International Macroeconomics & Trade
Marshall-Lerner Condition and J-Curve Trade-Balance Formula
Test the Marshall-Lerner condition (e_X + e_M > 1), Hirschman’s initial-deficit condition ((X_0 / M_0)*e_X + e_M > 1), and J-curve trade-balance response Delta_NX = dq * (X_0*e_X + M_0*(e_M - 1)).
Catch Hirschman’s deficit trap: when e_X + e_M > 1 STILL worsens the trade deficit!
Textbook macroeconomics states the Marshall-Lerner condition as e_X + e_M > 1: a currency depreciation of rate dq > 0 improves net exports iff export volume elasticity e_X plus import volume elasticity e_M exceeds 1. However, that simple sum assumes balanced initial trade (X_0 = M_0)! Starting from arbitrary exports X_0 and imports M_0, domestic-currency net-export change is Delta_NX = dq * (X_0 * e_X + M_0 * (e_M - 1)), which is positive iff (X_0 / M_0) * e_X + e_M > 1 (Hirschman, 1949)!
Verify all three regimes from initial exports X_0 = 800 (B2), imports M_0 = 1,000 (B3, initial deficit -200), and 10% depreciation dq = 0.10 (B6): (1) short-run J-curve inelasticity (e_X = 0.30 in B4, e_M = 0.40 in B5) widens the deficit by Delta_NX = 0.10 * (240 - 600) = -36.00; (2) medium-run elasticity (e_X = 0.80, e_M = 0.70) improves net exports by +34.00; (3) in Hirschman’s trap (e_X = 0.50, e_M = 0.55 -> sum 1.05 > 1), (800/1000)*0.50 + 0.55 = 0.95 < 1 so Delta_NX = -5.00!
| Elasticity Regime (X_0=800, M_0=1,000, dq=10%) | Sum e_X + e_M vs Hirschman Index | Trade-Balance Change Delta_NX |
|---|---|---|
| Short-Run J-Curve Trough (0.30, 0.40) | 0.70 vs 0.64 (< 1) | -36.00 (Deficit Widens) |
| Hirschman Deficit Trap (0.50, 0.55) | 1.05 (> 1!) vs 0.95 (< 1) | -5.00 (Still Worsens!) |
| Medium-Run Elastic Recovery (0.80, 0.70) | 1.50 vs 1.34 (> 1) | +34.00 (Trade Improves) |
Generalized Domestic-Currency Trade-Balance Response Delta_NX Formula
Evaluate dq * (X_0 * e_X + M_0 * (e_M - 1)) with positive baseline trade flows X_0 > 0, M_0 > 0 and non-negative price-elasticity magnitudes e_X, e_M >= 0.
=LET(x_0, B2, m_0, B3, ex_e, B4, em_e, B5, dq_fx, B6, IF(OR(x_0 <= 0, m_0 <= 0, ex_e < 0, em_e < 0), "Invalid Trade Input", ROUND(dq_fx * (x_0 * ex_e + m_0 * (em_e - 1)), 4)))Hirschman Coverage-Adjusted Elasticity Index (X_0 / M_0) * e_X + e_M Formula
Compare Hirschman’s exact index against 1.0000: when X_0 = M_0, it reduces identically to classic Marshall-Lerner e_X + e_M.
=LET(x_0, B2, m_0, B3, ex_e, B4, em_e, B5, IF(OR(x_0 <= 0, m_0 <= 0, ex_e < 0, em_e < 0), "Invalid Trade Input", ROUND((x_0 / m_0) * ex_e + em_e, 4)))Try the example
Use the original values or change them to test your understanding.