The H-O Theorem Graphical Depiction: Variable Proportions

The H-O model assumes that the two countries (United States and France) have identical technologies, meaning they have the same production functions available to produce steel and clothing. The model also assumes that the aggregate preferences are the same across countries. The only difference that exists between the two countries in the model is a difference in resource endowments. We assume that the United States has relatively more capital per worker in the aggregate than does France. This means that the United States is capital abundant compared to France. Similarly, France, by implication, has more workers per unit of capital in the aggregate and thus is labor abundant compared to the United States. We also assume that steel production is capital intensive and clothing production is labor intensive.

The difference in resource endowments is sufficient to generate different PPFs in the two countries such that equilibrium price ratios would differ in autarky. To see why, imagine first that the two countries are identical in every respect. This means they would have the same PPF (depicted as the blue PPF₀ in Figure 5.7 "Endowment Differences and the PPF"), the same set of aggregate indifference curves, and the same autarky equilibrium. Given the assumption about aggregate preferences—that is, U = C_CC_S—the indifference curve, I, will intersect the countries’ PPF at point A, where the absolute value of the slope of the tangent line (not drawn), P_C/P_S, is equal to the slope of the ray from the origin through point A. The slope is given by $\frac{C_S^A}{C_C^A}$. In other words, the autarky price ratio in each country will be given by

$${\left(\frac{P_C}{P_S}\right)}_{Aut}^0=\frac{C_S^A}{C_C^A}\text{.}$$

Next, suppose that labor and capital are shifted between the two countries. Suppose labor is moved from the United States to France, while capital is moved from France to the United States. This will have two effects. First, the United States will now have more capital and less labor, and France will have more labor and less capital than it did initially. This implies that K/L> K∗/L∗, or that the United States is capital abundant and France is labor abundant. Second, the two countries’ PPFs will shift. To show how, we apply the Rybczynski theorem.

The United States experiences an increase in K and a decrease in L. Both changes will cause an increase in output of the good that uses capital intensively (i.e., steel) and a decrease in output of the other good (clothing). The Rybczynski theorem is derived assuming that output prices remain constant. Thus if prices did remain constant, production would shift from point A to B and the U.S. PPF would shift from the blue PPF₀ to the green PPF in Figure 5.7 "Endowment Differences and the PPF".

Using the new PPF, we can deduce what the U.S. production point and price ratio would be in autarky given the increase in the capital stock and the decline in the labor stock. Consumption could not occur at point B because first, the slope of the PPF at B is the same as the slope at A because the Rybczynski theorem was used to identify it, and second, homothetic preferences imply that the indifference curve passing through B must have a steeper slope because it lies along a steeper ray from the origin.

Thus to find the autarky production point, we simply find the indifference curve that is tangent to the U.S. PPF. This occurs at point C on the new U.S. PPF along the original indifference curve, I. (Note that the PPF was conveniently shifted so that the same indifference curve could be used. Such an outcome is not necessary but does make the graph less cluttered.) The negative of the slope of the PPF at C is given by the ratio of quantities C_S′/C_C′. Since C_S′/C_C′ > C_S^A/C_C^A, it follows that the new U.S. price ratio will exceed the one prevailing before the capital and labor shift, that is, P_C/P_S > (P_C/P_S)⁰. In other words, the autarky price of clothing is higher in the United States after it experiences the inflow of capital and outflow of labor.

France experiences an increase in L and a decrease in K. These changes will cause an increase in output of the labor-intensive good (i.e., clothing) and a decrease in output of the capital-intensive good (steel). If the price were to remain constant, production would shift from point A to D in Figure 5.7 "Endowment Differences and the PPF", and the French PPF would shift from the blue PPF₀ to the red PPF′.

Using the new PPF, we can deduce the French production point and price ratio in autarky given the increase in the capital stock and the decline in the labor stock. Consumption could not occur at point D since homothetic preferences imply that the indifference curve passing through D must have a flatter slope because it lies along a flatter ray from the origin. Thus to find the autarky production point, we simply find the indifference curve that is tangent to the French PPF. This occurs at point E on the red French PPF along the original indifference curve, I. (As before, the PPF was conveniently shifted so that the same indifference curve could be used.) The negative of the slope of the PPF at C is given by the ratio of quantities C_S″/C_C″. Since C_S″/C_C″ < C_S^A/C_C^A, it follows that the new French price ratio will be less than the one prevailing before the capital and labor shift—that is, P_C∗/P_S∗ < (P_C/P_S)⁰. This means that the autarky price of clothing is lower in France after it experiences the inflow of labor and outflow of capital.

All of the above implies that as one country becomes labor abundant and the other capital abundant, it causes a deviation in their autarky price ratios. The country with relatively more labor (France) is able to supply relatively more of the labor-intensive good (clothing), which in turn reduces the price of clothing in autarky relative to the price of steel. The United States, with relatively more capital, can now produce more of the capital-intensive good (steel), which lowers its price in autarky relative to clothing. These two effects together imply that

$${\left(\frac{P_C}{P_S}\right)}_{Aut}^{US}>{\left(\frac{P_C}{P_S}\right)}_{Aut}^{FR}\text{.}$$

Any difference in autarky prices between the United States and France is sufficient to induce profit-seeking firms to trade. The higher price of clothing in the United States (in terms of steel) will induce firms in France to export clothing to the United States to take advantage of the higher price. The higher price of steel in France (in terms of clothing) will induce U.S. steel firms to export steel to France. Thus the United States, abundant in capital relative to France, exports steel, the capital-intensive good. France, abundant in labor relative to the United States, exports clothing, the labor-intensive good. This is the H-O theorem. Each country exports the good intensive in the country’s abundant factor.