时间：2022-09-10 10:45:51
【摘要】“Banach space”， “Arzela-Ascoli theory” …What are all the jargons and why should we care？ This paper surveys how functional analysis affects the economics study by reviewing the history. Evidence suggests that functional analysis is one of the most indispensable analytic tools in the economics study.
【关键词】functional analysis history of economics
Introduction
Almost all programs include a compulsory Math Camp before the start of the Ph.D. program. Those mathematic courses with a range from linear algebra， functional analysis to stochastic differential equations reveals the importance of mathematics on economic field， without referring to the dizzy symbols in top journals.
Is it entirely for "filter for the smartest candidates" suggested by Mankiw？ Or the jargons like "Contraction Mapping"， "fixed point theorems of Kakutani" and "Hahn-Banach theorem" have their places in economic analysis？ This paper suggests the latter one.
This paper surveys as follows： first I briefly review the history， and then show？the power of functional analysis in Optimization theory， General Equilibrium and Game theory before conclusion.
A brief History
Economics has an as etymology as management， which gives an innate connection to statistics and accounting. However， the using of advanced mathematic in economics is a recent story.
Dates back from 19th century， when economists started to model the world with calculus， the economics has become more and more mathematical throughout the 20th century. For the failure of calculus in delivering general results， more and more mathematic tools is introduced or developed.
Nowadays， an array of mathematical tools like stochastic calculus， convex analysis， and graphic theory are widely applied throughout this field， among which， the functional analysis holds the most profound impact. In particular， it gives birth to general equilibrium theory and game theory. Meanwhile， it’s indispensable tool in infinite-period optimization and asset pricing. Among its financial， the most remarkable ones are the martingale pricing by change of probability measure， and the equivalence of non-arbitrary and the stochastic discount kernel guaranteed by Hahn-Banach theory.
In the next section， I shall discuss two applications in detail.
Optimization Theory
"Economic is the science which studies human behavior as a relationship between ends and scarce means which have alternative uses."（L. Robbins， 1935）
What is economics？ Perhaps， there is no definite answer. Yet the most commonly accepted one is optimization under constraint. The consumers maximize their utility， the firms maximize the profits and the government maximizes the social welfare by influencing the constraints faced by the consumers and the firms with physic or monetary policy.
It’s tempting to regard multi-variable calculus as more than sufficient， for the Lagrangian method gives solution to any finite-period maximization problem without uncertainty. However， uncertainty， infinite period and continuous-time framework require the knowledge of infinite-dim spaces.
In particular， there are two major problems in tackling these infinite-dim programing. One is "Is the "sequential choice problem" reducible to a "value function problem？" "； and the other is "Does the functional problem have a solution？".
With the works by Richard Bellman and L. Pontryagin， the problem is solved. Utilizing the tool of Kakutani Fixed Point theory Nadler's Contraction Correspondence Theorem， the optimal control theory was used extensively in economics in addressing dynamic problems in economic growth and finance.
General Equilibrium
“...General equilibrium theory offers the best available answer to the fundamental questions of economics： What determines relative value？ Under what conditions do competitive markets lead to an efficient allocation of resources？" （Duffie， 1989）
With the power of dynamic programing， the economic agents can tackle their problem with given constraints. However， a more difficult puzzle arises： where do the constraints come from？ In particular， if the price is determined by agents’ decision， how can the agents make decisions without the price？
This egg-and-chicken problem requires the concept of equilibrium. However， the existence of it goes beyond the grasp of calculus for its topological frame and the infinite price-vector
In 1954， Arrow and Debreu proved the existence of equilibrium and its welfare property， which is regarded as the beginning of mathematical economics. Later， Debreu and Uzawa characterized its cardinality while Arrow and Hurwicz established its stability.
Game theory （the interdependent strategies）
"That's trivial， you know. That's just a fixed point theorem" von Neumann to Nash （1949）.
We now look into the celebrated Nash Equilibrium.
What Nash in 1951 proves is the existence of strategy in which no one would deviate from it given others does not. The elegant simplicity derives from the application of Kakutani's fixed-point theorem in the players’ best response correspondence set. That is the reason why von Neumann made that comment on Nash's work at his first glance.
In an early work （1994）， von Neumann and Morgenstern introduced the functional analytic methods to economic analysis. They use tools like convex sets and topological fixed-point theory rather than the contemporary dominating calculus. It is from this approach， that Nash establishes the foundation of non-cooperative games， which we mentioned above.
This result has huge impact on industrial organization， information economics， public economic and the monetary economic. It also gives both to mechanism design， which includes， but not restricted to the wage design， voting mechanism and firm structure.
Conclusion
Why does functional analysis matters？ On one hand， it works like a tool-kit， which facilitates our work and comfort ourselves by guaranteeing the existence of implicit solution. On the other hand， its conceptual structure calms us against the problems that beyond the finite demission world we're living in and provide a crystal understanding.
Reference
[1]Arrow， Kenneth And Debreu， Gerard. （1954）， Existence of an Equilibrium for a Competitive Economy. Econometrica， 22， 265-90.
[2]Nash， John F.， Jr. （1950）， The Bargaining Problem. Econometrica， 18. 155-162.
[3]Neumann， John von， and Oskar Morgenstern. （1944）， Theory of Games and Economic Behavior， Princeton.
[4]Pontryagin， （1962）. The Mathematical Theory of Optimal Processes. New York： Wiley.
[5]Robbins， Lionel C. （1932）， An Essay on the Nature and Significance of Economic Science. London： Macmillan.