## Factor Price Equalization

l2L V a2K j

We have already illustrated the gradient vectors (aiL, aiK) to the iso-cost curves in Figures 1.5

and 1.6. Now let us take these vectors and re-graph them, in Figures 1.7 and 1.8. In the simpler case of Figure 1.7, we have a single equilibrium for factor prices and a single set of labor and capital requirements (a1L, a1K) and (a2L, a2K). Multiplying each of these by the output of their respective industries, we obtain the total labor and capital demands y1(a1L, a1K) and y2(a2L, a2K). Summing these as in (1.8') we obtain the labor and capital endowments (L, K). But this exercise can also be performed in reverse: for any endowment vector (L, K), there will be a unique value for the outputs (y1, y2) such that when (a1L, a1K) and (a2L, a2K) are multiplied by these amounts, they will sum to the endowments.

How can we be sure that the outputs obtained from (1.8') are positive? It is clear from Figure 1.7 that the outputs in both industries will be positive if and only if the endowment vector

(L, K) lies in-between the factor requirement vectors (a1L, a1K) and (a2L, a2K). For this reason, the space spanned by these two vectors is called a "cone of diversification", which we label by cone A in Figure 1.7. In contrast, if the endowment vector (L, K) lies outside of this cone, then it is impossible to add together any positive multiples of the vectors (a1L, a1K) and (a2L, a2K) and arrive at the endowment vector. So if (L, K) lies outside of the cone of diversification, then it must be that only one good is produced. At the end of the chapter, we will show how to determine which good it is.9 For now, we should just recognize that when only one good is

9 See problem 1.5. Figure 1.7 produced, then factor prices are determined by the marginal products of labor and capital as in the one-sector model, and will certainly depend on the factor endowments. This is why the Lemma stated above requires that both goods are produced, or equivalently, that the endowments are inside the "cone of diversification."

Now consider the more complex case in Figure 1.8, where we have re-drawn the two sets of gradient vectors (a1L, a1K) and (a2L, a2K), and (b1L, b1K) and (b2L, b2K) from Figure 1.6, after multiplying each of them by the outputs of their respective industries. These vectors create two cones of diversification, labeled as cone A and cone B. Now we can answer the question of which factor prices will apply in any given country: a labor abundant economy, with a high ratio of labor/capital endowments such as (La,Ka) in cone A of Figure 1.8, will have factor prices given by (wA, rA) in Figure 1.6, with low wages; whereas a capital abundant economy with a high ratio of capital/labor endowments such as shown by (LB,KB) in cone B, will have

factor prices given by (w , r ), with high wages. Thus, factor prices will depend on the endowments of the economy. A labor-abundant country such as China will pay low wages and a high rental (as in cone A). In contrast, a capital-abundant country such as the United States will have high wages and a low rental (as in cone B).

In summary, the "single cone" illustrated in Figures 1.5 and 1.7 show how we solve the zero-profit conditions (1.7) when there is a unique solution for the factor prices, and then use this solution in the full-employment conditions (1.8) to evaluate the ay coefficients and solve for outputs. In comparison, the "multi-cone" as presented in Figures 1.6and 1.8 show that when there are multiple solutions for factor prices from the zero-profit conditions, then we also need to make use of the full-employment conditions to determine which factor prices prevail in each country. Despite the complexity of the latter case, many trade economists feel that countries do in fact produce in different cones of diversification, and taking this possibility into account is a topic of current research.10

Let us conclude this section by returning to the simple case of a single cone of diversification, in which the Lemma stated above applies. What are the implications of this result for the determination of factor prices under free trade? To answer this question, let us sketch out some of the assumptions of the Heckscher-Ohlin model, which we be study in more details in the next chapter. We assume that there are two countries, with identical technologies but different factor endowments. We continue to assume that labor and capital are the two factors of production, so that under free trade the equilibrium conditions (1.7) and (1.8) apply in each country with the same product prices (p1,p2). With a single cone of diversification, we can draw Figures 1.5 and 1.7 for each country. Allowing factor endowments to differ across the countries will not affect the factor prices provided that both countries stay within the cone of diversification. In other words, the wage and rental determined by Figure 1.7 is identical in the two countries. We have therefore proved the Factor Price Equalization Theorem, which is stated as follows:

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