a 3* )21 (F )21 ; a 6 5 [g /(1 2 g )] 2 r .
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run. In the short run a global money expansion affects positively both consumption and output in each country. A relative money expansion depreciates the exchange rate worsening the terms of trade. The latter effect shifts demand to the good produced in the home country. The elasticity of changes in home output to changes in the terms of trade is given by the parameter (1 2 g ), while (2g ) is the same elasticity for the foreign output. As g increases, the ‘economic’ size of the home country increases. It becomes more a ‘closed’ economy and thus less affected by terms of trade changes. On the opposite, the ‘economic’ size of the foreign country decreases, becoming then a more ‘open’ economy and thus more influenced by terms of trade changes.6 A domestic money expansion has a positive effect on home and foreign consumption, and on domestic output. However the effect on foreign output is ambiguous. There are two contrasting channels: on one side, the increase in aggregate consumption induces an increase in foreign output; on the other side, the worsening of the terms of trade tends to reduce it. The former effect is stronger when r , 1 while the second channel prevails when r . 1. As stressed by Corsetti and Pesenti (2001), this result is related to the complementary or substitutability of home and foreign goods in utility. Two goods are complements in utility if the marginal utility of one good increases as the consumption of the other good increases. With our specification of preferences, two goods are complements when the elasticity of intertemporal substitution is larger than the elasticity of intratemporal substitution. In the opposite case, they are substitutes. In this model the intratemporal elasticity of substitution is unitary while the intertemporal elasticity is 1 /r. It is then the case that a domestic money expansion increases foreign output if the two goods are complements in utility, decreases otherwise.
4. Non-cooperative solution to monetary interdependence The utility of the consumer is a natural welfare criterion for the monetary policymaker of each country. However, following the example of recent literature on monetary policy evaluation, we consider the contribution of the liquidity service of money to the utility of the consumers to be small in size.7 Monetary policy operates through changes in the money supply that, in an unanticipated way, can affect the equilibrium allocation. Given that money is neutral in the long run, we can focus only on the impact of this policy experiment on the short-run utility flow. The welfare criteria W and W*, for the home and foreign country, respectively, are defined as 6
In this model the degrees of openness of the home and foreign countries are inversely related. This is not necessarily the case in a model that includes both traded and non-traded goods. 7 We are considering a case in which x is approaching zero. Rotemberg and Woodford (1998), Obstfeld and Rogoff (1998), and Devereux and Engel (1999) use the same assumption.
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C 12 r k W ; ]] 2 ] Y 2 , 12r 2
(C*)12 r k W* ; ]]] 2 ] (Y*)2 . 12r 2
This model can be solved in a closed form, thus allowing the analysis of large unanticipated shocks. However, there are natural constraints on the amplitude of these shocks. To counteract this, we introduce some definitions. Definition 1. In a competitive allocation the real wages are equated to the marginal cost of labour in terms of utility. Definition 2. In a monopolistic allocation the real wages are above the marginal cost of labour in terms of utility. Thus, we assume that the shocks are bounded by the requirement that real wages should be at least equal to the marginal cost of labour in terms of utility. Formally these conditions can be stated, for the home and foreign country, as ≠C C 2 r ] $ k Y, ≠Y
≠C* (C*)2 r ]] $ k Y*, ≠Y*
which can be rewritten as 1 $ k C r 21 Y 2 ,
1 $ k (C*) r 21 (Y*)2 .
We start analyzing the interaction between the home and foreign country in the Nash equilibrium, where each policymaker maximizes its own country’s welfare with respect to its instrument, taking as given the policy of the other policymaker. Proposition 3. In a Nash equilibrium both countries are operating at a monopolistic allocation. ] The maximization of the home welfare W with respect to M yields the following first-order condition 1 5 k C r 21 Y 2 D,
(2)
where D has been defined as
S
D
12g D ; 1 1 ]] r . 1. g ] While maximization of W* with respect to M* yields 1 5 k (C*) r 21 (Y*)2 D*, where D* has been defined as
(3)
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g D* ; 1 1 ]] r . 1. 12g
S
D
The distance between the Nash allocation and the competitive allocation is given by the ‘open-economy’ distortions D and D*. These distortions are induced by the strategic interaction in an open-economy context. As the ‘economic’ size of a country increases, the closer it will operate to the competitive allocation. In approximating the closed-economy framework, we retrieve the result of Blanchard and Kiyotaki (1987): monetary policy should eliminate all the monopolistic distortions. However, in an open-economy model there is another distorting channel to consider: the terms of trade. A monetary expansion in the home country increases consumption and worsens the terms of trade. These two effects lead to an increase in home output. However, the increase coming from the ‘terms of trade’ channel is specific to the ‘openeconomy’ context, while the ‘aggregate consumption’ effect is also a characteristic of a ‘closed-economy’ model. Were both policymakers to expand the economy until the competitive allocation, as in a closed economy, each policymaker would have an incentive to contract its money supply. In fact, this contraction reduces in the same proportion consumption and the ‘aggregate consumption’ component of output. However, output is further reduced by the improvement in the terms of trade. The reduction in the utility derived from consumption is more than compensated by the reduction in the disutility of producing effort, because the ‘burden’ of production is shifted to the other country through the improved terms of trade. The incentive to a strategic use of the terms of trade is magnified when output reacts more to the ‘terms of trade’ channel than to the ‘aggregate consumption’ channel.8 This is the case when the 8 It happens that starting from the long-run equilibrium it can be optimal not only to maintain the monopolistic allocation but also to reduce the money supply. If we compute the derivative of the home utility with respect to its own money supply at the long run equilibrium, we observe that there is a contraction if and only if
˜ D , 0, 12F where we have defined
f 21 F˜ 5 ]] 5 kF. f In a closed economy D 5 1. It follows that the inequality above is never satisfied, because f . 1. As the ‘open-economy distortions’ increase, the inequality can be satisfied. Note that in the case in which the inequality is satisfied as equality — which might be the case only in an open economy — it is possible to define the existence of a time-consistent equilibrium for any arbitrary initial condition on the money supply. Time-consistent or sustainable equilibria have also been defined in closed economy models superimposing an upper bound on the money supply, as in Ireland (1997). In our context we leave wages as a predetermined, not forward looking variable, and we analyse the optimal stabilization policy of unanticipated monetary policy disturbances, as in Canzoneri and Henderson (1991).
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‘economic’ size of the country is small or the degree of substitutability in utility between the goods is high. As a consequence, each country’s distance from the monopolistic allocation is a decreasing monotone function of the ‘economic’ size of that country and of the degree of substitutability between the goods. Here we characterize the optimal choice of the money supplies in the Nash equilibrium. The relative unanticipated money innovation can be derived by taking the ratio of (2) and (3), using the reduced forms of Tables 1 and 2. We obtain ]N MR 5 MR 0sDRd 21 / 2 . (4) In the short run, both the exchange rate and the terms of trade (see Tables 1 and 2) are proportional to (4): so they are functions of the ‘open-economy’ distortions; ] moreover MR N is a monotone function of the relative ‘economic’ size of the countries, i.e. the larger the home country the more the terms of trade will be worsened and the more the home country will operate near the competitive allocation. Again, the larger country has no reasons to strategically use the terms of trade and the smaller country, while working less, can benefit from the higher consumption induced by the expansion in the larger economy. After substituting (4) into (2) it is possible to obtain the equilibrium level of money in the global economy as ]N 2 r / (11 r ) MW 5 MW0sF˜ d sDWd 2r / ( 11r ). The need for a coordinated action in monetary policy clearly emerges in this context. At a Nash equilibrium, the home policymaker optimizes its welfare taking as given the policy of the other country. Any further expansion in the money innovation decreases its welfare, because of the deterioration of the terms of trade. Were the policy expansion coordinated with the foreign authority, it would be possible to obtain an improvement in welfare relative to the Nash equilibrium. Proposition 4. The Nash equilibrium is not a Pareto efficient allocation. Here we search for an allocation that constitutes a Pareto improvement. We consider an expansion in the money supply of both countries that leaves the exchange rate and the terms of trade unchanged at the Nash equilibrium. In order to evaluate whether this allocation increases the welfare of both countries, we compute the derivative of the indirect utility of the home and foreign countries, ] ] ] ] with respect to M and M*, leaving MR 5MR N . We then evaluate this derivative at the Nash equilibrium. For the home country, we obtain sign
F G ≠W ] ] ≠M
] ]N M R 5M R
D21 5 sign [1 2 k (C N ) r 21 (Y N )2 ] 5 ]] . 0, D
(5)
where we have used (2). A similar condition holds for the foreign country. In the Nash equilibrium both countries restrict their monetary policy too much: they do not internalize the positive externalities given by a coordinated expansion. This is
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in the spirit of the contractionary bias result of the traditional literature on international policy coordination, but in a micro-founded setting (e.g. Canzoneri and Henderson, 1991). The incentive to contract is inherent in the strategic role of the terms of trade that enable the policymaker to shift the burden of production to the other country. It is then natural that the Nash equilibrium can be improved by internalizing the externality given by the terms of trade. Consider an equiproportionate money shock. This has no effect on the terms of trade. The only effect is on consumption and output. However, at the Nash equilibrium both countries operate under monopolistic competition and output is suboptimally low; it follows that this kind of expansion has a first order effect on the global welfare. It is then possible to improve the welfare of the Nash equilibrium even for larger deviations. Proposition 5. Any allocation in which both countries are simultaneously at the monopolistic allocation is not Pareto efficient. In fact (5) is always positive if evaluated at any allocation where a country is at a monopolistic level of production. It is worth stressing that there is room for a possible improvement only when agents in both countries maintain their monopolistic power; in fact, any expansion is bounded by the requirement that agents’ real wages do not fall below their marginal cost in terms of utility. When at least one country reaches the competitive allocation, it might not be possible to find an improvement. The constraint on the real wage will then be binding. The analysis above suggests as a possible efficient allocation the solution in which the policymakers restrain themselves from the possibility of influencing the terms of trade.
5. Cooperative solution in a decentralized setting Here we consider a solution where both countries agree on fixed terms of trade. In this solution both countries act in the same way: there is no leader and the relative money innovation is simultaneously determined by their best responses. Proposition 6. A cooperative solution in which both monetary policymakers internalize the externalities given by the terms of trade coincides with the competitive allocation. We compute the derivative of the indirect utility of the home and foreign countries with respect to their money supplies, when they take as given the relative prices. The first order conditions are:
F G ≠W ] ] ≠M
] MR
5 1 2 k (C) r 21 (Y)2 5 0,
(6)
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F G ≠W* ]] ] ≠M*
] MR
5 1 2 k (C*) r 21 (Y*)2 5 0.
(7)
Both policymakers find it optimal to increase their production until real wages are equated to the marginal costs in terms of utility, i.e. the competitive allocation. By internalizing the externalities given by the terms of trade, they can also eliminate the distortion induced by monopolistic competition. The ratio of (6) and (7) determines the optimal relative money innovation under such a cooperative agreement ]C MR 5 MR 0 . (8) Relative money innovation is fixed at its initial level and, moreover, it is independent of the relative size of the economies and of the interdependence parameter. In fact, the ‘open-economy’ distortions have been internalized. The optimal global money expansion is ]C 2 r / (11 r ) , (9) MW 5 MW0sF˜ d which is also independent of the ‘open-economy’ distortions. Using (8) and (9) we can rewrite the Nash equilibrium in relation to the cooperative allocation as ]N ]C MR 5MR sDRd 21 / 2 , (10) ] N ] C 2r / ( 11r ) MW 5MW D W .
(11)
The terms of trade of the two solutions are equal only if g 5 1 / 2; otherwise, in the Nash equilibrium they are proportional to g. The larger is the home country, the more its terms of trade are depreciated. Instead the global money expansion is always less in the Nash equilibrium because of the contractionary bias. The result — that cooperation increases the global money expansion — is consistent with the traditional literature on international policy coordination: as Rogoff (1985) pointed out, cooperation may remove the disincentive to inflate. However, as in Rogoff (1985), an increased cooperation does not necessarily increase welfare in both countries. In our context, it can be shown relying on a graphical simulation that the comparison between the Nash allocation and the cooperative solution can be related to the size of the country (g ) and to the elasticity of substitution ( r ). For each r, it is possible to determine a region of values of g centered around 1 / 2, (g L , g U ), in which the cooperative solution Pareto dominates the Nash equilibrium. That the cooperative solution is better than the Nash equilibrium for countries not so dissimilar in size can be understood in the following way. Consider a home country small in ‘economic’ size. At a Nash equilibrium, it operates at a monopolistic allocation. The distance, D, from the competitive allocation is large because of the improvement in the terms of trade. On the other
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hand, it benefits from higher consumption because the foreign country operates near the competitive level. The improvement of the terms of trade does not impair its purchasing power: ‘it consumes without working’. It follows that for low values of g, the home consumer prefers the Nash allocation to the cooperative allocation. This reasoning applies similarly to the foreign country. As r increases, the interval (g L , g U ) becomes larger. There is more room for cooperation to improve upon the Nash equilibrium, given that in the latter equilibrium the distance from the competitive allocation is large for both countries. Proposition 7. The cooperative solution is not credible. Starting from the cooperative solution, we analyze if one policymaker has the possibility to increase the welfare given that the other policymaker reacts with the best response. For the home policymaker, this problem can be written as C 12 r 1 ]] max U 5 2 ] k Y 2, ] 1 2 r 2 M s.t. k (C*) r 21 (Y*)2 5 1, where the constraint can be simplified to ] ] M* 5 [k (a *1 ) r 21 (a *2 )2 ] 2 r / (11 r )M gR(12 r ) / (11 r ) .
(12)
The foreign policymaker takes the terms of trade as given. The best response maps ] from MR to the choice of its money supply. After plugging the best response of the foreign policymaker into the indirect ] utility of the home policymaker, we maximize the latter with respect to M and evaluate the derivative at the competitive solution. The resulting expression is sign
F G
≠U F 12 r ] ] 5 sign f 2sC d s1 2 gdg , 0. ≠M
In order to improve the welfare of the competitive solution, the home policymaker must contract money supply, thus improving the terms of trade. Starting from the competitive allocation, this latter effect leads to an increase in real wages. It follows that this deviation is feasible and that the competitive solution is not an equilibrium (a symmetric reasoning can be applied to the foreign country). The derivatives above are also meaningful for the analysis of the ‘fixed exchange rate as a leadership commitment’ a´ la Canzoneri and Gray (1985). In this kind of solution the follower takes as given the exchange rate following the reaction function (12), while the leader implicitly decides the exchange rate. In our context the follower reaches the competitive level while the leader maintains the monopolistic power.
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6. Pareto frontier In this section we focus on a centralized solution where a common monetary policymaker decides on the money stance in both countries. We assume that this common institution conducts monetary policy by maximizing a linear combination of the welfares of the countries. In doing so, we are implicitly tracing the characteristics of the Pareto efficient allocations. The central policymaker maxi] ] mizes with respect to MR and MW the following welfare function WW ; a W 1 (1 2 a )W*, where a [ [0, 1] represents the weight attributed to the home welfare. The maximum is constrained by the usual condition that the equilibrium real wages of both representative agents do not fall below the marginal cost in terms of utility. Proposition 8. The Pareto frontier has the following characterization: (i) if sa / 1 2 ad ,sg / 1 2 gd r the home country is at the competitive allocation, while the foreign country is at a monopolistic allocation; (ii) if sa / 1 2 ad 5sg / 1 2 gd r both countries are at the competitive allocation; (iii) sa / 1 2 ad .sg / 1 2 gd r the home country is at the monopolistic allocation, while the foreign country is at the competitive allocation.9 An interesting implication of this proposition is that it is possible to implement the cooperative solution of the previous section, in which both countries are at a competitive allocation, by using a particular weight in the welfare of the central institution. In similar fashion, the cooperative solution is Pareto efficient. This is the case if a solves the following equation r a g ]] 5 ]] . 12a 12g
S
D S
D
When the goods are neither substitutes nor complements in utility, the cooperative solution can be implemented by assigning to each country a weight equal to its economic size. When the two goods are substitutes in utility ( r . 1), the cooperative solution is implemented by magnifying the size of the bigger country, while if the two goods are complements ( r , 1) the distance in the ‘economic’ size is reduced by the choice of the weight a. An implication of Proposition 8 is that the country with more bargaining power maintains the monopolistic power while the other reaches the competitive level. So far, we have described the set of Pareto efficient allocations. However, it can be the case that in some allocations one country would receive less welfare than what it would get under the Nash equilibrium. This happens when one country has little bargaining power in the centralized solution. It is then natural to restrict the 9
The proof is available in an Appendix under the homepage http: / / homepages.nyu.edu / |pb50 / work.html.
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set of weights a in a way to at least satisfy the two participation constraints given by the Nash equilibrium. Fig. 1 plots the set of values for a, for each value of g and for some r, that constitutes an improvement upon the Nash equilibrium. The lowest dashed line represents the constraint implied by the Nash equilibrium for the home country, while the highest dashed-dotted line is the constraint of the foreign country. Between these two lines, both countries are better off than they were in the Nash equilibrium. For any given pair (g, r ), it is possible to find an interval on a such that for all a belonging to that interval the welfare of each country under the Pareto-efficient allocation is better than the welfare under the Nash equilibrium. As the degree of substitutability in utility increases, the region of improvement increases. In fact, as substitutability increases, the strategic distortion induced by the terms of trade is exacerbated. The distance between the Nash allocation and the competitive allocation is large. There is more room for improving upon the non-cooperative allocation.
7. Optimal delegation In the previous section we demonstrated how the Nash allocation can be improved by a centralized solution in which a weighted average of the welfare of the countries is maximized. This centralized solution coincides with a Pareto efficient allocation. However, for each structure of the economy — in this case for each pair of g and r — there is a continuum of Pareto efficient allocations that represents an improvement upon the Nash equilibrium. In this section we select the Pareto efficient allocation that solves optimally a problem of delegation. The starting point for the analysis of delegation is the definition of the welfare of the society. For this purpose we use a population-weighted average of the welfares of the single countries defined as 1 1 WP ; ] W 1 ] W*, 2 2 where each country’s welfare receives a weight equal to its relative population size. Given its preferences, society appoints a central institution that maximizes an objective function represented by a weighted average of the welfare of the single countries, with weights a and 1 2 a Wa 5 a W 1 (1 2 a )W*. Given the structure of the economy and the welfare of society, the problem of delegation coincides with the optimal choice of the weight a that maximizes the welfare of society, WP , while satisfying the two individual rationality constraints
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Fig. 1. Nash-equilibrium constraints.
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given by the Nash equilibrium. We are implicitly assuming that each country has a reservation value equal to the value that it would have achieved in the noncooperative allocation. The problem of delegation is ‘trivial’, in the sense that the preferences of the central policymaker reflect the preferences of society, when both the participation constraints are not binding and the goods are neither complements nor substitutes in the preferences. In fact, if r 5 1 society would like to appoint a central institution with preferences that mirror its own preferences. In this case a is set equal to 1 / 2. Fig. 2 presents the numerical solution for some values of r and for a grid of values of g. The dashed and dashed-dotted lines represent the constraints given by the Nash allocation, the dotted line the values of a that implement the cooperative solution, while the solid line represents the values of a that solve the problem of delegation. In the area above the dotted line country H is at a monopolistic allocation while country F is at a competitive allocation. The opposite happens in the area below. The problem of delegation has the following characterization. In the case the two goods are complements in utility ( r , 1), the country with smaller ‘economic’ size receives a weight bigger than its population size. Instead, when the two goods are substitutes in utility ( r . 1), the country with larger ‘economic’ size receives a weight bigger than its population size. Moreover, as the degree of substitutability increases, society tends to appoint a central institution that implements the cooperative allocation. With a higher degree of substitutability, there is more room for cooperation. In fact, the distance between the Nash allocation and the competitive allocation is large for both countries. It is then possible to eliminate both the monopolistic distortion and the terms of trade distortion without impairing the participation constraints given by the Nash allocation. Instead, with lower degrees of substitutability, the smaller country needs to receive some rent from participating to the cooperative agreement. These are cases in which, in the Nash equilibrium, the smaller country can ‘consume without working’. We conclude that there is then a rationale for delegating monetary policy to an institution that internalizes the terms of trade distortion and conducts the economy to the competitive allocation. This possibility arises when the incentive to use the terms of trade strategically is strong.
8. Conclusions In this paper we have shown how a model of imperfect competition with nominal rigidities can be useful and powerful in addressing normative analyses. We have focused on monetary shocks with the aim of reviving the literature on international policy coordination in a micro-founded model. We have found several similarities: the ‘contractionary bias’ and the inefficient outcome of the Nash
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Fig. 2. Optimal delegation problem.
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equilibrium; the possibility that cooperation can be counterproductive; the rationale for delegating monetary policy to a central institution. We have offered different explanations: all the conclusions can be read in terms of the interaction between the monopolistic distortions and the terms of trade externalities, with a specific attention given to the different ‘economic’ sizes of the countries and to the degree of substitutability of the goods. As the degree of substitutability increases, the incentive to a strategic use of the terms of trade increases and the optimal delegation implies a common institution that leads the economies to the competitive allocation. There are two directions toward which further studies of international monetary policy coordination should be addressed. Firstly, in the model presented here, monetary policy is ‘passive’. Other sources of fluctuations, such as productivity or demand shocks, should be added in order to study the optimal stabilization policy under different monetary regimes. Secondly, the model presented here is ‘static’. A stochastic dynamic general equilibrium model can be more powerful in addressing the strategic interaction between monetary policymakers. Along these lines, a better characterization of the behavior of the monetary policymaker in its response to the shocks affecting the economy can be analyzed.
Acknowledgements I owe all my gratitude to Giancarlo Corsetti and Paolo Pesenti, in a special way to Paolo who helped me with useful and long conversations. I thank two anonymous referees for useful and detailed comments. I thank Patrizia Baudino, Gianluca Benigno, Ben Bernanke, Gita Gopinath, Kenneth Rogoff and Michael Woodford for useful suggestions. The usual disclaimers apply. Special thanks to Ary, Dino, Gian, Virginia and ‘Brat’. I gratefully acknowledge financial support from the Istituto Bancario San Paolo of Turin and from the Alfred P. Sloan Doctoral Dissertation Fellowship.
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