Online Supplement for Trade and Inequality: From Theory to Estimation

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1 Online Supplement for Trade and Inequality: From Theory to Estimation Elhanan Helpman Harvard University and CIFAR Oleg Itskhoki Princeton University and NBER Marc-Andreas Muendler UC San Diego and NBER Stephen Redding Princeton University and NBER May Contents A Introduction 3 B Reduced-form Empirical Findings for Other Countries 3 C Theory Appendix 4 C.1 Theoretical model C.2 Derivation of the empirical model C.3 Reduced-form coefficients and structural parameters C.4 Counterfactuals D Econometric Inference 14 D.1 Derivation of the likelihood function D.2 Maximum Likelihood estimation D.3 Overidentified GMM estimation D.4 GMM Bounds D.5 Identification E Extensions and Generalizations 29 E.1 Isomorphisms E.2 Two sources of heterogeneity E.3 Measurement error We thank the National Science Foundation for financial support. See the paper for our acknowledgements. 1

2 E.4 Estimation with sector-region heterogeneity E.5 Multiple destinations F Dynamic model 41 G Data Appendix 48 H Additional Empirical Results and Robustness 51 H.1 Export Participation H.2 Wage inequality within versus between sectors and occupations H.3 Non-manufacturing industries H.4 Worker observables and residual wage inequality H.5 Regional Robustness H.6 Returns to education and tenure H.7 Robustness test using more disaggregated education measures H.8 Between-sector wage differentials H.9 Between versus within-firm wage inequality H.10 Between versus within-firm wage inequality by occupation H.11 Between versus within-firm wage inequality by sector H.12 Between versus Within-Firm Wage Inequality for Exporters and Non-Exporters H.13 Firm-year rather than firm-occupation-year fixed effects H.14 Constant composition residual wage inequality H.15 Firm and Worker Fixed Effects Estimates H.16 Counterfactuals Robustness Test H.17 Robustness Test using Colombian Data H.18 Robustness Test Using Brazilian Household Survey Data I Data Supplement with Industry and Occupation Concordances 78 2

3 A Introduction This online supplement contains the technical derivations for the theoretical results in the paper and reports additional empirical results and other information. Section B discusses reduced-form empirical findings for other countries that are consistent with our stylized facts for Brazil in Section 3 of the paper. Section C provides a full characterization of the structural model and discusses the relationship between the reduced-form coefficients and structural parameters. Section D deals with econometric inference including the derivation of the likelihood function and the generalized method of moments (GMM) bounds analysis. Section G discusses the data sources and definitions. Section H contains additional empirical results and robustness checks referred to in the paper. B Reduced-form Empirical Findings for Other Countries In this section we discuss that our stylized facts for Brazil are consistent with empirical findings for a number of other countries including the United States. First our reduced-form results on wage inequality within and between sectors and occupations (subsection 3.1 of the paper) are consistent with the findings of a number of existing studies. Katz and Murphy (1992) find that much of the growth of wage inequality in the United States from occurred within industry-occupation cells. Berman Bound and Griliches (1994) and Berman Bound and Machin (1998) find substantial increases in the relative employment and relative wages of skilled workers within industries in the United States and other OECD countries respectively (see also the survey by Katz and Autor 1999). Second our evidence on worker observables and wage inequality (subsection 3.2 of the paper) is in line with the conclusions of the existing labor literature. Juhn Murphy and Pierce (1993) find that much of the growth in wage inequality in the United States from is explained by a growth in residual wage inequality within narrowly-defined education and labor market experience groups (see also Autor Katz and Kearney 2008 Lemieux 2006). Similarly Akerman Helpman Itskhoki Muendler and Redding (2013) Machin (1996) and Attanasio Goldberg and Pavcnik (2004) show that a substantial component of the level and growth of wage inequality is unexplained by observed worker characteristics in countries as diverse as Sweden the United Kingdom and Colombia respectively. Third our stylized facts on wage inequality between versus within firms (subsection 3.3 of the paper) are consistent with existing studies for other countries. For example Davis and Haltiwanger (1991) find that between-plant wage dispersion accounts for around one half of the level and growth of wage inequality in U.S. manufacturing from Using data for a later time period Barth Bryson Davis and Freeman (2011) find that more than 70 percent of the increased dispersion of U.S. earnings among individuals from occurred across establishments. Using West German data Card Heining and Kline (2013) find that increasing plant-level heterogeneity and rising assortativeness in the assignment of workers to establishments explain a large share of the rise in wage inequality from Finally using data on U.K. manufacturing Faggio Salvanes and Van Reenen (2010) find that most of the increase in individual wage inequality can be accounted for by an increase in inequality between firms within industries. 3

4 Finally as discussed in the paper our reduced-form results relating between-firm wage differences to firm size and export status are supported by a large empirical trade literature following Bernard and Jensen ( ). Our estimate of the employer-size wage premium using data on raw wages for Brazilian manufacturing of 0.12 compares to a value of 0.14 reported for U.S. manufacturing in Bayard and Troske (1999). Similarly using raw firm wages we estimate an exporter premium of 0.26 in Table 7 (after controlling for firm size) which compares to the value of 0.29 reported for U.S. manufacturing in Table 8 of Bernard Jensen Redding and Schott (2007). C Theory Appendix C.1 Theoretical model Consider the model in Helpman Itskhoki and Redding (2010 henceforth HIR) with the following two extensions: 1. Screening costs are heterogenous across firms and given by e η C δ ( ac ) δ where η varies across firms while C and δ are common to all firms. In the original HIR model η Fixed costs of exporting are heterogenous across firms and given by e ε F x where ε varies across firms and the fixed export cost F x is common to all firms. In the original HIR model ε 0. Simplifying HIR slightly we set the fixed cost of production to zero for all firms (f d = 0). Here we describe the solution to the problem of a given firm in an industry taking industry labor market tightness as given. The details of the general equilibrium can be found in HIR. A firm with a shock triplet (θ η ε) solves the following problem as explained in the text and HIR: where: 1 Π(θ η ε) = { 1 max Na cι {01} 1 + βγ R(N a c ι; θ) bn e η C } δ (a c) δ ιe ε F x (C.1) 1 Revenues given export status ι {0 1} are a function of total output of the firm defined by: { R(Y ι) = max A d Y β d + ιax(yx/τ)β} Y d +Y x Y where A d Y β are revenues from domestic sales and A x(y x/τ) β are revenues from exporting. The parameter τ reflects variable trade costs (including iceberg transport costs). The demand is derived from a CES aggregator with elasticity 1/(1 β) and A d and A x are demand shifters which depend on total industry expenditure and a price index as detailed in HIR. The optimal allocation of sales across markets for an exporter satisfies Y x/y d = τ β/(1 β) (A x/a d ) 1/(1 β) and therefore we can write 4

5 and R(N a c ι; θ) = [1 + ι(υ x 1)] 1 β A d Y (N a c ; θ) β Y (N a c ; θ) = kaγk min k 1 eθ N γ (a c ) 1 γk ( ) 1 Υ x = 1 + τ β Ax 1 β 1 β. A d As described in the text the employment of the firm is given by The bargained wage rate of the firm is given by: 2 H(N a c ) = N (a min /a c ) k. W (N a c ι; θ) = βγ 1 + βγ R(N a c ι; θ) H(N a c ) and the remaining share 1/(1 + βγ) of revenues goes to the firm. Taking the first-order conditions in (C.1) with respect to N and a c and using the expression for H and W above we arrive at expressions (9) (11) in the text: [ R = κ r 1 + ι (Υx 1) ] 1 β Γ [ H = κ h 1 + ι (Υx 1) ] (1 β)(1 k/δ) Γ [ W = κ w 1 + ι (Υx 1) ] k(1 β) δγ (e θ) β Γ ( e η) β(1 γk) δγ (e θ) β(1 k/δ) Γ βk (e θ) ( ( δγ e η) k δ (e η) β(1 γk)(1 k/δ) k δγ δ 1+ β(1 γk) δγ where the expressions for the constants can be found in HIR (in particular see the Appendix to HIR at The first-order conditions further imply that the firm s profits are Π(θ η ε; ι) = Γ 1 + βγ R(θ η; ι) ιeε F x Y d = Y/Υ x and Y x = (Υ x 1)Y/Υ x where Υ x is defined in the text. The resulting revenues from domestic sales are R d = A d (Y/Υ x) β = R/Υ x. 2 As detailed in HIR at the bargaining stage the firm s costs are sunk and the firm bargains with its workers to divide revenues from production R(N a c ι; θ). The firm s revenues are a power function of employment H with power γβ and the firm and the workers know only that each worker has an ability of at least a c and hence an expected ability of ā. The outside option of the worker is normalized to zero and the Stole and Zwiebel (1996a) bargaining condition under these circumstances is [ ] R(H) W (H)H = W (H) H where we emphasize that at the bargaining stage the revenues and the resulting wage rate depend only on employment of the firm. The solution to this differential equation in H is W = βγ/(1 + βγ) R/H. ) 5

6 where Γ = 1 βγ β(1 γk)/δ. Revenues are a function of the firm s export status ι {0 1}. Similarly the optimally chosen N and a c are a function of export status given (θ η). The firm chooses to export when Π(θ η ε; ι = 1) Π(θ η ε; ι = 0) which can be written as condition (12) in the text: ι = I {κ π ( ) ( Υ 1 β Γ x 1 e θ) β ( Γ e η) β(1 γk) δγ F x e ε } where κ π = Γκ r /(1 + βγ) and I{ } is the indicator function. The above statements describe the relevant equilibrium conditions of the model. The remaining general equilibrium conditions can be found in HIR. C.2 Derivation of the empirical model We start by taking logs of the expressions for H W and inside the indicator function in the expression for ι: where h = α h + ( 1 k ) [ ( δ ι 1 β Γ log Υ x + β Γ θ + β(1 γk) [ ( ) ] w = α w + k δ ι 1 β Γ log Υ x + β Γ θ β(1 γk) δγ η { ( ) } ι = I 1 β σ Γ θ + β(1 γk) δγ η ε f δγ k/δ 1 k/δ f = 1 ( [ ]) α π + log F x log Υ 1 β Γ x 1 σ ) ] η α s = log κ s (for s = h w π) and σ is the variance of [(β/γ) θ + β(1 γk)/(δγ) η ε] explicitly given in (C.2) below. It is convenient to introduce the following notation to simplify the expressions in subsequent derivations: χ = k/δ 1 k/δ λ 1 = β Γ λ 2 = β(1 γk) δγ µ h = 1 1 β 1 + χ Γ log Υ x and µ w = χ 1 β 1 + χ Γ log Υ x = χµ h. With this notation we can write the structural model simply as: Using the definition of Γ we show that: h = α h + µ h ι + λ 1 1+χ θ + λ 2 χ 1+χ η w = α w + µ w ι + χλ 1 1+χ θ + χ(1+λ 2) 1+χ η ι = I {( λ 1 θ + λ 2 η ε ) /σ f } λ 2 χ = χ Γ ( 1 β ) < 0 k 6

7 since the model s parameter restrictions are β < 1 < k. This implies that the effect of η on h is negative. Finally we make the distributional assumption on the shocks: (θ η ε) N (0 Σ) Σ = σ 2 θ σ θη σ 2 η σ θε σ ηε σ 2 ε The above four expressions for h w ι and the distribution of (θ η ε) together with the definitions of the parameters (χ µ h µ w λ 1 λ 2 f) fully describe the structural model. The model implies that the following parameter restrictions: 3 χ µ h µ w λ 1 λ 2 > 0 λ 2 < χ µ w = χµ h and µ h + µ w = log Υ (1 β)/γ x. We now derive the relationship between the structural parameters of the model and the reducedform coefficients of our econometric model. Our derivation involves an orthogonalization of the shocks in the wage and employment equations. Define the first reduced-form shock u = λ χ θ + λ 2 χ 1 + χ η. so that h = α h + µ h ι + u. Next rewrite the wage equation as w = α w + µ w ι + χu + χη = α w + µ w ι + χ(1 + π)u + χ ( η πu ) where π is the projection coefficient of η on u so that (η πu) is uncorrelated with u. Under our normality assumption we further have E{η u} = πu. We can now define the second reduced-form shock v = χ ( η πu ) so that cov(u v) = 0 and we can write w = α w + µ w ι + ζu + v where ζ = χ(1 + π) < χ. Note that the orthogonality of u and v is without loss of generality because it is a normalization. 3 The parameter restrictions are discussed in detail in HIR and we require Υ x > 0 β (0 1) 1 < k < δ 0 < γk < 1 and Γ > 0 which leads to these restrictions on the derived parameters. 7

8 Consider now the selection equation. Note that λ 1 θ + λ 2 η = (1 + ζ)u + v. Define where and z = 1 σ [ (1 + ζ)u + v ε ] σ 2 = (1 + ζ) 2 σ 2 u + σ 2 v + σ 2 ε 2(1 + ζ)σ uε 2σ vε σ 2 u = λ 2 1 (1 + χ) 2 σ2 θ + (λ 2 χ) 2 (1 + χ) 2 σ2 η σ 2 v = χ 2 ( σ 2 η + π 2 σ 2 u 2πcov(η u) ) = χ 2 ( σ 2 η π 2 σ 2 u) σ uε = λ χ σ θε + λ 2 χ 1 + χ σ ηε σ vε = χ ( σ ηε πσ uε ). (C.2) Therefore we have σ 2 z = var(z) = 1. We hence rewrite the selection equation simply as ι = I {z f}. The joint distribution of the reduced-form shocks (u v z) is given by: (u v z) N (0 Σ R ) Σ R = σu 2 0 σv 2 ρ u σ u ρ v σ u 1 (C.3) where ρ u σ u = 1 σ ( (1 + ζ)σ 2 u σ uε ) ρ v σ v = 1 σ ( σ 2 v σ vε ). This completes the derivation of the reduced-form system. We now discuss the model s restrictions on the reduced-form coefficients Θ = (α h α w ζ σ u σ v ρ u ρ v µ h µ w f). As derived above we have µ h µ w > 0 µ w /µ h = χ and ζ = χ(1 + π) < χ. As discussed in the paper we estimate the model under the structural identifying assumption of a covariance condition on the structural shocks (σ θη = 0) as is common in the structural econometrics literature following Koopmans (1949) Fisher (1966) and Wolpin (2013). Under this structural covariance restriction the expression for 8

9 the projection coefficient π is: π = cov(η u) var(u) = λ 2 χ 1+χ σ2 η λ 2 1 (1+χ) 2 σ 2 θ + (λ 2 χ) 2 (1+χ) 2 σ 2 η 0 (C.4) where π < 0 follows from the model s parameter restriction λ 2 < χ. The structural covariance restriction (σ θη = 0) helps to separately identify the market access and selection forces by placing bounds on the relative market access effects (µ h /µ w ): Lemma S.1 Under the structural covariance restriction σ θη = 0 the reduced-form coefficients must satisfy the inequality constraint: ζ µ w µ h < ζ + σ 2 v (1 + ζ)σu 2. (C.5) Proof: Consider the definition of π in (C.4) which combined with the definition of σ 2 u can be written as: Rearranging we can rewrite: π = (λ 2 χ)ση 2 (1 + χ)σu 2. ( 1 λ ) 2 χ 2 ση 2 χ (1 + χ)σu 2 Since 0 < λ 2 < χ we have 0 < 1 λ 2 /χ < 1 and therefore so that 0 ( 1 λ ) 2 χ 2 ση 2 χ (1 + χ)σu 2 = (χπ). 0 (χπ) < χ2 ση 2 (1 + χ)σu 2. Next we use the definition of σ 2 v to substitute in for χ 2 σ 2 η: < χ2 ση 2 (1 + χ)σu 2 Note that this is equivalent to: 0 (χπ) < σ2 v + (χπ) 2 σu 2 (1 + χ)σu 2. σ 2 v 0 (χπ) < (1 + ζ)σu 2 where we have used the fact that (1 + χ + χπ) = 1 + ζ from the definition of ζ. Finally note that χπ = ζ χ and χ = µ w /µ h. Hence the above inequalities are equivalent to the inequality constraint (C.5). We summarize the reduced-form empirical specification by: h = α h + µ h ι + u w = α w + µ w ι + ζu + v ι = I {z f} (C.6) 9

10 with the shocks (u v z) distributed according to (C.3). The data is x = (h w ι) and Θ = (α h α w ζ σ u σ v ρ u ρ v µ h µ w f) is the 10 1 vector of reduced-form coefficients. We estimate the reduced-form coefficients Θ subject to the inequality constraint (C.5) implied by our structural covariance restriction. We check whether the resulting parameter estimates ˆΘ satisfy the theoretical restriction that ˆµ h ˆµ w 0 (without imposing this additional restriction on the estimation). C.3 Reduced-form coefficients and structural parameters The model has 10 reduced-form coefficients Θ and 12 structural parameters Ξ = (α h α w µ h χ λ 1 λ 2 f σ θ σ η σ ε σ θε σ ηε ) where we have taken into account our structural covariance restriction that σ θη = 0. The coefficients (µ w σ π) are not part of the structural parameter vector Ξ because we can fully recover (µ w σ π) from Ξ using (C.4) (C.2) and the condition µ w = χµ h. The parameters (k δ β γ) are related to (χ λ 1 λ 2 ) as shown above and can be recovered after one of the parameters (k δ β γ) is calibrated. Some other parameters like b and C are related to the constants (α h α w ) as described in HIR but we do not attempt to recover them and hence do not discuss them here. We can relate the reduced-form coefficients Θ back to the structural parameters Ξ using the derivations in subsection C.2. We summarize the according relationships below: ζ = χ(1 + π) χ = k/δ 1 k/δ σu 2 = λ2 1 (1+χ) σ 2 2 θ + (λ2 χ)2 (1+χ) ση 2 λ 2 1 = β Γ σv 2 = χ ( ) 2 ση 2 π 2 σu 2 λ2 = β(1 γk) δγ ρ u = 1 σσ u ( (1 + ζ)σ 2 u σ uε ) (1+χ)(λ 2 χ)σ π = 2 η λ 2 1 σ2 θ +(λ2 χ)2 ση 2 ρ v = 1 σσ v ( σ 2 v σ vε ) σ 2 = (1 + ζ) 2 σ 2 u + σ 2 v + σ 2 ε 2(1 + ζ)σ uε 2σ vε µ h = 1 1 β 1+χ Γ log Υ x σ uε = λ1 1+χ σ θε + λ2 χ 1+χ σ ηε µ w = χµ h σ vε = χ ( ) σ ηε πσ uε. ( [ ]) f = 1 σ α π + log F x log Υ 1 β Γ x 1 while (α h α w ) are a part of both the reduced-form coefficients Θ and the structural parameters Ξ. The constants (α h α w ) market access effects (µ h µ w ) and export threshold (f) depend on Υ x which in turn depends on variable trade costs (τ) and relative market demands in the export and domestic markets (A x /A d ) that are likely to change over time. The coefficients (α h α w µ h µ w f) can be estimated directly with the reduced-form model. We can identify χ and hence k/δ from χ = µ w /µ h. Furthermore we can identify the market access effect from log Υ 1 β Γ x = µ h + µ w. From the reduced-form estimate of ζ we can recover π = ζ/χ 1 which is itself a derived pa- 10

11 rameter and provides information about structural parameters. Estimates of the covariance matrix (σ u σ v ρ u ρ v ) impose four conditions on the following five parameter combinations: ( λ1 σ θ (λ 2 χ)σ η σ ε λ 1 σ θε (λ 2 χ)σ ηε ). Therefore the structural parameters are under-identified. Nonetheless this does not constitute a limitation for our counterfactual exercises because the coefficients of the reduced-form model Θ are sufficient statistics for the impact of trade on wage inequality as discussed in detail below. We now briefly discuss what information about the structural parameters can nonetheless be obtained from the reduced-form estimates. Instead of 12 structural parameters in Ξ there are only 11 that can be identified in principle since λ 1 and σ θ always show up together multiplicatively (λ 1 σ θ ) even in the structural equations. We split the above relationships between the reduced-form coefficients and structural parameters into two blocks the first one defines the second moments and the second one defines the selection correlations given the second moments: Block 1 (variances) Estimates of (σ u σ v χ) allow to recover (λ 1 σ θ χσ η λ 2 /χ 1) where we treat χ µ w /µ h as an estimated parameter. Given that the inequality constraint (C.5) on the reducedform coefficients is satisfied this block always has a solution. Block 2 (selection) Estimates of (ρ u ρ v ) along with the parameter estimates from the previous block and definitions of the auxiliary parameters (σ σ uε σ vε ) provide information about (σ ε σ θε σ ηε ). This block is under-identified as we have two relationships tieing together three parameters. Therefore we need to calibrate one of the structural parameters here and this block imposes no additional restrictions on the reduced-form coefficients. Despite the under-identification in the second (selection) block we can nonetheless assess the strength of the selection effect in the estimated model. Indeed the knowledge of (ρ u ρ v ) is sufficient to quantify the overall contribution of productivity and screening shocks (θ η) to variation in export status. Note first that the amount of information in (θ η) is the same as in (u v) since the latter is a linear non-degenerate transformation of the former. Further given the joint distribution (C.3) the regression of z on (u v) is given by E{z u v} = ρ u u σ u + ρ v v σ v and its R-squared equals ρ 2 u+ρ 2 v. Therefore ρ 2 u + ρ 2 v is an overall measure of the selection correlation in the model and it can be calculated based on the reduced-form model. Note however that a particular value of this measure does not determine whether selection is mostly due to variation in (θ η) or due to the covariance between (θ η) and ε. C.4 Counterfactuals Consider an estimated reduced-form model characterized by (C.6) (C.3) ˆΘ. This allows us to simulate a counterfactual dataset of {(h i w i ι i )} i for a large number of firms i and calculate measures of 11

12 worker wage inequality in this simulated dataset. In the paper we carry out four types of counterfactuals: (i) autarky; (ii) variation in fixed exporting costs F x ; (iii) variation in variable trade cost τ (iv) variation in the dispersion of the employment and wage shocks (σ u and σ v ). We discuss each of these counterfactuals in turn. Autarky counterfactual This is the most immediate counterfactual to carry out. This counterfactual maintains the estimated coefficient vector ˆΘ and the distributional assumption (C.3) but substitutes the model in (C.6) with its special case that shuts down the effects of trade on the employment and wage distributions: { h i = α h + u i w i = α w + ζu i + v i. Model (C.7) relies on 5 coefficients (α h α w ζ σ u σ v ) which form a subset of Θ. The trade-related coefficients (ρ u ρ v µ h µ w f) are irrelevant for the autarky counterfactual. We use the autarky model (C.7) setting (α h α w ζ σ u σ v ) as in ˆΘ to simulate the counterfactual dataset {(h i w i )} i in autarky and calculate the measures of worker wage dispersion in autarky which are directly comparable to the inequality measures calculated for the full model (C.6) under ˆΘ. Note that (α h α w ) depend on general equilibrium variables and in general change between the autarkic and open economy equilibria. This however has no effect on measures of log wage dispersion since α h and α w introduce proportional shifts to the distributions of employment and wages which are inconsequential for wage inequality. Therefore our autarky counterfactual holds regardless of the general equilibrium changes in α h α w. Variation in the fixed exporting cost F x Recall from Subsection (C.3) that F x affects directly only the reduced-form export threshold f and no other coefficient in Θ. Therefore variation in F x translates into variation in the export threshold f (which is a linear monotonically increasing function of F x ). In our fixed exporting cost counterfactuals we consider a special case in which µ h and µ w are held constant which implicitly holds constant the relative export market demand A x /A d as is the case with symmetric countries when Υ x = 1 + τ β/(1 β). In our variable trade cost counterfactuals below we allow changes in τ to affect µ h µ w and f both directly and indirectly through changes in relative export market demand A x /A d. Any general equilibrium effects of changes in the fixed exporting cost (F x ) in the domestic market are captured by α h and α w which are again inconsequential for our inequality measures as discussed above. Therefore we maintain the model (C.6) (C.3) and all estimated coefficients ˆΘ except for f which we vary from infinity (when no firm exports) to minus infinity (when all firms export). For each counterfactual value of f we simulate a dataset of {(h i w i ι i )} i and calculate measures of worker wage dispersion and trade openness (the fraction of exporters and the employment share in exporting firms). We then plot measures of worker wage dispersion against measures of trade openness (see Figure 1 in the paper). Variation in the variable trade cost τ This is the most involved counterfactual that we consider. Recall that τ affects the market access variable Υ x = 1 + τ β 1 β (A x /A d ) 1 1 β which in turn directly determines the reduced-form market access coefficients µ h = 1 1 β Γ 1+χ log Υx and µ w = χ 1 β Γ 1+χ log Υx. (C.7) 12

13 Furthermore through the market access variable τ also affects the reduced-form coefficient f which decreases linearly in Υ 1 β Γ x. To summarize movements in τ directly affect three reduced-form coefficients (µ h µ w f) but in each case the effect of τ happens through the market access variable Υ 1 β Γ x which allows us to jointly move (µ h µ w f) in an internally-consistent way without any further knowledge of the structural parameters. Specifically we gradually move τ from infinity (which results in Υ x = 1) to 1 (at which point Υ x reaches high values) and change (µ h µ w f) accordingly keeping other coefficients in ˆΘ unchanged. For each value of τ (and hence Υ x ) we simulate a counterfactual dataset {(h i w i ι i )} i and calculate measures of worker wage dispersion and trade openness (the fraction of exporters and the employment share in exporting firms). We then plot measures of worker wage dispersion against measures of trade openness (see Figure 1 in the paper). There are two caveats that need to be discussed. First as discussed before movements in τ can have indirect general equilibrium effects on the intercepts (α h α w ) and on A x /A d where the latter also affects Υ x. This however does not lead to any loss of generality for our counterfactual since we plot wage inequality against measures of trade openness such as the share of employment in exporting firms and for both of these measures Υ x is a sufficient statistic so that knowledge of τ and A x /A d is not needed. Therefore as long as τ has a monotonically decreasing effect on Υ x in equilibrium our counterfactual in the right panel of Figure 2 in the paper holds regardless of the general equilibrium effect of τ on the relative demand shifter A x /A d. Second we reproduce the expression for f = 1 ( [ ]) α π + log F x log Υ 1 β Γ x 1. σ Note that the effect of Υ x on f depends on σ one of the derived parameters of the model that is not identified (see discussion in Subsection C.3). Therefore in order to carry out this counterfactual we need to calibrate σ. Recall that σ 2 = (1 + ζ) 2 σu 2 + σv 2 + σε 2 2(1 + ζ)σ uε 2σ vε. We make the natural assumption that the contribution of σu 2 and σv 2 to σ 2 is equal to ρ 2 u + ρ 2 v the R 2 in the regression of z on (u v). That is we assume: σ 2 = (1 + ζ)2 σu 2 + σv 2 ρ 2 u + ρ 2. v While this constitutes a natural benchmark we experiment with a wide range of smaller and larger values of σ 2 and find largely the same outcomes of the counterfactual. 13

14 D Econometric Inference D.1 Derivation of the likelihood function The likelihood function for observation j is the probability of observing a data vector x j = (h j w j ι j ) given the model (C.6) and coefficient vector Θ. Since we treat all observations in our cross-section as iid the likelihood function for the full sample X = {x j } j is a product of the conditional probabilities for individual observations: L(Θ X) = P{x j Θ}. j We now derive the expression for P Θ {x j } = P{x j Θ}. We omit Θ whenever the omission causes no confusion. Consider first an observation for a non-exporter ι = 0: P{h w ι = 0} = P{h w z < f} = P{u = h α h v = (w α w ) ζ(h α h ) z < f} = P{u = h α h v = (w α w ) ζ(h α h ) z = z}d z z<f Similarly for an exporter with ι = 1 we can write: P{h w ι = 1} = z f Consider now the joint density of (u v z): 4 From (C.3) it follows that: P{u = h α h µ h v = (w α w µ w ) ζ(h α h µ h ) z = z}d z. P{u v z} = P{z u v} P{u v} u N (0 σ 2 u) v N (0 σ 2 v) and z (u v) N ( ρ u u/σ u + ρ v v/σ v 1 ρ 2 u ρ 2 v) with (u v) jointly normal and independent. We can therefore write: P{u v z} = 1 ( ) u φ σ u σ u ( ) 1 u φ σ v σ v ( 1 φ 1 ρ 2 u ρ 2 v z ρ u u/σ u ρ v v/σ v 1 ρ 2 u ρ 2 v where φ( ) is the standard normal probability density function given by φ(t) = 1 2π e t2 /2. 4 We can base an alternative derivation on the complementary decomposition P{u v z} = P{u v z} P{z} to show that (( ) ( )) ρuσ uz (1 ρ 2 z N (0 1) and (u v) z N u )σu 2 ρ uσ uρ vσ v ρ vσ vz ρ uσ uρ vσ v (1 ρ 2 v)σv 2 and then rearrange P{u v z} P{z} to separate out terms with z. ) 14

15 Integrating this expression over z we obtain: P{u v z < f} = 1 ( u φ σ u σ u P{u v z f} = 1 ( u φ σ u σ u ) ) ( 1 v φ σ v 1 σ v φ σ v ( v σ v ) ( Φ ) [ ( 1 Φ f ρ u u/σ u ρ v v/σ v 1 ρ 2 u ρ 2 v ) f ρ u u/σ u ρ v v/σ v 1 ρ 2 u ρ 2 v where Φ( ) is the standard normal cumulative distribution function Φ(t) = t φ(s)ds. Finally we relate these expressions to the probability of the data: P{h w ι = 0} = P{u = h α h v = (w α w ) ζ(h α h ) z < f} ( ) = 1 φ (û(x)) 1 f ρ u û(x) ρ φ (ˆv(x)) Φ vˆv(x) σ u σ v 1 ρ 2 u ρ 2 v P{h w ι = 1} = P{u = h α h µ h v = (w α w µ w ) ζ(h α h µ h ) z f} [ ( )] = 1 φ (û(x)) 1 f ρ u û(x) ρ φ (ˆv(x)) 1 Φ vˆv(x) σ u σ v 1 ρ 2 u ρ 2 v )] where û(x) = h α h µ h ι σ ( u w αw µ w ι ) ζ ( h α h µ h ι ) ˆv(x) = σ v. Combining these two expressions into one we obtain our result (17) for P{x j Θ} in the text of the paper. D.2 Maximum Likelihood estimation Maximum likelihood (ML) estimates are obtained by numerically maximizing the log-likelihood function with respect to the coefficient vector given the dataset: ˆΘ ML = arg max log L(Θ X) Θ subject to the inequality constraint (C.5) implied by our structural covariance restriction. To ease the numerical optimization routine we make the following transformations: σ u σ v σ 2 u σ 2 v f ρ u ρ v f 1 ρ 2 u ρ 2 v ρ u 1 ρ 2 u ρ 2 v ρ v. 1 ρ 2 u ρ 2 v We numerically maximize the log-likelihood function with respect to the transformed coefficient vector and upon completion we make the reverse transformation to the original vector of coefficients. Note 15

16 that this transformation automatically ensures that σ u and σ v are positive and that (ρ u ρ v ) lies inside the unit circle that is ρ 2 u + ρ 2 v < 1. As a result we do not need to impose upper and lower bounds in the estimation of the transformed coefficients. We use ML asymptotic theory to calculate standard errors for ˆΘ ML. Standard asymptotics imply that under the correct specification N( ˆΘML Θ) d N (0 V ML ) where for robustness we use the sandwich form of the covariance matrix V ML = ( { }) 1 E 2 log L(Θ x) Θ Θ E which we consistently estimate with ˆV ML = 1 N j 2 log L( ˆΘ ML x j ) Θ Θ 1 N j { log L(Θ x) Θ 1 1 N 2 log L( ˆΘ ML x j ) Θ Θ } ( { }) 1 log L(Θ x) Θ E 2 log L(Θ x) Θ Θ j log L( ˆΘ ML x j ) log L( ˆΘ ML x j ) Θ Θ When a constraint binds in the estimation (e.g. when ζ = µ w /µ h which corresponds to the lower bound of (C.5)) we use the resulting equality to substitute out redundant parameters and thus reduce the problem to an unconstrained maximization with a smaller parameter vector. We then apply standard asymptotic theory to the reduced problem and recover the covariance matrix for the remaining parameters using the -method. D.3 Overidentified GMM estimation As discussed in subsection 4.3 of the paper we also consider an overidentified generalized method of moments (GMM) estimator using eleven conditional first and second moments reported in Table 5 of the paper. In the context of GMM estimation it is convenient to make the following substitutions: so that the new triplet of shocks is distributed ω = ζu + v 1 σ 2 ω = ζ 2 σ 2 u + σ 2 v ρ ω σ ω = ζρ u σ u + ρ v σ v (u ω z) N ( 0 Σ R) Σ R =. σ 2 u ζσ 2 u σ 2 ω ρ u σ u ρ ω σ ω 1. 16

17 Given the structure of the model (C.6) and the distributional assumption above we can calculate in closed form the first and second moments of employment and wages conditional on export status as well as the fraction of exporters: m h0 (Θ) = m h1 (Θ) = m w0 (Θ) = m w1 (Θ) = s 2 h0 (Θ) = s 2 h1 (Θ) = s 2 w0 (Θ) = s 2 w1 (Θ) = Eι j = 1 Φ(f) E{h j ι j = 0} = α h ρ u σ u λ( f) E{h j ι j = 1} = α h + µ h + ρ u σ u λ(f) E{w j ι j = 0} = α w ρ ω σ ω λ( f) E{w j ι j = 1} = α w + µ w + ρ ω σ ω λ(f) V{h j ι j = 0} = σ 2 u ρ 2 uσ 2 uλ( f) V{h j ι j = 1} = σ 2 u ρ 2 uσ 2 uλ(f) V{w j ι j = 0} = σ 2 ω ρ 2 ωσ 2 ωλ( f) V{w j ι j = 1} = σ 2 ω ρ 2 ωσ 2 ωλ(f) c 0 (Θ) = C{w j h j ι j = 0} = C(ω j ρ ω σ ω z j u j ρ u σ u z j ) + ρ ω σ ω ρ u σ u V{z j z j < f} = (ζσ 2 u ρ ω σ ω ρ u σ u ) + ρ ω σ ω ρ u σ u [ 1 Λ( f) ] c 1 (Θ) = = ζσ 2 u ρ ω σ ω ρ u σ u Λ( f) C{w j h j ι j = 1} = ζσ 2 u ρ ω σ ω ρ u σ u Λ(f) (D.1) where λ(f) =φ(f)/[1 Φ(f)] λ( f) =φ( f)/[1 Φ( f)] = φ(f)/φ(f) Λ(f) =λ(f)[λ(f) f] Λ( f) =λ( f)[λ( f) + f] and Θ is our 10 1 coefficient vector. In the derivation we have used the expressions for the first and second moments of a truncated standard normal variable: E{z j z j < f} = λ( f) E{z j z j f} = λ(f) V{z j z j < f} = 1 λ( f)[λ( f) + f] V{z j z j f} = 1 λ(f)[λ(f) f] for a standard normal variable z j. In the derivation of conditional covariances we used the fact that we can write w j = α w + µ w ι j + ω j = α w + µ w ι j + ρ ω σ ω z j + (ω j ρ ω σ ω z j ) h j = α h + µ h ι j + ρ u σ u z j + (u j ρ u σ u z j ) 17

18 where (ω j ρ ω σ ω z j ) and (u j ρ u σ u z j ) are both independent of z j. Efficient Overidentified GMM estimator For numerical GMM estimation it proves convenient to use the unconditional versions of these moments which are given by the following moment function: m(x j Θ) = ι j [1 Φ(f)] h j (1 ι j ) m h0 (Θ)Φ(f) h j ι j m h1 (Θ)[1 Φ(f)] h 2 j (1 ι j) (s 2 h0 (Θ) + m2 h0 (Θ))Φ(f) h 2 j ι j (s 2 h1 (Θ) + m2 h1 (Θ))[1 Φ(f)] w j (1 ι j ) m w0 (Θ)Φ(f) w j ι j m w1 (Θ)[1 Φ(f)] w 2 j (1 ι j) (s 2 w0 (Θ) + m2 w0 (Θ))Φ(f) wj 2ι j (s 2 w1 (Θ) + m2 w1 (Θ))[1 Φ(f)] h j w j (1 ι j ) (c 0 (Θ) + m h0 (Θ)m w0 (Θ)) Φ(f) h j w j ι j (c 1 (Θ) + m h1 (Θ)m w1 (Θ)) [1 Φ(f)] where we have an overidentified system of 11 moments and 10 parameters. The efficient GMM estimator solves ˆΘ GMM = arg max 1 m (x j Θ) W 1 m(x j Θ) Θ N N j subject to the reduced-form inequality (C.5) implied by our structural covariance restriction where at the first stage W = I 11 is an identity matrix and at the second stage the optimal weighting matrix is W E = ( E{m(Θ)m (Θ)} ) 1 j which we consistently estimate by Ŵ E ( ˆΘ) = 1 N m(x j ˆΘ)m (x j ˆΘ) j 1 using ˆΘ from the first stage. In a Monte Carlo study the efficient overidentified GMM estimator has similar properties to the ML estimator but is slightly inferior. In the sample the overidentified GMM estimates are quantitatively very close to the ML estimates and we do not report them for brevity. In section 5.1 of the paper we report the square root of the overidentified GMM objective function 18

19 where as the weighting matrix we use the diagonal (matrix) of ŴE( ˆΘ): 1 N m (x j ˆΘ) Ŵ D ( ˆΘ) 1 N j m(x j ˆΘ) j 1/2 11 = k=1 ( 1 N j m k(x j ˆΘ) ˆσ 2 mk ( ˆΘ) ) 2 1/2. D.4 GMM Bounds We provide here the derivations for the GMM Bounds in section 5.3 of the paper which were omitted from the main text. The underidentified system of moments used in the GMM bounds analysis includes a number of the same moments as the overidentified GMM estimation discussed above (in particular the fraction of exporters and the conditional first moments of wages and employment). We derive here the unconditional second moments of employment and wages: σ 2 h = var(α h + µ h ι + u) = σ 2 u + µ 2 h var(ι) + 2µ hcov(ι u) = σ 2 u + µ 2 h Φ(f)[1 Φ(f)] + 2µ hρ u σ u φ(f) where we used the fact that var(ι) = Φ(f)[1 Φ(f)] since ι = I{z f} is a Bernoulli zero-one random variable with P{ι = 1} = P{z f} = 1 Φ(f) and that: cov(ι u) = E{ιu} Eι Eu = E{ιu} = [1 Φ(f)] E{u ι = 1} = [1 Φ(f)] ρ u σ u φ(f) 1 Φ(f) = ρ uσ u φ(f) where the expression for E{u ι = 1} = E{u z f} was derived above. Following similar steps we derive the expressions for: σ 2 w = var(w) = σ 2 ω + µ 2 wφ(f)[1 Φ(f)] + 2µ w ρ ω σ ω φ(f) σ hw = cov(h w) = ζσ 2 u + µ h µ w Φ(f)[1 Φ(f)] + [µ h ρ ω σ ω + µ w ρ u σ u ]φ(f) where w = α w + µ w ι + ω and we used the fact that cov(u ω) = cov(u ζu + v) = ζσ 2 u. We also make use of the following result: cov(ι u + ω) = [ρ u σ u + ρ ω σ ω ]φ(f) = [(1 + ζ)ρ u σ u + ρ v σ v ]φ(f) which parallels the derivations for cov(ι u) above. We next discuss the coefficients in the regression of wages on employment and export status. 19

20 We have: E{w h ι = 1} = α h + µ h + = α h + µ h + z f z f E{ω u z} dφ(z) 1 ˆΦ E { ρ ω σ ω z + (ω ρ ω σ ω z) (u ρ u σ u z) z } dφ(z) 1 ˆΦ = [α h + µ h + ρ ω σ ω λ(f)] + b[h α h µ h ρ u σ u λ(f)] E{w h ι = 0} = α h + E{ω u z} dφ(z) ˆΦ z<f where b is the regression coefficient in: = [α h ρ ω σ ω λ( f)] + b[h α h + ρ u σ u λ( f)] E { ω ρ ω σ ω z u ρ u σ u z } = b(u ρ u σ u z) which is independent from z as (u ρ u σ u z) and (ω ρ ω σ ω z) are jointly normal and independent from z and the regression coefficient b is given by: b = cov( u ρ u σ u z ω ρ ω σ ω z ) var ( u ρ u σ u z ) Combining the above expressions together we can write: where = σ uω ρ u σ u ρ ω σ ω σ 2 u(1 ρ 2 u) E{w h ι} = λ o + λ s h + λ x ι λ o = [α h ρ ω σ ωˆλ( f)] b[αh ρ u σ uˆλ( f)] λ s = b λ x = (µ w bµ h ) + [ˆλ(f) ˆλ( f)](ρ ω σ ω bρ u σ u ). Additionally we calculate the R 2 in this regression: = ζ ρ uσ u ρ v σ v σ 2 u(1 ρ 2 u). R 2 = 1 var (w E{w h ι}) σ 2 w = 2cov (w E{w h ι}) var (E{w h ι}) σ 2 w 20

21 = 2 bσ uω + µ w (bµ h + λ x )ˆΦ(1 ˆΦ) + [bµ w ρ u σ u + (bµ h + λ x )ρ ω σ ω ] ˆφ σ 2 ω + µ 2 w ˆΦ(1 ˆΦ) + 2µ w ρ ω σ ω ˆφ b2 σ 2 u + (bµ h + λ x ) 2 ˆΦ(1 ˆΦ) + 2b[bµh + λ x ]ρ u σ u ˆφ σ 2 ω + µ 2 w ˆΦ(1 ˆΦ) + 2µ w ρ ω σ ω ˆφ = (2ζ b)bσ 2 u + (bµ h + λ x ) [ 2µ w bµ h λ x]ˆφ(1 ˆΦ) + 2 [ b(µw bµ h λ x )ρ uσ u +(bµ h +λ x )ρ ωσ ω σ 2 ω + µ 2 w ˆΦ(1 ˆΦ) + 2µ w ρ ω σ ω ˆφ ] ˆφ. With this we can prove the following result: Lemma S.2 λ s contains additional information beyond what is already known from the moments in (A7) in the appendix at the end of the paper while λ o λ x and R 2 provide no additional information beyond what is contained in λ s. Proof: The value of b cannot be reconstructed from the unconditional second moments because it depends on the covariance between h and w conditional on export status. However given b we can reconstruct the values of λ o λ x and R 2 from the first conditional and second unconditional moments of the data (h w ι). For example λ x = E{w bh h ι = 1} E{w bh h ι = 0} = E{w bh ι = 1} E{w bh ι = 0} = [ w 1 b h 1 ] [ w 0 b h 0 ] λ o = Ew beh λ x Eι and a similar result can be shown for the R 2 using the expression above. D.5 Identification We first report the results of a Monte Carlo exercise in which we show that our estimation procedure correctly recovers the true parameters when the data are generated according to the model. We next provide an analytical characterization of the relationship between the model s parameters and moments in the data. For our maximum likelihood estimator we derive closed-form expressions for the score of the likelihood function. For the overidentified GMM estimator considered in Section D.3 above we derive closed-form expressions for the relationship between the parameters of the model and the first and second moments of the wage and employment distributions conditional on export status. Monte Carlo: We assume the following data generation process: h = α h + µ h ι + u w = α w + µ w ι + ζu + v ι = I {z f}. (D.2) 21

22 (u v z) N (0 Σ R ) Σ R = σu 2 0 σv 2 ρ u σ u ρ v σ u 1. (D.3) We assume the following parameter values: α h = 0 α w = 0 µ h = 1 µ w = 0.5 σ u = 1.1 σ v = 0.8 ρ u = 0.4 ρ v = 0.2 ζ = 0.5 which satisfy the inequality constraint (C.5) implied by our structural covariance restriction: σ θη = 0 ζ µ w µ h < ζ + σ 2 v (1 + ζ)σu 2. We consider 50 replications of the model. For each replication we draw realizations of (u v z) for hypothetical firms from the joint normal distribution (D.3). Given these realizations for (u v z) we compute employment wages and export status (h w ι) using the structure of the reduced-form model (D.2). Given these values for (h w ι) we estimate the parameters of the model using maximum likelihood. We repeat this exercise for each of the 50 replications. In Figure D.1 we display the distribution of the estimated parameters across the 50 replications. Each panel corresponds to a different parameter. Each panel shows the true value of the parameter (as the red vertical line) and the histogram of the parameter estimates across the 50 replications. We find that the estimated parameters are tightly clustered around the true values of the model s parameters. Therefore our estimation procedure indeed correctly recovers the true values of the parameters when the data are generated according to the model. Score of the Likelihood Function: We now provide closed-form solutions for the first-order conditions of the log likelihood function with respect to the parameters (the score of the likelihood function). These closed-form solutions directly relate the estimated parameters of the model to moments in the data. The log likelihood function is: { ( )} ln L = ι=0 ln σ u + ln φ (û) ln σ v + ln φ (ˆv) + ln Φ ˆf + { ι=1 ln σ u + ln φ (û) ln σ v + ln φ (ˆv) + ln [ 1 Φ ( ˆf )]} 22

23 Frequency Frequency Frequency Frequency Frequency Frequency Frequency Frequency 50 Panel A : 7 h 50 Panel B : 7 w 50 Panel C : ; u Panel D : ; v 50 Panel D : < u 50 Panel E : < v Panel F : f 50 Panel G : Figure D.1: Monte Carlo Results where (û ˆv ˆf) are related to the observed data (h w ι) and model parameters (αh α w µ h µ w σ u σ v ρ u ρ v ζ) as follows: f = û = h α h µ h ι σ ( u w αw µ w ι ) ζ ( h α h µ h ι ) ˆv = f ρ u = 1 ρ 2 u ρ 2 v ˆf = f ρ u û ρ vˆv σ v ρ u ρ v = 1 ρ 2 u ρ 2 v ρ v 1 ρ 2 u ρ 2 v where φ ( ) is the standard normal density function and Φ ( ) is the standard normal cumulative distribution function. 23

24 First-order condition for σ u ln L σ u = ι=0 1 σ u + φ (û) / σ u φ (û) ( ) Φ ˆf / σ u + ( ) Φ ˆf + ι=1 1 σ u + φ (û) / σ u φ (û) ( ) Φ ˆf / σ u ( ) 1 Φ ˆf. where we have used φ (ˆv) / σ u = 0. φ (û) σ u φ (û) û = φ (û) û û σ u = ûφ (û) û = û σ u σ u ( ) ( ) Φ ˆf Φ ˆf = σ u ˆf ( ) Φ ˆf ˆf ( ) = φ ˆf ρ u û σ u First-order condition for σ v ln L σ v = ι=0 1 σ v + φ (ˆv) / σ v φ (ˆv) ( ) Φ ˆf / σ v + ( ) Φ ˆf + ι=1 1 σ v + φ (ˆv) / σ v φ (ˆv) ( ) Φ ˆf / σ v ( ) 1 Φ ˆf. where we have used φ (û) / σ v = 0. φ (ˆv) σ v φ (ˆv) ˆv = φ (ˆv) ˆv ˆv σ v = ˆvφ (ˆv) ˆv = ˆv σ v σ v ( ) ( ) Φ ˆf Φ ˆf = σ v ˆf ( ) Φ ˆf ˆf ( ) = φ ˆf ρ v ˆv σ v 24

25 First-order condition for ζ ln L ζ = ι=0 φ (ˆv) / ζ φ (ˆv) ( ) Φ ˆf / ζ + ( ) Φ ˆf + φ (ˆv) ζ φ (ˆv) ˆv = ι=1 φ (ˆv) ˆv ˆv ζ = ˆvφ (ˆv) ˆv ζ = (h α h µ h ι) σ v ( ) ( ) Φ ˆf Φ ˆf ˆv = ζ ˆf ρ v ζ ( ) Φ ˆf ˆf ( ) = φ ˆf. φ (ˆv) / ζ φ (ˆv) ( ) Φ ˆf / ζ ( ) 1 Φ ˆf. First-order condition for α h { ln L φ(û)/ α α h = h ι=0 φ(û) + φ(ˆv)/ α h φ(ˆv) + Φ( } { [ ˆf)/ α h Φ( ˆf) φ(û)/ α + h ι=1 φ(û) + φ(ˆv)/ α h φ(ˆv) ( ) Φ ˆf α h φ (û) α h φ (û) û φ (ˆv) α h = φ (û) û û α h = ûφ (û) û α h = 1 σ u φ (ˆv) ˆv = φ (ˆv) ˆv ˆv α h = ˆvφ (ˆv) ˆv α h = ζ σ v [ û ˆv = ρ u + ρ v α h α h ) Φ ( ˆf ˆf ( ) = φ ˆf. ( ) ] Φ ˆf ˆf Φ( ˆf)/ α h 1 Φ( ˆf) ]} 25

26 First-order condition for µ h { ln L φ(û)/ µ µ h = h ι=0 φ(û) + φ(ˆv)/ µ h φ(ˆv) + Φ( } { [ ˆf)/ µ h Φ( ˆf) φ(û)/ µ + h ι=1 φ(û) + φ(ˆv)/ µ h φ(ˆv) φ (û) µ h = φ (û) û φ (û) û û µ h = ûφ (û) û µ h = ι σ u φ (ˆv) µ h = φ (ˆv) ˆv φ (ˆv) ˆv ˆv µ h = ˆvφ (ˆv) ˆv = ζι µ h σ v ( ) Φ ˆf [ û ˆv = ρ u + ρ v µ h µ h ( ) Φ ˆf ˆf µ h ( ) = φ ˆf. ( ) ] Φ ˆf ˆf Φ( ˆf)/ µ h 1 Φ( ˆf) ]} First-order condition for α w ln L α w = ι=0 φ (ˆv) / α w φ (ˆv) ( ) Φ ˆf / α w + ( ) Φ ˆf + φ (ˆv) α w φ (ˆv) ˆv = φ (ˆv) ˆv ι=1 ˆv α w = ˆvφ (ˆv) ˆv = 1 α w σ v ( ) ( Φ ˆf ˆv Φ ˆf) = ρ v α w α w ˆf ( ) Φ ˆf ( ) ˆf = φ ˆf. φ (ˆv) / α w φ (ˆv) ( ) Φ ˆf / α w ( ) 1 Φ ˆf 26

27 First-order condition for µ w ln L µ w = ι=0 φ (ˆv) / µ w φ (ˆv) ( ) Φ ˆf / µ w + ( ) Φ ˆf + φ (ˆv) µ w = φ (ˆv) ˆv φ (ˆv) ˆv ι=1 ˆv µ w = ˆvφ (ˆv) ˆv = ι µ w σ v ( ) ( Φ ˆf ˆv Φ ˆf) = ρ v µ w µ w ˆf ( ) Φ ˆf ( ) ˆf = φ ˆf. φ (ˆv) / µ w φ (ˆv) ( ) Φ ˆf / µ w ( ) 1 Φ ˆf First-order condition for ρ u ln L ρ u = ι=0 ( ) Φ ˆf / ρ u ( ) Φ ˆf ι=1 ( ) ( ) Φ ˆf Φ ˆf = ˆf ρ u ˆf ρ u ˆf = û ρ u ( ) Φ ˆf ( ) ˆf = φ ˆf. ( ) Φ ˆf / ρ u ( ) 1 Φ ˆf First-order condition for ρ v ln L ρ v = ι=0 ( ) Φ ˆf / ρ v ( ) Φ ˆf ι=1 ( ) ( ) Φ ˆf Φ ˆf = ˆf ρ v ˆf ρ v ˆf ρ v = ˆv ( ) Φ ˆf / ρ v ( ) 1 Φ ˆf 27

28 First-order condition for f ln L f = ι=0 ( ) Φ ˆf ˆf ( ) Φ ˆf ( ) = φ ˆf. / f ( ) Φ ˆf ι=1 ( ) Φ ˆf / f ( ) 1 Φ ˆf ( ) Φ ˆf f = ( ) Φ ˆf ˆf ˆf f Exactly-identified GMM: ˆf = 1 f ( ) Φ ˆf ( ) ˆf = φ ˆf. We now characterize the relationship between the model parameters and the moments in the data used in the overidentified GMM estimator considered in Section D.3 above. We consider an exactly-identified specification of the GMM system (D.1) in terms of first and second moments of wages and employment conditional on export status. In this case the mapping between the model parameters and moments in the data is particularly transparent as the exactly-identified GMM system has a recursive structure in which we can sequentially solve for the model parameters using the moments in the data. In a first equation bloc the export threshold f can be determined from the observed fraction of firms that export: ῑ 1 = Eι j = 1 Φ(f) which implies: f = Φ 1 (f) Φ (f) = 1 ῑ 1 where a bar above a variable denotes a value observed in the data. In a second block of equations the market access standard deviation and correlation parameters (µ h σ u ρ u ) can be determined from the conditional and unconditional first and second moments of employment in (D.1) and the solution for f above which imply: ρ 2 uσu 2 s 2 0 = s2 1 Λ (f) Λ ( f) µ h = [ m h1 m h0 ] ρ u σ u [λ (f) λ ( f)] σu 2 = s 2 h µ2 h Φ (f) [1 Φ (f)] 2µ hρ u σ u φ (f). In a third block of equations the market access standard deviation and correlation parameters (µ w σ v 28

29 ρ v ) can be determined from the conditional and unconditional first and second moments of wages in (D.1) and the solution for f above which imply: ρ 2 ωσω 2 = s2 w0 s2 w1 Λ (f) Λ ( f) µ w = [ w 1 w 0 ] ρ ω σ ω [λ (f) λ ( f)] σ 2 ω = s 2 w µ 2 wφ (f) [1 Φ (f)] 2µ w ρ ω σ ω φ (f). In a fourth block of equations the covariance parameter ζ can be determined from the conditional covariance of wages and employment in (D.1) and the solutions for (f σ u σ ω ρ u ρ ω ) above which imply: ζ = c hw {µ h µ w Φ (f) [1 Φ (f)] + µ h ρ ω σ ω φ (f) + µ w ρ u σ u φ (f)}. Having determined {f α h α w µ h µ w σ u σ ω ρ u ρ ω ζ} from the above moments in the data we obtain (σ v ρ v ) from the definitions of (σ ω ρ ω ): σ v = [ σω 2 ζ 2 σu 2 ] 1 2 ρ v = ρ ωσ ω ζρ u σ u σ v. E Extensions and Generalizations E.1 Isomorphisms Class of theoretical models: We now provide a formal analysis of the class of models that are isomorphic in terms of their predictions for wages employment and export status to the Helpman Itskhoki Muendler and Redding (henceforth HIMR) model developed above. This class of models is defined by the following assumptions: (a) Revenues and employment are power functions of export status and two stochastic shocks (b) Profits and wage bills are constant shares of revenues (c) Fixed exporting costs are subject to a third stochastic shock (d) The three stochastic shocks are joint normally distributed. The class of models defined by these assumptions can be represented as follows: R = κ r [1 + ι (Υ x 1)] ψr (e θ) ξ r (e η ) φr (E.4) H = κ h [1 + ι (Υ x 1)] ψ h ( e θ) ξ h (e η ) φ h (E.5) W = κ w [1 + ι (Υ x 1)] ψw (e θ) ξ w (e η ) φw (E.6) ( ) ι = I {κ π Υ ψr x 1 (e θ) } ξ r (e η ) φr F x e ε (E.7) Υ x > 1 F x > 0 29

30 ξ h + ξ w = ξ r φ h + φ w = φ r ψ h + ψ w = ψ r. Reduced-form representation: Taking logarithms in (E.4)-(E.7) we obtain: h = α h + µ h ι + ξ h θ + φ h η (E.8) w = α w + µ w ι + ξ w θ + φ w η (E.9) { } 1 ι = I σ (ξ rθ + φ r η ε) f (E.10) α s = log κ s s {r h w π} µ s = ψ s log Υ x s {h w} f = 1 ( [ ]) α π + log F x log Υ ψr x 1. σ We now transform this empirical system by orthogonalizing the errors in the employment and wage equations. We define: v = u = ξ h θ + φ h η ( φ w ξ ) w φ h η πu ξ h ζ = ξ w ξ h + π z = 1 [(1 + ζ) u + v ε] σ where π is the projection coefficient of η on u. Using these definitions we can re-write the empirical system (E.8)-(E.10) as: h = α h + µ h ι + u (E.11) w = α w + µ w ι + ζu + v ι = I {z f}. (E.12) (E.13) Under the assumptions above we have the following theoretical restriction: µ h + µ w = log Υ x > 0. Under the additional assumptions that ψ h ψ w > 0 we also obtain the following additional theoretical restriction: µ h µ w > 0. 30

31 Under the assumption that {θ η ε} are joint normally distributed {u v z} are also joint normally distributed. The reduced-form equations (E.11)-(E.13) are identical to those in the HIMR model. Therefore any structural model that can be mapped to the mathematical structure in (E.4)-(E.7) has the same reducedform econometric specification as the HIMR structural model. Likelihood function: Since the reduced-form equations (E.11)-(E.13) take exactly the same form as in the HIMR model the likelihood function also takes exactly the same form as in the HIMR model. L(Θ X) = j P{h j w j ι j = 0} j P{h j w j ι j = 1} P{h w ι = 0} = P{u = h α h v = (w α w ) ζ(h α h ) z < f} ( ) = 1 φ (û(x)) 1 f ρ u û(x) ρ φ (ˆv(x)) Φ vˆv(x) σ u σ v 1 ρ 2 u ρ 2 v P{h w ι = 1} = P{u = h α h µ h v = (w α w µ w ) ζ(h α h µ h ) z f} [ ( )] = 1 φ (û(x)) 1 f ρ u û(x) ρ φ (ˆv(x)) 1 Φ vˆv(x) σ u σ v 1 ρ 2 u ρ 2 v where û(x) = h α h µ h ι σ ( u w αw µ w ι ) ζ ( h α h µ h ι ) ˆv(x) =. σ v Counterfactuals: For all models within this class the reduced-form coefficients Θ {α h α w ζ σ u σ v ρ u ρ v µ h µ w f} are sufficient statistics for wages employment and export status (and hence wage inequality). Therefore for all models within this class counterfactuals can be undertaken following exactly the same procedure as for the HIMR model above. Fair wages example: Finally we provide an example of another model within the class defined by assumptions (a)-(d) above that is isomorphic to the HIMR model. This example is based on an extension of the fair wages model of Egger and Kreickemeier (2012). As in the HIMR model each firm faces a fixed exporting cost (e ε F x ) and an iceberg variable trade cost (τ). A firm of productivity θ that employs a measure h of workers produces the following measure of output: y = e θ h. 31