The Margins of Trade

Layer 1 — Overview

Research Question

Eaton and Fieler seek to reconcile two literatures that have advanced in parallel but remained at odds: (i) general equilibrium models of bilateral trade flows (the “gravity” tradition) and (ii) empirical work on the margins of trade — the decomposition of bilateral trade into the extensive margin (number of products traded), the quantity margin (physical volumes), and the unit-value (price) margin. Standard GE models cannot accommodate two of the most robust empirical regularities: that richer importing countries pay higher unit values for the same product, and that richer exporting countries charge higher unit values. The paper builds a framework that captures all three margins jointly while still delivering the standard gravity equation and the Arkolakis-Costinot-Rodriguez-Clare (ACR) welfare formula.

Data

The analysis uses UN COMTRADE bilateral merchandise trade data for the 50 largest economies by GDP in 2007, the most disaggregated 6-digit HS product classification (HS6). The working sample covers 2,611,700 importer-exporter-HS6 triads representing US $9.62 trillion of trade. Country characteristics (GDP, population) come from the World Development Indicators; geographical variables from CEPII.

Empirical Regularities Addressed

In a standard gravity decomposition of bilateral trade, the elasticities of total trade value with respect to importer and exporter GDP are both approximately 1.1 and the distance elasticity is approximately −0.81. Decomposing total value into its extensive, quantity, and price margins reveals: (i) the extensive margin of exporters rises strongly with exporter GDP (elasticity 0.76) but the corresponding importer extensive margin is much smaller (0.34), contrary to what the Eaton-Kortum (2002) model predicts; (ii) the unit-value margin rises with both importer GDP per capita (elasticity approximately 0.13 in product-level regressions controlling for exporter-product fixed effects) and exporter GDP per capita (elasticity approximately 0.22 controlling for importer-product fixed effects); (iii) there is no significant interaction between importer and exporter per capita income in bilateral trade values (coefficient 0.002, statistically insignificant), rejecting the Linder-type prediction from one-dimensional quality models that rich countries disproportionately sell to other rich countries.

Model

Building on the Ricardian EK framework with a continuum of varieties, CES aggregation, and perfect competition, the paper introduces two dimensions of quality:

• Vertical quality (q) complements quantity: as spending on a variety increases, both physical quantity and vertical quality rise. This drives the positive relationship between importer per capita income and unit values, because buyers in richer (higher-wage) countries optimally demand higher vertical quality.
• Horizontal quality (Q) perfectly substitutes for quantity and is determined by the producing country’s endowment of intermediates per worker. Because a better-equipped worker produces higher horizontal quality, this dimension rises with the exporter’s wage, explaining why richer countries charge higher unit values.

The model uses Fréchet-distributed productivities as in EK. Despite the non-homothetic intricacies introduced by the two quality dimensions, the trade-share equation is identical to EK’s homothetic formulation, and the welfare formula takes the standard ACR form with the elasticity of real income with respect to the home trade share equal to −1/(α̃θ).

To accommodate the extensive margin, the paper introduces stochastic minimum shipment sizes: small-value flows are observed probabilistically, generating zeros in the trade matrix. Products are treated as bundles of varieties, and the number of varieties per product follows a discretized Weibull distribution.

Estimation and Key Parameter Values

Multilateral resistance terms (Φ) are estimated from bilateral trade flow regressions. Using product-level unit values and wages/Φ estimates, the authors estimate three structural parameters: γ = 0.13 (cost elasticity of vertical quality, governing how spending splits between quantity and price), ν = 0.22 (elasticity of horizontal quality with respect to intermediate use), and θ = 4 (Fréchet shape parameter, calibrated from the literature as it is imprecisely identified from prices). From IV regressions of product-level spending on unit values — instrumenting a given destination’s price with the same exporter’s average price to other destinations — the implied demand elasticity with respect to price is −2.83, and the corresponding β (governing the distribution of the structural error across varieties) is estimated at 0.65. Three shipment-size parameters (λ₁ = 2.26×10⁻⁷, λ₂ = 0.042, λ₃ = 0.48) are fitted to match the observed bilateral extensive margins (R-squared 0.79).

Main Findings

A simulation of five million varieties, aggregated into approximately 3,807 traded products, reproduces the key margins of trade in the data. The model with only seven parameters (γ, ν, θ, β, λ₁, λ₂, λ₃) captures: (i) the positive relationship between unit values and both importer and exporter per capita income; (ii) the concave relationship between GDP and the extensive margin (leveling off for large countries); (iii) the standard gravity elasticities of bilateral trade on GDP and distance. Two discrepancies remain: the model understates the effects of per capita income on the extensive margin (shifting them toward the quantity margin), and it does not generate the Alchian-Allen distance effect on unit values.

Disaggregation to the level of 15 HS sections confirms the pooled results: 80% of HS6 products show positive importer-income elasticities and 94% show positive exporter-income elasticities of unit values. Although section-specific γ and ν estimates are formally rejected to be equal (χ²(28) = 1,249 against a critical value of 41), the implied improvement in model fit is only 3.6% of the total sum of squared residuals, vindicating the aggregate approach.

Layer 2 — Q&A

Q1: What are the two dimensions of quality in the model, and why does the paper require both?

A1: Horizontal quality (Q) substitutes perfectly for physical quantity and is valued identically by all users — a worker equipped with more intermediates produces goods of higher horizontal quality. Vertical quality (q) complements quantity: the isobeneft surface requires a CES aggregator with ρ < 0 (elasticity of substitution below one between effective quantity and vertical quality) so that a buyer spending more on a variety optimally raises both the amount and the vertical quality dimension. One dimension of quality is insufficient: with a single dimension, if rich countries both produce and prefer higher-quality goods, market shares of rich exporters should be systematically higher in rich importing destinations than in poor ones. No such interaction is found in the data (the interaction coefficient in bilateral trade regressions is 0.002, statistically insignificant), requiring the two-dimension structure to break the link.

Q2: How does the model predict that unit values rise with importer per capita income?

A2: The optimal spending on vertical quality for any variety is governed by the elasticity γ/(1+γ): as a buyer’s wage rises, spending on any variety rises, and the fraction of that spending that goes to higher unit values (price) has elasticity γ/(1+γ) with respect to spending. The structural elasticity of unit values with respect to the importer wage is δ_{w,M} = γ/(1+γ). With γ = 0.13, this equals approximately 0.115, close to the empirically estimated importer per capita income elasticity of 0.12–0.13 from product-level regressions with exporter-product fixed effects. Richer importers also face a higher price index Φ (lower competition), contributing an additional negative elasticity δ_{Φ,M} = −1/[θ(1+γ)] on Φ, reinforcing the unit-value–income gradient.

Q3: How does the model predict that unit values rise with exporter per capita income?

A3: In the model, horizontal quality Q is produced by equipping workers with intermediates: Q = m^ν where ν > 0 and m is intermediate use per worker. Because m is determined by the optimal factor mix and rises with the wage (w), horizontal quality rises endogenously with the exporter’s wage. The structural elasticity of unit values with respect to the exporter wage is δ_{w,X} = ν/(1+γ). With ν = 0.22 and γ = 0.13, this equals approximately 0.195, consistent with the estimated exporter per capita income elasticity of 0.20–0.22. The exporter Φ contributes δ_{Φ,X} = ν/[θ(1+γ)] > 0.

Q4: Why does the model still deliver a standard gravity equation despite the non-homothetic quality structure?

A4: The key result is that the trade-share equation — the fraction of varieties that destination n sources from country i — takes the same Fréchet-based form as in Eaton-Kortum (2002): πni = Ti(dni C̃i)^{−θ} / Φn, where C̃i = Ci/Qi is the horizontal-quality-adjusted unit cost. Although quality is non-homothetic in individual variety demands, the distribution of the maximum effective inverse cost across sources conditional on country i being the cheapest is independent of the source country — the key aggregation property inherited from the Fréchet structure. As a result, country i’s share in total absorption by n equals its share in the number of varieties sourced from i, and aggregate bilateral trade flows satisfy a standard log-linear gravity equation. The gains from trade are given by the standard ACR formula Un = constant × (Tn d^{−θ}_{nn} / πnn)^{1/(α̃θ)}.

Q5: How does the paper handle the extensive margin empirically, and what does the data show?

A5: The extensive margin is defined as the fraction of HS6 product categories that destination n imports from source i. In panel regressions, the elasticity of the extensive margin with respect to exporter GDP is 0.76 (much less than the total trade value elasticity of 1.16, implying an intensive margin of 0.38), while the importer extensive margin elasticity is 0.34. Both elasticities display a concave relationship with GDP in levels: the range of products both exported and imported expands rapidly for small countries but levels off at high GDP. The standard EK model predicts an importer extensive margin elasticity that is zero or negative (larger importers source more domestically), inconsistent with the positive 0.34 found in the data.

Q6: How does the paper model the extensive margin, and what are the parameter estimates?

A6: The extensive margin arises from stochastic minimum shipment sizes. Trade flows for individual varieties exist according to model-implied values, but are only observed in a given year if the flow exceeds the stochastic shipment size drawn from an exponential distribution H(x) = 1 − exp(−λ₁x). Products are treated as bundles of varieties drawn from a discretized Weibull distribution f(M) parameterized by λ₂ and λ₃. The three parameters are estimated by minimizing squared differences between model-predicted and observed bilateral extensive margins across all country pairs. The estimates are λ₁ = 2.26×10⁻⁷ (SE 1.21×10⁻⁷), λ₂ = 0.042 (SE 0.020), λ₃ = 0.48 (SE 0.10), with an R-squared of 0.79. These imply a mean shipment size of $4.42 million (median $3.07 million) and a mean number of varieties per product of 1,597 (median 344).

Q7: How is β estimated, and what does it govern?

A7: β governs how spending is distributed across varieties — specifically, the elasticity of spending on a variety with respect to its effective inverse cost. It also governs the elasticity of physical demand with respect to price: the model implies that log spending on a product equals log value minus (β/(1−β)) times log unit price. To identify β, the authors regress product-level trade values on unit prices, instrumenting a given importer’s price for a product with the same exporter’s average price of that product to all other destinations. The IV estimate of −β/(1−β) is −1.83 (SE 0.019), compared with the OLS estimate of −0.25, indicating substantial simultaneity bias. The implied price elasticity of demand is −2.83 and the implied β is 0.65.

Q8: What are the price regularities documented at the product level, and how does the two-quality model explain price overlaps across country pairs?

A8: Regressions of unit values at the importer-exporter-HS6 level show that individual exporters charge systematically higher prices to richer importers for the same product (elasticity 0.12 with exporter-product fixed effects), and that buyers pay systematically higher prices for products from richer exporters (elasticity 0.22 with importer-product fixed effects). A one-dimensional quality model would predict no overlap between prices charged by a rich and a poor exporter across destinations: even Japan’s lowest-priced sales should exceed Malaysia’s highest-priced sales for the same product. Back-of-envelope calculations using the regression coefficients predict a Malaysian product should sell in Norway at 0.3 log points above a Japanese product in Pakistan — systematic overlap. The paper documents this overlap in the raw data using two HS6 examples: motorcycle hubs (HS871493) and washing machines under 10kg (HS845011). The two-quality model resolves this by making horizontal quality an exporter attribute that raises prices proportionally but leaves market share determination to the EK gravity equation, allowing rich and poor country exporters to coexist in all markets.

Q9: How does the model address the absence of a Linder-type income interaction effect in aggregate trade flows?

A9: In one-dimensional quality models (e.g., Fajgelbaum, Grossman, and Helpman 2011), rich countries produce high-quality goods appealing primarily to high-income households, so rich-to-rich bilateral trade flows should be systematically higher than rich-to-poor flows. A gravity regression of bilateral trade values on importer and exporter fixed effects, distance, and an interaction of log importer GDP per capita × log exporter GDP per capita yields a coefficient of 0.0020 (SE 0.016), which is small and statistically insignificant. The two-quality model is consistent with this: horizontal quality enters as an exporter fixed effect (it affects prices proportionally for all destinations) and the demand system is structured so that all destinations spend the same share of absorption on a given source’s varieties, regardless of income level.

Q10: How robust are the results to disaggregation by industry?

A10: For each of 4,786 HS6 products with more than 20 country pairs, separate price regressions are estimated. Across all products, 80% have positive importer per capita income elasticities and 94% have positive exporter per capita income elasticities. The 15 broad HS sections account for only 10% of the variance in importer-income elasticities and 13% of the variance in exporter-income elasticities across HS6 products, suggesting high within-industry heterogeneity. A quasi-likelihood ratio test formally rejects equal γ and ν across sections (χ²(28) = 1,249 against a critical value of 41), but the reduction in total sum of squared residuals from allowing section-specific parameters is only 3.6%, and the R-squared increases from 0.353 to 0.376. The authors conclude the aggregate approach is vindicated for the purpose of characterizing common patterns.

Q11: How does the model simulate trade, and how many products does it generate?

A11: The simulation draws productivities for 5,000,000 varieties across 50 countries using the estimated model parameters (θ = 4, γ = 0.13, ν = 0.22, β = 0.65, and the estimated gravity fixed effects). For each variety, the cheapest source is determined; trade values and unit values are computed using equations (28) and (29); censoring due to stochastic shipment sizes generates zeros. Varieties are aggregated into products by partitioning sequentially using the estimated Weibull distribution. The simulation yields 3,842 total simulated products of which 3,807 are traded between at least one country pair, compared with 4,973 HS6 products in the COMTRADE data. A Monte Carlo exercise confirms that the estimation procedure recovers parameter values close to the true values when applied to simulated data.

Key Concepts

Vertical quality (q): A dimension of quality that complements physical quantity. In the paper’s utility specification, vertical quality and effective quantity enter a CES aggregator with elasticity of substitution below one (ρ < 0). A buyer spending more on a variety raises both quantity and vertical quality simultaneously, in proportions governed by γ. Vertical quality rises endogenously with the importer’s wage because higher-income buyers optimally demand it; it is the mechanism behind the positive relationship between importer per capita income and unit values.

Horizontal quality (Q): A dimension of quality that substitutes perfectly for physical quantity (enters the aggregator multiplicatively with quantity). All buyers value an increase in Q equivalently regardless of income level, so it does not generate Linder-type income-matching in trade flows. Horizontal quality is produced by the exporter: better-equipped workers produce higher horizontal quality (Q = m^ν), so it rises with the exporter’s wage. It is the mechanism behind the positive relationship between exporter per capita income and unit values.

Extensive margin (E_{ni}): In the paper’s empirical framework, the fraction of HS6 product categories that destination n imports from source i in a given year. The paper shows this margin rises with both importer and exporter size but in a concave, nonlinear fashion. It is generated in the model by stochastic minimum shipment sizes that probabilistically censor small-value variety flows.

Intensive margin: Total bilateral trade value divided by the extensive margin. The paper further decomposes the intensive margin into a quantity margin and a unit-value (price) margin. The paper’s key contribution is to generate all three margins jointly from one parsimonious framework.

Stochastic minimum shipment size: A modeling device, drawn from distribution H(x) (parameterized as exponential with parameter λ₁), that determines whether a given variety’s trade flow is observed in any year. If the annual flow x_{ni}(ω) exceeds the drawn minimum size x̄, the shipment is observed with certainty; otherwise, it is observed with probability x_{ni}(ω)/x̄. This mechanism generates the concavity of the extensive margin with respect to GDP without departing from the standard gravity framework.

Effective inverse cost (v_{ni}): Defined as Z_i(ω)/[C̃_i d_{ni}], where Z_i is country i’s Fréchet-distributed productivity for variety ω, C̃_i = C_i/Q_i is the horizontal-quality-adjusted unit cost, and d_{ni} is the iceberg trade cost. A buyer in n sources variety ω from the country maximizing v_{ni}. This formulation ensures that horizontal quality differences across exporters are absorbed into the effective cost, preserving the EK aggregation result.

γ (vertical quality cost elasticity): The parameter governing how spending on a variety divides between physical quantity and vertical quality. Spending has elasticity 1/(1+γ) with respect to quantity and elasticity γ/(1+γ) with respect to unit price. The paper estimates γ = 0.13 from product-level unit value regressions.

ν (horizontal quality elasticity): The parameter governing how horizontal quality rises with intermediate use per worker: Q = m^ν. Combined with γ, it determines the structural elasticity of unit values with respect to exporter per capita income: δ_{w,X} = ν/(1+γ). The paper estimates ν = 0.22.