Economists have long been aware of the importance of increasing returns to scale in many activities, and particularly as the partial determinant of trade patterns. However, the technical problems which arise in attempting to formulate and solve economic models incorporating increasing returns-or, similarly, externalities-are formidable and have led to a concentration on other issues which can be analysed using convex theory. Some of the most interesting problems which are presented by economies of scale concern the nature and scheduling of investment, and thus one of the primary justifications of planning has been the need to take best advantage of increasing returns and externalities. Recently some work on these problems has been done by Weitzman [6], Dixit, Mirrlees and Stern [3], and in a quantitatively-oriented model by Westphal [8]. The present paper is a further attempt to examine planning with increasing returns to scale. It differs from those mentioned in its explicit introduction of trade and in the fact that the economies of scale do not depend on the size of plant, but on the scale of industrial production. This last feature means that the analysis does not focus on the lumpiness of investments, but rather on the inter-industry ramifications of increasing returns. The model specified depicts an underdeveloped country which is gradually investing to build up its industrial sector. Trade is vital to such a country both for the import of capital goods and in order to relieve it of the need to build up all industries in such a way as to balance supply with domestic demand for consumption and inter-industry input requirements. For convenience it is assumed that all capital goods are imported; this interpretation is not, in fact, restrictive because we could view the export surplus as measuring the domestic resources available for investment purposes. The model has been set up to focus our attention on the investment allocation problem in such an economy. For this reason we assume that capital is non-shiftable between industries; in other words once capital has been invested in industry i it cannot subsequently be transferred to industry j. This is not an unreasonable assumption when considering underdeveloped countries, and has the virtue that it emphasizes the inter-industry aspects of development rather than aggregative savings/ investment behaviour. A further feature of the model which emphasizes these issues is the assumption that there is no substitution in production processes, which are specified in terms of known input requirements per unit of output. These input needs fall as output is increased-hence the economies of scale. In fact, one could interpret the reasons underlying the decline in input coefficients rather broadly to incorporate a variety of external economies, learning, or some form of technical progress. Clearly these possibilities could be expanded upon for discussion in their own right, but it suffices for present purposes to use the all-embracing term of increasing returns. Though the usual analytical techniques of optimal control theory do not give sufficient conditions for the optimal path, they do provide ample information about necessary
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