On the Shape of the World Production Possibility Frontier with Three Goods and Two Primary Factors with and without Capital Mobility
研究在三种商品和两种初级要素（劳动不可流动、资本可流动或不可流动）的假设下，世界生产可能性边界的形状，包括平坦段和严格凹性等情形。
Recently there have been many articles discussing whether it is possible for two trading countries to specialize only incompletely when labor is immobile while capital is perfectly mobile between two countries. Kemp [1966] pioneered this topic, followed by Jones [1967] and Inada-Kemp [1969]. Chipman [1971] showed that, based on classical assumptions (two traded goods, two primary factors with labor immobile but capital freely mobile, neoclassical technologies, and perfect competition) the world production possibility frontier (i.e., the efficient set of world production) with complete diversification contains a flat segment under certain conditions. Uekawa [1972] furthered his results by investigating some sufficient conditions for the existence of such a flat segment of the world production possibility frontier (p.p.f.) with complete diversification. The aim of this paper is to investigate the shape of the world p.p.f. when there are three goods instead of two, and two primary factors with labor immobile and under alternative assumptions concerning capital mobility. The other assumptions remain classical; neoclassical production functions, fixed factor supply, and perfect competition in product markets as well as in factor markets. We first show that when production technologies are different and without capital mobility, if the world p.p.f. with complete diversification exists, then in general, it is flat. Next, we investigate some sufficient conditions for the existence as well as non-existence of such a world p.p.f. Then, assuming that capital is mobile, we show that if there is a good whose technology is the same between countries, and is produced by both countries, then the wage rate as well as rental price of capital is equalized between countries. Further, assuming that capital is mobile, if the technology of at least one good is different, then the world p.p.f. with complete diversification does not exist in general. Lastly we show that assuming the technologies of all goods are different, (1) if three goods are produced in the home country, and two goods are in the foreign country, the world p.p.f. is a part of a tangent developable surface without capital mobility, (2) the world p.p.f. is flat with capital mobility, (3) if each country produces two goods such that the world as a whole produces three distinct goods, then the world p.p.f. is strictly concave toward the origin without capital mobility, and (4) the world p.p.f. is a part of a tangent developable surface with capital mobility.