¹ Stolper, W. F. and Samuelson, P. A., “Protection and Real Wages,” Review of Economic Studies, 11 1941.Google Scholar

² Rybczynski, T. M., “Factor Endowments and Relative Commodity Prices,” Economica, 11 1955.Google Scholar

³ Jones, R. W., “Duality in International Trade: A Geometrical Note,” this Journal, 08 1965 Google Scholar, and “The Structure of Simple General Equilibrium Models,” Journal of Political Economy, 12 1965.Google Scholar See also Samuelson, P. A., “Prices of Factors and Goods in General Equilibrium,” Review of Economic Studies, 1953 Google Scholar; Kemp, M. C., “The Gain from International Trade and Investment: A Neo-Heckscher-Ohlin Approach,” American Economic Review, 09 1966.Google Scholar

⁴ See Chipman, J. S., “Factor Price Equalization and the Stolper-Samuelson Theorem” (abstract), Econometrica, 10 1964.Google Scholar

⁵ Gale, D. and Nikaido, H., “The Jacobian Matrix and Global Univalence of Mapping,” Mathematische Annalen 159, 1965.CrossRefGoogle Scholar

⁷ At the Cleveland meeting of the Econometric Society, Sept. 5, 1963, Professor Chipman stated that the following matrix

is not the P-matrix (since its principal minors of the second and third order are all negative) but all diagonal elements of the inverse matrix are greater than unity. But I pointed out to him in Rochester that in this example industry 2 does not use factor 2, and industry 3 does not use factor 3 at all. Therefore, these diagonal elements of the inverse matrix may not represent the changes in the prices of the “intensive” factor for each industry. It is quite all right to rearrange the name of goods or the name of factors in the economic sense. We know that interchanging row (or column) vectors in the original matrix replaces the corresponding column (or row) vectors in the inverse matrix. By applying our definition of intensity to the example, we obtain

The stochastic matrix on the left becomes the P-matrix and all diagonal elements of the inverse matrix are still greater than unity. This may show the real meaning of “renumbering” commodities or factors.

⁷ This kind of matrix has been used by Jones, R. W. in his article “Comparative Advantage and the Theory of Tariffs: A Multi-Country, Multi-Commodity Model,” Review of Economic Studies, 06 1961.Google Scholar By using this definition we can exclude at least the case where any zero element lies in the diagonal of the Jacobian matrix.

⁸ This proof has been suggested by an anonymous referee.

⁹ I am in debt to Professor R. W. Jones for this example.

¹⁰ Each element a_ij shows factor j's share in industry i which is given by (∂p_i /∂w_j )w_j /p_i = α_ijW_j/p_i , where α_ij denotes the quantity of factor j required to produce a unit of commodity i. When renumbering of rows and columns makes the diagonal element the largest in each column, the income share of factor i is larger in the ith industry than in any other industry. In this case each commodity is easily able to associate with it an intensive factor. Our example belongs to this case.

¹¹ Chipman has presented an example of the weak form of the Stolper-Samuelson theorem denned above ( Chipman, J. S., “A Survey of the Theory of International Trade: Part 3, The Modern Theory,” Econometrica, 01 1966, 39 Google Scholar). But our argument is much stronger than Chipman's because the price of the intensive factor still rises in his case, while it decreases in our case.

¹² We can obtain directly the same result from the above example by renumbering the dual nature of the relationship between the Stolper-Samuelson theorem and the Rybczynski effect. This is so because the inverse of the transpose matrix is the same as the transpose of the inverse of the original matrix.

¹³ See my article, “The Hecksher-Ohlin Theorem, the Leontief Paradox and Patterns of Economic Growth,” American Economic Review, LVI, no. 5 (12 1966), 1193–211.Google Scholar

¹⁴ Leontief's input-output data show that the capital-abundant country, America, imports capital-intensive commodities. The “Leontief paradox” might be caused by the existence of many commodities and many factors in the actual world. This may give us a new explanation of the Leontief paradox. It has been made consistent if either (a) factor intensity reversals are present, or (b) technology differs as between the two countries such that the capital-abundant country has a technological superiority in producing the labour-intensive commodity. About the latter point see my article in American Economic Review, Dec. 1966.

¹⁵ Shinkai, Y., “On Equilibrium Growth of Capital and Labour,” International Economic Review, 05 1960.Google Scholar

¹⁶ Robinson, J., The Accumulation of Capital (London, 1956).Google Scholar

¹⁷ Notice that Stolper's and Samuelson's 2×2 Jacobian matrix is a special case of the row-ordered matrix.

¹⁸ We obtain this by multiplying the first row with 1/(c ₁ k), and the second row, …, the nth row with 1/(c ₂ k), …, 1/(c_nk) respectively. Then add them to the first row, which becomes (1, 1,…,1).

¹⁹ P. A. Samuelson first suggested that a sufficient condition for global univalence is that the successive principal minors of the Jacobian determinant lying in the upper left-hand corner be non-vanishing by some renumbering of goods or factors. ( Samuelson, P. A., “Prices of Factors and Goods in General Equilibrium,” Review of Economic Studies, 1953.Google Scholar) McKenzie has presented a counter-example which shows that Pearce's conjecture that the non-vanishing of the Jacobian alone might be sufficient for global univalence may not be true. (L. W. McKenzie, Factor Price Equalization and the Jacobian Determinant,” unpublished, University of Rochester, 1962; Pearce, I. F., “A Further Note on Factor-Commodity Price Relationship,” Economic Journal, 12 1959.CrossRefGoogle Scholar) Gale and Nikaido were the first to provide a counter-example which shows that Samuelson's condition is not sufficient for global univalence. (Gale and Nikaido, “The Jacobian Matrix and Global Univalence of Mapping.”)

²⁰ McKenzie has pointed out that a dominant diagonal matrix is a P-matrix. A matrix A is said to have a dominant diagonal if DA has a dominant diagonal which means the absolute value of the diagonal element in each column exceeds the sum of the absolute values of the other elements in that column, where D is a diagonal matrix with positive diagonal elements. Thus we know two kinds of sufficient conditions for a P-matrix in which the weak form of the Stolper-Samuelson theorem and full factor price equalization hold, that is, the row-ordered matrix and the dominant diagonal matrix. McKenzie, L. W., “Matrices with Dominant Diagonals and Economic Theory,” Mathematical Method in the Social Science, edited by Arrow, K. J., Karlin, S., and Suppes, P. (Stanford, Calif., 1959), 47–62.Google Scholar

²¹ This method of proof has been suggested by Professor McKenzie. Y. Uekawa presented a necessary and sufficient condition for the original form of the Stolper-Samuelson theorem. The condition might be clear from the proof mentioned above, but his condition is represented as follows: (i) the determinant of the Jacobian matrix is positive, (ii) all principal (n − 1)×(n − 1) cofactors are positive and (iii) for any i has a non-negative solution, where represents a matrix obtained from the original matrix by deleting the ith row and ith column of A, and a_i(i) is the ith column vector of the matrix A from which the ith componentis deleted. As I have proved here, a matrix defined above satisfies all those conditions. Uekawa, Y., “Factor Intensity and Global Univalence in the Stolper-Samuelson Theorem,” unpublished, Kobe City University of Foreign Studies, 1965.Google Scholar

²² Chipman, “A Survey of the Theory of International Trade.”

²³ McKenzie, L. W., “Equality of Factor Prices in World Trade,” Econometrica, 23, no. 3 (07, 1955)CrossRefGoogle Scholar, his Theorem 4, p. 248. The set K_pw is defined on p. 244.

²⁴ When we drop an which is greater than the average of α _i1, α _i2, …, α _in , the average of them will decrease.

²⁵ Notice that all corresponding sets K_pw contain always the vector (1, 1, …, 1). The same thing is true for the row-ordered matrix as shown in the proof of theorem 1. That is, the row-ordered matrix satisfies the McKenzie theorem as well as the P-matrix.

²⁶ Notice that we can obtain the Jacobian income share matrix A by differentiating the corresponding factor input matrix α. Therefore we can easily see this conclusion.

²⁷ In the previous section, I normalized for convenience a unit factor input vector such that the sum of factor inputs becomes n in each industry.

²⁸ As shown in note 20, a dominant diagonal matrix is a P-matrix. But it satisfies the McKenzie theorem as shown here graphically in a simple 3×3 case. The triangle used here has been suggested in McKenzie's paper. The algebraic proof for a general n×n case has been given by him in his “Equality of Factor Prices in World Trade.”