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Two rare analyses on the theory of international economics with linear economics exist that have different lines of thought but similar model specifications. One is the analysis of (production-)efficient patterns of specialization that allows intermediate goods with the Ricardo–Leontief model and that belongs to the field of modern economics. The other is the Sraffa model extended to international economy, which does not belong to the field of modern economics. In the model setting, the difference between the two analyses is whether the rate of profit exists or not, although the meanings whether the rate of profit exists or not are very different. However, at least in the era of Deardorff (2005a), only the definition of comparative advantage, including intermediate inputs, is not determined and has been the focus since McKenzie (1954a, b, 1955) and Jones (1961) analyzed the pattern of specialization in the multi-country, multi-good Ricardo–Graham model. Shiozawa (2007) made progress on this subject by extending the Sraffa model internationally on the evolutionary economics front but not in modern economics. In this subject, the solution to the problem which these analyses focus on is the production-efficient pattern of specialization; however, there are two problems with this approach. First, in the case where the number of goods is larger than that of countries, the efficient pattern of specialization essentially does not exist. Focusing on this case, Shiozawa (2007) showed the extended concept of pattern of specialization, i.e., “shared pattern of specialization,” and pointed out the importance of the case in the real-world economy. Second, as in Higashida (2005a, Japanese), which uses illustrations of price and specialization traditionally presented in Amano (1966) and Ikema (1993, Japanese), the (production-)efficient pattern of solution is not unique in the case of Jones’ (1961) setting allowing intermediate goods. Jones (1961) focused on the “production assignment problem” between technological parameters to determine (production-)efficient pattern of specialization. To solve the problem, Jones (1961) uses the method of the Hawkins–Simon theorem, where the concept of Z-matrix is the easier treatable concept of the linear complementarity problem. Higashida’s (2005b) result means that the (production-)efficient pattern of specialization cannot be determined easily with only a simple extension to Jones’ (1961) way. Considering the solution in the case allowing intermediate inputs, the more difficult concept—the S-matrix—which does not have the equivalent concept, must be used. Thus, the final solution, i.e., the necessary and sufficient technological parameters’ condition that determines the (production-)efficient pattern of specialization, may not exist. Shiozawa (2007) showed the general existence of a solution, considering the case allowing the number of goods is larger than that of countries (which may become the last meaningful progress of the model analysis), if the simple and meaningful economic condition like Jones’ inequality does not appear. Shiozawa (2014, Japanese) saw through this and positioned the result as a “final solution,” giving historical meaning to evolutionary economics. However, this progress has implications for not only evolutionary economics but also modern economics. This chapter discusses the significance of Shiozawa’s progress in terms of modern economics and in the context of historical illustrations.
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For the difference between bilateral Ricardian comparative production cost theory and multilateral Ricardian comparative production cost theory, see Minabe ( 1971).
For linear economic models, see, e.g., Gale ( 1960).
For more about the Ricardo–Sraffa formulation, see, e.g., Takamasu ( 1991).
For more about Helly’s theorem in convex set theory, see, e.g., Ludwig et al. ( 1963).
For more about tropical algebra, see, e.g., Ilia et al. ( 2009).
On the basis of Ikema ( 1993), Minabe ( 2001) illustrated the wage and good produced in the three-country model. However, Minabe’s ( 2001) illustration is difficult to extend to the Ricardo–Leontief model allowing intermediate goods, at least in general. Minabe’s ( 2001) illustration is applied in Shiozawa ( 2015).
To treat similar conditions of Hawkins–Simon theorem with matrices allowing positive off-diagonal elements, the theory of linear complementarity problem is required. See the linear complementarity problem’s orthodox textbook, e.g., Cottle et al. ( 1992), but the same (i.e., necessary and sufficient) conditions as in the Hawkins–Simon theorem with matrices allowing positive off-diagonal elements do not exist in Cottle et al. ( 1992).
In reality, the difference of each coefficient may be explained as other factors that cannot move, such as the endowment of immobile capital or land.
More generally, the formulation of joint production allowing intermediate inputs requires multiple kinds of outputs, but neither the nonnegative condition of each coefficient nor the naming of production process in Ogawa ( 2011b) is required. However, this is a benchmark case, so in the context, the formulation is used. Moreover, this chapter uses this formulation which is simple but a little strong assumption because Ogawa’s ( 2013b) meaning is that the readers can generalize the general formulation only arguing the simple formulation essentially.
For more about relaxing the assumption that labor is necessary for production, see Hosoda ( 2007).
T indicates transposition.
In this section, the free disposal of goods with positive value is permitted but not for those with negative value.
The chapter represents vector inequality with ≥ , > , > >.
In Shiozawa ( 2007), patterns of specialization are extended as a “shared pattern of specialization” when the number of the countries is smaller than the number of goods.
Some words are complemented.
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