Journal of International Money and Finance 30 (2011) 587–603

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Journal of International Moneyand Finance

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Real exchange rate, productivity and labor market frictions

Yu Sheng a, Xinpeng Xu b,*

aCrawford School of Economics and Government, Australian National University, Australiab Faculty of Business, Hong Kong Polytechnic University, Kowloon, Hong Kong

JEL classifications:F16F31J64

Keywords:The Balassa–Samuelson modelSearch unemploymentLabor market efficiency

* Corresponding author. Tel.: þ852 2766 7139; fE-mail address: afxxu@polyu.edu.hk (X. Xu).

0261-5606/$ – see front matter � 2011 Elsevier Ltdoi:10.1016/j.jimonfin.2011.01.006

a b s t r a c t

We extend the classic Balassa–Samuelson model to an environ-ment with search unemployment. We show that the classicBalassa–Samuelson model with the assumption of full employ-ment emerges as a special case of our more generalized model. Inour generalized model, the degree of labor market matching effi-ciency affects the strength of the structural relationship betweenthe real exchange rate and sectoral productivity through influ-encing labor’s choice between employment and unemployment aswell as movement across sectors. When the relative labor marketmatching friction is high, search unemployment is high and thestandard Balassa–Samuelson effect may not hold. Empiricalevidence supports our theory: controlling for differences in labormarket frictions across countries provides a better fit in estimatingthe Balassa–Samuelson effect.

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1. Introduction

The relative price of a common basket of goods between two countries, the real exchange rate (RER),is one of the most important prices in an open economy. Balassa (1964) and Samuelson (1964) arguethat deviations from Purchasing Power Parity (PPP) may be due to international productivity differ-entials between the tradable and nontradable sectors. Balassa and Samuelson posit that, as the law ofone price holds only for tradable goods, in a fast-growing economy, higher productivity growth in thetradable sector will increase real wages in all sectors since employedworkers are mobile across sectors,

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Y. Sheng, X. Xu / Journal of International Money and Finance 30 (2011) 587–603588

which will lead in turn to an increase in the relative price of nontradable goods, resulting in an overallrise in the national price level.1

Although the Balassa–Samuelson model predicts a simple theoretical relationship between produc-tivity differences and real exchange rates, the empirical evidence ismixed. On the onehand, the evidenceof the Balassa–Samuelson effect is weak for developed economies. For example, an influential study byEngel (1999) finds that the relative price of nontradable goods cannot explain much of themovement ofOECD countries’ real exchange rates even after changes in the terms of trade (or the short-term effectsfrom tradable sectors) are well controlled. On the other hand, there is considerable evidence for theBalassa–Samuelson effect across developed and developing economies; see Bergin et al. (2006) andRogoff (1996), among others. Bergin et al. (2004) show that the Balassa–Samuelson effect is not visible inearly 1900s, but the effect grows steadily over time to rather large values in the more recent years.

The mixed evidence for the Balassa–Samuelson effect demands further investigation. One of theunderlying assumptions of the Balassa–Samuelson model is that labor markets are frictionless so thatthere is always full employment. Yet, it is well recognized that it takes time and other resources for anunemployed worker to find a job and for a firm to fill a vacancy so that there exist frictions in the labormarket at steady state (see for example, McCall, 1970; Diamond, 1982; Mortensen, 1982; Pissarides,1985; Mortensen and Pissarides, 1999; Pissarides, 2000; Rogerson et al., 2005) and that there aresignificant differences in the degree of labor market frictions across sectors and countries. In fact, usingsectoral data, Wacziarg and Wallack (2004) show that trade liberalization has far smaller effects onintersectoral labor shifts than is often presumed. The existence of labor market frictions affects theresponse of employed workers’ wage to changes in relative productivity, but it is not clear howdifferences in labor market institutions across sectors and countries can affect the relationshipbetween the real exchange rate and productivity differentials.

By introducing search unemployment into a standard Balassa–Samuelson model, we demonstratein this paper that the degree of labor market matching efficiency across sectors and countries affectsthe strength of the structural relationship between the real exchange rate and sectoral productivitydifferentials through influencing labor’s choice between employment and unemployment as well asmovement across sectors. When the relative labor market matching efficiency is low, search unem-ployment is high and the standard Balassa–Samuelson effects may not hold. Specifically, in a worldwith labor market frictions and unemployment, the standard Balassa–Samuelson mechanism mayhave to be revised to incorporate the fact that workers cycle between employment and unemploymentin each sector. In such an environment, employed workers no longer move instantaneously andcostlessly across sectors in response to changes in productivity and relative sectoral wages. Instead, it isunemployed workers that move freely across sectors in response to changes in expected lifetimeincome arising from changes in relative sectoral wages and associated frictional costs such as theprobability of finding a new job and of an existing job being destroyed. Thus, an increase in the relativeproductivity in the tradable (or nontradable) sector may lead to an increase in the relative wage in thatsector, but the extent of the increase would, in general, be lower compared with what is predicted bythe standard Balassa–Samuelson model, as part of the increase in the marginal product of labor will beused to cover frictional costs in the labor market. The increase in the wage of the tradable sector willlead to an increase in the expected lifetime income of unemployed workers searching in the tradablesector, attracting unemployed workers in the nontradable sector to move to the tradable sector. Thismovement of unemployed workers across sectors will continue until the expected lifetime income ofunemployed workers searching in each sector is equalized. The resulting increase in the expectedlifetime income of unemployed workers in the nontradable sector may bid up wages of employedworkers in the nontradable sector.

1 Recent empirical investigations of the effect of productivity shocks (which are real shocks) on the real exchange rate, the so-called Balassa–Samuelson effect, include, inter alia, Asea and Mendoza (1994), De Gregorio et al. (1994), Froot and Rogoff (1995),Lothian and Taylor (2008) and Canzoneri et al. (1999). Although a survey of empirical findings by Froot and Rogoff (1995) findsweak support for the Balassa–Samuelson effect, recent work by Lothian and Taylor (2008) using data from 1820 to 2001 for theUS, the UK and France in a nonlinear framework reports a statistically significant Balassa–Samuelson effect which explains 40%of the variation of sterling-dollar exchange rate. Asea and Mendoza (1994) and Canzoneri et al. (1999) also provide similarresults to support the proposition that productivity differentials determine the relative price of nontradables.

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However, the increase in the price of nontradable goods may be higher or lower than what ispredicted by the standard Balassa–Samuelson model, depending on the relative market matchingefficiency between the two sectors, as the effect of an increase in the nontradable sector’s wage on thenontradable price will be absorbed by the relative flexibility in its labor market, resulting in a higher orlower price of nontradable goods and hence, the national price level. Furthermore, if the relative labormarket matching efficiency in the tradable sector is too low, there is a potential that the relativeincrease in the tradable sector productivity may be more than offset by relatively high frictional costs,thus operating against the Balassa–Samuelson effect. The importance of labor market institutionshighlighted in this paper offers an added dimension for explaining differences in price levels betweencountries with different or similar income levels.

To anticipate our results, we show that: (1) the classic Balassa–Samuelson model emerges asa special case of our more generalized model with unemployment; (2) the effects of sectoralproductivity differentials on the real exchange rate have to be adjusted quantitatively for differences inlabor market matching efficiency across sectors and between countries. This highlights a new andpotentially important channel for the transmission of various shocks to labor market institutions to thereal exchange rate, i.e., labor market institutions matter; (3) when differences in productivity growthare insignificant among developed economies, differences in labor market frictions may dominate theBalassa–Samuelson effect, which make the Balassa–Samuelson effect less visible. As labor marketmatching efficiency improves over time, the Balassa–Samuelson effect will become more significant.Our empirical evidence confirms that controlling for labormarketmatching efficiency provides a betterfit in estimating the Balassa–Samuelson effect.

We view our work as complementary to the recent literature that emphasizes imperfections ingoods markets in extending the Balassa–Samuelson model. Several recent papers have intended toupdate or extend the Balassa–Samuelson model by focusing on imperfections in goods markets. Forexample, Fitzgerald (2003) revisits the classic Balassa–Samuelson model by dropping out the Balassa–Samuelson assumption that all countries produce the same tradable goods. Instead, Fitzgerald intro-duces production of differentiated goods across countries and increasing returns to scale in production,which leads to endogenous specialization and intra-industry trade. In this environment, the rela-tionship between the real exchange rate and sectoral productivity is shown to depend on the strengthof terms-of-trade effects. Ghironi and Melitz (2005) propose a model highlighting the importance ofendogenous firm entry and exit to both domestic and export markets in determining the movement ofnational price levels. Bergin et al. (2006) develop a model of endogenous tradability where insteadof assuming productivity gains concentrating by coincidence in the production of existing tradablegoods as in the Balassa–Samuelson model, productivity gains in the production of particular goods canlead to those goods becoming traded. They demonstrate that such a model can deliver endogenouslytime-varying correlations between incomes and prices. A recent paper by Helpman and Itskhoki (2010)extends Melitz’s (2003) model by introducing search unemployment in one sector to examine theinteraction of labor market efficiency and trade impediments in shaping the relationship betweenproductivity and price levels across countries and shows that the country with more flexible labormarkets has both higher productivity and a lower price level, which operates against the standardBalassa–Samuelson effect.

To the best of our knowledge, our paper is the first to develop a model that integrates the standardBalassa–Samuelson model with search theory so that the classic Balassa–Samuelson model can beexamined in an environment without the assumption of full employment.2 In contrast to the latestliterature that focuses on imperfections in goods markets, we center our analysis on frictions in thelabor market. More importantly, search frictions in labor markets exist in the long run, which differsfrom short-run models of labor market rigidities. We discuss the case of out-of-steady-state in theAppendix. Unlike other theoretical models, the standard Balassa–Samuelson channel is still operativein our generalizedmodel. A focus on labor market frictions reflects the rising importance of the issue ofunemployment in the age of globalization, whereby real exchange rate movements may interact withdomestic labor market institutions when countries integrate more and more with each other.

2 For an introduction to search theory, see Romer (2001), Ljungqvist and Sargent (2000) and Pissarides (2000).

Y. Sheng, X. Xu / Journal of International Money and Finance 30 (2011) 587–603590

The rest of the paper proceeds as follows. In the next section we develop a two-sector model thatdistinguishes between the tradable and nontradable sectors and endogenizes unemployment ina simple framework of search theory. This provides us with a setup that is useful in discussing therelationship between the real exchange rate, sectoral productivity and labor market flexibility acrosssectors and between countries. We summarize the main results in a proposition and several lemmasand corollaries. In Section 3, we discuss our empirical results. The final section concludes.

2. The model

2.1. The economy

Consider a small open economy that produces two composite goods, tradable goods (T) priced ininternational markets and nontradable goods (N) priced in the domestic market. The productiontechnologies of tradables and nontradables is characterized by constant returns to scale (CRS)production functions of the capital Ki (i¼ T,N) and labor Li employed, Yi¼ AiF(Ki,Li)¼ AiLif(ki), where Yi isoutput in the tradable and nontradable sectors and the Ai are productivity shifters. Tradable goods aretaken as numeraire.

Output per unit of labor, yih Yi/Li, can then be written as, yi¼ Aif(ki) which makes use of thecondition of CRS where kih Ki/Li is the capital–labor ratio in sector i. The production function F(�)exhibits positive and diminishing marginal products with respect to each input. Capital is perfectlymobile across sectors domestically and internationally.

We now depart from the standard full employment assumption as in the Balassa–Samuelsonmodel.In a frictional economy, it takes time and other resources for a worker to find a job and for a firm to filla vacancy. Since there are workers searching for a job and vacancies waiting to be filled, there is alwaysunemployment in the labor market. With labor market frictions and unemployment, it is nowunemployed workers that move freely across sectors and therefore, it is the expected income ofunemployed workers that will be equalized across sectors. In contrast, employed workers’ income wi

may be different as it has to take into account compensation differentials arising from labor marketfrictions. We now specify job matching, job creation, job destruction, and wage determination ingeneral equilibrium.

2.2. Matching

Suppose the number of matches between firms and workers in sector i depends on the number ofunemployed works (Ui) chasing the number of vacancies (Vi). Let Zi be the sectoral labor force, ui theunemployment rate (Ui/Zi) and vi be the number of vacant jobs as a fraction of the labor force (Vi/Zi). Wehave the number of matches, miZi¼m(Ziui,Zivi). A typical assumption of the functional form for thematching function has constant returns to scale (Blanchard and Diamond, 1989). Thus, we can expressall variables as a function of the tightness of the labormarket, qih vi/ui. The rate at which a vacant job isfilled is therefore q(qi), which is equal tomi/vi. The rate at which an unemployedworker finds amatch isqiq(qi), which is equal to mi/ui.

2.3. Firms

Following Pissarides (2000), a typical firm has jobs that are vacant and has to pay frictional cost g asan advertising and recruiting cost (‘frictional cost’ in what follows) in order to fill a vacancy. Duringhiring, a vacant job is filled at the rate q(qi) while an unemployed worker finds a job at the rate qiq(qi).When a firm and a worker meet and agree to an employment contract, a job is occupied. The firm thengoes on to rent capital ki for each worker and produces output, which is sold in competitive markets.

We consider the optimal decision of a typical firm in the tradable sector first. Let VT be the presentvalue of expected profit for the firm from a vacant job and JT the present value of expected profit for thefirm from an occupied job in the tradable sector. VT satisfies the Bellman equation,

rVT ¼ �gþ qðqT ÞðJT � VT Þ: (1)

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A job is an asset owned by the firm and is valued in a perfect capital market characterized by a risk-freeinterest rate r. The asset value of a vacant job, rVT, is exactly equal to the rate of return on the asset: thevacant job costs g but has the probability of q(qT) for the vacancy to turn into a filled jobwhichwill yieldthe net return JT� VT. In equilibrium, perfect competition and profit maximization require that thegains from job creation are always exhausted, so that jobs are created up to the point where VT¼ 0,implying that JT¼ g/q(qT). The implicit assumption here is that firms decide to create jobs whenever thevalue of a vacancy is positive and thus potential profits will be eroded quickly by free entry.

As the capital stock owned (or rented) by the firm becomes part of the value of the job, the assetvalue of an occupied job is given by JTþ kT. The job yields net return ATf(kT)�wT where wT is worker’swage. Similar to the valuation of a vacant job, the asset value of an occupied job, r(JTþ kT), satisfies thefollowing Bellman equation

rðJT þ kT Þ ¼ AT f ðkT Þ �wT � lðJT � VT Þ; (2)

where l is the job destruction ratewhich leads to the loss of JT but not kT.3 Intuitively, the annuity of thereturn to the asset of an occupied job is equal to the output ATf(kT), net of its cost (which is wage here ifwe assume no capital depreciation for simplicity), with a probability of l that the relationship maycome to an end so that the firm will lose JT.

Given the interest rate andwage rate, the firm rents capital kT tomaximize the value of the job JT. Wecan write the firm’s first-order condition with respect to capital as,

AT f0ðkT Þ ¼ r; (3)

which has the standard interpretation where firms rent capital kT up to the point where the marginalproduct of capital is equal to the market rental rate, r, as we assume that there is no friction in thecapital market.

Substituting the firm’s first-order conditionwith respect to capital and the equilibrium job creationcondition JT¼ g/q(qT) into the asset value equation of an occupied job yields the familiar equilibriumcondition for the firm’s employment of labor. The firm posts vacancies and hires workers up to thepoint where marginal benefit of an additional worker, the marginal product of labor, is equal tomarginal cost, i.e., the market wage, after adjusting for the frictional cost,

AT�f ðkT Þ � kT f

0ðkT Þ� ¼ wT þ ðr þ lÞg

qðqT Þ: (4)

If there is no frictional cost so that g¼ 0, the last term on the right hand side of Eq. (4) becomes zeroand (4) reduces to the familiar marginal productivity condition for labor in a full-information, fric-tionless labor market.

2.4. Workers

Workers search for jobs and once offered, have to make a decision to accept or reject the offer.Therefore workers’ decisions will influence equilibrium market wages. Similar to a firm described inthe above section, a typical worker makes an optimal decision to accept a job offer and receive wagewT

or to remain unemployed and receive unemployment benefits during search. To illustrate the decisionmade by a typical worker in the tradable sector, let UT and ET be the present value of the expectedincome streams of an unemployed worker and an employed worker in the tradable sector respectively.UT satisfies the Bellman equation

rUT ¼ bþ qTqðqT ÞðET � UT Þ: (5)

Eq. (5) says that the asset value of the unemployed worker’s human capital is made up of two

3 For simplicity, we assume that job destruction rates are exogenous and same across sectors but it can be relaxed easilywithout affecting our main results.

Y. Sheng, X. Xu / Journal of International Money and Finance 30 (2011) 587–603592

components: the unemployment benefits b and the expected capital gain from a change of states q(qT)(ET�UT).4 rUT can be interpreted as the annuity (permanent income) that an unemployed workerexpects to receive during search.

Similarly, the asset value of an employed worker’s human capital satisfies the following Bellmanequation

rET ¼ wT þ lðUT � ET Þ: (6)

Eq. (6) has a similar interpretation to (5). The permanent income of an employed worker is made up oftwo components: the wage wT and the expected capital loss from a change of state l(UT� ET).

Combining Eqs. (5) and (6), we can solve for permanent income of an unemployed and an employedworker as follows

rUT ¼ ðr þ gÞbþ qTqðqT ÞwT

r þ lþ qTqðqT Þ; (7)

rET ¼ lbþ ½r þ qTqðqT Þ�wT

r þ lþ qTqðqT Þ: (8)

In contrast to the standard Balassa–Samuelson model with full employment where employed workersmove instantaneously and costlessly across sectors in response to changes in relative sectoral wages, ina model with labor market frictions, it is the searching unemployed workers who move freely acrosssectors. In equilibrium, free mobility of unemployedworkers across sectors ensures that the asset valueof the unemployed workers in the tradable sector (rUT) is equal to that of the unemployed workers inthe nontradable sector (rUN). However, employed workers’ income may be different across sectors assectoral employment income may have to reflect wage differentials in compensating for the risksassociated with each sector, for example, the probabilities of finding a new job and of being fired.

2.5. Wage determination

As an occupied job yields returns that go beyond the sum of the expected returns of a searching firmand a searching worker, the pure economic rent needs to be shared between the firm and theworker. Asimple approach is to assume that the wage is determined by the generalized Nash bargaining solutionwith threat points UT and VT for each job-worker pair, wT

j ˛ argmax[(ETj �UT)b(JTj � VT)1�b], where b˛(0,1) is the worker’s bargaining power.

The solution to the above first-order maximization problem satisfies

E jT � UT ¼ b

�J jT þ E j

T � VT � UT

�; (9)

which says that the worker receives his threat point UT, plus a share of the pure economic rent createdby the job match. Eq. (9) can be solved for the representative worker’s wage in the tradable sector wT.By substituting Eqs. (2) and (6) in (9), and making use of the equilibrium condition VT¼ 0, we have

wT ¼ ð1� bÞrUT þ bðAT f ðkT Þ � rkT Þ: (10)

Substituting Eq. (9) in (5) and making use of JT¼ g/q(qT), we can derive another equation

rUT ¼ bþ b

1� bgqT : (11)

Substituting Eq. (11) in (10), we have

4 The unemployment benefits b can be interpreted more broadly to include the value of leisure and home production, net ofany cost of search. See Rogerson et al. (2005).

Y. Sheng, X. Xu / Journal of International Money and Finance 30 (2011) 587–603 593

wT ¼ ð1� bÞbþ bgqT þ bðAT f ðkT Þ � rkT Þ; (12)

where gqT is the average frictional cost for each unemployed worker. Note that with search frictions inthe labor market, the wage for employed workers is no longer solely determined by the sectoralcapital–labor ratio and productivity. It also depends on the value of employed workers’ outside option(unemployment benefits b), frictional costs and the probabilities of the existing job being destroyedand finding a new job in the future. As such, wages of employed workers wi may be different acrosssectors as they have to take into account compensation differentials arising from labor market frictions.This is in contrast to the standard Balassa–Samuelson model where wages are equalized across sectorsdue to free mobility of workers without frictions.

Finally, since job creation, uTqTqðqT ÞZT , should be equal to job destruction, l(1� uT)ZT in equilib-rium, the steady-state unemployment rate in the tradable sector can be written as:

uT ¼ l

lþ qTqðqT Þ: (13)

Eq. (13) shows that search generated unemployment rate in the tradable sector is positively related tothe job destruction rate (l) but negatively associated with the probability of an unemployed workerencountering a job opportunity (qTq(qT)). It describes a fundamental equilibrium relationship betweenunemployment and vacancy, which is often referred to as the Beveridge Curve. This relationship can beillustrated as a downward sloping locus of unemployment and vacancy combinations in the U–V spacethat are consistent with the steady state inwhich the total flow of workers into unemployment is equalto the total flow of workers out of unemployment (Pissarides, 2000, p. 32).

2.6. Equilibrium

We are now able to characterize the steady-state equilibrium. The equilibrium conditions in thetradable sector consist of firms’ profit maximization conditions with respect to capital and labor, Eqs.(3) and (4) respectively, the equilibrium in wage bargaining Eq. (12), and the labor market equilibriumcondition (13), which are re-written as (Eq. (4a) below is from Eqs. (4) and (10)).

AT f0ðkT Þ ¼ r; (3)

AT�f ðkT Þ � kT f

0ðkT Þ� ¼ rUT þ ðr þ lÞg

ð1� bÞqðqT Þ; (4a)

wT ¼ ð1� bÞbþ bgqT þ bðAT f ðkT Þ � rkT Þ; (12)

uT ¼ l

lþ qTqðqT Þ: (13)

Similarly, the equilibrium conditions for the nontradable sector can be written as follows.

pANg0ðkNÞ ¼ r; ð30Þ

pANðgðkNÞ � kNg0ðkNÞÞ ¼ rUN þ ðr þ lÞg

ð1� bÞqðqNÞ; ð4a0Þ

wN ¼ ð1� bÞbþ bgqN þ bðANgðkNÞ � rkNÞ; ð120Þ

uN ¼ l

lþ qNqðqNÞ; ð130Þ

Y. Sheng, X. Xu / Journal of International Money and Finance 30 (2011) 587–603594

where p is the relative price of nontradable to tradable goods. Both sets of four equations for thetradable and nontradable sectors consist of a recursive system that can be solved easily for the eightunknowns, i.e., ki, qi, wi and ui where i¼ T,N.5

2.7. The generalized Balassa–Samuelson effect

To examine the price effect of anticipated productivity shifts, as in the Balassa–Samuelson model,we take natural logs of both sides of Eq. (4a) and differentiate them, which yields

dAT

ATþ AT f 0ðkT ÞkT

AT f ðkT ÞdkTkT

¼ rkTAT f ðkT Þ

dkTkT

þ 6

AT f ðkT Þd66

þðr þ lÞg

ð1� bÞqðqT ÞdAT

f ðkT ÞAT; (14)

where the frictional cost is assumed to be proportional to workers’ productivity g ¼ ATg.6 We adopt

the convention that the circumflex denotes a logarithmic derivative: bXhdlog X ¼ dX=X for anyvariable X restricted to some positive values.

Let mLT ¼ 6=AT f ðkT Þ and mCT ¼ ðrþlÞg=ð1�bÞqðqT ÞAT f ðkT Þ be labor’s share and the frictional cost share out of

the income generated in the tradable sector respectively. Then Eq. (14) reduces to

ð1� mCT ÞbAT ¼ mLTb6: (15)

Increased productivity in the tradable sector will increase expected income of unemployed workers,similar to the real wage, in the Ballassa–Samuelson model, but the extent of the increase in unem-ployed workers’ expected income has to be adjusted by sectoral labor market efficiency, as definedbelow.

Definition 1. We define ð1� mCiÞ ¼ 1� ðrþlÞg=ð1�bÞqðqiÞAif ðkiÞ (where i¼ T, N) as an indicator of labor market

matching efficiency for sector i. The higher this index, the more efficient a country (sector)’s labormarket is.

There are three labor market variables that are important for determining the degree of labormarket efficiency.7 The first is the frictional cost (unit advertising and recruitment costs) g. The higherthe frictional cost, the less efficient are labor market institutions in facilitating workers’ search for jobsand firms’ hiring of workers. The second and third factors are the job destruction rate and the jobcreation rate. A higher job destruction rate, together with a lower job creation rate, implies a lessefficient labor market institution. We summarize the relationship between these three labor marketvariables and a country’s labor market matching efficiency as follows.

Lemma 1. If the productivity-adjusted recruitment cost of a new vacancy is not equal to zero ðgs0Þ, thelabor market inefficiency increases with respect to the job destruction rate (l) and decreases with respect tothe job creation rate (qiq(qi)).

Proof. From Definition 1, we have vmCi=vli_0 and vmCi=vqiqðqiÞ30 (since vqiqðqiÞ=vqi_0).

5 For example, in the tradable sector, Eq. (3) can be used to solve for kT and then Eqs. (4a), (11) and (12) can jointly be used tosolve for qT and wT. Finally, qT and Eq. (13) are to solve for uT. Similar derivation can be carried out for the nontradable sector. Theresults are then combined with the free flow condition for unemployed workers across sectors rUT ¼ rUN ¼ 6 to obtain thegeneral equilibrium for the economy.

6 The recruitment cost is made proportional to the productivity on the ground that it is more costly to hire more productiveworkers. See Pissarides (2000, Chapter 1). We offer an alternative rationalization to this assumption. Although most searcheconomists believe that it is the search and matching process that is causing labor market friction, we think that such a frictionmay also be affected by some exogenous institutional factors, which is the rationale behind our assumption that recruiting costis a function of exogenous factor, productivity in this context. For example, two countries with the same size and the same laborflow characteristics such as market tightness and search efforts may experience different recruiting cost for each vacancy due todifferent labor market institutional arrangements.

7 These three labor market variables are usually defined by three coefficients, used to quantify matching function in the labormarket and thus the labor market matching efficiency. Estimates of these three coefficients can be found for some developedcountries in the literature, for example Shimer (2005), Genda (1998), Blanchflower and Burgess (1996), Kano and Ohta (2003,2004) etc. See Table 1 for details.

Y. Sheng, X. Xu / Journal of International Money and Finance 30 (2011) 587–603 595

Similarly, the equilibrium condition of (4a0) for the nontradable sector, after log-differentiation, canbe written as follows

bp þ ð1� mCNÞbAN ¼ mLNb6: (16)

Substituting b6 ¼ ð1� mCT ÞbAT=mLT from Eqs. (15) in (16) yields

bp ¼ mLNmLT

ð1� mCT ÞbAT � ð1� mCNÞbAN : (17)

Eq. (17) implies that the relative price of nontradable goods depends on the labor-market-effi-ciency-adjusted productivity differential in the tradable and nontradable sectors. As a country’s priceindex is an average of the prices of tradable and nontradable goods, we thus have the following:

Lemma 2. National price levels are positively related to labor-market-efficiency-adjusted productivity inthe tradable sector and negatively related to labor-market-efficiency-adjusted productivity in the non-tradable sector.

Proof. From Eq. (17), we have vbp=vmCT30 and vbp=vmCN_0. As the home price level is a weightedaverage of the prices of tradable and nontradable goods, the change in the home price level will beproportional to the change of nontradable prices, i.e., vbP=vbp_0. Using the rule of chain, we thus havevbP=vmCT30 and vbP=vmCN_0.

A few notes are in order here. Lemmas 1 and 2 demonstrate that the degree of labor market effi-ciency matters for the relationship between productivity and wages in both the tradable and non-tradable sectors. When workers have to cycle between employment and unemployment and it isunemployed workers who move across sectors to equalize their expected income, the increase inemployed workers’ wages due to productivity shocks may be less than what the standard Balassa–Samuelson model predicts as employed workers only share part of the benefit from the productivityincrease (since unemployed workers also have a share). More specifically, there are two additionalforces that can affect the changes in the national price level that arise from a productivity shock in thetradable sector. On one hand, the degree of labor market matching inefficiency in the tradable sectorweakens the response of workers’ wages to productivity shocks, which tends to operate against thestandard Balassa–Samuelson effect (see Eq. (15)). On the other hand, labor market matching ineffi-ciency in the nontradable sector weakens the response of the price (or workers’wages) to productivityshocks, which tends to strengthen the standard Balassa–Samuelson effect (See Eq. (16)). Consequently,differences in the degree of labor market matching efficiency across sectors and countries may lead todeviations from the standard Balassa–Samuelson effect.

When there is no cost associated with recruiting workers in the labor market (i.e., full labor marketmatching efficiency), we have mCT¼ mCN¼ 0, so that Eq. (17) simplifies to

bp ¼ mLNmLT

bAT � bAN : (18)

Eq. (18) is the original Balassa–Samuelson formulation of the price effects of anticipated productivityshifts. The relative price of nontradable goods depends on the productivity differential between thetradable and nontradable sectors. Provided the inequality mLN/mLT� 1 holds, faster productivity growthin the tradable sector will push up the price of nontradable goods over time.

Let a star in the superscript of a variable denote foreign country variables. It is easy to show that theprice of nontradable goods in the foreign country is as follows

bp� ¼ m�LNm�LT

�1� m�CT

�bA�T � �1� m�CN

�bA�N : (19)

We define a country’s price index P (P*) as the geometric mean of the prices of tradable and non-tradable goods, with weights s and 1� s. Following Obstfeld and Rogoff (1996, p. 211), we assume thattradables have a uniform price in each of the two countries. Since we take tradables as the numeraire,with a common price of 1 in both countries, the home-to-foreign price ratio is simply proportional tothe ratio of the internal relative prices of nontradable goods:

Y. Sheng, X. Xu / Journal of International Money and Finance 30 (2011) 587–603596

bP � bP� ¼ ð1� sÞ�bp � bp��

¼ ð1� sÞ�mLNmLT

hð1� mCT ÞbAT � �1� m�CT

�bA�T

i�hð1� mCNÞbAN � �1� m�CN

�bA�N

i�: (20)

Provided that nontradable goods are at least as labor-intensive as tradable goods so that mLN/mLT� 1,the home country will experience real appreciation (a rise in its relative price level) if its labor-market-efficiency-adjusted productivity advantage in the production of tradables exceeds its labor-market-efficiency-adjusted productivity advantage in the production of nontradables. This can be summarizedin the following proposition:

Proposition 1. The greater a home country’s labor-market-efficiency-adjusted productivity advantage isin the production of tradable goods than labor-market-efficiency-adjusted productivity advantage in theproduction of nontradables, the larger will be a home country’s real exchange rate appreciation.

Proof. From Eq. (20), we have vðbP � bP�Þ=v½ð1� mCT ÞbAT � ð1� m�CT ÞbA�T �_0 and vðbP � bP�Þ=

v½ð1� mCNÞbAN � ð1� m�CNÞbA�N �30.

Proposition 1 highlights two conditions that have to be satisfied for a country to experience a realexchange rate appreciation (assuming that the tradable sector is capital-intensive): (1) there is a higherproductivity growth in the tradable sector; (2) with frictional costs in the labor market ðgs0Þ, relativefrictional costs in the tradable sector should not be too high. In contrast, the classic Balassa–Samuelsonmodel requires only satisfaction of the first condition to experience a real exchange rate appreciation.

Again, if there is no cost associated with recruiting workers in the labor market, we havemCT¼ mCN¼ 0, so that Eq. (20) reduces to

bP � bP� ¼ ð1� sÞ�bp � bp�� ¼ ð1� sÞ

�mLNmLT

hbAT � bA�T

i�hbAN � bA�

N

i�; (21)

which is the full employment version of the Balassa–Samuelson model.To further appreciate Proposition 1, we provide the following three corollaries based on Eq. (21).

Corollary 1. Even though there is unemployment in the labor market, as long as it takes no frictional costto fill a new vacancy ðg ¼ 0Þ, a country’s real exchange rate will appreciate over time if and only if thetradable sector has faster productivity growth.

The result of Corollary 1 is the same as that of the full employment version of the Balassa–Samuelsonmodel, though it is cast in an environment with frictional unemployment. It suggests that the impact oflabormarket flexibility on the real exchange rate is not because of frictions in labormarkets themselvesbut because there are related hiring and firing costs to firms caused by frictions in labor markets.

Furthermore, suppose both countries experience unbiased technological progress but the homecountry has a faster productivity growth, i.e., bAT ¼ bAN and bA�

T ¼ bA�N and bAT and bAN are higher than bA�

Tand bA�

N , the home country may not experience real exchange rate appreciation if the labor market inthe tradable sector is seriously distorted, as ð1� mCT ÞbAT may be smaller than ð1� m�CT ÞbA�

T , as suggestedby Eq. (20). This gives:

Corollary 2. If a home country experiences a faster rate of productivity growth but has a much lower levelof labor market efficiency in the tradable sector, the classic positive relationship between real exchange rateand sectoral productivity may be reversed, i.e., the home country’s real exchange rate may not experiencereal exchange rate appreciation over time.

Proof. Since bAT ¼ bAN ¼ bA and bA�T ¼ bA�

N ¼ bA�, and bA are higher than bA�

, Eq. (20) reduces tobP � bP� ¼ ð1� sÞfððmLN=mLT Þ � 1Þ½ð1� mCT ÞbA � ð1� m�CT ÞbA��g:ðbP � bP�Þ may not be positive ifð1� mCT ÞbAT � ð1� m�CT ÞbA�

T .

The next corollary focuses on the role of sectoral labor market efficiency in determining realexchange rate.

Corollary 3. If both countries experience the same rate of productivity growth and achieve the same levelof labor market efficiency in their tradable sectors, the real exchange rate between two countries depends

Table 1Parameters for labor market flexibility variable: U.S., Japan and UK.

Parameter U.S. Japan UK

Unit recruiting cost ðg ¼ g=Af ðkÞÞ 0.05 0.35 0.50Job destruction rate (l) 0.10a 0.04b 0.07c

Interest rate (r) One year discount rate

Federal Reserve Bank of Japan Bank of EnglandMatching functionMatching flexibility a 0.13f 0.08f 0.09f

Elasticity of the matching function w.r.t. unemployment (a) 0.72a 0.69d 0.71e

Labor market matching efficiency (1� mCi) 0.886g 0.301g 0.718g

a Shimer (2005).b Genda (1998).c Blanchflower and Burgess (1996).d Kano and Ohta (2003, 2004).e Pissarides (1986).f Authors’ estimate. The procedures that we used to estimate job matching efficiencies for the US, Japan and UK are as follows.

With data on numbers of vacancies, unemployed workers and job matching for each period (m(Uit,Vit)), we estimate annual jobmatching flexibility for each country at from the equation at¼m(Uit,Vit)/Ut

aVt1�a. We then take the average of at as the job

matching flexibility for each country. For the period 1979–1996, our estimate of job matching flexibility for US is 0.13 and forJapan 0.08. However, since the UK data on job matching between vacancies and unemployed workers are not available for theperiod under study, we use estimates from the latest labor market data as a proxy (see http://www.econstats.com/uk/uk_unem_14m.htm for the latest data on job vacancies, job replacement andmatching rates). We estimate that the averagematching rate q(qt) in 2005 is about 18.5 percent, which implies a job matching flexibility of 0.09.

g Estimates of labor market matching efficiency follow Definition 1 and are based on estimates on other labor marketparameters in this table. They are averages over the sample period.

Y. Sheng, X. Xu / Journal of International Money and Finance 30 (2011) 587–603 597

not only on their relative rate of productivity growth but also their relative labor market efficiency in theirnontradable sectors.

Proof. Since bAT ¼ bA�T and mCT ¼ m�CT , we have ð1� mCT ÞbAT ¼ ð1� m�CT ÞbA�

T . Eq. (19) becomesbP � bP� ¼ �ð1� sÞfð1� mCNÞbAN � ð1� m�CNÞbA�Ng.

3. Empirical evidence

In this section, we show regression results of the revised Balassa–Samuelson effect in the presenceof labor market frictions. The model we lay out in the previous section suggests that in estimating theBalassa–Samuelson effect, the productivity measure should be adjusted for labor market efficiency.Ideally, we should follow equation (20) to distinguish the labor market efficiencies between thetradable and nontradable sectors. However, due to data availability, we have to restrict our focus tocross-country differences in labor market efficiency while assuming that both tradable and non-tradable sectors in the same country have the same labor market efficiency. Under this assumption,countries with faster labor-market-efficiency-adjusted productivity will experience larger realexchange rate appreciations (from Corollary 1). We should expect the coefficient of labor-market-efficiency-adjusted productivity to be positive and to provide more accurate measurement of theBalassa–Samuelson effect than the equivalent regression coefficient of productivity without adjustingfor labor market efficiency.

Our empirical investigation is dictated by the availability of data for labor market efficiency. Twoexercises have been carried out, although neither of them can perfectly capture the complexity of ourtheoretical predictions. The construction of our first dataset follows closely with what our modelsuggests in measuring labor market matching efficiency, but it comes with a cost as data on searchcoefficients such as the job destruction rate, labor market matching efficiency and elasticity of thematching function with respect to unemployment are scarce. We are therefore restricted to focus onannual frequency for only three countries, namely, the United States, the United Kingdom and Japanwhere relatively high quality data are available. The real exchange rates between the US dollar andJapanese Yen and between the US dollar and UK pound are defined as the nominal exchange rate

Y. Sheng, X. Xu / Journal of International Money and Finance 30 (2011) 587–603598

multiplied by the ratio of CPI in each country. These data are available from Charles Engel’s personalwebsite (http://www.ssc.wisc.edu/wcengel/data.htm). Data on labor productivity, defined as annualGDP per hour worked for the U.S., Japan and UK, is collected from the OECD’s country statistical profiledatabase (available at http://stats.oecd.org/index.aspx). We summarize our collected and estimatedlabor market matching efficiency data in Table 1.

With raw data for three countries over the period 1979 and 1995, we construct variables for twocountries, Japan and the UK. We consider the following two empirical specifications:

ln RERiUS;t ¼ b0 þ b1 ln

yi;tyUS;t

!þ hi;t ; (M1)

ln RERiUS;t ¼ b0 þ b1 ln

y0i;ty0US;t

!þ hi;t ; (M2)

where RERiUS,t is the real exchange rate between country i (Japan or the UK) and the United States attime t, yi,t and yUS,t are the productivities of country i and the US at time t not adjusted for labor marketefficiency; y0i;t ¼ ðyi;tÞð1�mi;tÞ and y0US;t ¼ ðyUS;tÞð1�mUS;tÞ are the productivities of country i and the US attime t adjusted for labor market efficiency; labor market matching efficiency for country i is repre-sented by (1� mi,t) while for the US is (1� mUS,t), as defined in Section 2. In both (M1) and (M2), the USis the base country. The difference between (M1) and (M2) is that in the latter the productivity isadjusted by a country’s labor market efficiency.

Table 2 presents OLS regression results based on (M1) and (M2) for the Japan–US and the UK–USseries and also for the pooled sample. Columns (1), (3) and (5) of Table 2 report results withproductivity not adjusting for labor market efficiency, while columns (2), (4) and (6) present thecorresponding results with labor-market-efficiency-adjusted productivity for the Japan–US, UK–USand the pooled sample. The coefficients for productivity (with or without labor market efficiencyadjustment) are all positive and statistically significant at the one percent level. For the Japan–US case,our estimated Balassa–Samuelson effect is 0.536 for the model without adjusting for labor marketefficiency (M1) and 0.578 for the model that adjusts for labor market efficiency (M2). For the UK–UScase, our estimated Balassa–Samuelson effect is 0.920 for the model without adjusting for labor marketefficiency (M1). Using labor-market-efficiency-adjusted productivity, the coefficient is 0.983. For thepooled sample, the estimated Balassa–Samuelson effect is 0.576 for the model without adjusting labormarket matching efficiency and 0.612 for the model that adjusts for labor market matching efficiency.These elasticity estimates are comparable with those of Bergin et al. (2006) from cross-sectionalregression estimation (their estimated elasticity with respect to productivity is 0.41 in a cross section of

Table 2Regression results for (M1) and (M2).

Japan–US UK–US Pooled sample

(1) (2) (3) (4) (5) (6)

ln(yi/yUS) 0.536*** 0.920*** 0.576***(0.039) (0.122) (0.039)

ln(yi0/yUS

0) 0.578*** 0.983*** 0.612***

(0.041) (0.084) (0.044)Country dummies 5.265*** 5.149***

(0.026) (0.034)Constant �4.729*** �4.632*** 0.588*** 0.637*** �4.708*** �4.610***

(0.017) (0.023) (0.027) (0.025) (0.019) (0.030)Observations 17 17 17 17 34 34R2 0.917 0.938 0.733 0.850 0.999 0.999

Note: Dependent variable is the logarithm of the real exchange rate of country i (Japan or UK) with the United State as the basecountry. ***, ** and * represent significant at 1%, 5% and 10% level. Standard errors are in parentheses. The results are from OLSregressions.

Table 3Cross-section regression results for (M3) and (M4): 2004.

(1) (2)

ln(yi/yUK) 0.396*** 0.486***(0.032) (0.055)

Dummy �0.254*(0.146)

Dummy* ln(yi/yUK) �0.136**(0.068)

Constant 0.095 0.066(0.071) (0.116)

Observations 132 132R2 0.539 0.546

Notes: Dependent variable is the logarithm of the real exchange rate of country i with the UK asthe base country. Labor market efficiency dummy variable, Dummy, equals to 1 if a country hasa higher hiring and firing cost than that of the United Kingdom which is 1 by default. ***, ** and *represent significant at 1%, 5% and 10% level. Standard errors are in parentheses. The results arefrom OLS regressions.

Y. Sheng, X. Xu / Journal of International Money and Finance 30 (2011) 587–603 599

142 countries for the year 1995). Overall, the regressions of the real exchange rate on labor-market-efficiency-adjusted productivity perform somewhat better than those of the real exchange rate onunadjusted productivity, with R2 in (M2) somewhat higher than that in (M1). It lends support to ourprediction that in the presence of labor market frictions, productivity adjusted for labor market effi-ciency provides better estimates of the Balassa–Samuelson effect.

With data for only three countries, one might argue that the above results are of limited value. Wetherefore construct our second dataset which uses labor hiring and firing costs to proxy for the degreeof labor market matching efficiency. We make use of a newly available dataset published in DoingBusiness in 2006 by the World Bank (2006), which is also used by Helpman and Itskhoki (2010).Although data for labor hiring and firing costs are available for one year (2004) only in the World Bankdataset, it covers more than one hundred countries. More importantly, the dataset indicate that thereare vast differences in labor market efficiency across countries. For example, the sum of hiring andfiring costs for the US is 8 (weeks of salary) while that for the UK and China is 74 and 203 (weeks ofsalary) respectively. Data for other variables such as price and productivity (using GDP per capita asa proxy) are taken from Penn World Table (PWT) database.8 The PWT gives both income and pricevariables in terms of the US that are comparable across space. We perform a cross-sectional regressionanalysis of real exchange rates on productivity with and without adjustment for the degree of labormarket efficiency (M4).

Specifically, we perform two regressions, one without accounting for the effect of labor marketmatching efficiency (M3) and the other controlling for the degree of labor market efficiency (M4).

ln�

PiPUK

¼ b0 þ b1 ln

�yiyUK

þ hij; (M3)

ln�

PiPUK

¼ b0 þ b1 ln

�yiyUK

þ b2Dummy � ln

�yiyUK

þ b3Dummyþ hij: (M4)

Our dependent variable, ln(Pi/PUK), is the logarithm of country i’s real exchange rate against the UK.9 ln(yi/yUK) denotes the logarithm of country i’s GDP per capita relative to the UK. The labor market effi-ciency dummy variable, Dummy, equals 1 if a country has higher hiring and firing costs than those ofthe UK which is 1 by default. We also include an interaction between the labor market efficiency

8 The data is available at http://pwt.econ.upenn.edu/.9 The results are the same if the US is taken as the base country.

Y. Sheng, X. Xu / Journal of International Money and Finance 30 (2011) 587–603600

dummy variable and the productivity variable to control for the effect of differences in labor marketinstitutions across countries. We expect the coefficients of both the Dummy and the interaction term tobe negative. That is, the higher the labor market frictions (hiring and firing costs), the smaller theBalassa–Samuelson effects, other things being equal.

Table 3 reports the empirical results of the cross-sectional regressions. Without adjusting for labormarket firing and hiring costs, the Balassa–Samuelson effect is about 0.396 and is statistically signif-icant at the 1 percent level (column (1)). Again, these elasticity estimates are comparable with those ofBergin et al. (2006). The labor market efficiency dummy variable, proxied by the relative firing andhiring costs, is negative and statistically significant at the 10 percent level. The coefficient of the slopedummy is negative and statistically significant at the 5 percent level. For countries with higher labormarket hiring and firing costs (Dummy¼ 1), the Balassa–Samuelson effect is much smaller (about 0.350which is the difference between the coefficient of the productivity variable, 0.486, and that of theinteraction term, 0.136) than that obtained without adjusting for labor market frictions (0.396). Forcountries with lower labor market hiring and firing costs (Dummy¼ 0), the Balassa–Samuelson effect ismuch larger (0.486, which is higher than 0.396). These results suggest that, other things equal, thehigher the firing and hiring costs, the smaller the Balassa–Samuelson effect. Again, the evidence fromcross-sectional regressions lends support to the central notion of our paper that the relationshipbetween real exchange rate and productivity depends on labor market efficiency.

4. Conclusion

It is well known that there exist significant differences in labor market institutions across countries.Yet it is not clear how these differences in labor market efficiency affect the relationship between thereal exchange rate and productivity. This paper extends the classic Balassa–Samuelson model to anenvironment with search unemployment.We show that there is an important role for the labor marketinstitutional environment to play in determining the magnitude of the effects of sectoral productivitydifferentials on the real exchange rate. In particular, there is the potential that the standard Balassa–Samuelson effect may not hold. Accounting for a country’s and a sector’s degree of labor marketmatching frictions, the relationship between the real exchange rate and sectoral productivity may bemore complex than that predicted by the Balassa–Samuelson models.

To test our prediction that the relationship between the real exchange rate and productivitydepends on labor market frictions, we construct two datasets and perform regressions that includelabor market friction variables. Our empirical evidence supports our proposition that the degree oflabor market frictions is important in explaining the impact of productivity on the real exchange rate.

Although our model is derived at the steady state, the derivation of the model out-of-steady-state isprovided in the Appendix. The result from the out-of-steady-state model is that the short-run Balassa–Samuelson relationship may deviate from its long-run relationship, taking into account labor marketfrictions across sectors or countries. An important insight from our analysis is that models that ignorelabor market frictions may be inadequate in explaining the Balassa–Samuelson effect.

Acknowledgement

We are grateful to the editor, James Lothian, and two anonymous referees for very detailed andinsightful comments and suggestions which improve the paper substantially. We thank Jean Imbs,Peter Drysdale, Martin Richardson, Guillaume Rocheteau, Jane Golley, Dong He and Ligang Song forcomments and suggestions. Financial support from the Hong Kong Polytechnic University (G-U685) isgratefully acknowledged. Part of the research on this paper was conducted while Xinpeng Xu wasvisiting the Hong Kong Institute for Monetary Research and he thanks them for their hospitality andsupport. The views expressed in this paper are those of the authors and do not necessarily representviews and opinions of the HKIMR.

AppendixTo derive the dynamic equations for wages and market tightness for the two sector model with

search unemployment, we need to specify the expected returns of firms and workers out-of-steady-

Y. Sheng, X. Xu / Journal of International Money and Finance 30 (2011) 587–603 601

state. Following Pissarides (2000), the net worth of jobs and workers is an explicit function of time. Thearbitrage equations determining their value are similar to the ones that hold at the steady state, exceptthat there may be capital gains or losses from changes in the valuation placed by the market on jobsand workers. To conserve space, we use the tradable sector in home country as an example. Theanalysis can easily be extended to the nontradable sector and foreign countries.

Starting with the same model in Section 2, we again assume that VT denotes the asset value ofa vacant job in the tradable sector of home country. With a perfect capital market and perfect foresight,Eq. (1) can be re-written as:

rVT ¼ �gþ _VT þ qðqT ÞðJT � VT Þ: (A1)

The only difference between (1) and (A1) is that the (A1) takes into account expected capital gainsfrom changes in the valuation of the asset during adjustment ð _VT Þ.

Similarly, the value of a filled job, JT, satisfies the arbitrage condition. For simplicity, suppose that thevalue of physical capital does not change. Equation (2) can be re-arranged as:

rðJT þ kT Þ ¼ AT f ðkT Þ �wT þ _JT � lJT ; (A2)where _JT is the expected capital gain from changes in job value during adjustment.

The assumption that firms exploit all profit opportunities from new jobs, regardless of whether theyare in the steady state or out of it, implies that VT ¼ _VT ¼ 0. Thus the job creation process is deter-mined by two equations:

JT ¼ g=qðqT Þ; (A3)

_JT ¼ ðr þ lÞJT � ½AT f ðkT Þ � rkT �wT �: (A4)

The asset value of employed and unemployed workers can also be given by two arbitrage equationsthat are similar to (5) and (6) but with the changes in valuation out-of-steady-state.

rUT ¼ bþ _UT þ qTqðqT ÞðET � UT Þ (A5)

and

rET ¼ wT þ _ET þ lðUT � ET Þ: (A6)

All differential equations for asset values are unstable because of arbitrage and perfect foresight.We assume that wages are determined by the Nash solution to the bargaining problem, which as

before implies the sharing rule in (9). This condition ensures that (10) holds both in and out-of-steady-state. Since there is no lag in the wage bargaining process, the out-of-steady-state dynamics of wagesare driven entirely by the dynamics of labor-market tightness (15).

We are now ready to describe the dynamics of wages and tightness. From (A5) and (A6), thedynamics of unemployment can be given by the difference between job destruction:

_uT ¼ lð1� uT Þ � qTqðqT ÞuT : (A7)

From (A3), we have _JT ¼ �g=½qðqT Þ�2q0ðqT Þ _qT (cq0(qT)< 0) which is a monotonically increasing func-tion of _qT . Substituting (A3) and (12) into (A4), we have

�g=½qðqT Þ�2q0ðqT Þ _qT ¼ ðr þ lÞg=qðqT Þ þ bgqT þ ð1� bÞ½AT f ðkT Þ � rkT þ b� (A8)

which is an unstable equation of qT with no other unknowns in it.The sign pattern of a first-order linear approximation to the two differential equations (A7) and (A8)

is: �_uT_qT

¼�� �0 þ

�uTqT

: (A9)

Fig. 1. Phase diagram of unemployment and market tightness.

Y. Sheng, X. Xu / Journal of International Money and Finance 30 (2011) 587–603602

Since the determinant of differential equations (A9) is negative, we have both the necessary andsufficient conditions for a saddle-point equilibrium to be satisfied. The reason for this saddle-pointequilibrium is that unemployment is sticky and stable, while vacancy is forward-looking and unstable(Pissarides, 2000; p. 29). Thus, the dynamics of qT and uT in the neighbourhood of saddle-point equi-librium follows perfect-foresight paths which is determined by the initial condition of qT. If the initialcondition is such that qT ¼ qT where _qT ¼ 0, the dynamic path is qT-stationary and thus qT is constantand unemployment adjusts until there is convergence to equilibrium. However, if initially qTsqT , thedynamic path is divergent with qT becoming larger and larger as qT > qT and smaller and smaller asqT < qT .

Using the dynamics in Fig.1 to explain the inconsistency between the short-run and long-runwage-productivity relationship (and thus the Balassa–Samuelson effect) is straightforward. As is shown in(15), mCT is an increasing function of qT. If qT ¼ qT , the short-run relationship between wage and sec-toral productivity is consistent with that in the long run since the dynamic path is qT-stationary.However, if qTsqT , the short-run relationship betweenwage and sectoral productivity is not consistentwith that in the long run. In particular, when vacancies overshoot ðqT > qT Þ following productivityshocks, the labor market matching efficiency in the short runwould be lower than that in the long run.Thus, the impact of productivity shocks in the tradable sector on price would be weakened. Extendingthis mechanism to the nontradable sector and other countries, we have the result that the Balassa–Samuelson effect is less stable in the short run than in the long run.

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