Author's personal copy Journal of Macroeconomics 34 (2012) 1095–1110

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Is openness inflationary? Policy commitment and imperfect competition Richard W. Evans ⇑ Department of Economics, Brigham Young University, 167 FOB, Provo, UT 84602, United States

a r t i c l e

i n f o

Article history: Received 5 January 2011 Accepted 24 May 2012 Available online 10 September 2012 JEL classification: E52 E61 F41 F42

a b s t r a c t This paper proposes a channel through which increased openness to international trade can increase a country’s long-run incentive to create inflation. The theoretical justification for this channel is the well known ‘‘beggar thy neighbor’’ incentive, and its dominance relies on a monetary authority’s ability to commit to policy as well as the asymmetric effects of the underlying frictions in the model across domestic and foreign households. Consistent with previous work, the model predicts that the inflationary bias of openness is dampened by the degree of imperfect competition within a country. Ó 2012 Elsevier Inc. All rights reserved.

Keywords: Optimal monetary policy Imperfect competition International monetary policy Openness

1. Introduction Most of the empirical literature measuring the relationship between openness and inflation has found a negative relationship between the two or no relationship at all. These empirical studies use as their foundation a theory based on time-consistent discretionary monetary policy. Consistent with those theoretical assumptions, many of the empirical studies sample a period before the late 1980s in which discretionary monetary policy and lack of adherence to monetary rules was more common than today. Other empirical studies use a broad sample of countries which includes less developed countries that are more characterized by discretion or a lack of commitment. This study proposes a long-run analysis of a theoretical channel through which increased openness to international trade can have the opposite effect and can increase a country’s incentive to create inflation. The theoretical justification for this channel is the well known ‘‘beggar thy neighbor’’ incentive, and its dominance relies on a monetary authority’s long-run ability to commit to policy, the degree of imperfect competition within the country, and the asymmetric effects of the underlying frictions in the model across domestic and foreign households.1 The stylized theoretical environment is a two-country perfect foresight overlapping generations model in which monetary authorities precommit to their respective money growth rates in order to maximize the lifetime utility of the representative household. Consumers have preferences for both domestic and foreign goods, producers have some degree of market

⇑ Tel.: +1 801 422 8303; fax: +1 801 422 0194. E-mail address: [email protected] This ‘‘beggar thy neighbor’’ spillover from monetary policy has been studied in the trade literature in terms of fiscal policy as well. See Corsetti and Pesenti (2001) for a monetary example, and see Eaton and Grossman (1986) and Canzoneri (1989) for a fiscal policy example. 1

0164-0704/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jmacro.2012.05.002

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power, and monetary authorities have a degree of influence over the terms of trade because transactions must occur in the currency of the producer. Three recent papers provide the foundation for the theoretical approach taken here. In order to answer the question of whether monetary cooperation is optimal among countries, Cooley et al. (2003) and Cooper and Kempf (2003) propose similar models in which the optimal money growth rate in the presence of inelastic demand for foreign goods is inflationary. However, both papers characterize firms as being perfectly competitive and neither paper characterizes the degree of openness of each country. Arseneau (2007) adds imperfectly competitive firms to this type of framework and shows that the inflationary bias of non-cooperative monetary policy in an open economy is dampened by the degree of imperfect competition. I use the two-country overlapping generations with commitment approach of Cooper and Kempf (2003) and include imperfect competition as in Arseneau (2007). For an intuitive reason to be explained more in Section 2, I then parameterize the degree of openness of a country in the household preferences of that country. The positive effect of openness on inflation described in this paper runs counter to much of the previous work addressing this question. The theoretical paper that is the foundation for most research on openness and inflation is Rogoff (1985). His approach is to extend the Barro and Gordon (1983) time-consistent policy framework to a two-country Mundell–Fleming model. As in Barro and Gordon, a labor market friction causes the optimal time-consistent policy of the monetary authority to increase inflation in order to raise the level of employment. However, in Rogoff’s international model, the increased inflation has an extra cost in that optimal employment is a function of the real exchange rate and the real exchange rate depreciates with higher inflation. Thus the optimal time-consistent inflation rate chosen by a monetary authority is lower as the deteriorating effect on the exchange rate increases. More openness leads to a lower equilibrium inflation rate in this timeconsistent environment. Two main differences explain why the theory of this paper predicts that increased openness will have an inflationary effect on optimal monetary policy while Rogoff (1985) predicts a deflationary effect: asymmetric effects of the underlying frictions and the focus on optimal monetary policy with commitment. The first difference is that underlying frictions of Rogoff’s model have symmetric costs and benefits to both domestic and foreign consumers, while the underlying frictions of this paper have asymmetric effects. The underlying friction in Rogoff’s Mundell–Fleming framework is a labor market friction in which the privately supplied labor is exogenously less than the socially optimal level, based on a wage contracting model in which wages are negotiated at the end of the previous period, fixed for one period, partially indexed to consumer prices (which include foreign goods), and labor is supplied at whatever level is demanded at the contracted wage. As in Barro and Gordon (1983), Rogoff’s monetary authority has the incentive to create surprise inflation to temporarily increase labor supply closer to the socially efficient level. Because this increases output (without any disutility of work costs) and output is a tradable good, the benefits of this policy are symmetric to both domestic and foreign consumers. Also, because of the demand specification for both domestic and foreign goods, the resulting inflation tax from the depreciation of the real exchange rate is symmetric across consumers.2 This paper follows the convention of the New Open Economy Macroeconomic (NOEM) literature and other recent papers in which the benefit of domestic monetary inflation accrues only to the domestic country while the costs of the inflation are shared equally across domestic and foreign consumers—the foundation of the beggar-thy-neighbor effect. In this paper’s model, the existence of monopolistic competition in both countries lowers output below its socially efficient level. Instead of a pricing friction, this paper includes a money demand distortion in the form of a particular form of cash-in-advance constraint.3 Inflationary monetary intervention causes the return on savings to rise above the return on cash balances, thereby causing households to substitute away from labor into leisure. Because labor and leisure are non-tradable goods, this leisure subsidy is felt only by domestic households. However, increased price of domestic goods consumption is shared equally across domestic and foreign households. In this framework, the more open is an economy, the larger portion of the inflation costs can be shared abroad given a particular non-tradable leisure subsidy of inflation. Thus openness can be inflationary. The second main difference between Rogoff (1985) and this paper that leads to a more inflationary bias of monetary policy is this paper’s focus on long-run monetary policy to which a central bank can commit rather than discretionary policy. Arseneau (2012) shows in a New Keynesian Open Economy Model that discretionary monetary authorities have the incentive to create deflationary surprises if the private sector has expectations that are sufficiently low. That is, discretion can lead to a deflationary bias in an open economy setting.4 Terra (1998) argues empirically that less developed countries suffer more from the time inconsistency problem in monetary policy than do developed countries, and she shows empirically that the negative relationship between openness and inflation is found mostly among those less developed and ‘‘severely indebted’’ countries. Cooley et al. (2003) show theoretically that time-consistent discretionary policy is more inflationary than commitment

2

Rogoff (1985, p. 213) assumes that consumers hold only domestic currency but hold both foreign and domestic bonds which are perfect substitutes and are characterized by uncovered interest rate parity. This assumption abstracts from macroeconomic effects of sterilized intervention. 3 See Appendix B for a more detailed discussion of the effects of the cash-in-advance constraint. 4 Arseneau (2012) documents that ‘‘surprisingly little work has been done in the direction of providing a complete characterization of [equilibria] under discretion in an open economy.’’ His paper uses the same model as Arseneau (2007). Arseneau (2012) shows that a multiplicity of equilibria can arise in this open economy environment—both inflationary and deflationary—but that a deflationary global Friedman rule equilibrium results if private inflation expectations are below a threshold. His two papers together provide a comparison of the effects of discretion versus commitment in a NOEM framework.

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policy in a two-country environment. This paper takes the approach of studying the openness and inflation relationship among developed countries in which policy commitment is more appropriate.5 This paper’s theoretical prediction that more openness can lead to more inflation is contrary to the empirical findings of most previous empirical studies. Romer (1993) found a strong negative relationship between openness and inflation for the period 1973–1987, even when controlling for the development level of a country, central bank independence, and political stability. The two main follow-up studies are Lane (1997) and Terra (1998), both of whom also find a negative relationship between openness and inflation but find that the relationship can be lessened (not changed) by controlling for development and indebtedness. However, Romer (1993), Lane (1997), and Terra (1998) all look at the same time period.6 Other recent empirical papers, such as Daniels et al. (2005) and Badinger (2009), test the effect of openness on inflation as well as openness on the sacrifice ratio. They find that the effect of openness on inflation is robustly negative. However, they use a broad cross section of countries. When Badinger (2009) limits his sample to OECD countries, the relationship between openness and inflation goes away. The theoretical foundation for these papers is a Rogoff-style model in which a Phillips curve relationship is assumed (see, Daniels and VanHoose, 2006) and multi-sector imperfect competition is present. However, the long-run focus of the theory in this paper will ignore the short-run openness and sacrifice ratio tradeoff and will focus on the long-run relationship between openness and inflation. Daniels et al. (2006) and Cavallari (2001) present theoretical models in which the labor market of an open economy is characterized by union-style wage setting behavior and in which more openness leads to more inflation in countries with intermediate levels of unionization. However, their empirical tests using cross-country panel data find either a negative effect of openness on inflation or no effect. The hypothesis of this paper is that a long-run empirical study of more recent macroeconomic outcomes focusing on developed countries is likely to support the predictions of the theory based on monetary commitment outlined in this paper because world monetary authorities since the late 1980s have been more characterized by monetary commitment and adherence to rules.7 Section 2 presents the theoretical model, and Section 3 concludes. 2. Model The goal of the theoretical model is to provide the simplest possible general equilibrium framework that produces the proposed ‘‘beggar thy neighbor’’ channel through which openness affects inflation. Following Cooper and Kempf (2003), I use a two-country perfect foresight overlapping generations (OLG) general equilibrium framework with an independent monetary authority in each country whose objective is to maximize the welfare of its own citizens. In addition, this model includes imperfectly competitive producers in each country similar to Arseneau (2007). The OLG model described here is much simpler than an analogous infinite horizon model, but the many of the results are the same. The innovation of this model is its parameterization of the degree of openness and its predictions for the data. Most of the exposition in this section focuses on the problem of Home agents and the Home monetary authority, but the Foreign problem is symmetric. However, I allow Home and Foreign countries to differ in their respective levels of openness to international trade. Within a country, I assume the equilibrium is symmetric, so any subscripting of individuals is dropped. Home variables are denoted by either no superscript or an ‘‘h’’ superscript, and Foreign variables are denoted by either a ‘‘’’ superscript or a ‘‘f’’ superscript or both. This stylized economy is made up of two countries, each of which has a monetary authority, producers, and consumers. The overlapping generations of agents live for two periods. In the first period of their lives, they produce differentiated goods in a monopolistically competitive environment and sell the goods to both Home and Foreign consumers in exchange for the producer’s own-country currency. The producers then choose how much of their own-country currency to hold and how much to exchange for foreign currency given that they will use a portfolio of each respective currency to consume Home and Foreign goods in the second period of their lives. The role of each country’s monetary authority is to maximize the lifetime welfare of the representative agent in its own country by following a policy to which it has committed of giving a non-proportional transfer of own-country currency to the consumers of its own country in each period. Money is held in this economy because it is the only store of value and because of the two cash-in-advance constraints. Changes in the money supply are not neutral due to the transfers being nonproportional and only given to the monetary authority’s own citizens.8 The two cash-in-advance constraints and consumer preferences generate demand for both currencies by a given consumer. 2.1. Money The objective of the monetary authority in each country is to choose a fixed gross money growth rate policy xt ¼ x or xt ¼ x at the beginning of time in order to maximize the welfare of its own citizens. I assume here that the monetary authority commits to its money growth rate at the beginning of time and cannot deviate once it has chosen its money growth path. 5 Commitment also circumvents the issue of multiple equilibria in the time-consistent policy framework. See Barro and Gordon (1983), Ireland (1997), King and Wolman (2004), and Arseneau (2012). 6 For a more complete review of the openness and inflation literature, see Wynne and Kersting (2007). 7 See discussion in Section 2.1 for evidence of post 1980s developed countries’ monetary policy being more characterized by commitment. 8 See Azariadis (1981) for a proof that non-proportional monetary transfers are not neutral, even in a perfect foresight economy.

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15 10 5 0

annual inflation rate (%)

20

1098

1973

1980

1990

2000

2006

year 25th-75th percentile

Median

Source: International Monetary Fund International Finacial Statistics

Fig. 1. OECD median annual inflation rates: GDP deflator, 1973–2006.

This commitment assumption is a strong one. As was discussed in the introduction, most of the theoretical and empirical literature has focused on time periods and samples of countries that are more characterized by discretionary monetary policy. But Bernanke and Mishkin (1997) have shown that more recent monetary policy among developed countries has followed more closely the ideology of monetary policy rules and commitment.9 Fig. 1 shows that both the median inflation rate across OECD countries and its dispersion have decreased since the 1970s. Romer (1993) and Lane (1997) note that the negative relationship between openness and inflation diminishes among developed countries, and Terra (1998) shows that the time-inconsistency problem plays more of a factor with less developed countries than with developed countries. Let M t and M t be the aggregate nominal supplies of Home currency and Foreign currency, respectively, in period t. I normalize the initial supply of Home and Foreign currency to 1 and divide it equally among the period-1 consumers at the beginning of the period. 



M0 ¼ M0 ¼ 1 and mh0 ¼ mf0 ¼ mh0 ¼ mf0 ¼

1 2



where mh0 and mf0 are the individual holdings of Home currency by Home consumers and Foreign currency by Foreign consumers, respectively, at the beginning of period 1. Each country’s monetary authority makes non-proportional transfers of ðx  1ÞM t1 to each Home consumer and ðx  1ÞM t1 to each Foreign consumer in period t. So aggregate supply of currency in each country obeys the following laws of motion:

Mtþ1 ¼ xM t

and M tþ1 ¼ x Mt

ð2:1Þ

This implies that the following relationships for stþ1 and stþ1 represent the non-proportional transfer to each Home consumer and to each Foreign consumer by their respective monetary authorities.

stþ1 ¼ ðx  1ÞMt and stþ1 ¼ ðx  1ÞMt

ð2:2Þ

At the end of the first period of their lives, producers make a portfolio decision of how much of each type of currency to hold. They exchange some of their domestic currency balances from sales revenues for foreign currency balances at the nominal exchange rate et . Let mht and mft represent each Home producers’ portfolio choice between Home and Foreign currency, respectively, in period t. Because the monetary authority of each country only transfers currency to its own consumers, the laws of motion for individual currency balances in each country are the following:

mhtþ1 ¼ mht þ stþ1 



mftþ1 ¼ mft þ stþ1

mftþ1 ¼ mft 

mhtþ1 ¼ mht

ð2:3Þ







Because the equilibrium currency holdings within each country are symmetric, mht ; mft ; mft ; mht represent the aggregate   amounts of each currency (Mht ; M ft ; M ht ; M ft ) held in each country in each period. 9 Clarida et al. (2000) show that monetary policy in the United States became more sensitive to changes in inflation since 1987 and that the fed funds rate since 1987 has followed more closely a forward-looking Taylor rule. Crowe and Meade (2008) show that central bank independence and transparency have increased greatly since the 1980s. Levin and Piger (2002) find strong evidence for a structural break in inflation time series for a broad sample of countries in the late 1980s and early 1990s. See also Mishkin (2007) for more evidence that monetary policy across the globe has moved further toward commitment and rules since the mid 1980s.

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2.2. Individuals A unit measure of agents are born in each period in both the Home country (indexed by z) and the Foreign country (indexed by z ) and live for two periods. In the first period of their lives, individuals can either enjoy leisure lt or provide labor nt ðzÞ subject to their endowment of one unit of time.

lt þ nt ðzÞ ¼ 1 8t; z Each individual also has access to a linear production technology through which he can convert labor hours into a differentiated good yt ðzÞ indexed by the individual z for each Home producer and z for each Foreign producer.

yt ðzÞ ¼ nt ðzÞ 8t; z Supplying labor to the production process costs producers g ðnt ðzÞÞ in terms of utility, where g 0 > 0 and g 00 P 0. Producers in the Home country then sell their differentiated good yt ðzÞ to both Home and Foreign consumers for Home currency money balances at their profit maximizing prices pt ðzÞnt ðzÞ. At the end of the period, producers take their revenue in terms of Home currency and exchange some of it for Foreign currency. In the second period of life, producers become consumers. Their balances of home currency are augmented by the lump-sum transfer from the central bank as shown in (2.3). They then spend their currency balances on Home and Foreign goods at the prices prevailing in the second period of life. Lifetime utility U is additively separable in individual aggregate consumption C tþ1 and labor nt ðzÞ and is given by the following function.

UðC tþ1 ; nt Þ ¼ uðC tþ1 Þ  g ðnt ðzÞÞ ðC tþ1 Þ1r  1 1r n and g ðnt ðzÞÞ ¼ vðnt ðzÞÞ for where uðC tþ1 Þ ¼

for

ð2:4Þ

rP1

v > 0 and n P 1

I assume the coefficient of relative risk aversion r is greater than or equal to 1. Recent research has estimated the value of r to be somewhere between 2 and 10, so this assumption is not very restrictive.10 Also note that the parameter n in the cost of labor function in (2.4) is inversely related to the Frisch elasticity of labor supply. Individual aggregate consumption C tþ1 is a CES aggregator of individual aggregate consumption of Home goods C htþ1 and individual aggregate consumption of Foreign goods C ftþ1 .

C tþ1

 1hh  hh  C htþ1 C ftþ1

  1hf   hf C tþ1  C ftþ1 C htþ1

  1 for hh 2 0; 2   1 for hf 2 0; 2

ð2:5Þ

In this Cobb–Douglas form, the elasticity of substitution between individual aggregate consumption of Home goods C htþ1 and individual aggregate consumption of Foreign goods C ftþ1 equals one. The parameters hh and hf represent the exogenous degree of openness to international trade for the Home country and Foreign country, respectively. That is, higher values of hh represent increased preferences of Home consumers for Foreign goods. One stylistic advantage of the simple theoretical notion of openness characterized by hh and hf and the unit elastic functional form in (2.5) is that the model implies a constant import share of GDP in each country equal to hh and hf , respectively, as shown in (2.17). Import share of GDP is the most common proxy for openness in the empirical trade literature. This notion of openness is appropriate for this study’s long-run approach of assessing the effects of openness on inflation. But this would not be appropriate for explaining more high frequency fluctuations. I follow an international variation of the Dixit and Stiglitz (1977) model of monopolistic competition in characterizing the individual aggregate consumption of Home goods C htþ1 and the individual aggregate consumption of Foreign goods C ftþ1 as CES aggregators of the individual differentiated goods from each country,

C htþ1



Z 0

1

ctþ1 ðzÞ

e1 e

e e1 dz

and

C ftþ1



Z

1

ctþ1

e1  ðz Þ e dz

e e1

8t

ð2:6Þ

0

where e P 1 represents the elasticity of substitution among all the differentiated goods of a given country.11 The individual aggregate consumption levels for Foreign consumers are defined in the same way. The Dixit–Stiglitz differentiated goods aggregation assumptions in (2.5) and (2.6) deliver the following expressions for consumer demand and aggregate prices.12 10 For estimates of the coefficient of relative risk aversion r that lie between 2 and 10, see Mankiw and Zeldes (1991), Blake (1996), Campbell (1996), Kocherlakota (1996), and Brav et al. (2002). However, earlier estimates of r were lower. See Mehra and Prescott (1985). 11 Note that e P 1 has the intuitive implication that the elasticity of substitution among goods produced within a given country is at least as large as the elasticity of substitution between Home goods and Foreign goods. 12 A derivation for these demand and price equations is in the Technical Appendix and is available upon request.

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ptþ1 ðzÞ

ctþ1 ðzÞ ¼

C htþ1

Phtþ1

!e C htþ1

P htþ1

Ph ¼ ð1  hh Þ tþ1 Ptþ1

¼

Z

1

ptþ1 ðzÞ

and ctþ1 ðz Þ ¼

C tþ1

1e

11 e dz

and

C ftþ1

and

Pftþ1

1 ð1  hh Þ1hh hhh h

et Pftþ1 Ptþ1

¼ hh

¼

 1hh  hh P htþ1 et Pftþ1

!e C ftþ1

Pftþ1

!1

0

Ptþ1 ¼

ptþ1 ðz Þ



Z

1

8 t; z; z

ð2:7Þ

8t

ð2:8Þ

!1 C tþ1

 1e

ptþ1 ðz Þ



11 e

dz

8t

ð2:9Þ

0

8t

ð2:10Þ

where ptþ1 ðzÞ; P htþ1 , and P tþ1 are prices of individual consumption, aggregate country-specific consumption, and aggregate total consumption, respectively. Individuals seek to maximize lifetime utility derived from disutility of work in the first period of life in order to sell a differentiated production good for own-country currency balances that are carried over to the second period of life in which the individual can spend those balances on consumption of both Home and Foreign goods. Because the monopolistically competitive producers can set the quantity demanded by choosing price in order to clear their goods, the producer’s problem is characterized by choosing how much to charge for her differentiated good pt ðzÞ and then the portfolio decision of the amount of sales to keep in the form of Home currency mht and how much to exchange for Foreign currency mft .13

max

f

ct ðzÞ;ct ðz Þ;mht ;mt ;pt ðzÞ

uðC tþ1 Þ  g ðnt ðzÞÞ

ð2:4Þ

s:t: pt ðzÞnt ðzÞ ¼ mht þ et mft

ð2:11Þ

Phtþ1 C htþ1 ¼ mht þ stþ1

ð2:12Þ

Pftþ1 C ftþ1 ¼ mft

ð2:13Þ

where (2.11) is the budget constraint reflecting the portfolio decision and (2.12) and (2.13) are cash-in-advance constraints.14 An important expression in the solution of the individual’s problem is the total demand for the output of producer z. This is because producers set their price pt ðzÞ in order to equate production with demand. Total demand dt ðzÞ is defined as follows.

dt ðzÞ  ct ðzÞ þ

ct ðzÞ

¼

pt ðzÞ Pht

!e

xMt1 Pht

ð2:14Þ

Substituting the portfolio constraint and CIA constraints from 2.11, 2.12, and 2.13 into the objective function (2.4) and solving for mft and pt ðzÞ, the solution to the individual’s problem is characterized by the following two Euler equations15:

Phtþ1 C htþ1 et Pftþ1 C ftþ1

¼

1  hh hh

  e  1 pt ðzÞ  h ð1hh Þð1rÞ1  f hh ð1rÞ n1 ð1  hh Þ C tþ1 C tþ1 ¼ vnðnt ðzÞÞ e Phtþ1

ð2:15Þ

ð2:16Þ

where Eq. (2.15) equates the marginal cost of giving up a Home-currency unit of Home consumption for the marginal benefit of a Home-currency unit of Foreign consumption. Eq. (2.16) equates the marginal benefit of raising price to its marginal cost in terms of reduced demand, increased utility of leisure, and the change in income in the next period of life. Because each agent within a country is identical, other than for a differentiated production good, the resulting individual equilibrium price pt ðzÞ and the amount of total revenues held in Foreign currency mft will be symmetric across individuals in a given country. 13 An implicit assumption in this setup is that the producer will meet demand, whatever it is. Thus the producer sets price pt ðzÞ and then produces nt ðzÞ to meet the resulting demand. Some other interesting cases arise in a model with shocks when producers are not required to meet demand. 14 The two cash-in-advance constraints are a simplification of a richer environment in which governments or monetary authorities strategically choose what currencies to accept for exchange that takes place within their borders. See Appendix B for a detailed discussion of how this two CIA constraint setup is a special case of a richer theoretical environment. 15 The derivations of 2.14, 2.15, and 2.16 are given in the Technical Appendix, which is available upon request.

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The first order condition for mft in (2.15) is a direct result of the unit elasticity Cobb–Douglas form of the CES aggregator for individual aggregate consumption in (2.5). It implies that constant shares of income are spent on consumption of Home goods and on consumption of Foreign goods. Using the demand for individual aggregate consumption of Foreign goods from (2.8), the import share is exactly equal to the openness parameter hh .

et Pftþ1 C ftþ1 ¼ hh Ptþ1 C tþ1

8t

ð2:17Þ

This result is one of the benefits of using the simple characterization of openness hh because this paper focuses on the longrun relationship between openness and inflation and because import share of GDP is one of the most commonly used empirical measures of openness. 2.3. Market clearing conditions This economy has three markets that must clear—the goods market, the money market, and the currency exchange market. The following paragraphs describe each market and the respective market clearing condition. Goods market. Both Home and Foreign consumers demand goods from both countries. Producers meet that demand by construction in this model. Due to the linear production technology, nt ðzÞ represents the amount of production by each Home producer of differentiated good z. Goods market clearing requires that production equal the sum of all the Home demands ct ðzÞ and Foreign demands ct ðzÞ for differentiated good z.

nt ðzÞ ¼ dt ðzÞ  ct ðzÞ þ ct ðzÞ 8 t; z

ð2:18Þ

nt ðz Þ ¼ dt ðz Þ  ct ðz Þ þ ct ðz Þ 8 t; z

ð2:19Þ

where the right-hand side of each equation is characterized by (2.14) and its Foreign country analogue. Money market. Money market clearing simply requires that money supply equal money demand at the time that goods are purchased.

Mt ¼ mht þ mht Mt ¼ mft þ mft





8t

ð2:20Þ

8t

ð2:21Þ

M t

where M t and are the Home and Foreign aggregate money supplies, respectively, at time t. Currency exchange market. After trade has taken place in the goods market, period-t producers go to the currency market and make a portfolio decision of how much of each currency to hold. The nominal exchange rate et is the price that equates the amount of Foreign currency purchased with Home currency by Home producers with the amount of Home currency purchased by Foreign producers with Foreign currency.

et mft ¼ mht



8t

ð2:22Þ

It is important to note that the exchange rate here is not pinned down by the assumption of the law of one price as in models with a single cash-in-advance constraint, such as Corsetti and Pesenti (2001) and Arseneau (2007). I follow Cooper and Kempf (2003) in that the exchange rate is a price that clears the currency exchange market in period-t. Because of the two cash-in-advance constraints, the law of one price holds by definition. Using the cash-in-advance constraint (2.13) and its Foreign country analogue, it can be shown that currency exchange market clearing implies that the nominal value of imports equals the nominal value of exports. 

et Pftþ1 C ftþ1 ¼ Phtþ1 C htþ1

8t

ð2:23Þ

2.4. Equilibrium In this section, I define two equilibria. The first is the steady-state equilibrium given international monetary policies x and x . The second is the international monetary Nash equilibrium in which both monetary authorities implement their best responses given the steady-state equilibrium outcomes for those best responses. I define the steady-state international equilibrium given both Home and Foreign monetary policy (x; x ) as follows. Definition 1. Steady-state international equilibrium given x and x⁄. A steady-state international equilibrium, given Home and Foreign monetary policy (x; x ) is the set of Home consumption of both Home and Foreign aggregate goods C h and C f , Home production n, Home portfolio holdings of both Home and Foreign currency mh and mf , the Foreign counterparts   h f (C ; C ; n ; mh ; mf ), individual Home and Foreign prices pt ðzÞ and pt ðz Þ, and exchange rate et such that:

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 Individual optimization. Home and Foreign agents choose the price level of their differentiated good as well as their currency portfolio holdings in order to maximize lifetime utility in (2.4) and its Foreign counterpart subject to a budget constraint (2.11) and two cash-in-advance constraints (2.12) and (2.13). Therefore, the two first order conditions (2.15) and (2.16) hold.  Market clearing. The goods markets (2.18) and (2.19), money markets (2.20) and (2.21), and currency exchange market (2.22) all clear.

The following proposition gives the equilibrium prices and allocations from Definition 1 as functions of x and x . Proposition 1. Steady-state equilibrium prices and allocations given x and x⁄. If Definition 1 holds, the steady-state inflation rates for Home produced goods as well as the aggregate Home consumer price index are equal to the Home money growth rate x, and the inflation rates for the Foreign produced goods as well as the aggregate Foreign consumer price index are equal to the Foreign money growth rate x .

Phtþ1 Pht Pftþ1 Pft

¼

Ptþ1 ¼x Pt

ð2:24Þ

¼

Ptþ1 ¼ x P t

ð2:25Þ

Steady-state labor supply of both Home and Foreign producers have the following representation,

nðx; x Þ ¼ XH ðxÞDH ðx ÞRH

ð2:26Þ

n ðx ; xÞ ¼ XF ðxÞDF ðx ÞRF

ð2:27Þ

where the symbols XH ; XF ; DH ; DF ; RH , and RF are functions of the parameters and summarize the otherwise messy closed form expressions as shown in the proof. And steady-state consumption allocations are the following,

C h ¼ ð1  hh Þnðx; x Þ f



ð2:28Þ



C ¼ hf n ðx ; xÞ f



ð2:29Þ 

C ¼ ð1  hf Þn ðx ; xÞ C

h

ð2:30Þ



¼ hh nðx; x Þ

ð2:31Þ

where nðx; x Þ and n ðx ; xÞ are defined in (2.26) and (2.27), respectively. Proof. See Appendix A for proof. h The monetary authority in each country seeks to maximize the lifetime utility of a representative agent in this economy by choosing Home monetary policy x given Foreign monetary policy x . Define V ðx; x Þ as the lifetime utility of a representative agent given x and x . The objective of the Home monetary authority is then

max V ðx; x Þ ¼ max x

x

  h 1r ½ð1  hh Þnðx; x Þ1hh hf n ðx ; xÞ h 1 1r

 vnðx; x Þn

ð2:32Þ

The following definition characterizes the steady-state international monetary Nash equilibrium. Definition 2. Steady-state international monetary Nash equilibrium. A steady-state international monetary Nash equilibrium is the intersection of the monetary policy best response function for the Home monetary authority given Foreign monetary policy ^ xðx Þ and the monetary policy best response function for the Foreign monetary authority given Home monetary policy ^ x ðxÞ such that:  the individual steady state equilibrium conditions from Definition 1 hold for each country, x ðxÞ is a best response function to the other country’s monetary policy derived  each monetary policy function ^ xðx Þ and ^ from the maximization problem (2.32) and its Foreign analogue. Taking the derivative of (2.32), the resulting solution for the Home monetary best response function ^ xðx Þ is:16 16

The derivation for this result is in the Technical Appendix which is available upon request.

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R.W. Evans / Journal of Macroeconomics 34 (2012) 1095–1110

h

= 0.35,

= 3,

= 10,

= 0.5,

=2

xˆ = 1.56

-0.8 -1

V(x,x*)

-1.2 -1.4 -1.6 -1.8 -2 -2.2 3

0 2.5

1 2

1.5

x, domestic MP

2 1

0.5 3

x*, foreign MP

Fig. 2. Home lifetime utility V as a function of x and x .

^x ¼





ð1  hf Þð1  rÞ  n e1 e ð1  hh  hf Þð1  rÞ  ð1  hh Þn

ð2:33Þ

The analogous solution for the Foreign monetary authority ^ x ðxÞ is:

^x ¼





e1 ð1  hh Þð1  rÞ  n e ð1  hh  hf Þð1  rÞ  ð1  hf Þn

ð2:34Þ

The first characteristic to note about the optimal Home monetary policy function in (2.33) is that it is independent of Foreign monetary policy x . That is, the optimal level of the Home money growth rate does not change with changes in the Foreign money growth rate and is a dominant strategy equilibrium.17 This dominant strategy equilibrium is shown in Fig. 2, which plots the lifetime utility of a representative Home agent from (2.32) as a function of Home inflation x and Foreign inflation x . The parameters (h; r; e; v; n) are calibrated to reflect values estimated in the empirical literature in order to make a simple example. The dark line running across the top of Fig. 2 represents the Home monetary policy best response function from (2.33). The optimal Home inflation level at the selected parameter values is a constant ^ x ¼ 1:56, which is not overly high given that the duration of a period is a generation. Because each country’s best response function for monetary policy is a dominant strategy equilibrium, the world Nash monetary equilibrium is the same as the country partial monetary equilibrium. The optimal policy functions in (2.33) and (2.34) highlight the effects of openness hh and hf and imperfect competition e on a country’s equilibrium inflation rate. The following propositions detail the resulting channel through which increased openness can increase inflation, decreased competition among firms can actually lower equilibrium inflation, and increased elasticity of labor supply can increase inflation. Proposition 2. Monetary response to changes in opennessThe equilibrium Home money growth rate ^ x in (2.33) increases with more Home openness in the form of a higher level of hh and in response to more Foreign openness in the form of a higher level of hf . The argument for the Foreign country is symmetric. However, when hh increases, the increase in ^ x is greater than the increase in ^ x . Conversely, when hf increases, the increase in ^ x is greater than the increase in ^ x.

@ ^x @ x^ ; > 0 and @hh @hf

@ ^x @ ^x > @hh @hf

Proof. See Appendix A. h Because the Home country CPI growth rate (Ptþ1 =Pt ) is equal to the Home money growth rate x, an increase in hh increases Home country inflation as well as Foreign country inflation. From the perspective of the Home monetary authority, if the Home marginal utility of Home consumption decreases relative to the Home marginal utility of Foreign consumption, as is the case when hh increases while hf remains constant, Home country agents bear a smaller proportion of the inflation 17 The Technical Appendix details why ^x is independent of x and is available upon request. The main reason is the simplifying assumption of unit elasticity between Home consumption and Foreign consumption in the Cobb–Douglas utility aggregator (2.5).

Author's personal copy R.W. Evans / Journal of Macroeconomics 34 (2012) 1095–1110

.4 .2 0

Percent of total reserves

.6

1104

1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006

Year dollar yen euro

pound swiss franc other

Source: International Monetary Fund COFER data

Fig. 3. Currency reserves as a fraction of total: 1995–2006.

tax. In effect, higher hh increases the welfare benefits from higher money growth rates to the Home country and lowers the costs. Consequently, the optimal response by the Home monetary authority is to raise the Home money growth rate or the CPI inflation rate in response to a higher degree of openness. This is the classic ‘‘beggar thy neighbor’’ effect described in Corsetti and Pesenti (2001) and Arseneau (2007). The next two propositions further explain how the level of imperfect competition among producers in a country, as parameterized by the elasticity of substitution among a country’s differentiated goods e, and how the Frisch elasticity of labor supply, which is inversely related to n, influence the optimal money growth rate and equilibrium inflation rate ^ x. Proposition 3. Deflationary bias of imperfect competition and the Friedman rule. Both the optimal Home money growth rate ^ x and the optimal Foreign money growth rate ^ x decrease as the level of imperfect competition increases (as e decreases).

@ ^x @ ^x ; >0 @e @e Furthermore, there exist two critical within-country elasticities of substitution for the Home country and Foreign country (e; e ) such that ^ x ¼ 1 when e ¼ e and ^ x ¼ 1 when e ¼ e . That is, these two critical levels of the imperfect competition parameter implement the Friedman Rule (Friedman, 1969) in the Home and Foreign country, respectively.

ð1  hf Þð1  rÞ  n hh ð1  r  nÞ ð1  hh Þð1  rÞ  n e ¼ hf ð1  r  nÞ

e ¼

Proof. See Appendix A. h That the level imperfect competition induces a deflationary bias in monetary policy has been shown in an infinite horizon setting by Arseneau (2007). The intuition behind the deflationary bias of imperfect competition is the following. The degree of inelasticity of substitution between Home and Foreign goods in individual preferences (unit elastic in this case) creates a situation in which some monopoly rents are available in the international market. Under perfect competition e ¼ 1, none of those rents are captured by the optimal pricing of producers because the producers have no market power. In this case, the monetary authority can capture all of those rents by inflating at a high rate. But as the degree of producer market power increases, the producers in a country capture a larger percentage of the international rents available due to their increased pricing power. The monetary authority then captures whatever rents remain by using the inflation tax x. The degree of imperfect competition within a country ends up being a substitute for the inflation tax for taking advantage of monopoly rents available through international trade. The final proposition is in reference to the relationship between equilibrium inflation ^ x and the Frisch elasticity of labor supply, which is inversely related to the shape parameter n in the cost function in (2.4). Proposition 4 shows that increases in the Frisch elasticity of labor supply also increase the equilibrium inflation rate in a country in most specifications of the model.

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R.W. Evans / Journal of Macroeconomics 34 (2012) 1095–1110 Table 1 Equilibrium conditions given x and x . Home country (2.15)

P htþ1 C htþ1 et Pftþ1 C ftþ1

¼

Foreign country 

et P ftþ1 C ftþ1

1hh hh



P htþ1 C htþ1

¼

1hf hf

 pt ðzÞ ðC htþ1 Þð1hh Þð1rÞ1 ¼ ð1  hh Þ e1 h ð1rÞ e Phtþ1 ðC ftþ1 Þ h

 ð1hf Þð1rÞ1

 pt ðz Þ C ftþ1 ð1  hf Þ e1

h hf ð1rÞ ¼ e Pftþ1 C

(2.14), (2.18)

. . . vnðnt ðzÞÞn1  e xM t1 nt ðzÞ ¼ pt ðzÞ h h

. . . vnðnt ðz ÞÞn1  e  x M t1 nt ðz Þ ¼ pt ðzf Þ f

(2.11)

pt ðzÞnt ðzÞ ¼ mht þ et mft

(2.16)

Pt

Pt

(2.12), (2.3)

C htþ1 ¼

(2.13), (2.3)

C ftþ1

(2.5)

¼

C tþ1 ¼

tþ1

mft

C htþ1

1hh 

C ftþ1

hh

 mft



þ

mht et

f



mt þðx 1Þx Mt1



mht

C ftþ1 ¼

Phtþ1





pt ðz Þnt ðz Þ ¼

mht þðx1ÞxMt1

Pftþ1

Pt

Pt



Pftþ1 

C htþ1

¼

C tþ1

  1hf   hf ¼ C ftþ1 C htþ1

Phtþ1

Market clearing conditions nt ðzÞ ¼ ct ðzÞ þ ct ðzÞ nt ðz Þ ¼ ct ðz Þ þ ct ðz Þ

(2.18) (2.19) (2.20)

Mt ¼ mht þ mht



 ¼ mft þ mft  et mft ¼ mht

(2.21)

M t

(2.22)

Table 2 Properties of representative parameters.

Dh ¼ ð1  hh Þð1  rÞ  n

D

DH ¼ Dh Df fRh Rf

Df ¼ ð1  hf Þð1  rÞ  n

DF ¼

Dh Dh Df Rh Rf

Rh ¼ hh ð1  rÞ

RH ¼ Dh Df RRh h Rf

Rf ¼ hf ð1  rÞ

e  vn Xh ¼ e1 ð1hh Þð1hh Þð1rÞ ðhf Þhh ð1rÞ

e  vn Xf ¼ e1 ð1hf Þð1rÞ hf ð1rÞ

RF ¼ Dh Df Rf h Rf

ð1hf Þ

Symbol

Dh Df Rh Rf Xh Xf Dh Df  Rh Rf

R



XH ¼ ðXh ÞDH Xf

XF ¼ Xf

ðhh Þ

Sign () always () always () when r > 1; hh > 0 () when r > 1; hf > 0 (þ) when hf > 0 (þ) when hh > 0 (+) always

DF

RH

ðXh ÞRF

@ðÞ @hh

(þ) when

r>1

() when

r>1

() when (þ) when () when

r > 1; hf > 0 r > 1; hf > 0 r>1

@ðÞ @hf

(þ) when

r>1

() (þ) () ()

r>1 r > 1; hh > 0 r > 1; hh > 0 r>1

when when when when

Note: The results from this table are derived in the Technical Appendix and are available upon request.

Proposition 4. Inflationary bias of Frisch elasticity of labor supplyFor values of the coefficient of relative risk aversion strictly greater than one r > 1 and hh ; hf > 0, both the optimal Home money growth rate ^ x and the optimal Foreign money growth rate ^ x increase as the Frisch elasticity of labor supply increases (as n decreases).

@ ^x @ ^x ; <0 @n @n Proof. See Appendix A. h The inflationary bias of the elasticity of labor supply is an intuitive result. The costs of the inflation tax are mitigated when suppliers of labor can more easily substitute into leisure in the face of a depreciation of the terms of trade from inflation x. That is, the inflation tax has less of a negative effect on domestic utility when labor supply is more elastic. In summary, more openness leads to higher equilibrium inflation, more imperfect competition leads to lower equilibrium inflation, and more elasticity of labor supply leads to higher inflation in this simple general equilibrium model with commitment. A monetary authority has more incentive to inflate when a country is more open to international trade because foreigners hold more of the domestic currency and are thereby taxed disproportionately by any domestic

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inflation. That incentive to inflate is mitigated by degree to which domestic producers can raise their prices to take advantage of any inelasticity in foreign demand for domestic products. In essence, domestic market power substitutes for monetary inflation taxation. 3. Conclusion The previous empirical findings of Romer (1993), Lane (1997), and Terra (1998) that openness leads to lower long-term inflation rates may be restricted in application to economies for which a lack of commitment characterizes monetary policy. This study characterizes a theoretical channel through which openness is inflationary when monetary policy can be approximated by commitment. This result likely only applies to the period since the late 1980s and only to a more developed subset of countries. In addition, the model predicts that the inflationary bias of openness is dampened by the degree of imperfect competition within a country. The possibility that increased openness could lead to rising long-term inflation rates is important for national trade policy questions. This begs the question of what is the optimal degree of openness for a country. Solving for optimal openness is a conceptually simple exercise to perform in this framework, although it is analytically quite involved. The parameter for openness in this paper h incorporates both individual preferences and policy choices such as barriers to trade. But h could be endogenized as fiscal decision that takes place before the monetary authority chooses the money growth rate. Lastly, these predictions could shed light on the incentives faced by the monetary authorities of the world’s most widely held currencies, especially in the face of mounting sovereign debt. Acknowledgments I especially thank Russell Cooper and Mark Wynne for their helpful guidance on this project. Special thanks also go to Dean Corbae, Anthony Landry, Erwan Quintin, Jim Dolmas, and Kim Ruhl for their comments. I am also grateful to the Federal Reserve Bank of Dallas for financial support and many discussions with research department staff. Matthew Yancey provided excellent research assistance. All errors are mine. Appendix A. Proofs Proof of Proposition 1. Table 1 shows the conditions that must hold for the steady state-equilibrium in Definition 1. Following Cooper and Kempf (2003), let /t represent the share of revenues pt ðzÞnt ðzÞ kept in the form of Home currency in period t, and let 1  /t be the share of revenues exchanged for Foreign currency as characterized in the portfolio budget constraint (2.11). Then the Home portfolio budget constraint (2.11) and its Foreign analogue, combined with the Home and Foreign versions of the law of motion for individual money balances (2.3), and the first order condition (2.15) give the following unique non-autarkic steady-state equilibrium share of currency from sales held for own-country consumption.

  1 / ¼ 1  hh x 8 x 2 0; hh

ðA:1:1Þ

  1 1  / ¼ hh x 8 x 2 0; hh

ðA:1:2Þ



/ ¼ 1  hf x

8 x 2 0;

1  / ¼ hf x

8 x 2 0;



1 hf 1 hf

 ðA:1:3Þ  ðA:1:4Þ

From the aggregate money laws of motion in (2.1) and from the money market clearing conditions in (2.20) and (2.21), the non-autarkic steady-state equilibrium country-specific consumption inflation rates are equal to their respective money growth rates.

Phtþ1 Pht

¼ x and

Pftþ1 Pft

¼ x

ðA:1:5Þ

Furthermore, using the definition of the Home country CPI level P tþ1 from (2.10) and its Foreign country analogue, the expressions for the share Home country revenues traded for Foreign currency balances (A.1.2) and the share of Foreign country revenues traded for Home currency balances (A.1.4), and the currency exchange market clearing condition (2.22), the Home country CPI growth rate and the Foreign country CPI growth rates can be shown to be equal to their respective countries’ money growth rates as well.18 18

The derivation is given in the Technical Appendix, and is available upon request.

Author's personal copy R.W. Evans / Journal of Macroeconomics 34 (2012) 1095–1110

P tþ1 ¼ x Pt

Ptþ1 ¼ x and Pt

1107

ðA:1:6Þ

Using A.1.1, A.1.2, A.1.3, and A.1.4, as well as the equilibrium inflation rates from (A.1.5), steady-state equilibrium consumption can be derived in terms of steady-state employment from the cash-in-advance constraints as:

C h ¼ ð1  hh Þn

ðA:1:7Þ

f

C ¼ hf n

ðA:1:8Þ

f

C ¼ ð1  hf Þn C

h

ðA:1:9Þ

¼ hh n

ðA:1:10Þ

where the steady-state employment levels n and n are characterized below in Eqs. (A.1.13) and (A.1.14). The expressions for the steady-state international equilibrium employment are then found by solving the two equilibrium forms of the Home first order condition (2.16) and its Foreign analogue.

   h ð1rÞ e1 1 ð1  hh Þ ½ð1  hh Þnð1hh Þð1rÞ1 hf n h ¼ vnðnÞn1 e x 



ð1hf Þð1rÞ1 e1 1  ð1  hf Þn ½hh nhf ð1rÞ ¼ vnðn Þn1  e x

ð1  hf Þ

ðA:1:11Þ

ðA:1:12Þ

Solving (A.1.12) for n and plugging it into (A.1.11), and doing the symmetric operation for the Foreign country gives the expressions for Home and Foreign steady-state equilibrium labor supply:

nðx; x Þ ¼ XH ðxÞDH ðx ÞRH

ðA:1:13Þ

n ðx ; xÞ ¼ XF ðx ÞDF ðxÞRF

ðA:1:14Þ

where the symbols in (A.1.13) and (A.1.14) summarize otherwise messy closed form expressions of the steady state employment in terms of the parameters. The functions of the parameters represented by the symbols along with their derivatives with respect to hh and hf are shown in Table 2. For the Home country, (A.1.13) shows that the effect of Home monetary policy x on the real economy is governed by the sign of Df and that the effect of Foreign monetary policy x on the real economy is governed by the sign of Rh . Table 2 shows that Df < 0 always, so increases in Home money growth x decrease employment, output, and consumption of Home goods. This is the production or consumption tax (or leisure subsidy) of inflation. Because Rh < 0 for the most common parameterizations of risk aversion r > 1, the effect of increases in Foreign money growth x result in increases in real economic activity in the Home country. It is worth noting the theoretical implications of the cases in which Foreign monetary policy x has no effect on the real economy Rh ¼ 0 as shown in the second line of (A.1.15).



Rh ¼

< 0 if

h 2 ð0; 0:5 and

¼ 0 if

hh ¼ 0 or

r>1

r¼1

ðA:1:15Þ

Obviously, when the economies do not trade with each other, hh ¼ 0, Foreign monetary policy will be neutral. But it is interesting to note that the case of log utility (r ¼ 1) also induces the neutrality of Foreign monetary policy.19 Proof of Proposition 2. (Monetary response to changes in openness). Taking the derivative of the expression for ^ x in (2.33) with respect to hh and hf gives the following results. Some of the symbols from Table 2 from the previous proof are used to simplify the analysis.

^x ¼









Df ð1  hf Þð1  rÞ  n e1 e1 ¼ e ð1  hh ÞDf  hh Rf e ð1  hh  hf Þð1  rÞ  ð1  hh Þn 



Df ð1  r  nÞ e1 >0 e ð1  hh ÞDf  hh Rf 2   @ ^x e  1 hh ð1  rÞð1  r  nÞ ¼  2 > 0 @hf e ð1  hh ÞDf  hh Rf

@ ^x ¼ @hh

Taking the derivative of the expression for ^ x in (2.34) with respect to hf and hh gives the following results: 19 As another special case, when the two countries are symmetric in their degree of openness hh ¼ hf , the steady state equilibrium levels of employment n and n are independent of the level of imperfect competition e within both countries. See Technical Appendix (available upon request) for the derivation.

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R.W. Evans / Journal of Macroeconomics 34 (2012) 1095–1110

^x ¼









e1 Dh e1 ð1  hh Þð1  rÞ  n ¼ e ð1  hf ÞDh  hf Rh e ð1  hh  hf Þð1  rÞ  ð1  hf Þn 



e1 Dh ð1  r  nÞ  2 > 0 e ð1  hf ÞDh  hf Rh   @ ^x e  1 hf ð1  rÞð1  r  nÞ ¼ >0 @hh e ð1  hf ÞDh  hf Rh 2 @ ^x ¼ @hf

@ ð ^x Þ The proposition that when hh increases, the increase in ^ x is greater than the increase in ^ x , simply means that @h^x > 0.

^x 

@ ^x @hh

h

" #   ^x @ ^x 1 @ ^x  ¼ @hh ð^x Þ2 @hh ^x    e  ð1  h ÞD  h R Df ð1  r  nÞ e1 f h f h ... ¼  2 e  1 D e h ð1  hh ÞDf  hh Rf !    e  D ð1  h ÞD  h R 2 Rf ð1  r  nÞ e1 f f h f h   e ð1  hf ÞDh  hf Rh 2 e  1 D2h ð1  hh ÞDf  hh Rf Df ð1  r  nÞ½ð1  hf ÞDh  hf Rh  Df Rf ð1  r  nÞ  2 ¼ 2 Dh ½ð1  hh ÞDf  hh Rf  Dh ½ð1  hh ÞDf  hh Rf  Dh Df ð1  r  nÞ½ð1  hf ÞDh  hf Rh   Df Rf ð1  r  nÞ½ð1  hh ÞDf  hh Rf  ¼ D2h ½ð1  hh ÞDf  hh Rf 2 ! Dh ½ð1  hf ÞDh  hf Rh   Rf ½ð1  hh ÞDf  hh Rf  ¼ Df ð1  r  nÞ D2h ½ð1  hh ÞDf  hh Rf 2 ¼ Df ð1  r  nÞ

^x 

@ ^x ¼ Df ð1  r  nÞ @hh

Dh ½ð1  hh  hf Þð1  rÞ  ð1  hf Þn  Rf ½ð1  hh  hf Þð1  rÞ  ð1  hh Þn D2h ½ð1  hh ÞDf  hh Rf 2  ! ðDh  Rf Þð1  hh  hf Þð1  rÞ þ n Rf ð1  hh Þ  Dh ð1  hf Þ D2h ½ð1  hh ÞDf  hh Rf 2

!

>0

The last line is true because Dh  Rf < 0 and Rf ð1  hh Þ  Dh ð1  hf Þ > 0. Proof of Proposition 3. (Deflationary bias of imperfect competition). From (2.33) and (2.34):

  Df @ ^x 1 ¼ 2 >0 @e e ð1  hh ÞDf  hh Rf   @ ^x 1 Dh ¼ 2 >0 @e e ð1  hf ÞDh  hf Rh Then, to find the respective levels of e that induce the Home and Foreign monetary authorities, respectively, to set their money growth rates equal to 1 is found by solving (2.33) and (2.34) for e when ^ x ¼ 1 and when ^ x ¼ 1.

e : e :

  e  1 ð1  hf Þð1  rÞ  n e ð1  hh  hf Þð1  rÞ  ð1  hh Þn    e  1 ð1  hh Þð1  rÞ  n 1¼ e ð1  hh  hf Þð1  rÞ  ð1  hf Þn

1¼

Solving the two equations for e and e , respectively, gives the result.

Df ð1  hf Þð1  rÞ  n ¼ Rh  h h n hh ð1  r  nÞ D ð1  hh Þð1  rÞ  n h e ¼ ¼ hf ð1  r  nÞ Rf  hf n

e ¼

Proof of Proposition 4. (Inflationary bias of elasticity of labor supply). The parameter n from the cost of labor function in the individual’s objective function (2.4) is inversely related to the Frisch elasticity of labor supply. From the expression for the optimal equilibrium monetary policy (2.33), the derivative with respect to n is:

@ ^x ¼ @n



e1 e



2

3

1 7 2   5 ð1  h  h Þð1  r Þ  ð1  h Þn h f h ð1  hh  hf Þð1  rÞ  ð1  hh Þn |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

6

þ 4

|fflfflfflfflffl{zfflfflfflfflffl}

 ð1  hh Þ ð1  hf Þð1  rÞ  n

ðA:1:16Þ

Author's personal copy R.W. Evans / Journal of Macroeconomics 34 (2012) 1095–1110

1109

The inverse markup is always positive ðe  1Þ=e > 0. The numerator of the first term in brackets in (A.1.16) is always negative because ð1  hf Þð1  rÞ  n < 0, and the denominator of the first term in brackets is positive because it is squared. So the first term in the brackets is negative. The denominator of the second term in the brackets is always negative ð1  hh  hf Þð1  rÞ  ð1  hh Þn < 0. It only remains to show that the difference is negative. The term in brackets above can be rewritten in the following way.

2 3 ð1  h  h þ h h Þð1  r Þ  ð1  h Þn 1 h f h f h 4  1þ 5  ð1  hh  hf Þð1  rÞ  ð1  hh Þn ð1  hh  hf Þð1  rÞ  ð1  hh Þn |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} The term in brackets in (A.1.17) is positive when

ðA:1:17Þ

r > 1 because hh hf > 0.

@ ^x <0 @n Appendix B. Two cash-in-advance constraints and richer model The two cash-in-advance constraints from (2.12) and (2.13) are a strong form of a more general assumption that has both empirical and theoretical foundations. Theoretically, the cash-in-advance constraints are an extreme form of a transaction cost theory of money demand. Less restrictive versions of transactions theories include Khan et al. (2003) who model money as the least costly of multiple media of exchange for converting wealth into consumption and the shopping time model of money demand from McCallum (1983). As long as some cost exists for switching from money to a less liquid medium of exchange, the beggar-thy-neighbor incentive described in this paper will exist for a monetary authority. In addition, the two cash-in-advance constraints can be thought of as a simplification of one equilibrium outcome of a richer environment in which governments or monetary authorities strategically choose what currencies to accept for exchange that takes place within their borders. Matsuyama et al. (1993) present a random matching search model of money after the flavor of Kiyotaki and Wright (1989) in which blocks of agents (countries) choose which currencies to accept for local and international transactions based on the likelihood of that currency being accepted in future transactions. In one equilibrium, corresponding to the two cash-in-advance constraint environment of this paper, each block of agents (country) only accepts local currency for all local and international transactions. Another equilibrium in the Matsuyama et al. (1993) is the case in which vendors in both countries accept currency of both countries. This is analogous to the more standard approach in the open economy macroeconomic literature as exemplified by Corsetti and Pesenti (2001). Their environment is one characterized by a single cash-in-advance constraint in which producers sell their goods in both countries and charge a price in terms of Home currency and a price in terms of Foreign currency. The exchange rate is then pinned down by an assumption of the law of one price. The reason for choosing the two cash-in-advance constraints approach as shown in Eqs. (2.12) and (2.13) instead of the more standard Corsetti and Pesenti (2001) method of one cash-in-advance constraint and the law of one price is that the method employed here gives rise to a portfolio decision. The law of one price is implicit in the two cash-in-advance constraint assumption because, by definition, vendors only accept one currency and therefore only charge one price. As is shown in Section 2.3, the exchange rate here serves as a price that clears the currency exchange market rather than a mechanism for enforcing the law of one price. Furthermore, the currency portfolio decision is an interesting one that has not received much attention. Engel and Matsumoto (2006) and Evans and Lyons (2005) are good international portfolio papers. However, both the single CIA constraint with the law of one price method and the dual CIA constraints with currency exchange market clearing method deliver the same results for optimal monetary policy. Empirically, the beggar-thy-neighbor effect seems to be a real factor in policy decisions. As evidence that this effect is important in practice, (Cooper and Kempf (2003), p. 120) cite the European Commission report which stated in regard to monetary policy that ‘‘. . .the adoption of a common monetary policy. . .will remove the possibility of beggar-thy-neighbor monetary and exchange rate policies.’’ The IMF (2011) COFER data show that dollar, euro, and yen currency reserves in the year 2000 were about 75% of all currency reserves. Fig. 3 shows currency reserves as a percent of total reserves for 1995–2006 for the five major currencies and the ‘‘other’’ category. These currencies’ shares of total reserves have been relatively stable over the decade from 1995 to 2006, providing some justification for the exogeneity assumed in the. Much of international trade is financed with relatively short-term debt instruments from commercial banks, factors firms, commercial finance companies, and government institutions and not currency. However, the assets that finance international trade represent a wide portfolio of term structure and risk structure as well as being denominated in multiple currencies. Also, the existence of these intermediaries suggests a transactions cost motivation for international money demand. This paper does not try to model the decision between each type of currency or the relative exchange rate risk of each. But it does assume that engaging in international trade requires some transaction costs. References Arseneau, D.M., 2007. The inflation tax in an open economy with imperfect competition. Review of Economic Dynamics 10 (1), 126–147. Arseneau, D.M., 2012. Expectation traps in a new Keynesian open economy model. Economic Theory 49 (1), 81–112.

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