The Rybczynski Theorem:
Mathematical Derivation

The Rybczynski theorem demonstrates the effects of changes in the resource endowments on the quantities of outputs of the two goods in the context of the H-O model. One can apply the theorem anytime some change in the model causes a change in one of the endowments. This could occur as a country invests and thus raises its capital stock, if immigration or emigration occurs or as population growth or growth of the workforce occurs for other reasons.

We use the two resource constraint conditions which must be satisfied in an equilibrium.

 (5c) (5d)

The asterisks indicate that these unit-factor requirements are the optimal levels derived from the cost minimization exercise and are functions of the wage, w, and the rental rate on capital, r. We will assume that wages and rents remain fixed which implies that output prices remain fixed as well.

Differentiating (5a) and (5d) with respect to L yields,

 (8a) (8b)

Writing these in matrix form yields,

This expression can now be solved using Cramer's Rule to get,

 (9a) (9b)

Whether these partial derivatives are positive or negative depends on the signs of the denominator.

Assume the denominator of each expression is less than zero. Then,

implies

which is true if

or

This means that the denominator is negative if and only if production of good one is capital-intensive and production of good two is labor-intensive.

So, let's suppose that good one is capital-intensive (good two is labor-intensive). Then, since each unit factor requirement is positive,

and,

This implies, that if good one is capital-intensive (good two labor-intensive) and if the labor endowment rises, then the output of good one would fall and the output of good two would rise if output prices of both goods remained the same.

If we conducted the same exercise for changes in the capital endowment, and we continue to assume that good one is capital-intensive and good two labor-intensive, then we would show that,

If we assumed the converse, i.e., if good one were labor intensive and good two capital intensive, then the signs of all of the above derivatives would be reversed.

These results lead to the following general statement of the Rybczynski theorem.

If a factor endowment in a country rises (falls), and if prices of the outputs remain the same, then the output of the good that uses that factor intensively will rise (fall) while the output of the other good will fall (rise).