Transitivity

Abstract

Transitivity is formally just a property that a binary relation might possess, and thus one could discuss the concept in any context in economics in which an ordering relation is used. Here, however, the discussion of transitivity will be limited to its role in describing an individual agent’s choice behaviour. In this context transitivity means roughly that if an agent chooses A over B, and B over C, that agent ought to choose A over C, or at least be indifferent. On the surface this seems reasonable, even ‘rational’, but this ignores how complicated an agent’s decision making process can be. For an excellent discussion of this issue see May (1954). Given a model of agent behaviour, transitivity can be imposed as a direct assumption, or can be an implication of the model for choice behaviour. The standard model of agent behaviour in economics is that the agent orders prospects by means of a utility function, which in effect assumes transitivity. With appropriate continuity and convexity restrictions on utility functions, the model allows one to demonstrate that: (1) Individual demand functions are well defined, continuous, and satisfy the comparative static restriction, the strong axiom of revealed preference (SARP). (In the smooth case, this corresponds to the negative semi-definiteness and symmetry of the Slutsky matrix.) (2) Given a finite collection of such agents with initial endowments of goods, a competitive equilibrium exists. What will be discussed in the remaining part of this article is to what extent one can obtain results analogous to (1) and (2) above while using a model of agent behaviour which does not assume or imply transitive behaviour. To keep the discussion as simple as possible, we will only consider the situation in which the agent’s set of feasible commodity vectors is the non-negative orthant of n-dimensional Euclidean space, and the agent’s problem is to choose a commodity vector x when faced with positive prices and income. A vector p in the positive orthant of Euclidean n-space will denote the vector of price-income ratios, or a ‘price’ system.

This chapter was originally published in The New Palgrave Dictionary of Economics, 2nd edition, 2008. Edited by Steven N. Durlauf and Lawrence E. Blume