Eaton–Kortum Model

Eaton–Kortum Model is a multi-country general-equilibrium Ricardian model of international trade developed by Jonathan Eaton and Samuel Kortum in their landmark 2002 Econometrica paper, "Technology, Geography, and Trade." Classical Ricardian trade theory, initiated by David Ricardo (1817) and extended to a continuum of goods by Rudiger Dornbusch, Stanley Fischer, and Paul A. Samuelson (1977), demonstrated how international differences in relative labor productivity generate comparative advantage and mutually beneficial specialization. However, the Dornbusch–Fischer–Samuelson (DFS) framework proved intractable when generalized beyond two countries in the presence of bilateral geographic trade barriers, because ordering a continuum of goods by comparative advantage breaks down across N > 2 trading partners with asymmetric transport costs. Eaton and Kortum resolved this multi-country dimensionality barrier by replacing deterministic productivity schedules with a probabilistic representation in which each country draws its efficiency for every good on the unit interval from a Type II asymptotic maximum (Fréchet) distribution. Through the analytical closure properties of the Fréchet distribution, the model aggregates microeconomic lowest-cost sourcing across a continuum of goods into a tractable structural gravity model of trade that links bilateral trade shares directly to absolute technology parameters, input costs, geographic trade barriers, and a single structural trade elasticity governing the dispersion of comparative advantage.

Consider a world economy consisting of N countries indexed by i, n ∈ {1, 2, …, N} and a continuum of goods indexed by j ∈ [0, 1]. Consumers in each destination country n maximize a symmetric Constant Elasticity of Substitution (CES) utility function across all goods on the unit interval with elasticity of substitution σ > 0:

U_n = [ ∫₀¹ Q_n(j)^{(σ − 1) / σ} dj ]^{σ / (σ − 1)}

Under constant-returns-to-scale technology and perfect competition, country i produces good j with efficiency z_i(j), defined as the quantity of output generated per unit of a composite input bundle costing c_i. In the simplest labor-only specification, c_i = w_i represents the nominal wage rate in country i; in Eaton and Kortum's full roundabout-production specification, c_i = w_i^β p_i^{1 − β} is a Cobb–Douglas geometric average of the local wage w_i (with labor share β ∈ (0, 1]) and the local price index p_i of tradable intermediate inputs.

Delivering goods from source country i to destination country n incurs Samuelson iceberg trade costs d_ni ≥ 1: shipping 1 unit of any good j to country n requires dispatching d_ni units from country i, with d_nn = 1 and the triangle inequality d_ni ≤ d_nk d_ki holding for all country triples (n, k, i). Because commodity markets are perfectly competitive, the unit price at which producer i can supply good j to destination n equals unit production and delivery cost:

p_ni(j) = ( c_i d_ni ) / z_i(j)

Buyers in destination n purchase each good j ∈ [0, 1] from the lowest-cost supplier around the world, so the prevailing market price of good j in country n is:

p_n(j) = min_{i∈{1,…,N}} { p_ni(j) }

Rather than specifying deterministic functions z_i(j), Eaton and Kortum model country i's efficiency z_i(j) for each good j as the realization of an independent random variable drawn from a country-specific Fréchet distribution with cumulative distribution function:

F_i(z) = Pr[ Z_i ≤ z ] = exp( −T_i z^−θ ), z > 0

This distribution is governed by two structural parameters with distinct economic interpretations:

1. Absolute Advantage Parameter (T_i > 0): Represents country i's overall state of technology. A higher T_i shifts the efficiency distribution rightward (toward higher productivity values), increasing the proportion of goods for which country i achieves high efficiency draws across the continuum. Microeconomically, T_i is proportional to the accumulated stock of domestic research ideas and R&D knowledge.
2. Comparative Advantage Dispersion Parameter (θ > 1): Governs the inverse of productivity dispersion across goods and is common to all countries (restricted to θ > σ − 1 so the CES price index converges). Because the standard deviation of ln Z_i equals π / (θ√6), a low θ implies wide dispersion in relative efficiencies across goods—generating powerful forces of Ricardian comparative advantage that make trade flows relatively insensitive to input costs c_i and trade barriers d_ni. Conversely, a high θ implies tightly clustered efficiencies across goods, meaning small changes in c_i or d_ni cause large shifts in which country is the lowest-cost supplier.

Substituting z_i(j) = c_i d_ni / p_ni(j) into the Fréchet distribution F_i(z) determines the cumulative distribution G_ni(p) of prices offered by source country i in destination market n:

G_ni(p) = Pr[ P_ni ≤ p ] = 1 − F_i( c_i d_ni / p ) = 1 − exp[ −T_i ( c_i d_ni )^−θ p^θ ]

Because destination n buys each good from the minimum-price source p_n(j) = min_i p_ni(j), the probability that p_n(j) exceeds p is the product of the probabilities that every country i quotes a price above p. Consequently, the distribution G_n(p) of actual transaction prices in destination n is:

G_n(p) = 1 − ∏_i=1^N [ 1 − G_ni(p) ] = 1 − exp( −Φ_n p^θ )

where the multilateral price parameter Φ_n summarizes how global technologies, input costs, and geographic barriers govern the price level in destination n:

Φ_n = ∑_i=1^N T_i ( c_i d_ni )^−θ

This aggregation yields three foundational properties of the Eaton–Kortum equilibrium:

1. Closed-Form Bilateral Trade Share Equation

The probability π_ni that source country i supplies a given good j at the lowest price in destination n is obtained by integrating the probability that all other countries k ≠ i quote a price above p_ni(j) = p:

π_ni = Pr[ P_ni(j) ≤ min_k≠i P_nk(j) ] = ∫₀^∞ ∏_k≠i [ 1 − G_nk(p) ] dG_ni(p) = [ T_i ( c_i d_ni )^−θ ] / Φ_n

By the law of large numbers across the continuum of goods [0, 1], π_ni is also the fraction of goods that country n purchases from country i.

2. Source-Invariance of Conditional Price Distributions

A remarkable mathematical property of the Weibull minimum price distribution G_ni(p) is that the price distribution of goods that country n actually buys from country i—conditional on i being the lowest-cost supplier—is identical to the overall price distribution G_n(p) = 1 − exp(−Φ_n p^θ) for every source i. Intuitively, a distant or high-cost exporter i wins a smaller fraction π_ni of goods in market n, but on the select subset of goods where i does win, its efficiency draw z_i(j) is sufficiently high to offset its cost disadvantage, resulting in the same average transaction price as domestic suppliers.

3. Expenditure Shares Equal Good Shares and Ideal Price Index

Because every active source country i exhibits the same conditional price distribution G_n(p) in destination n, average expenditure per good in country n does not vary by source. Therefore, country n's bilateral expenditure share on imports from country i, X_ni / X_n, equals the probability share π_ni:

π_ni = X_ni / X_n = [ T_i ( c_i d_ni )^−θ ] / [ ∑_k=1^N T_k ( c_k d_nk )^−θ ]

Integrating p^{1 − σ} against dG_n(p) yields the CES ideal aggregate price index p_n in destination n:

p_n = γ Φ_n^{−1 / θ}, where γ = [ Γ( (θ + 1 − σ) / θ ) ]^{1 / (1 − σ)}

and Γ(·) denotes the Euler Gamma function.

Multiplying the bilateral expenditure share π_ni by total expenditure X_n in destination n and substituting Φ_n = (p_n / γ)^−θ yields the structural gravity model of trade:

X_ni = T_i ( c_i d_ni )^−θ ( p_n / γ )^θ X_n

Taking logarithms shows that bilateral trade flows decompose log-linearly into exporter capabilities (T_i c_i^−θ), importer market size and multilateral inward resistance (p_n^θ X_n), and bilateral trade barriers (d_ni^−θ), with the Fréchet dispersion parameter θ serving as the structural trade elasticity:

− ∂ ln( X_ni / X_nn ) / ∂ ln d_ni = θ

Unlike Armington or Krugman (1980) models where the trade elasticity equals σ − 1 (driven by the intensive margin of consumer substitution across fixed varieties), the trade elasticity in Eaton–Kortum is governed by θ via the extensive margin of goods: as d_ni falls, country i undercuts rival suppliers on a wider measure of goods j ∈ [0, 1], while the CES elasticity σ affects only the scalar constant γ.

Closing the general-equilibrium system requires labor market clearing in every country i ∈ {1, …, N}. With labor share β in production and balanced trade (X_n = w_nL_n / β), total labor income in country i must equal the labor content of global purchases from country i:

w_i L_i = ∑_n=1^N π_ni w_n L_n

In a domestic-expenditure normalization (d_nn = 1), country n's domestic trade share is π_nn = T_n c_n^−θ / Φ_n = T_n c_n^−θ (p_n / γ)^θ. Solving for the real wage W_n = w_n / p_n in the labor-only case (c_n = w_n, β = 1) yields:

W_n = w_n / p_n = γ⁻¹ T_n^{1 / θ} π_nn^{−1 / θ}

In the full roundabout intermediate-input model (c_n = w_n^β p_n^{1 − β}), input prices feed back into unit production costs, amplifying the real wage equation by the Leontief input-output multiplier 1 / β:

W_n = w_n / p_n = γ^{−1 / β} T_n^{1 / (βθ)} π_nn^{−1 / (βθ)}

Evaluating the ratio of the open-economy real wage (π_nn < 1) to the autarky real wage (π_nn = 1) produces the foundational Arkolakis–Costinot–Rodríguez-Clare (2012) sufficient-statistics welfare formula:

W_n / W_n,autarky = π_nn^{−1 / θ} (or π_nn^{−1 / (βθ)} with intermediate inputs)

Building on this structure, Robert Dekle, Jonathan Eaton, and Samuel Kortum (2007, 2008) introduced exact hat algebra (denoting the proportional change of any variable x between a baseline equilibrium and a counterfactual equilibrium as x̂ = x′ / x). Instead of estimating all unobserved technology levels T_i and bilateral trade cost levels d_ni, counterfactual equilibria for arbitrary tariff or productivity shocks (d̂_ni, T̂_i) can be computed directly from observed baseline expenditure shares π_ni:

π̂_ni = [ T̂_i ( ĉ_i d̂_ni )^−θ ] / [ ∑_k=1^N π_nk T̂_k ( ĉ_k d̂_nk )^−θ ]

Calibrating the model to 19 OECD manufacturing economies using international price dispersion and wage-adjusted gravity regressions, Eaton and Kortum (2002) estimated θ = 8.28 (with subsequent trade literature estimates by Simonovska & Waugh [2014] and Caliendo & Parro [2015] placing θ in the range of 4.0 to 8.5), demonstrating that removing all geographic trade barriers would increase real manufacturing wages by 15% to 25% for major economies while preserving strong domestic specialization patterns.

Why does the Eaton–Kortum model use the Fréchet distribution for productivity draws?

The Fréchet distribution F_i(z) = exp(−T_iz^−θ) is the extreme-value limit distribution for the maximum of independent heavy-tailed random variables (representing the best production technique discovered out of many historic innovations in country i). Mathematically, when unit efficiency z_i(j) is Fréchet-distributed, the unit price p_ni(j) = c_id_ni / z_i(j) follows a Weibull distribution, which is closed under minimization: the minimum price across N countries p_n(j) = min_i p_ni(j) also follows a Weibull distribution with aggregate parameter Φ_n = ∑_i T_i(c_id_ni)^−θ, enabling closed-form aggregation across arbitrary N countries.

What is the economic difference between the parameters T_i and θ?

The country-specific location parameter T_i governs absolute advantage: an increase in T_i raises average productivity in country i across all goods j ∈ [0, 1], boosting country i's real wage and market share worldwide. The common shape parameter θ governs comparative advantage: a lower θ means productivity draws vary more widely from good to good across countries, creating deeper efficiency differences that make bilateral trade more resilient to geographic barriers and increase the welfare gains from trade (W_n ∝ π_nn^−1/θ).

Why does the conditional price distribution in destination n not depend on the exporting country i?

Although high-cost or distant exporters (c_id_ni high) offer higher prices on average across all goods on [0, 1], destination n only imports from country i those specific goods j where country i happened to draw an exceptionally high productivity realization z_i(j) that beat every competitor k ≠ i. Under the Weibull price distribution, this selection effect exactly offsets the average cost disadvantage, so the distribution of prices conditional on winning the sale is G_n(p) = 1 − exp(−Φ_np^θ) for all sources i.

1. Eaton, J., & Kortum, S. (2002). Technology, geography, and trade. Econometrica, 70(5), 1741–1779. https://doi.org/10.1111/1468-0262.00352
2. Dornbusch, R., Fischer, S., & Samuelson, P. A. (1977). Comparative advantage, trade, and payments in a Ricardian model with a continuum of goods. The American Economic Review, 67(5), 823–839. https://www.jstor.org/stable/1828066
3. Arkolakis, C., Costinot, A., & Rodríguez-Clare, A. (2012). New trade models, same old gains? The American Economic Review, 102(1), 94–130. https://doi.org/10.1257/aer.102.1.94
4. Dekle, R., Eaton, J., & Kortum, S. (2008). Global rebalancing with gravity: Measuring the burden of adjustment. IMF Staff Papers, 55(3), 511–540. https://doi.org/10.1057/imfsp.2008.17
5. Caliendo, L., & Parro, F. (2015). Estimates of the trade and welfare effects of NAFTA. The Review of Economic Studies, 82(1), 1–44. https://doi.org/10.1093/restud/rdu035