PROOF OF THE STOLPER-SAMUELSON THEOREM

As a break from my three-part series on asset bubbles, I will show the proof to the Stolper-Samuelson theorem as shown in this paper.

To read about the Stolper-Samuelson theorem, click here.

Proof of the Stolper-Samuelson theorem

The Stolper-Samuelson theorem says that in free trade, given two goods, an increase in the price of one good will result in an increase in the return to that good, and a fall in the return to the other good.

There are two proofs to the Stolper-Samuelson theorem provided in this paper. I will be working through the first one.

Part 1: Introduction

Assume two homogenous goods, A and B, and two factors of production, labor and capital (denoted by L and K respectively). Both goods use a mix of labor and capital, but good A is capital-intensive, while good B is labor-intensive.

Part 2: The dynamics between capital and labor in two industries

1. Assume that the amount of labor and capital in the economy are fixed. This will be denoted by L bar and K bar.

The total labor used in the economy is the sum of the labor used in industry A and the labor used in industry B. The total capital used in the economy is the sum of the capital used in A and B as well, i.e.:

2. Now:

3. Now:

4. The dynamics between labor and capital:

Combining this with (1):

I.e. when all the labor moves towards sector A, all the capital will move towards sector A as well.

Part 3: Prices, marginal product, wages, and rent

1. Let all the production in the economy be M. Let the price of good A be PA and the price of good B be PB.

Thus:

Now, we find the marginal products of labor and capital for both goods by finding the partial derivatives.

Let the price of labor, which is wages, be denoted by w. Let the price of land, which is denoted by rent, be r. Now, if we assume total factor mobility, then wages and rent must be equal (if they are not equal, then all the labor or capital will go to one industry, and there will be no labor or capital in the other).

2. Thus, each industry can swap out capital for labor equally as well as the other industry, i.e. the ratio of the marginal products in both industries must be equal:

8. Solve for PA and PB:

Thus, the price ratio is:

9. Reworking statement 2:

Part 4: The Stolper-Samuelson Theorem

When free trade takes place, “trade would reduce the relative price of the import-competing good”, according to the paper. In other words, PB would decrease. As PB decreases, labor and capital moves towards good A, as it is more profitable.

When the price ratio increases, the marginal product of capital in industry A will increase more than the marginal product of capital in industry B, as industry A is capital-intensive, i.e.

Similarly, when the price ratio increases, the marginal product of labor in industry B will decrease more than the marginal product of labor in industry A, as industry B is labor-intensive, i.e.

The lower relative marginal products of labor in both industries mean a lower real wage and a higher rent. This is the Stolper-Samuelson theorem.