Nonsubstitution Theorems for a Small Trading Country

One of the elegant gems of modern economic theory is Paul Samuelson’s nonsubstitution theorem. Samuelson (1951) showed that a surprisingly rich class of closed economies behave in general competitive equilibrium as if production were of the Leontief fixed coefficient form. Though Samuelson’s assumptions allow the possibility of neoclassically smooth substitution among many inputs, his nonsubstitution theorem is thought to be of limited practical application because it must assume that there is only one non-produced factor of production and no joint production. The first assumption rules out any reasonable treatment of an economy with significant agricultural and mining sectors, while the lack of joint production makes it difficult to treat intertemporal issues adequately. In a later paper (1961), Samuelson pointed out that the nonsubstitution theorem extends to an economy with several non-produced factors so long as all factors but one can be purchased in arbitrary quantities at a fixed price. In international trade theory, a country is said to be “small” if its market activities can be assumed to have no effect on prices of internationally traded commodities. Samuelson’s 1961 nonsubstitution theorem could be interpreted directly as a theorem about small trading countries which have neither an agricultural nor a mining sector. Perhaps Hong Kong or Singapore is an example of such a country. Remarkably, the nonsubstitution theorem can be extended to a much larger class of small countries. In this paper, I will prove a nonsubstitution theorem for small countries in which there may be many nontradeable goods, including nonproduced factors of production, such as land and minerals. The assumptions allow joint production and the use of nontraded goods as inputs. Of course, some restrictive assumptions are required, but the class of economies included appears to be broad enough to permit a great deal of descriptive realism. In a closed economy where Samuelson’s nonsubstitution theorem applies, not only are factor proportions determined independently of final demand and of the endowment of labor, but so are the equilibrium prices of all produced goods relative to the wage rate of labor. In fact, the production possibility frontier is linear with slopes proportional to the equilibrium prices of the corresponding