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Step-by-step mathematical derivation with typeset equations and expandable detail.
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The 2x2x2 framework
The Heckscher-Ohlin model operates in a world with two countries (Home and Foreign), two goods (say cloth and steel ), and two factors of production (labor and capital ). Both countries share identical constant-returns-to-scale production technologies, but differ in their relative factor endowments. Home is capital-abundant if its capital-to-labor ratio exceeds Foreign's: . Steel production is capital-intensive if it uses a higher capital-to-labor ratio than cloth at any common factor prices: .
The model assumes perfect competition in all markets, free mobility of factors between industries within each country (but not across borders), identical homothetic preferences, and no transport costs or trade barriers. Under these conditions, free trade equalizes goods prices across countries and, through factor markets, pins down factor returns. The production possibilities frontier is concave because reallocating factors from one sector to another encounters diminishing returns as the factor mix departs from each industry's preferred ratio.
Factor abundance condition: Home is capital-abundant relative to Foreign when its aggregate capital-to-labor endowment ratio is higher.
Factor intensity condition: steel is capital-intensive if it always uses a higher capital-to-labor ratio than cloth, regardless of factor prices (no factor intensity reversal).
The Heckscher-Ohlin theorem
The central result is that each country exports the good that uses its abundant factor intensively. Home, being capital-abundant, exports steel (the capital-intensive good) and imports cloth (the labor-intensive good). The logic runs through autarky prices: in autarky, Home's relative abundance of capital makes capital cheap relative to labor, which in turn makes the capital-intensive good (steel) cheap relative to cloth. Foreign faces the opposite price structure. When trade opens, Home has a comparative advantage in steel because its autarky relative price of steel is lower.
Formally, the proof proceeds by showing that autarky relative prices reflect relative factor endowments under identical technologies and homothetic preferences. If is lower in Home than in Foreign under autarky, then free trade moves the world price to an intermediate level, inducing Home to produce more steel (where it has a cost advantage) and less cloth, exporting the surplus steel and importing cloth. The pattern of trade is thus entirely determined by differences in factor endowments, not by differences in technology (as in Ricardo) or in tastes.
H-O theorem: the capital-abundant country has a lower autarky relative price of the capital-intensive good, generating comparative advantage and the trade pattern.
Each country exports the good intensive in its abundant factor: Home exports steel, Foreign exports cloth.
Stolper-Samuelson: trade and factor prices
The Stolper-Samuelson theorem links goods prices to factor returns: an increase in the relative price of a good raises the real return to the factor used intensively in that good's production and lowers the real return to the other factor. When trade raises the relative price of steel in Home, the real rental rate on capital rises and the real wage falls. The result is sharp: the factor that gains sees its return rise by more than the price increase (magnification effect), and the losing factor's return falls in terms of both goods.
The mechanism works through zero-profit conditions. With competitive pricing, the unit cost of each good equals its price: and , where are unit input requirements. An increase in with fixed must raise and lower to maintain zero profits in both sectors. The magnification effect follows because the steel sector is capital-intensive: the cost share of capital in steel exceeds its share in cloth, so a given increase in has a larger impact on steel's cost, requiring to fall. This theorem explains why trade creates winners and losers within a country, even as it raises aggregate welfare.
Zero-profit conditions: competitive pricing equates unit cost to price in each sector, linking factor prices to goods prices.
Stolper-Samuelson magnification: the return to the factor used intensively in the rising-price good increases proportionally more than the goods price, while the other factor's return falls.