Center for Economic Research, Department of Economics, University of Minnesota
Classical economic agents perform arbitrarily complex operations on
arbitrarily complex magnitudes (real numbers). By contrast, real world agents
have bounds on their abilities to perceive, think about, calculate with, and communicate
magnitudes. There are many ways to model agents with bounded
abilities, and here we mention two - one through bounds on computational
abilities, and one through bounds on descriptive or definitional abilities.
In both cases, we propose a "uniformity principle" constraining in a parallel
fashion both the magnitudes (prices, quantities, etc.) and the operations (to perceive,
evaluate, choose, communicate, etc.) that agents can use. We focus on the
definitional bounds, deferring computational bounds to other papers (1996a,b).
The languages allowed are those of ordered rings, and certain expansions; the
structures are those of real closed ordered fields, and corresponding expansions.
It is not obvious that a theory of definable economies is possible, since there
may not be any definable structures that are reasonably close to the classical
one. And even if such structures existed, it is not obvious that the classical
theorems of economics would hold in them.
Our two main conclusions are positive: In many interesting cases mathematical
structures do exist with definability-bounded agents. Furthermore, many
classical theorems of economic theory survive in a definable context: existence
of demand and utility functions, existence of competitive equilibria, First and Second Welfare Theorems, characterization of aggregate excess demand, etc.
Our proofs rely on theorems of mathematical logic (completeness (Tarski),
model completeness (A. Robinson, Wilkie), o-minimality (van den Dries, Pillay
and Steinhorn, Wilkie)) that allow us to establish existence of definable models
and to transfer classical theorems to a definable framework.
Although superficially different, the concepts underlying (Blume and Zame,
1992) are fundamentally close to the ones we use here.
Richter, M.K. and Wong, K., (1996), "Bounded Rationalities and Definable Economies", Discussion Paper No. 295, Center for Economic Research, Department of Economics, University of Minnesota.
Richter, Marcel K.; Wong, Kam-Chau.
Bounded Rationalities and Definable Economies.
Center for Economic Research, Department of Economics, University of Minnesota.
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