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A P P E N D I X B Derivation of Equations Representing Trade Models I his appendix presents the assumptions and the analysis necessary to derive the relations between trade and endowments that are given in Chapter 3. The Heckscher-Ohlin Model This section derives equation (3.1) of Section 3.1. Because the HeckscherOhlin model is now the standard in trade theory, there is no need here for a complete derivation that painstakingly fills in all intermediate steps.' Instead, the focus here is on details that are particularly pertinent in obtaining equation (3.1). The analysis follows that of Learner (1984, pp. 2-10, 158-160). The basic assumptions are as follows: HO-I. There are fixed endowments of S primary factors, which are mobile within countries but cannot cross borders. Vkj denotes the 7th country's endowment of factor k. HO-2. Each country produces N goods. The amount produced of each good is a function of the total endowments used in the sector, where that function exhibits constant returns-to-scale. Q11 is the amount of good i produced by country j . HO-3. S = N, the number of goods equals the number of factors. In a more general analysis, one could assume that N^S. For the ways in which models with N > S can be converted to ones with N = S, one should consult either Learner (1984, pp. 16-18) or Anderson (1987, p. 147). HO-4. Technology is the same across all countries. A discussion of the meaning and validity of this assumption is in Section 3.3. HO-5. The structure of factor endowments varies among countries less than the structure of factor input intensities varies across industries.2 1 Such an analysis can be found in standard textbooks—for example, Bhagwati and Srinivasan 1983. 2 For the meaning of this assumption the reader is referred to either Bhagwati and Srinivasan 1983, pp. 56-58, or Learner 1984, pp. 5-7. Although the assumption is important in deriving 238 Appendix B HO-6. Production decisions are made as «/profits are being maximized. The conditions under which this is the case are considered in Section 3.1. HO-7. The structure of income distribution and demand functions is such that the consumption of any particular good, at given prices, is the same proportion of national income in all countries. HO-8. Transportation costs are zero. HO-9. Trade is balanced for all countries. HO-IO. There are no tariffs, export subsidies, or other trade impediments. The analysis begins with equations for the demand and supply of resources. Given identical technologies in all countries and factor price equalization, which is implied by the assumptions, input-output coefficients are identical across countries. Let ak, be the amount of resource k used to produce one unit of good i. Then: S ν*=Σ^δ»· (Β-D / = 1 Given factor-price equalization and constant returns-to-scale: N S Gj= Σ P.Q, = Σ JkVk1 , (B.2) I = I A = I where G1 is the national income of country j , p, is the price of good i, and yk is the price of endowment k. It is convenient here to use the "dot" notation to form vectors: an entity with a dot subscript indicates the column vector formed by listing all the variables obtained by varying the subscript that is replaced by the dot. For example V'%J = (V,,, . . . ,VN). The input-output coefficients can be taken to form the matrix A = {ak}. Then if Λ is invertible, (B. 1) becomes: Q-J = A1 V.,. (B.3) Net exports, W11, are the difference between production and consumption: W.y = Q.j — C.j, where C.t is the consumption vector of country j . Given assumption HO-7, and introducing the parameters c„ which represent the proportion of national income spent on good i, the following equation de­ scribes the cross-country pattern of consumption: C.j = c.Q1 . (B.4) Denote world values with a w subscript. Because the world production of every good must equal world consumption: A~l V C = ^ T ' the conclusions, detailed discussion, which would involve a greater diversion into the mechanics of production economics than is merited in the present context, is not appropriate here. Appendix B 239 then W., = Q., - C , = A-1 V., - c Gj (B.5) = A-'V.j - A-l V.J.Gj/Gw ) . Denoting the elements of Λ inverse by aCi and using equation (B.2): Wu = Σ k=\ Noting that the term in square brackets...


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