Skilled-Labor Intensity Differences Across Firms, Endogenous Product Quality, and Wage Inequality

Abstract

This paper proposes a theory to explain the relative wage-rate increase for skilled labor that results from trade liberalization that relies on within-sector reallocations of production resources (skilled and unskilled labor) across firms. Motivated by some stylized facts, in a model with firm heterogeneity, including firms that differ in their skill intensity even within a narrowly defined industry, firms with relatively high skill intensity that are more likely to be exporters, and a positive association between a firm’s skill intensity and its product quality, I develop a general equilibrium model where firms with a higher skill intensity endogenously choose a higher-quality product, and tend to be more profitable. In this framework, a reduction in trade costs allows members of the workforce to reallocate to more efficient firms that produce higher-quality products, using their skilled labor more intensively, resulting in a rising skill premium. The main sources of the increasing wage inequality that followed trade openness are a positive link between a firm’s skill intensity, its product quality, and quality competition.

Appendix

1.1 Proof of Proposition 1

To prove this, two curves defined by the equilibrium conditions Eqs. 13 and 19 must be uniquely intersected in the (s/w, 𝜃 ^∗) space. For convenience, I rewrite two equilibrium conditions (the free entry condition and the labor market clearing condition) as follows:

$$ FE\left( \frac{s}{w},\theta^{*}\right)= \left\{ \left( \frac{s}{w}\right)^{(\tilde{\theta}(\theta^{*})-\theta^{*})(1-\sigma)} \left( \frac{\tilde{\theta}(\theta^{*})}{\theta^{*}}\right)^{\frac{\alpha(\sigma-1)}{\phi}} -1 \right\} -\frac{\delta}{1-G(\theta^{*})}\frac{f_{e}}{f}=0, $$

(A.1)

$$ LE\left( \frac{s}{w}, \theta^{*}\right)=\frac{\left( \frac{w}{s}\right){\int}_{\theta^{*}}^{1} \left( \frac{s}{w}\right)^{\theta(1-\sigma)} \theta^{\frac{\alpha(\sigma-1)}{\phi}}\theta g(\theta)d\theta}{{\int}_{\theta^{*}}^{1} \left( \frac{s}{w}\right)^{\theta(1-\sigma)} \theta^{\frac{\alpha(\sigma-1)}{\phi}}(1-\theta)g(\theta)d\theta}-\frac{H}{L}=0, $$

(A.2)

where the weighted average skill intensity $\tilde {\theta }(\theta ^{*})$ is given by Eq. 10.

First, I show that the free entry condition (A.1) gives a negative association between s/w and 𝜃 ^∗ using the implicit function theorem.

$$\frac{\partial FE(s/w, \theta^{*})}{\partial (s/w)}=(1-\sigma)(\tilde{\theta}(\theta^{*}) -\theta^{*}) \left( \frac{s}{w}\right)^{(\tilde{\theta}(\theta^{*}) -\theta^{*})(1-\sigma)-1} \left( \frac{\tilde{\theta}(\theta^{*})}{\theta^{*}}\right)^{\frac{\alpha(\sigma-1)}{\phi}} <0. $$

where the inequality holds because the constant term, (1−σ), is negative. By the second fundamental theorem of calculus, taking the derivative with respect to 𝜃 ^∗ gives

$$\begin{array}{@{}rcl@{}} \frac{\partial FE(s/w, \theta^{*})}{\partial \theta^{*}}=& \xi(s/w, \theta^{*}) \left\lbrace \tilde{\theta}^{\prime}(\theta^{*}) \left[ \frac{\alpha}{\tilde{\theta}(\theta^{*})}-\textnormal{ln}(\frac{s}{w})\phi \right] - \left[\frac{\alpha}{\theta^{*}}-\textnormal{ln}(\frac{s}{w})\phi \right] \right\rbrace\\ &- \delta\frac{f_{e}}{f}\frac{g(\theta^{*})}{(1-G(\theta^{*}))^{2}}< 0, \end{array} $$

where $\xi (s/w, \theta ^{*})=\frac {\sigma -1}{\phi } \left (\frac {s}{w}\right )^{(\tilde {\theta }(\theta ^{*}) -\theta ^{*})(1-\sigma )} \left (\frac {\tilde {\theta }(\theta ^{*})}{\theta ^{*}}\right )^{\frac {\alpha (\sigma -1)}{\phi }}$ is positive. $\tilde {\theta }^{\prime }(\theta ^{*})$ denotes the derivative of the weighted average skill intensity with respect to 𝜃 ^∗. By the second fundamental theorem of calculus, taking the derivative of Eq. 10 with respect to 𝜃 ^∗ gives:

$$\tilde{\theta}^{\prime}(\theta^{*})=\frac{1}{1-\sigma} \left( {\int}_{\theta^{*}}^{1}\theta^{\sigma-1} \frac{g(\theta)}{1-G(\theta^{*})}d\theta\right)^{\frac{1}{\sigma-1}-1}(\theta^{*})^{\sigma-1} \frac{g(\theta^{*})}{1-G(\theta^{*})}.$$

Since σ > 1, $\tilde {\theta }^{\prime }(\theta ^{*})$ is negative. Thus, the inequality, $\frac {\partial FE(s/w, \theta ^{*})}{\partial \theta ^{*}}<0$, holds because of the assumption that I take in the previous section, $\alpha >\textnormal {ln}(\frac {s}{w})\phi$, which implies that the firms revenue/profitability increases with the skill intensity as well as with product quality. Therefore, two equilibrium variables (s/w and 𝜃 ^∗) have a negative relationship, which is established by using the implicit function theorem:

$$\frac{d(\frac{s}{w})}{d\theta^{*}}=-\frac{\partial FE(\frac{s}{w}, \theta^{*})/\partial \theta^{*}}{\partial FE(\frac{s}{w}, \theta^{*})/\partial (\frac{s}{w})} <0. $$

Second, the free entry condition (A.1) implies that as 𝜃 ^∗ goes to zero, the skill premium s/w has to increase as much as possible for F E(s/w, 𝜃 ^∗) = 0. As 𝜃 ^∗ increases, on the other hand, s/w must decrease as quickly as possible for F E(s/w, 𝜃 ^∗) = 0 so that s/w goes to zero.

To complete this proof, I need to show that the labor market clearing condition gives a positive link between 𝜃 ^∗ and s/w. Unfortunately, Eq. A.2 cannot be solved analytically using the implicit function theorem. To establish a positive link between 𝜃 ^∗ and s/w, I follow the method introduced by Harrigan and Reshef (2011) where they effectively prove it. I begin with the labor market clearing condition, Eq. 19:

$$ \frac{{\int}_{\theta^{*}}^{1} D_{h}\left( \theta, \frac{s}{w}\right)g(\theta)d\theta}{{\int}_{\theta^{*}}^{1} D_{l}\left( \theta, \frac{s}{w}\right)g(\theta)d\theta}=\frac{H}{L}. $$

Since $\frac {D_{h}(\theta , \frac {s}{w})}{D_{l}(\theta , \frac {s}{w})}=\frac {\theta }{1-\theta }\left (\frac {w}{s}\right )$, which is from Eq. 16, I can rewrite equation above as follows:

$$ \frac{\left( \frac{w}{s}\right){\int}_{\theta^{*}}^{1} \left( \frac{\theta}{1-\theta} \right) D_{l}(\theta, \frac{s}{w})g(\theta)d\theta}{{\int}_{\theta^{*}}^{1} D_{l}(\theta, \frac{s}{w})g(\theta)d\theta}=\frac{H}{L}. $$

where $\left (\frac {\theta }{1-\theta } \right )$ represents the ratio of skilled to unskilled labor employed by firms with 𝜃. By defining ${\Psi }(\theta , \frac {s}{w}, \theta ^{*})=\frac {D_{l}(\theta , \frac {s}{w})}{{\int }_{\theta ^{*}}^{1} D_{l}(\theta , \frac {s}{w})g(\theta )d\theta }$, the labor market clearing condition is

$$ \left( \frac{w}{s}\right){\int}_{\theta^{*}}^{1}\left( \frac{\theta}{1-\theta}\right) {\Psi}\left( \theta, \frac{s}{w}, \theta^{*}\right) g(\theta) d\theta=\frac{H}{L}. $$

where ${\Psi }(\theta , \frac {s}{w}, \theta ^{*})$ can be interpreted as the share of unskilled labor that works for firms with 𝜃. The above Equation indicates that the relative endowment of skilled labor (H/L) is equal to the average of the firm level skill ratios weighted by the firm’s unskilled labor share. Now I examine how the increase in 𝜃 ^∗ affects the skill premium, s/w. The increase in skill intensity 𝜃 ^∗ implies that the least profitable firms, which are less skill intensive than average, exit so that the weighted average of the skill ratio of the surviving firms increases (i.e., ${\int }_{\theta ^{*}}^{1}\left (\frac {\theta }{1-\theta }\right ) {\Psi }(\theta , \frac {s}{w}, \theta ^{*}) g(\theta ) d\theta \nearrow$). To keep the same level of the left hand side of equation above, the relative wage of skilled to unskilled labor (s/w) should be higher. Thus I confirm that the zero profit cut-off, 𝜃 ^∗, is positively associated with the skill premium s/w along the labor market clearing equation. The intuition behind the positive link between 𝜃 ^∗ and s/w is that an increase in 𝜃 ^∗ leads to an incipient relative excess in demand for skilled labor, which results in raising the relative skilled worker wage. Now I confirm that the equilibrium variables 𝜃 ^∗ and s/w are determined uniquely. ▪

Figure 6 illustrates the existence of the equilibrium skill intensity cut-off, 𝜃 ^∗, and the skill premium, s/w, at which the free entry condition (Eqs. 10 and 13) and the labor market clearing condition (19) are intercepted. For Fig. 6, I assume that all parameters are the same as noted in the numerical exercise (section 2.3.4). In addition, I assume that the relative abundance of skilled labor H/L is either 0.5 or 0.75 and that the fixed production cost, f, is either 1 or 2.

Fig. 6

Free entry and labor market equilibrium curves in autarky

As shown in Fig. 6, the decline of fixed cost f shifts the free entry curve to the left and the increase in the relative abundance of skilled labor H/L shifts the labor market curve to the right. These are all four possible equilibria, which depend on the value of H/L and f. The equilibrium 𝜃 ^∗ and $\frac {s}{w}$, in each equilibrium, are also shown in Table 1.

1.2 Proof of Proposition 2

Conveniently, I rewrite the costly trade equilibrium conditions under the symmetric assumption as follows: First Eq. A.3 links the equilibrium variables of interest (s/w, $\theta _{d}^{*}$ and $\theta _{x}^{*}$), which depend on trade costs. Equations A.4 and A.5 represent the free entry condition and labor market clearing condition respectively.

$$ \left( \frac{s}{w}\right)^{-(\theta_{x}^{*}-\theta_{d}^{*})}\left( \frac{\theta_{x}^{*}}{\theta_{d}^{*}}\right)^{\frac{\alpha}{\phi}}=\tau\left( \frac{f_{x}}{f}\right)^{\frac{1}{\sigma-1}}. $$

(A.3)

$$\begin{array}{@{}rcl@{}} FE\left( \frac{s}{w}, \theta_{d}^{*}\right)&=&[1-G(\theta^{*}_{d})]f\left[\left( \frac{s}{w}\right)^{(\tilde{\theta}_{d}(\theta_{d}^{*})-\theta^{*}_{d})(1-\sigma)}\left( \frac{\tilde{\theta}_{d}(\theta_{d}^{*})}{\theta^{*}_{d}}\right)^{\frac{\alpha(\sigma-1)}{\phi}} -1\right]+\\ &&[1-G(\theta_{x}^{*})]f_{x}\left[\left( \frac{s}{w}\right)^{(\tilde{\theta}_{x}(\theta_{x}^{*})-\theta^{*}_{x})(1-\sigma)}\left( \frac{\tilde{\theta}_{x}(\theta_{x}^{*})}{\theta^{*}_{x}}\right)^{\frac{\alpha(\sigma-1)}{\phi}} -1\right]\\&&-\delta f_{e}. \end{array} $$

(A.4)

$$\begin{array}{@{}rcl@{}} LE(\bullet)&=&\frac{\left( \frac{w}{s}\right)\left[{\int}_{\theta^{*}_{d}}^{1} \left( \frac{s}{w}\right)^{\theta(1-\sigma)}\theta^{\frac{\alpha(\sigma-1)}{\phi}}\theta g(\theta)d\theta+\tau^{-\sigma}{\int}_{\theta^{*}_{x}}^{1} \left( \frac{s}{w}\right)^{\theta(1-\sigma-1)}\theta^{\frac{\alpha(\sigma-1)}{\phi}}\theta g(\theta)d\theta \right]}{{\int}_{\theta^{*}_{d}}^{1} \left( \frac{s}{w}\right)^{\theta(1-\sigma)}\theta^{\frac{\alpha(\sigma-1)}{\phi}}(1-\theta) g(\theta)d\theta+\tau^{-\sigma}{\int}_{\theta^{*}_{x}}^{1} \left( \frac{s}{w}\right)^{\theta(1-\sigma)}\theta^{\frac{\alpha(\sigma-1)}{\phi}}(1-\theta) g(\theta)d\theta}\\&&-\frac{H}{L}. \end{array} $$

(A.5)

where the weighted average skill intensity for each market, $\tilde {\theta }_{d}(\theta _{d}^{*})$ and $\tilde {\theta }_{x}(\theta _{x}^{*})$ are given by Eqs. 27. Note that the export skill intensity cut-off is a function of both the skill premium and the zero-profit skill intensity cut-off, that is, $\theta _{x}^{*}(s/w, \theta _{d}^{*})$ given by Eq. A.3. From the Eq. A.3, it can easily be shown that $\frac {\partial \theta _{x}^{*}}{\partial (s/w)}>0$ and $\frac {\partial \theta _{x}^{*}}{\partial \theta _{d}^{*}}>0$.

In the first step, I show that the free entry condition given by Eq. A.4 has a downward slope in the $(s/w, \theta _{d}^{*})$ space using the implicit function theorem: $\frac {d(\frac {s}{w})}{d\theta _{d}^{*}}=-\frac {\partial FE(\frac {s}{w}, \theta _{d}^{*})/\partial \theta _{d}^{*}}{\partial FE(\frac {s}{w}, \theta _{d}^{*})/\partial (\frac {s}{w})} <0$. A tedious amount of manipulation gives

$$\begin{array}{@{}rcl@{}} &\frac{\partial FE(s/w, \theta_{d}^{*})}{\partial s/w}= [1-G(\theta_{d}^{*})]f (1-\sigma)[\tilde{\theta}_{d}(\theta_{d}^{*}) -\theta_{d}^{*}] \left( \frac{s}{w}\right)^{(\tilde{\theta}_{d}(\theta_{d}^{*}) -\theta_{d}^{*})(1-\sigma)-1} \left( \frac{\tilde{\theta}(\theta_{d}^{*})}{\theta_{d}^{*}}\right)^{\frac{\alpha(\sigma-1)}{\phi}} \\ &-g(\theta_{x}^{*})\frac{\partial \theta_{x}^{*}}{\partial (s/w)}f_{x}\left[\left( \frac{s}{w}\right)^{(\tilde{\theta}_{x}(\theta_{x}^{*})-\theta^{*}_{x})(1-\sigma)}\left( \frac{\tilde{\theta}_{x}(\theta_{x}^{*})}{\theta^{*}_{x}}\right)^{\frac{\alpha(\sigma-1)}{\phi}} -1\right] \\ &+\psi(\frac{s}{w}, \theta_{x}^{*}) \left\lbrace \tilde{\theta}_{x}^{\prime}(\theta_{x}^{*})\frac{\partial \theta_{x}^{*}}{\partial (s/w)}\left[ \frac{\alpha}{\tilde{\theta}_{x}(\theta_{x}^{*})}-\textnormal{ln}\left( \frac{s}{w}\right)\phi\right]- \right. \left. \frac{\partial \theta_{x}^{*}}{\partial (s/w)}\left[ \frac{\alpha}{\theta_{x}^{*}}-\textnormal{ln}\left( \frac{s}{w}\right)\phi\right]\right.\\&\left.-\left( \frac{s}{w} \right)(\tilde{\theta}_{x}(\theta_{x}^{*})-\theta_{x}^{*})\phi \right\rbrace<0, \end{array} $$

where $\psi (s/w, \theta _{x}^{*})=[1-G(\theta _{x}^{*})]f_{x} \left (\frac {\sigma -1}{\phi } \right ) \left (\frac {s}{w}\right )^{(\tilde {\theta }_{x}(\theta _{x}^{*})-\theta ^{*}_{x})(1-\sigma )}\left (\frac {\tilde {\theta }_{x}(\theta _{x}^{*})}{\theta ^{*}_{x}}\right )^{\frac {\alpha (\sigma -1)}{\phi }}$ is positive. Note that ϕ > 1, σ > 1, α > 0, and $\tilde {\theta }_{x}^{\prime }(\theta _{x}^{*})<0$. The inequality, $\frac {\partial FE(s/w, \theta _{d}^{*})}{\partial (s/w)}<0$, holds, due to the assumption of $\alpha >\textnormal {ln}(\frac {s}{w})\phi$.

$$\begin{array}{@{}rcl@{}} \frac{\partial FE(s/w, \theta_{d}^{*})}{\partial \theta_{d}^{*}}=&-g(\theta_{d}^{*})f \left[\left( \frac{s}{w}\right)^{(\tilde{\theta}_{d}(\theta_{d}^{*})-\theta^{*}_{d})(1-\sigma)}\left( \frac{\tilde{\theta}_{d}(\theta_{d}^{*})}{\theta^{*}_{d}}\right)^{\frac{\alpha(\sigma-1)}{\phi}}-1 \right]+ \\ & \upsilon(s/w, \theta_{d}^{*}) \left\lbrace \tilde{\theta}_{d}^{\prime}(\theta_{d}^{*})\left[ \frac{\alpha}{\tilde{\theta}_{d}} -\textnormal{ln}\left( \frac{s}{w} \right)\phi\right] -\left[\frac{\alpha}{\theta_{d}^{*}} -\textnormal{ln}\left( \frac{s}{w} \right)\phi\right] \right\rbrace \\ &-g(\theta_{x}^{*})\frac{\partial \theta_{x}^{*}}{\partial \theta_{d}^{*}} f_{x} \left[\left( \frac{s}{w}\right)^{(\tilde{\theta}_{x}(\theta_{x}^{*})-\theta^{*}_{x})(1-\sigma)}\left( \frac{\tilde{\theta}_{x}(\theta_{x}^{*})}{\theta^{*}_{x}}\right)^{\frac{\alpha(\sigma-1)}{\phi}}-1 \right]+ \\ &\psi(s/w, \theta_{x}^{*})\left\lbrace \tilde{\theta}_{x}^{\prime}(\theta_{x}^{*})\frac{\partial \theta_{x}^{*}}{\partial \theta_{d}^{*}} \left[ \frac{\alpha}{\tilde{\theta}_{x}} -\textnormal{ln}\left( \frac{s}{w} \right)\phi\right] -\frac{\partial \theta_{x}^{*}}{\partial \theta_{d}^{*}} \left[\frac{\alpha}{\tilde{\theta}_{x}} -\textnormal{ln}\left( \frac{s}{w} \right)\phi\right] \right\rbrace <0, \end{array} $$

where $\upsilon (s/w, \theta _{d}^{*})$ and $\psi (s/w, \theta _{x}^{*})$ are both positive.³¹

Since $\frac {\partial \theta _{x}^{*}}{\partial \theta _{d}^{*}}>0$ and $\alpha >\textnormal {ln}\left (\frac {s}{w} \right )\phi$, $\frac {\partial FE(s/w, \theta _{d}^{*})}{\partial \theta _{d}^{*}}<0$. By the implicit function theorem, the free entry condition implies a negative link between $\theta _{d}^{*}$ and $\frac {s}{w}$: $\frac {d(\frac {s}{w})}{d\theta _{d}^{*}}=-\frac {\partial FE(\frac {s}{w}, \theta _{d}^{*})/\partial \theta _{d}^{*}}{\partial FE(\frac {s}{w}, \theta _{d}^{*})/\partial (\frac {s}{w})} <0$. In addition, applying the same logic in the proof of the proposition 1, the free entry condition (A.4) implies that as $\theta _{d}^{*}$ goes to zero, the skill premium s/w has to increase as much as possible for F E(∙) = 0. As $\theta _{d}^{*}$ increases to one, on the other hand, s/w must decrease as quickly as possible for F E(∙)=0 so that s/w goes to zero.

Second, to complete the proof, I need to show a positive relationship between $\theta _{d}^{*}$ and s/w. Applying the same logic presented in the proof of proposition 1, the Eq. A.5 can be rewritten as:

$$\begin{array}{@{}rcl@{}} &&\left( \frac{w}{s}\right) \left[ {\int}_{\theta_{d}^{*}}^{1}\left( \frac{\theta}{1-\theta}\right) {\Psi}\left( \theta, \frac{s}{w}, \theta_{d}^{*}\right) g(\theta) d\theta+\tau^{-\sigma}{\int}_{\theta_{x}^{*}(\theta_{d}^{*})}^{1}\left( \frac{\theta}{1-\theta}\right) {\Psi}\left( \theta, \frac{s}{w}, \theta_{d}^{*}\right) g(\theta) d\theta \right]=\frac{H}{L}. \end{array} $$

Now it can be easily shown that the zero-profit skill intensity $\theta _{d}^{*}$ is positively associated with the skill premium s/w on the $(\theta _{d}^{*}, s/w)$ space using the same logic as in the proof of proposition 1. Thus equilibrium exists and is unique in the costly trade between symmetric countries. ▪

1.3 Proof of Proposition 3

Now, I prove that the skill premium s/w increases as trade costs (i.e., τ and/or f _x ) fall. To do this, I examine the shifts in each FE and LE curve in response to a reduction in trade costs. Without any loss of generality, I compare two curves in the autarky regime with ones in the costly trade condition. In this way, one can analyze the effect of lowering trade costs on the skill premium. First, I compare the free entry condition under autarky with the free entry condition with costly trade. Conveniently, I rewrite the free entry equation in costly trade, which is from Eq. 28.

$$\begin{array}{@{}rcl@{}} (1-G(\theta^{*}_{d}))f\left[\left( \frac{s}{w}\right)^{(\tilde{\theta}_{d}-\theta^{*}_{d})(1-\sigma)}\left( \frac{\tilde{\theta}_{d}}{\theta^{*}_{d}}\right)^{\frac{\alpha(\sigma-1)}{\phi}} -1\right]+~~~~~~~~~~~~~~~~~~~~~~~~~~\\(1-G(\theta^{*}_{x}))f_{x}\left[\left( \frac{s}{w}\right)^{(\tilde{\theta}_{x}-\theta^{*}_{x})(1-\sigma)}\left( \frac{\tilde{\theta}_{x}}{\theta^{*}_{x}}\right)^{\frac{\alpha(\sigma-1)}{\phi}} -1\right]=\delta f_{e}. \end{array} $$

(A.6)

Note that Eq. A.6 reduces to $(1-G(\theta ^{*}_{d}))f\left [\left (\frac {s}{w}\right )^{(\tilde {\theta }_{d}-\theta ^{*}_{d})(1-\sigma )}\left (\frac {\tilde {\theta }_{d}}{\theta ^{*}_{d}}\right )^{\frac {\alpha (\sigma -1)}{\phi }} -1\right ]=\delta f_{e}$, which is the first term on the left-hand side of Eq. A.6, in the closed economy. When trade costs are low enough for some firms to engage in exporting, the first term in the left-hand side of Eq. A.6 must be smaller than the one in autarky, because the second term in the left-hand side is positive and the right-hand side of Eq. A.6, δ f _e , does not depend on trade costs. Notice that the first term in the left-hand side of Eq. A.6 will decrease as s/w increases, while $\theta _{d}^{*}$ is fixed. Thus, the free entry curve (FE) will shift upward (that is, shifts to the right) as trade costs fall.

Second, the upward-sloping labor market curve (LE) also shifts upward (that is, shifts to the left) as a result of the reduction in trade costs. For convenience, I rewrite the labor market equilibrium condition under costly trade that is given in the proof of proposition 2:

$$ \left( \frac{w}{s}\right) \left[ {\int}_{\theta_{d}^{*}}^{1}\left( \frac{\theta}{1-\theta}\right) {\Psi}(\theta, \frac{s}{w}, \theta_{d}^{*}) g(\theta) d\theta +\tau^{-\sigma}{\int}_{\theta_{x}^{*}(\theta_{d}^{*})}^{1}\left( \frac{\theta}{1-\theta}\right) {\Psi}(\theta, \frac{s}{w}, \theta_{d}^{*}) g(\theta) d\theta \right]=\frac{H}{L}. $$

(A.7)

where ${\Psi }(\theta , \frac {s}{w}, \theta _{d}^{*})$ denotes the share of unskilled workers employed by firms with 𝜃 out of the total endowment of unskilled labor. Now consider either the movement from autarky to costly trade or the reduction of trade costs under the condition of costly trade. Note that autarky regime reduces Eq. A.7 to $\left (\frac {w}{s}\right ) \left [ {\int }_{\theta _{d}^{*}}^{1}\left (\frac {\theta }{1-\theta }\right ) {\Psi }(\theta , \frac {s}{w}, \theta _{d}^{*}) g(\theta ) d\theta \right ]=\frac {H}{L}$. Either the movement from autarky to the costly trade or the reduction of trade costs increases the second integral in the left-hand side of Eq. A.7. Thus $\left (\frac {w}{s}\right ) \left [ {\int }_{\theta _{d}^{*}}^{1}\left (\frac {\theta }{1-\theta }\right ) {\Psi }(\theta , \frac {s}{w}, \theta _{d}^{*}) g(\theta ) d\theta \right ]$ must be decreased with lowering trade costs because the right hand side of Eq. A.7, H/L, is constant. Holding $\theta _{d}^{*}$ fixed, the skill premium s/w must increase in order to keep the left hand side of Eq. A.7 constant because of the definition of ${\Psi }(\theta , \frac {s}{w}, \theta _{d}^{*})$: as the relative price of a skilled worker increases, ${\Psi }(\theta , \frac {s}{w}, \theta _{d}^{*})$ decreases. As a result, the LE curve also shifts up, as trade costs fall. Since both the FE and LE curves shift upward the relative wage of skilled labor s/w will be higher when trade costs fall.

Figure 7 illustrates that the skill premium increases as trade costs fall. When trade costs decrease, both the free trade FE and the labor market curve, LE, shift upward, which results in increasing wage inequality, s/w. ▪

Fig. 7

FE and LE curves in autarky and costly trade