International Trade & General Equilibrium
Stolper-Samuelson & Rybczynski Magnification Formula
Solve Ronald Jones’s (1965) 2x2 Heckscher-Ohlin hat-algebra equations to verify Stolper-Samuelson factor-price magnification and Rybczynski endowment-output magnification.
Prove why a 10% goods-price shock drives a >10% real wage gain AND an absolute rental decline!
Ronald W. Jones (1965) unified the two core Heckscher-Ohlin general-equilibrium theorems into transparent 2x2 matrix inversion ("hat algebra", where x_hat = dx / x)! First, under zero profits in Sector 1 (labor cost share theta_L1, capital share 1 - theta_L1) and Sector 2 (labor cost share theta_L2 <> theta_L1), goods-price shocks (p1_hat, p2_hat) map to factor-return changes via determinant Delta_theta = theta_L1 - theta_L2: Nominal Wage Change w_hat = ((1 - theta_L2) * p1_hat - (1 - theta_L1) * p2_hat) / Delta_theta and Capital Rental Change r_hat = (theta_L1 * p2_hat - theta_L2 * p1_hat) / Delta_theta — proving the Stolper-Samuelson Magnification Effect w_hat > MAX(p1_hat, p2_hat) > MIN(p1_hat, p2_hat) > r_hat when theta_L1 > theta_L2 and p1_hat > p2_hat! Second, by duality under full employment (sector labor employment share lambda_L1 and capital employment share lambda_K1, Delta_lambda = lambda_L1 - lambda_K1), factor-supply shocks (L_hat, K_hat) map to Sector-1 Output Change Y1_hat = ((1 - lambda_K1) * L_hat - (1 - lambda_L1) * K_hat) / Delta_lambda and Sector-2 Output Change Y2_hat = (lambda_L1 * K_hat - lambda_K1 * L_hat) / Delta_lambda (Rybczynski Theorem)!
Verify on exact decimals: Labor-Intensive Sector 1 theta_L1 = 0.70 (B2) vs Capital-Intensive Sector 2 theta_L2 = 0.20 (B3 -> Delta_theta = 0.50), Goods Price Shocks p1_hat = +0.10 (+10%, B4) and p2_hat = 0.00 (B5): Wage Change jumps to w_hat = (0.80 * 0.10 - 0) / 0.50 = +0.1600 (+16.00% > +10%!), while Capital Return drops to r_hat = (-0.20 * 0.10) / 0.50 = -0.0400 (-4.00%!). Dually, with Employment Shares lambda_L1 = 0.75 (B6), lambda_K1 = 0.25 (B7 -> Delta_lambda = 0.50), Labor Growth L_hat = +0.10 (B8), and Capital Growth K_hat = 0.00 (B9): Labor-Intensive Output jumps Y1_hat = +0.1500 (+15.00%!) while Capital-Intensive Output shrinks Y2_hat = -0.0500 (-5.00%!)!
| 2x2 Hat-Algebra Metric (Delta_theta = 0.50, Delta_lambda = 0.50) | Formula | Verified Output |
|---|---|---|
| Stolper-Samuelson Wage Hat (w_hat) | ((1 - theta_L2)*p1_hat - (1 - theta_L1)*p2_hat) / (th_L1 - th_L2) | +0.1600 (+16.0% > +10% Price!) |
| Stolper-Samuelson Rental Hat (r_hat) | (theta_L1*p2_hat - theta_L2*p1_hat) / (th_L1 - th_L2) | -0.0400 (-4.0% Real Loss!) |
| Rybczynski Sector-1 Output Hat (Y1_hat) | ((1 - lam_K1)*L_hat - (1 - lam_L1)*K_hat) / (lam_L1 - lam_K1) | +0.1500 (+15.0% Expansion!) |
| Rybczynski Sector-2 Output Hat (Y2_hat) | (lam_L1*K_hat - lam_K1*L_hat) / (lam_L1 - lam_K1) | -0.0500 (-5.0% Contraction!) |
Stolper-Samuelson Magnified Factor-Price Change w_hat Formula
Guard equal factor intensities theta_L1 = theta_L2 (singular determinant = 0 where factor-price insensitivity breaks down).
=LET(th_1, B2, th_2, B3, p1_h, B4, p2_h, B5, IF(OR(th_1 <= 0, th_1 >= 1, th_2 <= 0, th_2 >= 1, th_1 = th_2), "Singular Equal Factor Intensity", ROUND(((1 - th_2) * p1_h - (1 - th_1) * p2_h) / (th_1 - th_2), 4)))Rybczynski Magnified Sectoral Output Change Y1_hat Formula
Notice balanced proportional growth: when L_hat = K_hat = +0.10, both sectors expand by exact +0.1000 (+10.00%) with zero Rybczynski contraction!
=LET(lm_l1, B6, lm_k1, B7, l_ht, B8, k_ht, B9, IF(OR(lm_l1 <= 0, lm_l1 >= 1, lm_k1 <= 0, lm_k1 >= 1, lm_l1 = lm_k1), "Singular Equal Factor Intensity", ROUND(((1 - lm_k1) * l_ht - (1 - lm_l1) * k_ht) / (lm_l1 - lm_k1), 4)))Try the example
Use the original values or change them to test your understanding.