General Equilibrium Effects in Space: Theory and Measurement · In partial equilibrium, regional...

General Equilibrium Effects in Space:

Theory and Measurement∗

Rodrigo Adao

Chicago Booth

Costas Arkolakis

Yale

Federico Esposito

Tufts

June 2020

Abstract

How do international trade shocks affect spatially connected regional markets? We answer this

question by extending shift-share empirical specifications to incorporate general equilibrium

effects that arise in spatial models. In partial equilibrium, regional shock exposure has a

shift-share structure: it is the average shock weighted by regional exposure shares in revenue

and consumption. General equilibrium responses of employment and wages in each market

are the sum, across all regions, of these shift-share measures times bilateral reduced-form

elasticities determined by the economy’s spatial links. We use this reduced-form representation

of the model to efficiently estimate the bilateral elasticities exploiting exogenous variation

in shock exposure across markets. Finally, we study the general equilibrium impact of the

‘‘China shock’’ on U.S. CZs using our model-consistent generalization of the specification in

Autor et al. (2013). We find that indirect effects from the shock exposure of other markets

reinforce the negative impact of the market’s own shock exposure, leading to employment

and wage losses that are significantly larger than those reported in the existing literature.

∗We thank David Atkin, David Autor, Marta Bengoa, Martin Beraja, Varadarajan V. Chari, Lorenzo Caliendo,Arnaud Costinot, Jonathan Dingel, Dave Donaldson, Farid Farrokhi, Gordon Hanson, Rich Hornbeck, ErikHurst, Samuel Kortum, Andrew McCallum, Eduardo Morales, Elias Papaioannou, Amil Petrin, Steve Redding,Esteban Rossi-Hansberg, Jonathan Vogel, David Weinstein, as well as numerous participants at many seminars andconferences for helpful suggestions and comments. We also thank Ariel Boyarsky, Zijian He, Guangbin Hong, JackLiang, and Josh Morris-Levenson for excellent research assistance. Rodrigo Adao thanks the NSF (grant 1559015)for financial support. All errors are our own. A previous version of this paper circulated under the title ‘‘SpatialLinkages, Global Shocks and Local Labor Markets: Theory and Evidence.’’

1 Introduction

What are the labor market consequences of international trade shocks, such as a trade policy change

or a foreign productivity boom? Answering this question requires quasi-experimental variation

in trade shocks affecting different labor markets. A growing literature obtains such a variation

from regional measures of exposure to trade shocks that are constructed from the interaction

of aggregate shocks and associated region-specific exposure shares, as in the shift-share designs

in Autor et al. (2013) and Kovak (2013). These measures yield estimates of how labor market

outcomes differentially respond in regions with higher shock exposure. However, these differential

responses may not fully capture all the channels through which trade shocks affect regional labor

markets in general equilibrium. Such is the case if spatial connections imply that the shock exposure

of a region not only affects its own labor market, but also has spillover effects on other regions.

Without estimates of this type of spatial spillover effects, any analysis of the general equilibrium

impact of trade shocks on local labor markets is incomplete.1

In this paper, we analyze how trade shocks affect local labor markets by extending shift-share

empirical specifications to incorporate the general equilibrium effects that arise from spatial links

in a flexible model. We first show that a market’s shock exposure in partial equilibrium can be

written in terms of two shift-share variables based on market-specific exposure shares for revenue

and consumption. We then establish that, in general equilibrium, responses of employment and

wages in each market are the sum, across all regions, of these shift-share measures times bilateral

reduced-form elasticities determined by the economy’s spatial links. Thus, though the lens of our

spatial model, these reduced-form elasticities are sufficient for computing the general equilibrium

impact of observed measures of regional shock exposure on local labor market outcomes. We then

show how to efficiently estimate these elasticities using the model’s reduced-form representation for

employment and wage responses to exogenous variation in the shock exposure of different markets.

Finally, we study the impact of the ‘‘China shock’’ on U.S. Commuting Zones (CZs). Our theory

yields a generalization of the shift-share specification in Autor et al. (2013) that accounts for both

the direct effect of the CZ’s own shock exposure in revenue and consumption, as well as the indirect

effect of the shock exposures of other CZs. We find that indirect effects reinforce direct effects,

leading to employment and wage losses that are significantly larger than those reported in the

existing literature.

We consider a general equilibrium framework with three types of spatial links. Every market

has multiple sectors, each featuring a gravity-type demand for the goods from different markets.

Local labor supply is endogenous: it depends on wages and prices in all markets. Finally, we allow

for local economies of scale and spatial productivity spillovers by making local labor productivity a

1This is related to the well-known problem that difference-in-difference empirical strategies do not recover thegeneral equilibrium effect of the ‘‘treated’’ on ‘‘non-treated’’, as pointed out by Heckman et al. (1998) and, morerecently, by Muendler (2017) in the context of regional regressions.

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function of employment in all markets. Through the shape of the functions specifying spatial links

in the economy, our model encompasses several of the mechanisms in existing trade and spatial

frameworks – for example, the gravity trade models reviewed by Costinot and Rodrıguez-Clare

(2014) and the spatial models reviewed in Moretti (2011) and Redding and Rossi-Hansberg (2017).

We start by expressing equilibrium wages and employment in terms of each market’s excess

labor demand. This way studying the impact of trade shocks in our spatial framework becomes a

traditional comparative statics exercise in general equilibrium – see e.g. Arrow and Hahn (1971). In

partial equilibrium, for any given initial wage vector, trade shocks trigger shifts in the excess labor

demand of each market. In general equilibrium, wages and employment in all markets respond to

these partial equilibrium shifts to guarantee labor market clearing everywhere. Such responses

depend on the Jacobian matrix of the excess demand system with respect to wages. This is the

‘‘spatial links’’ matrix that summarizes the combined strength of different types of spatial links.

We then separately analyze these two components of the impact of trade shocks on local

labor markets. We first show that the partial equilibrium shifts in excess labor demand can be

written in terms of two shift-share variables, given by the sum of the product of trade shocks and

market-specific exposure shares for revenue and consumption. For sectoral foreign productivity

shocks, a market’s revenue exposure is the commonly used shift-share variable based on sectoral

employment shares – e.g., Autor et al. (2013) and Kovak (2013). In addition, our theory yields

a consumption exposure measure that is a shift-share variable where the ‘‘share’’ is instead the

sectoral spending share.

In general equilibrium, responses of employment and wages in each market are the sum, across

all markets, of their partial equilibrium excess demand shifts multiplied by bilateral reduced-form

elasticities. These general equilibrium reduced-form elasticities control how much the shift in

a market’s excess labor demand affects directly its own market and indirectly other markets.

Depending on their sign, indirect reduced-form elasticities may reinforce or attenuate the direct

effect of the market’s own excess demand shift. Thus, in our spatial model, these elasticities are

ex-ante sufficient statistics that allow the general equilibrium aggregation of excess demand shifts

across markets.

We open the black-box of spatial shock propagation in the economy by writing the reduced-form

elasticities as a series expansion of the spatial links matrix. This implies that bilateral reduced-form

elasticities are larger between markets with stronger spatial links, due to tighter bilateral or

third-market connections (e.g., similar compositions of trade partners and sectoral employment, or

tighter labor supply links).2 Only when spatial links are identical for all markets, bilateral indirect

effects are identical and yield a common endogenous variable that is absorbed by time fixed-effects.

2Intuitively, this arises from the several adjustment rounds triggered by the response of each market’s excesslabor demand to shock-induced wage changes elsewhere. This is similar to the channel creating percolation ofsectoral shocks in production networks, as in Acemoglu et al. (2012) and Acemoglu et al. (2016).

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We further show that, if trade demand links are strong enough, indirect and direct effects have the

same sign and, therefore, reinforce each other.

Taken together, our theoretical results point to two critical components of any empirical analysis

of how trade shocks affect local labor markets: (i) the partial equilibrium shock exposure in revenue

and consumption, and (ii) the general equilibrium reduced-form elasticities to the shock exposure

of different markets. Accordingly, we build on these two components to propose a new empirical

methodology to study the labor market consequences of trade shocks.

To this end, we use the reduced-form representation of our general equilibrium model to specify

how regional outcomes respond to an observed trade shock and other unobserved shocks. We

combine the observed trade shock with trade data to compute the revenue and consumption

exposures of each market. Our model yields two estimating equations linking observed changes

in employment and wages in each market to linear combinations of the product of reduced-form

elasticities and observed measures of shock exposure across markets, as well as additive unobserved

residuals. We proceed in two steps to estimate the reduced-form elasticities using these equations

and exogenous cross-regional variation in shock exposure.

First, we reduce the dimensionality of the reduced-form elasticity matrix by imposing that

spatial links are known functions of observables and parameters. Thus, we only need to estimate

the parameters regulating the strength of reduced-form effects associated with different observed

variables (e.g., bilateral flows in migration and sectoral trade).3 We obtain moment conditions

to estimate these parameters by assuming that the observed shock exposure is mean-independent

of unobserved residuals. This assumption is the same orthogonality condition underlying the

shift-share strategies in Kovak (2013) and Autor et al. (2013). Notice however that our estimating

equations generalize the specifications in this literature: they include direct and indirect reduced-

form elasticities to the shock exposures in both revenue and consumption.

Second, we characterize the efficient estimator of the parameters that non-linearly regulate the

general equilibrium reduced-form elasticities. We outline a class of consistent GMM estimators for

the parameter vector that differ in how they aggregate the shock exposure of different markets.

We follow Chamberlain (1987) to characterize the ‘‘optimal’’ variance-minimizing estimator within

this class and its two-step feasible implementation. For each market, the model-implied optimal

moment puts more weight on the observed exposure of markets whose bilateral reduced-form

elasticities are more sensitive to changes in the parameter of interest.4

In the last part of the paper, we apply our methodology to study the impact of the ‘‘China

3Our approach is similar to the common practice in demand estimation of projecting cross-price elasticities onproduct characteristics – see e.g. Berry (1994), Berry et al. (1995) and, for a review, Nevo (2000).

4The approach of Chamberlain (1987) has been used in partial equilibrium models by Berry et al. (1995),Petrin (2002), and Reynaert and Verboven (2014). Our contribution is, for a flexible spatial model in generalequilibrium, to formally establish a class of consistent estimators and among them the optimal estimator and itsfeasible implementation.

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shock’’ on U.S. Commuting Zones (CZs), as in Autor et al. (2013) (henceforth ADH). We follow

ADH by considering the exposure of each CZ to industry-level Chinese export growth to a group

of developed countries (excluding the U.S.). In this case, the model-consistent revenue exposure

is proportional to the shift-share instrument in ADH. However, our theory predicts that regional

outcomes also respond to (i) the CZ’s own consumption shift, and (ii) indirectly to the revenue and

consumption shifts of other CZs. Therefore, we begin our empirical analysis by estimating a simple

extension of the specification in ADH with the goal of qualitatively investigating the relevance

of these two additional channels. We impose that each market’s indirect effect is proportional to

the average shock exposure of all other markets, weighted by their inverse bilateral distance. We

find that, conditional on a CZ’s own revenue exposure, growth of both employment and wages

are significantly lower if nearby CZs suffer larger revenue shocks. In our theory, these reinforcing

indirect effects are consistent with strong trade demand links across CZs (relative to labor supply

links). In addition, we do not find evidence that regional outcomes significantly respond to shifts

in consumption cost. This suggests weak labor supply responses to lower import costs.5

We then implement our methodology to estimate the general equilibrium reduced-form elastici-

ties implied by our parametrization of the spatial links matrix. This allows general equilibrium

reduced-form elasticities to depend on bilateral as well as third-market connections that arise in

general equilibrium. In addition, it implies that, through the lens of our spatial model, neither

time fixed-effects nor residuals include any component of the endogenous responses of regional

outcomes to the observed sectoral average of the China shock. In line with the evidence discussed

above, we find that the direct and indirect effects of revenue exposure have the same sign. These

indirect reinforcing effects are mostly driven by spatial trade links. We estimate large reduced-form

elasticities that are consistent with strong amplification through local economies of scale. In

addition, a reduction in consumption cost has a positive direct and indirect impact on wages and

employment. However, the positive responses to consumption cost shifts are much weaker than

the negative responses to revenue shifts.

To gain confidence in our methodology, we propose a way of evaluating a spatial model’s

predictions based on their ability to match the impact of regional shock exposure on labor market

outcomes across CZs – in line with the evaluation strategy suggested by Kehoe (2005). Specifically,

we regress actual changes in employment and wages of U.S. CZs on the model-predicted responses

to the China shock. We find that the estimated fit coefficients are close to one for our baseline

predicted responses, indicating that their magnitude is consistent with the shock’s differential

5We construct CZ-level sectoral spending shares using the CZ’s sectoral employment shares interacted withnational input-output matrices, so they capture also the CZ’s input spending on different sectors. Thus, this resultindicates that we do not find evidence of stronger employment growth in CZs intensively using inputs from industrieswith larger increases in Chinese exports. This is consistent with the evidence in Pierce and Schott (2016) andAcemoglu et al. (2016) who do not find a significant response of an industry’s employment to stronger Chineseexport growth in the industries supplying its inputs.

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impact on employment and wages across CZs.6

Using the same methodology, we evaluate the fit implied by typical specifications of spatial

links in the quantitative spatial literature. We find that they perform poorly: the fit coefficient is

much higher than one and, depending on the specification, very imprecisely estimated. This finding

indicates a troubling disconnect between the results of empirical analyses using cross-regional

variation in shock exposure and those of quantitative analyses using general equilibrium spatial

models. It implies that common specifications of spatial links generate regional responses to the

China shock that are too small compared to –and often uncorrelated with– actual changes in

employment and wages across CZs. We show that these results follow mainly from specifying weak

agglomeration forces and strong sensitivity of employment to import prices.

Finally, we compute the impact of the China shock on U.S. CZs by aggregating their exposure

to the China shock using our estimates of the general equilibrium reduced-form elasticities. On

average, revenue losses dominate consumption gains, reducing employment and wage log-growth

respectively by 2.8 and 4 log-points between 1990 and 2007. The consumption gains however lead

to a small increase in the average real wage of 0.2 log-points. These responses vary substantially

across CZs: the standard deviation of log-changes is 1.3 for wages, 3.3 for employment, and 1.7 for

real wages. These employment and wage losses are significantly larger than those in the existing

literature. Empirical specifications ignoring indirect reinforcing effects yield smaller losses. In

addition, quantitative frameworks yield average responses that are close to zero because they rely

on reduced-form elasticities that are too small compared to those necessary to match the observed

cross-regional responses to the shock.

Building on the seminal works of Bartik (1991) and Blanchard and Katz (1992), a growing

literature uses shift-share strategies to estimate how regional markets respond to economic shocks,

in general, and trade shocks, in particular – see e.g. Topalova (2010), Autor et al. (2013), Kovak

(2013), Dix-Carneiro and Kovak (2017), Autor et al. (2016), Pierce and Schott (2020) and Burstein

et al. (2020). Moving beyond the differential responses documented in this literature, we propose a

generalization of this empirical approach that captures the indirect general equilibrium impact of

the shock exposures of different markets. Through the lens of our spatial model, such estimates

can be used to aggregate the shock exposure of different regions to obtain the general equilibrium

responses of local labor market outcomes.7

Our work provides a bridge between this empirical literature and the alternative popular

approach used to study the labor market consequences of trade shocks: quantitative analyses using

6While we use this methodology to evaluate regional outcomes, the same logic can be applied to evaluateoutcomes such as bilateral trade or commuting flows, see e.g. Dingel and Tiltenot (2020).

7Heterogeneity in spatial links also implies heterogeneity in the direct ‘‘treatment’’ effect of the own market’sshock exposure – as shown by Monte et al. (2018). In a recent related paper, Hornbeck and Moretti (2018) empiricallydocument indirect effects of regional productivity shocks through out-migration to other regions. Instead, we focuson the implications of spatial links for the measurement of shock exposure in partial equilibrium and the estimationof indirect reduced-form elasticities in general equilibrium.

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general equilibrium spatial models. Several papers point out the challenge of extrapolating general

equilibrium counterfactual predictions from estimated differential responses across regions – see

e.g. Moretti (2011); Kline and Moretti (2014); Beraja et al. (2019); Muendler (2017); Kehoe et al.

(2017); Redding and Rossi-Hansberg (2017). This motivated recent analyses based on quantitative

spatial frameworks studying the effects of import growth from China on local labor markets in

the U.S. – see e.g. Galle et al. (2017) and Caliendo et al. (2019). We depart in two ways from

this quantitative approach. First, we characterize the reduced-form representation of our general

equilibrium model that links actual changes in employment and wages to shift-share measures of

regional shock exposure in revenue and consumption. Second, we use this representation to obtain

estimating equations that can be used to identify parameters regulating the general equilibrium

reduced-form elasticities of local outcomes to the shock exposure of different markets. Thus, our

approach is an extension of existing shift-share empirical specifications that allows the aggregation

of regional shock exposure in a way that is consistent with the predictions of general equilibrium

spatial models.8 As a result, compared to common specifications of spatial links in quantitative

frameworks, our methodology yields a substantially better fit for the differential response of

labor market outcomes to shock exposure across regions, increasing the credibility of the model’s

counterfactual predictions.

Finally, our paper is related to the so-called ‘‘market access’’ approach in Redding and Venables

(2004) that has been recently used to study how regional labor markets respond to economic shocks

in general equilibrium – e.g., Donaldson and Hornbeck (2016); Alder et al. (2015); Bartelme (2018).

Our model-consistent exposure measures correspond to partial equilibrium versions of the changes

in producer and consumer market access in this literature, if the latter were computed holding

wages and employment constant. Our exposure measures can be immediately constructed using

trade data and can be aggregated in a model consistent-way using our estimates of the general

equilibrium reduced-form elasticities. Market access measures are instead endogenous variables

obtained from solving the general equilibrium model under restrictive assumptions and whose

aggregation requires additional general equilibrium shifters.

The rest of the paper is structured as follows. Section 2 describes our spatial model. Section 3

characterizes the partial equilibrium exposure measures and the general equilibrium reduced-form

elasticities. Section 4 presents the linear expressions for changes in wages and employment in

terms of the exposure measures. Section 5 describes our methodology to estimate the reduced-form

elasticities using these expressions. In Section 6, we estimate the impact of the China shock on

U.S. CZs. Section 7 concludes.

8Note that our methodology is different from that used in papers exploiting cross-regional variation in shockexposure to estimate parameters of particular structural equations of the model – e.g., Faber and Gaubert (2019);Fajgelbaum et al. (2018); Galle et al. (2017); Allen et al. (2020); Burstein et al. (2020). This is because they donot explicitly characterize nor estimate the model-implied reduced-form elasticities of regional outcomes to shockexposure.

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2 Model

We consider a spatial general equilibrium model in which N ‘‘markets’’ are linked through trade

flows, labor productivity, and labor supply. Each ‘‘market’’ i consists of a set of sectors, s ∈ Si,within a geographic unit r where producers face identical endogenous production costs.9 In the

rest of the paper, we use bold variables to denote stacked vectors of market outcomes, x ≡ xii,and bar bold variables to denote matrices with bilateral variables associated with origin market i

and destination market j, x ≡ [xij]i,j.

Labor Supply Labor is freely mobile within a market, so that wi is market i’s wage. Labor

supply in i, Li, is a function of the vectors of wages, w ≡ wj, and price indices, P ≡ Pj:

Li = Φi(w,P ), (1)

where Φi(· ) is strictly positive, differentiable, bounded, and homogeneous of degree zero.

We use the matrices of labor supply elasticities to changes in wages and prices to summarize

the economy’s spatial links in labor supply,

φwij ≡∂ ln Φi(w,P )

∂ lnwjand φpij ≡

∂ ln Φi(w,P )

∂ lnPj. (2)

The own-market elasticities, φwii and φpii, control the response of employment to changes in

the market’s wages and prices. They allow employment to respond to the regional exposure to

import competition – as documented in Autor et al. (2013) and Dix-Carneiro and Kovak (2017).

Our specification allows, but does not require, employment responses to wages and prices to be

different. As discussed below, this implies that our model can match different types of employment

responses to shocks in import competition and import prices. This is important given the evidence

in Acemoglu et al. (2016) and Pierce and Schott (2016) that these shocks have different implications

for employment growth across industries. Finally, the cross-market elasticities, φwij and φpij , regulate

employment responses in market i to wages and prices in other markets, capturing endogenous

employment changes across markets – as in the literature reviewed by Moretti (2011).

Our general labor supply specification Φi (w,P ) encompasses the labor supply function implied

by several micro-founded frameworks – for a formal discussion, see Online Appendix B. It can

replicate the labor supply functions in spatial models with homogeneous individuals facing housing

congestion forces, as in Helpman (1998), and heterogeneous individuals in market-specific amenity

preferences, as in Allen and Arkolakis (2014), Redding (2016), and Bryan and Morten (2015).10

9Each market in our model can be seen as a group of industries within a geographic unit at a point in time. Forexample, a market may be the set of manufacturing industries in a U.S. Commuting Zone.

10If a market is a group of industries in a region, then Φj(w,P ) can also replicate the industry-level labor supply

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Our specification also allows aggregate employment to respond to changes in wages and prices.

The fact that labor supply is a separate function of wages and price indices may arise from

the endogenous labor supply choice in the presence of unemployment benefits when preferences

either imply income effects on leisure demand, as in Shimer (2009) and Keane (2011), or entail

heterogeneity in the disutility to work across individuals, as in Rogerson (1988) and Chetty (2012).

In fact, without migration, these two settings imply that labor supply is more sensitive to the

market’s wage than to the price index, φwii > −φpii > 0. We show that this feature is key to match

employment responses to trade shock exposure in the empirical application in Section 6.

Production The region-sector pair (r, s) in market i has a competitive representative firm whose

production function uses only labor,

Qr,s = Ψi (L)Lr,s, (3)

where Ψi(· ) is a strictly positive differentiable function that governs the endogenous component of

labor productivity in all sectors and regions of market i. We use the matrix of labor productivity

elasticity to changes in employment to summarize the economy’s spatial links in productivity,

ψij =∂ ln Ψi(L)

∂ lnLj. (4)

The own-market elasticity ψii captures labor productivity gains caused by higher local employ-

ment – as documented, for example, by Greenstone et al. (2010), Kline and Moretti (2014) and

Peters (2019). The cross-market elasticity ψij regulates spatial spillovers in labor productivity.

Again, different micro-founded frameworks imply that labor productivity takes the general

form in Ψi(L). A setting with ψii = ψ and ψij = 0 arises from firm entry and increasing returns to

scale with homogeneous firms, as in Krugman (1980), or heterogeneous firms, as in Arkolakis et al.

(2008) and Chaney (2008).11 By properly defining markets, our specification allows for external

economies of scale in a region-sector, as in Ethier (1982), and technology diffusion between regions,

as in Fujita et al. (1999) and Lucas and Rossi-Hansberg (2003). When markets are industry groups

in a region, our environment accommodates differences across sectors in market structure and

economies of scale – e.g., Krugman and Venables (1995), Balistreri et al. (2010), Kucheryavyy et

al. (2016). In addition, the cross-market elasticity of labor productivity, ψij, may also incorporate

congestion forces implied by the spatial re-allocation of other factors of production (see Online

Appendix B). Finally, under combined parametric restrictions on Φi(w,P ) and Ψi(L), our model

is equivalent to existing quantitative gravity spatial models, such as those reviewed by Redding

implied by assignment models with heterogeneous individuals in terms of industry-specific productivity, as inBurstein et al. (2019); Galle et al. (2017); Adao (2015).

11Costinot and Rodrıguez-Clare (2014) establish this in the context of gravity trade models.

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and Rossi-Hansberg (2017).12

Under perfect competition, the linear production function in (3) and the labor supply equation

in (1) imply that the endogenous production cost in market i, pi, is

pi =wi

Ψi (Φ(w,P )). (5)

Demand We consider a multi-sector nested CES trade demand. The spending share of j on

goods of sector s is constant and given by ξj,s, with∑

s ξj,s = 1. Within sector s, the demand for

goods from different markets is a constant elasticity function. This implies that the spending share

of market j on goods produced in i is

xij(p|τ ) ≡∑s∈Si

xij,sξj,s =∑s∈Si

(τij,spi)−εs∑

o:s∈So (τoj,spo)−εs ξj,s, (6)

where εs > 0 is the trade elasticity in sector s, τ ≡ τij,sijs is the stacked vector of bilateral trade

costs, and the summation in the denominator is over markets that produce in sector s, o : s ∈ So.The sector-specific trade costs combine bilateral differences in tastes, iceberg trade costs, and

productivity. In our analysis below, we consider the consequences of exogenous changes in these

bilateral terms.

Our multi-sector gravity trade structure follows closely that of the recent quantitative gravity

literature reviewed by Costinot and Rodrıguez-Clare (2014). In fact, when labor supply is exogenous

(i.e., Φi(w,P ) = Li), our framework becomes a standard multi-sector gravity trade model with

perfect labor mobility across sectors within a market but no mobility across markets. With a single

sector, our trade demand specification is equivalent to that of the Armington model in Anderson

(1979), the Ricardian model in Eaton and Kortum (2002), and, more generally, the class of gravity

trade models in Arkolakis et al. (2012).

The revenue of i is Yi(p,E|τ ) =∑

j xij(p|τ )Ej where Ej is j’s total spending. To summarize

the economy’s spatial linkages in trade, we use the elasticity of revenue to production costs,

χij ≡∂ lnYi(p,E|τ )

∂ ln pj=∑s∈Si

∑k

(yikyik,s) (xjk,s − I[i=j])εs, (7)

where yij is the share of j in the revenue of i, and yij,s is the share of sector s in the sales of i

to j. Equation (7) has two properties that follow directly from our model’s multi-sector gravity

structure. First, all markets are substitutes in the trade demand: χij ≥ 0 for all i 6= j. Second,

markets i and j are closer substitutes whenever i gets more of its revenue from destinations and

sectors in which j’s market share is large: χij is increasing in∑

k,s(yikyik,s)(xjk,sεs).

12Online Appendix B shows that the formal equivalence with these settings requires specifying the regionalallocation of payments to non-labor factors (e.g., rent payments to land and capital).

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Lastly, the gravity trade demand in (6) implies that the price index in market j is

Pj(p|τ ) =∏s

[ ∑o:s∈So

(τoj,spo)−εs

] ξj,s−εs

. (8)

Equilibrium We assume that trade is balanced, so that market clearing requires wiLi =∑j xij (wjLj). Using the labor supply in (1) and the trade demand in (6), this market clearing

condition yields the excess labor demand in market i:

Di (w|τ ) ≡∑j

∑s∈Si

(τij,swi

Ψi(Φ(w,P (w|τ )))

)−εs∑

o:s∈So

(τoj,swo

Ψo(Φ(w,P (w|τ )))

)−εs ξj,swjΦj(w,P (w|τ ))− wiΦi (w,P (w|τ )) (9)

where P (w|τ ) is implicitly defined as the solution of (8) for all j with po given by (5).

We define the equilibrium as a wage vector, w ≡ wii, that satisfies

D (w|τ ) = 0. (10)

We now establish conditions for the existence and uniqueness of the equilibrium wage vector in

terms of the normalized Jacobian matrix of the excess demand system with respect to the wage

vector: γij = −Y −1i ∇lnwDi(w|τ ) where Yi is the total revenue in i. In the rest of the paper, we

refer to γ ≡ [γij] as the ‘‘spatial links’’ matrix. This guarantees that our counterfactual analysis

yields unambiguous predictions for the impact of trade cost shocks on local labor markets.

Assumption 1. [Uniqueness] There is a market m with limwm→0Ψm(Φ(w,P (w|τ )))

wm=∞. For any

equilibrium wage vector w, assume that (i) γii > 0 for all i, and (ii) hiγii >∑

j 6=i,m |γij|hj for all

i 6= m and some vector hii 6=m 0.

This assumption requires a weighted sum of the cross-market wage elasticities of excess labor

demand to be lower than the own-market wage elasticity of excess labor demand.13 Under this

assumption, we establish the following result.

Proposition 1. [Uniqueness] Suppose that Assumption 1 holds. There exists a unique wage

vector, w with wm ≡ 1, that satisfies (10).

Proof. See Appendix 8.1.

Imposing diagonal dominance on γ to achieve equilibrium uniqueness has a long tradition in

general equilibrium theory. In Proposition 1, we apply the tools in Arrow and Hahn (1971) to

13This assumption is weaker than the gross substitution property (i.e., γii > 0 and γij < 0 for all i 6= j) that yieldsuniqueness of gravity trade models (e.g., Alvarez and Lucas (2007)). This is despite the fact that our environmententails endogenous labor supply and economies of scale.

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establish uniqueness in our framework with spatial links in labor supply, labor productivity, and

trade demand. Note that, whereas proving existence in the presence of such links is straightforward

(see Mas-Colell et al. (1995) Chapter 17), characterizing uniqueness is typically harder and involves

specific parametric restrictions (e.g., see Miyao et al. (1980), Allen et al. (2020), Allen et al. (2015)).

3 Theory of General Equilibrium Effects in Space

We now study how exogenous changes in bilateral trade costs τij,s affect outcomes in different

markets. We use the superscript 0 to denote variables in the initial equilibrium, z0j , and hats to

denote log changes in variables between the initial and new equilibria, zj ≡ ln(zj/z0j ). Given the

normalization wm = 1, relative wage changes follow directly from the equilibrium definition in

terms of excess labor demand. By totally differentiating (10), we obtain

−(Y0)−1(∇lnwD

(w0|τ 0

))︸︷︷︸γ

w = (Y0)−1(∇ln τD

(w0|τ 0

))τ︸︷︷︸

η(τ )

, (11)

where Y0

is the diagonal matrix of initial market revenue.

The system in (11) frames our analysis as a traditional comparative statics exercises in general

equilibrium.14 The right hand side measures the partial equilibrium shift in the excess labor demand

of each market caused by the shock (holding wages constant). The left hand side is the general

equilibrium change in excess demand, which depends on the ‘‘spatial links’’ matrix γ, triggered by

relative wage responses that restore labor market clearing everywhere.

Remark 1. Following shocks to bilateral trade shifters, relative wage changes combine: (i) the

vector of partial equilibrium shifts in excess labor demand, η(τ ), and (ii) the Jacobian matrix of

the excess demand system, γ.

In the rest of this section, we first establish that the excess demand shift can be written in

terms of shift-share variables similar to those used to measure regional exposure to trade shocks

in a recent empirical literature – e.g., Kovak (2013) and Autor et al. (2013). We then show

that, because of the spatial links in γ, the general equilibrium elasticity of the wage in market i

aggregates the direct effect of i’s own demand shift and the indirect effect of the demand shift of

other markets.

It is worth noting that our analysis focuses on the elasticity of labor market outcomes to trade

cost shocks. As such, it is a good approximation for the response to small changes in trade costs.

14For example, see sections 10.2 in Arrow and Hahn (1971) and 17.G in Mas-Colell et al. (1995). Allen et al.(2020) use an analogous representation in a single-sector gravity economy with a logit function of effective labor

supply – in our notation, Φi(w,P ) = νi(wi/Pi)φ/∑j νj(wj/Pj)

φ and Ψi(L) = Lψi .

11

For large shocks however, it is necessary to compute the integral of our formulas accounting for

changes in trade outcomes along the adjustment path to the new equilibrium.

3.1 Partial Equilibrium Shifts in Excess Labor Demand

Our first result characterizes partial equilibrium shifts in the only two variables directly affected

by the shock: revenue Yi and consumption cost index Pi.

Proposition 2. Partial equilibrium shifts in i’s revenue, ηRi (τ ) ≡∑

j,o,s∂ lnYi(p

0,E0|τ0)∂ ln τoj,s

τoj,s, and

consumption cost, ηCi (τ ) ≡∑

o,s∂ lnPi(p

0|τ0)∂ ln τoi,s

τoi,s, are given by

ηRi (τ ) =∑j,o,s

(y0ijy

0ij,s

) (x0oj,s − I[i=o]

)εsτoj,s and ηCi (τ ) =

∑o,s

ξi,sx0oi,sτoi,s. (12)


Equation (12) shows that the partial equilibrium shifts, ηRi (τ ) and ηCi (τ ), have a shift-share

structure: they interact the shock τoj,s with i’s initial exposure in terms of revenue and consumption.

In particular, market i’s revenue is more exposed to τoj,s when i’s revenue share in sector s and

destination j is higher (i.e., y0ijy

0ij,s is higher) and the spending share of j on market o is higher (i.e.,

x0oj,sεs is higher). In addition, by Shepard’s lemma, i’s consumption cost is more exposed to τoi,s

when i spends more on sector s of market o (i.e., ξi,sx0oi,s is higher). Thus, given any trade shock,

the computation of ηRi (τ ) and ηCi (τ ) only requires bilateral trade flows and trade elasticities.15

We simplify the notation by denoting ηRi (τ ) and ηCi (τ ) as ηRi and ηCi .

To understand better ηRi and ηCi , consider a shock in a single origin F , τij,s = 0 for i 6= F and

τFj,s 6= 0, under the assumption of εs = ε for all s. For i 6= F , the exposure to the sectoral average

of the demand-adjusted shock, ζs ≡ N−1∑

j(εx0Fj,sτFj,s), is16

ηRi =∑s

y0i,sζs and ηCi = ε−1

∑s

ξi,sζs for i 6= F, (13)

where y0i,s ≡

∑j y

0ijy

0ij,s is the share of sector s in market i’s revenue. Notice that yi,s is also the

share of sector s in i’s total employment since our model features a single wage rate in each market.

15The measures ηRi (τ ) and ηCi (τ ) are the partial equilibrium (i.e. holding wages and employment constant in allmarkets) impact of trade shocks on the measures of firm and consumer market access introduced in Anderson andVan Wincoop (2003) and Redding and Venables (2004). However, while our measures can be readily recovered withdata in the initial equilibrium, the definition of market access requires solving the full general equilibrium, which istypically done under the assumption of a single industry whose bilateral trade costs are symmetric and observed(e.g., Redding and Sturm (2008), Donaldson and Hornbeck (2016), Bartelme (2018)).

16Without loss of generality, we can decompose εx0Fj,sτFj,s into a sector mean, ζs ≡ N−1∑j(εx

0Fj,sτFj,s), and a

sector-market residual, εj,s ≡ εx0Fj,sτFj,s − ζs.

12

In this case, ηRi is a shift-share variable based on sectoral employment shares. It is thus the

regional exposure to import competition used in a growing empirical literature. If F becomes more

productive (ζs < 0 for all s), then every other market suffers a negative revenue shift, ηRi < 0. The

effect is stronger in markets with a higher employment share in sectors (i.e., higher y0i,s) with a

stronger foreign shock (i.e., lower ζs). Our theory also entails a consumption cost shift that is

proportional to a shift-share variable based, instead, on sectoral spending shares. If ζs < 0, then

consumption cost falls everywhere, ηCi < 0. The decline is stronger in markets with higher spending

shares in sectors (i.e., higher ξi,s) with a stronger foreign shock (i.e., lower ζs).

The partial equilibrium excess demand shift can be now written in terms of the shifts in revenue

and consumption cost.

Theorem 1. The vector of excess labor demand shifts is

η(τ ) = ηR︸︷︷︸Demand shift

− αφpηC︸︷︷︸

Supply shift

, (14)

where α ≡(I − y0 + χ0ψ

) (I + φ

px0′ψ

)−1, y0 is the revenue share matrix with entries y0

ij, and

x0′ is the spending share matrix with entries x0ij.


The first component in (14), the revenue shift ηR, measures the shock’s impact on the demand

for goods from each market (holding wages and employment constant). The second component

measures the labor supply shift caused by the import price shock. It arises only when labor supply

is a function of consumption prices (φp 6= 0). The total effect on the excess labor demand is

amplified by the multiplier matrix α implied by the feedback effect of employment on productivity

and market size (holding wages constant).

We convey the intuition for the supply shift in the special case without cross-market links in

labor supply and productivity: ψ = ψI and φp

= −φpI. If 0 < φpψ < 1, then

− αφpηC =(I − y0

) (φpηC

)︸︷︷︸

Net labor supply

+ χ0ψ(φpηC

)︸︷︷︸

Productivity

+(I − y0 + ψχ

) ∞∑d=1

(φpψx0′)d

︸︷︷︸Amplification

(φpηC

). (15)

Consider the supply shift implied by an increase in foreign productivity that lowers consumption

cost everywhere, ηCi < 0. The shock causes a labor supply increase in i, pushing down excess

labor demand by φpηCi . However, labor supply increases in every other market j, increasing i’s

excess demand proportionally to its revenue share from j, −φp∑

j y0ij η

Cj . The net effect is the first

term in (15). Second, through the productivity response, the higher labor supply in i lowers its

good prices by ψφpηCi . The second term in (15) is this cost reduction times the demand elasticity

13

χ0. The third term in (15) arises from the several feedback rounds of changes in labor supply

and productivity. Because all entries of φpψx0′ are positive, every term in the series expansion is

positive and, therefore, amplifies the effect of the net labor supply and productivity terms.

Remark 2. The revenue shift ηR has a direct impact on excess labor demand η. The impact of the

consumption cost shift ηC is proportional to the price elasticity of labor supply φp.

3.2 Direct and Indirect Effects in General Equilibrium

We now investigate the general equilibrium impact of partial equilibrium shifts in excess labor

demand. We start by establishing that the spatial links matrix is a function of the spatial

connections in labor supply, productivity, and trade demand.

Proposition 3. The spatial links matrix is

γ = I −(y0 + χ0

)︸︷︷︸Demand substitution

+ α(φw

+ φpx0′)︸︷︷︸

Supply substitution

. (16)


The spatial links matrix combines two forces created by spatial connections. The first is the

demand substitution effect implied by changes in trade demand, as summarized in (y0, χ0). It

controls how much the wage change in one market affects demand for goods elsewhere (given initial

employment). The second is the supply substitution effect that regulates how much excess labor

demand changes due to employment responses. It is the sum of the wage’s direct impact on labor

supply, φw

, and its indirect impact on labor supply through changes in the price index, φpx0′.17

Again, the labor supply response is amplified by the multiplier matrix α defined in Theorem 1.

We now characterize the general equilibrium response of relative wages.

Theorem 2. Suppose that Assumption 1 holds. Consider shocks to bilateral shifters, τ , with an

associated vector of excess demand shifts given by η. Then,

wi = βiiηi︸︷︷︸Direct effect

+∑j 6=i

βij ηj︸︷︷︸Indirect effect

, (17)

where

βij =1

γii

(I[i=j] −

γijγjj

I[i 6=j]

)+∞∑d=2

γ(d)ij

γjj(18)

17This indirect response is an immediate implication of Shepard’s lemma: a wage change of wj in market j has aneffect on i’s price index that is proportional to i’s spending share on j, x0jiwj .

14

such that γ(d)ij is the i-j entry of (¯γ)

dwith γij ≡ −γij

γiiI[i 6=j; i,j 6=m].


Theorem 2 characterizes the reduced-form impact of shifts in excess labor demand on relative

wages (given the normalization wm ≡ 1). The term βij is the reduced-form elasticity of market

i’s relative wage to market j’s partial equilibrium excess demand shift. Equation (17) shows that

local shock exposure percolates to other markets: ηj has a direct effect of βjj ηj on the own market

j and an indirect effect of βij ηj on other markets i. Thus, i’s relative wage responses aggregates

the direct exposure to its own shock (ηi) and the indirect exposure to the shock of other markets

(ηj for j 6= i), where the aggregation weights are given by βij . Accordingly, conditional on knowing

the excess demand shifts η, the matrix of reduced-form elasticities β is sufficient to aggregate wage

responses in general equilibrium. Thus, expression (17) can be used to construct an ex-ante analog

of the type of sufficient statistic for welfare gains in Arkolakis et al. (2012) that only requires

outcomes in the initial equilibrium.18

The theorem shows that βij is a series expansion of the spatial links matrix γ. A stronger

bilateral link γij yields a larger indirect effect βij : the first term of βij in (18) is proportional to γij .

The change in i’s excess demand must also be corrected for changes in the wages of other markets,

triggering higher-round responses captured by the power series in (18), whose magnitude depends

on the combined strength of the connections across the network of markets.19

This result indicates that the spatial links matrix γ determines the reduced-form elasticity

matrix β. In fact, Appendix 8.6 shows that the connection between γ and β holds also in the

opposite direction since β is, up to a normalization, the inverse of γ.

Remark 3. Given the vector of partial equilibrium shifts η, the general equilibrium wage responses

aggregate the direct effect of the own market shock exposure, βiiηi, and the indirect effects of other

markets’ exposure,∑

j 6=i βij ηj. The indirect effects βij are stronger among markets with stronger

bilateral spatial links, γij, or stronger third-market connections,∑∞

d=2 γ(d)ij /γjj.

3.3 Understanding Indirect Effects in General Equilibrium

We now present two special cases to provide further intuition for the results above. They illustrate

how indirect effects shape the general equilibrium impact of trade shocks on local labor markets.

They also link the reduced-form representation in (17) to specifications used to investigate how

regional markets respond to trade shocks in a recent empirical literature.

18Our characterization also complements the connection between outcomes and market access across regionsintroduced by Donaldson and Hornbeck (2016). While the aggregation of the reduced-form responses based on (17)immediately yields wage changes in general equilibrium, the aggregation of the impact of market access also involvesan endogenous shifter implied by the full specification of the model.

19Similar power series of indirect effects arise in the percolation of shocks across the network of sectors e.g.,Acemoglu et al. (2012) and Acemoglu et al. (2016).

15

Our first special case provides a sufficient condition for the direct and indirect effects to have

the same sign. It imposes that γ satisfies the gross substitution property, which effectively restricts

demand substitution in γ to dominate opposing supply substitution forces.

Corollary 1. If γ satisfies gross-substitution (γij < 0 for i 6= j), then βij > 0 for all (i, j).


To understand this result, consider again the productivity increase in foreign market F . If the

negative revenue shift of i dominates the impact of its consumption cost shift (ηi < 0), then market

i’s excess demand shift leads to reinforcing indirect effects everywhere, lowering not only i’s wage

relative to F , but also that of all other markets. Intuitively, a negative revenue shift in a market

reduces the demand for goods produced in other markets, further decreasing revenues and wages

in those markets. In this case, ignoring indirect effects underestimates the impact of the foreign

productivity shock on local outcomes.

The second special case restricts spatial links to be identical.

Corollary 2. If the entries of γ satisfy γij = γI[i=j] − γj, then βij = βI[i=j] + βj ∀ (i, j).


This case gives rise to an ‘‘endogenous’’ fixed-effect, which contains the sum of the indirect

effects of all markets,∑

j βj ηj . This common component can be ignored only if the shock affects a

zero mass of markets. Notice that βj is positive whenever γ also satisfies gross substitution (i.e.,

γj > 0). In this case, a foreign productivity gain leading to ηi < 0 for all i 6= F creates a negative

fixed-effect that reinforces the wage decline caused by the market’s own shock exposure.

Remark 4. The off-diagonals of the spatial links matrix determine the sign and the heterogeneity

of the indirect effects of excess labor demand shifts in other markets.

Connection to gravity trade models. The previous two corollaries illustrate the predictions

of gravity trade models with exogenous labor supply (φw

= φp

= 0) – e.g., see Costinot and

Rodrıguez-Clare (2014). In fact, these models satisfy the gross-substitution property if the trade

elasticity εs is positive (Alvarez and Lucas, 2007). By Corollary 1, in this case, excess labor demand

shifts further propagate through the trade network, generating reinforcing indirect effects on other

markets. Theorem 2 indicates that these indirect effects are stronger in nearby markets for which

γij is likely to be higher.20

20A gravity structure of bilateral trade is also present in quantitative spatial frameworks used in a recentliterature (e.g. Allen and Arkolakis (2014); Redding and Rossi-Hansberg (2017)). However, in these models, thegross-substitution property may not hold due to the combination of endogenous responses in employment andproductivity across markets. This creates a force for attenuating indirect effects as negative shocks elsewhere leadto local employment increases that have positive impacts on local market size and labor productivity.

16

This special case allows us to tightly connect our results to existing evidence on the response of

regional wages to trade shocks – e.g., Kovak (2013); Autor et al. (2013). To see this, assume no

trade costs (i.e., yij,s = xij,s = xi), and consider again the shifts created by the sectoral component

of productivity gains in the foreign country F (i.e., ζs > 0 for all i 6= F ). By Corollary 2, equation

(17) for any i 6= F becomes

wi = βηRi + ηR where ηR ≡∑j

βj ηRj and ηRi ≡

∑s

y0i,sζs. (19)

This expression implies that the differential wage response of market i is proportional to that

market’s shift-share exposure (as measured by ηRi ). The elasticity β measures how much more

relative wages respond in markets experiencing stronger shifts in excess demand. The general

equilibrium response also includes the common ‘‘endogenous’’ fixed-effect ηR that incorporates the

indirect effects of labor demand shifts in other markets. This fixed-effect reinforces the impact of

the market’s own revenue shift caused by the foreign productivity gains (ηRi < 0 for all i 6= F ). Its

size is proportional to the size of the more severely exposed markets (see Appendix 8.8).

Our model thus generalizes shift-share specifications used in this recent empirical literature. In

the general case considered in Theorem 2, the indirect effect of the shock exposure of other markets

may reinforce or attenuate the direct effect of the local shock exposure. These indirect effects are

stronger between markets with stronger spatial links. In Section 5, we use the heterogeneity in

indirect effects to propose a strategy to estimate how local labor markets respond to the shock

exposure of different markets. The strategy exploits the one-to-one mapping between β and γ

to reduce the number of parameters in estimation. Before we do that, we derive our model’s

reduced-form expressions for changes in wages, employment, and real wages.

3.4 Extensions

In Online Appendix C, we derive versions of the results in this section for several extensions of

our baseline model. We highlight potential differences in terms of sources of spatial links, which

regulate the reduced-form general equilibrium elasticities, and the shock exposure measures, which

take the form of shift-share variables.

Trade imbalances. We follow Dekle et al. (2007) to incorporate exogenous trade imbalances for

each market (specified in terms of the world’s average wage). All results above remain valid with

a spatial links matrix that accounts for the effect of wage changes on transfers.

Bilateral migration. We follow Bryan and Morten (2015) to incorporate bilateral migration

flows into our model. All results above remain the same. The only difference is that this extended

model also yields predicted changes in bilateral migration flows following trade shocks.

Multiple worker groups. We introduce multiple worker groups, as in Cravino and Sotelo (2019).

17

In this case, the market definition includes worker groups and the spatial links matrix depends on

the elasticity of substitution in production across groups. We show that revenue shifts for each

group are based on the group’s employment distribution across sectors.

General trade demand. We relax the nested CES demand structure by considering an one-factor

economy with arbitrary within-sector heterogeneity in productivity across goods and markets, as

in Adao et al. (2017). The main difference in this case is that spatial links in trade demand χij

take a more general form and they also enter the definition of the revenue shift.

Input-Output linkages. We finally extend our model to incorporate input-output linkages in

production, as in Caliendo and Parro (2015). Bilateral trade demand for intermediate inputs

are reflected in the spatial links matrix. This economy entails an additional exposure measure

capturing shifts in the cost of imported inputs, which takes the form of a shift-share variable

based on shares of intermediate input usage across sectors for each market (similar to the ones in

Acemoglu et al. (2016)).

4 Reduced-Form Responses in General Equilibrium

In this section, we derive our theory’s reduced-form representation for the impact of trade shocks

on employment and wages across markets. In particular, we show how each labor market responds,

directly and indirectly, to the shock-induced shifts in revenue and consumption cost of different

markets. These reduced-form expressions are the cornerstone of our empirical strategy to measure

the labor market consequences of trade shocks in general equilibrium. Appendix 8.9 contains the

derivations for this section.

4.1 Wage and Employment Responses

The expressions for excess demand shifts in (14) and wage changes in (17) imply

w = βRηR + β

CηC such that β

R ≡ β and βC ≡ −βαφp. (20)

Wage changes combine (i) the vector of shock-induced shifts in revenue and consumption costs,

ηR and ηC , and (ii) the reduced-form elasticity matrices of wage changes to these shifts, βR

and βC

.

The elasticity of wages to shifts in revenue and excess labor demand are identical, so it has all the

properties discussed in Sections 3.2 and 3.3. The reduced-form elasticity of wages to consumption

cost shifts also includes the supply component of the excess demand shift discussed in Section 3.1

and, as such, it is proportional to φp.

Turning to employment, the labor supply function in (1) and the price index in (8) yield

18

L = ρ(φw + φ

pηC), (21)

where ρ ≡ (I + φpx0′ψ)−1 and φ ≡ (φ

w+ φ

px0′). This expression captures all the different

channels through which trade shocks affect employment in our model. ρ is the multiplier of

employment-induced changes in productivity and prices that feedback into further employment

changes. It is the matrix that introduces the amplifying series expansion in α – see equation (15).

The terms inside brackets capture the labor supply responses to changes in wages and prices: φw

is the impact of wages through the elasticity structure of labor supply in φ, and φpηC is the impact

of import prices through the price elasticity of labor supply in φp.

We use this expression to write employment changes as only a function of shifts in revenue and

consumption cost. The combination of (20) and (21) implies

L = ϕRηR + ϕCηC such that ϕR ≡ ρφβR and ϕC ≡ ρ(I − φβα

)φp. (22)

Employment responses depend again on ηR and ηC , but they are now multiplied by different

reduced-form elasticity matrices, ϕR and ϕC . The revenue elasticity combines the revenue shift’s

impact on wages β and the employment elasticity to wages ρφ. The consumption cost elasticity

is the sum of employment responses to the consumption cost shift, ρφp, and the wage change

induced by the cost shift, ρφβC

. Thus, ϕC is proportional to φp.

Through the lens of our model, (20) and (22) are the reduced-form responses of local labor

market outcomes to trade shocks in general equilibrium. Accordingly, they allow us to aggregate any

vector of partial equilibrium shifts ηR and ηC to obtain wage and employment responses in general

equilibrium. This aggregation only requires the reduced-form elasticity matrices (βR, β

C, ϕR, ϕC).

Remark 5. The reduced-form expressions in (20) and (22) connect responses of wage and em-

ployment to shock-induced vectors of shifts in revenue and consumption cost. The reduced-form

elasticities are sufficient to aggregate any vector of shifts ηR and ηC to obtain wage and employment

responses in general equilibrium.

4.2 Real Wage Responses

We conclude our theoretical analysis by establishing how trade shocks affect real wages, defined

as Wi = wi − Pi. We use the real wage change as measure of the shock’s impact on welfare. In

fact, Online Appendix B.3 shows that, under general preferences for consumption and leisure, the

equivalent welfare variation in each market implied by the shock is increasing in the shock’s impact

19

on the market’s real wage. The real wage changes can be decomposed into three terms:

W = (I − x0′)w︸︷︷︸Terms-of-trade gains

+ x0′ψL︸︷︷︸Efficiency gains

− ηC︸︷︷︸Consumption cost gains

. (23)

The terms in this expression represent the three main sources of welfare gains in the model: (i)

changes in terms-of-trade due to the wage changes characterized in (20), (ii) changes in labor

productivity due to the employment changes characterized in (22), and (iii) the partial equilibrium

changes in consumption cost, ηC in (12).

The main forces shaping real wage responses can be readily seen using again the example of the

gravity trade model and a foreign productivity shock. In such case, φw

= φp

= 0 and, therefore,

W =(I − x0′) βηR − ηC

where β > 0. For a foreign productivity gain, ηRi < 0 and ηCi < 0 for all i 6= C. This creates two

opposite forces on the real wage: it tends to fall because of adverse terms of trade movements, but

it tends to rise due to lower import prices. The overall effect depends on the relative intensity of

these forces as controlled by the reduced-form elasticity β.

5 Measurement of General Equilibrium Effects in Space

We now develop a methodology to estimate the general equilibrium reduced-form elasticities. We

specify the data generating process using our theory’s reduced-form representation for changes in

wages and employment across markets. We then outline parametric restrictions on spatial links

that make the reduced-form elasticity matrix a function of observable variables and unknown

parameters. We finally characterize a GMM estimator of the vector of parameters based on the

Optimal IV approach of Chamberlain (1987).

5.1 Data Generating Process

We observe changes in employment and wages between t and t0, Lt

and wt, as well as the sector-

level trade matrix at t0, X t0ij,s. We decompose trade cost changes into observed and unobserved

components such that τ t = τ tO + τ tU .21 Since ηRi (τ t) and ηCi (τ t) in (12) are linear combinations of

τ , we can define the shifts implied by each component of the shock:

ηR(τ t) = ηR(τ tO) + ηR(τ tU) and ηC(τ t) = ηC(τ tO) + ηC(τ tU). (24)

21The observed component may be sectoral productivity shocks in a foreign country or changes in import barriersin an importer country. The unobserved component includes all other sources of trade shocks.

20

Up to a first order approximation, expressions (20), (22) and (24) imply that[wti

Lti

]=

[βt

ϕt

]+∑j

[βRij

ϕRij

]ηRj (τ tO) +

∑j

[βCij

ϕCij

]ηCj (τ tO) +

[νw,ti

νL,ti

]. (25)

Without loss of generality, we define the time fixed-effects, βt and ϕt, to be the mean effect

on wage and employment of the unobserved trade shocks, ηR(τ tU) and ηC(τ tU). The terms νw,ti

and νL,ti are then the deviations from the mean for market i in wage and employment responses

to ηR(τ tU) and ηC(τ tU). Notice that we can easily allow νw,ti and νL,ti to also include the effect of

labor supply shocks – i.e., the reduced-form responses to the shifts in Li = Φi(vwj wjj, vpjPjj).

Assumption 5a. [DGP] Between periods t0 and t, we observe a trade shock vector τ tO. We

also observe changes in wages and employment, wti and Lti, that are given by equation (25) with

νw,ti and νL,ti denoting mean-zero unobserved residuals.

5.2 Dimensionality Reduction: Parametrizing Spatial Links

The estimation of equation (25) entails one important challenge: while we only observe employment

and wage changes for N markets, (25) has 4N2 unknown elasticities. To circumvent this problem,

we parametrize spatial links to write reduced-form elasticities as non-linear functions of observable

variables in the initial equilibrium and a small number of unknown parameters.22 Our parametric

assumptions yield a parsimonious specification while incorporating enough degrees of freedom to

flexibly capture different response patterns in the data.

Definition of a Market. Each market i is an integrated regional labor market, in which workers

are perfectly mobile across sectors. In the empirical analysis below, markets are U.S. Commuting

Zones (CZs), as in Tolbert and Sizer (1996).

Labor Supply. The labor supply elasticities to changes in wages and prices are given by

φwij = φwI[i=j] − φmm0ij − (φw + φp)bwj , and φpij = φpI[i=j] + φmm0

ij − (φw + φp)bpj , (26)

where m0ij is the observed share of i’s population born in j at t0, and bwj and bpj are observed

j-specific attributes such that∑

j(bwj + bpj) = 1.

The parameters φw and φp control, respectively, the sensitivity of regional employment to

the local wage and price index. As discussed in Section 4, φw is proportional to the response of

22This procedure effectively projects the reduced-form elasticities in (25) onto observable variables regulating thestrength of spatial links. It is similar to the common practice in demand estimation of specifying cross-price demandelasticities in terms of observable variables (Berry, 1994; Berry et al., 1995).

21

regional employment to the shock-induced change in the region’s wage. It captures the evidence in

Autor et al. (2013) and Dix-Carneiro and Kovak (2017) that both employment and wage growth

are lower in regions more exposed to import competition shocks. In contrast, φp is proportional

to i’s employment response to its own import price shock ηCi . This allows the model to flexibly

match the type of employment responses to lower import prices studied by Acemoglu et al. (2016)

and Pierce and Schott (2016). The parameter φm controls the employment response in i to wage

changes in the regions where a higher share of i’s population was born (i.e., m0ij is higher). It thus

captures the gravity migration links documented by Bryan and Morten (2015). For simplicity, we

impose identical absolute values for the cross-market labor supply elasticities associated with mij

for changes in prices and wages.

Lastly, bwj and bpj in (26) guarantee the homogeneity of the labor supply function, so that

employment changes are invariant to the numeraire choice. Our baseline specification uses bpj = 0

and bwj = Yj/∑

o Yo, which can be interpreted as setting non-employment benefits in terms of the

world’s average wage (the same numeraire used for international transfers).23

Productivity. The labor productivity elasticity to changes in employment is

ψij = ψI[i=j]. (27)

The parameter ψ is the elasticity of regional labor productivity to regional employment. As

discussed in Section 3, when combined with the labor supply elasticity in (26), a higher value of ψ

amplifies the reduced-form elasticities of wages and employment to the observed shifts in revenue

and consumption. Intuitively, it regulates the strength of the feedback effect of employment on

productivity and, therefore, excess labor demand. It captures the agglomeration forces documented

by Greenstone et al. (2010), Kline and Moretti (2014) and Peters (2019). It is important to note

that such channel is absent in recent quantitative spatial frameworks based on the Ricardian model

of Eaton and Kortum (2002) – e.g. Caliendo et al. (2018), Caliendo et al. (2019) and Galle et al.

(2017).

Trade Demand. The trade demand elasticity is

εs = ε ∀s, such that χij = −εI[i=j] + ε∑s,k

(y0iky

0ik,sx

0jk,s

). (28)

The parameter ε controls the sensitivity of good’s demand to production costs. By increasing

the demand substitution elasticity (i.e., higher χij), a higher ε strengthens spatial links and,

23Online Appendix A.2.1 evaluates how our results change when we use alternative labor supply specificationswith the normalization in terms of the average price index.

22

consequently, indirect effects among regions specialized in similar sectors or destinations (i.e.,

higher∑

s,k y0iky

0ik,sx

0jk,s). Thus, ε captures demand spillovers implied by the gravity trade links

extensively documented in the literature reviewed by Head and Mayer (2014).

Estimating Equation. For θ ≡ (φw, φp, φm, ψ, ε), equation (25) becomes, after imposing

(26)–(28): [wti

Lti

]=

[βt

ϕt

]+∑j

[εβRij(θ)

εϕRij(θ)

]ηR,tj +

∑j

[βCij (θ)

ϕCij(θ)

]ηC,tj +

[νw,ti

νL,ti

](29)

where, by equation (12),

ηR,tj ≡∑j,o,s

(yt0ij y

t0ij,s

) (xt0oj,s − I[i=o]

)τ toj,s and ηC,ti ≡

∑o,s

ξt0i,sxt0oi,sτ

toi,s. (30)

We summarize these parametric restrictions in the following assumption.

Assumption 5b. [Parametrization of spatial links] Assume that (26)–(28) hold. Condi-

tional on θ ≡ (φw, φp, φm, ψ, ε), equation (29) relates observed changes in wage and employment to

the observed shifts in revenue and consumption cost in (30) across markets.

5.3 Model-Implied Optimal IV

We now derive an estimator of θ. We start by imposing the following orthogonality condition.

Assumption 5c. [Exogeneity] Let νti ≡[νw,ti , νL,ti

]′. In every period t, E

[νti |(ηR,t, ηC,t

)]= 0.

Assumption 5c imposes that the unobserved residuals in (29) are mean-independent from the

observed shifts in revenue and consumption. This type of exogeneity condition is necessary for the

causal interpretation of estimates of how regional labor markets respond to international trade

shocks – e.g. Topalova (2010), Kovak (2013), Autor et al. (2013).24

Consider a function Hi

(ηR,t, ηC,t

)with dimension dim(θ)× 2. By the law of iterated expecta-

tions, Assumption 5c implies

E[Hi(ηR,t, ηC,t)νti ] = 0, (31)

which yields the following class of GMM estimators of θ.

Definition 1. Let Hi

(ηR,t, ηC,t

)be a dim(θ)× 2 function and νti (θ) be the residual implied by

(29) under Assumptions 5a and 5b. Define the GMM estimator:

24Since τ tU generates the structural residuals νw,ti and νL,ti , Assumption 5c is implied by the independence betweenτ tU and τ tO (given the initial trade matrix). This is similar to the requirements for identification and consistency inshift-share designs (see Borusyak et al. (2018) and Adao et al. (2019)).

23

θH ≡ argminθ

[∑i,t

Hi(ηR,t, ηC,t)νti (θ)

]′ [∑i,t

Hi(ηR,t, ηC,t)νti (θ)

]. (32)

The estimator in (32) is consistent for θ whenever Hi(·) satisfies the usual rank conditions

establishing identification in GMM estimators.25 Notice that a necessary condition for identification

is that all elements of θ are associated with heterogeneous responses to the shock exposure across

markets (instead of an identical effect on all markets). In fact, the results in Section 3 show that

bilateral reduced-form elasticities are increasing in bilateral spatial links, implying that identification

relies on heterogeneity in the bilateral variables governing the spatial links associated with θ.

The implementation of the GMM estimator in (32) requires specifying Hi(·). Although any

function yields a consistent estimator of θ, functions vary in terms of asymptotic variance – that

is, the estimators differ in precision.26 The following proposition uses the approach in Chamberlain

(1987) to characterize the function H∗i (·) that minimizes the asymptotic variance of θH .

Proposition 4. Under Assumptions 5a–5c, the function that minimizes the asymptotic variance

of the class of estimators in Definition 1 is

H∗i(ηR,t, ηC,t

)≡ E

[∇θνti (θ) |ηR,t, ηC,t

] (Ωti

) −1 (33)

where Ωti ≡ E

[νti (θ) νti (θ)′ |ηR,t, ηC,t

]and

E[∇θνti (θ) |ηR,t, ηC,t

]= −

∑j

[∇θεβRij (θ)

∇θεϕRij (θ)

]ηR,tj −

∑j

[∇θβCij (θ)

∇θϕCij (θ)

]ηC,tj . (34)

Proof. Appendix 8.10.

The optimal function H∗i(ηR,t, ηC,t

)has two components. The matrix Ωt

i attributes larger

weight to observations with a lower variance of unobserved residuals – under homoskedasticity,

Ωti = Ωt is the GMM optimal moment weight matrix. The second component is the predicted

response of the endogenous variables associated with θ. Intuitively, through the Jacobian matrix

in (34), the Optimal IV of each parameter in θ puts more weight on the observed shifts of markets

whose reduced-form effects on market i are more sensitive to changes in that parameter.

The Optimal IV in Proposition 4 is a function of the unknown vector θ. To simplify its implemen-

tation, we characterize an asymptotically equivalent two-step estimator: the Model-implied Optimal

25See Theorems 2.6 and 2.7 in Newey and McFadden (1994).26Newey and McFadden (1994) provide regularity conditions for normality of GMM estimators of the form in

(32) – see Theorem 3.4. This relies on the central limit theorem and, therefore, requires some sort of independenceassumption. We assume that residuals are i.i.d across markets or clusters of markets. Notice that the number ofmoments is equal to the number of unknown parameters in θ. This does not imply any loss of generality, since it isalways possible to define Hi(· ) to include the optimal moment weighting matrix.

24

IV (MOIV). In the first-step, we use an arbitrary θ0 to compute H0i = E[∇θνti (θ0)|ηR,t, ηC,t](Ωt

i)−1

and obtain θ from (32). Since the instrument is a function of (ηR,t, ηC,t), the first-step estimator is

consistent but not optimal. Thus, in the second-step, we use the consistent estimator θ to compute

a consistent estimator of the Optimal IV, H∗i = E[∇θνti (θ)||ηR,t, ηC,t](Ωti)−1. We then use it to

obtain θMOIV

from (32).

Proposition 5. The efficient estimator H∗i (ηR,t, ηC,t) in (33)–(34) is asymptotically equivalent to

the Model-implied Optimal IV (MOIV) estimator obtained with the following two-step procedure.

Step 1. Using a guess θ0, estimate θ from (32) using H0i = E[∇θνti (θ0)|ηR,t, ηC,t](Ωt

i)−1.

Step 2. Using θ, estimate θMOIV

from (32) using H∗i = E[∇θνti (θ)|ηR,t, ηC,t](Ωti)−1.

Proof. Appendix 8.11.

6 Application: Measuring the Effects of The China Shock

In the last part of the paper, we use the results above to study how the rise in Chinese manufacturing

exports affected U.S. CZs. We follow Autor et al. (2013) (henceforth ADH) to specify the observed

sectoral intensity of the ‘‘China Shock.’’ We first show that the methodology of Section 5 yields a

generalization of the empirical specification in ADH. We then estimate our theory’s reduced-form

representation for employment and wage responses to the shock and use it to compute the shock’s

general equilibrium impact on U.S. CZs.

6.1 A General Equilibrium Extension of ADH

We start by connecting our theory’s estimating equation (29) to the main specification in ADH.

We focus on the labor market consequences of the sector-level average of the Chinese cost shock,

ζts ≡ N−1∑

j

(εx0

China,j,sτChina,j,s). We project ζts onto ADH’s sector-level shift: the per-worker

Chinese exports growth to developed countries (excluding the U.S.), ∆M o,ts .

Assumption 6a. [ADH Shock] Assume that

ζts = κ∆M o,ts + εts such that ∆M o,t

s ⊥ εtk ∀s, k. (35)

In equation (35), κ is the pass-through coefficient that connects the increase in per-worker

Chinese exports ∆M o,ts to changes in Chinese production costs ζts. This parameter is negative because

a sector with stronger cost reduction should have stronger export growth. Under Assumption

6a, equation (30) implies that ηR,ti and ηC,ti are proportional to shift-share variables based on

25

employment and spending shares across sectors:

ηR,ti =κ

ε

∑s

y0i,s∆M

o,ts︸︷︷︸

≡IPW ti

+εR,ti and ηC,ti =κ

ε

∑s

ξ0i,s∆M

o,ts︸︷︷︸

≡IPCti

+εC,ti , (36)

where εR,ti and εC,ti are regional residuals implied by εts.

The revenue shift is a linear function of the shift-share instrumental variable used in ADH – see

equation (3) in ADH. In addition, the consumption cost shift yields a new shift-share measure, IPCti ,

where the shift ∆M o,ts is interacted with the CZ’s spending share on sector s. Under Assumption

6a, εR,ti and εC,ti are orthogonal to IPW ti and IPCt

i . From (29) and (36),[wti

Lti

]=

[βt

ϕt

]+∑j

[βRij(θ)

ϕRij(θ)

] (κIPW t

j

)+∑j

[ε−1βCij (θ)

ε−1ϕCij(θ)

] (κIPCt

j

)+

[vw,ti

vL,ti

](37)

where vw,ti and vL,ti are structural residuals that contain the direct and indirect effects of all

unobserved trade cost shocks, including εR,ti and εC,ti .

Equation (37) is a strict generalization of ADH’s main specification. In fact, ADH only allow

for a common direct effect of the CZ’s own revenue shift and the endogenous fixed-effect of revenue

shifts: βRij(θ) = βI[i=j] + βj , ϕRij(θ) = ϕI[i=j] + ϕj , and βCij (θ) = ϕCij(θ) = 0. As discussed in Section

3.3, such a specification arises under two restrictions. First, consumption cost shifts can only be

ignored when labor supply does not respond to prices (i.e., φp = φm = 0). Second, common indirect

effects only arise when we restrict further the spatial links matrix to have identical off-diagonal

elements in all rows – as in the case of equal spending shares (xij = yji = xi for all j). Notice that,

even in this special case, the specification in ADH cannot separately identify the common indirect

effect from other national shocks included in the time fixed-effect. Hence, whenever this common

indirect effect is not zero, it delivers only part of the general equilibrium impact of the China shock

on U.S. CZs.

In general, our theory implies that we must estimate how employment and wages in CZ i

respond to both the IPW tj and IPCt

j of the own CZ as well as those of other CZs. As discussed in

Section 3.2, these reduced-form elasticities vary across markets because of the bilateral spatial links

that we parametrize in terms of sector specialization, trade flows, and migration flows. Even in the

presence of time fixed-effects, we identify each element of θ from the response of outcomes in a CZ

to the shock exposure of the CZs with higher values of the bilateral variables associated with each

parameter. Thus, in the rest of this section, we rely on equation (37) to estimate the reduced-form

elasticity matrices, βRij(θ), ϕRij(θ),ε−1βCij (θ),ε−1ϕCij(θ)ij, using the MOIV estimator described in

Section 5.3. To adjust for the scale of the shock, we separately estimate the pass-through parameter

κ using the linear expression in (35).

26

Figure 1: Exposure to Chinese export growth, 1990-2007

Notes: For each CZ, the left panel reports IPW ti and the right panel reports IPCti . IPW

ti has standard deviation of 2.52, IPCti has

standard deviation of 1.22. The spatial correlation between IPW ti and IPCti is 0.34.

6.2 Data Construction

We now provide an overview of the steps to construct the data used in estimation. Appendix 9

presents a detailed discussion of the data construction methodology.

We follow ADH by considering the 722 CZs in mainland U.S. over 1990-2000 and 2000-2007.

We use the procedure in the Online Appendix of ADH to construct the number of employed

individuals and the average weekly log-wage in each CZ using the Census Integrated Public Use

Micro Samples in 1990 and 2000 and the American Community Survey in 2006-2008. We also use

the U.S. Census data in 2000 to measure mij as the share of working-age individuals in CZ i that

report to be living in CZ j 5 years ago.

We construct the shift-share variables IPW ti and IPCt

i as follows. We compute IPW ti by

interacting ADH’s sector-level shift ∆M o,ts and the CZ’s ten-year-lagged employment share in

that sector. For each CZ, we use the imputation methodology in ADH to obtain employment

by 4-digit SIC industry from the County Business Patterns. Thus, our IPW t is identical to the

instrumental variable in ADH. To construct IPCti , we interact ADH’s sector-level shift ∆M o,t

s

and the CZ’s sectoral spending share. We follow Gervais and Jensen (2019) to construct CZ-level

spending shares by 4-digit SIC industries using the national input-output table. Specifically, for

each industry, we combine the final consumption share from the national input-output table with

the CZ’s predicted input purchase share obtained by interacting national industry spending shares

and regional industry employment.27 Figure 1 reports the spatial variation in IPW ti and IPCt

i . The

two measures have a good degree of variation across space, and their spatial correlation is 0.34.

To implement our strategy, we also construct sectoral trade matrices between the 722 U.S. CZs

and 52 foreign countries in 1990 and 2000. We assume a single labor market in each foreign country.

First, we use trade data from UN Comtrade to construct a country-to-country matrix of trade flows

27In Appendix 9.3, we evaluate our procedure to construct CZ-level spending shares across 4-digit SIC industries.We run a regression of state-STCG shipment inflows in the Commodity Flow Survey (CFS) on the state-SCTGspending shares implied by the aggregation of our dataset, where SCTG is the commodity classification used in theCFS. We obtain a coefficient close to 1 and a R2 of 0.95.

27

in 368 industries. We use the gravity structure of our model and data on domestic sales from Eora

MRIO to impute domestic spending shares in each industry. Second, we distribute U.S. domestic

and international trade flows across CZs using again the gravity structure of our model. We first

split U.S. Census data on imports and exports for each industry-country across CZs using each

CZ’s share in that industry’s national spending and production. We then impute bilateral trade

shares across CZs using an industry-level gravity specification estimated with bilateral shipment

data from the Commodity Flow Survey (CFS). Since our baseline model imposes trade balance, we

adjust market sizes to balance trade flows given the bilateral trade shares.28

6.3 Simple Extension of ADH

We begin our analysis with a simpler extension of the specification in ADH. Our goal is to provide

qualitative evidence for our main novel channels: how regional outcomes respond (i) directly to

the CZ’s own consumption shift, and (ii) indirectly to the revenue and consumption shifts of other

CZs. To this end, we consider an intuitive approximation of the reduced-form elasticity matrices

in equation (37):

Y ti = αt + αRIPW t

i + αCIPCti + αIR

∑j 6=i

zijIPWtj + αIC

∑j 6=i

zijIPCtj +X t

iλ+ νti (38)

where, for CZ i in period t, Y ti is the change in log-employment or average weekly log-wage, αt is a

time fixed-effect, and X ti is a set of regional controls.29 We equally weight all CZs when estimating

(38) because our theory’s unit of observation is a market.

As in ADH, αR is the direct impact of higher exposure to Chinese import competition. We

also include αC to measure the direct effect of the CZ’s own consumption cost shift, as well as αIR

and αIC to measure the indirect effects of the revenue and consumption shifts of other CZs. We

use zij to parametrize cross-regional variation in these indirect effects. Since general equilibrium

reduced-form elasticities are stronger between CZs with stronger spatial links, we build on our

model’s gravity structure to set zij ≡L0jD−δij∑

k L0kD−δik

, where Dij is the bilateral distance between i and j,

so that CZ i responds more to shocks in nearby and larger CZs. We set the parameter δ controlling

the relative importance of distance to five, following typical estimates of the trade elasticity.30

28Table 11 in Appendix 9.3 reports validation tests using the CFS data for 1997, 2002 and 2007. Regressions ofactual on predicted trade flows across states and SCTGs yield coefficients close to 1 and R2 around 0.5.

29We include the largest set of regional controls in ADH – i.e., column (6) of Table 3 in ADH. We alsoinclude controls for the CZ’s exposure to the secular manufacturing decline in the period: the CZ’s spending andemployment shares in manufacturing (y0i,M and ξ0i,M ), as well as the weighted average of these shares across other

CZs,∑j 6=i zijy

0j,M and

∑j 6=i zijξ

0j,M . Finally, we follow Greenland et al. (2019) by including the CZ’s lagged

population growth to absorb the effect of persistent confounding shocks.30Donaldson and Hornbeck (2016) rely on a similar specification to compute a proxy for their market access

measure in partial equilibrium. Online Appendix A shows that we obtain similar qualitative results for differentvalues for δ and functional forms for zij .

28

Table 1: Impact of the China Shock on Labor Market Outcomes across U.S. CZs

(1) (2) (3) (4) (5) (6)

Panel A: Change in avg. log weekly wage

IPW ti -0.437*** -0.417** -0.329** -0.405** -0.319** -0.319**

(0.156) (0.170) (0.138) (0.155) (0.152) (0.152)

IPCti -0.092 -0.043 -0.043

(0.214) (0.208) (0.197)∑j 6=i zijIPW

tj -1.039*** -1.036*** -1.032***

(0.309) (0.309) (0.333)∑j 6=i zijIPC

tj -0.556 -0.010

(0.452) (0.473)

R2 0.529 0.529 0.536 0.530 0.536 0.536

Panel B: Change in log of employment

IPW ti -0.561** -0.593** -0.423** -0.555** -0.468** -0.467**

(0.216) (0.238) (0.206) (0.212) (0.223) (0.225)

IPCti 0.149 0.212 0.154

(0.416) (0.419) (0.423)∑j 6=i zijIPW

tj -1.315*** -1.330*** -1.598***

(0.340) (0.345) (0.425)∑j 6=i zijIPC

tj -0.101 0.701

(0.462) (0.544)

R2 0.472 0.472 0.476 0.472 0.476 0.477

Notes: Pooled sample of 1,444 Commuting Zones in 1990-2000 and 2000-2007. Indirect effects computed with zij ≡ L0jD−δij /

∑k L

0kD−δik

where δ = 5, Dij is the distance between CZs i and j, and L0j is the population of CZ j in 1990. All specifications include the following

three sets of controls. Regional controls in ADH: period dummies, college-educated population share in 1990, foreign-born populationshare in 1990, employment share of women in 1990, employment share in routine occupations in 1990, average offshorability in 1990, andCensus division dummies. Initial manufacturing exposure: CZ’s share of employment and spending in manufacturing (

∑s y

t−20i,s and∑

s ξt−20i,s ), CZ’s indirect exposure to manufactruing employment and spending (

∑j 6=i zij

∑s y

t−20j,s and

∑j 6=i zij

∑s ξt−20j,s ). Lagged

population growth from Greenland et al. (2019): growth of population with 15-34 years old and 35-64 years old in the previous 10-yearperiod. Robust standard errors in parentheses are clustered by state. *** p < 0.01, ** p < 0.05, * p < 0.10

Table 1 reports the estimation of (38). Column (1) qualitatively replicates the main findings in

ADH. Regions initially specialized in industries subject to stronger Chinese import competition

exhibit slower growth in employment and wages. In contrast, column (2) shows that the shock-

induced shift in consumption cost does not have a significant impact on both employment and

wage growth across CZs. Notice that this is driven by lower point estimates with relatively tight

confidence intervals – especially for the wage response in Panel A.31

31Notice that input-output connections can be incorporated in the reduced-form of our model as an additionalshift-share variable, as discussed in Section 3.4. Since ξi,s includes sectoral intermediate input spending in the CZ,our results indicate no systematic difference in the outcomes of CZs that intensively source inputs from industrieswith larger increases in Chinese export growth. This result is consistent with those in Pierce and Schott (2016) and

29

Columns (3) and (4) investigate the indirect effect of the shock exposure of other CZs. Given

the CZ’s own revenue exposure, column (3) indicates that employment and wage growth are weaker

if nearby and larger CZs are more exposed to Chinese import competition. Compared to column

(1), the point estimates for the direct effect are one-fourth lower in column (3), suggesting that

the CZ’s own revenue shift is partially correlated with the revenue shifts of closer and larger CZs.

Column (4) shows however that the indirect effect of consumption exposure is not significant for

both wages and employment – again, confidence intervals have the same magnitude as those of

the indirect effect of revenue exposure. Finally, columns (5) and (6) indicates that results are

qualitatively similar when we use the specification with all measures of shock exposure.

Our estimates show that revenue shifts percolate through the spatial link’s network, amplifying

the negative direct impact of the CZ’s own exposure to Chinese import competition. The point

estimates in column (5) indicate that an increase of $1000 dollars in Chinese imports per U.S.

worker in all nearby CZs is associated with a weaker growth of 1.3 log-points in employment and

1.0 log-points in wages. These negative indirect effects are three times larger than the differential

direct impact of increasing the CZ’s own import competition exposure by $1000.

We can interpret these estimates using the theoretical insights presented in Section 3. First,

the negative direct impact of exposure to the China shock on employment and wages suggests

that revenue shifts have strong impacts on both employment and wages (i.e., φw is large). Second,

the negative indirect effect of revenue shifts indicates that demand substitution dominates supply

substitution in the spatial links matrix, as in Corollary 1. Lastly, the weak impact of consumption

cost shifts on employment and wages suggests that labor supply does not respond much to import

prices (i.e, φp is low relative to φw).

Robustness. Online Appendix A.1.1 investigates the robustness of the results in Table 1. We

focus on our preferred specification in column (5). We first analyze the importance of the baseline

controls. While the wage responses are not sensitive to the control set, the estimated employment

responses are larger and more precise when we control for the lagged population growth, as in

Greenland et al. (2019).32 In addition, results are qualitatively similar when we compute zij (i)

with other distance parameters δ ∈ (1, 8), (ii) without CZ size, and (iii) only with CZs in the same

state. The indirect effects are negative and statistically significant when we weight CZs by their

1990 population (as in ADH), use the inference procedure in Adao et al. (2019), and use other

measures of the CZ’s spending shares ξi,s.

Acemoglu et al. (2016) that find no evidence of differential employment growth in industries using more intensivelyinputs from sectors in which the China shock was stronger.

32This is consistent with the results in ADH who find a significant negative effect of Chinese import exposureonly on manufacturing employment. In their baseline specification, the response of total employment is negativebut non significant at usual levels (see also Bloom et al. (2019)).

30

Additional results. Online Appendix A.1.2 complements our baseline estimates. We find

that the indirect effect is negative and statistically significant for employment responses in both

manufacturing and non-manufacturing. In contrast, similar to ADH, wage responses are mainly

driven by the non-manufacturing sector. Finally, we document that results are qualitatively similar

when the sector-level shifter is the NTR gap used by Pierce and Schott (2016).

6.4 Reduced-Form Elasticities to Revenue and Consumption Shifts

We now turn to the implementation of the empirical strategy described in Section 5. In comparison

to the simple linear specification presented above, this strategy imposes that general equilibrium

reduced-form elasticities are implied by a parametrization of the spatial links matrix. This has

two main advantages. It guarantees that the reduced-form elasticities capture both bilateral and

third-market connections among CZs that arise in general equilibrium (as parametrized by the

observable variables in our specification). In addition, it implies that, through the lens of our

spatial model, neither time fixed-effects nor residuals include any component of the endogenous

responses of regional outcomes to the observed sectoral average of the China shock. For these

reasons, the predicted reduced-form responses implied by our elasticity estimates can be used to

properly aggregate regional shock exposures when computing the general equilibrium impact of the

China shock on U.S. CZs.

We present our results in three steps. We first estimate the pass-through coefficient κ in

equation (35). We then estimate equation (37) and present the implied reduced-form elasticities.

Lastly, we evaluate the fit of our model by comparing its predicted responses to the China shock

to actual changes in employment and wages across CZs.

6.4.1 Pass-through of Chinese Export Growth to Chinese Cost Shock

We start by estimating the pass-through coefficient κ in equation (35). Since the reduced-form

elasticities in (37) are a non-linear function of θ, κ is necessary to adjust the scale of the sectoral

import changes in ADH to be consistent with the sectoral cost shocks in our theory. Thus, the

estimation of (35) can be seen as the first-stage in the estimation of θ.

We measure ζts in two steps. We first estimate destination-sector-period fixed-effects, ρtj,s, in

a gravity equation of changes in bilateral trade shares across countries. Using the gravity trade

demand in (6), we approximate ζts using ζts ≈ N−1∑

j

(xt0China,j,sρ

tj,s −∆xtChina,j,s

).33 We consider

the same periods and countries used in the construction of ∆M o,ts .

33The gravity trade demand in (6) implies that ∆xtChina,j,s ≈ −εsxt0China,j,s(w

tChina + τ tChina,j,s − P tj,s) if

ΨChina

(Lt)≡ Ψt. By setting the Chinese wage to be the numeraire (wtChina ≡ 1), the definition of ζts implies that

ζts = N−1∑j

(xt0China,j,sεsP

tj,s −∆xtChina,j,s

). We obtain the expression above by noting that ρtj,s ≡ εsP tj,s is the

destination-sector-period fixed-effect in the sector-level gravity equation for changes in bilateral trade shares.

31

Table 2 presents the estimation of (35). It indicates that industries with stronger growth of

per-worker Chinese exports experienced stronger decline in Chinese production costs (as measured

by ζts). Specifically, a sector with $1000 higher per-worker Chinese export growth had a decline

in Chinese production cost of 0.38 percentage points in 1990-2000 and 0.10 percentage points in

2000-2007. We then set κ to the average estimate over the two periods: κ = −0.0024.

Table 2: Estimation of the Pass-through Parameter κ

Dependent variable: Chinese production cost shock, ζts(1) (2)

∆M o,ts -0.0038*** -0.0010***

(0.0012) (0.0003)

Period: 1990− 2000 2000− 2007

Notes: Sample of 368 4-digit SIC manufacturing industries. ζs is the Chinese cost shock described in the main text, and ∆Mo,ts is the

growth in Chinese exports to non-U.S. developed countries normalized by the initial U.S. sector-level employment (as in ADH). Allspecifications also include a constant. Standard errors in parenthesis clustered by 3-digit industry. *** p < 0.01

6.4.2 Estimation of the Reduced-form Elasticity Matrices

We now turn to the estimation of the reduced-form elasticities in estimation of (37). Table 3 reports

our baseline estimates of θ obtained from the estimation of (37) with the two-step procedure in

Proposition 5. We consider the same set of baseline controls in Table 1.

In Panel A, we present the estimation results without migration links in labor supply (i.e.,

φm = 0). We estimate an elasticity of labor productivity to local employment of 0.56. This is a

direct consequence of the large employment and wage responses to revenue shifts that we document

in Table 1. In order to rationalize such large reduced-form responses, the model requires strong

agglomeration forces that are higher than typical calibrations in the quantitative spatial literature

(we return to this point below). It is roughly twice the agglomeration elasticity implied by firm

entry in Krugman (1980) (as specified in Monte et al. (2018)) and much higher than the elasticity

of zero in Ricardian models such as Eaton and Kortum (2002) (as specified in Galle et al. (2017),

Caliendo et al. (2018) and Caliendo et al. (2019)). Instead, our estimate is closer to the elasticity

of manufacturing productivity to population density estimated from regional demand shocks in the

U.S. – for instance, Kline and Moretti (2014) estimate this elasticity to be around 0.4.

Second, we estimate a large elasticity of labor supply to wages, but a lower labor supply

elasticity to consumption prices. In fact, our estimate is closer to estimates based on the aggregate

employment responses to the business cycle, being three times higher than the median micro-

estimate reviewed by Chetty et al. (2013). This high value is necessary to rationalize the strong

responses of employment to revenue shifts, both directly and indirectly. In contrast, we estimate

32

Table 3: Estimates of the Structural Parameters

ψ φw φp ε φm

Panel A:

0.56 2.11 -1.36 3.94 -(0.07) (0.25) (0.24) (0.41) -

Panel B:

0.55 2.11 -1.36 3.94 -0.06(0.22) (1.26) (0.72) (1.03) (0.05)

Notes: Estimation of θ using the reduced-form expressions in (37) for the pooled sample of 1,444 Commuting Zones in 1990-2000 and2000-2007. Estimation uses the two-step procedure in Proposition 5. All specifications include the set of baseline controls in Table 1.Standard errors in parentheses are clustered by state.

a lower labor supply elasticity to prices, φp = −1.36. This parameter controls the response of

employment and wages to consumption cost shifts across CZs. Accordingly, the low φp relative to

φw follows from the weak responses of wages and employment to IPCtj in Table 1. Notice that, in

the full structural estimation, we obtain a small confidence interval for φp because estimation uses

all the channels through which this parameter affects reduced-form elasticities in the model.34

Third, we find that gravity trade links across regions are important, implying a trade elasticity

of roughly four. This lies within the range of estimates in the literature – e.g., see Simonovska and

Waugh (2014) and Costinot and Rodrıguez-Clare (2014). This parameter captures the negative

indirect effects of revenue shifts on both employment and wages that we document in Table 1.

Panel B presents results when we also estimate the parameter controlling migration links in

labor supply across markets. Our estimate of φm is not statistically different from zero at usual

significance levels. This implies that employment in a CZ, conditional on its own wage change,

does not respond much to revenue shifts in CZs with stronger migration links. This is consistent

with the evidence in ADH of weak responses in the CZ’s working-age population to its own revenue

shift.35 Notice that the point estimates of all other parameters are almost the same in Panels A

and B. However, relative to Panel A, standard errors are two to five times higher in Panel B. This

34In Online Appendix B.1.2, we show that two rudimentary formulations of the intensive and extensive marginsof labor supply imply that employment is more sensitive to wages than to prices (i.e., φw > φp). In a setting with arepresentative household deciding the number of hours worked, this arises from the fact that a higher wage affectslabor supply only through changes in the opportunity cost of leisure (i.e., the real wage), whereas a lower priceindex has ambiguous effects on labor supply because it increases both the real wage and the real value of lump-sumtransfers. In a setting with heterogeneous individuals in terms of disutility to work, the labor supply functionemerges from the comparison between the nominal wage and the home sector’s payoff in each region and, therefore,it is not a direct function of the region’s price index.

35Greenland et al. (2019) find that, in response to the China shock, population responses are weak for allworking-age individuals, but are much stronger among young individuals aged below 30 years old. Our results arealso consistent with the evidence in Cadena and Kovak (2016) who find weak migration responses of U.S. nativeworkers to regional labor demand shocks, especially for non-college graduates.

33

follows from the high correlation between bilateral migration and trade shares across CZs, which

makes it hard to separately estimate the different parameters in the model.36 Given these results,

our preferred specification uses the estimates in Panel A. In Appendix A.2.1, we investigate the

sensitivity of reduced-form responses to different values of φm.

We now turn to the estimates of the reduced-form elasticities implied by the parameters in

Table 3. Table 4 reports percentiles of the empirical elasticity distribution for 2000. The top panel

reports the direct effects and the bottom panel reports the indirect effects.

The direct and indirect effects of revenue shifts are positive. Thus, a negative demand shock

in a CZ triggers reductions in wages and employment in that CZ as well as in other CZs. Since

the labor supply elasticity to wages is around two, revenue shifts affect more employment than

wages. In addition, indirect effects are typically smaller than direct effects: while the median direct

wage elasticity is 0.67, the median indirect wage elasticity is 0.002 (for employment, the median

elasticities are 1.46 and 0.003, respectively). This reflects the fact that there are 721 CZs indirectly

affecting each CZ.

Moreover, our estimates indicate that both wages and employment respond less to consumption

cost shifts than to revenue shifts. The median direct elasticity to ηCi is -0.35 for wages and -1.21

for employment. For indirect effects, the difference is even starker: the median elasticity to

consumption shifts is close to 0 for both wages and employment.

Importantly, the difference between the 90th and 10th percentiles of the estimated elasticities

suggests a large dispersion in direct and indirect effects across U.S. CZs. As discussed in Section 3,

this arises from the observed heterogeneity in the variables controlling bilateral spatial linkages.

Online Appendix A.2.2 shows that the indirect effects are increasing in the intuitive measure of

gravity links zij used in Table 1, but zij explains only a small fraction of the variation in indirect

effects. Instead, the elements of the spatial links matrix, yij and χij, explain roughly 50% of the

variation in indirect effects across pairs of CZs.

Robustness. Online Appendix A.2.1 investigates the robustness of the estimated reduced-form

elasticities to the assumptions in the baseline parametrization of spatial links in Section 5.2. In

particular, we re-estimate the specification in Panel A of Table 3 when we (i) allow for trade

imbalances, (ii) use a calibration of migration links from the literature, and (iii) impose labor

supply homogeneity in terms of the national price index (rather than the world’s average wage).

For all alternative specifications, despite the estimated parameters being different, the estimated

reduced-form elasticities are highly correlated with our baseline estimates.

36It may be possible to improve on the estimation of spatial links in labor supply by extending our empiricalstrategy to include one additional estimating equation for reduced-form responses in bilateral migration flows – asin the extension presented in Online Appendix C.2.

34

Table 4: Percentiles of the Empirical Distribution of Reduced-form Elasticities

Revenue Consumption Cost10th 50th 90th 10th 50th 90th

Panel A: Direct elasticitiesWages 0.436 0.665 1.666 -0.899 -0.349 -0.190Employment 0.924 1.461 3.965 -2.776 -1.206 -0.757

Panel B: Indirect elasticitiesWages 0.000 0.002 0.021 -0.002 0.000 0.001Employment 0.000 0.003 0.039 -0.008 -0.001 0.000

Notes: Percentiles of the 2000 empirical distribution of reduced-form elasticities implied by the estimates in Panel A of Table 3.

6.4.3 Model Fit

Finally, we investigate how observed changes in employment and wages across CZs relate to our

baseline predicted responses to the China shock. As a benchmark, we compare this relationship to

that obtained with alternative specifications of spatial links motivated by the existing literature.

Specifically, we consider the following linear regression:

Y ti = αt + ρY t

i (IPW , IPC|θ) +X tiλ+ νti (39)

where, in CZ i in period t, Y ti (IPW , IPC|θ) is the response to the China shock implied by our

model with parameter vector θ, and X ti is the same control set used in Table 1.

The coefficient ρ summarizes the relationship between actual and predicted changes in labor

market outcomes across CZs. A coefficient of one means that predicted reduced-form responses

have the correct magnitude to match cross-regional variation in wage and employment growth. In

contrast, an estimated coefficient much larger than one implies that reduced-form responses in the

model need to be multiplied by a large re-scaling coefficient ρ to match the observed variation in

wage and employment growth between CZs. In this case, the predicted responses in the model are

too small compared to the differential effect of higher shock exposure observed in the data. Finally,

a non-significant coefficient indicates that the model’s predicted responses are not correlated with

the observed changes in labor market outcomes.

Table 5 reports the estimates of (39) under alternative parameterizations of our model. The

coefficients close to one in column (1) indicate that our baseline estimates yield reduced-form

responses to the China shock that are aligned with observed changes in wages and employment for

U.S. CZs. This is a consequence of the fact that we estimate θ precisely from the effect of shock

exposure on employment and wages, as specified in equation (37). Column (2) reports similar fit

35

Table 5: Predicted Impact of China Shock and Actual Labor Market Outcomes across U.S. CZs

Structural Estimates Alternative Calibrations(1) (2) (3) (4) (5) (6)

Panel A: Change in avg. weekly log wage

Predicted response 0.67** 0.66** 3.56** 3.97** 3.70** 3.72**(0.27) (0.26) (1.50) (1.76) (1.57) (1.57)


Predicted response 0.90*** 0.84*** 6.60*** 8.95*** 10.42 9.60(0.14) (0.16) (1.74) (2.45) (6.55) (6.29)

Parameters:

ψ 0.56 0.55 0.20 0.00 0.00 0.00φw 2.11 2.11 2.11 2.11 0.70 0.70φp -1.36 -1.36 -1.36 -1.36 -0.70 -0.70ε 3.94 3.94 3.94 3.94 3.94 3.94φm 0.00 -0.06 0.00 0.00 0.00 0.25

Notes: Pooled sample of 1,444 Commuting Zones in 1990-2000 and 2000-2007. All specifications include the set of baseline controls inTable 1. Robust standard errors in parentheses are clustered by state. *** p < 0.01, ** p < 0.05, * p < 0.10

coefficients when we consider the alternative specification with migration links.37

Columns (3) and (4) show how the fit coefficients change when we calibrate agglomeration

forces to be weaker. In column (3), we set ψ = 0.2 as in the increasing returns framework of

Krugman (1980) and, more recently, Monte et al. (2018) and, in column (4), we set ψ = 0 as in

the Ricardian framework of Eaton and Kortum (2002) and, more recently, in Galle et al. (2017),

Caliendo et al. (2018) and Caliendo et al. (2019). The fit coefficient is substantially higher when ψ

is lower. As discussed in Section 3, higher values of ψ amplify the reduced-form responses in the

model. Thus, as we reduce ψ, a higher ρ is necessary to match the large differential responses to

shock exposure that we observed across U.S. CZs.

Columns (5) and (6) investigate how the specification of labor supply links affects the model

fit. In column (5), we set φw = −φp = 0.7, so that the CZ’s labor supply responds to changes

in the local real wage with an elasticity given by the median estimate reported by Chetty et al.

(2013). This parametrization is consistent with that in Caliendo et al. (2019) who specify the

non-employment benefit in terms of the local price index – see equation (60) in their Appendix

C5. In this case, the fit coefficient for employment growth is very imprecise, indicating that this

specification greatly reduces the model’s ability to match cross-regional variation in employment

responses to shock exposure. As discussed in Section 3, by setting φw = −φp, we increase the

relative magnitude of the positive employment response to cheaper Chinese imports (relative to

37As a robustness check, Table A.10 in Online Appendix A.2.1 shows that our estimates of the fit coefficients aresimilar when we consider the alternative specifications of spatial links described in Section 6.4.2.

36

Figure 2: Predicted Impact of China Shock and Actual Labor Market Outcomes across U.S. CZs

Notes: Bin scatter plot of predicted and actual changes in avg. weekly log wages (left panel) and log of employment (right panel)after partialling out the baseline controls in Table 1. Pooled sample of 1,444 Commuting Zones in 1990-2000 and 2000-2007. Plotsreport average predicted and actual changes in percentile bins based on predicted changes. Baseline predicted changes are computedwith the reduced-form responses in equation (37) using estimates in Panel A of Table 3. Alternative calibration computed with thereduced-form responses in equation (37) using parameters in the corresponding columns of Table 5.

the negative response to the decline in revenue). However, as Table 1 shows, the impact of higher

consumption exposure on regional outcomes is not statistically significant at usual level. Lastly,

column (6) sets the elasticity of migration links in labor supply to 0.25 – an elasticity similar to the

(annual) estimate of the migration elasticity in Caliendo et al. (2019). We can see that allowing for

a larger migration elasticity does not affect much the fit of the model.

Figure 2 graphically illustrates the results reported in Table 5 using bin scatter plots of actual

changes in labor market outcomes and predicted responses to the China shock across U.S. CZs.

In line with the estimates in column (1) of Table 5, the black circles show that our baseline

predicted responses are around the 45o degree line, indicating that they are not only correlated

with actual changes in employment and wages across CZs, but also their magnitude is consistent

with the differential regional responses in labor market outcomes. We also report the relationship

for the alternative specifications in columns (3) and (6) of Table 5. In both cases, the points are

concentrated around zero, indicating that the magnitude of these predicted responses is too small

compared to differential changes in labor market outcomes across U.S. CZs.38

38Online Appendix A.2.2 investigates whether different specifications of spatial links can match the findings inADH: namely, the magnitude of the differential employment response to higher regional exposure to Chinese importcompetition. Our baseline estimates yield differential responses that are similar to those in column (1) of Table 1.In contrast, the alternative specifications in columns (3) and (6) of Table 5 yield much smaller differential responsesthat are less than one-fourth of those reported in column (1) of Table 1.

37

6.5 The Impact of the China Shock in General Equilibrium

We conclude our analysis by quantifying the impact of the China shock on U.S. CZs. Based on

the theoretical results in Sections 3 and 4, we directly aggregate our estimates of equation (37) to

obtain responses of employment and wages in general equilibrium. We then use expression (23) to

compute the real wage response in each CZ. Table 6 reports the average and standard deviation of

shock-induced changes in wages (relative to the Chinese wage), employment and real wages.39 We

consider the predicted responses over the entire period between 1990 and 2007.40

The first column of Table 6 shows that, on average, the wage of U.S. CZs fell by 4 log-points

relative to the Chinese wage. Most of the wage decline was driven by the indirect effect of negative

revenue shifts in other CZs. This is a consequence of our finding that indirect effects reinforce

direct effects. The impact of lower consumption costs is positive, but it is not sufficient to offset

the impact of the revenue decline. This follows from the fact that consumption elasticities are

smaller than revenue elasticities. Interestingly, our results indicate large differential effects across

CZs: all component of wage responses exhibit large cross-regional dispersion.

We then turn to the effect of the China shock on employment. Again, revenue shifts lead to

average employment losses, but consumption cost shifts create partially offsetting employment

gains. The sum of all components yields an average employment loss of 2.8 log-points between

1990 and 2007. The impact of the shock on employment varied greatly across U.S. CZs as can be

seen by the large standard deviation of employment responses.

The last two columns report real wage responses across CZs. On average, the China shock

generated a small real gain of 0.16 log-points. This gain was mostly driven by the reduction in

import prices that is captured by the large positive effect of the CZ’s own consumption cost shift.

This more than compensates for the loss in terms-of-trade that followed from the wage reduction

induced by the revenue decline. Notice that, again, the large standard deviation indicates that

these consumption gains varied substantially across CZs. In fact, 39% of U.S. CZs experienced

declines in real wages due to the China shock.

Robustness. Online Appendix A.3 investigates the robustness of our counterfactual predictions

to the alternative specifications of spatial links discussed in Section 6.3. All alternative specifications

yield predicted responses that have a high correlation with the baseline responses. Similar to

the baseline, the average wage decline is close to 4 log-points in all cases. However, the average

employment decline may be stronger or weaker depending on the specification. When we allow for

trade imbalances and migration links, the average employment losses are respectively -4.1 and -5.4

39Recall that, by Walras’ law, only relative wages are determined in general equilibrium (see discussion in Sections2 and 3). Without loss of generality, we specify the Chinese wage as the numeraire. This implies that our predictedwage changes must be interpreted as relative to the Chinese wage.

40Online Appendix A.3 reports the same statistics for 1990-2000 and 2000-2007. We find that most of the impactof the China shock happened in the second period, after China’s accession to WTO in 2001.

38

Table 6: Effect of the China Shock on U.S. CZs, 1990-2007

Response in the log ofWage Employment Real wage

Avg. St. Dev. Avg. St. Dev. Avg. St. Dev.

Total effect -3.98 1.30 -2.78 3.31 0.16 1.75

Direct effect of ηR -0.81 1.79 -1.94 4.75 -0.98 2.53Direct effect of ηC 0.98 1.36 3.18 3.92 3.14 2.23Indirect effect of ηR -4.24 1.71 -4.95 4.59 -2.88 2.44Indirect effect of ηC 0.09 1.18 0.93 3.38 0.88 1.84

Notes: Predicted changes in employment and wages computed with the reduced-form responses in equation (37) using estimates inPanel A of Table 3. Prediced real wage change computed with expression (23).

log-points. When we specify the labor supply normalization in terms of the U.S. price index, the

employment decline is only 0.5 log-points because this specification entails stronger employment

gains to lower import prices.

6.5.1 Comparison to other specifications of spatial links

In Table 7, we compare our baseline estimates of the effect of the China shock on U.S. CZs

to those obtained from two types of alternative specifications of spatial links in the literature.

Panel A presents changes in employment and wages implied by the simple specifications of

spatial links embedded in the linear regressions presented in Table 1. That is, we aggregate the

predicted responses implied by the linear specification in ADH and its extension with intuitive

parametrizations of indirect effects in equation (38).41 Panel B reports changes in employment and

wages that we compute using the alternative calibrations of spatial links used in Table 5.

The first row of Panel A shows that the simple aggregation of ADH’s specification (column (1)

of Table 1) implies average reductions of 1.2 log-points in wages and 1.5 log-points in employment.

These average changes are less than half of those implied by our baseline specification. The second

row of Panel A shows that, when we extend ADH’s specification by including intuitive measures

of indirect effects (column (3) of Table 1), the average predicted response is much closer to our

baseline average response. This is a consequence of the fact that both our baseline estimates and

the extension of ADH’s specification yield indirect effects that reinforce the negative impact of

the local exposure to import competition. Notice however that the correlations with our baseline

predicted responses are below 0.5, indicating that these simple extensions still miss an important

fraction of the heterogeneity in predicted responses of wages and employment across CZs.

41We only consider specifications that ignore consumption cost shifts since the effect of these variables is notstatistically significant (see Table 1). Thus, researchers using the reduced-form results in Table 1 would concludethat consumption cost shifts are not relevant drivers of labor market outcomes across CZs.

39

In Panel B, we compare our baseline results to those implied by two alternative calibrations

of spatial links used in Table 5. In the first row, we eliminate agglomeration forces, but keep

other parameters unchanged. In this case, predicted responses remain highly correlated with

our baseline responses. However, they become substantially smaller: the standard deviation of

predicted responses falls by more than 80%. In fact, the average employment loss is close to zero.

This results from the removal of the amplification channel of agglomeration forces (see Section 3).

This is also consistent with the model fit coefficients in column (4) of Table 5, which suggest that

employment responses were too small, such that they had to be multiplied by a factor of nine to

match the cross-regional variation in the data.

In the last row of Panel B, we further change the labor supply specification imposing that

−φp = φw = 0.7, as in column (5) of Table 5. In this case, the positive impact of lower import

prices on employment dominates in the aggregate, implying a small increase in average employment.

Across CZs, the correlation with our baseline predicted responses falls substantially – it was 0.65

in the first row, but it is only 0.35 in the second row. This is because this parametrization induces

responses to import prices that are too strong compared to those in the data.

Table 7: Effect of the China Shock on U.S. CZs, 1990-2007 – Alternative Approaches

Response in the log ofWage Employment

Avg. St. Dev. Corr. Avg. St. Dev. Corr.

Baseline -3.98 1.30 1 -2.78 3.31 1

Panel A: Aggregation of ADH extensions

Table 1 column (1) -1.17 1.10 0.47 -1.50 1.42 0.42Table 1 column (3) -3.55 2.12 0.47 -4.51 2.70 0.42

Panel B: Alternative calibrations

Table 5 column (4) -3.17 0.21 0.78 -0.40 0.24 0.65Table 5 column (5) -3.28 0.23 0.77 0.23 0.08 0.35

Notes: Baseline predicted changes are computed with the reduced-form responses in equation (37) using estimates in Panel A of Table 3.In panel A, predicted changes are computed with the linear specification in (38) using estimates in the corresponding columns of Table1. In panel B, predicted changes are computed with the reduced-form responses in equation (37) using parameters in the correspondingcolumns of Table 5. For each row, column ‘‘Corr.’’ indicates the correlation between predicted responses implied by the specificationin the row and our baseline predicted changes.

7 Conclusions

For a general class of spatial models, we show that changes in labor market outcomes, as a result

of foreign shocks, can be written as the product of two components. First, the partial equilibrium

impact of trade shocks on markets, which takes the form of shift-share variables. Second, the

general equilibrium reduced-form elasticities that summarize how local outcomes respond directly

40

to the local shock exposure and indirectly to the shock exposure of other markets through spatial

links. We use these results to propose a novel estimation strategy that extends existing empirical

shift-share specifications to measure the general equilibrium impact of trade shocks on local labor

markets. Our empirical strategy recovers the general equilibrium reduced-form elasticities using

the response of labor market outcomes to the exposure of different regions to observed trade

shocks, such as the China shock in ADH. These elasticities can then be used to aggregate the

shock exposure of different markets to compute the general equilibrium impact of trade shocks on

employment, wages, and real wages.

While our methodology differs from existing quantitative approaches, our empirical findings are

ultimately ones that quantitative analyses need to reckon with. For that purpose, we devise a new

model validation procedure that makes additional use of cross-regional variation in shock exposure.

The procedure uncovers the ability of different variants of the model to come to grips with empirical

estimates of the differential response of labor market outcomes in regions with higher exposure

to observed trade shocks. We feel, to some extent, that this procedure achieves the standards

set by Kehoe (2005): ‘‘Ex-post performance evaluations of applied GE models are essential if

policymakers are to have confidence in the results produced by these models. Such evaluations also

help make applied GE analysis a scientific discipline in which there are well-defined puzzles with

clear successes and failures for competing theories’’. We hope, therefore, that our approach will

lead to a better understanding of the role of spatial links in shaping the impact of trade shocks on

regional labor markets.

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48

8 Proofs

8.1 Proof of Proposition 1

We now establish the existence and uniqueness of the equilibrium wage vector of our economy. Considerthe solution w∗ of the system Di(w

∗|τ ) = 0 for all i, and the following two lemmas.

Lemma 1. [Arrow and Hahn (1971) T.1.3 (p. 33)] Suppose that Fi(·) is a function defined for everyw ∈ RN++ such that Fi(·) is (i) differentiable, (ii) homogeneous of degree 0, (iii) satisfies Walras’ law,∑N

i=1wiFi(w) = 0 for all w, (iv) there exists a scalar s such that Fi(w) < s for every w, and (v) ifwn → w with wm = 0, then

∑i Fi(w

n)→ −∞. Then, there exists w∗ ∈ RN++ such that Fi(w∗) = 0 for

all i.

Lemma 2. [Arrow and Hahn (1971) T.9.12 (p. 234)] Suppose that Fi(·) satisfies the conditions in

Lemma 1. Assume that, for any w∗ ∈ RN+ with Fi(w∗) = 0 for all i, fij(w

∗) ≡ ∂Fi(w∗)

∂wjsatisfies (i)

fii(w∗) > 0 and (ii) ∃hi(w∗)i,m 0 such that hi(w

∗)fii(w∗) >

∑j 6=i,m |fij(w∗)|hj(w∗) for all i 6= m.

Then, there is a unique w∗ ∈ RN++ such that Fi(w∗) = 0 for all i.

We consider the function Fi (w) ≡ − 1wiDi(w|τ ) with Di(w|τ ) defined by (9) and P (w|τ ) implicitly

defined as the solution of (8) with po given by (5). In this proof, we simplify notation by denotingP (w|τ ) as P (w). Notice that the definition of P (w) implies that P (κw) = κP (w). We now establishthe existence and uniqueness of the equilibrium wage vector by verifying that Assumption 1 implies allconditions in Lemmas 1 and 2.

1. The function Fi(w) is differentiable because it only combines differentiable functions.

2. We now verify that the system is homogeneous of degree 0. Since Fi(κw) = − 1κwi

Di(κw|τ ),

Fi(κw) = Φi(κw,P (κw))−∑j

∑s∈Si

(τij,s

κwiΨi(Φ(κw,P (κw)))

)−εsξjs∑

o

(τoj,s

κwoΨo(Φ(κw,P (κw)))

)−εs κwjΦj(κw,P (κw))

κwi.

As discussed above, P (κw|τ ) = κP (w|τ ). Thus,

Fi(κw) = Φi(κw, κP (w))−∑j

∑s∈Si

(τij,s

wiΨi(Φ(κw,κP (w)))

)−εs∑

o

(τoj,s

woΨo(Φ(κw,κP (w)))

)−εs ξjswjΦj(κw, κP (w))

wi.

This implies that Fi(κw) = Fi(w) because Φi(κw, κP ) = Φi(w,P ) for all i.

3. We now verify that Walras’ law holds:∑iwiFi(w) =

∑i

[wiΦi(w,P (w))−

∑j xij

(wo

Ψo(Φ(w,P (w)))

o|τ)wjΦj(w,P (w))

]=

∑iwiΦi(w,P (w))−

∑j

[∑i xij

(wo

Ψo(Φ(w,P (w)))

o|τ)]wjΦj(w,P (w))

=∑

iwiΦi(w,P (w))−∑

j wjΦj(w,P (w))

= 0

49

4. By assumption, Φi(w,P (w)) is bounded from above, so Fi(w) ≤ Φi(w,P (w)) < Φ for all i.

5. Let w be a real vector with wm = 0. Since Φi(w,P (w)) is bounded from above by assumption,

limwn→w Φi(wn,P (wn)) ∈ [0, Φ] for all i. Notice also that the assumption of limwm→0

Ψm(Φ(w,P (w)))wm

=∞ implies that

limwn→w

(τmj,s

wmΨm(Φ(w,P (w)))

)−εs∑

o

(τoj,s

woΨo(Φ(w,P (w)))

)−εs = limwn→w

(Ψm(Φ(w,P (w)))

τmj,swm

)εs∑

o

(Ψo(Φ(w,P (w)))

τoj,swo

)εs = 1.

So,

limwn→w

∑j

∑s∈Si

(τmj,s

wmΨm(Φ(w,P (w)))

)−εs∑

o

(τoj,s

woΨo(Φ(w,P (w)))

)−εs ξjswjΦj(w,P (w)) = limwn→w

∑j

wjΦj(w,P (w)) > 0

and, therefore,

limwn→w

1

wm

∑j

∑s∈Si

(τmj,s

wmΨm(Φ(w,P (w)))

)−εs∑

o:s∈So

(τoj,s

woΨo(Φ(w,P (w)))

)−εs ξjs (wjΦj(w,P (w))) =∞.

Because Fi(w) is bounded from above for all i (see condition 4 above), this implies that limwn→w Fm(w) =−∞ and, therefore, limwn→w

∑i Fi(w) = −∞.

6. For any equilibrium wage vector w∗,

fij(w∗) ≡ ∂Fi(w

∗)

∂wj= I[i=j]

1

(w∗i )2Di(w

∗|τ )− 1

w∗i

∂Di(w∗|τ )

∂wj=

Y ∗iw∗iw

∗j

γij

where the last equality follows from Di(w∗|τ ) = 0 and the definition γ ≡ −[Y −1

i ∇lnwDi(w|τ )] inAssumption 1. Since Assumption 1 imposes that γii > 0, we have that fii(w

∗) > 0. In addition,let hii 6=m be the vector such that hiγii >

∑j 6=i,m |γij |hj for all i 6= m (by Assumption 1). If

hi(w∗) ≡ hiw∗i , then

hi(w∗)fii(w

∗) =Yiw∗ihiγii >

Yiw∗i

∑j 6=i,m

|γij |hj =∑j 6=i,m

|fij(w∗)|hj(w∗).


Consumption price shift. From the price index definition in (8), ∂ lnPi(p|τ )∂ ln τoj,s

= 0 for all i 6= j and

∂ lnPi(p|τ )

∂ ln τoi,s= −ξi,s

εs

∂

∂ ln τoi,s

∑o′:s∈So′

(τo′i,spo′

)−εs = ξi,s(τoi,spo)

−εs∑o′:s∈So′

(τo′i,spo′

)−εsBy the definition of ηCi (τ ),

ηCi (τ ) ≡∑o,s

∂ lnPi(p0|τ 0)

∂ ln τoi,sτoi,s =

∑o,s

ξi,sx0oi,sτoi,s,

50

which is equivalent to ηCi (τ ) in (12).

Revenue shift. The definitions Yi(p,E|τ ) ≡∑

k xik(p|τ )Ek and xik(p|τ ) in (6) imply that ∂ lnYi(p,E|τ )∂ ln τoj,s

=

0 for s /∈ Si. The same definitions also imply that, for all s ∈ Si,

∂ lnYi(p,E|τ )∂ ln τoj,s

=∑

k∂ lnxik(p|τ )∂ ln τoj,s

xik(p|τ )Ek∑k′ xik′ (p|τ )Ek′

= (−εs)(I[i=o] − xoj,s(p|τ )

) xij,s(p|τ )ξj,sxij(p|τ )

xij(p|τ )Ej∑k xik(p|τ )Ek

where the second equality uses the fact that ∂ lnxik(p|τ )∂ ln τoj,s

= 0 for all k 6= j.

From the definition of ηRi (τ ),

ηRi (τ ) =∑j,o,s

∂ lnYi(p0,E0|τ 0)

∂ ln τoj,sτoj,s =

∑j,o,s

x0ij,sξj,s

x0ij

x0ijE

0j

Y 0i

(x0oj,s − I[i=o]

)εsτoj,s.

We obtain ηRi (τ ) in (12) by noting that the share of j in the revenues of i is y0ij ≡ x0

ijE0j /Y

0i and the

share of sector s in the sales of i to j is y0ij,s ≡ x0

ij,sξj,sE0j /x

0ijE

0j = x0

ij,sξj,s/x0ij .

8.3 Proof of Theorem 1

In order to characterize the shift in excess labor demand, we define the following two functions. First, wedefine the price index of i as a function of w and τ :

Pj = Πs

[ ∑o:s∈So

(τoj,s

woΨo (Φ(w,P ))

)−εs] ξj,s−εsfor all j. (40)

Second, we define the revenue of i as a function of w and τ :

Yi(w|τ ) ≡ Yi

(wj

Ψj(Φ(w,P (w|τ )))

j,wjΦj(w,P (w|τ ))

j|τ)

=∑

j

[∑s∈Si

(τij,swi

Ψi(Φ(w,P (w|τ)))

)−εs∑o:s∈So

(τoj,swo

Ψo(Φ(w,P (w|τ)))

)−εs ξj,s]wjΦj(w,P (w|τ )).

(41)

Using these definitions, the excess demand function in market i can be written as

Di(w|τ ) = Yi(w|τ )− wiΦi(w,P (w|τ )). (42)

We now characterize∂lnPj(w|τ )∂lnτod,s

using the implicit function theorem and the system in (40). By defining

pj(w|τ ) ≡ wj/Ψj(Φ(w,P )),

∂lnPj(w0|τ 0)

∂lnτod,s= I[d=j]ξj,sx

0oj,s +

∑o′

(∑k

ξj,kx0o′j,k

)∂ ln po′(w

0|τ 0)

∂ ln τod,s

where∂ ln po′(w

0|τ 0)

∂ ln τod,s= −

∑j′

∂lnΨo′(L0)

∂ lnLj′

∑k′

∂lnΦj′(w0,P 0)

∂ lnPk′

∂lnPk′(w0|τ 0)

∂lnτod,s. (43)

51

Thus, the elasticity matrices defined in (2) and (4) imply that

∂lnPj(w0|τ 0)

∂lnτod,s= I[d=j]ξj,sx

0oj,s −

∑o′

∑j′

x0o′jψo′j′

∑k′

φpj′k′∂lnPk′(w

0|τ 0)

∂lnτod,s,

where we have used the fact that x0o′j =

∑k ξj,kx

0o′j,k.

Define the vector xod,s with d-th entry equal to ξd,sx0od,s and all other entries equal to zero. In matrix

notation,∇ln τod,s lnP (w0|τ 0) = xod,s − x0′ψφ

p∇ln τod,s lnP (w0|τ 0)

and, therefore,

∇ln τod,s lnP (w0|τ 0) =(I + x0′ψφ

p)−1xod,s.

The combination of this expression and the definition of ηCi in (12) yields the total change in the priceindex vector:

∑o,d,s

(∇ln τod,s lnP (w0|τ 0)

)τod,s =

(I + x0′ψφ

p)−1[∑o,s

ξj,sx0oj,sτoj,s]j =

(I + x0′ψφ

p)−1ηC . (44)

We now use the total change in the price index vector to derive the excess labor demand shift. Theexpression of Di(w|τ ) in (42) implies that

∂Di(w|τ )

∂lnτod,s

1

Yi(w|τ )=∂ ln Yi(w|τ )

∂ ln τod,s− wiΦi(w,P (w|τ ))

Yi(w|τ )

∑j

∂lnΦi(w,P )

∂ lnPj

∂lnPj(w|τ )

∂lnτod,s(45)

We then combine this expression with the definition of Yi(w|τ ) in (41) to obtain

∂ ln Yi(w|τ )

∂ ln τod,s=∂ lnYi(p,E|τ )

∂ ln τod,s+∑j

∂ lnYi(p,E|τ )

∂ ln pj

∂ ln pj(w|τ )

∂ ln τod,s+∑j

∂ lnYi(p,E|τ )

∂ lnEj

∂ lnEj(w|τ )

∂ ln τod,s, (46)

where Ej(w|τ ) ≡ wjΦj(w,P (w|τ )) and

∂ lnEj(w|τ )

∂ ln τod,s=∑j′

∂lnΦj(w,P )

∂ lnPj′

∂lnPj′(w|τ )

∂lnτod,s.

Consider the elasticity matrices defined in (2), (4) and (7). By combining these definitions withexpressions (43) and (46), we obtain that

∂ ln Yi(w0|τ0)

∂ ln τod,s= ∂ lnYi(p

0,E0|τ )∂ ln τod,s

−∑

j χij∑

j′ ψjj′∑

k φpj′k

∂lnPk(w0|τ0)∂lnτod,s

+∑

j y0ij

∑j′ φ

pjj′

∂lnPj′ (w0|τ0)

∂lnτod,s.

(47)

Since Yi(w0|τ 0) = Y 0

i = w0iL

0i = w0

i Φi(w0,P (w0|τ 0)), (45) implies that

∑o,d,s

∂Di(w0|τ 0)

∂lnτod,s

τod,sY 0i

=∑o,d,s

∂ ln Yi(w0|τ 0)

∂ ln τod,sτod,s −

∑o,d,s

∑j

φpij∂lnPj(w

0|τ 0)

∂lnτod,sτod,s

52

and, by (47),∑o,d,s

∂Di(w0|τ0)

∂lnτod,s

τod,sY 0i

=∑

o,d,s∂ lnYi(p,E|τ )∂ ln τod,s

τod,s −∑

j χij∑

j′ ψjj′∑

k φpj′k

(∑o,d,s

∂lnPk(w0|τ0)∂lnτod,s

τod,s

)+∑

j y0ij

∑j′ φ

pjj′

(∑o,d,s

∂lnPj′ (w0|τ0)

∂lnτod,sτod,s

)−∑

j φpij

(∑o,d,s

∂lnPj(w0|τ0)

∂lnτod,sτod,s

).

We use the definition of ηRi in (12) to write the stacked vector of excess demand shifts:

η = ηR −(I − y0 + χψ

)φp∑o,d,s

(∇ln τod,s lnP (w0|τ 0)

)τod,s,

which, by the expression in (44), is equivalent to

η = ηR −(I − y0 + χψ

)φp (I + x0′ψφ

p)−1ηC .

Finally, note that

φp (I + x0′ψφ

p)−1=

((I + x0′ψφ

p) (φp)−1

)−1

=((φp)−1 (

I + φpx0′ψ

))−1

=(I + φ

px0′ψ

)−1φp.

Thus,

η = ηR −(I − y0 + χψ

) (I + φ

px0′ψ

)−1φpηC ,

which corresponds to equation (14) given the definition of α ≡(I − y0 + χψ

) (I + φ

px0′ψ

)−1.


We start by characterizing∂lnPj(w|τ )∂lnwo

using the implicit function theorem and the system in (40):

∂lnPj(w0|τ 0)

∂lnwo=∑o′

(∑k

ξj,kx0o′j,k

)∂ ln po′(w

0|τ 0)

∂ lnwo=∑o′

x0o′j

∂ ln po′(w0|τ 0)

∂ lnwo

where

∂ ln po′(w|τ )

∂ lnwo= I[o′=o] −

∑j′

∂lnΨo′(L)

∂ lnLj′

(∂lnΦj′(w,P )

∂ lnwo+∑k′

∂lnΦj′(w,P )

∂ lnPk′

∂lnPk′(w|τ )

∂lnwo

). (48)

Thus, the elasticity matrices defined in (2) and (4) imply that

∂lnPj(w0|τ 0)

∂lnwo= x0

oj −∑o′

∑j′

x0o′jψo′j′

(φwj′o +

∑k′

φpj′k′∂lnPk′(w

0|τ 0)

∂lnwo

).

Define the vectors xo ≡ [x0oj ]j and φo ≡ [φwjo]j . In matrix notation,

∇lnwo lnP = xo − x0′ψφo − x0′ψφp∇lnwo lnP .

53

Thus, the Jacobian of the price index with respect to lnw is

∇lnw lnP (w0|τ 0) =(I + x0′ψφ

p)−1x0′ (I − ψφw) . (49)

We now derive the Jacobian of the excess labor demand with respect to lnw. The expression ofDi(w|τ ) in (42) implies that

∂Di(w0|τ 0)

∂ lnwo

1

Y 0i

=∂ ln Yi(w

0|τ 0)

∂ lnwo−

I[i=o] +∂lnΦi(w

0,P 0)

∂ lnwo+∑j

∂lnΦi(w0,P 0)

∂ lnPj

∂lnPj(w0|τ 0)

∂ lnwo

. (50)

The definition of Yi(w|τ ) in (41) yields

∂ ln Yi(w|τ )

∂ lnwo=∑j

∂ lnYi(p,E|τ )

∂ ln pj

∂ ln pj(w|τ )

∂ lnwo+∑j

∂ lnYi(p,E|τ )

∂ lnEj

∂ lnEj(w|τ )

∂ lnwo, (51)

where∂ lnEj(w|τ )

∂ lnwo= I[j=o] +

∂lnΦj(w,P )

∂ lnwo+∑j′

∂lnΦj(w,P )

∂ lnPj′

∂lnPj′(w|τ )

∂lnwo.

Consider the elasticity matrices defined in (2), (4) and (7). By combining these definitions withexpression (51), we obtain that

∂ ln Yi(w0|τ0)

∂ lnwo= χio −

∑j χij

∑j′ ψjj′

(φwj′o +

∑k φ

pj′k

∂lnPk(w0|τ0)∂lnwo

)+ y0

io +∑

j y0ijφ

wjo +

∑j y

0ij

∑j′ φ

pjj′

∂lnPj′ (w0|τ0)

∂lnwo.

(52)

The combination of (50) and (52) implies that

∂Di(w0|τ0)

∂lnwo1Y 0i

= χio −∑

j χij∑

j′ ψjj′(φwj′o +

∑k φ

pj′k

∂lnPk(w0|τ0)∂lnwo

)+ y0

io +∑

j y0ijφ

wjo

+∑

j y0ij

∑j′ φ

pjj′

∂lnPj′ (w0|τ0)

∂lnwo−(I[i=o] + φwio +

∑j φ

pij∂lnPj(w

0|τ0)∂ lnwo

).

Since γ ≡ [−∂Di(w0|τ0)

∂lnwo1Y 0i

]i,o, we have that

γ = −χ0 + χ0ψ(φw

+ φp∇lnw lnP (w0|τ 0)

)− y0 − y0φ

w

−y0φp∇lnw lnP (w0|τ 0) +

(I + φ

w+ φ

p∇lnw lnP (w0|τ 0)),

which can be written as

γ = I −(y0 + χ0

)+(I − y0 + χ0ψ

) (φw

+ φp∇lnw lnP (w0|τ 0)

).

We now substitute for ∇lnw lnP (w0|τ 0) using (49) to obtain

γ = I −(y0 + χ0

)+

(I − y0 + χ0ψ

)φp (I + x0′ψφ

p)−1x0′

+(I − y0 + χ0ψ

) (φw − φp

(I + x0′ψφ

p)−1x0′ψφ

w).

As shown in Appendix 8.3, φp (I + x0′ψφ

p)−1=(I + φ

px0′ψ

)−1φp. This relationship and the

54

definition of α imply that

γ = I −(y0 + χ0

)+ αφ

px0′(

I − y0 + χ0ψ) (φw −

(I + φ

px0′ψ

)−1φpx0′ψφ

w).

By re-arranging the term in the second row, we obtain

γ = I −(y0 + χ0

)+ αφ

px0′

α((I + φ

px0′ψ

)− φpx0′ψ

)φw,

which is equivalent to the expression for γ in (16).

8.5 Proof of Theorem 2

We re-define the system in (11) to set the change in the wage of market m to zero. Consider the matrixM obtained by deleting the m-th row from the identity matrix with dimension equal to the number ofmarkets. If MγM

′is nonsingular, then we can write

Mw =(MγM

′)−1

Mη,

which yields the representation in (17) when we define β ≡ M ′(MγM

′)−1M .

In the rest of the proof, we first show that MγM′

is nonsingular and then establish that β admitsthe series representation in (18). To simplify exposition, we abuse notation by defining

γ ≡ MγM′, w ≡ Mw and η ≡ Mη.

This modified system does not include the row associated with the market clearing condition of marketm and imposes that wm = 0. To obtain a characterization for the solution of this system, let κ be thediagonal matrix with the diagonal elements of γ: κ = [κij ] s.t. κii = γii and κij = 0 for i 6= j. Thus,we can write the system as

γ = κ(I − ¯γ

)st ¯γ ≡ I − κ−1γ,

which implies that γii = 0 and γij = −γij/γii.Consider the vector hii 6=m 0 that guarantees the diagonal dominance of γ in the initial equilibrium.

Let h be the diagonal matrix such that hi is the diagonal entry in row i. Thus, the system in (11) isequivalent to

κ(I − ¯γ

) (hh−1)w = η

κ(h− ¯γh

)h−1w = η(

κh) (I −

(h−1 ¯γh

))h−1w = η

which implies that

w = h(I − ¯γ

)−1 (κh)−1

η, ¯γ ≡ h−1 ¯γh. (53)

Notice that, for all i, ˜γii = 0 and ˜γij = −γijhj/γiihi.First, we show that (I − ¯γ) is non-singular, so that we can write the expression in (53). We proceed

by contradiction. Suppose that (I − ¯γ) is singular, so λ = 0 is an eigenvalue of (I − ¯γ). Take theeigenvector x associated with the zero eigenvalue and normalize it such that xi = 1 and |xj | ≤ 1. Notice

55

that (I − ¯γ)x = 0, so that the i-row of this system is

1 +∑j 6=i,m

−˜γijxj = 0 =⇒ 1 +∑j 6=i,m

γijγii

hjhixj = 0

Thus,

γiihi = −∑j 6=i,m

γijhjxj ≤ |∑j 6=i,m

γijhjxj | ≤∑j 6=i,m

|γij ||hj ||xj | ≤∑j 6=i,m

|γij |hj

where the last inequality holds because |xj | ≤ 1 and hj > 0. Thus, γiihi ≤∑

j 6=i,m |γij |hj , whichcontradicts Assumption 1.

Second, we show that (I − ¯γ)−1 admits the series representation in (18). This is true whenever the

largest eigenvalue of ¯γ is below one. To show this, we proceed by contradiction. Assume that the largesteigenvalue λ is weakly greater than one. Take the eigenvector x associated with the largest eigenvalueand normalize it such that xi = 1 and |xj | ≤ 1. Notice that λx = ¯γx so that the i-row of this system is

1 ≤ λ =∑j 6=i,m

−γijγii

hjhixj

Since γii and hi are positive,

γiihi ≤ −∑j 6=i,m

γijhjxj ≤ |∑j 6=i,m

γijhjxj | ≤∑j 6=i,m

|γij ||hj ||xj |

Since |xj | ≤ 1 and hj > 0,∑

j 6=i,m |γij ||hj ||xj | ≤∑

j 6=i,m |γij |hj . Thus, γiihi ≤∑

j 6=i,m |γij |hj ,which contradicts Assumption 1. Thus, the largest eigenvalue of ¯γ is below one, allowing us to write(I − ¯γ)−1 =

∑∞d=0(¯γ)d. Substituting this series expansion into (53) yields

w =∞∑d=0

(h(

¯γ)dh−1)κ−1η.

Finally, to establish the result, we now show that h(¯γ)dh−1

= (¯γ)d. We proceed by induction. For

d = 1, it is trivial to see that h(¯γ)dh−1

= ¯γ. Then,

h(

¯γ)d+1

h−1

=

(h(

¯γ)dh−1)(h¯γh

−1)

= (¯γ)d(h(h−1 ¯γh

)h−1)

= (¯γ)d+1

.

Thus,

w =∞∑d=0

(¯γ)dκ−1η,

which immediately implies the result.

8.6 One-to-one mapping between β and γ

From Appendix 8.5, we know that β = M′(MγM

′)−1

M . Since MM′

= IN−1,(MβM

′)

=(MγM

′)−1

and, therefore, MγM′=(MβM

′)−1

. This implies that knowledge of β yields knowledge

of γij for all i, j 6= m.To recover γim, recall that, from Appendix 8.1, Di(w|τ ) is homogeneous of degree one on w, implying

56

that∑

j γij = − 1Y 0i

∑j∂Di(w

0|τ0)∂wj

wj = −Di(w0|τ0)

Y 0i

. Since w0 is an equilibrium wage vector, Di(w0|τ 0) = 0

and∑

j γij = 0 for all i. So, γim = −∑

j 6=m γij .To recover γmj , recall that

∑iDi(w|τ ) = 0 for all w due to trade balance in the world economy.

Thus, for every j,∑

i Y0i γij = −

∑i∂Di(w

0|τ0)∂wj

wj = 0, implying that γmj = −∑

i 6=mY 0iY 0mγij .

8.7 Proof of Corollary 1

We start by showing that the gross substitution property implies the diagonal dominance in Assumption1. As shown in Appendix 8.6,

∑j γij = 0 for all i, which implies that γii = −

∑j 6=i γij for all i. Thus,

γij < 0 for all i 6= j implies that γii =∑

j 6=i |γij | >∑

j 6=i,m |γij | for any numeraire m. Thus, Assumption1 holds and Theorem 2 immediately implies the expression in (18). Since γij < 0 for all i 6= j, γij ≥ 0 for

all i and j and, therefore, γ(d)ij ≥ 0 for all i, j and d.

8.8 Proof of Corollary 2

We abuse notation by defining the matrix γ after dropping the numeraire market: γ ≡ γ(I − Iγ ′

),

where I is a column vector of ones and γ ≡ γjj 6=m is column vector. We verify that γ−1 =γ−1

(I + (1 + γm)−1Iγ ′

).

γ−1γ = I + (1 + γm)−1Iγ ′ − Iγ ′ − (1 + γm)−1Iγ ′Iγ ′= I + (1 + γm)−1Iγ ′ − Iγ ′ + (1 + γm)−1γmIγ ′= I

where the second equality follows from γ ′I =∑

j 6=m γj = −γm (because∑

j γij = 0 for all j).

Overall, we can write β = γ−1I + γ−1(1 + γm)−1Iγ ′. Notice that the second term in case of a gravitymodel with symmetric trade costs can be written as γ−1x−1

m xj where xj = yj is the import share of marketj, or, equivalently, the size of the market. In that case, it can be shown that βj ≡ γ−1x−1

m xj for j 6= mand βm = 0, so that

∑j βj ηj =

∑j 6=m γ

−1x−1m xj ηj . Therefore, conditional on the size of the numeraire

country, if the countries that are large have larger shocks, then the fixed effect term will be larger.

8.9 Derivation of Expressions (21) and (23)

Expression (21). The combination of the labor supply equation in (1) and the price index in (8)implies that

Li =∑j

φwijwj +∑j

φpijPj =∑j

φwijwj +∑j

φpij

(ηCj +

∑o

x0oj po

)Using the unit cost expression in (5),

Li =∑j

φwijwj +∑j

φpij

(ηCj +

∑o

x0oj

(wo −

∑o′

ψoo′Lo′

))In matrix form,

L = φww + φ

p(ηC + x0′

(w − ψL

))(I + φ

px0′ψ

)L = φ

ww + φ

p(ηC + x0′w

),

57

which immediately yields the expression in (21) with the definition ρ ≡(I + φ

px0′ψ

)−1and φ ≡(

φw

+ φpx0′).

Expression (23). By definition, Wi = wi− Pi = wi−(ηCi +

∑o x

0oipo

). Using the unit cost expression

in (5),

Wi = wi −

(ηCi +

∑o

x0oi

(wo −

∑o′

ψoo′Lo′

)).

In matrix notation,

W = w − ηC − x0′(w − ψL

),

which immediately yields Expression (23).


The asymptotic variance of the GMM estimator for any function Hi(· ) is

V (H) =(E[HtiG

ti

])−1 (E[Htiνtiνi

t′Ht′i

]) (E[HtiG

ti

])−1′(54)

where Gti ≡ E[∇θνti (θ)|ηR,t, ηC,t

]and Ht

i ≡ Hi

(ηR,t, ηC,t

).

From (29), the gradient of νti (θ) with respect to θ is

Gti ≡ E[∇θνti (θ) |ηR,t, ηC,t

]= −

∑j

[∇θεβRij (θ)

∇θvϕRij (θ)

]ηR,tj −

∑j

[∇θβCij (θ)

∇θϕCij (θ)

]ηC,tj . (55)

We now show that function that minimizes the asymptotic variance is

H∗i

(ηR,t, ηC,t

)≡ Gti

(Ωti

)−1 (56)

where Ωti ≡ E

[νti (θ) νti (θ)′ |ηR,t, ηC,t

]. For this function, the asymptotic variance is

V (H∗) =(E[Gi

t′ (Ωti

)−1Gti

])−1. (57)

To establish the result, we show that V (H)− V (H∗) is positive semi-definite for any Hi(· ):

V (H)− V (H∗) =(E[HtiG

ti

])−1(E[(Htiνti

) (Htiνti

)′]) (E[HtiG

ti

])−1 ′ −(E[Gi

t′ (Ωti

)−1Gti

])−1

=(E[HtiG

ti

])−1(E[(Htiνti

) (Htiνti

)′]− E [HtiG

ti

] (E[Gi

t′ (Ωti

)−1Gti

])−1E[HtiG

ti

]′)(E[HtiG

ti

])−1 ′.

Let us define

U ti ≡ Htiνti − E

[(Htiνti

) (Gi

t′ (Ωti

)−1vti

)′](E[Gi

t′ (Ωti

)−1Gti

])−1Gi

t′ (Ωti

)−1vti ,

which implies that

E[U tiUi

t′] = E[(Htiνti

) (Htiνti

) ′]− E [HtiG

ti

] (E[Gi

t′ (Ωti

)−1Gti

])−1E[HtiG

ti

] ′.Therefore,

58

V (H)− V (H∗) =(E[HtiG

ti

])−1 (E[U tiUi

t′]) (E [HtiG

ti

])−1′.

Since E[U tiUi

t′] is positive semi-definite, V (H) − V (H∗) is also positive semi-definite. Therefore, theasymptotic variance is minimized at H∗.


We use the strategy in Section 6.1 of Newey and McFadden (1994) to establish asymptotic properties oftwo-step estimators. To this end, we define the joint moment equation for the two estimating steps:

(θ2, θ1

)≡ arg min

θ2,θ1

∑i,t

eti (θ2,θ1)

′∑i,t

eti (θ2,θ1)

(58)

where

eti (θ2,θ1) ≡[H∗i (ηR,t, ηC,t|θ1)vti(θ2) H∗i (ηR,t, ηC,t|θ0)vti(θ1)

]We have that

(θ2, θ1

)p→ (θ,θ), with an asymptotic variance given by

V ar(θ2, θ1

)=(G′Ω−1G

)−1

where G ≡[∇(θ2,θ1)e

ti (θ2,θ1)

]and Ω ≡ E

[(eti (θ2,θ1)

) (eti (θ2,θ1)

)′].

Define hti ≡ H∗i (ηR,t, ηC,t|θ)eti(θ) and hti ≡ H∗i (ηR,t, ηC,t|θ0)eti(θ). Thus, G and Ω are given by

Ω = E

[htih

t′i hih

t′i

htiht′i htih

t′i

]and G =

[G G1

0 G2

]where

G ≡ E[H∗i (ηR,t, ηC,t|θ)∇θvti(θ)

]G1 ≡ E

[∇θH∗i (ηR,t, ηC,t|θ)vti(θ)

]G2 ≡ E

[H∗i (ηR,t, ηC,t|θ0)∇θvti(θ)

].

By Assumption 4c, any function of (ηR,t, ηC,t) is orthogonal to vti(θ), which implies that G1 = 0.Thus, (G′Ω−1G)−1 is block diagonal and the marginal distribution of θ2 is asymptotically normal withvariance

V ar(θ2

)=(G′Ω−1G

)−1,

which is equivalent to the asymptotic distribution of the Optimal IV in (57).

59

9 Data Construction

This appendix describes the procedure to construct bilateral trade flows among 722 US CZs and 52countries in 1990 and 2000 for 367 4-digit SIC manufacturing industries and one non-manufacturing sector.

9.1 Summary Statistics

We start by reporting in Table 8 summary statistics for the main variables in our empirical application.The dispersion of employment changes is higher than that of wage changes in both periods. Our main shockexposure measures IPW t

i and IPCti have larger average and dispersion in 2000-2007 than in 1990-2000.This arises because the increase in imports from China was stronger after China’s accession to WTO in2001. Finally, the measures of the CZ’s indirect shock exposure have lower dispersion than the measure ofthe CZ’s own exposure.

Table 8: Summary Statistics

1990-2000 2000-2007 1990-2007Mean St. Dev. Mean St. Dev. Mean St. Dev.

(1) (2) (3) (4) (5) (6)

100 x Change in log wi 12.39 4.65 3.84 5.51 16.23 6.47100 x Change in log Li 11.73 11.64 8.85 10.21 20.58 18.70IPW t

i 0.91 0.96 1.76 1.78 2.67 2.52IPCti 2.27 0.52 5.19 0.92 7.45 1.22∑

j 6=i zijIPWtj 0.84 0.59 1.73 1.00 2.57 1.51∑

j 6=i zijIPCtj 2.33 0.32 5.35 0.63 7.68 0.85

Notes: Sample of 722 Commuting Zones. Indirect effects computed with zij ≡ L0jD−δij /

∑k L

0kD−δik where δ = 5, Dij is the distance

between CZs i and j, and L0j is the population of CZ j in 1990.

9.2 Methodology

Country-to-country bilateral trade matrix. We start by creating a country-to-country matrix oftrade flows at the 4-digit SIC classification. We consider the countries listed in Table 9.

We obtain international trade flows at the product-country level from the BACI dataset, assembled byCEPII, which we aggregate at the 4-digit SIC level. Since the starting year of the BACI dataset is 1995,we use the trade flows for 1995 and 2000.42 To obtain domestic spending shares for each country, we notefirst that our gravity model implies that Xt

ij,s = τ tij,s(pti)−εs(P tj,s)

εsEtj,s. We define two aggregate sectors:manufacturing and non-manufacturing. For any sector s within an aggregate sector S, assume that, fori 6= j, τ tij,s = ζti,sτ

O,ti,S τ

D,tj,S e

τ tij,s with Ei[τtij,s] = 0 and Ej [τ

tij,s] = 0. For i = j, τ tii,s = ζti,s, implying that only

productivity affects domestic sales. So,

lnXtij,s = τ tij,s + αti,s + ϕtj,s, (59)

where αti,s ≡ ln τO,ti,S ζti,s(p

ti)−εs and ϕtj,s ≡ ln τD,tj,S (P tj,s)

εsEtj,s.

42Although there is trade data available for 1990 from UN Comtrade, the data is quite sparse across countriesand industries.

60

Table 9: Sample of Countries

Argentina MalaysiaAustralia MexicoAustria NetherlandsBaltic Republics New ZealandBelarus NorwayBenelux PakistanBrazil PhilippinesBulgaria PolandCanada PortugalChile Rest of WorldChina RomaniaColombia RussiaCroatia Saudi ArabiaCzech Republic SingaporeDenmark SlovakiaFinland South AfricaFrance South KoreaGermany SpainGreece SwedenHungary SwitzerlandIndia TaiwanIndonesia ThailandIreland UkraineItaly United KingdomJapan UruguayKazakhstan Venezuela

Notes: Baltic Republics includes Estonia, Lithuania and Latvia.

To get the domestic trade flows, notice that Xtii,s = ζti,s(p

ti)−εs(P ti,s)

εsEti,s =(eα

ti,seϕ

ti,s

)/(τO,ti,S τ

D,ti,S

).

Since Xtii,S =

∑k∈S X

tii,k,

Xtii,s = Xt

ii,S

eαti,seϕ

ti,s∑

k∈S eαti,keϕ

ti,k

(60)

We use (60) to compute Xtii,s. In each year t, we obtain αti,s and ϕtj,s from the estimation of (59) with

bilateral trade flows by sector, and Xtii,S from the domestic sales in manufacturing and non-manufacturing

in the Eora MRIO dataset.

CZ employment share. We use the same imputation procedure of ADH to compute employment ineach 4-digit SIC manufacturing industry for 1980, 1990 and 2000 using the County Business Pattern(CBP). In year t, we use Lti,s to denote employment in CZ i and 4-digit SIC industry s and yti,s = Lti,s/L

ti

to denote the associated employment share.

CZ spending shares. We construct spending by sector and CZ, ξti,s, using

ξti,s ≡Eti,sEti

=γts +

∑k θ

tskb

tkyti,k

1 +∑

k btkyti,k

. (61)

61

where, in year t, θtsk is the share of spending on intermediates of sector s by sector k (common to all CZs),btk is the ratio of intermediate cost to labor cost of sector k (common to all CZs), and γts is consumers’

spending share on final goods of sector s (common to all CZs). We compute θtsk ≡Mtsk∑

s′Mts′k

where M tsk

is the spending of industry k on industry s in 1992 in the BEA 1992 U.S. Input-Output table used inAcemoglu et al. (2016). For manufacturing SIC-4 industries, we compute btk using total material costsdivided by payroll in the NBER manufacturing database for year t. For non-manufacturing industries, wecompute btk as average the material to payroll ratio across all U.S. non-manufacturing industries in theWIOD database. Finally, we obtain γts from the BEA 1992 U.S. Input-Output table.43

CZ exports and imports. We follow three steps to create exports and imports for each CZ and industry.First, we compute the CZ spending on sector s as Eti,s = ξti,sL

ti where ξti,s is the sectoral spending share

described above and Lti is the total employment in the CZ. Second, for each sector s, we compute the shareof CZ i in national spending, ξti,s = Eti,s/

∑j E

tj,s, and in national employment, yti,s = Lti,s/

∑j L

tj,s. Third,

we use the US Census data at the state-sector level for 1997 to compute the share of exports/imports ofeach state for each foreign country in a SCTG category, which is the 40-sector classification used by the USCensus.44 This yields βstate,i,s =

Xstate,i,sXUS,i,s

, where i is any of 52 foreign importer, and βi,state,s =Xstate,i,sXUS,i,s

,

where i is any of 52 foreign exporters. We use the same share βstate,i,s and βstate,i,s for all SIC-4 industrieswithin the same SCTG category. Finally, in each year t, we take US imports Xt

i,US,s and US exports

XtUS,i,s in each sector s and foreign country i, and split them across CZs using the following expression:

Xtij,s =

ξtj,s∑j∈state ξ

tj,s

βi,state,sXti,US,s and Xt

ji,s =ytj,s∑

j∈state ytj,s

βstate,i,sXtUS,i,s.

CZ-to-CZ bilateral trade matrix. We follow three steps to impute trade flows across CZs using thegravity trade structure of our model. First, for each SCTG category, we use state-to-state shipment datafrom the Commodity Flow Survey in 1997 to estimate

lnXij,s = δs + β1lnDij + β2 lnEj,s + β3 lnYi,s + β4di=j + εij,s (62)

where i is the origin state, j is the destination state, s is the SCTG sector, Dij is the distance between iand j, Ej,s are expenditures, Yi,s are production, di=j is a dummy that equals 1 when i = j.

Second, we use the estimated coefficients to impute trade flows across CZs with the following gravityspecification:

lnXtij,s ≡ β1lnDij + β2 ln ξtj,s + β3 ln yti,s + β4dstate(i)=state(j) (63)

where Dij is the distance between CZ i and j, and dstate(i)=state(j) is a dummy equal 1 if i and j belong tothe same state.

Lastly, we re-scale the imputed CZ-to-CZ trade flows to sum to the total US domestic sales in eachSIC sector (as in the country-to-country trade matrix).

43The final consumption shares vary across sectors but not across CZs. We take this approach because we are notaware of any comprehensive data on final consumption shares by CZ. The Consumer Expenditure Survey producedby the BLS only covers 26 selected MSA and does not vary by manufacturing sectors. We verify that, across theseMSAs, consumption shares in the CEX display little variation – for instance, the average share of consumption onfood and apparel is 19%, with a standard deviation of only 2% across MSAs.

44We construct state-sector exports and imports as follows. First, we use the US Merchandise Trade Data for1997 released by the US Census to create a mapping from each of the 44 US districts to the 50 US states, interms of share of imports and exports to each foreign country. Note that this is done at the aggregate level as thisinformation is not available at the industry-level. We then use US Census data to create district-level exports andimports at the HS-6 level for 1997. Finally, we use the mapping previously constructed to obtain state-HS6, andthen state-SIC 4 digit, trade flows with our sample of foreign countries.

62

Trade balance. Finally, we impose that trade is balanced at the regional level, as in the baseline model.We use the trade flows obtained above to compute matrix xt whose entries correspond to the share ofspending of each region j on another region i. Under trade balance, the vector of total revenue in theworld economy, Y t, must satisfy xtY t = Y t and, therefore, (I − xt)Y t = 0. Notice that it is alwayspossible to find a vector Y t that satisfies this system since (I − xt) is singular (

∑i x

tij = 1 for every j).

Thus, we find the vector Y t as the eigenvector of (I − xt) associated with the eigenvalue of zero. Wethen normalize it such that world GDP is one,

∑i Y

ti = 1.

9.3 Validation Tests

We first evaluate the correlation between the expenditure shares ξti,s constructed in equation (61) andthe spending shares implied by the shipment data for US states. To this end, for each of the 40 SCTGcategories, we compute state-level total shipment inflow in the Commodity Flow Survey (CFS) for 1997.We then aggregate our expenditure shares at the SCTG level using a crosswalk between SIC-4 and SCTGcategories, and compute total spending by state-SCTG using the size of the CZs in the state. Table 10reports the result of a regression of the expenditure shares in the CFS on our constructed spending sharesin 1990 and 2000. We can see that they are positively and significantly correlated, with an OLS coefficientclose to 1 and a R2 of 0.95.

We then proceed to assess whether our constructed CZ-level trade matrix reproduces the patterns ofobserved trade flows for US states. We use the CFS to measure bilateral shipments between US states ineach SCTG category for 1997, 2002 and 2007. To obtain comparable data, we use the methodology inOnline Appendix 9.2 to construct the trade matrix for CZs and SIC industries for 1997, 2002, and 2007.We then aggregate this data at the state-SCTG level in each year.

Table 11 reports the results of regressing actual shipment data on the corresponding trade flowobtained from our trade matrix. Column (1) considers domestic flows between US states, column (2)considers export flows from US states to foreign countries, and column (3) considers import flows fromforeign countries to US states. All specifications include sector fixed-effects. We can see that the predictedtrade flows are significantly and positively related to the actual flows, with coefficients close to 1. Noticealso that our imputed data captures a large share of the variation in bilateral trade flows. The R2 isabove 0.8 for exports and imports of US states, and around 0.5 for domestic flows between US states.

Table 10: Validation Test – Predicted Expenditure Shares

Dependent variable: Observed expenditure shares, 1997(1) (2)

Constructed expenditure shares, 1990 1.275***(0.01)

Constructed expenditure shares, 2000 1.265***(0.01)

Constant -0.009*** -0.009***(0.00) (0.00)

Observations 1,392 1,392R2 0.95 0.95

Notes: Sample of 1,392 state-SCTG pairs, where SCTG is the industry classification used in the CFS. Dependent variable is theobserved expenditure share in 1997 computed from the CFS. The regressors are the expenditure shares computed in equation (61),aggregated at the state-SCTG level. Robust standard errors in parentheses. *** p < 0.01, ** p < 0.05, * p < 0.10

63

Table 11: Validation Test – Bilateral Trade Flows

(1) (2) (3)

Panel A: Log of Actual Flows in 1997

Log of Predicted Flows in 1997 1.068*** 0.973*** 0.993***(0.01) (0.00) (0.00)

Observations 64,512 68,544 68,544

R2 0.512 0.950 0.950

Panel B: Log of Actual Flows in 2002


Observations 64,512 68,544 68,544

R2 0.509 0.816 0.837

Panel C: Log of Actual Flows in 2007


Observations 64,512 68,544 68,544

R2 0.477 0.806 0.827

Flow type:U.S. state to U.S. state Yes No NoU.S. state to Country No Yes NoCountry to U.S. state No No Yes

Notes: The dependent variable in column (1) is the actual shipment flow reported in the CFS for state-state-SCTG triples. Thedependent variables in columns (2) and (3) are trade flows constructed from the US Census trade data for state-country-SCTG triples.The regressors are the trade flows constructed using our methodology for the years 1997, 2002 and 2007, aggregate at the state-state-SCTG or state-country-SCTG level. All regressions include sector fixed effects. Robust standard errors in parentheses. *** p < 0.01,** p < 0.05, * p < 0.10

64

A Online Appendix: Empirical Application

This appendix presents additional empirical results that complement the baseline estimates presentedin Section 6. Section A.1 complements the estimates of the simple extension of ADH’s specification inSection 6.3. Section A.2 presents extensions of the baseline estimates of the reduced-form elasticities inSection 6.4.2. Section A.3 complements the results on the baseline predicted effects shown in Section 6.5.

A.1 Simple Extension of ADH

A.1.1 Robustness of Baseline Estimates in Table 1

This section investigates the robustness of the baseline estimates in Table 1. Given the non-significantresponses to consumption cost shifts, we focus on our preferred specification in column (5) where weinclude only the indirect effects of revenue shifts.45

Table A.1 investigates the importance of the three sets of controls used in our baseline specification.Column (1) only includes the baseline controls in ADH: period dummies, college-educated populationshare in 1990, foreign-born population share in 1990, employment share of women in 1990, employmentshare in routine occupations in 1990, average offshorability in 1990, and Census division dummies. Wecan see that, while the effect of revenue shifts is negative and significant on wages, it is not significant onemployment. This is consistent with the results in ADH who find a significant negative effect of Chineseimport exposure only on the number of employed individuals in the manufacturing sector. In their baselinespecification, the impact of shock exposure on the total number of employed individuals in the CZ isnegative but it is not statistically significant at usual levels – see also Bloom et al. (2019) for a discussionabout this point.

In column (2), we add controls for the total exposure of the CZ to shocks in the manufacturing sector,both directly and indirectly. In particular, we include the following controls: manufacturing employmentshare (

∑s y

t0i,s), manufacturing spending share (

∑s ξ

t0i,s), indirect exposure to manufacturing employment

share (∑

j 6=i zij∑

s yt0i,s) and manufacturing spending share (

∑j 6=i zij

∑s ξ

t0i,s). This set of controls absorbs

the impact of the secular decline in manufacturing employment. Results show that estimated coefficientsare very similar to those in column (1).

Finally, in column (3), we follow Greenland et al. (2019) by controlling for the 10-year lagged growthof the population that is 15-34 and 35-64 years old. These controls capture potential confounding effectsof slow moving trends in population growth. This control set has almost no effect on the estimatedcoefficients for wage responses. However, for the employment responses, controlling for population trendsyields higher point estimates and smaller standard errors. This is consistent with the evidence in Greenlandet al. (2019) that obtain more precise population responses to trade shocks when controlling for pastpopulation growth.

Table A.2 investigates how our baseline results change when we modify the function zij specifyingthe indirect effects. In all columns, we normalize

∑i 6=j zijIPW

tj so that the reported coefficient is the

impact of changing indirect exposure by one standard deviation. In columns (1)–(4), we maintain thesame functional form for zij , but modify the value of δ. For all specifications, the indirect effects arenegative and statistically significant, but the point estimates are higher when δ is lower. In column (5),we show that point estimates are slightly lower when we ignore the CZ size in the computation of zij .Finally, in column (6), we specify zij to only assign positive weight to other CZs in the same state. In thiscase, the point estimates are slightly higher, reflecting again the fact indirect effects decay with distance.

45For the specification in column (6), results are typically similar, but standard errors are larger. This is becausethe correlation of the indirect exposure to revenue and consumption shifts is high, 0.53.

1

Table A.3 reports results obtained with two alternative specifications. Columns (1) and (3) replicatethe baseline estimates in columns (5) of Table 1. In column (2), we follow ADH by weighting CZs bytheir 1990 population. Note that this weighting scheme is not consistent with our theory in which theunit of analysis is a market. Results suggest similar estimates of the indirect effect for both employmentand wages. However, conditional on the indirect exposure, the direct effects are not significant. Noticethat, in this case, responses to stronger declines in consumption prices are negative. So, the consumptionexposure may be partially capturing the revenue exposure of large CZs. This further confirms that thereis no evidence of a positive impact of lower prices on employment. Columns (3) and (6) report standarderrors computed with the inference procedure in Adao et al. (2019) that accounts for spatial correlation ofthe residuals. Although confidence intervals are wider, the indirect effects are still significant at 10%.

Table A.4 investigates how our estimates vary with the procedure to compute the spending shares ξt0i,s.In column (2), we show that all coefficients are similar when we ignore final spending in the computationof ξt0i,s. In this case, the cross-regional variation in IPCtj is entirely driven by intermediate spending shares.This suggests that local outcomes have only weak responses to import shocks that reduce the input pricesfor the industries in the CZ. One potential concern with our measure of the consumption shift is that, byincluding the own-industry spending, ξt0i,s may capture part of the effect of Chinese import competition inthe industry. To address this concern, column (3) shows that our estimates are similar when we computeξt0i,s ignoring input spending on the own sector.

2

Table A.1: Impact of the China Shock on Labor Market Outcomes across U.S. CZs, AlternativeControl Sets

(1) (2) (3)


IPW ti -0.531*** -0.362** -0.319**

(0.138) (0.161) (0.152)

IPCti 0.146 -0.021 -0.043(0.208) (0.210) (0.208)∑

j 6=i zijIPWtj -1.051*** -1.127*** -1.036***

(0.257) (0.369) (0.309)

R2 0.518 0.522 0.536


IPW ti -0.301 -0.359 -0.468**

(0.262) (0.220) (0.223)

IPCti 0.357 0.506 0.212(0.476) (0.500) (0.419)∑

j 6=i zijIPWtj -0.100 -0.417 -1.330***

(0.456) (0.509) (0.345)

R2 0.236 0.237 0.476

Control set:Regional controls in ADH Yes Yes YesInitial manufacturing exposure No Yes YesLagged population growth No No Yes

Notes: Sample of 1,444 Commuting Zones in 1990-2000 and 2000-2007. Indirect effects computed as in Table 1: zij ≡L0jD−δij /

∑k L

0kD−δik where δ = 5, Dij is the distance between CZs i and j, and L0

j is the population of CZ j in 1990. Regionalcontrols in ADH: period dummies, college-educated population share in 1990, foreign-born population share in 1990, employment shareof women in 1990, employment share in routine occupations in 1990, average offshorability in 1990, Census division dummies. Initialmanufacturing exposure: CZ’s share of employment and spending in manufacturing (

∑s y

t0i,s and

∑s ξt0i,s), CZ’s indirect exposure to

manufactruing employment and manufacturing spending in other CZs (∑j 6=i zij

∑s y

t0js and

∑j 6=i zij

∑s ξt0j,s). Lagged population

growth from Greenland et al. (2019): growth of population with 15-34 years old and 35-64 years old in the previous 10-year period.Robust standard errors in parentheses are clustered by state. *** p < 0.01, ** p < 0.05, * p < 0.10

3

Table A.2: Impact of the China Shock on Labor Market Outcomes across U.S. CZs, AlternativeIndirect Effect Specification

(1) (2) (3) (4) (5) (6)


IPW ti -0.551*** -0.525*** -0.494*** -0.513*** -0.489*** -0.568***

(0.147) (0.139) (0.131) (0.134) (0.128) (0.145)

IPCti 0.188 0.164 0.126 0.130 0.112 0.154(0.191) (0.193) (0.196) (0.195) (0.195) (0.192)∑

j 6=i zijIPWtj -6.928** -1.772** -0.903*** -0.726*** -0.731*** -1.690**

(2.753) (0.815) (0.292) (0.195) (0.196) (0.819)

R2 0.534 0.530 0.532 0.530 0.531 0.534


IPW ti -0.391 -0.360 -0.449** -0.470** -0.506** -0.534**

(0.235) (0.221) (0.213) (0.208) (0.207) (0.216)

IPCti 0.145 0.138 0.155 0.116 0.172 0.223(0.417) (0.406) (0.403) (0.401) (0.400) (0.375)∑

j 6=i zijIPWtj -14.551*** -3.968*** -1.253*** -0.833*** -0.822*** -2.089**

(2.968) (0.692) (0.324) (0.244) (0.283) (0.904)

R2 0.488 0.480 0.474 0.472 0.473 0.479

Indirect Effect Specification:

Definition of zijL0jD−1ij∑

k L0kD−1ik

L0jD−2ij∑

k L0kD−2ik

L0jD−5ij∑

k L0kD−5ik

L0jD−8ij∑

k L0kD−8ik

D−5ij∑

kD−5ik

L0jStij∑k L

0kStik

Notes: Sample of 1,444 Commuting Zones in 1990-2000 and 2000-2007. All specifications include the set of baseline controls in Table1. Indirect effects computed as specified in each column where Dij is the distance between CZs i and j, L0

j is the population of CZ j

in 1990, and Stij is a dummy that equals one if CZs i and j belong to the same state. In all columns, we normalize∑j 6=i zijIPW

tj to

have a standard deviation of one. Robust standard errors in parentheses are clustered by state. *** p < 0.01, ** p < 0.05, * p < 0.10

4

Table A.3: Impact of the China Shock on Labor Market Outcomes across U.S. CZs, AlternativeSpecifications

100 × Change inAvg. log weekly wage Log of employment

(1) (2) (3) (4) (5) (6)

IPW ti -0.319** -0.143 -0.319* -0.468** -0.735 -0.468**

(0.152) (0.219) (0.176) (0.223) (0.464) (0.225)

IPCti -0.043 -0.796** -0.043 0.212 -0.662 0.212

(0.208) (0.328) (0.348) (0.419) (0.743) (0.453)∑j 6=i zijIPW

tj -1.036*** -1.036*** -1.036* -1.330*** -1.844*** -1.330*

(0.309) (0.304) (0.596) (0.345) (0.613) (0.743)

Specification:Weight by population No Yes No No Yes NoAdao et al. (2019) SEs No No Yes No No YesState-clustered SEs: Yes Yes No Yes Yes No

Notes: Sample of 1,444 Commuting Zones in 1990-2000 and 2000-2007. All specifications include the set of baseline controls in Table1. Indirect effects computed as in Table 1: zij ≡ L0

jD−δij /

∑k L


j is

the population of CZ j in 1990. *** p < 0.01, ** p < 0.05, * p < 0.10

5

Table A.4: Impact of the China Shock on Labor Market Outcomes across U.S. CZs, AlternativeSpending Shares

(1) (2) (3)


IPW ti -0.319** -0.365** -0.369**

(0.152) (0.150) (0.148)

IPCti -0.043 0.087 0.128(0.208) (0.104) (0.131)∑

j 6=i zijIPWtj -1.036*** -1.049*** -1.057***

(0.309) (0.310) (0.310)

R2 0.536 0.536 0.537


IPW ti -0.468** -0.470** -0.449**

(0.223) (0.217) (0.216)

IPCti 0.212 0.110 0.082(0.419) (0.177) (0.211)∑

j 6=i zijIPWtj -1.330*** -1.328*** -1.327***

(0.345) (0.344) (0.346)

R2 0.476 0.476 0.476

Construction of IPCti :Drop final spending No Yes YesDrop own industry spending No No Yes


jD−δij /

∑k L


j is

the population of CZ j in 1990. In column (2), we compute ξi,s ignoring final spending. In column (3), we compute ξi,s ignoring alsothe own industry spending. Robust standard errors in parentheses are clustered by state. *** p < 0.01, ** p < 0.05, * p < 0.10

A.1.2 Additional Results

This section provides additional results that complement our baseline estimates. We again focus on ourpreferred specification in column (5) of Table 1.

We start by investigating how other labor market outcomes respond directly and indirectly to regionalshock exposure. Table A.5 shows that a negative revenue shock yields indirect negative effects onemployment in both the manufacturing and the non-manufacturing sectors of other nearby CZs. Thisnegative effect on employment translates into higher rates of non-participation and unemployment.Notice however that the impact on the unemployment rate is much smaller than the impact on thenon-participation rate. In addition, column (2) reports a negative and significant effect of IPCti onmanufacturing employment. Table A.6 investigates the responses of wages in the manufacturing andnon-manufacturing sectors. As in ADH, column (2) indicates that responses in manufacturing wages arenot significant. The estimates in column (3) reveal that the direct and indirect effects of the shock onwages are concentrated in the non-manufacturing sector.

Finally, Table A.7 estimates how labor market outcomes responded to the removal of uncertainty ontariffs created by the NTR gap studied in Pierce and Schott (2016). To this end, we use an alternativeshift to compute the shift-share variables measuring exposure in terms of revenue and consumption cost.

6

Instead of ∆Mo,ts , we use the simple average of the NTR gaps by 4-digit SIC industry. The results in

Table A.7 show similar qualitative patterns to those in Table 1. First, both the direct and the indirecteffects of higher revenue exposure are negative for wages and employment. Second, the impact of higherconsumption exposure is non-significant for wages and employment – in this case, confidence intervals onthe estimated coefficients are wider.

Table A.5: Impact of the China Shock on Employment Outcomes across U.S. CZs

Employed Not employedAny sector Manuf. Non-Manuf. Unemp. NILF

(1) (2) (3) (4) (5)

Panel A: Change in the share of working-age population by category

IPW ti -0.312*** -0.143*** -0.169*** 0.094*** 0.218***

(0.070) (0.053) (0.053) (0.028) (0.054)

IPCti -0.171 -0.189*** 0.018 0.054 0.118

(0.131) (0.057) (0.116) (0.045) (0.103)∑j 6=i zijIPW

tj -0.844*** -0.431*** -0.413*** 0.308*** 0.536***

(0.159) (0.090) (0.143) (0.072) (0.121)

R2 0.321 0.556 0.236 0.313 0.288

Panel B: Change in the log of working-age population by category

IPW ti -0.468** -1.021** -0.281 1.966*** 0.861***

(0.223) (0.406) (0.209) (0.667) (0.248)

IPCti 0.212 0.151 0.444 2.058* 0.917**

(0.419) (0.648) (0.412) (1.030) (0.412)∑j 6=i zijIPW

tj -1.330*** -3.157*** -0.751** 7.158*** 2.037***

(0.345) (0.884) (0.343) (1.885) (0.730)

R2 0.476 0.388 0.499 0.344 0.529


jD−δij /

∑k L


j is

the population of CZ j in 1990. Robust standard errors in parentheses are clustered by state. *** p < 0.01, ** p < 0.05, * p < 0.10

7

Table A.6: Impact of the China Shock on Wages across U.S. CZs

100 x Change in avg. log weekly wageAll Manuf. Non-Manuf.(1) (2) (3)

IPW ti -0.319** 0.234 -0.427***

(0.152) (0.216) (0.147)

IPCti -0.043 0.473 -0.009(0.208) (0.500) (0.200)∑

j 6=i zijIPWtj -1.036*** -0.114 -1.128***

(0.309) (0.523) (0.306)

R2 0.536 0.216 0.530


jD−δij /

∑k L


j is

the population of CZ j in 1990. Robust standard errors in parentheses are clustered by state. *** p < 0.01, ** p < 0.05, * p < 0.10

8

Table A.7: Impact of the Removal of NTR Gaps on Labor Market Outcomes across U.S. CZs

(1) (2) (3) (4) (5) (6)


NTRPW ti -2.852*** -2.653*** -1.565*** -2.425*** -1.410*** -1.410***

(0.415) (0.470) (0.368) (0.412) (0.388) (0.388)

NTRPCti -1.419 -1.156 -1.214

(1.181) (1.193) (1.099)∑j 6=i zijNTRPW

tj -2.439*** -2.425*** -2.488***

(0.526) (0.529) (0.654)∑j 6=i zijNTRPC

tj -5.278** 0.502

(2.467) (2.477)

R2 0.574 0.574 0.589 0.578 0.589 0.589


NTRPW ti -3.281*** -3.525*** -1.452** -3.031*** -1.735** -1.728**

(0.592) (0.725) (0.573) (0.648) (0.673) (0.667)

NTRPCti 1.743 2.120 1.451

(2.002) (1.928) (1.876)∑j 6=i zijNTRPW

tj -3.466*** -3.492*** -4.227***

(0.721) (0.719) (0.672)∑j 6=i zijNTRPC

tj -3.084 5.850*

(2.564) (3.086)

R2 0.493 0.493 0.504 0.493 0.504 0.505


jD−δij /

∑k L


j isthe population of CZ j in 1990. We compute regressors using the shift-share definitions in Section 6.3 where the sector-level shifter isthe NTR tariff gap in Pierce and Schott (2016) (instead of ADH’s per-worker Chinese ∆Mo,t

s ). Robust standard errors in parenthesesare clustered by state. *** p < 0.01, ** p < 0.05, * p < 0.10

9

A.2 Reduced-Form Elasticities to Revenue and Consumption Shifts

A.2.1 Robustness

This section investigates the robustness of our baseline estimates of the reduced-form elasticities tothe assumptions embedded in the parametrization of spatial links described in Section 5.2. Table A.8presents the estimates of the structural parameters using different specifications of spatial links. PanelA reports the baseline parameters in Table 3. In Panel B, we report the estimates obtained using analternative specification of spatial links that allows for trade imbalances in each CZ. In this case, we usethe reduced-form elasticities defined in Online Appendix C.1 and trade matrix implied by our imputationbefore adjusting market-level income to guarantee balanced trade everywhere. In Panel C, we use adifferent normalization of labor supply. Rather than using the world’s average wage, we use the nationalprice index: in the labor supply specification (26), we impose bwj = 0 and bpj = Yj/

∑j∈C(j) Yo with C(j)

denoting the country of region j. Lastly, in Panel D, we calibrate the spatial elasticity of labor supplyusing a value similar to the one in Caliendo et al. (2019). That is, we set φm = 0.25 in the labor supplyspecification (26).

In all specifications, we obtain an estimate of ε close to four. All panels report high values of φw andψ, but point estimates vary across specifications. In Panels C and B, the labor supply elasticity to wagesis higher, but productivity elasticity to employment is lower. Finally, notice that our estimate of φp ismuch lower in Panels B and C. This is because the estimation of this parameter relies more on the generalequilibrium mechanism of the model, whose strength varies with the specification of trade imbalances andlabor supply.

We then evaluate how these assumptions affect our estimates of the reduced-form elasticities. Thisis important because our theoretical results imply that these are the sufficient objects to compute themodel’s predicted responses for any given vector of shifts in revenue and consumption costs. For thisreason, Table A.9 reports the correlation between the reduced-form elasticities implied by our baselinespecification and the alternative specifications shown in Table A.8. We can see that the correlationsare very high for all alternative specifications, attesting the robustness of our baseline estimates of thereduced-form elasticities.

Finally, we report the fit coefficients estimated with equation (39) for the alternative specifications inTable A.8. As in our baseline, Table A.10 shows that the fit coefficients are not statistically differentfrom one for all alternative specifications.

10

Table A.8: Estimates of the Structural Parameters, Robustness

ψ φw φp ε

Panel A: Baseline

0.56 2.11 -1.36 3.94(0.07) (0.25) (0.24) (0.41)

Panel B: Allowing for trade imbalances

0.62 1.92 -0.73 3.98(0.07) (0.22) (0.45) (0.40)

Panel C: Labor supply normalization with national price index

0.50 2.92 -0.12 3.96(0.07) (0.60) (0.41) (0.35)

Panel D: Exogenous labor supply links, φm = 0.25

0.35 3.59 -1.28 4.42(0.05) (0.54) (0.99) (0.82)

Notes: Estimation of θ using reduced-form expressions in (37) for the pooled sample of 1,444 Commuting Zones in 1990-2000 and

2000-2007. Estimation uses the two-step procedure in Proposition 5. All specifications include the set of baseline controls in Table 1.

Each panel corresponds to one specification of spatial links (as described in the main text). Standard errors in parentheses.

Table A.9: Correlation between Baseline and Alternative Estimates of Reduced-form Elasticities

(1) (2) (3)

Panel A: Elasticity to revenue shifts

βRii 0.93 0.94 0.96ϕRii 0.92 0.94 0.95βRij 0.57 0.92 0.44ϕRij 0.87 0.92 0.78

Panel B: Elasticity to consumption cost shifts

βCii 0.75 0.67 0.90ϕCii 0.78 0.57 0.90βCij 0.80 0.61 0.72ϕCij 0.82 0.58 0.59

Specification

Panel B of Table A.8 x

Panel C of Table A.8 x

Panel D of Table A.8 x

Notes: Table of correlations between reduced-form elasticities computed with alternative specifications of spatial links and estimatedparameters reported in Table A.8. Sample of 521,284 pairs of CZs in 2000.

11

Table A.10: Predicted Impact of the China Shock and Labor Market Outcomes across U.S. CZs,Robustness

(1) (2) (3) (4)

Panel A: Change in avg. weekly log wage

Predicted Effect 0.67** 1.08*** 0.82** 1.07*(0.27) (0.32) (0.34) (0.57)


Predicted Effect 0.90*** 1.24*** 0.86*** 1.07***(0.14) (0.31) (0.20) (0.29)

Specification

Baseline x

Panel B of Table A.8 x

Panel C of Table A.8 x

Panel D of Table A.8 x

Notes: Pooled sample of 1,444 Commuting Zones in 1990-2000 and 2000-2007. All specifications include the set of baseline controls inTable 1. Robust standard errors in parentheses are clustered by state.


Cross-regional variation in reduced-form elasticities. We now investigate the determinantsof the cross-section variation in the estimated indirect effects. We first regress βRij(θ) and ϕRij(θ) on thebilateral index zij that we use to construct the indirect effects in the simple extension of ADH in equation(38). Columns (1) and (2) in Table A.11 indicate that regions that are larger or closer to each other havelarger reduced form elasticities. That is, they induce stronger indirect effects in general equilibrium. Thelow R2, however, suggest that the gravity bilateral index zij is not able to capture the vast heterogeneityin the reduced-form elasticities implied by our baseline specification.

Next, we investigate how the different observed components of the spatial links matrix determine

the matrices βRij(θ) and ϕRij(θ) . We build on the series expansion shown in Theorem 2 and look at how

the observed bilateral components of γ – namely, the revenue share yij and the revenue elasticity χij –

predict the estimated reduced-form elasticities. Table A.11 indicates that a large share of the dispersion

of indirect effects is explained by these two variable: the R2 is around 0.5 for both elasticities. For both

variables, the indirect effect is increasing in χij , indicating that indirect effects are stronger between

regions with a similar sectoral and destination composition.46

46Another determinant of the spatial matrix γ is the matrix of trade shares x. However, since the correlationbetween yij and xij is close to 1, we do not add xij to the regression.

12

Table A.11: Estimates of Indirect Effects and Observable Spatial Links

Reduced-Form Elasticity Wage (βRij) Employment (ϕRij) Wage (βRij) Employment (ϕRij)(1) (2) (3) (4)

Size/Distance (zij) 0.027*** 0.055***(0.00) (0.00)

Revenue share (yij) -2.916*** -1.866***(0.02) (0.03)

Gravity competition (χij) 8.630*** 12.540***(0.02) (0.03)

Constant 0.005*** 0.009*** 0.002*** 0.003***(0.00) (0.00) (0.00) (0.00)

R2 0.003 0.003 0.583 0.524

Notes: Sample of 468,506 bilateral pairs of Commuting Zones. All the variables are constructed using data for 2000 and the baselineestimates in Panel A of Table 3. We trim the top and bottom 5% of the observations. *** p < 0.01, ** p < 0.05, * p < 0.10

Model Fit: Predicted responses and regional shock exposure. Figure A.1 investigatesfurther the ability of the different specifications of spatial links to match the findings in ADH– namely, the magnitude of the differential employment response to higher regional exposure toChinese import competition. The black circles replicate the relationship in column (1) of Table 1:regions with $1000 dollars more of per-worker exposure to Chinese import competition (i.e., higherIPW t

i ) suffer an employment growth reduction of 0.56 log-points. The red squares illustrate therelationship between regional shock exposure and employment changes predicted by our baselineestimates. We can see that our baseline model yields a negative differential response to local shockexposure whose magnitude is very similar to that in the data. Notice that the squares are notperfectly aligned with IPW t

i because employment changes in our model are also driven by regionalconsumption exposure and indirect effects created by the shock exposure of other CZs. Finally,the blue diamonds indicate that the alternative specifications in columns (3) and (6) of Table 5are not able to replicate the magnitude of the differential employment response to higher revenueexposure across CZs. They predict that regions in which IPW t

i is $1000 dollars higher experiencereductions in employment growth of -0.089 and -0.017 log-points according to the specifications incolumns (3) and (6), respectively.

13

Figure A.1: Employment Changes and Exposure to Chinese Import Competition across U.S. CZs

Notes: Bin scatter plot of changes in log of employment and expposure to Chinese import competition (IPW ti ) after partialling out

the baseline controls in Table 1. Pooled sample of 1,444 Commuting Zones in 1990-2000 and 2000-2007. Plots report average log-employment change and IPW t

i by percentile bins based on IPW ti . Baseline predicted changes are computed with the reduced-form

responses in equation (37) using estimates in Panel A of Table 3. Alternative calibration computed with the reduced-form responsesin equation (37) using parameters in the corresponding columns of Table 5.

14

A.3 The Impact of the China Shock in General Equilibrium

A.3.1 Robustness

Table A.12 computes the effect of the China shock using the alternative specifications of the spatial linksdiscussed in Section A.2.2. The first row of each panel reports the predicted responses implied by ourbaseline specification. For each alternative specification, the remaining rows present the average, standarddeviation and correlation with baseline predicted responses across CZs.

Our results indicate that all alternative specifications yield predicted responses that have a highcorrelation with our baseline predicted responses. The correlations reported in the third column are alwaysbetween 0.79 and 0.89. However, the first column indicates that alternative specifications generate averageresponses that may be higher or lower than our baseline predicted responses. The average negative effectof the shock is stronger when we allow for trade imbalances or specify stronger labor supply links. This isbecause of the different estimated parameters reported in Table 3 that are necessary to match the directand indirect impacts of shock exposure on labor market outcomes. We obtain a higher agglomerationelasticity in the case of trade imbalances and a higher labor supply elasticity in the case of exogenousmigration links. These tend to strengthen the amplification channel created by the combination ofendogenous responses in employment and productivity. Lastly, the second row of Panel B shows that theaverage employment response is smaller when we specify the labor supply normalization in terms of theU.S. price index. This is because the national employment change is now more sensitive to the decline inimport prices.

Table A.12: Effect of the China Shock on U.S. CZs, 1990-2007 – Robustness

National Standard CorrelationAverage Deviation w/ baseline

Panel A: Wages

Baseline -3.98 1.30 1.00Panel B of Table A.8 -5.64 1.34 0.83Panel C of Table A.8 -3.37 1.12 0.89Panel D of Table A.8 -3.69 0.65 0.88

Panel B: Employment

Baseline -2.78 3.31 1.00Panel B of Table A.8 -4.08 2.71 0.79Panel C of Table A.8 -0.52 3.30 0.84Panel D of Table A.8 -5.44 2.39 0.84

Panel C: Real Wages

Baseline 0.16 1.75 1.00Panel B of Table A.8 -0.79 1.57 0.79Panel C of Table A.8 1.30 1.40 0.86Panel D of Table A.8 -0.12 0.81 0.85

Notes: Predicted changes in employment and wages computed with the reduced-form responses in equation (37). Prediced real wagechange computed with expression (23). Reduced-form elasticities computed with alternative specifications of spatial links and estimatedparameters reported in Table A.8.

15


We first report the national effect of the China shock on wages, employment and real wages, separatelyfor the two time periods. Tables A.13 and A.14 document that in the first decade of our analysis, between1990 and 2000, the effect of the rise of China has been much smaller than the effect in the second decade.This is a consequence of the larger magnitude of the shock in the second period since China increased itsexports to high-income countries more after its accession to WTO in 2001. The tables also show thatthe indirect effects reinforce the direct effects of both revenue and consumption shifts in both periods.Lastly, we can see that the predicted responses of employment were more dispersed than the ones ofwages throughout both decades.

Table A.13: Effect of the China Shock on U.S. CZs, 1990-2000

Wage Employment Real wageAverage St. Dev. Average St. Dev. Average St. Dev.

Total effect -1.14 0.70 0.06 1.91 0.51 1.02

Direct effect of ηR -0.31 1.03 -0.76 2.82 -0.39 1.51Direct effect of ηC 0.32 0.65 1.03 1.86 0.99 1.03Indirect effect of ηR -0.92 0.95 0.28 2.70 0.03 1.47Indirect effect of ηC -0.23 0.58 -0.49 1.66 -0.13 0.90


Table A.14: Effect of the China Shock on U.S. CZs, 2000-2007

Wage Employment Real wageAverage St. Dev. Average St. Dev. Average St. Dev.

Total effect -2.84 0.82 -2.85 2.03 -0.33 1.06

Direct effect of ηR -0.50 0.84 -1.18 2.15 -0.59 1.13Direct effect of ηC 0.66 0.77 2.15 2.23 2.15 1.30Indirect effect of ηR -3.32 1.05 -5.23 2.72 -2.91 1.43Indirect effect of ηC 0.32 0.68 1.41 1.94 1.01 1.05


16

B Online Appendix: Micro-foundations and Welfare with

Endogenous Labor Supply

In this appendix, we establish micro-foundations for the functions governing the spatial links in laborsupply and labor productivity in our model. We then use one of these micro-foundations to derive theequivalent welfare variation implied by a shock as a function of the shock’s impact on the real wage ineach market.

B.1 Labor Supply Function

In this section, we lay out three settings that imply labor supply functions of the form in equation (1).In the first framework, we derive the labor supply function from a setting in which a representativehousehold decides the number of hours worked (intensive margin) in each region of the country. Thesecond framework yields the labor supply function from the choice of heterogeneous individuals of workingand residing in different regions of the country (extensive margin). In the third framework, we derive aspecial case of equation (1) in a Roy model with a generic distribution of individual-specific amenities andefficiencies in different sector-region pairs.

B.1.1 Representative Household with Intensive Labor Supply Margin

We lay out a setting in which a representative household decides the number of hours worked in eachregion of the country. This is the same mechanism present in business cycle models with elastic laborsupply, see e.g. Shimer (2009) and Keane (2011). We consider an extended version of this frameworkwhere the representative household allocates individuals across different regions within the country.

Preferences. Country c has a representative household with preferences over consumption and laborsupply in different markets. The representative agent’s utility function is given by

U (NiVi(Ci, Hi)i) , (B.1)

where Ni is the total number of individuals in region i and Vi is the utility of individuals in region i. Weassume that Uc(· ) is twice differentiable, and strictly quasi-concave.

We assume that Vi takes the following separable form:

Vi(Ci, Hi) =(Ci)

1−η

1− η−H

1+ 1φ

i

1 + 1φ

(B.2)

where Hi is the per-worker hours worked in region i and Ci is a consumption aggregator. This utilityfunction has two parameters: η > 0 regulates income effects, while φ > 0 is the Frisch elasticity of laborsupply. This specification has been used by Mankiw et al. (1985), Domeij and Floden (2006), Corsetti etal. (2007) and Keane (2011) among others. In the extreme case of η = 0, income effects are absent andthus labor is increasing in real wage, while if η > 1 the income effect dominates the substitution effectand labor supply is decreasing in the real wage. The limiting case of η = 1 is often used in the macroliterature, such as Kimball and Shapiro (2008), Shimer (2009) and Ohanian and Raffo (2012), because inthis way preferences are consistent with a balanced growth path. We further assume that Ci takes thenested CES form that yields the spending shares in (6) and the price index in (8).

We impose that country c has Nc individuals such that

17

∑i∈c

Ni = Nc.

Budget Constraint. In each region, consumption is financed by labor income and a lump-sum transferof Ti units, expressed in terms of a numeraire b(w,P ). There is also an income tax of t% of total income(including the transfer). Thus, the budget constraint in region i is

NiCiPi = (1− t)(wiHiNi + Tib(w,P )Ni

).

where we impose that b(w,P ) is homogeneous of degree one in (w,P ).We impose budget balance at the country-level, so that

t∑i∈c

wiHiNi =(1− t

)∑i∈c

Tib(w,P )Ni

This can be rearranged to obtain the optimal tax rate

t =b(w,P )

∑i∈c TiNi

b(w,P )∑

i∈c TiNi +∑

i∈cwiHiNi

Intensive margin of labor supply in each region. We first solve the first-stage problem of decidingthe number of hours worked in each region i conditional on Ni:

Vi

((1− t)wi

Pi, Ti

b(w,P )

Pi

)≡ max

Hi

1

1− η(1− t)

(wiPiHi + Ti

b(w,P )

Pi

)1−η−H

1+ 1φ

i

1 + 1φ

. (B.3)

The first order condition of this problem implies that((1− t)wi

Pi

) 1η

H− 1ηφ

i − (1− t)wiPiHi = (1− t)Ti

b(w,P )

Pi

which implies that

Hi = ΦH

((1− t)wi

Pi, (1− t)Ti

b(w,P )

Pi

)(B.4)

It is easy to verify that, if η < 1, then ∂ΦH

∂(1−t)wiPi

> 0 and ∂ΦH

∂(1−t)Ti b(w,P )Pi

< 0. We obtain the indirect

utility in region i by substituting Hi in (B.4) into the objective function in (B.3).

Number of individuals in each region. Finally, the second-stage problem is

maxNi

Uc

(Vi

((1− t)wi

Pi, (1− t)Ti

b(w,P )

Pi

)Ni

i

)s.t.

∑i

Ni = Nc

We can equate the first-order condition for any two regions:

∂U

∂N1V1

((1− t)w1

P1, (1− t)T1

b(w,P )

P1

)=

∂U

∂NiVi

((1− t)wi

Pi, (1− t)Ti

b(w,P )

Pi

)∀i ∈ c (B.5)

18

where the tax rate ti is given by

t =

∑i∈c

b(w,P )Pi

TiNi∑i∈c

b(w,P )Pi

TiNi +∑

i∈cwiPiHiNi

, (B.6)

and the total population constraint is given by∑i∈c

Ni = Nc. (B.7)

In equilibrium, Ni and t solve the system in (B.5)–(B.7). Notice that the system implies Ni and t arefunctions of wi/Pi and Tib(w,P )/Pi. Thus, we can write

Ni = ΦN

(wiPi, Ti

b(w,P )

Pi

i

)and t = Φtax

(wiPi, Ti

b(w,P )

Pi

i

). (B.8)

Labor supply. Labor supply in region i is Li = HiNi. The combination of equations (B.4) and (B.8)implies that

Li = ΦLi

(wiPi, Ti

b(w,P )

Pi

i

)(B.9)

and, therefore, it has the general form in equation (1).

Notice that, as long η < 1, both Hi and Ni respond to Tib(w,P )Pi

, so that φwij 6= −φpij . This becomes

clear in the case of exogenous population in each region where labor supply is given by (B.4). In fact, inthis case, we have that

∂ lnHi

∂ lnPi= −∂ lnHi

∂ lnwi+∂ ln ΦH

∂ lnTiTib(w,P )

PiHi

(1 +

∂ ln b

∂ lnPi

),

so that −∂ lnHi∂ lnPi

< ∂ lnHi∂ lnwi

whenever ∂ ln b∂ lnPi

≥ 0.

B.1.2 Spatial Assignment with Extensive Labor Supply Margin

We now present a micro-foundation for the general specification of the labor supply equation in Section 2based on the choice of heterogeneous agents to work and reside in different regions of a country. Oursetting combines the extensive margin decision to work of heterogeneous individuals in terms of disutilityto work, as in Rogerson (1988) and Chetty (2012), and the discrete choice of residing in different regions,as in Bryan and Morten (2015). A similar framework has been used in Adao, Kolesar and Morales (2019)to motivate shift-share empirical designs.

Preferences. Consider a country c composed of regions (indexed by i and j). We assume that allindividuals in the country have identical preferences for consumption goods, which take the nested CESstructure that generates the trade shares introduced in Section 2. Each region i is endowed with Ni

individuals that have heterogeneous preferences for the amenities of the different regions of the country,a ≡ ajj . If an individual moves from region i to region j, her amenity value is discounted by ζij .Individuals residing in region j may decide to work at a wage rate wj or to stay at home in exchange fora government transfer of bj . All residents of region j receive a lump-sum transfer tj = χbi and pay anad-valorem income tax χ. Then, once an individual resides in region i, she learns her work disutility νand decides whether or not to participate in the labor market. Thus, the utility of an individual born in

19

region i living in region j is

Uij(a, ν) =

ζijaj

νwjχ+tjPj

if work

ζijajbiχ+tiPi

if do not work

where Pi is the price index in region j.We assume that individuals take i.i.d. draws of their amenity vector and their work disutility such

thataj ∼ e−a

−φmand ν ∼ 1− (φe − 1)v−φ

e,

where φm > 0, φe > 1, and ν ≥ (φe − 1)1/φe .

Budget Constraint. In equilibrium, we assume that the income tax rate χ guarantees budget balancein the country. Let Nj be the population of region j and ej the employment share in region j. The budgetbalance condition requires that

χ =

∑i bi(1− ei)Ni∑i (wi − bi) eiNi

.

Extensive margin of labor supply in each region. We now derive the labor supply equation ineach region j. Among the Nj residents of region j, only individuals with νwj > bj decide to work. Thus,the employment share in region j is

ei = Pr [νwj > bj ] = (φe − 1)

(wjbj

)φe. (B.10)

Number of individuals in each region. This implies that the expected utility of an individual bornin region i and residing in region j is Uij(a) ≡ E[Uij(a, ν)], which is equivalent to

Uij(a) =ζijajPj

(χwj(φ

e − 1)∫∞bj/wj

φev−φedv + χbj(φ

e − 1)∫ bj/wj

(φe−1)1/φe φev−φ

e−1dv + tj

)=

ζijajPj

(−χwjφe

(limv→∞ (v)1−φe −

(bjwj

)1−φe)− χbj(φe − 1)

((bjwj

)1−φe− (φe − 1)−1

)+ tj

)=

ζijajPj

((χφe − χ(φe − 1))wφ

e

j b1−φej − χbj + tj

)=

(χζijw

φe

j b1−φe

j

Pj

)aj

where the last row uses the fact that we specify the lump-sum transfer to be tj = χbi. This implies thatthe share of individuals born in region i residing in region j is

nij = Pr

[j = argmax

j′

(χζij′w

φe

j′ b1−φej′

Pj′

)aj′

]=

(ζijw

φe

j b1−φe

j

Pj

)φm∑

j′

(ζij′w

φej′ b

1−φej′

Pj′

)φm . (B.11)

Thus, the number of individuals residing in region j is Nj =∑

i nijNi

20

Labor supply. Thus, given wj , bj , Pjj , total employment in region j is Lj = ei∑

j nijNi, which isequivalent to

Lj = (φe − 1)

(wjbj

)φe∑i

(ζijw

φe

j b1−φe

j

Pj

)φm∑

j′

(ζij′w

φe

j′ b1−φej′

Pj′

)φm Ni.

To derive the labor supply equation in terms of wages and prices, it is necessary to specify the monetarynon-employment benefit bi. We assume that the benefit in region i has a component that depends onlocal outcomes and another that is common to all regions:

bj = bjwαwj P

αpj b(w,P )1−αw−αp ,

where b(w,P ) is homogeneous of degree one in (w,P ). In this specification, bi is an exogenous shifterof the benefit and, therefore, captures the generosity of transfers available in each region. The othercomponents specify how the transfer responds to the outcomes in the own region and those in the overalleconomy. Notice that, by imposing restrictions that guarantee that the benefit is homogeneous of degreeone in (w,P ), we prevent the specification of the economy’s numeraire from affecting employment. Underthis assumption, labor supply in region i is

Lj = vj

(wφ

e,w

j P φe,p

j

b(w,P )φe,w+φe,p

)∑i

ζijwφw

j P φp

j∑j′ ζij′w

φw

j′ Pφp

j′

Li, (B.12)

where vj ≡ (φe − 1)b−φe

j , φe,w ≡ (1− αw)φe, φe,p ≡ −αpφe, φw ≡ (φe + (1 − φe)αw)φm, φp ≡ −(1 +αp(φ

e − 1))φm.This expression implies that labor supply takes the general form specified in equation (1) where

φwij 6= −φpij whenever αp < 1. In the case of αp = 0 and φm = 0,

Lj = vj

(wj

b(w,P )

)φe(1−αw)

,

so that ∂ lnLi∂ lnPi

= −φe(1 − αw) ∂ ln b∂ lnPi

and ∂ lnLi∂ lnwi

= φe(1 − αw)(

1− ∂ ln b∂ lnwi

). Thus, 0 ≤ −∂ lnLi

∂ lnPi< ∂ lnLi

∂ lnwi

whenever 0 ≤ ∂ ln b∂ lnPi

< 1− ∂ ln b∂ lnwi

.

B.1.3 Spatial Assignment Models with General Distribution of Amenities and Effi-ciency

In this section, we consider a general class of assignment models in which a market in defined as a groupof industries in a region. Individuals are heterogeneous in terms of both market-specific productivityand preferences. Our setting covers the seminal setting in Roy (1951) and its recent applications ininternational trade – e.g., Adao (2015), Burstein et al. (2019), Galle et al. (2017), and Lee (2020).

Preferences. Suppose that countries are populated by a continuum of individuals, ι ∈ Ic, that areheterogeneous in terms of preferences and efficiency across markets (i.e, sector-region pairs). We assumeindividual ι has market specific preferences, aj(ι), and market specific efficiency, ej(ι). In particular, ifemployed in market j, we assume that individual ι has homothetic preferences given by

Uj(ι) = aj(ι) + Cj ,

21

where Cj is a consumption aggregator that is identical to all individuals. We assume that it takes thenested CES structure that generates the trade shares introduced in Section 2. Notice that, compared tothe models presented in the previous two sections, the agents cannot choose to work for the home sector.Thus the model is simpler in that we do not need to specify the unemployment benefit function and thetax system.

We further assume that individuals take independent draws of (aj(ι), ej(ι)) from a common fullsupport distribution:

aj(ι), ej(ι)j ∼ F (a, e).

Notice that, compared to the framework in the previous section, we allow here for a general distributionF (a, e) that governs cross-section heterogeneity in both market-specific amenities and efficiency acrossindividuals. It covers as special case of extreme value distributions used in Burstein et al. (2019), Galle etal. (2017), and Lee (2020). It also covers the setting in Adao (2015) that only entails heterogeneity insector-specific efficiency.

Budget Constraint. The budget constraint of individual ι is given by

Cj =wjej(ι)

Pj

where Pj is the price index in (8). To simplify the notation, we let ωj = wj/Pj be the real wage in marketj.

Labor supply. The solution of this problem implies that, for individual ι, the utility of being employedin j is Uj(ι) = aj(ι) + ωjej(ι). Thus, the set of individuals choosing j is

Ij (ωii) ≡ (a, e) : aj + ejωj ≥ ai + eiωi ∀i .

Thus, the labor supply is a function of the real wage vector in the economy:

Lj = Φj(ω) ≡∫Ij(ωii)

ej dFc(a, e). (B.13)

Notice that the function Φj(· ) is homogeneous of degree zero with∂Φj∂ωj

> 0 and∂Φj∂ωi

< 0.47 As shown in

Adao (2015), these properties imply that Φ(.) is invertible up to scalar. This expression is a special caseof the labor supply function in (1) where φwij = −φpij for all i and j.

B.2 Labor Productivity

In this section, we provide a micro-foundation for the labor productivity function in (5) from a spatialmodel with capital and land in in production.

B.2.1 Model with Other Factors in Production

Preferences. Consider a country c composed of regions and sectors. We assume that all individualsin the country have identical preferences for consumption goods, which take the nested CES structure

47The homogeneity of Φj(· ) follows immediately from the definition of Ij . To see that∂Φj∂ωj≥ 0 and

∂Φj∂ωi≤ 0, notice that Ii (ωc) ⊂ Ii (ωc) and Ij (ωc) ⊂ Ij (ωc) whenever ωj > ωj and ωi = ωi.

22

that generates the trade shares introduced in Section 2. We assume that labor supply is generated by theclass of spatial assignment modes in Section B.1.3. In this case, labor supply is a function of the vector ofafter-tax real wages in the country:

Li = Φi

((1 + ρj)

wjPj

j

),

where ρi is an ad-valorem income tax rate in region i.

Production. Each sector has a production function that is linear in a regional composite good:

Qi,s = ζi,sQi.

The production function of the composite good in each region uses labor, capital, and land:

Qi = ΨQi (L)

(Ki

αKi

)αKi ( LiαLi

)αLi ( Ti

αTi

)αTi(B.14)

where αLi + αKi + αTi = 1.

Endowments of capital and land. Land is immobile across regions. Each region has an endowmentTi of land. In contrast, we assume that capital is fully mobile across regions. The country has a capitalendowment of Kc.

Similar to Caliendo et al. (2018), there is a national mutual fund that owns all the land and capitalendowments of the country. We assume that the central government distributes dividends to workers indifferent regions such that

1 + ρi = Piχ.

This implies that the labor supply in region i is

Li = Φi (χwii) . (B.15)

The tax rate χ guarantees budget balance:∑i

(χPi − 1)wiLi =∑i

RiTi +∑i

riKi. (B.16)

Labor productivity function. Profit maximization implies that the price of the composite intermediategood in region i is

pi =1

ΨQi (L)

(wi)αLi (Ri)

αTi (ri)αKi (B.17)

To obtain the equilibrium level of Ri, consider the market clearing condition for the land in region i.Using the fact that the regional spending shares on land and labor are αTi and αLi , we have that

Ti = Ti =αTiαLi

wiLiRi

and, therefore,

Ri =αTiαLi

wiLiTi

(B.18)

23

Since capital is fully mobile across regions, its rental price is identical everywhere: ri = rc for all i.The rental rate must satisfy the capital market clearing condition, rcK =

∑i rcKi. Using the fact that a

share αKi of the regional revenue is spent on capital, we get the following expression for the equilibriumrent:

rc =1

Kc

∑i

αKiαLi

wiLi. (B.19)

By plugging the expressions for Ri and rc in (B.18)–(B.19) into the expression for pi in (B.17), we getthat

pi =wi

ΨQi (L)

(αTiαLi

LiTi

)αTi 1

K

∑j

αKj

αLj

wjwiLj

αKi

As argued in Section B.1.3, the labor supply equation in (B.15) is invertible up to a scalar, so we canwrite

χwjχwi

= Φ−1i,j (L). Thus,

pi =wi

ΨQi (L)

(αTiαLi

LiTi

)αTi 1

K

∑j

αKj

αLjΦ−1i,j (L)Lj

αKi

,

which implies that

pi =wi

Ψi (L)

where

Ψi (L,P ) ≡ ΨQi (L)

(αTiαLi

LiTi

)−αTi 1

K

∑j

αKj

αLjΦ−1i,j (L)Lj

−αKi .For completeness, we can also derive the tax rate χ from the budget balance condition in (B.16).

Using the fact that shares αTi and αK are spent on land and capital, we can write

χ =

∑iwiLiαLi∑

i PiwiLi.

B.3 Welfare Gains with Endogenous Labor Supply

B.3.1 Equivalent variation and Real Wages

We consider an environment similar to that of Section B.1.1, but assume that each market has anexogenous number of individuals. Specifically, we consider a representative household in each marketi whose homothetic preferences over consumption and hours worked are given by the utility functionVi(C,H). We assume that Vi(C,H) is differentiable, strictly quasi-concave, and strictly increasing in(C,−H). We further assume that C is a consumption aggregator that takes the nested CES form leadingto the spending shares in (6) and the price index in (8).

In general, given a price index Pi, a nominal wage wi and an exogenous transfer Ti, the budgetconstraint of the representative household in market i is PiCi = wiHi + Ti. Thus, the representativehousehold in market i solves the following utility maximization problem:

V ∗i

(wiPi,TiPi

)≡ max

HVi

(wiPiH +

TiPi, H

). (B.20)

24

We consider the welfare consequences of an arbitrary change in prices and wages from (w0i , P

0i ) to

(wi, Pi) in an economy without transfers (as in our baseline model, T 0i = Ti = 0). Following Mas-Colell et

al. (1995), we define the Equivalent Variation (EV) as the transfer that equates the welfare in the twoequilibria:

V ∗i

(w0i

P 0i

,EV

P 0i

)= V ∗i

(wiPi, 0

). (B.21)

We now show that the equivalent variation is increasing in the real wage change. Let us define thereal wage change as ωi = (wi/Pi)/(w

0i /P

0i ) and the following function whose root defines the equivalent

variation,

F

(EV

P 0i

, ωi

)= V ∗i

(w0i

P 0i

,EV

P 0i

)− V ∗i

(w0i

P 0i

ωi, 0

).

By the Implicit Function Theorem, there exists a local function EV (ωi) such that

∂EV

∂ωi= −

∂F∂ωi∂F∂EV

=

∂V ∗i

(w0i

P0i

ωi,TiPi

)∂(w/P )

w0i

P 0i

∂V ∗i

(w0i

P0i

,Ti+EV

P0i

)∂(T/P )

1P 0i

=∂Vi(Ci,Hi)

∂C

∂Vi(CEVi ,HEVi )

∂C

Hiw0i ,

where the last equality follows from the Envelope Theorem applied to (B.20). Since Vi(C,H) is increasingin C, this establishes that the equivalent welfare variation is increasing in the real wage change for anarbitrary change in wages and prices.

B.3.2 Special Case with Quasi-Linear Preferences

We now consider a special case in which we derive in closed form the equivalent variation as a share ofinitial income for an arbitrary change in wages and prices. We consider the same environment as above,but preferences are quasi-linear in consumption:

Vi(Ci, Hi) = Ci −H

1+ 1φ

i

1 + 1φ

.

In this case, the utility maximization problem implies that

Hi =

(wiPi

)φand Ci =

(wiPi

)1+φ

+TiPi,

so that the indirect utility in market i is

V ∗i

(wiPi,TiPi

)=

1

1 + φ

(wiPi

)1+φ

+TiPi.

Thus, after some manipulation, the definition of the equivalent variation in (B.21) implies that

EV

w0iH

0i

=1

1 + φ

[(ωi)

1+φ − 1], (B.22)

which is increasing in the real wage change.

25

C Online Appendix: Extensions

We now derive the excess demand shift and the spatial links matrix under extended versions of the baselinemodel. We first allow for trade imbalances in our baseline model, as in Dekle et al. (2007). Second, weallow for bilateral migration, as in Bryan and Morten (2015). Third, we allow for multiple labor typesin each region. Fourth, we extend our demand structure to cover a more general class of one-factorneoclassical models, as in Adao et al. (2017), but augmented to have spatial links in labor supply andlabor productivity. Lastly, we consider a model with input-output linkages in production, as Caliendoand Parro (2015), but again extended to have spatial links in labor supply and labor productivity.

C.1 Model with Trade Imbalances

C.1.1 Environment

We extend the baseline model in Section 2 by allowing for exogenous trade imbalances, as in Dekle et al.(2007). Specifically, the trade balance condition is

Ej = wjLj + Tj , ∀j, (C.1)

where the sum of all transfers in the world economy is equal to zero,∑

j Tj = 0. We assume that transfersare set in terms of the world’s average wage. As pointed out by Dekle et al. (2007), this implies thattransfer changes are invariant to the numeraire choice. Specifically, we impose that Tj = T (w)Tj with Tjand T (w) denoting respectively the number of transfer claims and the world’s output.

In this case, the market clearing condition is

wiLi =∑j

xij (wjLj + Tj) .

Equilibrium. To define the equilibrium wage vector, we combine the market clearing condition withequations (1), (5) and (6). This implies that the equilibrium wage vector must satisfy

∑j

∑s∈Si

(τij,swi

Ψi(Φ(w,P (w|τ )))

)−εs∑

o:s∈So

(τoj,swo

Ψo(Φ(w,P (w|τ )))

)−εs ξj,s (wjΦj(w,P (w|τ )) + Tj) = wiΦi (w,P (w|τ )) (wjLj) .

C.1.2 Counterfactual analysis

We now extend the counterfactual analysis of Section 3. We present the derivations below and focushere on the main implications of trade imbalances for our results. Notice that the equilibrium system hassimilar structure to that of our baseline model with the only difference being the exogenous transfer Tj .Thus, we obtain a similar system to characterize changes in relative wages:

γw = η (C.2)

whereη ≡ ηR − αφpηC (C.3)

andγ ≡ I −

(yι+ %+ χ

)+ α

(φw

+ φpx′)

(C.4)

26

with α ≡(I − yι+ χψ

)ρ and ρ ≡

(I + φ

px′ψ

)−1. The matrix ι is the diagonal matrix of output-

spending ratios, ι0ii ≡ Y 0i /E

0i . The elements of the matrix % measure the elasticity of i’s demand to

changes in transfers to j′ driven by wage changes in different markets j: %ij =∑

j′ y0ij′(1− ι0j′j′)y0

j where

y0j ≡ Y 0

j /∑

j′ Y0j′ is j’s share in world output.

Compared to the expressions in the baseline model, we can see that the revenue shares in the multiplierα and in the spatial links matrix γ are now multiplied by the imbalance ratio diagonal matrix ι. Thisrepresents the change in other markets’ spending, i.e. y0

ijijj , triggered by wages and trade costs changesholding constant transfers. The matrix %, instead, is the change in spending coming from transfers, whichmay change because the world’s average wage changes in response to the shock (relative to the numerairewm ≡ 1).

C.1.3 Derivation of equation (C.2)

We start by totally differentiating the market clearing equation:

wi + Li = ηRi +∑j

xijEjYi

Ej +∑j

∂ log Yi∂ log pj

pj

where ηRi is defined in Proposition 2. Notice that (5) implies that pi = wi −∑

j ψijLj , and Ei = Yi + Ti

implies that Ei = YiEi

(wi + Li) + TiEiTi. Thus, in matrix form:

w + L = ηR + yι(w + L

)+ %w + χ

(w − ψL

). (C.5)

By rearranging this expression, we get(I − yι− %− χ

)w +

(I − yι+ χψ

)L = ηR

Applying the result in equation (21) into this expression, we get(I − yι− %− χ

)w +

(I − yι+ χψ

) (ρφ

ww + ρφ

p(ηC + x′w

))= ηR,

which implies that[(I − yι− %− χ

)+(I − yι+ χψ

)ρ(φw

+ φpx)]w = ηR −

(I − yι+ χψ

)ρφ

pηC .

The definitions of α and γ imply that this expression is equivalent to (C.2).

C.2 Model with Bilateral Migration

C.2.1 Environment

Labor Supply We assume that bilateral migration flows from i to j are given by the following function:

Mij = Φij(w,P ). (C.6)

As discussed in Section B, this general specification covers the environment in Bryan and Morten(2015) where individuals born in region i have heterogeneous region-specific efficiency and make a discretechoice of which region j to reside and work. We allow bilateral labor supply to be a function of the vectorof wages and prices in all markets. In this case, spatial links can be summarized by the elasticity of the

27

bilateral labor supply to changes in wages and prices in different markets:

φwij,o ≡∂ ln Φij(w,P )

∂ lnwoand φpij,o ≡

∂ ln Φij(w,P )

∂ lnPo.

In equilibrium, labor supply in region j is the sum of the labor supply originated in different markets:

Lj = Φj(w,P ) ≡∑i

Φij(w,P ). (C.7)

Thus, spatial links in labor supply are determined by the elasticity structure of the bilateral laborsupply function:

φwjo ≡∂ ln Φj(w,P )

∂ lnwo=∑i

mijφwij,o and φpjo ≡

∂ ln Φj(w,P )

∂ lnPo=∑i

mijφpij,o (C.8)

where mij is the share of the labor force of j coming from origin market i.

Productivity and Trade Demand We assume that the production technology and demand for goodsare identical to those in the baseline model, so that equations (3)–(8) still hold.

Equilibrium Conditional on the labor supply function in (C.7), the excess labor demand in (9) remainsvalid. The equilibrium wage vector then solves the excess demand system in (10).


Given that the equilibrium remains the same, all the results in Section 3 still hold for the labor supplyelasticity matrices in (C.8). However, this extended version of the model yields predictions regardingchanges in bilateral migration flows. Equation (C.6) immediately implies that

Mij =∑o

φwij,owo +∑o

φwij,oPo,

where, as in the baseline model, P = ηC + x0′(w − ψL

), w is given by (17), and L is given by (21).

C.3 Model with Multiple Labor Types

C.3.1 Environment

We assume that the production in region r and sector s of market i uses multiple types of workers, indexedwith g. There are N markets and G groups. In this section, we use the notation x to indicate a stackedvector of size M × 1, where M = GN , such that the first N rows are the variables for group g = 1, therows from N + 1 to 2N are the variables for group g = 2, and so on. We use the notation ¯x to indicatematrices of length M with the same stacked configuration.

Trade Demand. We maintain the same nested gravity trade demand of our baseline model such thatthe spending share of market j on goods produced in i is

xij(p|τ ) =∑s∈Si

(τij,spi)−εs∑

o:s∈So (τoj,spo)−εs ξj,s, (C.9)

28

and the price index in market j is

Pj(p|τ ) =∏s

[ ∑o:s∈So

(τoj,spo)−εs

] ξj,s−εs

. (C.10)

This assumption effectively imposes that all worker groups in market i have identical preferences, sothat they have the same spending shares and price indices in a given market. Thus, the stacked vector ofprice indices P has identical entries for all groups in the same market.

Production. We assume that the production function is a CES across different worker groups:

Qr,s = Ψi

(L)(∑

g

ϑ1ρ

ig (Lrsg)ρ−1ρ

) ρρ−1

,

where ρ is the elasticity of substitution between labor types. The cost minimization problem of therepresentative firm implies that the zero profit condition is

pi =

[∑g (wig)

1−ρ] 1

1−ρ

Ψi

(L) . (C.11)

Notice that we have now labor productivity links that depend on group-level employment. However,the production function imposes that the effect is the same on all worker groups employed in a region.Accordingly, the matrix summarize spatial links in labor productivity has dimension N ×M :

ψijg ≡∂ ln Ψi(L)

∂ lnLjg.

Labor Supply. The labor supply function of each worker type g is

Lig = Φig(w, P ), (C.12)

where Φig(· ) is strictly positive, differentiable, bounded from above, and homogeneous of degree zero in(w, P ). We use again the matrices of labor supply elasticities to changes in wages and prices to summarizethe economy’s spatial links in labor supply,

φwig,jb ≡∂ ln Φig(w, P )

∂ lnwjband φpigj ≡

∂ ln Φig(w, P )

∂ lnPj.

We will use the notation¯φw to denote the matrix M ×M of labor supply elasticity to wages. We

allow the supply of a worker group to depend on relative wages across groups. In addition, we denote¯φp

as the M ×N matrix of labor supply elasticity to changes in the price index. It has N columns becausethe price index is the same for all groups in the same market.

Market Clearing. Since all sectors have the same CES labor aggregator across worker types, thespending share on group g for each sector s is

µig ≡ϑig (wig)

1−ρ∑b ϑib (wib)

1−ρ . (C.13)

29

Thus, the market clearing condition for each group is

wigLig = µigYi(p|τ ) = µig∑j

xij(p|τ )Ej . (C.14)

Finally, we assume again trade balance so that total expenditures in market j are also equal to thesum of the labor income of all groups in that markets:

Ej =∑g

wjgLjg.


We now extend the counterfactual analysis of Section 3. We present the derivations below and focus hereon the main implications of trade imbalances for our results. In this case, we obtain a similar systemdetermining relative wage changes across both groups and markets:

¯γ ˆw = ˆw (C.15)

whereˆη ≡ ˆηR − ¯α

¯φpηC

¯γ ≡ ρI −((ρ− 1)I + ¯y + ¯χ

)¯µ+ ¯α

(¯φw +

¯φpx′ ¯µ

)with ¯α ≡

(I − ¯y ¯µ+ ¯χ

¯ψ)(I +

¯φpx′

¯ψ)−1

.

It is important to notice that the economy with multiple labor groups has a similar shift in excesslabor demand. Here, the ˆηR is just the stacked vector of revenue shifts with the entry of group g inmarket i given by the same ηRi defined in Proposition 2 and ηC is the same stacked vector of consumptioncost shifts ηCi defined in Proposition 2.

Relative to the baseline model, the economy with multiple labor groups entails two modifications.First, the substitution effect in the spatial links matrix also accounts for the substitution across workergroups in the same market. This is captured by the terms multiplied by the elasticity of substitution ρ.Second, changes in production costs and market size also account for the initial share of each group in themarket’s spending, µjg. This is captured by the N ×M matrix ˜µ whose row j has entries µjg for columnin the j-th entry for the sub-vector of group g and zero in all other entries.


Using the definition of ηRi in Proposition 2, the total differentiation of the market clearing condition (C.14)implies

wig + Lig − µig = ηRi +∑j

xijEjYi

Ej +∑j

∂ log Yi∂ log pj

pj . (C.16)

From (C.13), the change in the labor share of g is

µig = (1− ρ)wig − (1− ρ)∑b

µibwib, (C.17)

30

and, from (C.11), the change in production cost of i is

pi =∑g

µigwig −∑j,g

ψijgLjg. (C.18)

Plugging equation (C.17) and (C.18) into (C.16), we obtain

wig + Lig − (1− ρ) (wig −∑

b µibwib) = ηRi +∑

j yij∑

b µjb

(wjb + Ljb

)+

∑j χij

(∑b µjbwjb −

∑j′,b′ ψjj′b′Lj′b′

).

We can write this system in matrix form:

ˆw + ˆL− (1− ρ)(

ˆw − ¯µ ˆw)

= ˆηR + ¯y ¯µ(

ˆw + ˆL)

+ ¯χ(

¯µ ˆw − ¯ψ ˆL),

where (i) ˆηR is the M × 1 vector with ηiR for all groups g in market i, (ii) ˜µ is the N ×M matrix where

row j has entries µjg for column in the j-th entry for the sub-vector of group g and zero in all otherentries, and (iii) ˜y and ˜χ are the M × N matrices where the row of group g in market i has entriesyijNj=1 and χijNj=1, respectively. By rearranging this expression, we get that(

I − ¯y ¯µ+ ¯χ¯ψ)

ˆL+(I − (1− ρ)

(I − ¯µ

)− ¯y ¯µ− ¯χ¯µ

)˜w = ˆηR (C.19)

We then turn to the change in employment implied by the shock. Using the definition of ηCi inProposition 2, the total differentiation of the labor supply equation in (C.12) and the price index in (C.10)imply that

Lig =∑j,b

φwigjbwjg +∑j

φpigj

(ηCj +

∑o

xoj po

).

Using the expression for production cost change in (C.18), this expression can be written as

Lig =∑j,b

φwigjbwjg +∑j

φpigj

ηCj +∑o

xoj

∑b

µobwob −∑j′,b

ψoj′bLj′b

.

Again, using the same definitions, we write this system in matrix form:

ˆL =¯φw ˆw +

¯φp(ηC + x′

(¯µ ˆw − ¯

ψ ˆL))

where, as in the baseline model, ηC is the N × 1 stacked matrix of consumption cost shifts across marketsand x is N ×N matrix of spending shares. By rearranging this expression, we obtain

˜L = ρ

¯φw ˆw + ¯ρ

¯φp(ηC + x′ ¯µ ˆw

)(C.20)

where ¯ρ ≡(I +

¯φpx′

¯ψ)−1

is the M ×M matrix that captures the multiplier employment effect of

endogenous changes in productivity, prices, and labor supply.The combination of the employment change in (C.20) and the market clearing condition in (C.19)

implies that(I − ¯y ¯µ+ ¯χ

¯ψ)(

¯ρ¯φw ˆw + ¯ρ

¯φp(ηC + x′ ¯µ ˆw

))+(ρI + (1− ρ)¯µ− ¯y ¯µ− ¯χ¯µ

)˜w = ˜ηR

31

Let us define ¯α ≡(I − ¯y ¯µ+ ¯χ

¯ψ)

¯ρ. By rearranging this expression, we get that[ρI −

((ρ− 1)I + ¯y + ¯χ

)¯µ+ ¯α

(¯φw +

¯φpx′ ¯µ

)]˜w = ˜ηR − ¯α

¯φpηC ,

which gives equation (C.15).

C.4 Model with Generalized Demand

We now generalize the nested CES trade demand in the model of Section 2. We follow Adao et al.(2017) by considering a general class of one-factor economies without restrictions on preferences andtechnologies. However, in contrast to Adao et al. (2017), we also allow for endogenous labor supply andlabor productivity in each market. As in Adao et al. (2017), it is useful to further refine the definition ofa market to include regions and sectors whose trade cost shocks change proportionally. In this case, wecan completely omit the sector subscript and think of a market i as a collection of regions and sectorswith identical changes in bilateral trade shifts, τij .

In order to model this general environment, we consider again a set of aggregate functions thatsummarize the implications of this alternative production structure for trade demand. We establish belowthe equivalence of this aggregate specification and a general one-factor Ricardian model with externaleconomies of scale.

C.4.1 Environment

Representative Household. We assume that each country has a representative household thatdecides the allocation of consumption and employment across markets. We denote the representativehousehold’s utility function as

Uc (C,L) ,

where C ≡ Cii and L ≡ Lii are respectively vectors of consumption and labor supply in all markets.We assume that Uc(· ) is twice differentiable, increasing in C, and quasi-concave in (C,L). We alsoassume that Cj is an index that aggregates quantities consumed of the differentiated goods produced indifferent origin markets,

Cj ≡ Vj (cj) ,

where cj ≡ ciji with cij denoting the consumption in market j of the good produced in market i. Weassume that the function Vj (· ) is twice differentiable, increasing, and quasi-concave in all arguments.Importantly, we also restrict Vj (· ) to be homogeneous of degree one, so that we can separate the problemof allocating spending shares across origin markets from the problem of determining labor supply acrossmarkets in the country.

We only allow for exogenous transfers across markets, so that the representative household faces thefollowing budget constraint: ∑

j

pijcij = wjLj + Tj ,

where wj is the wage, Pj is the price of the homogeneous good, and Tj is an exogenous transfer.

Trade demand. The homogeneity of Vj (· ) implies that, conditional on prices, the solution of thecost minimization problem yields the price index in market j:

32

Pj = Pj(pj)≡ min

cj

∑o

pojcoj s.t. Vj (cj) = 1, (C.21)

where pij is the price of the good produced in market i sold in market j.This problem yields a spending share on goods from origin i given by

xij = Xij

(pj). (C.22)

The price index and spending share functions inherit the usual properties of demand implied by utilitymaximization. The price index Pj (· ) is homogeneous of degree one, concave, and differentiable. Inaddition, Xij (· ) is a convex set, with a single element if Vj (· ) is strictly quasi-concave.

Labor Supply. We consider a competitive environment where, in deciding consumption and laborsupply, the representative agent takes as given prices and wages. Given the solution of the spendingminimization problem above, the budget constraint is PjCj = wjLj + Tj . Thus, the utility maximizationproblem yields the labor supply in market j:

Lj = Φj (w,P ) . (C.23)

Production. In each market, there exists a representative firm that operates under perfect competition.Production requires only labor and it is subject to external economies of scale. Market i’s productionfunction is

Yi = Ψi (L)Li,

where Ψi(· ) is a strictly positive real function. As in Section 2, the profit maximization problem impliesthat

pi =wi

Ψi (L). (C.24)

We also impose that there are iceberg trade costs to ship goods between markets, so that

pij = τijpi. (C.25)

Market Clearing. To close the model, we specify the labor market clearing condition. In each market,

wiLi =∑j

xijEj . (C.26)

Equilibrium. From equations (C.22) and (C.25), we can define the revenue of market i as

Yi(p,E|τ ) ≡∑j

Xij

(τijpij

)Ej .

This is the main change relative to our baseline model. The gravity structure of our model yields aspecific functional form for the revenue function. Instead, the general preference structure Vj(·) implies thatrevenue depends on the demand function Xij(·). The spatial links in trade demand are still summarizedby the elasticity of the revenue function to changes in production costs, which now has the followinggeneral form

33

χij ≡∂ lnYi(p,E|τ )

∂ ln pj=∑d

yidλidj , (C.27)

where λidj ≡ ∂ logXid∂ log pdj

is a general specification of the trade elasticity.

As in Section 2, to define the equilibrium of the economy, we need to solve for the price indexas a function of the wage vector w, Pj = Pj(w|τ ). From equations (8), (C.23), (C.24) and (C.25),Pj ∈ Pj(w|τ ) if, and only if,

Pj = Pj

(τij

wiΨi (Φ (w,P ))

j

)∀j. (C.28)

Finally, we can write the market clearing condition in terms of the wage vector. Using equations(C.23) (C.24) and (C.25), the market clearing condition in (C.26) is

∑j

Xij

(τij

wiΨi (Φ (w,P (w|τ )))

j

)(wjΦj (w,P (w|τ )) + Tj) = wiΦi (w,P (w|τ )) . (C.29)

Given the normalization that wm ≡ 1 for an arbitrary market m, equilibrium wage vector w mustsatisfy the market clearing condition in (C.29) for all i.


We now extend the counterfactual analysis of Section 3 for changes in the bilateral trade shifts τij . Tothis end, notice that the equilibrium of the economy has a similar structure as that of the baseline modelin Section 2. The main difference is that it entails general functions for the price index in (C.28) and thetrade demand in (C.29). Accordingly, we now show that the results of Section 3 still hold with modifieddefinitions for the revenue shift and the trade elasticity matrix.

In this case, the revenue shift is

ηRi (τ ) ≡∑j,o

∂ lnYi(p0,E0|τ 0)

∂ ln τojτoj =

∑o,j

y0ijλijoτoj . (C.30)

The more general demand function Xij(pj) in Yi(p,E|τ ) yields a different functional form for therevenue shift. In this case, the revenue shift is a function of the sensitivity of the demand for goods from iin different markets when the cost of exporting to those markets change. From Shepard’s Lemma, theprice index in equation (C.21) implies

P = ηC + x′p,

where ηC is defined in Proposition 2. Note that ηC does not depend on the assumptions on the demandsystem because it follows directly from applying the envelope theorem to the minimization problem in(C.21). Conditional on the new definitions in (C.27) and (C.30), we have exactly the same set of equationsas in the baseline model:

γw = η (C.31)

whereη ≡ ηR − αφpηC (C.32)

andγ ≡ I −

(yι+ %+ χ

)+ α

(φw

+ φpx′)

(C.33)

34

with α ≡(I − yι+ χψ

)ρ and ρ ≡

(I + φ

px′ψ

)−1.

C.4.3 Equivalence with a general one-factor Ricardian model with external economiesof scale

For completeness, we now show that the environment above is observationally equivalent to a generalone-factor Ricardian model with external economies of scale.

One-factor Ricardian model with external economies of scale. We consider an economy withan arbitrary number of goods indexed by s. We assume that each country has a representative agentwith preferences for consumption and labor supply in different markets, with utility function given by

Uc

(CNjj,LNjj

)such that CNj ≡ V N

(cNij,s

i,s

).

where V N (· ) is twice differentiable, quasi-concave, homothetic, and increasing in all arguments. Noticethat V N (· ) allows for the possibility that goods from different origins are imperfect substitutes.

Let pij,s be the price of good s from i in market j. The representative household’s budget constraint is∑i

∑s

pNij,scNij,s = wNj L

Nj + Tj .

There are many perfectly competitive firms supplying each good in any market. The productiontechnology uses only labor and entails external economies of scale at the market level. In particular, thetechnology of producing good s in market i and delivering to j is given by

Y Nij,s = Ψi

(LN) LNij.s

τijαNij,s,

where αNij,s is good-specific productivity of producing in i and delivering in j. We restrict trade costs tobe identical to all goods being traded between markets i and j: τij,s = τij for all s.

Equilibrium. We use the fact that V N (· ) is homothetic to derive the price index in market j:

PNj = PNj

(pNoj,k

o,k

)≡ minck,ojk,o

∑k,o

pNoj,kcoj,k s.t. V N(coj,ko,k

)≥ 1 (C.34)

where the associated spending share on good s from i is

xNij,s = XNij,s

(pNoj,k

o,k

). (C.35)

Conditional on prices, the representative household solves the utility maximization problem that yieldsthe labor supply in market j:

LNj = Φj(wN ,PN ). (C.36)

Profit maximization implies that

pNij,s = τijpNi α

Nij,s (C.37)

35

where

pNi =wNi

Ψi

(LN) . (C.38)

Finally, the labor market clearing condition is

wNi LNi =

∑j

∑s

xNij,s(wNj L

Nj

). (C.39)

The equilibrium can be written aspNi , w

Ni , L

Ni , P

Ni

i

solving (8)–(C.39) with Φj(· ), Ψj(· ), and

XNij

(τojp

No

o

)≡

xNij =

∑s

xNij,s : xNij,s = XNij,s

(τojp

No α

Noj,k

o,k

)(C.40)

such thatPNi

(τojp

No

o

)= PNi

(τojp

No α

Noj,k

o,k

).

Equivalence with environment in Section C.4.1. We now construct an equivalent economywhose with equilibrium vector satisfies a condition identical to (C.29). Production in market i is

Qi = Ψi (L)Li.

To deliver in market j, producers of i face iceberg trade costs so that

pij = τijpi.

In addition, each country has a representative agent with preferences for consumption and labor supplyin different markets, with utility function given by

Uc

(CNjj,LNjj

).

with

Vj(ciji

)≡ maxcij,si,s V

N(cij,si,s

)s.t.

∑s α

Nij,scij,s = cij ,

and associated spending shares given by

xij ∈ Xij

(τojpoo

).

It is straight forward to see that this alternative economy has the same structure as the economyintroduced in Section C.4.1. Thus, to formally establish the equivalence with the one-fact Ricardinaneconomy above, it is sufficient to show that

Xij

(τojpoo

)= XN

ij

(τojpoo

)∀ τojpoo , (C.41)

where XNij (·) is the function defined in (C.40).

Intuitively, the preference structure above implies that, if the representative household acquires cijunits of i′s composite good for j’s consumption, then it optimally allocates the composite good intothe production of different goods, given the exogenous weights αNz,ij that are now embedded into the

representative agent’s preferences. Since the relative price of goods in market i only depends on αNz,ij , this

36

decision yields allocations that are identical to those in the competitive equilibrium of the decentralizedeconomy.

First, we show that xij ∈ Xij

(τojpoo

)=⇒ ∃xNij,s ∈ XN

ij,s

(τojpoα

Noj,k

o,k

)with xij =

∑s x

Nij,s.

Let cz,ijz,i be the solution of the good allocation problem in the definition of Vj (cij). We proceed

by contradiction to show that cij,si,s implies spending shares, xij,si,s =τojpoα

Nij,scij,s

i,s

, such that

xz,ij ∈ XNij,s

(τojpoα

Noj,k

o,k

). Suppose there exists a feasible allocation

cNij,s

i,s

such that

V N(cNij,s

i,s

)> V N

(cij,si,s

)and

∑i

∑s

τijpiαNij,sc

Nij,s ≤ 1. (C.42)

Notice that∑

i

∑s τijpiα

Nij,sc

Nij,s ≤ 1, which implies that the allocation cNij ≡

∑z α

Nij,sc

Nij,s is feasible in

the equivalent economy. Thus,

V N(cij,si,s

)= Vj (cij) ≥ Vj

(cNij)≥ V N

(cNij,s

i,s

),

which is a contradiction of inequality (C.42).

Second, we show that xij =∑

s xNij,s, with xNij,s ∈ XN

ij,s

(τojpoα

Noj,k

o,k

), and cNij =

∑z α

Nij,sc

Nij,s imply

that xij ∈ Xij

(τojpoo

). We start with cNij =

∑z α

Nij,sc

Nij,s implied by the solution of the consumer’s

problem in the one-factor Ricardian economy. We proceed by contradiction to show thatcNij

i

is optimal

in the equivalent economy given prices τijpii. Suppose there exists a feasible allocation ciji in theequivalent economy such that

Vj (cij) > Vj(cNij)

and∑i

pijcij ≤∑i

pijcNij = 1.

Let cz,ijz,i be the be the solution of the good allocation problem in the definition of Vj (cij). Thus,∑i

τijpi∑z

αNij,scNij,s =

∑i

τijpicij ≤ 1

and, by revealed preference,

Vj(cNij)≥ V N

(cij,si,s

)≥ V N

(cij,si,s

)= Vj (cij) .

This is a contradiction, which establishes the result.

C.5 Model with Intermediate Goods in Production

C.5.1 Environment

As in our baseline model, we assume that all producers in a market face a single wage rate. To simplifyexposition, we now assume that all sectors are present in a market, so that markets can be interpreted asgeographic units where labor is perfectly mobile across sectors. We use S to denote the number of sectorsand N to denote the number of markets.

Labor Supply. As in the baseline model, the labor supply is market i is given by equation (1).

37

Production. Following Caliendo and Parro (2015), we assume a Cobb-Douglas production functionbetween labor and an intermediate inputs aggregator in sector s of market i:

Qi,s = Ψi (L)

(Li,s$i,s

)$i,s ( Mi,s

1−$i,s

)1−$i,s, such that Mi,s = Πk

(Qi,ksθi,ks

)θi,kswith

∑k θi,ks = 1. In each sector s, the demand for inputs from different origin markets is given by the

following constant elasticity function:

Qi,ks =

∑j

(Qji,ks)εk

1+εk

1+εkεk

where Qji,ks is the good produced by sector k in market j consumed in sector s of market i.

Input Trade Demand. The cost minimization problem of the representative firm implies that thezero profit condition is

pij,s =τij,spi,sΨi (L)

(C.43)

wherepi,s = (wi)

$i,s(PMi,s

)1−$i,s(C.44)

and

PMi,s =∏k

[∑o

(τoi,kpo,kΨo (L)

)−εk] θi,ks−εk

. (C.45)

The cost minimization problem above implies that the share of spending on sector k of market j bysector s of market i is

xMji,ks = xji,kθi,ks(1−$i,s)

where

xji,k =

(τji,kpj,kΨj(L)

)−εk∑

o

(τoi,kpo,kΨo(L)

)−εk . (C.46)

Final Trade Demand. As in the baseline model, we assume that final consumption follows a nestedgravity structure where the share of j′s final spending on sector k of market i is

xCji,k = xji,kξi,k,

where xji,k is the share of j’s spending in sector k on goods from i (as defined above). The shares xji,k arethe same in both final and intermediate spending shares because the elasticity of substitution across goodsof different origins is the same as the one in the input demand. Caliendo and Parro (2015) introduce thisassumption to circumvent the need of separately measuring trade flows for final and intermediate uses.

As in the baseline model, the consumption price index in market j is

Pj =∏s

[ ∑o:s∈So

(τoj,spo,sΨo (L)

)−εs] ξj,s−εs. (C.47)

38

Market clearing. The total spending by market i on sector k of market j is

Eji,k = xCji,k (wiLi) +∑s

xMji,ksYi,s.

Thus, total revenue of sector k of market j is

Yj,k =∑i

Eji,k,

and, therefore,

Yj,k =∑i

xji,k

[ξi,k (wiLi) +

∑s

θi,ks(1−$i,s)Yi,s

]. (C.48)

The labor market clearing condition is

wjLj =∑k

$j,kYj,k. (C.49)

Equilibrium. We use the production cost in (C.44) and the labor supply equation in (1) to write theprice indices in (C.45) and (C.47) as

Pj =∏s

∑o:s∈So

(τoj,s

(wo)$i,s

(PMo,s

)1−$i,sΨo (Φ(w,P ))

)−εsξj,s−εs

, (C.50)

PMi,s =∏k

∑o

τoi,k (wo)$o,k

(PMo,k

)1−$o,k

Ψo (Φ(w,P ))

−εk

θi,ks−εk

. (C.51)

We rewrite the sectoral market clearing condition in (C.48) using the production cost in (C.44), thegravity sectoral demand in (C.46), and the labor supply equation in (1):

Yj,k =∑i

(τji,k

(wj)$j,k(PMj,s)

1−$j,k

Ψj(Φ(w,P ))

)−εk∑

o

(τoi,k

(wo)$o,k(PMo,k)

1−$o,k

Ψo(Φ(w,P ))

)−εk[ξi,k (wiΦi(w,P )) +

∑s

θi,ks(1−$i,s)Yi,s

]. (C.52)

Finally, we use the labor supply equation in (1) to write the labor market clearing condition:

wjΦj(w,P ) =∑k

$j,kYj,k. (C.53)

The equilibrium is defined as vectors of wage rates w ≡ wii, consumption price indices P = Pjj ,sectoral input price indices PM = Pj,kjk, and sectoral revenues Y = Yj,kjk satisfying equations(C.50)–(C.53) for a given numeraire wage wm ≡ 1.

39


We now present the main implications of allowing for non-labor inputs in production. All derivations andmatrix definitions are presented below. We obtain a similar system determining relative wage changesacross markets:

γw = η. (C.54)

However, in this case, the shift in excess labor demand has the following form

η ≡ αRˆηR − αCφpηC + αM ′ˆηM .

Relative to the baseline model, there are three main differences in the excess labor demand shift. First,revenue shift is defined at the market-sector level:

ηRj,k ≡ −εk∑i

yji,k

(τji,k +

∑o

xoi,kτoi,k

).

This is because sectors within the same region have a different composition of intermediate inputs andthus a different production structure. The matrix αR simply weights the sector-level shifts using theLeontief inverse matrix that translates revenue shocks in on sector to shocks for its suppliers.

Second, the consumption cost shift is the same as in Proposition 2, since the trade demand for finalconsumption is the same as in the baseline. The labor supply multiplier αC , however, takes into accountalso for connection across sectors arising from production linkages.

Third, there is a shift in intermediate input cost for each sector-market:

ηMi,s ≡∑o,k

θi,ksxoi,kτoi,k. (C.55)

The matrix αM is the demand multiplier implied by changes in the cost of intermediate inputs throughthe production network.


We use bold variables with tilde to denote stacked vectors of sector-market variables, y ≡ [yi,s]is, boldvariables with two dots to denote matrices with sector-market to sector-market variables, y ≡ [yij,ks]is,ks,and bold variables with checks to denote matrices with market to sector-market variables y ≡ [yij,k]ik,j .

The production cost equation in (C.44) implies that

pi,s = wi$i,s + PMi,s (1−$i,s). (C.56)

Using this expression and the input price index in (C.45), we have that

PMi,s = ηMi,s +∑o,k

θi,ksxoi,k

wo$o,k + PMo,k(1−$o,k)−∑j

ψojLj

where

ηMi,s ≡∑o,k

θi,ksxoi,kτoi,k. (C.57)

40

In matrix form and invert it as

ˆPM = ˆηM + xM($w + (I − $) ˆPM − iψL

)where xM ≡ [θi,ksxoi,k]is,ok. We define $ as a NS ×N matrix with row of sector k in market o given by$o,k in the column o and zero in other columns, and i as a NS ×N matrix whose rows of sector s inmarket o has a one in the column of market o and zero in other columns.

Thus,ˆPM = λ

(ˆηM + xM

($w − iψL

))(C.58)

where λ ≡(I − xM (I − $)

)−1.

The consumption price index change is similar to what we had before:

Pi = ηCi +∑o,k

ξi,kxoi,k

wo$o,k + PMo,k(1−$o,k)−∑j

ψojLj

where

ηCi ≡∑o,k

ξi,kxoi,kτoi,k. (C.59)

Define the N × SN matrix xC′ ≡ [ξi,kxoi,k]i,ok. The price index equation above can be written in matrixform:

P = ηC + xC′($w + (I − $) ˆPM − iψL

).

By applying equation (C.58) into this expression,

P = ηC + xC′(I − $)λˆηM + xC′((I + (I − $)λxM

)$w −

(I + (I − $)λxM

)iψL

). (C.60)

From the labor supply equation in (1),

L = φww + φ

pP .

Using (C.60),

L = φww+ φ

pηC + φ

pxC′(I − $)λˆηM + φ

pxC′

((I + (I − $)λxM

)$w −

(I + (I − $)λxM

)iψL

)Thus,

L = ϕww + φp(ηC + xC′(I − $)λˆηM

)(C.61)

whereϕw ≡

(φw

+ φpxC′

(I + (I − $)λxM

)$)

ρ ≡(I + φ

pxC′

(I + (I − $)λxM

)iψ)−1

.

Together, expressions (C.56), (C.58) and (C.61) imply that

ˆp =[(I + (I − $)λxM

)$ − (I − $)λxM iψϕw

]w

+ (I − $)λ[xM iψφ

pηC +

(I + xM iψφ

pxC′(I − $)λ

)ˆηM].

(C.62)

41

From equation (C.48),

Yj,k =∑i

Eji,kYj,k

[xji,k +

xCji,kwiLi

Eji,k

(wi + Li

)+∑s

xMji,ksYi,s

Eji,kYi,s

],

which implies that

Yj,k =∑i

yji,k

[xji,k + πji,k

(wi + Li

)+ (1− πji,k)

(∑s

µji,ksYi,s

)]

where yji,k ≡Eji,kYj,k

is the revenue share of market i from sales to market j in sector k; πji,k ≡xCji,k(wiLi)

Eji,kis

the fraction of final consumption in total spending of market i on sector k of market j; µji,ks ≡xMji,ksYi,s∑s x

Mji,ksYi,s

is the fraction of intermediate inputs purchases of sector s in all intermediate input flows from sector k ofmarket j to market i.

From equation (C.46),

xji,k = −εk

(τji,k + pj,k −

∑d

ψjdLd

)+∑o

xoi,kεk

(τoi,k + po,k −

∑d

ψodLd

).

Let us define the revenue shift of sector k of market i as

ηRj,k ≡ −εk∑i

yji,k

(τji,k +

∑o

xoi,kτoi,k

)

and the revenue elasticity of sector k of market j to changes in the production cost of sector k in marketo as

χoj,k =∑i

yji,k(xoi,kεk − I[i=j]

)εk.

Applying these definitions to the equations above, we get that

Yj,k = ηRj,k +∑o

χoj,k

(po,k −

∑d

ψodLd

)+∑i

yji,kπji,k

(wi + Li

)+∑s

∑i

yji,k (1− πji,k)µji,ksYi,s.

We can then write this system in matrix form:

ˆY = ˆηR + χ(

ˆp− iψL)

+ π(w + L

)+ µ ˆY ,

where χ is the SN × SN matrix with row of market j sector k given by χoj,k for the columns of marketso in sector k and zeros for the columns of other sectors; µ ≡ [yji,k (1− πji,k)µji,ks]jk,is is the SN × SNmatrix with input revenue shares; andπ is the SN ×N matrix with row of market j sector k given byyji,kπji,k for the columns of market i.

Thus,ˆY =

(I − µ

)−1 (ˆηR + χ

(ˆp− iψL

)+ π

(w + L

)). (C.63)

Finally, we need to solve for the labor market clearing condition. Define `jk ≡$j,kYj,kwjLj

as the share of

42

labor income coming from sector k in market j. From C.53,

wj + Lj =∑k

`j,kYj,k,

which implies that

w + L = ˇ′ ˆY

where ˇ′ is the N ×NS matrix with row j with entries `j,k in the columns of market j for different sectorsk and zeros in the columns of other markets. By replacing equation (C.63) into the expression above, wehave

w + L = ˇ′(I − µ

)−1 (ˆηR + χ

(ˆp− iψL

)+ π

(w + L

)),

so that(I − ˇ′

(I − µ

)−1π

)w +

(I − ˇ′

(I − µ

)−1 (π − χiψ

))L = ˇ′

(I − µ

)−1 (ˆηR + χˆp

).

Substituting expressions (C.61) and (C.62) into this expression,

γw = η

where

γ ≡ I +

(I − ˇ′

(I − µ

)−1 (π − χiψ

))ϕw

− ˇ′(I − µ

)−1 (π + χ

[(I + (I − $)λxM

)$ − (I − $)λxM iψϕw

])η ≡ αRˆηR − αCφpηC + αM ′ˆηM

and

αR = ˇ′(I − µ

)−1

αC = I − ˇ′(I − µ

)−1 [π − χ

(I − (I − $)λxM

)iψ]

αM ′ = ˇ′(I − µ

)−1χ(I − $)λ

(I + xM iψφ

pxC′(I − $)λ

)−

(I − ˇ′

(I − µ

)−1 (π − χiψ

))φpxC′(I − $)λ.

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