Bernard–Eaton–Jensen–Kortum Trade Model

Bernard–Eaton–Jensen–Kortum Trade Model (widely known as the BEJK model) is a foundational quantitative framework of international trade, industrial organization, and firm-level heterogeneity in business and economics theory, developed by American economists Andrew B. Bernard, Jonathan Eaton, J. Bradford Jensen, and Samuel Kortum in their landmark 2003 American Economic Review article "Plants and Productivity in International Trade." Formulated simultaneously with Marc J. Melitz's (2003) monopolistic-competition trade model to explain plant-level export stylized facts from the U.S. Census of Manufactures, the BEJK model embeds head-to-head Bertrand price competition into the multi-country probabilistic Ricardian framework of the Eaton–Kortum trade model (2002). In each country i and for each continuum commodity j ∈ [0, 1], both the most productive domestic producer Z_1i(j) and the second-most productive producer Z_2i(j) are drawn from a joint multivariate Fréchet order-statistic distribution governed by comparative-advantage dispersion parameter θ and technology level T_i. Rather than charging a constant Dixit–Stiglitz markup σ/(σ − 1) over marginal cost C_1n(j), the lowest-cost supplier to destination market n engages in Bertrand limit pricing, setting price equal to the marginal delivered cost of the second-lowest-cost global competitor C_2n(j) (capped at the monopoly markup m̄ = σ/(σ − 1)). This order-statistic structure yields an endogenous variable markup M_n(j) = min{C_2n(j) / C_1n(j), *m̄*} whose unconditional Pareto distribution P(M_n(j) ≤ m) = 1 − m^−θ (for 1 ≤ m < m̄) is invariant to destination market size and bilateral trade barriers, while explaining why exporting plants are larger and more productive than non-exporters and why trade liberalization induces pro-competitive markup compression alongside Ricardian creative destruction.

Five Microeconometric Facts from the U.S. Census of Manufactures

During the mid-1990s, Andrew B. Bernard and J. Bradford Jensen (1995, 1999) analyzed longitudinal microdata on over 200,000 manufacturing plants from the U.S. Census of Manufactures, uncovering five empirical regularities that challenged both neoclassical representative-firm comparative advantage models and symmetric monopolistic-competition trade models (Krugman, 1980):

1. Large Within-Industry Productivity Dispersion: Within 4-digit Standard Industrial Classification (SIC) industries, value added per worker varies by a factor of more than 5 between the 10th and 90th percentiles of plants.
2. Rarity of Exporting: Only 21% of U.S. manufacturing plants report any export sales, and over two-thirds of exporters sell less than 10% of their output abroad.
3. The Exporter Productivity Premium: Exporting plants have 33% higher measured labor productivity (value added per worker) and 15% to 20% higher revenue total factor productivity (TFPR) than non-exporting plants in the same industry.
4. The Exporter Size Premium: Exporting plants ship 4.8 times more total output than non-exporting plants in the same industry.
5. Within-Industry Reallocation Under Trade Liberalization: Falling trade costs cause low-productivity plants to contract or exit as import penetration rises, while high-productivity domestic plants expand into export markets.

Reconciling Measured Revenue Productivity (P · Z) with Variable Markups

Statistical agencies rarely observe physical output quantities Q(j) directly for differentiated manufacturing plants; instead, they measure nominal revenue per unit of input, which equals price times physical productivity:

In any model with constant Dixit–Stiglitz markups (such as the Melitz trade model, 2003), a plant with higher physical productivity Z(j) charges a proportionally lower price P(j) = [σ/(σ − 1)] (w/Z(j)), so revenue per unit of variable input P(j) · Z(j) / w = σ / (σ − 1) is constant across all plants. By replacing monopolistic competition with Bertrand limit pricing, the BEJK model generates an endogenous positive relationship between a plant's physical efficiency lead over its nearest rival (Z₁/Z₂) and its equilibrium price-cost markup M_n(j) = P_n(j)/C_1n(j).

Continuum of Goods and Iceberg Trade Barriers

Consider a world economy of N countries (i, n ∈ {1, …, N}) and a continuum of distinct goods j ∈ [0, 1]. Representative consumers in destination n aggregate quantities Q_n(j) via a symmetric constant-elasticity-of-substitution (CES) utility function with elasticity σ > 1:

where P_n = [ ∫₀¹ P_n(j)^{1 − σ} dj ]^{1 / (1 − σ)} is the CES price index in country n. Shipping one unit of good j from country i to destination n incurs a Samuelson iceberg geographic trade cost d_ni ≥ 1 (d_nn = 1), and w_i denotes the unit cost of the composite input bundle in country i.

Probabilistic Technology: Top Two Producers (Z_1i(j), Z_2i(j))

In each source country i and for each good j ∈ [0, 1], only the most efficient producer (with physical productivity Z_1i(j)) and the second-most efficient producer (with physical productivity Z_2i(j) ≤ Z_1i(j)) matter for Bertrand pricing. Generated by an underlying Poisson process of technological ideas with country intensity T_i > 0 (absolute advantage) and Pareto tail exponent θ > σ − 1 (comparative-advantage parameter), the joint distribution of (Z_1i(j), Z_2i(j)) over 0 ≤ z₂ ≤ z₁ < ∞ is the bivariate Fréchet order-statistic distribution:

Setting z₂ = z₁ = z recovers the univariate Eaton–Kortum Fréchet distribution F_i(z, z) = e^−T_iz−θ.

Distribution of the Lowest Cost C_1n(j) and Second-Lowest Cost C_2n(j) in Market n

Given input cost w_i and iceberg barrier d_ni, the delivered marginal costs in market n of the top two producers from country i are C_1*ni*(j) = w_id_ni / Z_1i(j) and C_2*ni*(j) = w_id_ni / Z_2i(j). Across all N countries competing to supply destination n, let C_1n(j) = min_i C_1*ni*(j) denote the lowest delivered marginal cost in market n, and let C_2n(j) ≥ C_1n(j) denote the second-lowest delivered marginal cost in market n. By Poisson additivity across independent countries i = 1, …, N, the joint distribution of (C_1n(j), C_2n(j)) for 0 < c₁ ≤ c₂ is:

where Φ_n ≡ ∑_i=1^N T_i ( w_i d_ni )^−θ, and the probability π_ni that country i is the lowest-cost supplier to destination n (as well as the conditional cost distribution) is independent of source i:

Limit Pricing and the Universal Pareto Distribution of Markups

In destination market n, potential producers of good j compete head-to-head in prices (Bertrand competition). To prevent the second-lowest-cost rival C_2n(j) from undercutting it while never exceeding the unconstrained CES monopoly price m̄ C_1n(j) (where m̄ ≡ σ / (σ − 1)), the lowest-cost supplier charges the Bertrand limit price:

Integrating the bivariate density of G_n(c₁, c₂) yields Lemma 1 and Proposition 1 of BEJK (2003):

1. Invariance and Independence of the Efficiency Ratio: Conditional on C_2n(j) = c₂, the cost ratio U_n(j) = C_2n(j) / C_1n(j) follows a Pareto distribution with exponent θ on [1, ∞) that is independent of C_2n(j) and independent of Φ_n:

1. Truncated Pareto Markup Distribution: Consequently, the equilibrium markup M_n(j) in every country n follows the universal distribution:

1. Exact Gravity and Price Index: Because M_n(j) and C_2n(j) are independent, the CES price index is P_n = γ_BEJK Φ_n^−1/θ, aggregate bilateral expenditure shares equal X_ni / X_n = π_ni, and the variable profit share of aggregate revenue is 1 / (1 + θ) when m̄ is non-binding.

Selection via Iceberg Barriers and Quantitative Calibration

Unlike the Melitz trade model (2003), which requires a fixed export overhead cost f_x > 0 per market, the BEJK model generates selection into exporting and the exporter productivity and size premia purely from iceberg barriers d_ni > 1 and Bertrand competition:

1. Why Exporters Have Higher Domestic Markups and Measured Productivity: For a domestic leader in country i to overcome the iceberg barrier d_ni > 1 and beat foreign competitors in market n, its physical productivity Z_1i(j) must lie in the upper tail of the domestic distribution—which implies via Bayes' rule that its efficiency lead Z_1i(j) / Z_2i(j) over the second-best domestic producer is also stochastically larger. A wider domestic efficiency gap translates into a higher domestic Bertrand markup M_ii(j) = min{Z_1i/Z_2i, *m̄*} and higher measured value added per worker.
2. Quantitative Fit (θ = 3.60, σ = 3.79): Calibrated to 47 countries and the 1992 U.S. Census of Manufactures with θ = 3.60 and σ = 3.79, the BEJK simulation reproduces the observed exporter value-added-per-worker premium (24%–33%), the 4.8x exporter size premium, and a 4.7% aggregate manufacturing productivity gain from a 5% reduction in global trade barriers.

How can the unconditional distribution of markups P(M ≤ m) = 1 − m^−θ remain unchanged when trade barriers fall, even though competition intensifies for individual plants?

For any surviving domestic plant, lower import barriers d_ni reduce the delivered cost of foreign rivals, lowering C_2n(j) and compressing that plant's markup M_n(j) = C_2n(j)/C_1n(j). However, for goods where a foreign rival undercuts the domestic leader (C_1*ni* < C_1*nn*), the domestic plant exits and the new foreign winner enters with its own cost lead over the former domestic leader. Under the Fréchet distribution, pro-competitive markup compression on survivors and creative-destruction entry of new leaders balance so that the cross-sectional distribution H(m) = 1 − m^−θ remains invariant to Φ_n.

How does the BEJK (2003) model differ from the Melitz (2003) model in explaining why only a minority of plants export?

In the Melitz trade model (2003), every firm has a monopoly on its own horizontally differentiated variety and would export to every country if there were only iceberg costs, so Melitz introduces a fixed export cost f_x > 0. In BEJK (2003), plants in different countries compete head-to-head to supply the same good j; a domestic plant fails to export good j to country n because the iceberg shipping cost d_ni > 1 raises its delivered marginal cost w_id_ni/Z_1i(j) above the marginal cost C_1n(j) of local producers in country n.

Why does measured revenue productivity (value added per worker) vary across plants in BEJK when it is constant in a basic Melitz model?

Measured revenue productivity in census data equals P_n(j)Q_n(j) / [C_1n(j)Q_n(j)] = M_n(j) (the plant's gross markup). Under Dixit–Stiglitz monopolistic competition, every firm charges the constant markup σ/(σ − 1), generating identical revenue per unit of variable input. Under BEJK's Bertrand limit pricing, a plant with a wider efficiency lead over its second-best rival (Z₁/Z₂) charges a higher markup M_n(j) = min{C_2n/C_1n, *m̄*}, directly generating cross-plant dispersion in measured revenue per worker.

What role does the parameter θ play in the BEJK model?

The Fréchet shape parameter θ > 0 governs two forces simultaneously: (1) Comparative Advantage and Trade Elasticity, where a smaller θ implies wider productivity dispersion and a lower trade elasticity (−θ); and (2) Market Power, where a smaller θ implies larger average efficiency leads C_2n/C_1n, higher average Bertrand markups, and a higher aggregate profit share 1/(1 + θ).

1. Bernard, A. B., Eaton, J., Jensen, J. B., & Kortum, S. (2003). "Plants and Productivity in International Trade." American Economic Review, 93(4), 1268–1290. https://doi.org/10.1257/000282803769206296
2. Eaton, J., & Kortum, S. (2002). "Technology, Geography, and Trade." Econometrica, 70(5), 1741–1779. https://doi.org/10.1111/1468-0262.00352
3. Melitz, M. J. (2003). "The Impact of Trade on Intra-Industry Reallocations and Aggregate Industry Productivity." Econometrica, 71(6), 1695–1725. https://doi.org/10.1111/1468-0262.00467
4. Bernard, A. B., & Jensen, J. B. (1999). "Exceptional Exporter Performance: Cause, Effect, or Both?" Journal of International Economics, 47(1), 1–25. https://doi.org/10.1016/S0022-1996(98)00027-000027-0)
5. Atkeson, A., & Burstein, A. (2008). "Pricing-to-Market, Trade Costs, and International Relative Prices." American Economic Review, 98(5), 1998–2031. https://doi.org/10.1257/aer.98.5.1998