Consumer theory with bounded rational preferences Georgios Geras´ ımou * Faculty of Economics AND King’s College, University of Cambridge June 28, 2010 Abstract Building on the work of Shafer (1974), this paper provides a continuous bivariate representation theorem for preferences that need not be complete or transitive. Applying this result to the problem of choice from com- petitive budget sets allows for a proof of the existence of a demand correspondence for a consumer who has preferences within this class that are also convex. Similarly to the textbook theory of utility maximization, this proof also uses the Maximum Theorem. With an additional mild convexity axiom that conceptually parallels uncertainty aversion, the correspondence reduces to a function that satisfies WARP. Keywords: incomplete & intransitive preferences, representation, demand. JEL: D0 1 Introduction This paper is concerned with the consumer’s problem of choosing the best among the fea- sible alternatives when her preferences over the latter are not assumed complete or transi- tive. It is motivated by experimental evidence that favors bounded rationality in this sense of limitations to the consumer’s decisiveness and consistency 1 and builds on the literature generated by the seminal work of Sonnenschein (1971), and particularly on the innovative paper of Shafer (1974). The demand theory presented here is one that highlights weighing as the fundamental behavioral principle in competitive choice environments where deviations from the utility maximization paradigm of completeness and transitivity are in place. The paper’s building block is a generalization in the direction of incomplete preferences of a continuous representation theorem due to Shafer (1974), which delivers a bivariate preference function that fully characterizes strict preferences in a general metric space. The properties of this representation are then compared to those of other results in the relevant literature and it is argued that the preference function generally performs better than weak- utility (Peleg (1970) and multi-utility (Kochov (2007), Evren and Ok (2007)) when the criteria of generality, tractability and characterization of strict preferences are considered jointly. We then restrict the consumption set to the neoclassical R n + , assume that preferences are hemicontinuous and convex, employ this representation theorem to maximize the prefer- * I am indebted to Robert Evans for very kind and helpful advice. I also thank Jose Apesteguia, Eddie Dekel, Jayant Ganguli, Nikos Gerasimou, David K. Levine, R. Duncan Luce, Fabio Maccheroni, S¨ onje Reiche, Ariel Rubinstein, Aldo Rustichini, Nicholas Yannelis, a referee and participants at SAET 2009 (Ischia) and RES 2010 (Surrey) for comments or questions. Any errors are my own. 1 See, for example, Anderson (2003) and Kivetz and Simonson (2000). 1
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