The Stolper-Samuelson Theorem: Mathematical Derivation

The Stolper-Samuelson theorem was originally derived to analyze the effects of a tariff on factor prices in the context of the H-O model. Since a tariff will raise the domestic price of a country's import competing good, if we know what happens in the model when a price changes, then we can deduce the effect of a tariff.

More generally, of course, the theorem tells us the effects on factor prices for any change in the price of an output good for whatever reason. Thus, one can apply the theorem anytime some change in the model causes a change in one of the output prices. This could occur as a country opens up to free trade, as growth and investment affects a country's endowments, and many other things.

We use the two zero profit conditions which must be satisfied in an equilibrium.

(5a)
(5b)

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.

Differentiating (5a) with respect to p₁ yields,

Rearranging terms gives,

The last two terms in this expression

Why?

Recall that

which implies,

Substituting expressions (4a) and (4b) from the cost minimization exercise yields,

which in turn implies,

Thus, we can reduce the above expression to

(6a)

Similarly, by differentiating equation (5b) above and following similar procedures we can get,

(6b)

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.

Differentiating (5a) with respect to p₁ yields,

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

(7a)
(7b)

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 labor-intensive). Then, since each unit factor requirement is positive,

and,

This implies, that if good one is capital-intensive and if the price of good one rises, then the equilibrium wage will fall for all workers and the equilibrium rental rate will rise for all capital owners.

If we conducted the same exercise for changes in the price of good two, 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., that good one is 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 Stolper-Samuelson theorem.

If the price of a good rises (falls) then the price of the factor used intensively in that industry will also rise (fall) while the price of the other factor will fall (rise).

International Trade Theory and Policy - Chapter 115-2: Last Updated on 3/10/98