International Trade & Quantitative Macroeconomics
Eaton-Kortum Trade Share & ACR Gains-from-Trade Formula
Compute multi-country Eaton-Kortum Fréchet import shares pi_ni = T_i*(c_i*d_ni)^(-theta) / Phi_n and Arkolakis-Costinot-Rodríguez-Clare welfare gain pi_nn^(-1/theta).
Unify Ricardian comparative advantage and sufficient-statistic welfare gains
Eaton and Kortum (2002) extended Ricardian comparative advantage to N countries using Fréchet productivity draws with technology level T_i > 0, unit input cost c_i > 0, iceberg trade barrier d_ni >= 1 (with domestic d_nn = 1), and dispersion / trade elasticity theta > 0: country i captures expenditure share pi_ni = T_i * (c_i * d_ni)^(-theta) / SUM_k(T_k * (c_k * d_nk)^(-theta))! Even more remarkably, Arkolakis, Costinot, and Rodríguez-Clare (ACR, 2012) proved that across Armington, Krugman, Melitz-Pareto, and Eaton-Kortum models, real-income welfare relative to autarky equals an exact two-number sufficient statistic: W_n / W_aut = pi_nn^(-1 / theta)!
Audit a 3-country sourcing table (A2:E4) at Fréchet trade elasticity theta = 4.0 (H2): Home (T_1 = 32, c_1 = 2, d_11 = 1 -> potential 32 * 2^(-4) = 2.0000), Partner 2 (T_2 = 81, c_2 = 2, d_12 = 1.5 -> 81 * 3^(-4) = 1.0000), and Partner 3 (T_3 = 16, c_3 = 1, d_13 = 2 -> 16 * 2^(-4) = 1.0000). Total price-index potential is Phi_1 = 4.0000, Home’s domestic expenditure share is pi_11 = 2 / 4 = 0.5000 (50% import penetration!), and ACR real-wage welfare relative to autarky is 0.50^(-1/4) = 1.1892 (+18.92% gain from trade!).
| Source Country i (theta = 4) | T_i / Cost c_i / Iceberg d_1i | Potential & Expenditure Share pi_1i |
|---|---|---|
| Home (i = 1, Domestic) | 32 / 2.0 / 1.0 | 2.0000 -> pi_11 = 0.5000 (50%) |
| Partner 2 (i = 2) | 81 / 2.0 / 1.5 | 1.0000 -> pi_12 = 0.2500 (25%) |
| Partner 3 (i = 3) | 16 / 1.0 / 2.0 | 1.0000 -> pi_13 = 0.2500 (25%) |
| ACR Welfare Ratio W / W_aut | pi_11^(-1 / theta) | 1.1892 (+18.92% Real Income) |
Eaton-Kortum Bilateral Sourcing Potential T_i * (c_i * d_1i)^(-theta) Formula
Compute each supplier country’s Fréchet cost-adjusted technology potential in E2:E4 and normalize by Phi_1 = SUM($E$2:$E$4).
=IF(OR(B2 <= 0, C2 <= 0, D2 < 1, $H$2 <= 0), "Invalid EK Input", B2 * POWER(C2 * D2, -$H$2))ACR Sufficient-Statistic Real-Income Welfare Ratio pi_11^(-1 / theta) Formula
Evaluate POWER(pi_11, -1 / theta) directly from observed domestic expenditure share pi_11 and trade elasticity theta > 0.
=LET(th_ek, H2, p_dom, F2, IF(OR(ISTEXT(E2), ISTEXT(E3), ISTEXT(E4), th_ek <= 0, p_dom <= 0, p_dom > 1), "Invalid EK Input", ROUND(POWER(p_dom, -1 / th_ek), 4)))Try the example
Use the original values or change them to test your understanding.