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Appendix 4.1
Alternate Proofs of Selected HO Theorems
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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.
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FIGURE A4.1 Isoquant Map for the S Industry
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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).
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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).
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FIGURE A4.2 Proof of the HO Theorem (Price Definition)
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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.
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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.
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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’.
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FIGURE A4.3 Proof of the Rybczynski Theorem
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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.
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FIGURE A4.4 Proof of the Stolper–Samuelson Theorem
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Appendix 4.2
The Specific Factors Model
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FIGURE A4.5 Equilibrium in the Specific Factors Model
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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 0S (0T). The vertical axes measure wage rate in A.
• The VMPS curve shows the S industry’s demand for labor; the industry will hire labor until W =PS x MPLS. Likewise for VMPT curve.
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Equilibrium in Specific Factors Model
• Labor market equilibrium occurs at the intersection of the VMPS and VMPT curves.
• 0SD workers are employed in the S industry and D0T workers in the T industry.
• Wage rate paid to workers in both sectors is W0.
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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.
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FIGURE A4.6 Effects of an Increase in PS