Real Exchange Rates and Productivity
Closed-Form Solutions and Some Empirical Evidence
Author: Jahanara Begum

Contributor Notes

Author’s E-Mail Address: Jbegum@imf.org

This paper examines the impact of productivity shocks on real exchange rate fluctuations in a dynamic international general equilibrium model with nontraded goods. The model predicts a close association between relative technology shocks and bilateral real exchange rate movements. Empirical results based on the data for Group of Seven countries are consistent with the predicted theoretical correlations. Using Johansen and Juselius (1990) multivariate cointegration tests the study finds that a statistically significant relationship exists between bilateral real exchange rates and international productivity differentials in the traded and nontraded sectors.

Abstract

This paper examines the impact of productivity shocks on real exchange rate fluctuations in a dynamic international general equilibrium model with nontraded goods. The model predicts a close association between relative technology shocks and bilateral real exchange rate movements. Empirical results based on the data for Group of Seven countries are consistent with the predicted theoretical correlations. Using Johansen and Juselius (1990) multivariate cointegration tests the study finds that a statistically significant relationship exists between bilateral real exchange rates and international productivity differentials in the traded and nontraded sectors.

I. Introduction

Deviations from purchasing power parity (PPP)2 and real exchange rate movements have been a major focus of empirical research since the return to floating exchange rates by major currencies in the early 1970s. A substantial body of evidence (for example, Isard (1977), Roll and Solnik (1979), Frankel (1981), and Mussa (1986)) indicates that nominal price indexes and foreign exchange rates display large – and permanent – departures from PPP Several explanations are put forward for the observed volatility of real exchange rates. One explanation for this phenomenon is the evidence of exogenous shocks to the world economy. To explain exchange rate movements under the float, many economists resort to models in which real shocks play the dominant role. Some of the important channels that allow real shocks to affect exchange rate movements are sectoral relative prices (i.e., relative price of nontraded to traded goods), technology shocks, changes in taste, and fiscal policy shocks.

The motivation for this study comes from the empirical evidence that large movements in real exchange rates and the failure of PPP to hold in the 1970s can be explained by the predominance of real shocks. A number of recent empirical studies (e.g., Dutton and Strauss (1997), Mark and Choi (1997), Bahmani and Rhee (1996) and Strauss (1996)) find evidence of a significant relationship between the real exchange rate and real variables (e.g., productivity differentials between domestic and foreign economy in the traded and nontraded sectors, relative price of nontraded goods, and taste shocks). Other empirical studies (Amano and Van Norden (1992), Lastrapes (1992), and Manzur (1991)) also find support for real shocks causing exchange rate movements. A number of theoretical models (Balvers and Bergstrand (1997), Kollmann (1995), Backus and Smith (1993), Cardia (1991), and Stulz (1987)) also establish the link between real factors and exchange rate movements. For example, the recent work by Balvers and Bergstrand (1997) presented closed-form solutions of real exchange rate in terms of relative productivity of non-tradeables and the rate of time preference. Backus and Smith (1993) used a dynamic exchange economy with nontraded goods to account for deviation from PPP and to explain real exchange rate movements.

A goal of this paper is to examine the role of relative productivity shocks in explaining real exchange rate movements in a two-country dynamic, general equilibrium framework with nontraded goods. Compared to previous studies, in particular those which focus on an exchange economy (e.g., Balvers and Bergstrand (1997) and Backus and Smith (1993)), there are two major contributions of the present study First, it incorporates production technologies, which depend on both labor and physical capital. This is an important extension because the introduction of technological processes allows the adjustments in both consumption and output, whereas in exchange economies without production, the level of output is exogenous.

Secondly, the model can predict explicitly the impact of technology shocks on real exchange rate movements.

While the existing literature suggests a number of real variables may have significant effects on the real exchange rate, there are a number of reasons for focusing on relative productivity differentials. First, total factor productivity (measured in terms of the Solow residual) is an important determinant of the long-term economic growth and prosperity of a country. Second, productivity shocks are exogenous variables that may play a key role in explaining movements in the real exchange rates. In recent years, the extension of real business cycle models (RBC) to the open economy has stimulated interest in technology shock effects.3 Third, productivity differentials between domestic and foreign economy have long been emphasized as a significant determinant of real exchange rate movements (Balassa (1964)).

To study the relationship between sectoral productivity differentials (or ratios) (between home and foreign country) and the real exchange rate we rely on a two-country, two-sector model with nontraded goods. The presence of nontraded goods allows for permanent deviations from aggregate PPP4 Assuming logarithmic preferences, full depreciation of capital, and some restrictions on the technology shock process, closed-form theoretical solutions are derived for optimal policy functions for consumption, investment, and employment. The main reason for focusing on analytical solutions is that the analytical method5 is capable of generating explicit expressions for equilibrium real exchange rates which are easy to interpret and allow empirical testing of the model. Numerical solution techniques (which provide quantitative predictions of the model using a set of parameter values) are unable to produce such simple and interpretable expressions. However, analytical method may require some restrictive assumptions (e.g., full depreciation of capital and a stationary shock process) and thereby may cause loss of many of the interesting cross country dynamics.

The model predicts a close link between bilateral real exchange rates and productivity differentials (ratios) (defined as domestic productivity in traded (nontraded) goods relative to foreign productivity in traded (nontraded) goods) in the traded and nontraded sectors. In particular, the theory suggests a positive relationship between the real exchange rate and productivity ratio in the nontraded sector and conversely, a negative relationship with the corresponding ratio in the traded sector. The empirical results, using time series data for the G-7 countries, are consistent with the predicted theoretical signs. The cointegration tests suggest that a statistically significant cointegrating relationship exists between domestic and foreign productivity differentials in each sector and the real exchange rate. We conclude that relative productivity shocks are an important determinant of real exchange behavior.

The remaining sections of the paper are organized as follows. Section II presents a brief review of related literature. Section III provides a two-country dynamic general equilibrium model with nontraded goods. Quantitative implications of the exchange rate dynamics and effects of relative productivity shocks are examined in Section IV Finally, some concluding remarks are made in section V.

II. Literature Review

There have been numerous studies, both theoretical and empirical, attempting to examine what accounts for real exchange rate fluctuations. This section briefly reviews some recent literature on fundamental determinants of real exchange rate. The analysis is divided into two parts. First, it focuses on the theoretical models of exchange rate determination which predict a close relationship between real exchange rate and real factors, i.e., terms of trade, relative price of nontraded goods, productivity differentials, and the like. Second, the study critically analyzes some recent empirical research devoted to quantifying the real exchange rate dynamics.

A. Theoretical Predictions

PPP is the foundation of many long-run exchange rate models and has a long history of theoretical support. Some commonly cited causes for PPP violations and permanent movements in real exchange rates are differences in productivity between traded and nontraded sectors. Balassa (1964) and Samuelson (1964) first identified productivity differentials between two countries as a major factor contributing to the deviation of the PPP-based exchange rate from the equilibrium exchange rate.6 They argued that when productivity differentials are greater in the production of traded goods between two countries, the differences in wages and prices will also be larger, which will lead to a larger gap between PPP and the equilibrium exchange rate. This has come under the heading of the “productivity-bias hypothesis” in the PPP theory.

Recent work of Balvers and Bergstrand (1997) generated closed-form theoretical solutions for real exchange rates (in terms of relative non-tradeable productivity, taste shocks and rates of time preference) in a two-country stochastic dynamic general equilibrium model. Their solutions are consistent with equilibrium exchange rate theories and the productivity-differentials models of Balassa (1964) and Samuelson (1964). The empirical studies of this group of models using classical estimation techniques are rare, as noted in Taylor (1995). Backus and Smith (1993) use the moments of growth rates of relative consumption levels and of real exchange rates to examine the implications from a dynamic, stochastic, two-country equilibrium model. Their theoretical model suggests a positive relationship between relative consumption (between the home and foreign) of nontraded goods and bilateral real exchange rates. Empirical evidence for eight OECD countries, however, finds little support for this pattern. Kollmann(1995) examines the relationship among per capita consumptions and real exchange rates implied by a similar model, but in the absence of non-tradable goods.

Over the past decade and a half, several equilibrium models of exchange rate determination have explained the movements in real exchange rates and PPP departures in terms of relative productivity shocks, e.g., Stockman (1980 and 1987), Lucas (1982), Helpman and Razin (1982), Stulz (1987), Stockman and Svensson (1987), and Stockman and Delias (1989). In optimizing frameworks, most of these models employ the “perfect pooling equilibrium” attributable to Lucas (1982). Stockman (1987) asserts that exchange rate volatility reflects the greater variance of real shocks in the flexible exchange rate era. Some open economy RBC models (e.g., Backus,Kehoe, and Kydland (1994), Stockman and Tesar (1995), and Costello and Pranschik (1993) and Mendoza (1995)) attempted to capture the impact of real shocks on exchange rate movements. Empirical support for these models, however, has been very limited.

B. Empirical Studies

A number of recent empirical studies find evidence of a significant relationship between the real exchange rate and its fundamental determinants, including terms of trade, relative price of nontraded goods, productivity differentials etc. For example, Balvers and Bergstrand (1997), Bahmani and Rhee (1996), and Strauss (1996) have provided unanimous support to the productivity-differentials model of Balassa (1964) and Samuelson (1964). Using Johansen conintegration tests, they all find that a statistically significant cointegrating relationship exists between domestic and foreign productivity differentials and the real exchange rate. Dutton and Strauss (1997) and Strauss (1995) presented additional evidence that changes in the relative price of non-tradeable and productivity differentials between economies are significant in explaining real exchange rate movements in the short-run and in the long-run, and thus explaining the source of violations of PPP All these studies use OECD data and, generally speaking, they all focus on the existence of relationships between productivity differentials and real exchange rate, rather than on the explicit pattern of correlations over time. Chinn (1997) documented the evidence in support of a stable, long-run relationship between the real dollar/yen exchange rate, productivity and government spending differentials, and the real price of oil. The study finds limited support for sectoral productivity differentials due to the high multicollinearity between the two productivity variables. Using monthly data for some OECD countries, Mark and Choi (1997) reported that differentials in productivity, real interest rates, and per capital income display some predictive power in explaining the deviation of the real exchange rate from its long-run equilibrium value. Stein (1995) provided evidence that the fundamental determinants of the evolution of the medium-to-longer-run of the real effective exchange rate of the United States relative to the rest of the G-7 countries are productivity and thrift in the sample countries.

Another strand of literature has focused on the terms-of-trade shocks to explain fluctuations in real exchange rates and deviations from PPP since the collapse of the Bretton Woods system. Mendoza (1995) provided evidence for G-7 countries and a group of developing countries that terms-of-trade shocks account for nearly half of actual GDP variability, and cause real appreciations and positive interest differentials. De-Gregorio and Wolf (1994) present empirical evidence for a sample of 14 OECD countries that faster productivity growth in the tradable relative to the nontradeable sector and an improvement in the terms of trade induce a real appreciation. Amano and Van Norden (1992) also reported that terms of trade shocks can potentially explain much of the variations in the Canadian-US. exchange rate over the past 19 years. Decomposing the overall terms-of-trade index into two components: the price of exported energy and the price of exported non-energy commodities (each divided by the price of imported manufactured goods) Amano and Van Norden find that changes in energy prices played a critical role in predicting changes in real exchange rates. For example, they find that the cumulative effect of the rise in energy prices from early 1970 to mid-1981 was to move the real exchange rate by 35 percent. Other empirical studies (Lastrapes (1992); Manzur (1991)) also find support for real shocks causing exchange rate movements. Comparing the relative importance of monetary and real sources of fluctuations in exchange rates over the current flexible rate period, they find that much of the long-run movement in real exchange rates is the result of real disturbances.

In light of the theoretical predictions and the empirical evidence of a close association between the real exchange rate and real variables, this seems an important exercise to explore further the impact of relative productivity shocks on real exchange rate fluctuations. Compared to earlier studies, the closed-form theoretical solutions in this paper for equilibrium real exchange rates (in terms of the parameters of preference and technology, and the sectoral productivity ratios) are easy to interpret and are potentially estimable.

III. A Two-Country Equilibrium Model with Nontraded Goods

In the following section we first describe the model and then derive the intertemporal equilibria for logarithmic preference. Estimable closed form solutions are obtained under some restrictive assumption regarding the technology shock process.

This is a simple two-country, two-sector equilibrium real business cycle model that is capable of being solved for analytical decision rules. The model is formulated in discrete time with an infinite horizon. In the world economy, each country produces the same internationally traded good and one nontraded good for domestic consumption and investment in each period. The countries have identical preferences and follow identical production techniques, but each country’s techniques are subjected to country-specific productivity shocks. Let the countries be labeled as “home” and “foreign” and denote all foreign variables with an asterisk.

The preferences of a representative household in home country are defined by a logarithmic time separable utility function,

U=t=0βtEt[γlog(CtT)+(1γ)log(CtN)+μlog(Lt)](1)

where CtT is consumption of the traded good, CtN is consumption of the nontraded good, Lt is leisure; β ∈ (0,1) is the subjective discount factor, γ(1 − γ) is the share of traded good(nontraded good) in the aggregate consumption bundle, μ is the weight given to leisure in the utility function; Et is the expectation operator.

Similarly, the preferences of a representative agent in the foreign country is defined as

U*=t=0βtEt[γlog(CtT*)+(1γ)log(CtN*)+μlog(Lt*)](2)

The technologies used to produce the traded and nontraded goods are assumed to be constant-returns-to-scale Cobb-Douglas, that is,

Traded Sector:

YtT=θtTKtTαTNtT(1αT)(3)
YtT*=θtT*KtT*αTNtT*(1αT)(4)

nontraded Sector:

YtN=θtNKtNαNNtN(1αN)(5)
YtN*=θtN*KtN*αNNtN*(1αN)(6)

where YtT,YtT*,YtN,YtN* are output levels for home and foreign country in the traded and nontraded sector respectively; KtT,KtT*,KtN,KtN* are capital stocks at the beginning of period t; NtT,NtT*,NtN,NtN* are the amount of work supplied at period t; ZtT,ZtT*,ZtN,ZtN* are technology shocks; αT and αN are the shares of capital in each sector. It is assumed that capital is perfectly mobile across countries in the traded sector, but there is no inter-sectoral mobility within the country.

In each country, the rate of physical depreciation of capital stock in each sector is assumed to be unity. Therefore, the laws of motion are: ItT=K(t+1)T,ItT*=K(t+1)T*, It/N =K(t+1)N, and It/N*=K(t+1)N*. This assumption is required to derive the closed-form analytical solution for optimal decision rules.

In equilibrium the goods and labor markets clear. In the market for traded goods, the aggregate resource constraint requires that the world supply must be exhausted by world consumption and investment demand,

CtT+CtT*+ItT+ItT*=YtN+YtN*(7)

The equilibrium conditions for the nontraded goods sectors imply that the domestic supply of the goods be equal to domestic consumption and investment demand,

CtN+ItN=YtN (8)
CtN*+ItN*=YtN*(9)

It is assumed that labor is perfectly mobile between the traded and nontraded sectors within a country, but immobile internationally. We normalize each country’s population and the endowment of time of the representative household in each country, to one. Therefore, the labor market clearing conditions are given in,

NtT+NtN+Lt=1(10)
NtT*+NtN*+Lt*=1(11)

The technology shocks are modeled as a quadrivariate autoregressive process, AR(1),

θt+1=Ωθt+εt(12)

where θt=[θtT,θtN,θtT*,θtN*] is me vector of exogenous disturbances, Ω is a 4x4 matrix describing the autoregressive components of the disturbances. Finally, εt is a vector [εtT,εtN,εtT*,εtN*] describing the contemporaneous components of the shocks which are independently and identically distributed with mean zero and variance σε2.

A. Equilibrium Allocation

We can define and derive a competitive equilibrium for this two-country world economy. Since there are no distortions in the economy, and production exhibits constant returns to scale, the equivalence between competitive equilibrium and a social planning optimum allows us to solve instead a social planner’s problem in which a weighted sum of national utilities is maximized, subject to technologies and the aggregate resource constraint. Treating each country symmetrically (so that the weights in the social planner’s objective function are equal) leads us to formulate the social planning problem as follows:

Choose(CtT,CtN,CTN*,NtT,NtN,NtT*,Nt/N*,K(t+1)T,K(t+1)N,K(t+1)*,K(t+1)N*)to

maximize

t=0βtEt[U(CtT,CtN,Lt)+U*(CtT*,CtN*,Lt*)]

subject to the production technologies in equations (3)-(6) and the market clearing conditions given in equations (7)-(11) and the technology shock process in equation (12).

Let the state vector for the planning problem be,

st=[KtT,KtN,KtT*,KtN*,θtT,θtN,θtT*,θtN*]

To solve the planning problem we can define the value function V(st) as

V(st)=max[γlog(CtT)+(1γ)log(CtN)+μlog(Lt)+γlog(CtT*)+(1γ)log(CtN*)+μlog(Lt*)+βEtV(s(t+1))](13)

Substituting for CtT using equation (7), for CtN using (8), for CtN* using (9), for Lt using (10) and for Lt* using (11), the value function is expressed as

V(st)=max[γlog(YTt+YTt*CTt*K(t+1)TK(t+1)T*)+(1γ)log(YNtCNtK(t+1)N)+μlog(1NTt+NNt)+γlog(CtT*)+(1γ)log(YNt*CNt*+K(t+1)N*)+μlog(1NTt*+NNt*)+βEtV(s(t+1))]

subject to resource constraints and technology.

The first-order-conditions are:

CtT*:γCtT=γCtT*(14)
K(t+1)T:γCtT=βEtVk(t+1)T(15)
K(t+1)N:1γCtN=βEtVk(t+1)N(16)
K(t+1)T*:γCtT*=βEtVk*(t+1)T(17)
K(t+1)N*:1γCtN*=βEtVk*(t+1)N(18)
NtT:μ(1NtTNtN)=γ(1αT)YtTCtTNtT(19)
NtN:μ(1NtTNtN)=(1γ)(1αN)YtNCtNNtN(20)
NtT*:μ(1NtT*NtN*)=γ(1αT)YtT*CtT*NtT*(21)
NtN*:μ(1NtN*NtN*)=(1γ)(1αN)YtN*CtN*NtN*(22)

These FOCs implicitly define optimal decision rules for consumption, investment and hours.

B. Closed-Form Solutions

To derive closed-form analytical solutions for optimal policy functions, we add some restrictive assumptions to the basic model. However, the simple restrictive models that are capable of being solved for analytical decision rules may lose many of the interesting cross-country dynamics, particularly those arising from the persistence of shocks. In particular, we can derive closed-form solutions under three alternative restrictions: first, technology shocks are independently and identically distributed (i.i.d.) processes; second, shocks processes are persistent but (a) there is no capital and technology is linear and (b) agents have perfect foresight.7

1. Productivity shocks are i.i.d. process

Under this assumption the technology shock processes are serially independent across time, and identical and independent across countries. Thus, there is no persistence in the shock processes. In this situation, home country’s relative share of world output of the traded good is constant.8 Employment is constant too, because income and substitution effects cancel out The optimal decision rules (see Appendix II for details of solution method) for consumption and investment are now:

CtT=CtT*=(1βαT)2HT(1αT)[θtTKtTαT+θtT*KtT*αT](23)
K(t+1)T=C(t+1)T*=βαT2HT(1αT)[θtTKtTαT+θtT*KtT*αT](24)
CtN=(1βαN)HN(1αN)θtNKtNαN(25)
CtN*=(1βαN)HN*(1αN)θtN*KtN*αN(26)
K(t+1)N=βαN2HN(1αN)θtNKtNαN(27)
K(t+1)N*=βαN2HN*(1αN)θtN*KtN*αN(28)

Where N¯T=N¯T*=Hr and N¯N=N¯N*=HN and

HT=γ(1αT)μ(1βαT)+γ(1αT)+(1γ)(1βαT)(1βαN)1(1αN)HN=(1γ)(1αN)μ(1βαN)+(1γ)(1αN)+γ(1βαN)(1βαT)1(1αT)

2. Productivity shocks are persistent

The assumption made above that technology shocks are i.i.d. is very strong and empirically not plausible due to the evidence of large persistence in the technology shocks. The absence of persistence in the shock processes implies that current shocks provide no signal about the rate of return on current investment, nor do they provide information about where to invest. With country-specific persistent shocks, current shocks do provide information about the profitability of investment. In this case, with perfect mobility of capital, the occurrence of shocks in one country will lead to a redirection of investment expenditure towards that country and away from the other country. Therefore, with persistence in the shock processes, the home country’s relative share of world output of traded goods is no longer constant. It also allows for fluctuations in employment in equilibrium. Thus, the persistence of shocks leads to interesting cross-country dynamics. However, we are unable to derive closed-form analytical solutions under persistence, unless we put two alternative restrictions on the standard RBC model described above.

(i) Labor is the only factor of production, and the technology is linear:.

Now, the technologies take the simple form

YtT=θtTNtTYtN=θtNNtNYtT*=θtT*NtT*YtN*=θtN*NtN*

In the absence of physical capital or investment, traded consumption in each country is half of the world output, and nontraded consumption is equal to the domestic production of the nontraded good. In this case the optimal policy functions for consumption and hours in the traded and nontraded sector are

CtT=CtT*=12[θtT1+A(δt)1+θtT*1+A(1δt)1](29)
CtN=θtN1+B+Dδt(30)
CtN*=θtN*1+B+D(1δt)(31)
NtT=11+A(δt)1(32)
NtT*=11+A(1δt)1(33)
NtN=11+B+D(δt)1(34)
NtN*=11+B+D(1δt)1(35)

where,

A=μ+(1γ)2γ;δt=1+A(1θtT*θtT1)1+θtT*θtT1B=μ1γ;D=2γ1γ

(ii) The model is deterministic, i.e., the agents have perfect foresight.. Perfect foresight implies that the evolution of the productivity disturbances is known with certainty. Under this assumption the utility function takes the following deterministic form

U=t=0βt[γlog(CtT)+(1γ)log(CtN)+μlog(Lt)]

Now the optimal decision rules for consumption, investment, and hours are:

CtT=CtT*=1βαT2[θtTKtTαT(11+Gωt1)(1αT)+θtT*KtT*αT(11+G(1ωt)1)(1αT)](36)
CtN=(1βαN)θtNKtNαN[11+Q+Rωt](1αT)(37)
CtN*=(1βαN)θtN*KtN*αN[11+Q+R(1ωt)](1αN)(38)
K(1+t)T=βαTωt+1[θtTKtTαT(11+Gωt1)(1αT)+θtT*KtT*αT(11+G(1ωt)1)(1αT)](39)
K(1+t)T*=βαT(1ωt+1)[θtTKtTαT(11+Gωt1)(1αT)+θtT*KtT*αT(11+G(1ωt)1)(1αT)](40)
K(t+1)N=βαNθtNKtNαN[11+Q+Rωt](1αN)(41)
K(t+1)N=βαNθtNKtNαN[11+Q+R(1ωt)](1αN)(42)
NtT=11+Gωt1(43)
NtT*=11+G(1ωt)1(44)
NtN=11+Q+Rωt(45)
NtN*=11+Q+R(1ωt)(46)

where

ωt=1+G[1(θtT*θtT1)11αT]1+(θtT*θtT1)11αTG=μ(1βαT)+(1γ)(1αN)(1βαT)(1βαN)12γ(1αT)Q=μ(1βαT)(1γ)(1αN);R=2γ(1βαN)(1αT)(1γ)(1αN)(1βαT)

The closed-form solutions derived above can be used to relate real exchange rate to the primitives of the underlying economic structure. For example, the parameters of utility, technology and the productivity shock processes.

C. Determination of Real Exchange Rate

The real exchange rate is defined as the nominal exchange rate adjusted for changes in the domestic and foreign price levels, and represents the real price of a foreign basket of goods relative to domestic one:

etR=etPt*Pt(47)

where etR is the real exchange rate, et is the nominal exchange rate (defined as the price of foreign currency in terms of a unit of domestic currency), and Pt* is the foreign general price level and Pt is the general price level in the domestic economy Under this definition, an increase in nominal exchange rate (real exchange rate) is equivalent to a depreciation (real depreciation) of domestic currency. Conversely, a fall in the nominal exchange rate (real exchange rate) is equivalent to an appreciation (real appreciation) of domestic currency.

The general price level (typically given by the consumer price index,9 is embodied in the expenditure function associated with the solution to the consumers problem) in both economies is comprised of traded goods prices, PtT, and nontraded goods prices, PtN, as follows:

Pt=PtTγPtN(1γ)(48)
Pt*=PtT*γPtN*(1γ)(49)

Combining equations (48) and (49), the ratio of traded goods prices can be expressed as

PtTPtT*=[PtN*PtT*PtNPtT](1γ)PtPt*(50)

The purchasing power parity (PPP) hypothesis states that the nominal exchange rate adjusts one-for-one to movements in domestic and foreign price levels. PPP relies on the existence of arbitrage of goods and services between the domestic and foreign economy when the prices (denominated in a common currency) in the two economies diverge. However, if large price divergences can be arbitrated only for traded goods not for the nontraded goods, PPP will exist only for traded goods in the long run. We model the assumption that PPP holds only for traded goods,

etPtT*=PtT(51)

Combining (47), (50) and (51) yields the following

etR=[PtN*PtT*PtNPtT](1γ)PtPt*(52)

or, in logarithms,

q=(1γ)ln(PtN*PtT*)(1γ)ln(PtNPtT)(53)

where q is the real exchange rate in logarithms. Equation (53) states that the real exchange rate is a function of the relative price of non-tradeable in the domestic and foreign economy An increase in the domestic relative price of nontradeables decreases q (the price of foreign currency in real terms), appreciating the domestic currency in real terms. Economies that experience an increase in the relative prices of nontraded goods are predicted to have an appreciating real exchange rate.

One method to model and explain the price levels of traded and nontraded goods is in terms of the consumer’s optimization conditions.10 The setup of the optimization problem in section III suggests that the competitive equilibrium is equivalent to the social planner’s optimum, and hence prices are simply the Lagrange multipliers on the resource constraints. Therefore the first-order conditions of consumer’s optimization can be used to solve for prices,

PtT=U(CtT,CtN)CtTPtN=U(CtT,CtN)CtNPtT*=U(CtT,CtN)CtT*PtN*=U(CtT,CtN)CtN*

The first-order conditions illustrate some of the features of the equilibrium. In general PPP does not hold because output and, hence, prices of nontraded goods differ across countries. However, because utility functions are additively separable between traded and nontraded goods, and nomothetic and identical across countries, marginal utilities of consumption, and hence the prices of traded goods will be equalized across countries. So bilateral real exchange rates between home and foreign countries depend only on the relative consumption of nontraded goods

etR=(CtNCtN*)(1γ)(54)

Equation (54) is similar to the exchange rate equation reported by Backus and Smith (1993) in that it shows a direct positive relation between bilateral real exchange rates and the ratio of nontraded consumption between domestic and foreign economies. Now, substituting the analytical solutions to optimal policy functions for consumption of nontraded goods derived above we can express real exchange rates in terms of the parameters of preference, technology, and the productivity differentials (or ratios) between home and foreign country in the traded and nontraded sectors. The exchange rate equations under the three alternative restrictive assumptions discussed earlier are derived as follows.

1. Technology shocks are i.i.d:

Substituting the closed-form solution to optimal consumption of nontraded goods from equations (24) and (25) into the exchange rate equation (54) we get,

etR=(θtNKtNαNθtN*KtN*αN)(1γ)(55)

In this case real exchange movements depend only on the productivity differentials(ratios) between home and foreign country in the nontraded sector.

2. Persistence of shock processes: Case 1: No capital and a linear technology:

Substituting the analytical solution from equations (33) and (34) into equation (54), we get the following exchange rate equation,

etR=[θtN[(1+BAD)+(1+B+D+AD)(θtT*θtT1)]θtN*[(1+B+D+AD)+(1+BAD)(θtT*θtT1)]](1γ)(56)

3. Case 2: Perfect foresight i.e. no uncertainty:

Substituting equations (45) and (46) into equation (54) yields the following expression for real exchange rate,

etR=[θtNKt/NαT[{(1+QGR)+(1+Q+R+GR)(θtT*θtT1)}11αT]1αNθtN*Kt/NαT[{(1+Q+R+GR)+(1+QGR)(θtT*θtT1)}11αT]1αN](1γ)(57)

In equations (55)-(57), real exchange rate depends on the parameters of preference and technology and the productivity differentials (ratios) in the traded and nontraded sectors. Therefore the implications of our simple model are striking because they provide expressions for equilibrium prices and real exchange rates which are easy to interpret and do not require restrictions on parameter values.

D. Relationship Between Productivity Shocks and Real Exchange Rate

Some comparative static exercises will now be performed to analyze the impacts of the relative productivity shocks in the traded and nontraded sectors on the bilateral real exchange rate movements. By partially differentiating the exchange rate equations (55)-(57), the effects of these exogenous disturbances on the level of real exchange rate can be derived.

1. Effects of productivity differentials in the nontraded sectors

In this case, we assume that shocks to the traded sector are constant. For all the exchange rate equations, we find that

etR(θtNθtN*)>0

Thus, there exists a direct positive relationship between bilateral real exchange rates and the relative productivity shocks (between home and foreign country) in the nontraded industries. The economic intuition is clear: increases in domestic productivity growth in the nontraded sector, [(lnθtNlnθtN*)>0] (relative to foreign nontraded productivity and to domestic traded productivity) cause workers to move from the traded to the more productive nontraded sector, so that domestic output of the nontraded goods, YtN, increases and home firms are able to reduce PtN. The fall in domestic relative price of nontraded goods, [(PtN − PtT) < 0] leads to a rise in etR, implying a real depreciation of home currency. Therefore, a rise in the relative productivity differentials (or ratios) in the nontraded sector will cause a real appreciation of foreign currency and a domestic depreciation of the real exchange rate.

2. Effects of productivity differentials in the traded sectors:

In this model, the productivity shocks in the traded sector appear as a determinant of real exchange rates only under the persistence of shock processes. From the exchange rate equations (56) and (57) we find that

etR(θtTθtT*)<0

which indicates an inverse relation between the relative productivity shocks in the traded sector and bilateral real exchange rates. (See Appendix II for proof.) The economic intuition is straightforward: an increase in domestic productivity of traded goods, [(lnθtTlnθtT*)>0] (relative to foreign traded productivity) causes both labor and capital to become more productive in that sector, so that domestic output of the traded good YtT increases,11 which allows domestic firms to reduce PtT. The consequent rise in the relative price of non-tradeables, [(PtN − PtT) > 0] will cause an appreciation of the real exchange rate, implying a fall in etR. Therefore, an increase in the productivity differentials (or ratios) in the traded sector implies a depreciation of the foreign currency in real terms and a real domestic appreciation.

IV. Empirical Results

This section presents empirical results using quarterly data for G-7 countries for the period 1960:QI-1997:QIII, The following discussion contains a brief description of the definitions, construction, and sources of the data; the summary statistics (standard deviation, autocorrelation and correlation) of nominal and real exchange rate and the productivity differentials; the results of Augmented Dickey-Fuller tests for unit roots of relevant variables; and the examination of the impact of relative productive shocks on the real exchange rate using the multivariate cointegration methodology proposed by Johansen (1988) and Johansen and Juselius (1990 and 1992).

A. Data Descriptions

This paper constructs total factor productivity12 for the traded and nontraded sectors for the G-7using OECD data from the International sectoral Database. The sector breakdowns follow the International Standard Industrial Classifications (ISIC) currently used in the OECD National Accounts (ANA) publication. Under this sectoral decomposition, nontraded goods account for half of the aggregate output. Following the work by the OECD (1988), the open or traded good sectors comprise only manufacturing, and the nontraded goods are the service sectors comprising: (1) electricity, gas and water, (2) construction, (3) wholesale and retail trade, restaurants and hotels, (4) transport, storage and communications, (5) financial service and insurance, (6)community, social and personal services and (7) government services.13 Productivity data for the traded(nontraded) sector are obtained by dividing the OECD’s figures for GDP in constant prices for the traded (nontraded) sector by total labor employment for that sector, yielding labor productivity or real output per worker.14

Home and foreign productivity differentials (or ratios) are estimated relative to the United States i.e. using the United States as the benchmark. Productivity differentials (or ratios) in the traded sector is defined as: ln(θtTiθtTUS) or (lnθtTilnθtTUS) where θt is the total factor productivity and i = Canada, France, Germany, Italy, and United Kingdom. Similarly, for the nontraded sector: ln(θtNiθtNUS) or (lnθtNilnθtNUS).

The GDP price deflators are constructed by dividing the nominal GDP for the traded (nontraded) by the real GDP for that sector. The nominal exchange rate for each country is the bilateral exchange rate vis-a-vis the U.S dollar.15 Real exchange rates are measured using the nominal exchange rates and individual country consumer price indexes(CPI). Prices and the exchange rate data are from the International Financial Statistics, IFS. Definitions of the variables and the sources are given in Appendix I.

B. Summary Statistics

Table 1 presents of Hodrick-Prescott filtered quarterly observations of nominal and real exchange rates and productivity differentials for G-7countries. Three regularities are evident. First, for each country standard deviation of both nominal and real exchange rates are much higher during the recent flexible exchange rate period (1973-1997) compared to the estimates for the entire sample period (1960-97) and for the fixed exchange rate period (1960-73). This result extends support to one of the stylized facts of exchange rate behavior well documented by empirical studies16. Second, in all cases (except for Canada) the volatility of nominal and real exchange rates arebetween two to five times higher than the volatility of productivity differentials in traded and nontraded sectors17. Third, exchange rate movements are highly persistence as indicated by first-order autocorrelation coefficients for real and nominal exchange rate of approximately 0.84. On the average, productivity differentials are less persistent than those of real exchange rate The estimates of sample correlations suggest the following features of association among exchange rates and productivity differentials; first, nominal and real exchange rates are significantly positively correlated with the correlation coefficient close to one, second, there is positive relationship between productivity differential s (between home and foreign country) in the nontraded sector and the real exchange rate; third, the correlations between real exchange rate and productivity differentials in the traded sector is negative. The direction of correlations are consistent with the predictions of the theoretical model regarding the relationship between real exchange rate and home and foreign productivity differentials.

Table 1:

Summary Statistics of Exchange Rates and Productivity Differentials, Quarterly data, 1960-1997

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Notes:

I: 1960:01-1997:03; II: 1960:01-1973:03; III: 1973:03-1997:03.

Persistence measure is the first-order autocorrelation coefficient.

Source: OECD Quarterly National Accounts and International Financial Statistics.

C. Test for Stationarity

The first step in the cointegration analysis is to test the unit roots in each variable.18 To this end we apply Augmented Dickey-Fuller (ADF) stationarity tests on the real exchange rate (q), productivity differential in the traded sector (lnθtTilnθtTUS) and differential in the nontraded sector (lnθtNilnθtNUS). The ADF tests are performed assuming a constant, one difference and a time trend. Table 2 reports the results of the ADF test for the level as well as for the first-difference of the relevant variables. For all countries, the tests fail to reject at the 5 percent level the null hypothesis of a unit root in the data-generating process. The null hypothesis is rejected when the series are first-differenced i.e., all variables are first-differenced statationary. This implies that all the series are integrated of order one, I(1) variables.

Table 2:

Augmented Dickey Fuller Tests 1/

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Notes:

The Mackinnon (1991) critical value of the ADF test at the 5% level of significance is -3.49.

Variables:q = log of real exchange rate.ln(θTiθtTUS)=relative productivity in the traded sector.ln(θNiθtNUS)=relative roductivity in the non-traded sector.Source: OECD Quarterly National Accounts and International Financial Statistics.

D. Cointegration tests

We examine the relationship between the real exchange rate and the sectoral productivity differentials between home and foreign country using the multivariate cointegration methodology proposed by Johansen (1988) and Johansen and Juselius (1990 and 1992).

The Johansen maximum likelihood approach has some advantages over the traditional Engle-Granger procedure: (i) it allows testing in a multivariate framework, (ii) considers the error structure of the data processes, (iii) allows for interactions in the determination of the relevant economic variables independent of the choice of the endogenous variable and (iv) allows explicit hypotheses tests of parameter estimates and rank restrictions using likelihood ratio tests that employ Chi-Square statistics.19

Johansen tests can determine the rank of exchange rate equation (56) by testing the following vector

J=[q(lnθtTilnθtTUS)(lnθtNilnθtNUS)](58)

where all variables are in logarithms. Johansen tests suggest that if a linear combination of vector (58) is a rank one (i.e., a linear combination of relevant vector is stationary), then a single cointegrating relationship exists between productivity differentials (between home and foreign country) in the traded and nontraded sector and the real exchange rate. Chi-Square restrictions tests are then imposed to test hypotheses regarding the cointegration space of the vector.

Table 3 reports Johansen cointegration and Chi-Square restriction tests for real exchange rates and domestic and foreign productivity differentials in the traded and nontraded sectors. Column ΙII indicated that the null of no cointegrating linear combinations between real exchange rate and sectoral productivity differentials between domestic and foreign country can be rejected at the 1percent confidence level for all countries except Germany, where the null is rejected at 5 percent significance level. Cointegrating ranks of less than one or two cannot be rejected. Therefore, at least one cointegrating relationship is supported by the data generating process.

Table 3:

Johansen Cointegration and Hypothesis Tests

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Notes: ** = significance at the 1% level, * = significance at the 5% level.Variables:q = log of real exchange rateln(θTiθtTUS)=domestic traded productivity relative to foreign traded productivy.ln(θNiθtNUS)=domestic non-traded productivity relative to foreign non-traded productivy.Column II: Ho: No unit root.Source: OECD Quarterly National Accounts and International Financial Statistics.

The next step is to show that the cointegrating linear combination arises not solely between the productivity differentials, but between real exchange rates and domestic and foreign productivity differentials. Column IV reports the reduced rank restriction tests [0 β2 β3] for the cointegrating vector [β1qβ2(lnθtTilnθtTUS)β3(lnθtNilnθtNUS)]. The test results reject the no significance of real exchange on the cointegrating vector. This implies that the cointegrating results are not due to linear combination between productivity differentials between domestic and foreign country in the traded sector and in the nontraded sector. Therefore, the cointegrating results reported in column III arise due to a linear combination between real exchange rates and sectoral productivity differentials between home and foreign country.

Columns V and VI report the test statistic for individual significance of productivity differentials in each sector on the real exchange rate. These columns contain the magnitude, sign, and adjusted Johansen standard errors for the productivity differentials in the traded and nontraded sectors. In all cases, the coefficients are significantly different from zero with the correct sign, which is consistent with the predictions of the theory. In all six economies, higher productivity differentials in the traded sector (increases in domestic traded productivity relative to foreign traded productivity, i.e., (lnθtTilnθtTUS)>0) implies a significant increase in the relative price of nontraded goods in the home country, which is equivalent to a real appreciation of the exchange rate (i.e., a decline in the price of foreign currency). Conversely, in all cases, increases in the productivity differentials in the nontraded sectors (lnθtNilnθtNUS)>0 implies a decline in the relative price of nontraded goods and consequently a real depreciation of the exchange rates.

V. Conclusion

In this paper a two-country dynamic general equilibrium model with nontraded goods has been developed to examine the impact of productivity differentials (between domestic and foreign country) in the traded and nontraded sector on the real exchange rate movements. The model yields closed-form analytical solutions for optimum consumption, investment, and hours and for equilibrium real exchange rate. The real exchange rate obtained depends on the parameters of preference, technology and the sectoral productivity differentials between home and foreign country. The theory predicts a close relationship between the real exchange rate and domestic and foreign productivity differentials. Increases in the domestic productivity of traded goods (relative to the foreign productivity of traded goods) lead to an appreciation of real exchange rates. Conversely, increases in the domestic productivity of nontraded goods cause a real depreciation of exchange rates.

Empirical results using data for G-7 countries suggest excess volatility of exchange rates and positive comovements between real and nominal exchange rates during the recent floating exchange rate period. The estimates of sample correlations (between real exchange rates and productivity differentials) coefficients are consistent with the sign predicted by the theoretical model. Johansen cointegration tests also show that a significant cointegrating relationship exists between relative (between domestic and foreign) productivity shocks in each sector and the real exchange rate. Therefore, sectoral productivity differentials (or ratios) between home and foreign country may be important determinants of real exchange rate fluctuations.