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 is concerned with Fisher's tests for index numbers. In particular, uniqueness and inconsistency theorems are proved. Beyond that, Fisher's system of tests is weakened considerably. Without any regularity assumption (such as differentiability or continuity) it is shown that every subset of the system of weakened tests is consistent while the whole system is inconsistent. The question of how far the whole system must be weakened to obtain a consistent set of tests is also considered.]