Proof of the Heckscher-Ohlin Theorem

Note that supply of good 1, , is positively sloped, and supply of good 2, and supply of good 2, , is inversely related to . Thus, relative supply of good 1, is positively sloped.

First step: Labor-intensive output/capital-intensive output is higher in the labor-abundant country.

By assumption HC is labor abundant. First, remove ΔL workers from the HC so that the two countries have the same capital-labor ratios, i.e., Then, the output ratios will be the same in both countries.

In this case, there would be no trade. See Figure 23.

Next, add ΔL to the labor supply of the HC. By the Rybczynski theorem, y₁ increases and y₂ decreases. Thus,

or dropping the middle term, we have

See Figure 24.

Second step

The world output ratio lies somewhere between output ratios of the two countries.

Third Step

Consumers in both countries have identical and homothetic preferences. Their income levels may differ, but their consumption ratios are the same, equal to the world output ratio. Thus, the middle term can be replaced by .

or simply

which implies the labor-abundant HC exports good 1, which is labor-intensive.