THE EXISTENCE of increasing returns to scale in production poses a number of difficult problems. In general equilibrium theory the problems are obvious. Standard results on the competitive equilibrium guarantee that when an equilibrium exists it is contained in the core of an economy. When the competitive equilibrium fails to exist, as is likely under increasing returns, the core may provide a useful equilibrium concept of its own. However, very little is known about the existence of a core except under conditions of constant returns to scale. One of the objectives of the present paper is to demonstrate the existence of the core in a general economy in which certain types of increasing returns are present. The second objective of this paper concerns partial equilibrium analysis. When it is asserted that a particular industry is an increasing returns industry it is not at all obvious what is being said, particularly when the industry produces more than one output. One aspect of increasing returns is the property of scale economies, which exist if by multiplying all inputs by a factor A > 1 it is possible to increase all outputs by at least a factor of A. A more primitive concept of increasing returns behavior is the concept of subadditivity of a cost function (or superadditivity of the production possibility set). Presumably the properties of subadditivity and economies of scale are both assumed of a given industry when it is said to have increasing returns. One particular problem involving increasing returns which has generated much recent research is the question of sustainability or supportability of natural monopoly.2 For this problem the existence of a core is of great interest, but at present, a set of necessary and sufficient conditions for a core to exist in a completely general model is not known. This paper is an attempt to advance the theory of natural monopoly, and of increasing returns industries in general, by considering a more general model of equilibrium than has previously been used. Thus, the paper is both a very general model of an essentially partial equilibrium problem and a rather specialized model of general economic equilibrium. In the next section, the basic model will be developed and its interpretation for both general and partial equilibrium will be discussed in greater detail.

[This paper considers a problem faced by players who are involved in a sequence of games: not necessarily the same games, not necessarily with the same opponents, and not necessarily under conditions of complete information. The players are assumed to act in response to stationary Markovian hypotheses which they form about the actions of their opponents. Conditions are explored which require that these hypotheses be correct on average and that the players actions be optimal in response to their hypotheses.]

This paper provides a complete set of local duality results for a utility maximizing consumer (or single output cost minimizing firm). Given a continuous local expenditure function defined on a compact, convex set of positive prices we establish the existence of continuous local direct, indirect utility and distance functions. This procedure avoids troublesome continuity problems at the boundary of IR N. In addition it is shown that if two utility functions are second order approximations at some point, then their respective expenditure, distance, and indirect utility functions are also second-order approximations to each other at some point. This latter result provides additional impetus for using duality theory and substantial justification for the use of flexible functional forms which can provide second-order differential approximations to any twice continuously differentiable function at a point.

Opportunities for individual learning in multi-period insurance contexts introduce fundamental economic aspects not present in conventional static models. Using a twoperiod model in which there are two states (accident and no accident), it is shown that more precise prior probability assessments lead to increased insurance coverage and reduced self-protection. These dynamic adverse incentive problems can be diminished by merit rating, which has a backwards influence on earlier actions. Self-protection and insurance purchases in the initial period respond in opposite fashion to changes in insurance prices in the second period, the interest rate, and parameters of the prior probability assessment.

This paper examines various theorems of trade and general equilibrium in a generalized framework involving arbitrary numbers of goods and factors. It develops structural relations among the changes in outputs, commodity prices, factor rewards, and factor endowments. By finding a way of inverting a bordered matrix with a singular Hessian, the paper derives explicit expressions for the following matrices: the Stolper-Samuelson matrix; the Rybczynski matrix; the matrix which measures the effect of a change in factor endowments upon factor rewards at constant commodity prices; and the matrix which measures the effect of a change in commodity prices upon outputs at constant factor endowments. Various properties of these matrices are used to obtain, among other results, the reciprocity relations and general results on factor-price equalization. The paper also ey.amines the problem of indeterminacy in production when the number of commodities exceeds the rank of the input-coefficient matrix and presents the correct specifications of the supply functions of outputs. Finally a new theorem on the degree of flatness of the production transformation surface is derived.

THIS PAPER CONSIDERS how certain theoretical results on consistent commodity aggregation can be applied to the problem of the estimation of a complete system of demand equations. The method proposed here builds upon the classic results of Gorman [16], particularly the case he calls perfect price aggregation. These results are well known in the literature on two-stage budgeting but have not been widely applied to problems of empirical demand analysis. This relative neglect of perfect price aggregation is regrettable because several of their features recommend their use in econometric applications. In the perfect price aggregate approach to demand analysis, demand equations are characterized as a two-level system consisting of a system of group expenditure functions and a number of systems of conditional demand functions. This two-level system provides a manageable way of introducing greater detail into a complete system of demand equations. Information about particular commodities may be introduced in the specification of conditional demand functions. As will be discussed below, the two-level system can be estimated by an iterative estimation method in which the maximum number of demand equations which can be estimated is greatly increased over the number found in past estimates of complete systems of demand equations. Even when one is principally interested in estimating group expenditures the perfect price aggregate approach is attractive because the intragroup substitutions due to detailed price changes can affect total group expenditures and this effect is taken into consideration through the computation of the perfect price indices. Thus it may be possible that a perfect price aggregate model will yield better estimates of group expenditures than would an approach which did not explicitly treat aggregation problems. Another pleasing feature of the perfect price aggregate approach to demand analysis is that it avoids the misspecification which results when the commodity

[This paper studies the effects of irreversibility of capital investment upon optimal exploitation policies for renewable resource stocks. It is demonstrated that although the long-term optimal sustained yield is not affected by the assumption of irreversibility (except in extreme cases), the short-term dynamic behavior of an optimal policy may depend significantly upon the assumption. It is suggested that the results may have profound implications for problems of rehabilitation of overexploited fisheries and other renewable resource stocks.]