INSTITUT NATIONAL DE LA STATISTIQUE ET DES ÉTUDES ÉCONOMIQUES

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de la Direction des Etudes et Synthèses Économiques

G 2000 / 02

# The real exchange rate as the relative price of nontradables in terms of tradables: theoretical investigation and empirical study on French data

Céline ALLARD-PRIGENT *

Hélène GUILMEAU **

Alain QUIINFT ***

AVRIL 2000

We are grateful to Guy Laroque, Françoise Maurel and Bernard Salanié, and to the audience of the D3E seminar, especially Nicolas Sobczak who discussed the paper on that occasion.

<footnote>* Faisait partie du Département des Etudes Economiques d'Ensemble au moment de la rédaction de ce document</footnote>

<footnote>** ENSAE</footnote>

<footnote>*** Direction de la Prévision</footnote>
Résumé

Les larges fluctuations des parités, tant réelles que nominales, entre les différents pays industrialisés depuis la fin du système de Bretton Woods ont très tôt posé la question des déterminants effectifs des taux de change. L'approche que nous développons ici considère le taux de change comme le prix relatif des biens non échangeables en terme de biens échangeables. Elle permet de dériver le prix relatif d'un comportement de maximisation des firmes et des consommateurs dans un modèle d'équilibre général. A cet effet, nous nous appuyons sur le modèle d'Obstfeld-Rogoff (1996), modèle d'équilibre général intertemporel adapté à une petite économie ouverte à deux secteurs. Néanmoins, nous enrichissons leurs résultats, obtenus en concurrence parfaite, en introduisant une forme de concurrence parfaite dans le secteur des biens non échangeables.

Les résultats sur données françaises semblent rejeter le modèle pur d'Obstfeld-Rogoff qui ne fait dépendre les taux de change que de facteurs d'offre, c'est-à-dire des chocs de productivité et le taux d'intérêt réel. Le modèle enrichi d'éléments de demande semble davantage validé : il est possible de dégager une relation de cointégration entre le taux de change et d'un côté des facteurs d'offre, et de l'autre des facteurs de demande, en l'occurrence les dépenses publiques.

Mots-clés : Productivité, taux de change réel, inflation sectorielle, Relation de cointégration

Abstract

Large fluctuations in both nominal and real parities between G7 countries raise the issue of their determinants. This paper offers a formal description of a model of the real exchange rate, defined as the relative price of nontradables in terms of tradables, explicitly based on microeconomic optimization. Our approach relies on a version of the general equilibrium model first presented by Obstfeld-Rogoff (1996). But we enrich the model by introducing monopolistic competition in the nontradables sector. This allows for demand factors to influence the real exchange rate in the long run.

Our results on French data do not corroborate the Obstfeld-Rogoff supply-side model. However, they provide some support for the extended version of the model: a cointegration relationship can be found between the real exchange rate, sectoral productivities and the « world » real interest rate, once public expenditure is also taken into account.

Keywords: Productivity, Real Exchange Rate, Sectoral Inflation, Public Spending, Cointegration Relationship

Classification JEL : D40 - F41 - O33
# I - Introduction

Large fluctuations in both nominal and real parities between G7 countries raise the issue of the appropriate level of exchange rates. In the very short term, monetary conditions and more specifically interest rate differentials seem to play a major role. Beyond that horizon, however, it is hard to detect any kind of convergence towards an equilibrium level.

The most traditional approach of exchange rate determination is based on the purchasing power parity: the nominal exchange rate would only compensate in the medium run for differences between domestic and foreign inflations, without any interference of other macroeconomic variables. This theory explains movements of the nominal exchange rates only over a century (Froot-Rogoff, 1994), but seems unable to explain fluctuations of the real exchange rate on the medium term.

On the contrary, according to the more recent equilibrium exchange rate theory, the real exchange rate does depend on the macroeconomic context. More precisely, the real equilibrium exchange rate guarantees both domestic and external "equilibrium" of a given economy. This macroeconomic approach allows to estimate quite easily equilibrium parities, once potential GDP and external current account targets have been explicitly defined. Following pioneer work by Williamson (1994), Wren Lewis-Driver (1998), among others, have presented some evaluations for the major currencies. However, this equilibrium exchange rate is not based on explicit optimization by economic agents but on subjective judgements on the state of the economy. Besides, this approach does not say anything on the error correction mechanisms towards the so-called equilibrium level.

An alternative approach, based on rigorous microeconomic optimization, specifies the exchange rate as the relative price of nontradables in terms of tradables in a general equilibrium framework. Such a definition has been used by Edwards (1989) for emerging countries or by Obstfeld and Rogoff (1996). These models explicitly derive the exchange rate as the result from an optimization program by both firms and consumers, and describe the allocation of resources between the tradables and nontradables sectors, in response to productivity or taste shocks. For instance, an appreciation of the real exchange rate - a fall in the relative price of tradables - may trigger a reallocation of resources towards the nontradables sector and even a de-industrialization of the economy, a phenomenon known as the "Dutch disease". Although the relative price of the nontradables can differ from its equilibrium value in the short run, it tends to converge towards it as prices progressively adjust to accommodate the shocks.

Balassa (1964) initiated this analysis in terms of nontraded goods to explain the convergence process of the price level in developing countries vis-à-vis the price level of advanced countries. Such an approach should benefit from renewed interest in the context of EMU. In the euro zone, indeed, the macroeconomic real exchange rate becomes largely irrelevant, as intra-zone nominal exchange rates disappear and as tradables prices progressively converge. Variations in the relative price of nontradables remain the only indicator to assert whether domestic inflation reflects convergence of the general price levels or specific inflationary pressures.

The paper tries to identify the factors behind the trend increase of the relative price of nontradables in France over the last decades. We distinguish three potential determinants:

* "natural" domestic factors (sectoral total factor productivities),

* the real interest rate,

* direct public spending.

The paper is divided into three sections. In section I, we discuss the theoretical framework underlying our analysis, an enriched version of the general equilibrium model first presented by Obstfeld and Rogoff (1996). We do not assume perfect but rather monopolistic competition in the nontradables sector, which allows for demand factors to influence the
real exchange rate in the long run. In section II, we describe the data, paying special attention to the total factor productivity shocks in both sectors. Section III deals with the econometrics on French data. We find a role for demand as well as supply factors in explaining movements of the real exchange rate.
## II - The theoretical framework

Our model is based on the interaction between supply and demand: supply is provided by firms, demand comes from both private consumers and the public sector. The tradable and nontradable sectors are distinguished, as the conditions in which each market clears differ.

In the tradable sector, the price is determined on the world market ("law of one price"), and firms operate in a perfectly competitive environment. However, domestic production and consumption do not necessarily equate, as part of the supply can be exported and part of the demand imported.

By contrast, foreign competition plays no role in the nontradables sector<fnref n="1" /> and *ex post* price equates domestic supply and demand. We assume that firms in that sector face monopolistic competition and explicitly take into account the way demand reacts to price. As demand factors enter the firms' maximization program, demand can influence, even in the medium term, the relative price of nontradables in terms of tradables. The more rigid markets are, the more demand is expected to play a role.

Our model assumes an infinite horizon. At each period, firms optimize the level of both productive factors, namely labor and capital. Assuming constant-returns production, the factor-price frontier entails a relationship between the real cost of capital and the real wage: as capital mobility is supposed to be perfect, the real interest rate is fixed on world financial markets, and the relationship can be written as a mapping from the interest rate to the domestic real wage.

Relative price determination results from the interaction between supply and demand in both markets. Given that firms in the nontradable sector take into account the influence of prices on demand, we first present the demand side of the market. The optimization program of consumers can be solved in two steps. They first determine the intertemporal path of consumption, and then, period by period, the way consumption is devoted to traded and nontraded goods. We then present the supply side of the market and conclude with the determination of the real exchange rate.

### 2.1 Consumers' program

#### 2.1.1 Setting up the program

The representative consumer chooses the optimal intertemporal path of consumption as well as its allocation, in each period, to traded and nontraded goods. He maximizes, under budget constraint, an infinite intertemporal utility function, which is assumed to be CES (Constant Elasticity of Substitution).

His wealth is composed of wages and firms' profits, as he rents capital to firms. He pays lump taxes, and inherits a debt (or a dotation) at the very beginning of the first period. He therefore has to solve the following program:

<footnote n="1">¹ such as building sector or services that cannot be exported. The popular example quoted in the literature is a hair-dresser. More generally, we imply by nontradables services all the services on which foreign competition can clearly play no role.</footnote>
$$ (1) \left\{ \begin{array}{l} \underset{C_s}{\operatorname{Max}} \sum_{s=t}^{\infty} \beta^{s-t} \frac{C_s^{1-\frac{1}{\sigma}}}{1-\frac{1}{\sigma}} \\ \\ \text{subject to} \quad \sum_{s=t}^{\infty} \left(\frac{1}{1+r}\right)^{s-t} P_s C_s = E_t \end{array} \right. $$

where E is the consumer's permanent income

$$ E_t = \sum_{s=t}^{\infty} \left(\frac{1}{1+r}\right)^{s-t} w_s L_s + \sum_{s=t}^{\infty} \left(\frac{1}{1+r}\right)^{s-t} r K_s + \sum_{s=t}^{\infty} \left(\frac{1}{1+r}\right)^{s-t} p_s - \sum_{s=t}^{\infty} \left(\frac{1}{1+r}\right)^{s-t} \tau_s + (1+r)B_t $$

and where:

* $C_s = \left( \gamma^{\frac{1}{\theta}} C_{T,s}^{\frac{\theta-1}{\theta}} + (1-\gamma)^{\frac{1}{\theta}} C_{N,s}^{\frac{\theta-1}{\theta}} \right)^{\frac{\theta}{\theta-1}} \quad \forall s$ is the CES intratemporal utility function, $C_{T,s}$ and $C_{N,s}$ being respectively the consumption levels in tradables and nontradables of period s,

* $\sigma$ represents the intertemporal substitution elasticity ($\sigma>0$),

* $\beta$ is the discount factor ($\beta<1$),

* $P_s$ is the price index associated to total consumption of period s (see section 2.1.2 for a comprehensive calculation),

* $r$ is the world nominal interest rate,

* $B$ is the external debt inherited at the beginning of the first period<fnref n="2" /> and $rB_t$ the interest payments,

* $\tau_s$ refers to the lump taxes in period s required to finance public expenditures,

* $w_s$ is the nominal wage in period s,

* $L_s$ represents the level of employment in period s,

* $K_s$ the capital stock of capital in period s,

* $\pi_s$ the firms' profit in period s.

As $P_s Y_s = r K_s + w_s L_s + \pi_s \quad \forall s$, the intertemporal budgetary constraint can be rewritten so as to equalize the present value of lifetime discounted consumption to the present value of lifetime discounted wealth, minus the discounted value of lump taxes and repayments of the inherited debt. Therefore:

$$ E_t = \sum_{s=t}^{\infty} \left(\frac{1}{1+r}\right)^{s-t} P_s Y_s - \sum_{s=t}^{\infty} \left(\frac{1}{1+r}\right)^{s-t} \tau_s + (1+r)B_t $$

Because the utility function is time-separable, the resolution of the program entails two separate choices:

* first, the determination of the allocation of consumption at each period,

* second, the allocation of consumption in period s, between tradables and nontradables.

<footnote n="2">² In the case where consumers inherit a debt, B is negative. One could also see B as an initial dotation ; in that case B would be positive.</footnote>
### 2.1.2 Intertemporal consumption optimum

The Euler condition derived from program (1) yields the expression of the desired intertemporal consumption ratio, depending on the price index, the discount factor and the interest rate:

$$ (2) \quad \frac{C_{s+1}}{C_s} = \left[ \frac{P_s}{P_{s+1}} \beta (1+r) \right]^{\sigma} \quad \forall s $$

Replacing the value of consumption given by the budget constraint provides a relationship between consumption in period s and the permanent income:

$$ (3) \quad C_s^* = \frac{E_s}{P_s} \frac{1}{\left[ 1 + \sum_{i=1}^{\infty} \left( \frac{P_s}{P_{s+i}} \right)^{\sigma-1} \beta^{i\sigma} (1+r)^{i(\sigma-1)} \right]} $$

These expressions can be simplified in the particular case of a Cobb-Douglas intertemporal utility function ($\sigma = 1$). The optimal consumption allocation in period s then is (4)

$$ C_s^* = \frac{E}{P_s}(1-\beta). $$ In order to make our calculations simpler, and without any prejudice to the generality of our results, we shall keep that hypothesis in the rest of the presentation.

### 2.1.3 Intratemporal consumption optimum

In each period $C^* = \frac{E}{P}(1-\beta)$ (in this section, the time index will be omitted, as the problem can be solved period by period). Under the assumption of a CES intratemporal utility function, the representative agent has to determine the optimal level of tradable and nontradable consumptions, by solving:

$$ (5) \quad \begin{cases} \underset{C_T, C_N}{\operatorname{Max}} C^* = \left( \gamma^{\frac{1}{\theta}} C_T^{\frac{\theta-1}{\theta}} + (1-\gamma)^{\frac{1}{\theta}} C_N^{\frac{\theta-1}{\theta}} \right)^{\frac{\theta}{\theta-1}} \\ \text{subject to} \quad Z = p_T C_T + p_N C_N \end{cases} $$

where:

* $\theta$ is the intratemporal (constant) substitution elasticity ($\theta > 0$),

* $\gamma \in [0; 1]$,

* Z is the level of revenue allocated to consumption during the period. That level Z follows directly from the intertemporal maximization previously described.

* $p_T$ refers to the tradable consumption price,

* $p_N$ to the nontradables price.
Solving the program gives the relative shares of consumption for both types of goods depending on their relative price, i.e. the real exchange rate $p = \frac{p_N}{p_T}$:

$$ (6) \qquad \frac{C_N}{C_T} = p^{-\theta} \frac{1-\gamma}{\gamma} $$

We use the budget constraint over the period $Z = p_T C_T + p_N C_N$ to derive the optimal levels of consumption in each sector, for a given level of expenditures:

$$ (7) \qquad \begin{cases} C_T^* = \frac{\gamma Z}{p_T [\gamma + (1-\gamma)p^{1-\theta}]} \\ C_N^* = \frac{(1-\gamma)p^{-\theta} Z}{p_T [\gamma + (1-\gamma)p^{1-\theta}]} \end{cases} $$

At this stage, we need to clarify the expression of $Z$, the share of wealth allocated to current spending. This expenditure can be linked to the previously-used price index $P$ and to the index of total consumption $C^*$ through (8): $Z = PC^{*3}$. We use equations (7) to express the price index as a function of the real exchange rate.

$$ (9) \qquad P = p_T (\gamma + (1-\gamma) p^{1-\theta})^{\frac{1}{1-\theta}} $$

The tradable and nontradable consumption can then be simply linked to both indexes $P$ and $C^*$, as well as their respective prices:

$$ (10) \qquad \begin{cases} C_T^* = \gamma \left( \frac{P}{p_T} \right)^{\theta} C^* \\ C_N^* = (1-\gamma) \left( \frac{P}{p_N} \right)^{\theta} C^* \end{cases} \quad \text{where } P = p_T (\gamma + (1-\gamma)p^{1-\theta})^{\frac{1}{1-\theta}} \text{ and } C^* = \frac{E}{P} (1-\beta). $$

Everything happens as if the consumer, after deciding the level of income he allocates to the period $C^*$, chooses between the two types of goods by comparing their price to the total price index: the elasticity of substitution $\theta$ appears in the ratio $P/p_i$ ($i = N, T$) but not directly in the relative price, namely the real exchange rate. The reason is quite straightforward: besides the substitution effect between the two goods, changes in the relative price induce a wealth effect. The expression is therefore complicated when directly written with the real exchange rate.

## 2.2 Firms' program

The model helps describing the long term determinants of the real exchange rate. Therefore, we forget about possible short term deviations, and assume that firms face no adjustment

<footnote n="3">³ In such a definition, $P$ measures the minimal spending required to purchase a unit of the total consumption index.</footnote>
costs. Each period, productive factors and produced goods correspond to the optimal levels. In particular, firms optimize their level of capital. Under these assumptions, the problem can be solved period by period, even if firms have an infinite horizon.

In both tradable and nontradable sectors, output is given by Cobb-Douglas production functions $Y_j = A_j L_j^{\alpha_j} K_j^{1-\alpha_j}$ where $j = N, T$, depending on whether the firm belongs to the tradables or the nontradables sector, and $A_j$ is the total factor productivity in sector $j$.

Productive factors differ in terms of mobility:

* Labor is mobile between the two sectors of the economy but not between countries. Besides, labor supply is taken as constant. As a consequence, employment adjusts instantly in the economy, and wages are identical in the two sectors;

* Capital is internationally mobile. The domestic interest rate is equal to the world interest rate, which is here taken as exogenous.

### 2.2.1 The tradable sector

In the tradable sector, firms face a perfectly competitive environment. Moreover, the price level does not result from the interaction between supply and demand but is fixed by world markets ("law of one price"). This context fits a small open economy, where firms in the tradables are completely price-takers.

Within each period, firms maximize the discounted value of profit subject to their technical constraint. Only tradables can be used as capital, so that all firms, both from tradables and nontradables sectors, buy their additional capital at the price of tradables. In each period, firms inherit the whole stock of capital from the previous period<fnref n="4" /> and buy the surplus of capital they need, the cost of capital including both principal and interest payments. Solving the optimization program given by:

$$ (11) \quad \left\{ \begin{array}{ll} \underset{Y_{T,s}}{\operatorname{Max}} & \displaystyle \sum_{s=t}^{\infty} \left(\frac{1}{1+r}\right)^{s-t} [p_{T,s} Y_{T,s} - w_s L_{T,s} - (1+r)p_{T,s} (K_{T,s} - K_{T,s-1})] \\ \text{subject to} & Y_{T,s} = A_{T,s} L_{T,s}^{\alpha_T} K_{T,s}^{1-\alpha_T} \\ & p_{T,s} \text{ given} \end{array} \right. $$

yields the capital-labor ratio $\left( k_{T,s} = \frac{K_{T,s}}{L_{T,s}} \right)$ in the tradable sector and the real wage in terms of tradables as a function of the real interest rate and total factor productivity in the tradable sector, both taken as exogenous.

$$ (12) \quad \begin{cases} A_{T,s} \alpha_T (k_{T,s}^*)^{1-\alpha_T} = w_{T,s}^* \\ A_{T,s} (1-\alpha_T) (k_{T,s}^*)^{-\alpha_T} = r_{T,s} \end{cases} $$

where:

<footnote n="4">⁴ We made the simplifying, but not essential assumption, that the depreciation rate is zero.</footnote>
* $w_{T,s} = \frac{w_s}{p_{T,s}}$ is the real wage in terms of tradables,

* $r_{T,s} = r - \dot{p}_{T,s+1}$ is the real interest rate deflated by price inflation in the tradable sector, $\dot{p}_{T,s+1}$ at period s+1.

### 2.2.2 The nontradable sector

Firms in the nontradable sector do not have to face foreign competition, and are more likely to adopt a monopolistic behavior: we assume that they explicitly take into account the way public and private demand respond to price fluctuations. Besides, public expenditures are assumed to consist exclusively of nontraded goods.

We saw previously how private demand of nontradables depended on the price charged by producers. Insofar as public authorities do not substitute traded for nontraded goods in response to an appreciation of the real exchange rate, we assume that their demand depends on the nontradables price, but with a much smaller elasticity than the private demand. Therefore, we expect firms to be more inclined to set higher prices when the share of public expenditures increases.

Finally, price and consumption indexes have been taken as exogenous for each firm in its individual optimization program, as we considered each firm had a negligible influence on the total indexes.

Each firm, therefore, solves the program:

$$ (13) \left\{ \begin{array}{l} p_{N,s} \underset{K_{N,s}}{\overset{Max}{\sum}} L_{N,s} \sum_{s=1}^{\infty} \left(\frac{1}{1+r}\right)^{s-1} [p_{N,s} Y_{N,s} - w_s L_{N,s} - (1+r)p_{T,s} (K_{N,s} - K_{N,s-1})] \\ \text{subject to} \quad Y_{N,s} = A_{N,s} L_{N,s}^{\alpha_N} K_{N,s}^{1-\alpha_N} \\ \qquad Y_{N,s} = C_{N,s} + G_{N,s} \\ \qquad p_{T,s}, C_s \text{ and } P_s \text{ given} \end{array} \right. $$

where:

* $C_{N,s} = (1 - \gamma) \left(\frac{P_s}{P_{N,s}}\right)^{\theta} C_s$ is the private demand in the nontradable sector for period s,

* $G_{N,s}$ refers to public expenditures (exclusively composed of nontradables):
$$ G_{N,s} = G_s (p_{N,s})^{-\theta_G} $$

* again, capital is made exclusively of traded goods.
The first-order condition for nontradables price yields the mark-up monopolistic competition allows firms to set on labor and capital costs: $M = \frac{1}{1 - \frac{1}{\theta - (\theta - \theta_G)\frac{G_{N,s}}{Y_{N,s}}}}$. With $\theta_G < \theta$, M is clearly greater than unity.

$$ \text{We obtain two relationships, similar to (12):} \quad (14) \quad \begin{cases} p_s^* A_{N,s} \alpha_N (k_{N,s}^*)^{1-\alpha_N} = M w_{T,s}^* \\ p_s^* A_{N,s} (1-\alpha_N) (k_{N,s}^*)^{-\alpha_N} = M r_{T,s} \end{cases} $$

The mark-up charged by firms in the nontradable sector is higher, and so are their prices, if public expenditure as a share of total nontradables demand, g, is significant. This result can be easily understood: a price increase depresses domestic private demand much more than public spending. For private demand, the elasticity of substitution is θ. Aggregated with public demand, nevertheless, this elasticity is reduced by the multiplicative factor (1-g). On the whole the price-elasticity taken into account by the firms is no more than $(1-g)\theta + g\theta_G$. In other words, when the public spending share increases, firms are more inclined to set higher prices, as a greater part of aggregate demand will not react very much to that increase.

### 2.2.3 Long term equilibrium

Finally, the four first-order conditions (12) and (14) allow us to determine completely the equilibrium of the economy, i.e. the level of the four exogenous variables: the real exchange rate, the sectoral capital-labor ratios and the real wages as function of exogenous parameters of the model, namely the world real interest rate, total factor productivities, price elasticities of private and public demand as well as the share of public expenditures on nontraded goods and services:

$$ \begin{cases} A_{T,s} \alpha_T (k_{T,s}^*)^{1-\alpha_T} = w_{T,s}^* \\ A_{T,s} (1-\alpha_T) (k_{T,s}^*)^{-\alpha_T} = r_{T,s} \end{cases} \quad \begin{cases} p_s^* A_{N,s} \alpha_N (k_{N,s}^*)^{1-\alpha_N} = M w_{T,s}^* \\ p_s^* A_{N,s} (1-\alpha_N) (k_{N,s}^*)^{-\alpha_N} = M r_{T,s} \end{cases} $$

Two of these relationships yield the capital-labor ratios. A third one corresponds to the price-factor frontier, which links factor costs as soon as returns to scale are constant (see Appendix). In a small open economy, where the interest rate is given by world financial markets, this relationship determines real wages in the economy. With the last equation, we derive the expression of the real exchange rate as a function of a set of exogenous parameters, among which technological shocks and labor shares in both tradable and nontradable sectors, the real interest rate, and, through the mark-up, public expenditures as well as the intratemporal elasticity of substitution:

$$ (15) \quad p_s^* = \left[ \frac{(1-\alpha_T)^{\frac{\alpha_N(1-\alpha_T)}{\alpha_T}} (\alpha_T)^{\alpha_N}}{(1-\alpha_N)^{1-\alpha_N} \alpha_N^{\alpha_N}} \right] M \left[ \frac{(A_{T,s})^{\frac{\alpha_N}{\alpha_T}}}{A_{N,s}} \right] r_{T,s}^{1-\frac{\alpha_N}{\alpha_T}} $$
Assuming perfect competition in both sectors, perfect international capital mobility, intersectoral mobility of productive factors and constant returns to scale, Obstfeld and Rogoff (1996) conclude that only supply factors, namely productivity shocks and the real interest rate, are likely to influence the real exchange rate. Here, the nontradable firms face monopolistic competition, so that demand patterns may influence the real exchange rate. More specifically, the intratemporal elasticity of substitution between tradables and nontradables for private consumers, θ, plays a role: the higher it is, the lower the mark-up is (and the closer we get to perfect competition). The real exchange rate then depreciates so as to equate supply and demand for nontradables, the equilibrium on the market for tradables remaining unaltered.

### 2.2.4 Impact of supply shocks

In this context, we can analyse how different shocks - positive productivity shocks in each sector and an increase of the real interest rate - impact the real exchange rate.

A positive productivity shock in tradables raises the *ex ante* marginal productivity of capital of the sector. Therefore, the capital-labor ratio increases in tradables, so as to equal *ex post* the marginal productivity of capital and the marginal cost $r_f$, given by the world real interest rate. The productivity shock, combined with the increase of the capital-labor ratio in tradables raises the marginal productivity of labor as well, inducing an increase in the real wage in terms of tradables. This wage increase spreads through the whole economy. In the nontradable sector, as the relative cost of labor increases, the capital-labor ratio rises and the marginal productivity of capital declines. The only way to get marginal productivity back to the real cost of capital is for the real cost in terms of nontradables to fall as well. Equilibrium is obtained through an increase of nontradables prices in terms of tradables, namely an appreciation of the real exchange rate, a phenemonon known as the “Harrod-Balassa-Samuelson effect” (Balassa, 1964).

Productivity shocks in nontradables do not have any impact on the tradables: capital-labor ratio and real wages in tradables, set by the factor-price frontier, remain unaltered. The relative cost of labor and the capital-labor ratio in nontradables remain unchanged as well. However the real exchange rate depreciates so as to allow *ex post* equalization between the factor real cost (multiplied by the mark-up) and marginal productivity<fnref n="5" /> for both productive factors.

Finally, increasing the real interest rate in terms of tradables leads to an *ex ante* decline of the marginal productivity of capital and the capital-labor ratio. This reduces the marginal productivity of labor and the real wage in the sector. Wage moderation spreads to the nontradable sector, inducing a substitution in favour of labor. Two mechanisms influence the real exchange rate: on the one hand, the interest rate hike pushes capital marginal cost up; on the other hand, capital productivity in tradables increases because of lower wages. According to the production process, the capital productivity rise may or may not exceed the initial interest rate hike. Nevertheless, when the nontradable sector is relatively more labor-intensive than the tradable sector ($α_N > α_T$), the capital productivity increase does exceed the interest rate hike. In this context, the real exchange rate depreciates to restore the balance between capital marginal productivity and cost. Nontradables are actually more labor-intensive than tradables in industrialized countries, and in particular in France (Prigent, 1999).

The table hereafter synthesizes the response of the real exchange rate to different shocks:

<footnote n="5">⁵ increased *ex ante* by the positive productivity shock in nontradables.</footnote>
<table><thead><tr><th></th><th>ΔA<sub>T</sub> > 0</th><th>ΔA<sub>N</sub> > 0</th><th>Δg > 0</th><th>Δθ > 0</th><th>Δr<sub>T</sub> > 0</th></tr></thead><tbody><tr><th>p</th><td>+</td><td>-</td><td>+</td><td>-</td><td>-</td></tr></tbody></table>

Apart from a modification in the shares of private and public demand of nontraded goods, the demand side does not play any role in the determination of the real exchange rate. More precisely, a rise in total demand (e.g. a decrease of the discount rate $\beta$) does not alter the long-run price of nontradables in terms of tradables. In the short term, the initial rise of the nontradables price following a demand shock fades away as domestic firms move from the tradable to the nontradable sector. The reallocation of resources in favour of nontraded goods forces the exchange rate back to its initial value. *Ex post*, factor marginal productivities, as well as capital-labor ratios, remain unchanged. The structure of activity, on the contrary, is modified: more domestic firms produce nontraded goods, the national tradable sector shrinks, and a larger part of traded goods domestically consumed is supplied by imports.
### III - The data

To test the model derived in the previous section on the French economy, we use annual data covering the period 1970-1997, taken from the macroeconomic NOUBA data base of INSEE.

#### 3.1 The sectoral desaggregation between tradables and nontradables sectors

Most empirical work on the real exchange rate consider the manufacturing and the services sectors as reasonable proxies of the tradable and nontradable sectors respectively. However, as highlighted by De Gregorio and alii (1994), the distinction between manufacturing industries and services is blurred by the development of traded services. These authors use a breakdown industry by industry based on the degree of openness. More specifically, the tradable sector is defined as all industries whose share of exports in the overall production is above 10 % since 1990.

We follow the same approach but our criterion of tradability includes imports as well. We define the degree of openness as the ratio [(exports + imports) /2 value added]. Table 1 shows the results for the 15 industries considered. When comparing the results over the period 1990-1997 with the results over 1970-1997, one difference appears: the services of financial institutions, once a nontradable industry, has recently shifted to the tradable part of the economy. As our econometric work covers the whole period (1970-1997) we have considered the services of financial institutions as nontradables.

<table><caption>Table 1: tradables and nontradables sectors</caption><thead><tr><th rowspan="2">Industry</th><th>average level of</th><th rowspan="2">Type of industry according to the 10% criterion</th></tr><tr><th>over the period 1990-1997 (en %)</th></tr></thead><tbody><tr><td>[U1] Agricultural, forestry, and fishery products</td><td>34.5</td><td>T</td></tr><tr><td>[U2] Agricultural and food industries</td><td>81.2</td><td>T</td></tr><tr><td>[U3] Energy</td><td>57.1</td><td>T</td></tr><tr><td>[U4] Intermediate goods</td><td>115</td><td>T</td></tr><tr><td>[U5A] Producer durables</td><td>108.1</td><td>T</td></tr><tr><td>[U5B] Consumer durables</td><td>151.6</td><td>T</td></tr><tr><td>[U5C] Automotive vehicles and other land transport equipment</td><td>158.7</td><td>T</td></tr><tr><td>[U6] Consumer non-durables</td><td>80</td><td>T</td></tr><tr><td>[U7] Products of building and construction, civil and rural engineering</td><td>0</td><td>N</td></tr><tr><td>[U8] Wholesale and retail trade</td><td>0</td><td>N</td></tr><tr><td>[U9] Transport and telecommunications services</td><td>12.3</td><td>T</td></tr><tr><td>[U10] Market services</td><td>7.4</td><td>N</td></tr><tr><td>[U12] Insurance services</td><td>7.4</td><td>N</td></tr><tr><td>[U13] Services of financial institutions</td><td>21.8</td><td>T</td></tr><tr><td>[U14] Non market services</td><td>0</td><td>N</td></tr></tbody></table>

As in the previous section the letter T refers to tradables, the letter N to nontradables.
Finally, we retain the following classification:

<table><thead><tr><th>Tradables</th><th>Nontradables</th></tr></thead><tbody><tr><td>[U1] Agricultural, forestry, and fishery products</td><td>[U7] Products of building and construction, civil and rural engineering</td></tr><tr><td>[U2] Agricultural and food industries</td><td>[U8] Wholesale and retail trade</td></tr><tr><td>[U3] Energy</td><td>[U10] Market services</td></tr><tr><td>[U4] Intermediate goods</td><td>[U12] Insurance services</td></tr><tr><td>[U5A] Producer durables</td><td>[U13] Services of financial institutions</td></tr><tr><td>[U5B] Consumer durables</td><td>[U14] Non market services</td></tr><tr><td>[U5C] Automotive vehicles and other land transport equipment</td><td></td></tr><tr><td>[U6] Consumer non-durables</td><td></td></tr><tr><td>[U9] Transport and telecommunications services</td><td></td></tr></tbody></table>

## 3.2 Solow residuals

In our model, technological shocks, among others, determine the real exchange rate in the long run. We have calculated Solow residuals assuming a constant returns to scale Cobb-Douglas production function: $Y_i=A_iL^{\alpha}K_i^{(1-\alpha)}$ (i=N,T). Technical progress can be calculated in each sector as the difference between the growth rate of value added and the growth rate of the production factors weighted by their income shares<fnref n="6" />. We obtain the following results:

<table><thead><tr><th></th><th>tradables</th><th>nontradables</th></tr></thead><tbody><tr><td>Solow residuals growth rate</td><td>2.5 %</td><td>- 0.2 %</td></tr><tr><td>Labour share</td><td>0.624</td><td>0.695</td></tr></tbody></table>

For the economy as a whole, the Solow residual growth rate stands at 1.1 %, a figure consistent with previous work on French data (Accardo et alii, 1999).

## 3.3 The other data used

The nominal long term interest rate r (on 10 years bonds) is deflated with the inflation rate in the tradable sector: $r_{T,s} = r - \dot{p}_{T,s+1}$;

The real exchange rate is defined as the ratio of the value added deflator of the nontradable sector vis-à-vis the tradable sector ($p_N/p_T$). As shown in graph 1 below, the real exchange rate exhibits a positive trend over the 1959-1997 period.

<footnote n="6">⁶ This result is classic under perfect competition. Under monopolistic competition, it still holds as long as the growth rates of the productive factors are weighted with income shares, which then differ from parameters α and 1-α.</footnote>
<figure><img src="image_1.png" /><figcaption>Graph 1: real exchange rate as the relative price of nontradables</figcaption></figure>

Public spending consists of public consumption (P30) and public investment (P41). Two series are used:

- public expenditures as a share of the value added of the non tradables sector (g) ;

- the level of public spending in nontraded goods and services (G<sub>N</sub>).
# IV - An empirical analysis on French data

The model presented in the second section yields a long-term relationship between:

- the real exchange rate (p) on the one hand;

- total factor productivities in both sectors ($A_T$ and $A_N$)$^7$, the interest rate deflated by inflation in the tradable sector ($r_T$), public spending on nontraded goods and services ($G_N$), the elasticity of substitution between tradables and nontradables on the other hand. The elasticity of substitution - a structural parameter - is considered as a constant term.

This relation has been tested on French data over the period 1970-1997. We have also investigated the Obstfeld-Rogoff model (1996) excluding demand factors. Stationarity tests (Dickey-Fuller (1979), Phillips Perron (1988), Kwiatowski, Phillips, Schmidt and Shin (1992)) show that all series are nonstationary, i.e their stochastic properties are not invariant with respect to time. We then test the model in a cointegration form.

## 4.1 Stationarity Analysis

Carrying out a sequential Dickey-Fuller test procedure, all series, in logarithms, appear to be nonstationary, as shown in the table 2. We have also carried out Phillips-Perron (1988) tests which differ from Dickey-Fuller as they propose a non parametric correction of the residual in order to cope with both autocorrelation and heteroscedasticity. We also conclude that all series are nonstationary.

<table><caption>Table 2: Dickey-Fuller tests</caption><thead><tr><th>Serie</th><th>Integrated of order one<br>around a trend</th><th>Integrated of order one</th></tr></thead><tbody><tr><td>ln p</td><td>X<sup>*</sup></td><td></td></tr><tr><td>ln A<sub>T</sub></td><td>X<sup>*</sup></td><td></td></tr><tr><td>ln A<sub>N</sub></td><td></td><td>X</td></tr><tr><td>ln <span style="font-size: 2em; vertical-align: middle;">(</span><table style="display: inline-table; vertical-align: middle; text-align: center;"><tbody><tr><td style="border-bottom: 1px solid black; padding: 0 0.2em;">α<sub>N</sub></td></tr><tr><td style="padding: 0 0.2em;">A<sub>T</sub>α<sub>T</sub></td></tr><tr><td style="border-bottom: 1px solid black; padding: 0 0.2em;">A<sub>N</sub></td></tr></tbody></table><span style="font-size: 2em; vertical-align: middle;">)</span></td></td><td>X<sup>*</sup></td><td></td></tr><tr><td>r<sub>T</sub></td><td></td><td>X</td></tr><tr><td>ln g</td><td></td><td>X</td></tr><tr><td>ln G<sub>N</sub></td><td>X<sup>**</sup></td><td></td></tr></tbody></table>

* deterministic trend

** quadratic trend

<footnote n="7">⁷ Sectoral total factor productivities can enter the regression separately or as a ratio. In the latter case, we may constrain the coefficient to one as predicted by the model, by using the variable $\frac{A_T^{\alpha_T}}{A_N}$ as an explanatory variable.</footnote>
However, Dickey-Fuller and Phillips-Perron tests have low power as they tend to accept too frequently the null hypothesis of non-stationarity in finite samples. We also carried out Kwiatowski, Phillips Schmidt and Shin (1992) tests, based on the null hypothesis of stationarity. Results are similar, except for the real exchange rate, which appears as stationary around a deterministic time trend. Contradictory conclusions for the real exchange rate may be due to the low power of those tests in finite samples. For homogeneity sake, we eventually consider all series as non-stationary, which allows us to attempt to identify a cointegration relation, i.e a stationary linear combination of the variables.

## 4.2 Estimation of a long-term relation

### 4.2.1 The empirical models

We study four models: one model based on supply-side factors exclusively, three other models including demand factors. Table 3 below presents the specification of each model. Model 1 is the supply-side Obstfeld-Rogoff model: assuming perfect competition, the real exchange rate depends on sectoral total factors productivities and the real interest rate. Model 2 includes the same variables as well as public spending, as a share of value added of the nontradable sector. Model 3a includes public spending in level terms. In model 3b we constrain the coefficients for sectoral productivities.

We test each model in the form of a cointegrating relation. More precisely, we use an univariate test<fnref n="8" />: each equation is estimated with the ordinary least squares procedure and we test the stationarity of the residual with an augmented Dickey-Fuller test. We use Engle-Yoo statistics to conclude. A smaller number of explanatory variables in model 3b (2 instead of 3 in model 3a) allows us to test their significance, following the Stock and Watson procedure (1993).

<table><caption>Table 3: The four models tested (dependent variable: lnρ)</caption><thead><tr><th></th><th colspan="6">Explanatory variables</th></tr><tr><th></th><th>r<sub>T</sub></th><th>lnρ</th><th>ln G<sub>N</sub></th><th>ln<br><span style="font-size: 2em; vertical-align: middle;">(</span><table style="display: inline-table; vertical-align: middle;"><tbody><tr><td style="border-bottom: 1px solid black; padding: 0 0.2em;">α<sub>N</sub></td></tr><tr><td style="padding: 0 0.2em;">A<sub>T</sub>α<sub>T</sub></td></tr><tr><td style="border-bottom: 1px solid black; padding: 0 0.2em;">A<sub>N</sub></td></tr></tbody></table><span style="font-size: 2em; vertical-align: middle;">)</span></th><th>ln A<sub>T</sub></th><th>ln A<sub>N</sub></th></tr></thead><tbody><tr><th>Expected sign</th><td>-</td><td>+</td><td>+</td><td>+</td><td>+</td><td>-</td></tr><tr><th>Model 1</th><td>X</td><td></td><td></td><td></td><td>X</td><td>X</td></tr><tr><th>Model 2</th><td>X</td><td>X</td><td></td><td></td><td>X</td><td></td></tr><tr><th>Model 3a</th><td>X</td><td></td><td>X</td><td></td><td>X</td><td></td></tr><tr><th>Model 3b</th><td>X</td><td></td><td>X</td><td>X</td><td></td><td></td></tr></tbody></table>

The role of interest rates deserves special attention in the French context.

Although interest rates play a major role in the theory of investment and consumption decisions, it is traditionally very difficult to exhibit an interest rate effect on French historical data. Up to the mid-eighties, the monetary and financial framework was based on regulatory measures to stem the growth of bank lending (quantitative credit controls), interest rate subsidies and foreign exchange controls. In this context, market long term interest rates could not be regarded as relevant for private sector decisions.

<footnote n="8">⁸ We first implemented the Johansen multivariate procedure but due to the small number of observations, results turned out to be inconclusive.</footnote>
1986 marked the end of this area of financial repression. That year, interest rates became the chief instrument through which the monetary authorities sought to achieve their objectives. For this purpose, the financial markets were reorganized and deregulated, quantitative credit controls were removed and foreign exchange controls were eased considerably. Since then, financial deepening has allowed long term market interest rates to play a major role. Disintermediation and standardization in lending to companies have undoubtedly made the impact of monetary policy and international developments on capital markets on long term interest rates more direct. This is why long-term interest rates enter our estimation from 1986 onwards.

### 4.2.2 The results

<table><caption>Table 4: Real exchange rate equations - Estimation results and cointegrating tests</caption><thead><tr><th>Estimating period: 1970-1997</th><th>Model 1</th><th>Model 2</th><th>Model 3a</th><th>Model 3b</th></tr></thead><tbody><tr><td><u>Explanatory variables</u></td><td></td><td></td><td></td><td></td></tr><tr><td>ln A<sub>T</sub></td><td>0.94</td><td>0.76</td><td>0.16</td><td>-</td></tr><tr><td>ln A<sub>N</sub></td><td>0.35</td><td>0.90</td><td>-0.63</td><td>-</td></tr><tr><td>Productivities ratio</td><td>-</td><td>-</td><td>-</td><td>0.27</td></tr><tr><td>r<sub>T</sub></td><td>-0.79</td><td>-0.79</td><td>-0.61</td><td>-0.72</td></tr><tr><td>ln g</td><td>-</td><td>0.81</td><td>-</td><td>-</td></tr><tr><td>ln G<sub>N</sub></td><td>-</td><td>-</td><td>0.79</td><td>0.64</td></tr><tr><td>Constant</td><td>-0.02</td><td>0.71</td><td>-10.6</td><td>8.55</td></tr><tr><td><u>Cointegrating test</u></td><td></td><td></td><td></td><td></td></tr><tr><td>ADF statistic</td><td>4.22</td><td>3.86</td><td>4.53</td><td>4.60</td></tr><tr><td>Critical value at the 5% (resp. 1%)</td><td>3.98</td><td>4.15</td><td>4.15</td><td>3.98</td></tr><tr><td>confidence level - 50 observations</td><td>(3.67)</td><td>(3.85)</td><td>(3.85)</td><td>(3.67)</td></tr><tr><td>Existence of a cointegrating relation</td><td>X</td><td></td><td>X</td><td>X</td></tr></tbody></table>

Reading: The augmented Dickey-Fuller test includes one lag in the equation testing the stationarity of the residual. The critical values are non standard and depend on the number of observations. They have been tabulated by Engle-Yoo (1987). For example, for the model 1, the test statistic is -4.22, and the critical value at the 5% level of confidence is -3.98, which leads us to reject the null hypothesis of the test, that is to say the non stationarity of the residual. As a consequence, we accept the existence of a cointegrating relation.

**The estimates do not appear to be entirely consistent with the predictions of the Obsfeld-Rogoff supply side model.**

In the supply side model presented by Obstfeld and Rogoff (model 1), the real exchange rate exclusively depends on sectoral productivities and on the real interest rate. A cointegrating relation can be found on French data for this model (see table 4), but the coefficient of the productivity in the nontradable sector exhibits the wrong sign: it appears positive whereas it is expected to be negative, as a slower productivity growth in the nontradable sector should lead to an appreciation of the real exchange rate. This apparently puzzling result may be due to missing variables: as suggested in our theoretical model, demand factors may also play a role if imperfect competition is assumed in the nontradable sector.
Public spending also plays a role, as highlighted by a cointegrating relation.

In models 2 and 3 we try to identify the possible role of public expenditures on nontraded goods and services. Public spending has been tested in two slightly different forms: as a share of nontradables value added and in volume terms. Although our model highlights the role of public spending as a share of nontraded goods and services value added, no cointegration relation can be found when including this variable (model 2). This may be due to the very limited variance of the ratio over the period.

By contrast, the ADF test supports a cointegrating relation between the real exchange rate, the productivity ratio, the real interest rate and public spending in level terms at a 5 % confidence level (model 3a). Our estimates are in line with the predictions of the model. A productivity shock in the nontradable sector, a rise in public spending or a fall in the real interest rate lead to an appreciation of the real exchange rate.

To support that evidence, we test the significance of the coefficients. As the possible endogeneity of the explanatory variables does not allow us to carry out standard tests of significativity, we use the Stock and Watson (1993) methodology, adding in the regression leads and lags of the explanatory variables in difference terms. Given the limited number of observations, we constrain the coefficients for sectoral productivities, in line with the theoretical model (model 3b) and introduce only one lead and one lag for each explanatory variable<fnref n="9" />. This gives the following cointegrating relation:

$$ \ln p = 0.25 \ln \left( \frac{\frac{\alpha_N}{A_T}}{A_N} \right) - 1.27 r_T + 0.77 \ln(G_N) - 10.3 \quad (2.78) \quad (-12.0) \quad (11.2) $$

The estimated coefficients all fit the theoretical model and are significant. Either a rise in domestic public spending, a quicker pace of productivity gains in the tradable sector or a fall in the “world” interest rate lead to an appreciation of the real exchange rate.

<footnote n="9">⁹ If the residuals are not autocorrelated, this simple transformation is sufficient for the Student statistics to asymptotically follow a standard normal. In the frequent case of autocorrelation, however, the Student statistics have to be multiplied by a factor which corrects for the autocorrelation of the residuals, which we have done there.</footnote>
Conclusion

Our analysis of the real exchange rate, defined as the relative price of nontradables in terms
of tradables, is based on an equilibrium approach. We extend the Obstfeld-Rogoff model by
assuming imperfect competition in the nontradable sector, therefore allowing demand
factors to play a role in the determination of the real exchange rate. Our evidence on French
data provides support for the extended version of the Obstfeld-Rogoff model. One
contribution of this paper is to show that a long-term relationship can be found between the
real exchange rate, the real interest rate and sectoral productivities, once public spending is
also taken into account.

Some further work could be considered. It would in particular be useful to establish a link
between the relative price of nontradables and the macroeconomic real exchange rate,
defined as the relative price of domestic tradables vis-à-vis foreign tradables in a common
currency. This requires taking into account the relative price of nontradables vis-à-vis
tradables for different countries.
References

Accardo J., Bouscharain L., Jlassi M. (1999), « Le progrès technique a-t-il ralenti depuis 1990 ? » Economie et Statistique n°323, 53-72

Balassa B. (1964), "The purchasing power doctrine: a reappraisal", *Journal of Political Economy* 72

Borowski D., Couharde C., Thibault F. (1997), "Sensibilités des taux de change d'équilibre aux output gaps et aux cibles de balance courante", *Document de travail n°97-3*, Direction de la Prévision.

De Gregorio J., Giovannini A., Wolf H. C. (1993), " The behavior of Nontradable Goods Prices in Europe: Evidence and Interpretation ", *International Monetary Fund Working Paper*

De Gregorio J., Giovannini A., Krueger T. H. (1994), " International evidence on tradables and non tradables inflation ", *European Economic Review* 38 (6), 1225-44

Dickey D.A., Füller W.A. (1979), " Distribution of the estimators for autoregressive time series with a unit root ", *Journal of the American Statistical Association* 74, 427-31

Edwards S. (1989), Real Exchange Rates, Devaluation and Adjustment: Exchange Rate Policy in Developing Countries, *MIT Press*

Engle R. F., Yoo B. S. (1987), "Forecasting and testing in cointegrated systems", *Journal of Econometrics* 35, 143-59

Froot K.A., Rogoff K. (1994), " Perspectives on PPP and Long-Run Real Exchange Rates ", *Working Paper*, n° 4952, NBER.

Hamilton J. D. (1994), " Time series analysis ", Princeton University Press

Johansen S., Juselius K. (1990), " Maximum likelihood estimation and inference on cointegration –with application to the demand of money ", *Oxford Bulletin of Economics and Statistics* 52 (2), 169-210

Joly H., Prigent C., Sobczak N. (1996), " Le taux de change réel d'équilibre: une introduction ", *Economie et Prévision* 123-24, 1-21

Kwiatkowski D., Phillips P. C. B., Schmidt P. et Shin Y. (1992), " Testing the null hypothesis of stationarity against the alternative of a unit root ", *Journal of Econometrics* 54, 159-78

Obstfeld M., Rogoff K. (1996), " Foundations of international macroeconomics ", *MIT Press*

Obterwald-Lenum M. (1992), " A note with quantiles of the asymptotic distribution of the maximum likelihood cointegration rank test statistics ", *Oxford Bulletin of Economics and Statistics* 54 (3), 461-72

Phillips P. C. B., Perron P. (1988), " Testing for a unit root in time series regression ", *Biomètrika* 75 (2), 335-46

Prigent C. (1999), " La part des salaires dans la valeur ajoutée en France : une approche macroéconomique ", *Economie et Statistique n°323*, 73-94.
**Solow R. M. (1957)**, " Technical change and the aggregate production function", *Review of Economic and Statistics* 39, 312-20

**Stock J., Watson M. (1993)**, " A simple estimator of cointegrated vectors in high order integrated systems ", *Econometrica* 61, 783-820

**Williamson J. (1994)**, " Estimates of FEERs ", in *Estimating Equilibrium Exchange Rates*, ed. John Williamson, Washington Institue for International Economics.

**Wren-Lewis S., Driver R.L., (1998)**, " Real Exchange Rates for the year 2000 ", *in Policy analyses in international economics*, Washington Institue for International Economics.
# Appendix

## The Factor Price Frontier

In the long run, firms can adjust both capital and labour. First order conditions for profit maximization equate marginal value products of labour and capital to the current wage and interest rates:

$$ egin{cases} p_s^* A_{N,s} \alpha_N (k_{N,s}^*)^{1-\alpha_N} = M w_{T,s}^* \\ p_s^* A_{N,s} (1-\alpha_N) (k_{N,s}^*)^{-\alpha_N} = M r_{T,s} \end{cases} $$

Making use of these first order conditions gives the capital-labour ratio in the traded goods sector:

$$ k_{N,s}^* = \frac{1 - \alpha_N}{\alpha_N} \frac{w_{T,s}^*}{r_{T,s}} \quad (1) $$

Substituting $k_{N,s}^*$ in one of the first order conditions for $k_{N,s}^*$ in equation (1) defines a negative relationship between the real interest rate and the real wage.

$$ \left( \frac{w_{T,s}^*}{p_s^*} \right)^{\alpha_N} \left( \frac{r_{T,s}}{p_s^*} \right)^{1-\alpha_N} = A_{N,s} \frac{1}{M} (\alpha_N)^{\alpha_N} (1-\alpha_N)^{1-\alpha_N} $$

The assumption of perfect capital mobility ties domestic interest rate to the world interest rate. Given the interest rate, the price factor frontier can therefore be written as a mapping from the interest rate to the domestic wage.

<figure><img src="image_2.png" /></figure>

Likewise, a factor price frontier can be drawn in the non traded goods sector:

$$ (w_{T,s}^*)^{\alpha_T} (r_{T,s})^{1-\alpha_T} = A_{T,s} (1-\alpha_T)^{1-\alpha_T} (\alpha_T)^{\alpha_T} $$