General Equilibrium Effects in Space: Theory and Measurement

Layer 1: Overview

How do international trade shocks propagate through spatially connected regional labor markets, and how large are the general equilibrium effects that standard shift-share specifications miss? Adão, Arkolakis, and Esposito address this question by extending shift-share empirical designs to incorporate general equilibrium (GE) effects arising from spatial links between markets. Their motivation is that the difference-in-difference logic of standard shift-share regressions recovers only the differential response of treated versus control regions, not the level response that includes indirect (spillover) effects propagating through trade, labor supply, and agglomeration links. Ignoring these indirect effects biases estimates of trade shocks’ aggregate labor market consequences.

The theoretical framework is a multi-sector general equilibrium spatial model with N markets linked through three channels: (i) gravity-type trade demand, (ii) endogenous labor supply that depends on wages and price indices in all markets, and (iii) local labor productivity that depends on employment (agglomeration). The key theoretical result is that wage and employment responses to trade shocks decompose into two shift-share exposure vectors — a revenue exposure (proportional to the ADH import penetration measure, weighted by sectoral employment shares) and a consumption cost exposure (weighted by sectoral spending shares) — multiplied by bilateral reduced-form elasticity matrices (βij and φij). These elasticities are sufficient statistics for GE aggregation and can be expressed as a series expansion of the “spatial links” matrix, which is itself a function of trade demand substitution, labor supply substitution, and agglomeration elasticities. When demand substitution dominates (gross substitution property holds), indirect effects reinforce direct effects: a negative revenue shock in one CZ reduces demand for goods from other CZs, propagating wage and employment losses outward.

The authors apply the framework to the China shock, using 722 U.S. Commuting Zones (CZs) over 1990–2007, following Autor, Dorn, and Hanson (2013) (ADH). The revenue exposure measure is identical to the ADH instrumental variable (employment-share-weighted Chinese export growth to non-U.S. developed countries); the consumption exposure is analogously constructed using sectoral spending shares from input-output tables. Structural parameters are estimated using a Model-implied Optimal IV (MOIV) two-step GMM estimator derived from Chamberlain (1987).

Main quantitative findings: (1) In a simple extension of ADH, the indirect revenue spillover effect on neighboring CZs is roughly three times larger in magnitude than the direct effect of a CZ’s own import competition exposure — an increase of $1,000 in Chinese imports per U.S. worker in nearby CZs is associated with 1.3 log-point lower employment growth and 1.0 log-point lower wage growth in a given CZ. (2) Consumption cost shifts (cheaper imports) have no statistically significant direct or indirect effect on employment or wages, consistent with a weak price elasticity of labor supply relative to the wage elasticity. (3) Structural parameter estimates yield: labor productivity–employment elasticity ψ = 0.56 (agglomeration), labor supply–wage elasticity φw = 2.11, labor supply–price elasticity φp = −1.36, trade elasticity ε = 3.94. (4) In GE aggregation, the China shock reduced average U.S. CZ wages by approximately 4.0 log-points and employment by approximately 2.8 log-points between 1990 and 2007, with the indirect revenue channel (−4.24 log-points for wages, −4.95 log-points for employment) dominating the direct revenue effect (−0.81 and −1.94 respectively) and being partially offset by positive consumption cost effects (+0.98 wages, +3.18 employment). Average real wages rose by 0.16 log-points on net, but 39% of CZs experienced real wage declines. Standard deviations of responses were 1.30 for wages, 3.31 for employment, and 1.75 for real wages, indicating large cross-CZ heterogeneity. (5) Model fit: the baseline estimated model yields fit coefficients close to 1 (0.67 for wages, 0.90 for employment), whereas quantitative models calibrated with Ricardian/standard parameters yield fit coefficients of 3.56 to 10.42, indicating their predicted responses are too small by factors of 4–10. Simple aggregation of the ADH specification implies employment losses of only 1.5 log-points — less than half the authors’ baseline estimate.

The key mechanism driving the amplification is strong agglomeration (ψ ≈ 0.56), which roughly doubles typical calibrations from Krugman-type models and is absent in Ricardian frameworks. Demand-side trade links propagate revenue shocks across CZs with similar sectoral composition and trade partners. The policy implication is that analyses of trade shocks using standard shift-share regressions — which absorb common indirect effects in time fixed effects — systematically understate aggregate employment and wage losses.

Layer 2: Deep Dive

What is the identification strategy and what are the main threats to it?

Identification rests on the same orthogonality condition used by ADH and Kovak (2013): observed shock exposure (revenue and consumption shift-share measures) is mean-independent of unobserved residuals. This is implied by independence between the observed Chinese export shock and unobserved trade cost shocks, given the initial trade matrix. The authors use the ADH instrument (Chinese export growth to non-U.S. developed countries) to construct exogenous sectoral shifts, exploiting cross-CZ variation in initial industry composition. The main threats are: (i) unobserved shocks correlated with pre-existing industry composition (e.g., concurrent automation), addressed by controlling for lagged population growth (following Greenland et al. 2019) and the full ADH control set; (ii) spatial correlation of residuals, addressed by clustering standard errors at the state level and by robustness using the inference procedure in Adão et al. (2019); (iii) simultaneity, since the MOIV estimator instruments the non-linear functions of shock exposure with model-implied moment functions that are functions of the observed shifts only.

What are the two shift-share exposure measures and how do they differ?

The revenue exposure (IPW) is the standard ADH shift-share variable: the product of Chinese export growth to other developed countries and the CZ’s initial employment share in each sector, summed across sectors. It captures the shock to the demand for a CZ’s goods. The consumption cost exposure (IPC) is an analogous variable where the share is the CZ’s sectoral spending share (including intermediate inputs, constructed using national input-output tables interacted with regional employment shares) rather than employment share. It captures the shock to the CZ’s cost of living and input costs. The two measures have a spatial correlation of 0.34. Standard deviations across CZs are 2.52 for IPW and 1.22 for IPC.

What are the main mechanisms and how are they distinguished empirically?

Three spatial channels determine GE reduced-form elasticities: (1) trade demand links — markets with similar sectoral composition and trade partners are closer substitutes, so a revenue shock in one CZ propagates negatively to CZs competing for the same export destinations; (2) labor supply links — employment responses in one CZ to wage/price changes in another, captured through migration (parametrized by bilateral birth-state shares) and the local wage and price elasticities of labor supply; (3) agglomeration — local labor productivity responds positively to local employment, amplifying both direct and indirect effects. Empirically, the authors distinguish these by estimating separate parameters (ψ for agglomeration, φw for wage elasticity of labor supply, φp for price elasticity, φm for migration links, ε for trade elasticity), with identification coming from cross-CZ heterogeneity in bilateral trade shares, sector specialization, and migration shares. The weak IPC effect (statistically insignificant) points to a small φp, while the large employment and wage responses to IPW point to large φw and ψ.

What are the estimated structural parameters and how do they compare to existing literature?

Panel A estimates (without migration): ψ = 0.56 (s.e. 0.07), φw = 2.11 (s.e. 0.25), φp = −1.36 (s.e. 0.24), ε = 3.94 (s.e. 0.41). Panel B (with migration): nearly identical point estimates but standard errors two to five times larger due to high collinearity of bilateral migration and trade shares; φm = −0.06 (s.e. 0.05), not statistically significant. The agglomeration elasticity ψ = 0.56 is roughly twice the Krugman (1980) implied value (~0.2) used by Monte et al. (2018) and far above zero (used in Ricardian frameworks by Galle et al. 2017, Caliendo et al. 2018, 2019). It is closer to Kline and Moretti (2014)’s estimate of ~0.4 from regional demand shocks. The labor supply elasticity φw = 2.11 is three times the median micro-estimate in Chetty et al. (2013) and is consistent with aggregate employment responses. The trade elasticity ε ≈ 4 is within standard literature ranges.

What heterogeneity in spatial effects is documented?

There is substantial heterogeneity in both direct and indirect reduced-form elasticities across CZs. For revenue shifts, the 10th/50th/90th percentiles of direct wage elasticities are 0.44/0.67/1.67, and for employment 0.92/1.46/3.97. For indirect effects, median values are 0.002 (wages) and 0.003 (employment), but the 90th percentile is 0.021 and 0.039 respectively. The simple gravity proxy zij (inverse distance weighted by population) explains only a small fraction of variation in indirect effects; instead, the elements of the full spatial links matrix (bilateral revenue shares yij and trade demand substitutability χij) explain roughly 50% of variation in indirect effects across CZ pairs. Both manufacturing and non-manufacturing employment show significant indirect effects; wage responses are mainly driven by the non-manufacturing sector (consistent with ADH). 39% of CZs experienced real wage declines despite a small average real wage gain.

What robustness checks are run?

For the simple ADH extension (Table 1): (i) varying the distance decay parameter δ ∈ (1,8); (ii) using CZ size vs. no size weighting in zij; (iii) restricting to same-state CZs for indirect effects; (iv) weighting CZs by 1990 population; (v) using the Adão et al. (2019) inference procedure; (vi) alternative spending share constructions. For the structural estimation: (i) allowing for trade imbalances (following Dekle et al. 2007); (ii) calibrating migration links from external estimates; (iii) alternative numeraire for labor supply homogeneity (national vs. world price index). In all cases, indirect effects remain negative and significant, and reduced-form elasticities are highly correlated with baseline estimates. Counterfactual employment losses range from −0.5 to −5.4 log-points depending on the labor supply normalization and migration specification, with average wage decline remaining close to 4 log-points across specifications. The NTR gap (Pierce and Schott 2016) as the sector-level shifter also yields qualitatively similar results.

How does the paper evaluate the fit of quantitative spatial models?

The authors propose regressing actual changes in CZ employment/wages on model-predicted responses (equation 39) and checking whether the slope coefficient ρ is close to 1. A coefficient much greater than 1 means the model’s predicted responses are too small relative to actual cross-CZ variation. The baseline structural estimates yield fit coefficients of 0.67 (wages) and 0.90 (employment) — close to 1. Alternative calibrations from quantitative frameworks yield coefficients of 3.56–10.42 for wages and 6.60–10.42 for employment, indicating those models underpredict differential responses by factors of 4–10. The main driver is weak agglomeration forces: setting ψ = 0 (Ricardian) vs. ψ = 0.56 (baseline) dramatically degrades fit. Setting φw = −φp (labor supply responding to real wages only, as in Caliendo et al. 2019) makes employment fit estimates very imprecise because the consumption price channel becomes too strong relative to its empirical counterpart.

What is the quantitative GE impact of the China shock on average U.S. CZ wages and employment, and how does it decompose?

Over 1990–2007: average wage fell by 3.98 log-points (s.d. 1.30), average employment fell by 2.78 log-points (s.d. 3.31), average real wage rose by 0.16 log-points (s.d. 1.75). Decomposition of wage change: direct revenue effect −0.81 (s.d. 1.79), direct consumption cost effect +0.98 (s.d. 1.36), indirect revenue effect −4.24 (s.d. 1.71), indirect consumption cost effect +0.09 (s.d. 1.18). The indirect revenue channel dominates; consumption gains are not large enough to offset revenue losses. For real wages, the main components are: terms-of-trade loss from wage decline (−0.98, s.d. 2.53), productivity/efficiency gains (+3.14, approximately), and consumption cost gains. Most impact occurred in the 2000–2007 sub-period after China’s WTO accession.

How do these GE estimates compare to estimates from the existing literature?

Simple aggregation of the ADH specification (ignoring GE indirect effects) implies average wage losses of 1.17 log-points and employment losses of 1.50 log-points — less than half the authors’ GE estimates. Including intuitive distance-weighted indirect effects (ADH extension in Table 1 column 3) brings employment estimates closer (−4.51 log-points) but with correlation below 0.5 with baseline cross-CZ heterogeneity predictions. Quantitative spatial models calibrated with standard parameters (Ricardian, weak agglomeration) generate average responses near zero and are often uncorrelated with actual CZ outcomes. The key reason quantitative models underperform is that they specify agglomeration forces as too weak (ψ ≈ 0 versus the estimated 0.56) and labor supply sensitivity to import prices as too strong relative to wage sensitivity.

What is the role of the consumption cost (IPC) channel and why does it matter less than the revenue channel?

The IPC captures the welfare gain from cheaper Chinese imports: as Chinese productivity rises, import prices fall, increasing real purchasing power and potentially stimulating labor supply. However, the estimated labor supply price elasticity (φp = −1.36) is substantially smaller in absolute value than the wage elasticity (φw = 2.11), so the positive employment and wage response to lower import prices is weaker than the negative response to falling demand for local output. Empirically, both the direct and indirect effects of IPC are statistically insignificant in the simple ADH extension (Table 1, columns 2 and 4), consistent with weak φp. The structural estimation exploits all channels to pin down φp precisely. Input-output linkages (CZs using inputs from sectors with stronger Chinese export growth) are incorporated in IPC and are also found to have no significant employment effect, consistent with Pierce and Schott (2016) and Acemoglu et al. (2016).

How does the paper connect to the shift-share and market access literatures?

The paper generalizes standard shift-share designs (Bartik 1991, Blanchard and Katz 1992, ADH 2013, Kovak 2013) in two ways: it adds a consumption cost shift-share (spending shares instead of employment shares) and it adds indirect exposure from other CZs’ shift-share measures, weighted by model-implied bilateral reduced-form elasticities. Unlike standard designs, time fixed effects in the authors’ estimating equation absorb only the mean unobserved shock, not any GE indirect effects (since the latter are heterogeneous across CZ pairs). The paper connects to the market access approach (Redding and Venables 2004; Donaldson and Hornbeck 2016) by showing that the authors’ revenue and consumption exposure measures are partial-equilibrium versions of producer and consumer market access, holding wages and employment constant. The key advantage is that the authors’ measures can be constructed from initial-equilibrium data without solving the full GE model.

What are the policy implications and their scope conditions?

The paper implies that trade shock analyses ignoring GE spillovers substantially understate aggregate employment and wage losses for U.S. workers. The gross substitution condition (trade demand links dominating labor supply links) is required for indirect effects to reinforce rather than attenuate direct effects; this is consistent with the empirical evidence but could fail in settings with very mobile labor markets. The real wage calculation shows that, on average, cheaper imports provide a small net welfare gain (+0.16 log-points), but 39% of CZs experienced net real wage losses, pointing to substantial distributional consequences within the U.S. The framework’s scope is first-order (linearization around initial equilibrium), so it is a good approximation for moderate shocks; large shocks require integrating over the adjustment path. The methodology is applicable beyond the China shock to any trade policy with measurable regional exposure variation.

What is the MOIV estimator and why is it efficient?

The Model-implied Optimal IV (MOIV) is a two-step feasible implementation of the Chamberlain (1987) efficient GMM estimator. The class of consistent GMM estimators for the spatial link parameters θ = (φw, φp, φm, ψ, ε) differs only in how they weight the observed exposure of different markets. The optimal weighting function H*i assigns more weight to markets whose reduced-form elasticities (βij and φij) are most sensitive to changes in the parameter being estimated — i.e., markets that provide the most information about a given parameter. In step 1, an arbitrary initial θ0 is used to obtain a consistent but non-optimal first-stage estimate. In step 2, the consistent estimate is used to compute the optimal instrument, and a second-stage GMM is run. The MOIV is asymptotically equivalent to the Chamberlain efficient estimator. The paper’s contribution is to derive the optimal moment conditions for a flexible spatial GE model with non-linear parameter-dependent elasticities.

Key Concepts

Spatial Links Matrix: The Jacobian of the excess labor demand system with respect to wages, denoted γ-bar, summarizing the combined effect of trade demand substitution (how wage changes in one market shift demand from other markets) and supply substitution (how wage changes affect labor supply across markets, amplified by agglomeration). It governs the propagation of partial equilibrium excess demand shifts to general equilibrium wage and employment responses, and determines the sign and heterogeneity of indirect effects.

Bilateral Reduced-Form Elasticity: The element βij (for wages) or φij (for employment) measuring how much market i’s outcome responds to a unit shift in market j’s excess labor demand, after all GE adjustment rounds. It is a series expansion of the spatial links matrix and is larger for market pairs with stronger bilateral or third-market spatial connections. These elasticities are sufficient statistics for aggregating regional shock exposures to compute GE impact.

Revenue Exposure (IPW): The shift-share variable capturing a CZ’s partial equilibrium revenue shift from a foreign productivity shock: the employment-share-weighted average of sectoral export growth shocks. Identical to the ADH instrument. Measures how much a CZ’s producer revenues (and thus labor demand) fall when Chinese costs decline.

Consumption Cost Exposure (IPC): A novel shift-share variable capturing the partial equilibrium consumption cost shift: the spending-share-weighted average of sectoral export growth shocks, constructed using national input-output tables interacted with regional employment. Measures how much cheaper Chinese imports reduce the cost of living and inputs in a CZ, with a positive effect on real wages and labor supply.

Model-Implied Optimal IV (MOIV): A two-step feasible GMM estimator that achieves the Chamberlain (1987) efficiency bound for estimating the vector of structural spatial link parameters θ. In the first step any consistent estimator is used; in the second step the first-step estimates are used to compute the optimal moment function — which places more weight on CZs whose reduced-form elasticities are most sensitive to changes in the parameter being estimated — and a second-stage GMM yields the efficient estimate.

Gross Substitution Property: A condition on the spatial links matrix (γij < 0 for all off-diagonal pairs) under which all bilateral reduced-form elasticities βij are positive, so indirect effects of excess demand shifts always reinforce direct effects. The condition is satisfied when trade demand substitution dominates labor supply substitution in the spatial links matrix. Empirically supported for U.S. CZs: negative revenue shocks spread negatively to other CZs rather than triggering offsetting employment inflows.

Agglomeration Elasticity (ψ): The elasticity of local labor productivity to local employment in the production function, governing the feedback of employment changes on production costs and thus on excess labor demand. The authors estimate ψ = 0.56 for U.S. CZs — roughly twice the Krugman (1980) value and far above the zero assumed in Ricardian frameworks — and show it is the key parameter that amplifies both direct and indirect responses to trade shocks and determines model fit.

Endogenous Fixed Effect: A common component of GE indirect effects that arises when spatial links are identical across markets (Corollary 2). In this special case all indirect effects collapse to a common term absorbed by time fixed effects in standard regressions, making those regressions unable to separately identify the indirect effect from aggregate time trends. In the general case with heterogeneous spatial links, indirect effects differ across CZ pairs and are not absorbed by time fixed effects.