Trade liberalization and wage inequality: new insights from a dynamic trade model with heterogeneous firms and comparative advantage

Abstract

The authors develop a dynamic general equilibrium trade model with comparative advantage, heterogeneous firms, heterogeneous workers and endogenous firm entry to study wage inequality during the adjustment after trade liberalization and potential policy responses to reduce wage inequality. They calibrate the model to simulate trade liberalization between the US and China. In the short run, inter-sectoral wage inequality is high but then recedes as more and more workers move to the expanding exporting sector. The skill premium does not change much in the short run but increases substantially in the medium and long run. Training subsidies are more powerful than sector-migration subsidies in reducing the wage inequality induced by trade liberalization.

A Proof of existence and uniqueness of the steady state

We start the proof by focusing on the steady state of the simplest version of our framework where workers do not have access to training and discuss the equilibrium under training and out-of-steady-state at the end. The following table contains the list of equations that define the steady state under no training. Asterisk denotes variables pertaining to F. For simplicity we have omitted the tilde sign on top of firm average productivities, prices and profits. Note that due to the absence of fixed costs of production $z_{d}=(\frac{k}{k-(\theta -1)})^{\frac{1}{\theta -1}}z_{min}$ is given. Equations 36–54 hold for the Home country for each sector i where $i=1$ is the skill intensive sector and $i=2$ is the unskilled intensive sector. This results in a total of 38 sector specific equations for H. Aggregate equations (55–59) hold for H, which results in a total of 43 equations for H. Equivalent equations hold for F. The equations for both countries in addition to the balanced trade condition (60) form a system of 87 equations that uniquely identifies the world steady state equilibrium, described by a vector of the following variables: $N_{xi},N_{ei},N_{di},z_{xi},d_{di},d_{xi},v_{i},d_{i},\rho _{d,i},\rho _{xi},\psi _{i},S_{i},L_{i},w_{i}^{s},w_{i}^{l},v_{i}^{s},v_{i}^{l},S_{ei},L_{ei},c_{i}^{s},c_{i}^{l}$ for each sector in H (42 in total for both sectors) and the equivalent variables in F. There remain the aggregate variables which are: C, Q,and $C^{*}$ for a total of 87 variables that describe the world economy (Table 3).

Table 3 Steady state equations

The subsequent discussion of the existence and uniqueness of the steady state equilibrium is similar to the proof of proposition 3 in the appendix of Bernard et al. (2007). We focus our discussion on the H economy. Equivalent considerations hold for the F country.

In the long run we have assumed that workers are perfectly mobile across sectors (Eqs. 55, 56). This implies that the wages of skilled and unskilled workers are equalized across sectors, i.e., $w_{1}^{s}=w_{2}^{s}\equiv w^{s}$ and $w_{1}^{l}=w_{2}^{l}\equiv w^{l}$. The same is true for consumption of workers across sectors such that $c_{1}^{s}=c_{2}^{s}\equiv c^{s}$ and $c_{1}^{l}=c_{2}^{l}\equiv c^{l}.$

Following Bernard et al. (2007) we suppose for the moment that the equilibrium wage vector is known. Defining factor intensities in the skill-intensive sector as $\lambda _{1}^{s}=S_{1}/S$ and $\lambda _{1}^{l}=L_{1}/L$, the factor-clearing Eqs. 57, 58 can be reformulated as

$$\begin{aligned} \frac{L}{S}= & {} \frac{L_{1}+L_{2}}{S}\\= & {} \frac{L_{1}}{S_{1}}\frac{S_{1}}{S}+\frac{L_{2}}{S_{2}}\frac{S_{2}}{S}\\= & {} \frac{L_{1}}{S_{1}}\lambda _{1}^{s}+\frac{L_{2}}{S_{2}}\left( 1-\lambda _{1}^{s}\right) \end{aligned}$$

and

$$\begin{aligned} \frac{S}{L}= & {} \frac{S_{1}+S_{2}}{L}\\= & {} \frac{S_{1}}{L_{1}}\frac{L_{1}}{L}+\frac{S_{2}}{L_{2}}\frac{L_{2}}{L}\\= & {} \frac{S_{1}}{L_{1}}\lambda _{1}^{l}+\frac{S_{2}}{L_{2}}\left( 1-\lambda _{1}^{l}\right) \end{aligned}$$

Using Eq. 47 for both sectors these become

$$\begin{aligned} \frac{L}{S}= & {} \frac{(1-\beta _{1})}{\beta _{1}}\frac{w^{s}}{w^{l}}\lambda _{1}^{s}+\frac{(1-\beta _{2})}{\beta _{2}}\frac{w^{s}}{w^{l}}\left( 1-\lambda _{1}^{s}\right) \\ \frac{S}{L}= & {} \frac{\beta _{1}}{(1-\beta _{1})}\frac{w^{l}}{w^{s}}\lambda _{1}^{l}+\frac{\beta _{2}}{(1-\beta _{2})}\frac{w^{l}}{w^{s}}\left( 1-\lambda _{1}^{l}\right) \end{aligned}$$

Given wages and exogenous endowments these two equations uniquely define the factor intensities in the skill-intensive sector $\lambda _{1}^{s}$ and $\lambda _{1}^{l}$, and thereby $S_{1}$ and $L_{1}$. Then the factor-clearing conditions uniquely define $S_{2}$ and $L_{2}$. In particular, $\lambda _{1}^{s}=-\frac{\beta _{1}(\beta _{2}w^{l}L-(1-\beta _{2})w^{s}S)}{(\beta _{1}-\beta _{2})w^{s}S}$and $\lambda _{1}^{l}=\frac{(\beta _{1}-1)((\beta _{2}-1)w^{s}S+\beta _{2}w^{l}L)}{(\beta _{1}-\beta _{2})w^{l}L}$. An important restriction for the existence of the steady state is that $\beta _{1}\ne \beta _{2}$ since otherwise the share of workers in the skill intensive sector is not well defined. Note that this restriction is satisfied as we assume that $\beta _{1}>\beta _{2}$. Similarly, other restrictions necessary for existence are that $L>0$ and $S>0$. The entry conditions for workers in each sector (Eqs. 49, 50) deliver the number of skilled and unskilled workers entering each sector at the steady state $Se_{i}$ and $Le_{i}$ as a function of sector employment ($S_{i}$ and $L_{i}$). Note that another important parameter restriction for the existence of positive worker entry at the steady state is $s>0$.

The free entry condition (Eq. 42) pins down the average value of the firm $v_{i}$ as a function of the wages and model parameters. For firms to have positive value, we require that $f_{e}>0$. Otherwise we have unlimited firm entry. Combining the recursive form of firm value (Eq. 46) with the free entry condition (Eq. 42) yields

$$\begin{aligned} f_{e}\left( w^{s}\right) ^{\beta _{i}}\left( w^{l}\right) ^{1-\beta _{i}}\frac{1-\gamma \left( 1-\delta \right) }{\gamma \left( 1-\delta \right) }=d_{i} \end{aligned}$$

which pins down average profits for each sector. Here we see other important parameter restrictions for the existence of positive firm profits at the steady state: $\gamma >0$ and $\delta <1$. Note that given a positive skill premium at the steady state, $\frac{w^{s}}{w^{l}}>1$, and our assumption about factor intensities $1>\beta _{1}>\beta _{2}>0$, the average profits and firm value in sector 1 will be higher than the average profits and firm value in sector 2.

We can use Eq. 43 to derive the average profits for firms that export $d_{xi}$. Note that $k>\theta -1$ is another necessary restriction for the existence of the steady state export profit. In addition, note that only when $f_{x}>0$, only a fraction of the existing firms export. If, $f_{x}=0$, then all existing firms will be exporters.

Next, it is useful to define average domestic revenue $r_{di}=\left( \frac{\rho _{d,i}}{\psi _{i}}\right) ^{1-\theta }\alpha _{i}C$, average export revenue for H $r_{xi}=\left( \frac{\rho _{xi}}{\psi _{i}^{*}}\right) ^{1-\theta }\alpha _{i}C^{*}$, and F $r_{xi}^{*}=\left( \frac{\rho _{xi}^{*}}{\psi _{i}}\right) ^{1-\theta }\alpha _{i}C$. Using this definition for average domestic revenue in Eq. 39 and the fraction of exporters in A9 yields $d_{i}=\frac{r_{di}}{\theta }+\left( \frac{z_{d}}{z_{xi}}\right) ^{k}d_{xi}$. Note that if $k=0,$ all domestic producers export and we require that $k>0$ for only a fraction of the firms to export. The definitions of domestic and export revenues together with the pricing Eqs. 37, 38 imply that $\frac{r_{di}}{r_{xi}^{*}}=\frac{\left( \rho _{d,i}\right) ^{1-\theta }}{\left( \rho _{xi}^{*}\right) ^{1-\theta }}$ and $\frac{r_{di}^{*}}{r_{xi}}=\frac{\left( \rho _{d,i}^{*}\right) ^{1-\theta }}{\left( \rho _{xi}\right) ^{1-\theta }}=\frac{\left( (w^{s*})^{\beta _{i}}(w^{l*})^{1-\beta _{i}}\right) ^{1-\theta }}{\left( \frac{\tau }{Q}(w^{s})^{\beta _{i}}(w^{l})^{1-\beta _{i}}\right) ^{1-\theta }}\left( \frac{z_{d}}{z_{xi}}\right) ^{\theta -1}$. Next, Eqs. 41, 43 imply that $r_{xi}=f_{x}\left( w^{s}\right) ^{\beta _{i}}\left( w^{l}\right) ^{1-\beta _{i}}\frac{k}{k-(\theta -1)}\frac{\theta }{Q}$ and $r_{xi}^{*}=f_{x}^{*}\left( w^{s*}\right) ^{\beta _{i}}\left( w^{l*}\right) ^{1-\beta _{i}}\frac{k}{k-(\theta -1)}\theta Q$. Thus, domestic revenues are only a function of the export cutoffs, wages, the real exchange rate and parameters

$$\begin{aligned} r_{di}=\left[ \frac{\left( (w^{s})^{\beta _{i}}(w^{l})^{1-\beta _{i}}\right) ^{1-\theta }}{\left( Q\tau ^{*}(w^{s*})^{\beta _{i}}(w^{l*})^{1-\beta _{i}}\right) ^{1-\theta }}\left( \frac{z_{d}}{z_{xi}^{*}}\right) ^{\theta -1}\right] \left[ f_{x}^{*}\left( w^{s*}\right) ^{\beta _{i}}\left( w^{l*}\right) ^{1-\beta _{i}}\frac{k}{k-(\theta -1)}\right] \theta Q \end{aligned}$$

and

$$\begin{aligned} r_{di}^{*}=\left[ \frac{\left( (w^{s*})^{\beta _{i}}(w^{l*})^{1-\beta _{i}}\right) ^{1-\theta }}{\left( \frac{\tau }{Q}(w^{s})^{\beta _{i}}(w^{l})^{1-\beta _{i}}\right) ^{1-\theta }}\left( \frac{z_{d}}{z_{xi}}\right) ^{\theta -1}\right] \left[ f_{x}\left( w^{s}\right) ^{\beta _{i}}\left( w^{l}\right) ^{1-\beta _{i}}\frac{k}{k-(\theta -1)}\right] \frac{\theta }{Q}. \end{aligned}$$

Substituting these equations for domestic revenue in Eq. 39 delivers

$$\begin{aligned} d_{i}=\frac{1}{\theta }\left[ \frac{\left( (w^{s})^{\beta _{i}}(w^{l})^{1-\beta _{i}}\right) ^{1-\theta }}{\left( Q\tau ^{*}(w^{s*})^{\beta _{i}}(w^{l*})^{1-\beta _{i}}\right) ^{1-\theta }}\left( \frac{z_{d}}{z_{xi}^{*}}\right) ^{\theta -1}\right] \left[ f_{x}^{*}\left( w^{s*}\right) ^{\beta _{i}}\left( w^{l*}\right) ^{1-\beta _{i}}\frac{k}{k-(\theta -1)}\right] \theta Q+\left( \frac{z_{d}}{z_{xi}}\right) ^{k}d_{xi} \end{aligned}$$

and

$$\begin{aligned} d_{i}^{*}=\frac{1}{\theta }\left[ \frac{\left( (w^{s*})^{\beta _{i}}(w^{l*})^{1-\beta _{i}}\right) ^{1-\theta }}{\left( \frac{\tau }{Q}(w^{s})^{\beta _{i}}(w^{l})^{1-\beta _{i}}\right) ^{1-\theta }}\left( \frac{z_{d}}{z_{xi}}\right) ^{\theta -1}\right] \left[ f_{x}\left( w^{s}\right) ^{\beta _{i}}\left( w^{l}\right) ^{1-\beta _{i}}\frac{k}{k-(\theta -1)}\right] \frac{\theta }{Q}+\left( \frac{z_{d}}{z_{xi}^{*}}\right) ^{k}d_{xi}^{*}. \end{aligned}$$

Note that from the latter condition and 43, we obtain, $\left( \frac{z_{d}}{z_{xi}^{*}}\right) =\left\{ \frac{d_{i}^{*}}{f_{x}^{*}\left( w^{s*}\right) ^{\beta _{i}}\left( w^{l*}\right) ^{1-\beta _{i}}\frac{\theta -1}{k-(\theta -1)}}-\frac{1}{\theta }\frac{1}{f_{x}^{*}\left( w^{s*}\right) ^{\beta _{i}}\left( w^{l*}\right) ^{1-\beta _{i}}\frac{\theta -1}{k-(\theta -1)}}\right.$

$\left[ \frac{\left( (w^{s*})^{\beta _{i}}(w^{l*})^{1-\beta _{i}}\right) ^{1-\theta }}{\left( \frac{\tau }{Q}(w^{s})^{\beta _{i}}(w^{l})^{1-\beta _{i}}\right) ^{1-\theta }}(\frac{z_{d}}{z_{xi}})^{\theta -1}\right]$$\left. \left[ f_{x}\left( w^{s}\right) ^{\beta _{i}}\left( w^{l}\right) ^{1-\beta _{i}}\frac{k}{k-(\theta -1)}\right] \frac{\theta }{Q}\right\} ^{\frac{1}{k}},$ and substituting for $\left( \frac{z_{d}}{z_{xi}^{*}}\right)$ into the first condition, we obtain an equation only in terms of the export cutoff:

$$\begin{aligned} A_{1}\left( \frac{z_{d}}{z_{xi}}\right) ^{k}+A_{2}A_{3}^{1-\theta }\left[ \frac{A_{4}-A_{5}\left( A_{6}z_{xi}\right) ^{1-\theta }}{(\theta -1)f_{x}^{*}}\right] ^{\left( \frac{\theta -1}{k}\right) }=d_{i}, \end{aligned}$$

where $A_{1}=f_{x}(w_{i}^{s})^{\beta _{i}}(w_{i}^{l})^{1-\beta _{i}}\frac{(\theta -1)}{k+1-\theta }$, $A_{2}=\frac{Qkf_{x}^{*}\left( w^{s*}\right) ^{\beta _{i}}\left( w^{l*}\right) ^{1-\beta _{i}}}{k+1-\theta }$, $A_{3}=\frac{(w^{s})^{\beta _{i}}(w^{l})^{1-\beta _{i}}}{Q\tau ^{*}(w^{s*})^{\beta _{i}}(w^{l*})^{1-\beta _{i}}}$, $A_{4}=\frac{f_{e}^{*}(1-(1-\delta )\gamma )\left( k+1-\theta \right) }{(1-\delta )\gamma }$, $A_{5}=\frac{f_{x}\left( w^{s}\right) ^{\beta _{i}}\left( w^{l}\right) ^{1-\beta _{i}}k}{Q\left( w^{s*}\right) ^{\beta _{i}}\left( w^{l*}\right) ^{1-\beta _{i}}}$, $A_{6}=\frac{Q\left( (w^{s*})^{\beta _{i}}(w^{l*})^{1-\beta _{i}}\right) }{\left( \tau (w^{s})^{\beta _{i}}(w^{l})^{1-\beta _{i}}\right) z_{d}}$. Note that $A_{1}>0$, $A_{2}>0$$A_{3}>0$, $A_{4}>0,\,A_{5}>0,$$A_{6}>0$ under the parameter restrictions discussed so far. Given positive wages and a positive real exchange rate, the left-side is a hyperbola for $z_{xi}>0$ which guarantees existence and uniqueness for $z_{xi}$. Ghironi and Melitz (2005, TA) employ a similar strategy to prove uniqueness and existence of the steady state.

Now that we have obtained the export cutoffs equation, Eq. 44 allows us to pin down the fraction of exporting firms. We have also obtained average domestic revenue $r_{di}$ and profits $d_{di}$.

Substituting for the $N_{ei}$ from Eq. 45, and for average domestic and export revenue allows to write Eq. 48 as: $w_{i}^{s}S_{i}+w_{i}^{l}L_{i}=N_{di}\left( v_{i}\frac{\delta }{(1-\delta )}-d_{i}+r_{di}+\frac{N_{xi}}{N_{di}}Qr_{xi}\right)$, which allows us to pin down the number of producing firms $N_{di}$. Then Eqs. 44, 45 deliver the number of exporters as $N_{xi}=(\frac{z_{d}}{z_{xi}})^{k}N_{di}$ and new entrants $N_{ei}=\frac{\delta N_{di}}{(1-\delta )}$. Note that to obtain positive firm entry at the steady state, we require that $0<\delta <1.$

The domestic and export prices are obtained from Eqs. 37, 38 as a function of wages and Q and Eq. 36 pins down the sector price index $\psi _{i}$. Note that Eq. 37 implies another important restriction for the existence of a positive steady state domestic price, namely $\theta >1$ and $z_{d}>0$ (which holds as long as $z_{min}>0$ and $k>1-\theta$). In addition, we can write Eq. 36 as $N_{di}r_{di}+N_{xi}^{*}r_{xi}^{*}=\alpha _{i}C$ and obtain total revenue C in each country. Note that $1>\alpha >0$ in order to have positive demand in both sectors.

Finally, we can use Eq. 48 for each sector in each country to pin down the wage vector and the balanced trade condition to pin down the real exchange rate as a function of relative exports: $Q=\frac{N_{x1}^{*}r_{x1}^{*}+N_{x2}^{*}r_{x2}^{*}}{N_{x1}r_{x1}+N_{x2}r_{x2}}$.

This concludes the proof of equilibrium in the steady state. Out of steady state additional equations are required to pin down the allocation of workers across sectors, the value functions of the workers (9), the cutoff values and rates of reallocation of incumbents (7, 8), the cutoff values and shares of entry of newly entering workers (11, 12), and the law of motion (19). These equations uniquely pin down the allocation of workers out of steady state, since the rates of reallocation are strictly increasing in the wage differential, while the marginal cost of reallocation is strictly increasing in the rates of reallocation. Put differently, wage differentials motivate workers to switch sectors, while migration costs reduce the incentives to switch. In equilibrium, both aspects balance and workers flows are uniquely pinned down.

Extending the model to incorporate training involves adding equations to ensure that the steady state supply of skilled and unskilled workers is identified. To asses the relative value of skilled versus unskilled entry, we define the average value of a skilled, and unskilled workers as $V^{s}=\frac{Se_{1}}{Se}V_{1}^{s}+\frac{Se_{2}}{Se}V_{2}^{s}$ and $V^{l}=\frac{Le_{1}}{Le}V_{1}^{l}+\frac{Le_{2}}{Le}V_{2}^{l}$, where $Se=Se_{1}+Se_{2}$ and $Le=Le_{1}+Le_{2}$ are the total skilled and unskilled workers entering the labor force. Note that Eqs. 49, 50 imply that for a stable steady state $Se=sS$, $Le=sL$ and for total worker entry $We=Se+Le=sENDOW$. Then, the level of the threshold training cost where a worker is indifferent between entering as skilled versus unskilled is $\bar{\varepsilon }^{T}=V^{s}-V^{l}$. In order to ensure a positive and unique probability of training at the steady state, we have to assume a probability distribution for the training cost $\Gamma (\varepsilon _{t}^{T})$ that is only defined for non-negative values and gives zero probability to negative values, like the exponential distribution. Then, the probability of training is pinned down as $\eta ^{T}=\Gamma \left( \bar{\varepsilon }^{T}\right) =1-\exp \left( -scaleT\bar{\varepsilon }^{T}\right)$ and the share of skilled workers is obtained by $\frac{S}{ENDOW}=\frac{Se}{We}=\eta ^{T}$.