# Growth, Trade and Inequality in an Efficiency-Wage Framework

Rodrigue Mendez

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Distributed under a Creative Commons Attribution 4.0 International License
CAHIERS ECO & MATHS

98.10

GROWTH, TRADE AND INEQUALITY
IN AN EFFICIENCY-WAGE FRAMEWORK

Rodrigue MENDEZ

MAD
# Growth, Trade and Inequality in an Efficiency-Wage Framework

Rodrigue Mendez

MAD, 106-112 boulevard de l'Hôpital, 75647 Paris Cedex 13
and Université du Littoral

Email:mendez@univ-Paris1.fr.

February 1998

## Abstract

The last twenty years have seen a rise of inequalities and a slowdown of growth in the U.S. and Europe. It is common wisdom to interpret the rise of inequalities as the result of a skill-biased labor demand shift. While there is strong evidence that such a shift has happened, this overlooks the fact that inéquality has also markedly risen between individuals with the same qualification. Moreover, endogenous growth models usually predict that this kind of labor demand shift should have been growth enhancing.

The purpose of this paper is to build an endogenous growth model able to account for the rise of inequality among identical individuals as well as the growth slowdown. Inequality is introduced in the model through the hypothesis of a dual labor market where the primary sector workers must be paid efficiency-wages in order to induce effort. The analysis of the long run equilibrium yields then interesting results. We show that growth may increase inequality between otherwise identical individuals and that shifts in the parameters of the efficiency wage relation may account for the observed pattern of growth and inequality. Eventually, we show that international trade increases growth and inequality.

JEL classification: D33, F43, 033.

Keywords: Efficiency wages, endogenous growth, dual labor markets, inequality, international trade.

## Résumé

Les vingt dernières années ont été caractérisées, aux Etats Unis comme en Europe, par une hausse des inégalités et une baisse de la croissance. Il est assez habituel d'interpréter cette hausse des inégalités comme le résultat d'une baisse de la demande travail peu qualifié. S'il existe de nombreuses indices tendant à justifier une telle approche, il convient de souligner que la hausse des inégalités a aussi concerné des individus, à priori, identiques. Il est, en outre, difficile d'expliquer ces deux faits stylisés dans le cadre des modèles de croissance, puisque ceux-ci ont plutôt tendance à associer une accélération de la croissance à ce type d'évolution de la demande du travail.

L'objectif de cet article est de construire un modèle de croissance endogène qui puisse rendre compte à la fois du creusement des inégalités entre des individus identiques et de la baisse du taux de croissance. Nous faisons l'hypothèse d'un marché du travail dual où les travailleurs du secteur primaire sont payés des salaires d'efficience. Nous montrons qu'une accélération de la croissance peut entraîner une hausse des inégalités entre des individus identiques et que des changements dans les paramètres de la relation d'efficience peuvent rendre compte de l'évolution observée de la croissance et des inégalités. L'étude des conséquences de l'ouverture des échanges nous permet de mettre en évidence un lien entre commerce extérieur, croissance et inégalités.

Classification JEL: D33, F43, 033.

Mots clés: Salaires d'efficience, croissance endogène, dualité du marché du travail, inégalités, commerce international.
# Growth, Trade and Inequality in an Efficiency-Wage Framework

Rodrigue Mendez *

February 1998

## Abstract

The last twenty years have seen a rise of inequalities and a slowdown of growth in the U.S. and Europe. It is common wisdom to interpret the rise of inequalities as the result of a skill-biased labor demand shift. While there is strong evidence that such a shift has happened, this overlooks the fact that inequality has also markedly risen between individuals with the same qualification. Moreover, endogenous growth models usually predict that this kind of labor demand shift should have been growth enhancing.

The purpose of this paper is to build an endogenous growth model able to account for the rise of inequality among identical individuals as well as the growth slowdown. Inequality is introduced in the model through the hypothesis of a dual labor market where the primary sector workers must be paid efficiency-wages in order to induce effort. The analysis of the long run equilibrium yields then interesting results. We show that growth may increase inequality between otherwise identical individuals and that shifts in the parameters of the efficiency wage relation may account for the observed pattern of growth and inequality. Eventually, we show that international trade increases growth and inequality.

JEL classification: D33, F43, 033

Keywords: Efficiency wages, endogenous growth, dual labor markets, inequality, international trade.

## 1 Introduction

The 1980s and early 1990s can be characterized as a period of slow growth and rising inequality in the U.S. and Europe. Most of the rise of inequalities has been interpreted as the result of a skill-biased labor demand shift <fnref n="1" />. Two main explanations have been advanced to account for the alleged demand shift. The first holds that technological change has been biased towards highly skilled workers <fnref n="2" />. The second attributes the demand shift to the rise of inter-

<footnote>*MAD, Université de Paris I, Panthéon-Sorbonne, 106-112 boulevard de l'Hôpital, 75647 Paris Cedex 13 and Université du Littoral. Email:mendez@univ-Paris1.fr.</footnote>

<footnote n="1">¹It is, indeed, well documented that: (a) the wage premium of College education rose dramatically in the U.S. from a low of 31% in 1979 to 53% in 1993 and (b) the experience premium (i.e. the wage premium given by ten years of labor experience) also rose markedly in the 1980s (Gootschalk, 1997).</footnote>

<footnote n="2">²See Berman, Bound, and Griliches (1994) and Krueger (1993) for direct evidence on the relationship between skill-biased technological change and changes in the wage structure.</footnote>
national trade with low-wage countries <fnref n="3" />. Ironically trade theorists have been vocal advocates the technological explanation of the rise of inequalities. Their main argument is that, so far, estimates of the consequences of trade on wages have yield very low values. Therefore trade can't account for the bulk of the rise of inequality (Krugman and Lawrence, 1994; Lawrence and Slaughter, 1993) <fnref n="4" />. This may explain why skill-biased technological change is widely considered as the main engine of the recent rise of inequality. Is this approach fully satisfying? Several objections can be raised.

Skill-biased technological change and the growing integration of the world economy are not two independent phenomenon. It has been well documented that technological progress in transport and telecommunications have increased the range of tradable goods and allowed a greater, finer slicing-up of value-added. A number of industries have in effect became "footloose". Footwear is a good example. Production is entirely outsourced in low-wages less developed countries, while Headquarters and other high skill high value added activities such as R&D and marketing remains the U.S. or Europe. Conversely, skill biased technological change may be driven by the rise international trade. Sachs and Shatz (1996) points that many skill intensive technologies have high fixed cost. Thus adoption of those technology is conditioned by market size and can be triggered by economic integration.

The logic of skill-biased technological change is, yet, poorly understood. Endogenous growth model à la Romer suggest a plausible link between technological change and the wage structure (Lai, 1995; Mendez, 1997). But these models can't account for the evolution of growth and inequality in the last 20 years. They predict that a skill-biased technological shift must be growth enhancing on the premiss that the high technology sectors that drive growth are skill intensive. However, these model suggest another possible linkage between economic integration and inequality through induced skill-biased technological change. They state that the increase of market size and technological diffusion brought by economic integration may rise growth and the relative wage of the high skilled.

The attention given to the rise of the wage differential between high-skilled and low-skilled workers has concealed the fact that inequality has increased not only among individuals with different education and experience, but also within groups of workers with the same education and experience (Gootschalk, 1997). This can't be explained by the usual framework.

The purpose of this paper is therefore to build an endogenous growth model able to account for the rise of inequality among identical individuals as well as the growth slowdown. Our model belongs (loosely) to the classic Grossman and Helpman (1991) family. An aggregate consumption good is produced using a continuum of intermediate goods. Growth is driven by the invention of new varieties of intermediate goods by rational, profit seeking entrepreneurs. Inequality is introduced in the model through the hypothesis of a dual labor market where the primary sector workers have to be paid efficiency-wages à la Shapiro-Stiglitz in order to induce effort <fnref n="5" />. The arrival of new varieties leads to labor reallocations in the primary sector: jobs are destroyed in the "old" intermediate sectors and created in the "new" intermediate sectors. Therefore growth generates a perpetual movement of "creative destruction" as "old" jobs are

<footnote n="3">³See Borjas and Ramey (1994) and Wood (1994, 1995) for (high) estimates of the impact of trade on the wage structure</footnote>

<footnote n="4">⁴Their methodology is contradicted, by Sachs and Shatz (1994, 1996)</footnote>

<footnote n="5">⁵This structure of the labor market is taken from Jones (1987) and Agénor and Aizeman (1997).</footnote>
replaced by "new" jobs. Efficiency wages create - through creative destruction - a linkage between the rate of growth of the number of intermediates and the relative wage of primary and secondary sector workers. Otherwise identical workers are paid an efficiency bonus in the primary sector. This bonus increases with growth, since the penalty for shirking (the utility cost of the loss of a primary job) decreases with the rate of job destruction.

*A note on the bonding critique.* Efficiency wages have been subject to severe criticism. Carmichael (1985) argues that a firm could avoid the payment of an efficiency-wage bonus by forcing workers to post a bond when hired, on which they would earn some annuity while employed, and that they would forfeit if caught shirking. Is the argument convincing? Katz (1986) notes that bonding is rarely observed in practice and is subject to severe legal limitations (except, maybe, in countries where slavery is allowed...). This problem could be solved with the intervention of a third party. But such a contract would be costly to enforce and subject to frequent litigations. Also, the risk of collusion between the firm (or the worker) and the third party must be taken into account. Paying efficiency wages may be a better alternative. Cahuc and Zylberberg (1996) argue that firms can induce effort using a wage profile that increases with experience. While this seems more feasible than the bonding mechanism, most of the Katz critique can be applied (indeed, both mechanism are formally equivalent). The slope of the wage schedule is limited both by law (minimum wages) and by custom (wage increase not only with experience, but also with the age). Indeed, the relevance of this mechanism is inversely related to the extent of job destruction for the slope of the wage profile increases with the extent of job reallocation.

The remainder of the paper proceeds as follows. Section 2 describes the structure of the dual labor market and the efficiency-wage relation. Section 3 outlines the growth model. We specify the link between growth and creative destruction and show the existence of a unique long run steady state equilibrium growth path. This steady state equilibrium is represented as the intersection of two curves in the (growth, relative wage) plane: the LMC curve, which summarizes the technology side of the model and the NS curve, which summarizes the efficiency-wage side of the model. Section 4 explains the slope of the NS and LMC locus. The long run consequences of changes in the technology and efficiency-wage parameters of the models on the growth and inequality are studied as shifts of these two curves. Section 5 describes the impact of international trade on long-run growth and inequality.

# 2 The dual labor market

## 2.1 Workers and rentiers

The model is set in continuous time. There are two types (or classes) of individuals : workers and rentiers. The former supply labor (which entails a disutility cost) while the latter hold all the asset of the economy. We thus assume that workers don't save <sup>6</sup>. Individuals within each class are infinite lived and risk neutral.

Hence, the rentiers choose savings so as to maximize the following expected discounted

<footnote n="6"><sup>6</sup>This Kaldorian assumption may be true on a steady growth path even if workers are allowed to save. This is shown by Bertola (1993) in the context of the Romer (1990) model.</footnote>
utility:

$$V = E \int_{0}^{\infty} c_t \exp(-\rho t) dt$$

where c is consumption and ρ the (constant) discount rate. Their instantaneous budget constraint is thus, taking the consumption good as numeraire: <fnref n="7" />

$$\dot{b} = rb - c$$

where b denotes assets and r the real interest rate.

It is straightforward to show that the consumption path must equalize the real interest rate and the discount rate.

$$r = \rho \quad (1)$$

Workers choose effort so as to maximize the following expected discounted utility:

$$V = E \int_{0}^{\infty} (w_t - e_t) \exp(-\rho t) dt$$

where w denotes the real wage and e the utility cost of effort.

## 2.2 The dual labor market

The basic structure is derived from The labor market is divided in two sectors: the primary and the secondary sector. Both sectors use the same kind of labor, but effort is not observable in the primary sector. We will show below that this implies that the primary sector must pay wages above marginal productivity. Therefore we may say that the primary sector offers "good jobs" i.e. high wage jobs that require a lot of motivation and responsibility whereas the secondary sector offers "cashier jobs" i.e. low wage, low responsibility jobs. Workers first look for a "good job", then go for a "cashier job" if they fail to get one. Let us describe the working of the efficiency wage sector.

A worker employed in the primary sector either works, and suffer a utility loss of e > 0, or she does not, in which case e = 0. Those who work supply one unit of labor. The opportunity to shirk is due to the fact that firms can't monitor perfectly on-the job effort in the primary sector. The monitoring technology is such that there exists a constant probability q that a worker engaged in shirking is caught. If caught, the worker is fired. Therefore firms set primary sector workers' wage so as to deter them from shirking and induce them to provide the required level of effort.

Worker employed in the secondary sector can't shirk (since effort is controllable at no cost). They always supply one unit of labor. In both sectors, the utility cost of effort is proportional to the real wage (e = cw, 0 < c < 1).

Let m be the turnover rate per unit of time, a the hiring rate, $L_P$ and $L_S$ the number of workers employed in the primary sector and secondary sectors. Both transition probabilities are endogenous and will be explicited later. Assuming that nobody shirk in equilibrium, the following condition must be verified in steady state:

$$mL_P = aL_S \quad (2)$$

<footnote n="7">⁷Here and henceforth we suppress time arguments where no confusion arises from doing so</footnote>
## 2.3 The no-shirking condition

We use the asset equations approach to derive the efficiency wage $w_P$ any employee in the primary sector must be offered to elicit positive effort. An agent that supplies $e > 0$, has an expected lifetime utility $V_E$ that obeys

$$rV_E = w_P(1-c) + m(V_0 - V_E) + \dot{V}_E \quad (3)$$

where $V_0$ is the expected lifetime utility of a worker employed in the secondary sector. If the agent shirks, its expected lifetime utility $V_S$ satisfies

$$rV_S = w_P + (m+q)(V_0 - V_E) + \dot{V}_S \quad (4)$$

Let's assume there exists a steady state growth path where consumption, wages and the expected lifetime utility of any agents all grow at the same rate. Let $g_c$ be this common growth rate. Equation (3) and (4) can now be rewritten in the following form:

$$(r - g_c)V_E = w_P(1 - c) + m(V_0 - V_E) \quad (5)$$

$$(r - g_c)V_S = w_P + (m+q)(V_0 - V_E) \quad (6)$$

These two valuation equations are very similar to those obtained in a static framework. The dividend plus the expected capital gain (from a change in employment status) is equal to the asset value times the interest rate minus the growth rate.

We can now derive the usual no-shirking condition (NS). A worker chooses to supply $e > 0$ as long as $V_E \ge V_S$. This implies that:

$$w_P \ge \frac{V_0(r - g_c)}{(1 - c) - (r - g_c + m)c/q} \quad (7)$$

So the NS condition requires that the expected loss from a transition out of the secondary sector at least balance the instantaneous dividend gain that accrues to a shirker.

The following step in the specification of equilibrium is to solve for $V_0$. Let $w_S$ be the wage in the secondary sector. Assuming that $w_S$ and $V_0$ both grow at the rate $g_c$, the relevant asset equation is:

$$(r - g_c)V_0 = w_S(1 - c) + a(V_E - V_0) \quad (8)$$

We can now compute $V_0$. By (5) and (8):

$$V_0 = (1 - c) \frac{w_S(r - g_c + m) + a w_P}{(r - g_c)(r - g_c + a + m)} \quad (9)$$

(7) and (9) give the steady state NS:

$$q \frac{1-c}{c} (1 - \omega) - (r - g_c + a + m) \ge 0 \quad (10)$$

where $\omega = w_S/w_P$ is the relative wage.

Equation (10) implies that people working in the primary sector receive a wage premium, which is needed to induce them to supply effort (since $a, m > 0$ and $r - g_c > 0$ $^8$, we must have $\omega < 1$).

To go further, we must specify the endogenous transition probabilities between employment and unemployment ($a$ and $m$) and the relative wage ($\omega$). This will be done in the next section.

<footnote>$^8$This condition is necessary for the expected lifetime value to converge.</footnote>
# 3 The growth model

## 3.1 The basic framework

The basic framework of the growth model is (loosely) derived from Grossman and Helpman (1991) and Barro and Sala-I-Martin (1995). Perfectly competitive firms manufacture a final good using a set of horizontally differentiated intermediate goods. Total factor productivity in final production increases with the number of available differentiated inputs. New varieties become available once they have been developed in the research lab. Labor is homogeneous and may be employed either in the efficiency wage sector or in the secondary sector.

The production function of the consumption good is derived from Ethier (1982):

$$Y = \left( \int_{0}^{A} x(i)^{\alpha} di \right)^{1/\alpha}$$

where $0 < \alpha < 1$, $Y$ is output, $x(i)$ is the quantity employed of the $i$th type of nondurable intermediate good, $A$ is the number of intermediate goods.

We assume perfect competition between final good producers. The demand of intermediate $i$ can be found using Shephard's lemma:

$$x(i) = \left( \frac{p(i)}{P} \right)^{-\epsilon} Y \quad (11)$$

where $p(i)$ is the price of intermediate $i$, $\epsilon = 1/1 - \alpha > 1$ is the constant elasticity of substitution between varieties and $P$ is the price of the final good (note that $P$ can also be interpreted as an aggregate price index of the intermediates). The choice of numeraire implies that:

$$P^{1-\epsilon} = \int_{0}^{A} p(i)^{1-\epsilon} di = 1 \quad (12)$$

Intermediates goods are produced with the following technology:

$$x(i) = \left( \frac{L_P(i)}{\mu} \right)^{\mu} \left( \frac{L_S(i)}{1-\mu} \right)^{1-\mu} \quad (13)$$

where $0 < \mu < 1$. $L_P(i)$ and $L_S(i)$ denotes, respectively, primary and secondary labor. We assume perfect competition in the secondary labor market. Intermediate goods producers choose $w_P$, $L_P(i)$, $L_S(i)$ to minimize cost and induce effort. This implies that the relative wage $w_S/w_P$ must verify the NS condition as well as the following relation:

$$\frac{w_S L_S(i)}{w_P L_P(i)} = \frac{1-\mu}{\mu} \quad (14)$$

The intermediate goods are manufactured by a continuum set of monopolistic competitors, each one of whom holds the patent to a countable number of varieties. The patents are the outgrowth of prior investment in R&D. It is easy to show that each monopolist sets a price that is a constant mark-up over its marginal cost. Hence the price of intermediate $i$ is:

$$p(i) = \frac{CU}{\alpha} = p \quad (15)$$
Where $CU = w_P^\mu w_S^{1-\mu}$ is the constant unit cost

Resources devoted to research generate "blueprints" that expand the measure of intermediates. Research outlays are made by private, profit-maximizing entrepreneurs, who appropriate some of the benefits from a new technological innovation in the form of a stream of oligopoly profits. We assume that innovators receive indefinite patent protection. Let $\pi$ be the profit of an intermediate firm, $v$ the value of an innovation. The following no-arbitrage condition must hold:

$$r = \frac{\pi}{v} + \frac{\dot{v}}{v} \qquad (16)$$

We specify the R&D technology as follows. R&D firms use the final good to produce new blueprints. The cost of each new blueprint is $fA^\lambda$, where $\lambda = 1/(\epsilon - 1) - 1 < 0$ (assuming that $\epsilon > 2$). Hence, the cost of each new blueprint decreases with the number of intermediate goods discovered. This can be justified by assuming that the number of blueprints $A$ is a proxy of the non-rival knowledge accumulated in R&D sector. Free entry into R&D implies that the value of each innovation must not exceed its unit cost, so that:

$$v \begin{cases} \le fA^\lambda & \text{if } \dot{A} = 0 \\ = fA^\lambda & \text{if } \dot{A} > 0 \end{cases} \qquad (17)$$

The model is closed with the labor market clearing condition. Let $u$ denote the percentage of workers employed in the secondary sector. Hence:

$$L_S = \int_0^A L_S(i) di = uL \qquad \text{and} \qquad L_P = \int_0^A L_P(i) di = (1-u)L \qquad (18)$$

(14) and (18) yield the relative wage and the participation condition:

$$\omega = \frac{w_S}{w_P} = \frac{1 - \mu}{\mu} \frac{1 - u}{u} \le 1 \quad \Leftrightarrow \quad u \ge 1 - \mu \qquad (19)$$

Given that all available intermediate enter symmetrically into the production function and all bear the same price, each one is demanded to the same extent $x = x(w_P, w_S)$. This, (13) and (14) implies that aggregate production can be simplified to

$$Y = A^{1/(\epsilon-1)} X = A^{1/(\epsilon-1)} \left(\frac{L_P}{\mu}\right)^{\mu} \left(\frac{L_S}{1-\mu}\right)^{1-\mu} \qquad (20)$$

where $X \equiv Ax$ is the aggregate quantity of intermediates used as an index of the amount of labor embodied in these intermediates. This implies that as long as the intersectoral labor allocation remains constant (which is the case in steady state), the growth rate of $Y$ is $g/(\epsilon - 1)$ (where $g \equiv \dot{A}/A$ is the rate at which new varieties are introduced to the economy). It is easy to prove that this is also the growth rate of wages, aggregate profits as well as the one of the lifetime utilities of all agents. Hence $g_c = g/(\epsilon - 1)$.
## 3.2 The steady state labor market clearing condition

We may now derive a relation between unemployment and growth that must hold in the steady state. By (12), (15), (17) and (20) the dividend rate of any firm producing intermediates is $\pi/v = (1 - \alpha)px/v = X/\epsilon f$. By (17) the value of blueprints increases at rate $\dot{v}/v = \lambda g$. This, (1), (16), (18) and (19) gives us the steady state labor market clearing (LMC) condition:

$$g = \frac{\frac{\omega^{\mu}}{1 - \mu(1 - \omega)} \frac{L}{\epsilon f} - \rho}{1 - \frac{1}{\epsilon - 1}} \quad (21)$$

## 3.3 Creative destruction and the transition rates between employment and unemployment

We can now compute the transition rates. The arrival of new varieties leads to a reallocation of labor within the primary sector and secondary sectors. Let $l_P = L_P/A$ be the number of primary jobs per firm. During an interval of length $dt$, $-Al_P dt$ "old" intermediate sector jobs are destroyed and $Al_P dt$ "new" intermediate sector jobs are created. Only secondary sector workers look for a new job in the primary sector $^9$. Assuming that all primary sector workers have the same probability of losing their job, the transition probability into the secondary sector is (using $x = L_P \omega^{\mu-1}/\mu A$, by (13) and (14)):

$$m = - \frac{Al_P}{L_P} = - \frac{l_P}{l_P} = - \frac{\dot{x}}{x} = g \quad (22)$$

In steady state the flows between the primary and the secondary sector are the same. This gives the transition probability into the primary sector:

$$a = \frac{L_P}{L_S} m = \frac{1 - u}{u} g \quad (23)$$

## 3.4 The steady state no-shirking condition

(1), (10), (19), (22) and (23) give the steady state NS condition:

$$g = \frac{q \frac{1-c}{c} (1-\omega) - \rho}{1 - \frac{1}{\epsilon-1} + \frac{\omega\mu}{1-\mu}} \quad (24)$$

## 3.5 Existence and Uniqueness of the steady state equilibrium growth path

A steady state long run equilibrium can be defined as any couple of feasible relative wage and blueprint growth rate that satisfies the LMC and NS conditions (i.e. $\omega \in ]0, 1]$ and $g \in ]0, \rho(\epsilon-1)[$, such that (21) and (24) are checked. The steady state equilibrium is depicted in figure 1.

<footnote>$^9$Since "good jobs" are homogeneous, primary sector workers have no incentive for on-the-job searching.</footnote>
**Proposition 1** The existence and uniqueness of the steady state equilibrium are guaranteed by the following hypothesis:

$$
\left\{
\begin{array}{ll}
H1 & \epsilon > 2 \\
H2 & q \frac{1-c}{c} - \rho > 0 \\
H3 & \frac{\left(1 - \frac{\rho}{q(1-c)/c}\right)^{\mu}}{1 - \mu \frac{\rho}{q(1-c)/c}} \frac{L}{\epsilon f} > \rho \\
H4 & \frac{(\bar{\omega}_{NS})^{\mu}}{1 - \mu(1 - \bar{\omega}_{NS})} \frac{L}{\epsilon f} < \rho(\epsilon - 1) \quad \text{if} \quad q \frac{1-c}{c} > \rho(\epsilon - 1)
\end{array}
\right.
$$

$$
\text{where } \bar{\omega}_{NS} = \frac{q \frac{1-c}{c} - \rho(\epsilon-1)}{q \frac{1-c}{c} + \rho(\epsilon-1) \frac{\mu}{1-\mu}}.
$$

*Proof*: see the appendix for a formal proof.

<figure><img src="image_3.png" /><figcaption>Figure 1</figcaption></figure>

4 Wage inequality and growth

Efficiency wages create an endogenous linkage between the rate of growth of the number of intermediates (*g*) and relative wages (*ω*). This link is embedded in the LMC curve, which summarizes the technology side of the model, and the NS curve, which summarizes the efficiency-wage side of the model. The shape of those curve is critical to the analysis that follows. As depicted in figure 1, the LMC curve slopes upward, while the NS curve slopes downward.
Let us explain the rising slope of the LMC curve. The maximum growth is obtained when both sectors pay the same wage (i.e. $\omega = 1$), which is the case when effort is perfectly observable in the primary sector. In that case, growth is entirely determined by the technology side of the model. The LMC curve yields $g = (\frac{L}{\epsilon f} - \rho)(\epsilon - 1)/(\epsilon - 2)$. Whenever $\omega \neq 1$, there is a misallocation of labor (if, for example, efficiency consideration leads to $\omega < 1$, too much labor goes to the secondary sector). This static inefficiency reduces production and profits. The fall of profits leads to a fall of growth. The farther the relative wage from 1, the bigger the fall of profits and growth.

Let us explain the decreasing slope of the NS curve. A rise of the growth rate has two types of effects on the relative wage. On one hand, it reduces the effective discount rate ($r - g_c$), which entails a rise of the value of future wages and, consequently, an increase of the penalty for shirking. This allows the firms to reduce the efficiency bonus they have to pay to induce effort. On the other hand, a rise of the growth rate leads to an increase of labor turnover (since both the rate of job destruction ($m = g$) and the hiring rate ($a$) go up). This reduces the penalty for shirking and entails a rise of the efficiency bonus. The second effect always dominates the first one.

We now turn to the analysis of the long run consequences of changes in the technology and efficiency-wage parameters of the model. We analyze the impact on growth and inequality of: (i) an increase of labor supply ($L$) or R&D efficiency ($1/f$); (ii) a decrease of the shirkers detection probability ($q$). Our measure of inequality will be the relative wage ($\omega$). Any fall (rise) of $\omega$ will be read as an increase (decrease) of inequality. This can be done in our model because factor inequality and functional inequality move about in the same direction. Thus, a decrease of the relative wage of the secondary sector workers will always come with an increase of their number ($uL$).<fnref n="10" />

Labor supply and R&D efficiency have the same impact on growth and inequality. Both bring about an increase of the dividend rate, since the rise of labor supply entails an increase of market size, and the fall of $f$ brings about a reduction of R&D cost. This leads, other things being equal, to an increase of the growth rate. The LMC curve shifts upward (as depicted in figure 2). The rise of $g$ brings about a rise of inequality. The cause of it is the conjunction of the twin effects described in the discussion about the shape of the NS curve. Lets recall the dominant one: the acceleration of innovation leads to a rise of labor turnover, which reduces the penalty for shirking and force employers to increase the efficiency bonus.

A fall of the detection probability leads to a reduction of the penalty for shirking, forcing the employers to increase the efficiency bonus. The NS curve shifts downward (as depicted in figure 3). Thus inequality rises and growth fall. The mechanism behind the fall of the growth rate is the one described in the discussion about the shape of the LMC curve. The fall of $\omega$ move away the economy farther from the static optimum ($\omega = 1$). This leads, through the reduction of the aggregate production and the profit rate, to a fall of investment and growth.

<footnote n="10">¹⁰There is, of course, another source of inequality in our model since the population is divided between rentiers and workers. It is easy to prove that the income share of rentier is constant and does not depend on the parameters mentioned above.</footnote>
<figure><img src="image_4.png" /><figcaption>Figure 3</figcaption></figure>

<figure><img src="image_5.png" /></figure>
# 5 The impact of International Trade on Growth and Inequality

We analyze the impact of international trade on long run growth and inequality. The world economy is made of two countries *home* and *foreign*. We use a star (*) as exponent to characterize the foreign prices and quantities. Both economies have the same structure but may differ on labor dotation (*L*), R&D efficiency (1/*f*), the number of intermediate goods that have been discovered before the opening of international trade (*A*) or the shirkers detection probability (*q*).

The first subsection show how the opening of international trade brings about a rise of growth and inequality. The growth effect of trade is linked to the increase of market size brought by trade. Bigger markets means, more profits per intermediate good and, therefore, a rise of dividend rates. This increases the incentives for R&D which leads to more growth. The inequality effect is linked to the rise of the growth rate. The mechanism is the one detailed in section 4 : the acceleration of innovation leads to a rise of labor turnover, which reduces the penalty for shirking and force employers to increase the efficiency bonus. The second subsection studies the equilibrium of an integrated world economy made of two countries. First, we show the existence and uniqueness of the steady state long run equilibrium growth path. Then we study the consequences of changes in the technology and efficiency-wage parameters of the model.

## 5.1 International Trade, market size and profitability

Here, we show how international trade changes long run growth and inequality through the increase of market size. Lets see how trade modifies the dividend rates ($\pi/v$) of the intermediate goods monopolists. Since each intermediate good is now sold in both markets at the same price, demand for intermediate *i* is:

$$x(i) = \left(\frac{p(i)}{P}\right)^{-\epsilon} (Y + Y^*) \quad (25)$$

Monopoly pricing, implies that all intermediate goods produced in the same country bear the same price ($p = CU/\alpha$ and $p^* = CU^*/\alpha^*$). Thus equilibrium is symmetric within each country. We take, as before, the final good as numeraire, so that:

$$P^{1-\epsilon} = Ap^{1-\epsilon} + A^* (p^*)^{1-\epsilon} = 1 \quad (26)$$

where *A* and *A*\* denotes the *home* and *foreign* varieties <fnref n="11" />. This and (25) yield the budget share to goods produced in *home*:

$$s = \frac{Apx}{Apx + A^*p^*x^*} = Ap^{1-\epsilon} \quad (27)$$

We assume that R&D technology is not modified by the opening of the economy. Hence, the cost of R&D is $fA^\lambda$ in *home* and $f^*A^{*\lambda}$ in *foreign* <fnref n="12" />.

<footnote n="11">¹¹Thus we assume that the sets of intermediate goods produced in *home* and *foreign* before the opening of trade do not overlap. Taking account of overlap would complicate the model without modifying the long run equilibrium.</footnote>

<footnote n="12">¹²Most authors (Grossman and Helpman, 1991; Rivera Batiz and Romer, 1991) assume that international trade produces international knowledge spillovers. The role of those spillovers is crucial in the Grossman and Helpman</footnote>
(17), (20) and (27) yield the dividend rate $\pi/v = s^{1/(1-\epsilon)}X/\epsilon f$. Since, $s < 1$, economic integration always leads to an increase of the dividend rate ($s = 1$ in autarky). The open economy LMC condition is thus:

$$g = \frac{\frac{\omega^{\mu}}{1 - \mu(1 - \omega)} s^{\frac{1}{1-\epsilon}} \frac{L}{\epsilon f} - \rho}{1 - \frac{1}{\epsilon - 1}} \quad (28)$$

The NS condition is not modified. Thus, equations (24) and (28) yield the long run growth rate and the home labor allocation as a decreasing (implicit) function of home budget share. We can label this the HH curve ($g = g_{HH}(s)$, $g'_{HH}(\cdot) < 0$ and $g_{HH}(1) = 0$) is the autarky growth rate). Since the opening of the economy entails always an increase of the dividend rate, economic integration always leads to an upward shift of the LMC curve (as depicted in figure 2). Therefore, both long run growth, secondary sector employment ($u$) as well as inequality ($1/\omega$) must go up.

Let summarize the main mechanisms. Trade brings about a rise of profits per firm. This increases the incentives for R&D, which is, of course, growth enhancing. As described in section 4, the increase of creative destruction leads to more inequality since the rise of labor turnover reduces the penalty for shirking and force employers to raise the efficiency bonus.

## 5.2 Existence and uniqueness of the steady state equilibrium growth path

To go further we must compute the home budget share. We can solve the foreign "partial" long-run equilibrium the same way we did for home. The foreign LMC and NS conditions yield the long run growth rate and the foreign labor allocation as an increasing function of home budget share. This curve will be labeled the FF curve ($g = g_{FF}(s)$, with $g'_{FF}(\cdot) > 0$). Figure 4 depicts the long run equilibrium growth and budget share as the intersection of the HH and FF curves.

Hypothesis H1-H4 are not sufficient to guarantee existence and uniqueness of equilibrium. We must distinguish two cases. The easy case is the one where the shirkers detection probability are the same in home and foreign ($q = q^*$). Then, the NS conditions are the same in both countries, which implies the equalization of the long run relative wages ($\omega = \omega^*$). The home budget share is obtained by equalizing the long run dividend rate ($\pi/v$) of both countries, hence:

$$s = \frac{\left(\frac{L^*}{f^*}\right)^{1-\epsilon}}{\left(\frac{L^*}{f^*}\right)^{1-\epsilon} + \left(\frac{L}{f}\right)^{1-\epsilon}} = \bar{s} \quad (29)$$

Then, the long run steady state equilibrium of the world economy is the vector $(g, \omega, s)$ where $s = \bar{s}$ and $g$ and $\omega$ are the solution of (24) and (21).

<footnote>framework since the long-run growth impact of international trade is entirely due to them, since trade without knowledge spillovers has no impact on long-run growth. It would be easy to introduce knowledge spillovers in our model. We could, for instance, assume that both economies share after economic integration the same body of knowledge (research and development cost are now $f(A + A^*)^\lambda$ in home and $f^*(A + A^*)^\lambda$ in foreign). This would certainly amplify the impact of trade on growth and inequality but would not change their direction.</footnote>
<figure><img src="image_6.png" /><figcaption>Figure 4</figcaption></figure>

**Proposition 2** When $q = q^*$, the existence and uniqueness of the steady state equilibrium of the world economy are guaranteed by hypothesis H1-H3 (stated in 3.5) and H5, where:

$$H5 \qquad \frac{(\bar{\omega}_{NS})^{\mu}}{1 - \mu(1 - \bar{\omega}_{NS})} \frac{\bar{s}^{\frac{1}{1-\epsilon}} L}{\epsilon f} < \rho(\epsilon - 1) \quad \text{if} \quad q \frac{1-c}{c} > \rho(\epsilon - 1)$$

*Proof:* hypothesis H1-H3 ensure that the existence of a positive growth rate, since the lower bound of the function $g = g_{HH}(s)$ is the autarky growth rate $g^A = g_{HH}(1)$. Furthermore, the new LMC condition (eq.(28)) is the same as the old when we replace $L$ by $\bar{s}L$. Thus, it suffice to replace $L$ by $\bar{s}L$ in hypothesis H4 to guarantee that $g(\bar{s}) < \rho(\epsilon - 1)$.

(29), entails that the *home* budget share increases with *home* relative size ($L/L^*$) and R&D relative efficiency ($f^*/f$). It is easy to show that the *home* dividend rate increase with *home* and *foreign* absolute size (since profits per firm rise with market size). The same applies to an increase to R&D efficiency (which entails a reduction of research cost). Thus, growth and inequality rise with *home* and *foreign* market size and R&D efficiency (as depicted in figure 2 and 4).

The resolution is trickier when $q \neq q^*$. In the appendix, we give conditions that ensure the existence and uniqueness of the long-run equilibrium and show that the relative labor allocation depends on the relative detection probability. *Home* will employ more (less) workers in the primary sector if her firms can detect shirkers with a greater (lesser) probability. This entails that:

$$\left. \begin{array}{l} s > (<) \bar{s} \\ \omega > (<) \omega^* \end{array} \right\} \quad \text{iff} \quad q > (<) q^* \qquad (30)$$

The consequences of market size or technology shifts are the same. However, changes in the efficiency-wage parameters may have unexpected consequences. For instance, a fall of the
shirker's detection probability in the *home* brings about a fall of worldwide economy growth and a reduction of *home* budget share (as depicted in figure 5). Inequality rises, as expected, in the *home* economy but falls in the *foreign* economy. This paradoxical effect is linked to the slowdown of worldwide innovation. The fall of labor turnover in the *foreign* economy increases the penalty for shirking. This allows *foreign* employers to reduce the efficiency bonus.

<figure><img src="image_7.png" /><figcaption>Figure 5</figcaption></figure>

# 6 Summary and conclusions

This paper has studied the consequences of technological shocks and economic integration on growth and inequality. The first two sections presented the analytical framework. We have built an endogenous growth model with a dual labor market with efficiency-wages in the primary sector and market clearing wages in the secondary sector. We have showed that efficiency wages generate a linkage between growth and inequality among otherwise identical individuals. In this, we have departed from existing literature, which usually interprets rising inequality as the result of skill biased technological change. Our model proves that this need not be the case. In an efficiency-wage framework, growth enhancing shocks such as a reduction of the R&D fixed costs or the deepening of economic integration increase inequality among identical individuals.

Moreover, contrary to most endogenous growth model we can account for the simultaneity of the inequality rise and the growth slowdown. Indeed, shocks on the parameters of the efficiency-wage relation, such as a fall of the shirkers detection probability, may account for this kind of evolution. Is this kind of parameter shift likely? Many employers put a premium on flexibility, autonomy and responsibility. This may reflect the fact that the new technologies make the
measure of individual input more difficult. Anyway, more empirical evidence on the strength of
efficiency-wage factors is needed.

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# A Appendix

## A.1 Existence and uniqueness of the steady state equilibrium

We prove existence using the intermediate value theorem. Let $g_{NS}: \omega \in ]0,1] \to g \in R$ be the function that maps the NS curve. Let $g_{LMC}: \omega \in ]0,1] \to g \in R$ be the function that maps the LMC curve. Lets define the function $\phi$ as $\phi(\omega) = g_{NS}(\omega) - g_{LMC}(\omega)$. $\phi(0) > 0$ and $\phi(1) < 0$ (by H1 and H2) imply that there must exist $\hat{\omega} \in ]0,1]$ such that $\phi(\hat{\omega}) = 0$.

Uniqueness comes from $\phi'(\omega) < 0, \forall \omega \in ]0,1]$ (by (H1)).

(H1) and (H3) ensures that the growth rate ($\hat{g} = \phi(\hat{\omega})$) is positive. Let $\omega_{NS}$, such that $g_{NS}(\omega_{NS}) = 0$ and $\omega_{LMC}$, such that $g_{LMC}(\omega_{LMC}) = 0$ denote the intercept of the NS curve and LMC curve with the horizontal axis. (H1) and (H3) imply $\omega_{LMC} < \hat{\omega} < \omega_{NS}$ and $0 = g_{LMC}(\omega_{LMC}) < \hat{g} < g_{LMC}(\omega_{NS})$, since $g'_{NS}(\omega) < 0, \forall \omega \in ]0,1]$.

(H1) and (H4) ensure that the consumption growth rate does not exceed the interest rate (i.e. $\hat{g} < \rho(\epsilon - 1)$). Let $\bar{\omega}_{NS}$, such that $g_{NS}(\bar{\omega}_{NS}) = \rho(\epsilon - 1)$ and $\bar{\omega}_{LMC}$, such that $g_{LMC}(\bar{\omega}_{LMC}) = 0$ denote the intercept of the NS curve and LMC curve with the $g = \rho(\epsilon - 1)$ curve. There are two cases: (i) if $q(1-c)/c \le \rho(\epsilon - 1)$, then $\bar{\omega}_{NS} \ge 1$ and $\hat{g} < \rho(\epsilon - 1)$, since $g'_{NS}(\omega) < 0, \forall \omega \in ]0,1]$ (ii) if $q(1-c)/c > \rho(\epsilon - 1)$, then (H1) and (H4) imply $\bar{\omega}_{NS} < \hat{\omega} < \bar{\omega}_{LMC} < 1$ and $g_{LMC}(\bar{\omega}_{NS}) < \hat{g} < g_{LMC}(\bar{\omega}_{LMC}) = \rho(\epsilon - 1)$, since $g'_{LMC}(\omega) > 0, \forall \omega \in ]0,1]$.

## A.2 The steady state equilibrium of the world economy

The long run equilibrium of the world economy is the vector of feasible $(g, s, \omega, \omega^*)$ that verifies the *home* and *foreign* LMC and NS conditions:

$$g = \frac{\frac{s^{\frac{1}{1-\epsilon}} \omega^{\mu}}{1-\mu(1-\omega)} \frac{L}{\epsilon f} - \rho}{1 - \frac{1}{\epsilon-1}} \quad (31)$$
$$g = \frac{(1-s)^{\frac{1}{1-\epsilon}} \frac{(\omega^*)^\mu}{1-\mu(1-\omega^*)} \frac{L^*}{\epsilon f^*} - \rho}{1 - \frac{1}{\epsilon-1}} \quad (32)$$

$$g = \frac{q^{\frac{1-c}{c}} (1-\omega) - \rho}{1 - \frac{1}{\epsilon-1} + \frac{\omega\mu}{1-\mu}} \quad (33)$$

$$g = \frac{q^{\frac{1-c}{c}} (1 - \omega^*) - \rho}{1 - \frac{1}{\epsilon-1} + \frac{\omega^* \mu}{1-\mu}} \quad (34)$$

This system is easy to solve when the labor efficiency parameters are the same in both economies ($q = q^*$). Then (33) and (34) imply that the steady-state relative wage is the same in both economies. This, and the equalization of the dividend rate, by (33) and (34), yield the long run home budget share:

$$\frac{s}{1-s} = \left( \frac{L^*/f^*}{L/f} \right)^{1-\epsilon} \quad (35)$$

The resolution is trickier when home and foreign are allowed to differ with respect to the efficiency parameters. Let's prove inequality (30). By (33) and (34):

$$q > (<)q^* \quad \Rightarrow \quad \omega > (<)\omega^* \quad (36)$$

By (31) and (32):

$$\left(\frac{s}{1-s}\right)^{1/1-\epsilon} = \frac{L^*/f^* \psi(\omega^*)}{L/f \psi(\omega)} \quad (37)$$

where $\psi(\omega) = \frac{\omega^\mu}{1-\mu(1-\omega)} \frac{L}{\epsilon f}$. (36), (37) and $\psi'(.) > 0$ prove inequality (30).

Assume $q < q^*$. Let $g^{A*} = g_{FF}(0)$ denote the autarky growth rate of the foreign economy. Existence and uniqueness of the steady state equilibrium growth path are guaranteed by the hypothesis H1-H3 (stated in 3.5) and hypothesis H6-H7, where:

$$\begin{cases} H6 & g^{A*} < g_{HH}(0) = \frac{q^{\frac{1-c}{c}} - \rho}{1 - \frac{1}{\epsilon-1}} \\ H7 & q^{\frac{1-c}{c}} < \rho(\epsilon - 1) \end{cases}$$

Hypothesis H1-H3 ensure the existence of an autarky solution in both countries. H6-H7 ensure the existence of the open economy solution.