# Rybczynski Theorem

# Rybczynski Theorem

The Rybczynski theorem, along with the Stolper-Samuelson, factor-price equalization, and Heckscher-Ohlin theorems, is one of four key propositions describing the properties of the standard Heckscher-Ohlin model with two goods and two factors. The Polish-born economist Tadeusz M. Rybczynski’s (1923–1998) paper, “Factor Endowment and Relative Commodity Prices” (1955), relates changes in an economy’s factor supplies to resulting changes in equilibrium outputs and prices.

The simplest version of the Heckscher-Ohlin model assumes that two goods, say autos and textiles, are produced using the same two factor inputs, say labor and capital, but in proportions that differ across the two industries. If the auto industry uses a higher ratio of capital to labor, it is termed the *capital-intensive* industry, while the textile industry is termed *labor-intensive*. Under the usual assumptions of the model, an increase in the supply of either factor, holding constant the supply of the other, results in an outward shift of the production possibility frontier. Thus, the economy can now produce more of both goods, so that most economists prior to Rybczynski’s contribution assumed that this kind of biased growth would result in higher equilibrium outputs of each good, though with relatively greater growth of the industry that uses more of the growing factor.

Rybczynski’s surprising result is that a given percentage increase in the supply of one factor, say capital, holding constant the supply of the second factor (labor) as well as the relative price of the two goods, must result in a still larger percentage increase in the equilibrium output of the good that is capital-intensive in production (autos), and an absolute *decrease* in the equilibrium output of the good that is labor-intensive (textiles). The result also requires that the economy produce some of each good in both the equilibrium prior to the change in factor supply and the final equilibrium following the change. Rybczynski’s proof makes use of the box diagram as applied for the first time to production by Wolfgang Stolper and Paul A. Samuelson in their landmark paper, “Protection and Real Wages” (1941), which presented what is now known as the Stolper-Samuelson theorem. Moreover, Rybczynski builds on Stolper and Samuelson’s key insight concerning the basic two-good, two-factor version of the Heckscher-Ohlin model—that the relative price of the two goods uniquely determines factor prices and thus factor proportions.

In general, an economy can adjust to an increase in the supply of capital, holding constant the supply of labor, through some combination of capital-deepening and a changed mix of outputs. This alternative formulation of the Rybczynski theorem was provided by Ronald Jones in his 1965 paper, “The Structure of Simple General Equilibrium Models,” which offers the first integrated treatment, as well as a number of generalizations, of the four key theorems of the Heckscher-Ohlin model. To the extent that more capital results in a reduction in its relative cost to producers, firms in both sectors will now opt for a higher ratio of capital to labor. But the Stolper-Samuelson theorem implies that with output prices fixed and both goods produced, factor prices and thus cost-minimizing capital-labor ratios must remain unchanged in the new equilibrium. Thus, the adjustment is achieved entirely through a change in the mix of outputs. Expansion of the capital-intensive industry allows additional capital to be employed, but more labor is also required to maintain that industry’s unchanged ratio of capital to labor. This labor (plus additional capital) must be obtained through a contraction in the equilibrium output of the labor-intensive industry, which releases a relatively higher ratio of labor to capital than is required for expansion of the capital-intensive industry. This allows the additional capital to be absorbed at constant factor prices.

The Rybczynski theorem, by indicating what would be produced at unchanged output prices, has clear implications for effects of biased factor growth on actual prices in the resulting equilibrium. Assuming that both goods are normal in consumption—that is, that as incomes rise more of each will be demanded—growth in the supply of capital necessarily means a drop in the relative prices of autos for a closed economy and, not surprisingly, an associated decline in the return to capital, the auto industry’s intensively used factor. Moreover, as Stolper and Samuelson demonstrated earlier, the decline is not merely relative; capital’s return must fall in real terms. Likewise, the return to labor, the factor now relatively scarcer, must rise in real terms. For an open economy, the Rybczynski theorem allows predictions about the resulting changes in a country’s equilibrium trade volume and terms of trade. As the stock of capital grows, desired trade at given terms of trade will increase (decrease) if the country is capital-abundant (labor-abundant) relative to its trading partners. An expansion of the capital stock will thus lead to deterioration (improvement) in the country’s terms of trade. Corresponding results hold for an expansion of labor with capital held constant.

Despite its stark assumptions, the Rybczynski theorem, especially as later generalized by Jones, provides powerful insights into the likely consequences of biased factor growth. Like the supply and demand curves of partial-equilibrium analysis, the simple Heckscher-Ohlin model provides the first back-of-the-envelope attack on a wide range of questions. Together with the other theorems of the Heckscher-Ohlin model, the stripped-down basic version of the Rybczynski theorem has become an essential part of the intellectual toolkit of every international economist.

**SEE ALSO** *Heckscher-Ohlin-Samuelson Model; Stolper-Samuelson Theorem*

## BIBLIOGRAPHY

Jones, Ronald W. 1965. The Structure of Simple General Equilibrium Models. *Journal of Political Economy* 73 (6): 557–572.

Rybczynski, T. M. 1955. Factor Endowment and Relative Commodity Prices. *Economica* 23 (88): 352–359.

Stolper, Wolfgang, and Paul A. Samuelson. 1941. Protection and Real Wages. *Review of Economic Studies* 9 (1): 58–73.

*Rachel McCulloch*

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**Rybczynski Theorem**