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 choses 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 semidefiniteness 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 essay 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 implv transitive behaviour.
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Bibliography
Allen, R.G.D. 1932. The foundations of a mathematical theory of exchange. Economica 12, 197–226.
Debreu, G. 1954. Representation of a preference ordering by a numerical function. In Decision Processes, ed. R.M. Thrall, C.H. Combs and R.L. Davis, New York: Wiley, 159–65.
Frisch, R. 1926. Sur un problème d’économie pure. Norsk mathematisk forenings skrifter 16, 1–40.
Gorgescu-Roegen, N. 1936. The pure theory of consumer’s behavior. Quarterly Journal of Economics 50, August, 545–93.
Georgescu-Roegen, N. 1954. Choice and revealed preference. Southern Economic Journal 21, October, 119–30.
Katzner, D. 1971. Demand and exchange analysis in the absence of integrability conditions. In Preferences, Utilit, and Demand, ed. J. Chipman et al., New York: Harcourt, Brace, Jovanovich, 254–70.
Kim, T. and Richter, M. 1986. Nontransitive–nontotal consumer theory. Journal of Economic Theory 38, April, 324–63.
Mas-Colell, A. 1974. An equilibrium existence theorem without complete or transitive preferences. Journal of Mathematical Economics 1, 237–46.
May, K. 1954. Intransitivity, utility, and aggregation in preference patterns. Econometrica 22, January, 1–13.
Shafer, W. 1974. The nontransitive consumer. Econometrica 42, 913–19.
Sonnenschein, H. 1971. Demand theory without transitive preferences, with applications to the theory of competitive equilibrium. In Preferences, Utilit, and Demand, ed. J. Chipman et al., New York: Harcourt, Brace, Jovanovich, 215–23.
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© 1990 Palgrave Macmillan, a division of Macmillan Publishers Limited
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Shafer, W. (1990). Transitivity. In: Eatwell, J., Milgate, M., Newman, P. (eds) Utility and Probability. The New Palgrave. Palgrave Macmillan, London. https://doi.org/10.1007/978-1-349-20568-4_37
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DOI: https://doi.org/10.1007/978-1-349-20568-4_37
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