International Trade & Price Theory
Alchian-Allen "Shipping Good Apples Out" Formula
Prove Armen Alchian and William Allen’s (1964) relative-price compression theorem (P_H + f) / (P_L + f) < P_H / P_L under per-unit freight or specific charges and predict export quality upgrading.
Why Washington exports premium apples while keeping standard apples at home: unit vs ad-valorem cost!
Armen Alchian and William Allen (1964) resolved a famous trade puzzle: why do distant export markets disproportionately consume higher-grade wine, fruit, timber, and semiconductors than home-region shoppers right next to the orchard? Suppose High-Grade FOB price is P_H > Low-Grade FOB price P_L > 0, and both grades pay the SAME additive per-unit freight / container / babysitter charge f > 0 per physical unit! Locally (f = 0), 1 premium unit sacrifices P_H / P_L standard units; in the distant destination market, Delivered Relative Price falls strictly to (P_H + f) / (P_L + f) < P_H / P_L because derivative d/df[(P_H + f)/(P_L + f)] = (P_L - P_H) / (P_L + f)^2 < 0! Under CES substitution elasticity sigma > 0 between quality grades, Export High-to-Low Quantity Ratio shifts upward by factor ((P_H / P_L) / ((P_H + f) / (P_L + f)))^sigma > 1!
Verify on Orchard_Premium_FOB P_H = 12.00 (B2), Orchard_Standard_FOB P_L = 4.00 (B3 -> Local Relative Price = 12 / 4 = 3.0000x!), Cold-Chain_Unit_Freight f = 4.00/box (B4), and Quality_Substitution_Elasticity sigma = 2.0 (B5): Delivered Destination Prices are 16.00 vs 8.00, compressing Destination Relative Price from 3.0000x down to 2.0000x (-33.33% cheaper in relative terms!), and Export Premium-to-Standard Volume Ratio jumps by (3.00 / 2.00)^2 = 2.2500x (2.25x higher premium mix abroad than at home — whereas proportional ad-valorem cost leaves 12*(1+t)/(4*(1+t)) = 3.0000x completely unchanged!)!
| Alchian-Allen Metric (P_H = 12, P_L = 4, f = 4, sigma = 2) | Formula | Verified Output |
|---|---|---|
| Local Origin Relative Price (f = 0) | P_H / P_L = 12 / 4 | 3.0000 standard boxes / premium |
| Destination Delivered Relative Price | (P_H + f) / (P_L + f) = 16 / 8 | 2.0000 (-33.33% Compression!) |
| Relative Premium Discount (%) | ((P_H+f)/(P_L+f)) / (P_H/P_L) - 1 | -0.3333 (-33.33%) |
| Export Quality Mix Shift Multiplier | ((P_H / P_L) / ((P_H + f)/(P_L + f)))^sigma | 2.2500x Premium-to-Low Ratio |
Delivered Destination Relative Price Ratio Formula
Validate P_H > P_L > 0, additive unit freight f >= 0, and substitution elasticity sigma >= 0.
=IF(OR(B3 <= 0, B2 <= B3, B4 < 0), "Require P_H > P_L > 0 & f >= 0", ROUND((B2 + B4) / (B3 + B4), 4))CES Export Quality Upgrading Volume Multiplier Formula
When additive freight f = 0 (or freight is purely proportional to value), relative price stays 3.0000 and the quality-mix shift multiplier equals 1.0000.
=LET(p_hi, B2, p_lo, B3, f_un, B4, sg_el, B5, IF(OR(p_lo <= 0, p_hi <= p_lo, f_un < 0, sg_el < 0), "Require P_H > P_L > 0 & f >= 0", LET(r_loc, p_hi / p_lo, r_dst, (p_hi + f_un) / (p_lo + f_un), ROUND((r_loc / r_dst) ^ sg_el, 4))))Try the example
Use the original values or change them to test your understanding.