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Copyright © 2010 Pearson Addison-Wesley. All rights reserved. Appendix 4.1 Alternate Proofs of Selected HO Theorems
Copyright © 2010 Pearson Addison-Wesley. All rights reserved Production Isoquant An isoquant shows the various combinations of labor and capital required to produce a fixed quantity of a product. The curvature of an isoquant indicates the ease of subsitutability between the two inputs, holding output constant. A straight line isoquant indicates that the inputs are perfect substitutes; right angles indicate that inputs are not substitutable.
Copyright © 2010 Pearson Addison-Wesley. All rights reserved FIGURE A4.1 Isoquant Map for the S Industry
Copyright © 2010 Pearson Addison-Wesley. All rights reserved Heckscher-Ohlin Theorem (Price Definition) If country A (B) is relatively abundant in K (L) and if good S (T) is relatively K (L)– intensive in its production, then country A (B) should have a comparative advantage in the production of good S (T).
Copyright © 2010 Pearson Addison-Wesley. All rights reserved Proof of HO Theorem See Figure A4.2 There are two isoquants, each representing the production of one unit of good S (T). The S isoquant is closer to the K-axis indicating that S is more K-intensive. The least costly input combination for producing a desired output level occurs at the tangency of an isocost line (such as GH) and an isoquant (such as point R for good S).
Copyright © 2010 Pearson Addison-Wesley. All rights reserved FIGURE A4.2 Proof of the HO Theorem (Price Definition)
Copyright © 2010 Pearson Addison-Wesley. All rights reserved HO Theorem Proof (cont.) If isocost line GH is tangent to both S and T isoquants (at points R and Q), then the cost of producing each product must be identical. The slope of isocost line GH is equal to country A’s autarky wage/rent ratio; GH cannot apply to country B. Since B is more labor abundant than A, its wage/rent ratio is lower than A’s. The isocost line to produce good S in country B is higher than the isocost line to produce T; thus, B has a comparative advantage in good T.
Copyright © 2010 Pearson Addison-Wesley. All rights reserved Proof of the Rybczynski Theorem Refer to Figure A4.3 Given isoquants representing $1 each of goods S and T and an isocost line tangent to both, the tangency points F and D represent optimal input combinations. The slopes of the rays from the origin passing through F and D indicate the optimal capital/labor use ratios.
Copyright © 2010 Pearson Addison-Wesley. All rights reserved Rybczynski Theorem (cont.) Given factor endowments represented by point E, draw a parallelogram connecting E to the two rays from the origin. Adding the factor combination OG (OH) to point H (G) will result in total endowment level E. When the country’s labor rises (capital and prices constant), the endowment level moves from E to E’. As a result, the output of S falls to G’ while T rises to H’.
Copyright © 2010 Pearson Addison-Wesley. All rights reserved FIGURE A4.3 Proof of the Rybczynski Theorem
Copyright © 2010 Pearson Addison-Wesley. All rights reserved Proof of Stolper-Samuelson Theorem Refer to Figure A4.4 The initial optimal input combinations are indicated by the tangency points F and D. If the price of T rises, then a $1 worth of this good is now on a lower isoquant T’. A new isocost line is tangent to the isoquants S and T’. A comparison of the isocost lines shows that wages have risen while rents have fallen. As a result, labor (capitalists) can purchase more (less) of both goods.
Copyright © 2010 Pearson Addison-Wesley. All rights reserved FIGURE A4.4 Proof of the Stolper– Samuelson Theorem
Copyright © 2010 Pearson Addison-Wesley. All rights reserved. Appendix 4.2 The Specific Factors Model
Copyright © 2010 Pearson Addison-Wesley. All rights reserved FIGURE A4.5 Equilibrium in the Specific Factors Model
Copyright © 2010 Pearson Addison-Wesley. All rights reserved Specific Factors (Ricardo-Viner) Model Same assumptions as HO Model except capital is immobile between industries Refer to Figure A4.5 The horizontal axis measures labor input in A, with labor units in S (T) industry measured from point 0 S (0 T ). The vertical axes measure wage rate in A. The VMP S curve shows the S industry’s demand for labor; the industry will hire labor until W =P S x MP LS. Likewise for VMP T curve.
Copyright © 2010 Pearson Addison-Wesley. All rights reserved Equilibrium in Specific Factors Model Labor market equilibrium occurs at the intersection of the VMP S and VMP T curves. 0 S D workers are employed in the S industry and D0 T workers in the T industry. Wage rate paid to workers in both sectors is W 0.
Copyright © 2010 Pearson Addison-Wesley. All rights reserved Effects of a Rise in Price of Good S Country A has a comparative advantage in S. When trade opens up, the price of S rises. Demand for labor will increase in industry S; employment in S rises while employment in the T sector falls. Wages also increase. Capital owners in industry S are better off as their rental payments rise.
Copyright © 2010 Pearson Addison-Wesley. All rights reserved FIGURE A4.6 Effects of an Increase in PS
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