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

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increase in their real purchasing power. The opposite result holds for theowners of the other factor.Previous diagrams suffice to establish that an increase in the price of good 1 raiseswages and reduces the price of capital, holding the price of good 2 constant.In terms of growth rates,Pˆ1= wˆθ ˆ ˆ ˆL1+ rθK1+ aL1θL1+ aK1PˆθK12 = wˆθ L2+ rˆθ K 2 + aˆL2θL2+ aˆK 2where θijare the share of factor i in total earnings of good j. These equationssimplify because of an Envelope theorem: although the a’s adjust to factor pricechanges, the first-order effects of these adjustments on costs are zero because theyare starting from their optimal values.θK 2Each firm takes factor prices as given and adjusts its factor proportions (a’s) tominimize cost per unit output. From the first-order conditions, at any costw FL− daKminimizing point on an isoquant: = = , where the secondr FKdaQ=1Linequality follows from, dQ = 0 = F da + F da . Hence⎛ da⎜⎝ aKK⎞⎟ra⎠K⎛ da+⎜⎝ aLL⎞⎟wa⎠lL= 0 , or aˆ θ aˆθ = 0 .LjLKLj+Kj KjKSubstituting aˆ θ aˆθ = 0 into the price growth equations above yieldsLjLj+Kj KjPˆ1 = wˆθ L1+ rˆθ K1and Pˆ2 = wˆθ L2+ rˆθ K 2 , which implies that the commodity pricechanges are convex combinations of the factor price changes. Further, if the twocommodity prices move in different directions, the one that increases must movemore at a larger rate than the associated price.⎛ Pˆ⎞ ⎡θ⎜ 1 L⎟= ⎢⎝ Pˆ2 ⎠ ⎣θL12θθK1K 2⎤⎛wˆ⎞⎥⎜⎟ , so solving for factor price growth, we have:⎦⎝rˆ⎠⎛ wˆ⎞ ⎡ θ K 2⎜ ⎟ = ⎢⎝ rˆ⎠ ⎣−θL2−θθK1L1⎤⎥D⎦−1⎛ Pˆ⎞⎜ 1 ⎟, where D = θ L1θK 2 −θL2θK1>0 by ourˆ⎝ P2⎠assumption that good 1 is labor intensive. Setting Pˆ2 = 0 to normalize, the effectof an increase in P 1 on wages is positive and the effect on the return to capital is8
negative. Further the coefficientθ ’s lie between zero and one.θ K2D−1must be greater than unity because allImplication: if opening to trade drives down the price of the good that usesintensively the factor you supply, you stand to lose. (Lobbying groups should lineup according to factor rather than industry.)Remember for future discussions that both the factor price equalization theoremand the Stolper-Samuelson theorem depend heavily on some special assumptions.D. The Rybcznski TheoremRybcznski theorem(A dual to the Stolper-Samuelson theorem.) Expanding one factor stock whileholding output prices constant leads to a more than proportionate expansion inthe good that uses that factor intensively, and a contraction in the other good.Log-differentiating the factor market clearing conditions, we have:Lˆ= aˆL1λ + aˆλ + λ Xˆ+ λL1L2L2L11L2Xˆ2Kˆ= aˆK1λ + aˆλ + λ Xˆ+ λK1K 2K 2K11K 2Xˆ2Further, if we assume that the conditions for factor price equalization hold, factorintensities are pinned down by global prices:Lˆ= λ Xˆ+ λL11L2Xˆ2Kˆ= λ Xˆ+ λK11K 2Xˆ2Where ij λ the fraction of factor i allocated to good j. The logic is simple. Supposethe labor force expands, while capital is held fixed. With factor proportions fixedby global prices and factor price equalization, both goods cannot expand. If 2were to expand and 1 contract, the result would increase demand for capital andreduce demand for labor, exacerbating the disequilibrium. Instead, 1 expands and2 contracts to free up the necessary capital. (Think of the expansion in L asputting incipient downward pressure on the relative price of good 1, attracting asurge of demand for good 1 in global markets.)9
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