1 J. Tybout Economics 507a: International Trade Lecture 1 I ...

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Define:⎡aL1aL2⎤A = ⎢ ⎥ = matrix of input requirements per unit output. (Note that these⎣aK1aK2 ⎦depend only upon k j in each sector.)Factor market clearing condition:L = aL 1X1+ aL2X 2 and K = aK 1X1+ aK2 X 2 ,Competitive output pricing:P = a w a r and P = a w a r .1 L1+ K12 L2+ K 2Profit maximization:Competitive firms maximize profits by setting the price of each input equal to thevalue of its marginal product. Thus:w PjMPLjf j ( k j ) − k j f j '(k j )= =. The numerator is a monotonic positiver PjMPKjf j '(k j )function of k and the denominator is a monotonic negative function of k, given aconcave production technology ( f j ''(k j ) < 0.) Thus the wage-rental ratiouniquely determines the profit-maximizing capital-labor ratio in each sector,aKjk j = , and the factor requirements per unit output:aLjL j 1K j 1a Lj = = ; akj= = , thereby fixing theL j f ( k j ) f ( k j ) L j f ( k j ) k j f ( k j )minimum costs of producing a unit of output: C = a w a rB. The Factor Price Equalization Theoremj Lj +The claim:When a set of countries faces common global prices, and each countryproduces both goods, identical factor prices will prevail in all countries.Kj4
Geometric proofUnder the assumption of no factor intensity reversals, one good is always moreintensive in a particular factor (relative to the other good), regardless of factorprices. Let’s assume that good 1 is relatively labor intensive, by which we meanaL1aL2> .a aK1K 21rKP Q $1 country’s endowment ratio2 2 =• P Q $ 11 1 =•• Aan endowment outside the cone1LwWorld prices dictate the position of these unit revenue isoquants, and theassociated, unique wage-rental ratio that is consistent with zero profits in bothsectors. In turn, this factor price ratio implies factor proportions in each sector, asrepresented by the heavy rays from the origin.A caveat: If the country only produces one of the goods, its factor prices are notdetermined by international trade alone. When will it produce only one good?When its factor endowments lie outside the “cone of diversification” defined bythe heavy arrows. (Refer to the lower dotted arrow).Suppose that the country’s capital intensity is given by the lower dotted ray. Thenat the factor prices implied by global product prices, there is excess demand forcapital. To see why, note that the country’s factor endowment must be a convexcombination of the cone borders if it is to produce both goods:KL⎛= ⎜⎝L1⎞⎛K⎟⎜L ⎠⎝L11⎞ ⎛+ L2⎟ ⎜⎠ ⎝ L⎞⎛K⎟⎜⎠⎝LAt an endowment like the lower dotted vector, excess demand for capital at thefactor price ratio shown will drive up the relative price of capital until productionat point A is cheap enough to eliminate losses (refer to the factor price ratioassociated with the short dotted line).22⎞⎟⎠5
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1 J. Tybout Economics 507a: International Trade Lecture 1 I ...

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