8/16/2019 Border (2000) Notes on the Arrow–Debreu–McKenzie Model of an Economy. California Institute of Technology

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CALIFORNIA INSTITUTE OF TECHNOLOGY

Division of the Humanities and Social Sciences

Brief Notes on the

Arrow–Debreu–McKenzie Model

of an Economy

KC Border

January 2000v. 2012.08.28::12.50

1 CommoditiesThe first primitive concept is that of a   commodity. A commodity is anygood or service that may be produced, consumed, or traded. Commoditiesmay distinguished by date, location, and state of the world. For mathemat-ical simplicity we usually assume there is a finite number  ℓ  of commodities.The   commodity space   is thus R ℓ.

2 Tastes

The next concept is that of an idealized   consumer  or   household. A con-sumer is partially described by a consumption set  X , which is a subset of the commodity space. Elements   x   of  X   are ordered lists of quantities of commodities consumed. If  xk  

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KC Border Arrow–Debreu–McKenzie Model 3

4 Resources

The third element in the description of an economy is the aggregate endow-ment  ω  ∈ R ℓ. We typically assume  ω  ≧ 0, but that is mainly a definition of what it means to be a resource.

5 Allocations

An  economy  is thus summarized by a list

E  =

(X i,≽i)mi=1, (Y  j)

n j=1, ω

.

An  allocation  for the economy  E   is a list

(x1, . . . , xm, y1, . . . , yn)

satisfyingxi ∈ X i   i = 1, . . . ,m

y j ∈ Y  j   j  = 1, . . . , n

m

i=1

xi = ω  +n

 j=1

y j.

A natural question is whether allocations exist at all. Let  X  = ∑m

i=1X i.

The question is whether  X  ∩ (Y   + ω)  ̸=  ∅. There are a couple of ways toguarantee this. One is to assume 0 ∈  Y  j (possibility of inaction) for each pro-ducer and that each consumer satisfies  ω i ∈ X i. Then (ω1, . . . , ωm, 0, . . . , 0)is an allocation. If we don’t wish to assume   ωi ∈  X i, we might assumethe existence of  x̂i ∈  X i with  x̂i ≪   ωi, and assume that   Y   exhibits freedisposability. (There are other reasons we may make this assumption. Doyou see why it guarantees the existence of allocations?)

6 Efficiency

An allocation   (x̄1, . . . , x̄m, ȳ1, . . . , ȳn)   is   inefficient3 if there is some other

allocation  (x1

, . . . , xm

, y1

, . . . , yn

)  such that

xi ≽i  x̄i for all  i,

3Or  Pareto dominated

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[6] . 1962. New concepts and techniques for equilibrium analysis.

International Economic Review   3(3):257–273.www.jstor.org/stable/2525394

[7] B. Ellickson. 1993.  Competitive equilibrium: Theory and applications .Cambridge and New York: Cambridge University Press.

[8] T. C. Koopmans. 1957.  Three essays on the state of economic science .New York: McGraw-Hill.

[9] L. W. McKenzie. 1955. Competitive equilibrium with dependent con-sumer preferences. In H. A. Antosiewicz, ed.,  Proceedings of the Second Symposium in Linear Programming , pages 277–294, Washington, D.C.

National Bureau of Standards and Directorate of Management Analy-sis, DCS/Comptroller, USAF.

[10] . 1959. On the existence of general equilibrium for a competitivemarket.   Econometrica  27:54–71.   www.jstor.org/stable/1907777

[11] . 1961. On the existence of general equilibrium: Some correc-tions.   Econometrica   29(2):247–248.   www.jstor.org/stable/1909294

[12] . 1981. The classical theorem on existence of competitive equi-librium.   Econometrica  49:819–841.   www.jstor.org/stable/1912505

[13] T. Negishi. 1960. Welfare economics and existence of an equilibrium

for a competitive economy.  Metroeconomica   12(2–3):92–97.DOI: 10.1111/j.1467-999X.1960.tb00275.x

[14] H. Nikaidô. 1956. On the classical multilateral exchange problem.Metroeconomica   8(2):135–145.

DOI: 10.1111/j.1467-999X.1956.tb00097.x

[15] J. P. Quirk and R. Saposnik. 1968.  Introduction to general equilibriumtheory and welfare economics . New York: McGraw–Hill.

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http://dx.doi.org/10.1111/j.1467-999X.1956.tb00097.xhttp://dx.doi.org/10.1111/j.1467-999X.1960.tb00275.xhttp://www.jstor.org/stable/1912505http://www.jstor.org/stable/1909294http://www.jstor.org/stable/1907777http://www.jstor.org/stable/2525394