## 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.

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## Notes

Burstein and Melitz (2013) show that positive fixed costs of domestic production would eliminate all transitional dynamics in GM. This is not the case in our model due to the slow adjustment of workers. We nevertheless prefer to use the GM assumption that fixed costs of domestic production are zero, due to tractability and the numerical problems discussed by Chaney (2005). In Sect. 7 we discuss the role of firm adjustment in more detail.

A recent literature analyzes the effects of trade liberalization on unemployment [see, e.g., Egger and Kreickemeier (2009), Felbermayr et al. (2011), Helpman and Itskhoki (2010), Helpman et al. (2010) or Larch and Lechthaler (2011)] and stresses within-group wage inequality as a contributor to overall wage inequality [see, e.g., Helpman et al. (2010)]. Given the already complicated structure of our model we concentrate on just two measures of wage inequality and leave the analysis of unemployment and within-group wage inequality for future research.

As in Dix-Carneiro (2014) we assume that the sector migration cost is paid in terms of utility, which has the benefit that the sector migration cost need not be traded in the market.

Note that the probability of switch from sector

*i*to sector*j*is \(\eta _{ijt}^{s}=F(\max ( -\overline{\varepsilon _{t}}^{s},\varepsilon _{\min }^{s}) )\).Allowing for negative sector migration costs would imply positive gross flows across sectors, which are relevant empirically. However, for wage inequality only the relative supply of workers and thus net flows are relevant.

Reversing the timing assumption would not have any implications for our results.

This assumption is in line with empirical evidence in Ryoo and Rosen (2004) who find that the fraction of college graduates who are engineers is closely related to a measure of relative earnings prospects in engineering.

Without this sector entry costs the choice of sector would not be well defined in the steady state, because workers are indifferent between the two sectors in the absence of wage differentials. Additionally, there would be no mechanism assuring that the steady state is hit, potentially implying overshooting and oscillatory dynamics.

Note that the expected value of the sector-preference is zero and therefore drops out of this equation.

The iceberg trade costs are proportional to the value of the exported product and represent a number of different barriers to trade. These include both trade barriers that can be influenced by policy, like restrictive product standards or slow processing of imports at the border, and trade barriers that cannot be influenced by policy, like the costs of transportation. We follow the standard practice in the literature and model trade liberalization as a decrease in the iceberg trade cost.

In contrast to Melitz (2003) it is not the households that invest themselves in new firms due to the credit constraints we are assuming.

Under the income-pooling assumption, we ran simulations allowing for trade in international bonds and unbalanced trade but the movements in the trade balance were negligible.

The data can be accessed at http://www.nber.org/nberces/.

We prefer this measure over the more traditional measure of revealed comparative advantage introduced by Balassa because it not only takes account of the export structure of the sector but also of its import structure. In addition, data on US exports and imports at the NAICS industry classification is readily available at Peter Schott’s website. In contrast the Balassa measure of RCA requires data on world export shares at the NAICS level which has to be converted from other industry classification systems.

We choose the exponential distribution because it has only one parameter, the scale parameter, and its minimum is always zero, which provides simple analytic expressions for integrals.

We obtain the steady states of the model with numerical methods in MATHEMATICA and use DYNARE in order to simulate the transition path of the economy from the pre- to the post-liberalization steady state.

Technically Eq. 17 becomes \(\frac{Se_{t}}{Se_{t}+Le_{t}}=\frac{Se_{0}}{Se_{0}+Le_{0}}\), so that the share of skilled workers remains constant.

Some variables such as the index for inter-sectoral wage inequality are reported as absolute deviations from their pre-liberalization steady state value rather than percent deviations because they are zero at the pre-liberalization steady state.

In our model firms only produce one product and cannot actively switch sectors but the implications are the same.

Dauth et al. (2014) show that the same is true for Germany.

Introducing fixed costs of production into our dynamic model is not tractable but it is possible to compute the steady states of such a model. Similarly to BRS, we assume that the fixed cost of production equals the fixed export cost. All other parameters remain identical to our benchmark model except for the trade costs which we re-calibrate to ensure that the share of international trade to GDP is the same as in the benchmark model (6% before and 27% after liberalization). For the model with worker training this implies a decline in trade costs from 2.586 to 1.663. Then, the share of exporting firms rises by 10.4 percentage points versus 9.5 percentage points without fixed costs, the skill premium rises by 6.5 percentage points versus 3 percentage points without fixed costs, and the Gini index rises by 4.3% versus 2.8% without fixed costs. Thus, the model with fixed costs of domestic production yields even larger changes in inequality in the long run.

See GM for a discussion of welfare-based price indices.

The Gini coefficient before taxes and transfers increased by \(6\%\) from 0.44 to 0.468, while the Gini after taxes and transfers increased by \(8\%\) from 0.347 to 0.375

Although there is a further blip just before the subsidy runs out.

A likewise migration subsidy to skilled workers would have the opposite effects, but the effects would be quantitatively smaller since skilled workers are already more mobile to begin with.

Again we assume that the subsidy is financed by a proportional wage tax.

Note, however, that this policy reduces the welfare of skilled workers, since the increase in the number of skilled workers puts downwards pressure on their wage (not depicted). Their wage still goes up, but by less than in the benchmark without policy response.

Again we restrict the analysis to our benchmark model. Results concerning the other scenarios are available upon request.

For this exercise we simulate a decrease in the fixed cost of exporting by 52% and a decrease in the iceberg trade cost by 4.7%. Both imply an increase in the share of trade in GDP by 1%. For most variables the lines exectly overlap so that only one can be seen.

This is of course different from a small open economy but serves to demonstrate the relevance of adjustability in international prices.

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### Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Wolfgang Lechthaler thanks the Thyssen Stiftung for the financial support of his research visit at the UC Santa Cruz, during which much of this research was conducted. Wolfgang Lechthaler also thanks the staff of the UC Santa Cruz for its hospitality. Mariya Mileva thanks the European Commission for financial support through the project WWWforEurope. We are grateful to Danvee Floro for her excellent research assistance. We also thank Sebastian Brown, Jesus Crespo Cuaresma, Jorgen Elmeskov, Mario Larch and Jennifer Poole for their insightful comments. All remaining errors are our own.

## A Proof of existence and uniqueness of the steady state

### 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).

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

and

Using Eq. 47 for both sectors these become

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

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

and

Substituting these equations for domestic revenue in Eq. 39 delivers

and

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:

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 \(\frac{S}{ENDOW}=\frac{Se}{We}=\eta ^{T}\).

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### Cite this article

Lechthaler, W., Mileva, M. Trade liberalization and wage inequality: new insights from a dynamic trade model with heterogeneous firms and comparative advantage.
*Rev World Econ* **155, **407–457 (2019). https://doi.org/10.1007/s10290-019-00347-0

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DOI: https://doi.org/10.1007/s10290-019-00347-0

### Keywords

- Trade liberalization
- Wage inequality
- Adjustment dynamics

### JEL Classification

- E24
- F11
- F16
- J31
- J62