FUNDAMENTAL ECONOMICS Vol. I - Walrasian and Non-Walrasian Microeconomics - Anjan Mukherji WALRASIAN AND NON-WALRASIAN MICROECONOMICS

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1 FUNDAMENTAL ECONOMICS Vol. I - Walrasian and Non-Walrasian Microeconomics - Anjan Mukerji WALRASIAN AND NON-WALRASIAN MICROECONOMICS Anjan Mukerji Center for Economic Studies and Planning, Jawaarlal Neru University, New Deli, India Keywords: walrasian, non-walrasian, microeconomics, transactions, effective demand, excess demand. Contents 1. Introduction 1.1 Decision Makers, Economic Agents and Markets 1.2 Walrasian and Non-Walrasian Approaces 1.3 Overview 2. Walrasian Transactions: Excess Demand 2.1 Demand 2.2 Supply 2.3 Excess Demand Functions and teir Properties 2.4 Walrasian Equilibrium 3. Non-Walrasian Transactions: Effective Demand 3.1 Introduction: A Two Commodity Example 3.2 Non-Walrasian Equilibria 3.3 Optimality of Non-Walrasian Equilibria 4. Applications 4.1 Stability Reconsidered 4.2 Unemployment Equilibria Glossary Bibliograpy Biograpical Sketc Summary Tis article addresses te question of ow economic agents suc as consumers and firms formulate teir plans on te basis of prices. Te concern is wit economic agents operating under perfect competition, i.e. wit agents wo are unable to influence market prices. Te compatibility of suc plans is considered, as is te nature of mutually compatible plans. Tis is te Walrasian part of te discussion. Non-Walrasian metods are also considered, and te transactions tat agents make on te basis of prices and quantity constraints are discussed. Te question of weter quantity constraints (rationing scemes may be devised, for any configuration of fixed prices, so tat te constrained plans of agents are compatible is discussed following tis. Te approac adopted is used to analyze te transactions tat take place wen agents respond not only to prices, but also to quantity constraints. Tis metod is applied to two areas were Walrasian metods ave been relatively unsuccessful. Te first is te question of ow te market computes te Walrasian configuration. Te second is te analysis of unemployment.

2 FUNDAMENTAL ECONOMICS Vol. I - Walrasian and Non-Walrasian Microeconomics - Anjan Mukerji 1. Introduction 1.2 Decision Makers, Economic Agents and Markets Tis article examines te beavior of certain kinds of decision-makers and ow tey plan out teir activities under various scenarios. Te decision-makers to be considered are usually called economic agents, and are distinguised on te basis of teir activities. An activity tat uses goods and/or services (factors to produce oter goods or services is called a production activity. A decision-maker wo controls suc activities is called a firm. An activity engaged in by agents to meet teir own needs is usually called a consumption activity; agents wo engage in suc activities are referred to as consumers or individuals. Tere may be oter agents wo need to decide on te levels of taxation or te amount of money to be spent on national security; suc agents are usually identified wit te government of te country. An economy is made up of tese different types of economic agents. Te economic agents, as sould be clear from te above description, generally ave different objectives; for instance, firms are usually taken to be interested in te maximization of profits; consumers (or ouseolds or individuals are interested in maximizing teir own well being, or utility; te government as its own typical concerns. Because tese agents ave suc very different objectives, te task of coordinating teir activities is difficult. In tis article, we assume agents operate in markets; tat is, agents turn to te markets to make any transactions tey migt wis. Furter, we assume te markets are competitive; i.e. tere is no agent in any market wo can control te price to is or er own advantage. (See capter Strategic Beavior 1.2 Walrasian and Non-Walrasian Approaces Two related but distinct approaces to te analysis of te issues outlined above will be discussed. In te Walrasian Approac, te different agents take only prices as signals, and decide on teir plans on te basis of tese. As we sall see, if te signals are appropriate te plans tat te different agents make will be mutually compatible. Te properties of suc configurations will form a major part of te discussion below. Walrasian metods consider only tat configuration were plans of agents are compatible and te only transactions tat are made are tose tat matc. In te course of te discussion it will become clear tat te configuration in wic all plans are mutually compatible may be difficult to reac; tis raises te question of wat appens wen te plans made by agents are not compatible, or, to put it differently, wen market clearing is not acieved. One approac to tis issue as been to introduce various quantity constraints, at some given configuration of prices, wic te agents are asked to respond to. If tese constraints are appropriate, te constrained plans sould be mutually compatible. Non-Walrasian approaces consider te kinds of rationing or quantity constraints tat will make te constrained plans mutually compatible. Te second part of tis article will consider te nature of tese quantity constraints and te property of transactions tat would be implied by te constraints. In addition to te one mentioned above, a Non-Walrasian approac as to be considered

3 FUNDAMENTAL ECONOMICS Vol. I - Walrasian and Non-Walrasian Microeconomics - Anjan Mukerji for anoter reason. In te Walrasian approac, as pointed out above, all plans made by agents are found to be compatible. All persons wo wis to work, for example, will find jobs. Tere is tus no scope to analyze unemployment. To analyze any problem in economics tat involves an observed under-utilization of some available resource, it is essential to step outside te Walrasian paradigm. An example of suc an approac to unemployment will be discussed below. Anoter problem to be addressed is te question of ow competitive markets reac a configuration were te plans are mutually compatible. A view is adopted tat competitive agents realize tat te plans of agents are not compatible only wen tey try to transact and are unable to do so. Based on tis observation, a process is described wereby a Walrasian configuration may be ultimately acieved. 1.3 Overview As outlined above, in tis article we confine ourselves to te framework of microeconomics, and bot Walrasian and Non-Walrasian metods will be discussed. So far as te former is concerned, te notion of demand and supply, and ence, excess demand will be introduced. We begin from te origins of demand and supply, sowing ow plans are made by agents suc as consumers and firms. Te desired transactions by tese agents, te notion of excess demand, will be taken up, next. In eac case, te properties of tese constructs will be discussed and te usual assumptions made in suc contexts will also be provided. Te main point of empasis is ow tese diverse plans are mutually compatible. For te Non-Walrasian Approac, on te oter and, it is te notion of effective demand tat is te main construct. Tere is, owever, a misuse of terms ere since te term effective demand is not really a different kind of demand; it sould peraps be termed effective excess demand, since te effective demand is actually a kind of constrained transaction tat an agent may wis to make. Te basic point of departure from te Walrasian approac is tat if markets do not clear, i.e. te plans are not mutually compatible, is tere any way of devising constraints on transactions so tat once tese constraints are taken in to account, te constrained transactions matc? Naturally te constraints must be meaningful and te article will consider ow best to introduce suc considerations. Finally, as discussed above, two applications of te above approac will be considered. One of tem is in te context of price formation--a process, wereby agents temselves bid te prices up or down wen tey fail to make te desired transaction, will be considered. Te oter application is in te area of unemployment equilibria; te constructs of effective demand for labor and output are analyzed witin te context of an aggregative model to sow te properties of te resulting unemployment equilibria. Because of constraints on space, tis cannot be an exaustive treatment. We provide ere only a statement of te major results and te assumptions under wic tey old. Wat ave been left out are primarily te demonstrations and proofs. Readers wising to explore tese issues will find suitable references in te bibliograpy.

4 FUNDAMENTAL ECONOMICS Vol. I - Walrasian and Non-Walrasian Microeconomics - Anjan Mukerji 2. Walrasian Transactions: Excess Demand 2.1 Demand Individual Demand Functions and teir Properties Demand for goods and services originates from te plans made by individuals wen tey know teir own incomes and te prices of te items tey wis to buy and sell. Tis section considers te formation of suc plans and ow tey cange wen prices cange. Suppose tere are n goods and/or services; an individual decision maker is assumed to possess a set of all possible consumption possibilities, X R n, te n-dimensional Euclidean space. It is usual to assume tat Axiom 1. X R is non-empty, closed, convex, and bounded from below Te individual as some initial resources e 0, e X, te endowment; since te individual must be able to compare various possibilities in X we sall assume tat Axiom 2. X is completely ordered by a binary relation R Tus te binary relation R is assumed to be reflexive (i.e. for any x A, xrx, complete (i.e. for any x, y A, eiter xry or yrx, transitive (i.e. for any x, y, z A, xry & yrz xrz. R is usually interpreted to be te no worse tan relation. Two oter binary relations are often derived from it. Tey are defined as follows. Strict preference P : if xry and ~ yrx ten xpy. Indifference I: if xry & yrx ten xiy. "~ yrx " is to be read as: it is not te case tat yrx. Te following axiom, continuity, is also usually imposed on te binary relation R : x Axiom 3. R = { y X : yrx }, = { : } x X. x Under te above axioms, it can be sown tat: R y X xry are closed subsets of X for every Tere is a continuous real valued function : R Te function ( U X suc tat ( ( U x U y xry U is called a utility function. Te individual s coice over te set X is constrained by is purcasing power, wic is measured by M = p e, were p R++ n denote te market prices. Te set of affordable consumption possibilities is provided by B pm, = y X: p y M ; alternatively, te budget set may be te budget set: ( { } denoted by B ( p, e. Te individual s coice problem may now be caracterized by te following problem: Max U( x

5 FUNDAMENTAL ECONOMICS Vol. I - Walrasian and Non-Walrasian Microeconomics - Anjan Mukerji subject to x B( p, M Tis problem is referred to as te Maximum Utility Problem (MUP; te solution to n MUP provides us wit te demand at p, M or at p, e ; note tat p R++ B( p, M is a compact subset of X ; consequently, a continuous function will always attain its bounds in suc a set, and ence tere will always be a solution to MUP. In general, suc a solution need not be unique. Consider, owever, Axiom 4. x, y X, x y, ( ( xry λ x + 1 λ y P y λ, 0< λ < 1 If R satisfies te above, te resulting U ( is said to be strictly quasi-concave; it is easy to ceck tat wit tis restriction, te solution to MUP is unique; tis unique solution is represented by f ( pm, or by a sligt abuse of notation, by f ( p, e ; tis is te demand function. Te binary relation R is said to satisfy local non-satiation if: Axiom 5. For x X and for any neigborood ( N x of x, ( y N x X, y P x Tis ensures tat in solving MUP, te decision maker must spend all or tat p f ( p, e = p e; note also tat f ( λ p, e = f ( p, e for any λ > 0 : omogeneity of degree zero in te prices; tis follows since multiplying all te prices by some constant does not alter te Budget set B ( p, M (recall, M = pe. Consider next, te continuity of te function f ( p, e. A ceaper point exists at ( p, e if tere is x X suc tat < p, e, one may sow tat te p.x p.e. Given te existence of a ceaper point at ( n B R X X is lower emi-continuous at (, (, s s s s,,, budget map : ++ z B p e, a sequence { p e } suc tat { } ( sequence { s s s s z } suc tat z B( p, e s and z s p e ; i.e. for any p e p e and tere is a z. Tis property is crucial for te demonstration of te fact tat f ( pe, is a continuous function of (, tere is a ceaper point at ( p, e. p e wenever Te demand function may be seen to ave some furter properties. Tese are best demonstrated troug te consideration of te following minimization problem: n ++ For a given p R, x X Minimize py. subject to y R x Tis problem is te Minimum Expenditure Problem (MEP; since te constraint set is

6 FUNDAMENTAL ECONOMICS Vol. I - Walrasian and Non-Walrasian Microeconomics - Anjan Mukerji closed (by Axiom 3 and bounded below (by Axiom 1, te MEP as a solution; let te,, U = U x. Te minimum value attained be denoted by E( p x or by E( p U were ( function E( p, x called te expenditure function determines te minimum expenditure required at prices p to attain te same level of utility as at x. Te following property is crucial: Te function E( p, x is a non-decreasing, concave function of p for any moreover, E( λp, x = λ E( p, x for any λ > 0. x X ; One may now define B ( p, E( p, x and f ( p, E( p, x exactly as above, replacing M by E( p, x ; it is to be noted tat U f ( p, E( p, x = U x ; tus f ( p, E( p, x ( ( is te compensated demand function. Muc of modern teory of te consumer is based on te relationsip between te problems MUP and MEP, often referred to as te duality teory. It may be sown, under te assumptions employed, if ˆx solves MUP for a given p and M, ten te problem MEP wit te same p and Rx ˆ in te constraint will be solved by ˆx ; conversely, if given p and some x X, MEP is solved E x, p = p. y = M, ten MUP for te same p and M will be solved by some y and ( by te bundle y. Tis is te duality link. Most of te results of demand teory relate to properties of te compensated demand function: (i f ( p, E( p, x = f ( p, E( p, x (ii in prices. Wenever derivatives exist: ( E p,x p And i = f (iii Te matrix f i λ λ for any λ > 0 : omogeneity of degree zero ( p,e( p,x ( i ( p,e( p,x 2 E p,x f ; = p p i ( p,e( p,x p j p i j j is negative semidefinite. Note tat te desired or planned transaction of te individual is tus (,. z = f p pe e; and given te fact tat e is fixed, we ave, wenever derivatives exist: ( ( fi p,e( p,f ( p,p.e ( z fi p,p.e fi p,m i = = ( f j ( p,p.e ej p p p M j j j (1 Te last step, te Slutsky Equation, as been called te Fundamental Equation of Value Teory. It may be noted tat te results of demand teory concern te matrix of partial

7 FUNDAMENTAL ECONOMICS Vol. I - Walrasian and Non-Walrasian Microeconomics - Anjan Mukerji derivatives of te compensated demand function, te first set of terms in te above expression. Tis resolution indicates tat te effect of a price cange may be broken down into te effect of a price cange togeter wit a cange in income to compensate te individual for te price cange (te substitution effect, te first term on te rigt and side and te effect of a pure cange in income (te second term on te rigt and side, te income effect; te second set of terms could be of any sign. Tis is te source of indeterminacy in many areas in microeconomics Market Demand Functions Wit many individuals or ouseolds, indexed by = 1, 2,, N, eac wit an utility function U ( defined over te individual s consumption possibility set X and an endowment e, one may define te demand function for eac as in te previous f p, p. e, and te market demand function may now be defined as section, ( { } = (,. (,. X p p e f p p e. Te aggregate demand, or te market demand owever cannot, in general, be considered to be derived from some optimization 1 1 = 1,,. f p, p. e and ence te exercise. If, for example, ( properties of te function (, N X p coincides wit ( X p coincide wit te property of te demand function analyzed in te previous section, namely, tat it is obtained by solving a problem suc as MUP described earlier. If E = e, te above may be reduced to an enquiry weter te function (, {. } X p p e may be taken to be f (,. p p E wic is obtained from te maximization of some aggregate welfare subject to an aggregate income constraint. In tis connection, te following may be noted: If for eac ouseold, ( U is omogeneous of degree one, and if endowments are proportional, i.e. e = δ E,, δ 0, and δ = 1 ten te market demand X ( p is generated from te following MUP: = ( ( Max U( x U x δ e subject to p. x = p.e For future reference, one sould note tat it is only under some special condition suc as mentioned above, tat te market demand may satisfy Weak Axiom of Revealed Preference (WARP; any demand function derived from a problem of te type MUP,

8 FUNDAMENTAL ECONOMICS Vol. I - Walrasian and Non-Walrasian Microeconomics - Anjan Mukerji satisfies te following: if = (,. x f p p e and = (, = (,. (, y B p p e ; ten y f p p e x B p p e. It is tis property wic is called WARP; owever, te market demand X ( p need not satisfy tis rationality property in general, i.e. X = X ( p, p. y p. E and y = X ( p need not imply tat p. X ( p > p. E Even toug te market demand function may, in general, fail to satisfy WARP, tere are also oter results were aggregation is seen to provide elpful regularizing effects. Tis is seen in terms of te average (per-consumer demand wen tere is a continuum of ouseolds Bibliograpy TO ACCESS ALL THE 39 PAGES OF THIS CHAPTER, Visit: ttp:// Arrow K. J. (1950. An Extension of te Basic Teorems of Classical Welfare Economics. Proceedings of te Second Berkeley Symposium in Matematical Statistics and Probability (ed. J. Neyman, pp Berkeley: University of California Press. [Tis contribution contains te first general treatment of te Fundamental Teorems of Welfare Economics. Section ] Bala V. and Majumdar M. (1992. Caotic Tatonnement. Economic Teory 2, [Te paper contains an analysis of complex dynamics wic may result from a discrete tatonnement process. Section ] Benassy J. P. (1982. Te Economics of Market Dis-Equilibrium. 241 pp. New York: Academic Press. [A detailed treatment of te various topics discussed in Section 3.] Debreu G. (1959. Teory of Value. 114 pp. New York: Jon Wiley. [A concise and terse state of te art treatment of Walrasian Equilibrium, Section 2.] Dreze J. H. (1975. Existence of an Equilibrium under Price Rigidity and Quantity Constraints. International Economic Review 16, [Te notion of Dreze Equilibrium was first introduced and analyzed in tis paper; it sould be pointed out tat te treatment in tis paper allowed for some price flexibility, witin some range in contrast to te absolute rigidity discussed in Section ] Dreze J. H. and Muller H. (1980. Optimality property of Rationing Scemes. Journal of Economic Teory 23, [Te paper provides an analysis of optimality properties of fixed price equilibria wit quantity constraints (Section 3.3.] Han F. H. and Negisi T. (1962. A Teorem on Non-Tatonnement Stability. Econometrica 30, [Tis paper introduces te non-tatonnement process wic is now referred to as te Han-Negisi Process. Section ] Hicks J. R. (1946. Value and Capital. 340 pp. Oxford: Clarendon Press. [Tis is te classic treatment of Walrasian Analysis; many refinements of te metod, for example, te Walrasian Temporary Equilibrium, originates from tis contribution. Section 2, ] Malinvaud E. (1977. Te Teory of Unemployment Reconsidered. 128 pp. Oxford: Basil Blackwell.

9 FUNDAMENTAL ECONOMICS Vol. I - Walrasian and Non-Walrasian Microeconomics - Anjan Mukerji [Tis book contains te first systematic treatment of unemployment equilibria of te type considered in Section ] Mas-Collel A., Winston M. D., and Green J. R. (1995. Microeconomic Teory. 998 pp. New York: Oxford University Press. [A state of te art treatment of an exaustive list of topics from modern microeconomics and contains detailed analysis of te material of Section 2 and related topics.] McKenzie L. W. (1981. Te Classical Teorem on te Existence of Competitive Equilibrium. Econometrica 49, [A reconsideration of te original contribution by McKenzie tat, togeter wit te contribution by Debreu and Arrow, formed te starting point of te modern treatment of Walrasian economics.] Mukerji A. (1990. Walrasian and Non-Walrasian Equilibria, An Introduction to General Equilibrium Analysis. 245 pp. Oxford: Clarendon Press. [Te approac of tis book as been followed in te discussion above, in particular te material used in Sections 3.2 and 3.4.] Safer W. and Sonnenscein H. (1982. Market Demand Functions and Excess Demand Functions. Handbook of Matematical Economics. (eds. K. J. Arrow and M. Intrilligator, Vol. 3, pp , Amsterdam: Nort Holland. [Tis paper provides te basic material for te properties of te aggregate demand function and te market excess demand function Sections and 2.3.] Uzawa H. (1962. Walras Existence Teorem and Brouwer s Fixed Point Teorem. Economic Studies Quarterly 8, [Tis paper contains an elegant demonstration of te equivalence between te Existence Teorem for a Walrasian Equilibrium and Brouwer s Fixed Point Teorem. Section ] Younes Y. (1975. On te role of Money in te Process of Excange and te existence of a Non- Walrasian Equilibrium. Review of Economic Studies 42, [Tis paper introduces te notion of Younes Equilibria as well as te equilibrium in one round of trading considered in Section 3.4 and is basic to te study of Non-Walrasian Equilibria.] Biograpical Sketc Anjan Mukerji was born July 31, 1945 in Calcutta, India. Education: B.A (1964 Presidency College Calcutta, M.A. (1966, Calcutta University, P.D. (1973, University of Rocester, Rocester, N.Y. Current Position: Professor of Economics, Center for Economic Studies and Planning, Jawaarlal Neru University, New Deli (since Also taugt at te London Scool of Economics and Political Science ( , Cornell University ( , University of Tsukuba, Japan ( , Autor of: Walrasian and Non-Walrasian Equilibria, An Introduction to General Equilibrium Analysis, Clarendon Press, Oxford (1990. Current Researc Interests: Non-Linear Dynamics in Economic Models.