Preference and Choice

Takayama oriented his Mathematical Economics stressing the set-theoretic approach to behavioral modeling, in contrast to the traditional calculus approach. In this Oxford book however, it is the rationality assumption (completeness and reflexibility of preference relations), present in earlier times including Takayama's for ease of modeling, that the authors now try to substitute with the new concept of choice rule, which basically weakened the assumption of complete and transitive rationality to instantaneous and pairwise rationality (WARP). How would such a simplification change the mathematical formulations? Maybe it changed nothing, but just extended the application of these formulations to less rational conditions, like the extension of KTCQ of optimization which is calculus based thus requiring differentiability, to non-differentiable cases using non-linear programming.

Like differentiability, completeness and transitivity are also properties of functions / correspondences from X, to >() if not a function (not continuous), and to R if represented by utility. The structure (X, >()) is a correspondence commonly restricted by the axiom of rationality, while the structure (B, C()) is a correspondence commonly restricted by the axiom of WERP (consistency in behavior rather than in the intellect, i.e. people are, assumed, to be more motivated or even forced to behave consistently in a social context, which reflects some rationality of mind in the decision making process, but such a process can also be skipped to arrive at the same behavior.).

However, less assumptions usually means more information to be considered and more complex analysis. Preference relations relaxed differentiability but generally went back to utility functions (assuming continuity and very often even back to differentiability) in analyzing practical problems. The choice approach will also do the same. 

Theories are simplifications of the world, and relaxing assumptions generally brings theories down by introducing a more complex world, which however is also a way of testing how powerful and brilliant the theories are. Everyone can have their own theories, just some theories are based on relatively less information and more assumptions in the view point of those who have more information, and these theories are usually just called prejudice, presumption, or cliche if information come with time. In this sense those economic theories that do not pass the test of 'choice rule' might then be considered as prejudice, presumption, or cliche, but can still be true and influential within their circles or within their time.

What then, do the assumptions of continuity and differentiability mean exactly, if preference relations means rationality and choice rule means consistent behavior? It seems the two calculus related concepts have something to do with the comparative statics and thus the thermodynamical equilibrium concept.

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