Melitz Export Productivity Cutoff & Exporter Share Formula

Model why only top-productivity firms export — and how trade reallocates industry TFP

Marc Melitz (2003) revolutionized trade theory by explaining why most manufacturers serve only their domestic market while exporters are systematically larger and more productive: with CES elasticity sigma > 1, domestic overhead f_D > 0, per-destination export fixed cost f_X > 0, and iceberg cost tau >= 1, the export productivity threshold exceeds the domestic survival threshold by ratio phi_X* / phi_D* = tau * (f_X / f_D)^(1 / (sigma - 1))! Under a Chaney (2008) Pareto productivity distribution G(phi) = 1 - (phi_min / phi)^k (k > sigma - 1), only fraction p_exp = MIN(1, (phi_D* / phi_X*)^k) of surviving firms export, while competition for labor across n symmetric partners raises the domestic survival cutoff by (1 + n * tau^(-k) * (f_X / f_D)^(1 - k / (sigma - 1)))^(1 / k)!

Verify with sigma = 4 (B2 -> sigma - 1 = 3), Pareto shape k = 4 (B3 > 3), domestic overhead f_D = 1 (B4), export fixed cost f_X = 8 (B5 -> 8^(1/3) = 2.00), iceberg cost tau = 1.25 (B6), and partner count n = 5 (B7): export productivity hurdle is 1.25 * 2.00 = 2.5000x the marginal domestic survivor, exporter share is 2.50^(-4) = 0.0256 (only 2.56% of firms export!), and opening trade with 5 partners elevates the domestic survival cutoff by (1 + 5 * 0.4096 * 0.50)^0.25 = 2.024^0.25 = 1.1928 (+19.28% industry productivity gain via exit of low-TFP firms!).

Melitz Export-to-Domestic Productivity Cutoff Ratio phi_X* / phi_D* Formula

Enforce finite Pareto aggregate sales condition k > sigma - 1 > 0 alongside f_D > 0, f_X > 0, and tau >= 1.

=LET(sig_e, B2, k_par, B3, f_dom, B4, f_exp, B5, tau_c, B6, IF(OR(sig_e <= 1, k_par <= sig_e - 1, f_dom <= 0, f_exp <= 0, tau_c < 1), "Require k > sigma - 1 > 0 and tau >= 1", ROUND(tau_c * POWER(f_exp / f_dom, 1 / (sig_e - 1)), 4)))

Trade-Induced Domestic Cutoff Elevation Ratio phi_D* / phi_aut* Formula

At n = 0 trading partners (autarky), the cutoff ratio equals an exact 1.0000; lowering tau or f_X raises labor demand at top exporters and forces the least productive domestic firms to exit.

=LET(sig_e, B2, k_par, B3, f_dom, B4, f_exp, B5, tau_c, B6, n_pt, B7, IF(OR(sig_e <= 1, k_par <= sig_e - 1, f_dom <= 0, f_exp <= 0, tau_c < 1, n_pt < 0), "Require k > sigma - 1 > 0 and tau >= 1", LET(exp_t, n_pt * POWER(tau_c, -k_par) * POWER(f_exp / f_dom, 1 - k_par / (sig_e - 1)), ROUND(POWER(1 + exp_t, 1 / k_par), 4))))

Further reading

• Arte Sheets Lesson — Krugman Love-of-Variety Gain & Home-Market Effect Formula
• Microsoft Support — POWER function