Indivisibilities

Indivisible goods and factors constitute a subject of economic theory associated with a series of unsolved problems. Even advanced works on microeconomic theory like Varian (1992) or Jehle (1991) refrain from the consideration of indivisible goods and factors to provide a structure for the analysis where relatively simple mathematical methods can be applied. The Handbook of Mathematical Economics also does not contain any approach treating explicitly the integer problem. While Green, Heller (1981) present the instrument of convex analysis with respect to economic applications in the first chapter, a corresponding work dealing with the indivisibility of goods and factors is missing. Even in Brown (1991), who introduces an equilibrium analysis with nonconvex technologies in Chapter 36, the problem of indivisibility is merely of minor importance.

In an economy composed of many economic agents two groups of individual economic decision units may be stressed: households and firms. The theory of the household deals with the question how to satisfy the household’s needs, whereas the theory of the firm concentrates on the production of new goods.

The notion of duality is used differently and is often misleading.¹ Usually, the attention is drawn to certain symmetry properties of statements or optimization problems. For example, we can formulate in accordance with mathematical logic the subsequent duality principle for the statements A and B which are either “true” or “false”: if the word “and” is exchanged everywhere with the word “or” in the two semantically equivalent statements H₁ and H₂ which are only composed of the functional words “and”, “or”, and “not”, then the resulting dual statements H₁^* and H₂^* are semantically equivalent, too.² When the text stresses this dual view of different statements, then such a substitution principle is intended bearing the symmetry properties in mind.³

In every economy it is necessary to know who can dispose of what quantities of goods and for what purpose. The examination of these problems assumes that the property rights of goods are exerted personally by individual persons, i.e. common property is ruled out.

In treating indivisible goods and factors we follow a path which is based on well known approaches in microeconomic theory.