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CB046/Starr 2010CHAP-5 April 25, 2010 16:28

5

Existence of general equilibrium in an economy

with an excess demand function

General equilibrium theory focuses on finding market equilibrium prices for

all goods at once. Since there are distinctive interactions across markets (for

example, between the prices of oil, gasoline, and the demand for SUVs) itis important that the equilibrium concept include the simultaneous joint de-

termination of equilibrium prices. The concept can then represent a solution

concept for the economy as a whole and not merely for a single market that

is artificially isolated. General equilibrium for the economy consists of an

array of prices for all goods, where simultaneously supply equals demand for

each good. The prices of SUVs and of gasoline both adjust so that demand

and supply of SUVs and of oil are each equated.

Let there be a finite number N of goods in the economy. Then a typical

array of prices could be represented by an N-dimensional vector such as

p = (p1, p2, p3, . . . , pN1, pN) = (3, 1, 5, . . . , 0.5, 10).

The first coordinate represents the price of the first good, the second the

price of the second good, and so forth until the Nth coordinate represents

the price of the Nth good. This expression says that the price of good 1 is

three times the price of good 2, that of good 3 is five times the price of good

2, ten times that of good N 1, and half that of good N.

We simplify the problem by considering an economy without taking ac-

count of money or financial institutions. Only relative prices (price ratios)

matter here, not monetary prices. This is an assumption common in microe-

conomic modeling in which the financial structure is ignored. There would

be no difference in this model between a situation where the wage rate is $1

per hour and a car costs $1000 and another where the wage rate is $15 and

the same car costs $15,000.

Since only the relative prices matter, and not their numerical values, we

can choose to represent the array of prices in whatever numerical values

1

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2 Existence of general equilibrium in an economy with an excess demand function

are most convenient. We will do this by confining the price vectors to a

particularly convenient set known as the unit simplex. The unit simplex

comprises a set ofN-dimensional vectors fulfilling a simple restriction: Each

coordinate of the vectors is nonnegative, and together the N coordinates sumup to 1. We think of a point in the simplex as representing an array of prices

for the economy. There is no loss of generality in this formulation. Any

possible combination of (nonnegative) relative prices can be represented in

this way. To convince yourself of this, simply take any vector of nonnegative

prices you wish. Take the sum of the coordinates and divide each term in the

vector by this quantity. The result is a vector in the unit simplex reflecting

the same relative prices as the original price vector. Hence, without loss of

generality we can confine attention to a price space characterized as the unit

simplex.

Formally, our price space, the unit simplex in RN, is

P =

p | p RN, pi 0, i = 1, . . . , N ,

Ni=1

pi = 1

. (5.1)

The unit simplex is a (generalized) triangle in N-space. For N = 2, it is

a line segment running from (1,0) to (0,1); for N = 3, it is the triangle

with angles (vertices) at (1,0,0), (0,1,0), and (0,0,1); for N = 4, it is a

tetrahedron (a three-sided pyramid with triangular sides and base) with

vertices at (1,0,0,0), (0,1,0,0), (0,0,1,0), and (0,0,0,1); and so forth in higher

dimensions.

A households demand for consumption or a firms supply plans are rep-resented as an N-dimensional vector. Each of the commodities is repre-

sented by a coordinate. We will suppose there is a finite set of households

whose names are in the set H. For each household i H, we define a de-

mand function, Di(p), as a function of the prevailing prices p P, that is,

Di : P RN. There is a finite set of firms whose names are in the set

F, each with a supply function Sj(p), which also takes its values in real

N-dimensional Euclidean space: Sj : P RN. The economy has an initial

endowment of resources r RN+ that is also supplied to the economy.

We combine the individual demand and supply functions to get a market

excess demand function representing unfulfilled demands (as positive coordi-

nates) and unneeded supplies (as negative coordinates). The market excessdemand function is defined as

Z(p) =iH

Di(p) jF

Sj(p) r, (5.2)

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Existence of general equilibrium in an economy with an excess demand function 3

Z : PRN (5.3)

Each coordinate of the N-dimensional vector p represents the price of the

good corresponding to the coordinate. The price vector p is (p1, p2, p3, . . . , pN),where pk is the price of good k. Z(p) is an N-dimensional vector, each coordi-

nate representing the excess demand (or supply if the coordinate has a nega-

tive value) of the good represented. Z(p) (Z1(p), Z2(p), Z3(p), . . . , Z N(p)),

where Zk(p) is the excess demand for good k. When Zk(p), the excess de-

mand for good k, is negative, we will say that good k is in excess supply.

We will assume the following properties on Z(p):

Walras Law: For all p P,

p Z(p) =Nn=1

pn Zn(p) =iH

p Di(p)jF

p Sj(p) p r = 0.

The economic basis for Walras Law involves the assumption of scarcity andthe structure of household budget constraints.

iHp D

i(p) is the value

of aggregate household expenditure. The term

jF p Sj(p) + p r is the

value of aggregate household income (value of firm profits plus the value of

endowment). The Walras Law says that expenditure equals income.

Continuity:

Z : P RN, Z(p) is a continuous function for all p P.

That is, small changes in p result in small changes in Z(p).

Continuity of Z(p) reflects continuous behavior of household and firm de-

mand and supply as prices change. It includes the economic assumptions

of diminishing marginal rate of substitution (M RS) for households and di-

minishing marginal product of inputs for firms.

We assume in this chapter that Z(p) is well defined and fulfills Walras

Law and Continuity. As mathematical theorists, part of our job is to derive

these properties from more elementary properties (so that we can be sure

of their generality) and to develop models and the mathematical structure

needed to deal with the many situations in which Z(p) is not well defined1.

In Chapters 11 through 18 and 23 to 25, we will use a formal axiomatic

method: describing the economy as a mathematical model, stating economic

assumptions in formal mathematical form, and finally deriving results like

Walras Law and Continuity and the existence of market equilibrium as thelogical result of these more elementary assumptions.

1 For example, when the price of a desirable good is zero, there may be no well-defined value forthe demand function at those prices (since the quantity demanded will be arbitrarily large).Nevertheless, it is important that we be able to deal with free goods (zero prices).

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The economy is said to be in equilibrium if prices in all markets adjust so

that for each good, supply equals demand. When supply equals demand, the

excess demand is zero. The exception to this is that some goods may be free

and in excess supply in equilibrium.2 Hence, we characterize equilibrium bythe property that for each good i, the excess demand for that good is zero

(or in the case of free goods, the excess demand may be negative an excess

supply and the price is zero).

Definition po P is said to be an equilibrium price vector if Z(po) 0 (0

is the zero vector; the inequality applies coordinatewise) with pok = 0 for k

such that Zk(po) < 0. That is, po is an equilibrium price vector if supply

equals demand in all markets (with possible excess supply of free goods).

We will now state and prove the major result of this introduction, that under

the assumptions introduced above, Walras Law and Continuity, there is anequilibrium in the economy. To do this we will need one additional piece of

mathematical structure, the Brouwer Fixed-Point Theorem:

Theorem 5.1 (Brouwer Fixed-Point Theorem) Let f() be a continuous func-

tion, f : P P. Then there is x P so that f(x) = x.

The Brouwer Fixed-Point Theorem is a powerful mathematical result. We

will use it again in later chapters. It takes advantage of the distinctive

structure of the simplex. It says that if we have a continuous function that

maps the simplex into itself, then there exists some point on the simplex that

is left unchanged in the process. The unchanged point is the fixed point. Wecan now use this powerful mathematical result to prove a powerful economic

result the existence of general economic equilibrium.

Theorem 5.2 3 Let Walras Law and Continuity be fulfilled. Then there is

p P so that p is an equilibrium.

Proof The proof of the theorem is the mathematical analysis of an economic

story. We suppose prices to be set by an auctioneer. He calls out one price

vector p, and the market responds with an excess demand vector Z(p). Some

goods will be in excess supply at p, whereas others will be in excess demand.

2 Of course, a price of zero is hard to distinguish from no price at all. Goods that are free maynot even be thought of as property. Examples of free goods include rainwater, air, or access tothe oceans for sailing.

3 Acknowledgement and thanks to John Roemer for help in simplifying the proof.

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Existence of general equilibrium in an economy with an excess demand function 5

The auctioneer then does the obvious. He raises the price of the goods in

excess demand and reduces the price of the goods in excess supply. But

not too much of either change can be made; prices must be kept on the

simplex. How should he be sure to keep prices on the simplex? First, theprices have to stay nonnegative. When he reduces a price, he should be sure

not to reduce it below zero. When he raises prices, he should be sure that

the new resulting price vector stays on the simplex. How? By adjusting

the new prices so that they sum up to one. Moreover, we would like to use

the Brouwer Fixed-Point Theorem on the price adjustment process; so the

auctioneer should make price adjustment a continuous function from the

simplex into itself. This leads us to the following price adjustment function

T, which represents how the auctioneer manages prices.

Let T : P P, where T(p) = (T1(p), T2(p), . . . , T k(p), . . . , T N(p)). Tk(p)

is the adjusted price of good k, adjusted by the auctioneer trying to bring

supply and demand into balance. Let k > 0; k has the dimension, 1/k .The adjustment process of the kth price can be represented as Tk(p), defined

as follows:

Tk(p) max[0, pk + kZk(p)]

Nn=1

max[0, pn + nZn(p)]

. (5.4)

The function T is a price adjustment function. It raises the relative price

of goods in excess demand and reduces the price of goods in excess supply

while keeping the price vector on the simplex. The expression pk + kZk(p)

represents the idea that prices of goods in excess demand should be raised

and those in excess supply should be reduced. The operator max[0, ] rep-resents the idea that adjusted prices should be nonnegative. The fractional

form ofT reminds us that after each price is adjusted individually, they are

then readjusted proportionally to stay on the simplex. In order for T to be

well defined, we must show that the denominator is nonzero, that is,

Nn=1

max[0, pn + nZn(p)] = 0. (5.5)

We omit the formal demonstration of (5.5), noting only that it follows from

Walras Law. For the sum in the denominator to be zero or negative, all

goods would have to be in excess supply simultaneously, which is contrary

to our notions of scarcity and it turns out to Walras Law as well. Re-

call that Z() is a continuous function. The operations of max[], sum, and

division by a nonzero continuous function maintain continuity. Hence, T(p)

is a continuous function from the simplex into itself.

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By the Brouwer Fixed-Point Theorem there is pP so that T(p)=p.

Because T() is the auctioneers price adjustment function, this means that

p is a price at which the auctioneer stops adjusting. His price adjustment

rule says that once he has found p the adjustment process stops.Now we have to show that the auctioneers decision to stop adjusting the

price is really the right thing to do. That is, wed like to show that p is not

just the stopping point of the price adjustment process, but that it actually

does represent general equilibrium prices for the economy. We therefore

must show that at p, all markets clear with the possible exception of a few

with free goods in oversupply.

Since T(p) = p, for each good k, Tk(p) = pk. That is, for all k =

1, . . . , N ,

pk =max[0, pk +

kZk(p)]

Nn=1

max[0, pn + nZn(p

)]

. (5.6)

Looking at the numerator in this expression, we can see that the equation

will be fulfilled either by

pk = 0 (Case 1) (5.7)

or by

pk =pk +

kZk(p)

N

n=1

max[0, pn + nZn(p

)]

> 0 (Case 2). (5.8)

CASE 1 pk = 0 = max[0, p

k + kZk(p

)]. Hence, 0 pk + kZk(p

) =

kZk(p) and Zk(p

) 0. This is the case of free goods with market clearing

or with excess supply in equilibrium.

CASE 2 To avoid repeated messy notation, let

=1

Nn=1

max[0, pn + nZn(p

)]

(5.9)

so that Tk(p) = (pk +

kZk(p)). Since p is the fixed point of T we have

p

k = (p

k + k

Zk(p

)) > 0. This expression is true for all k with p

k > 0,and is the same for all k. Lets perform some algebra on this expression.

We first combine terms in pk:

(1 )pk = kZk(p

), (5.10)

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5.1 Bibliographic note 7

then multiply through by Zk(p) to get

(1 )pkZk(p) = k(Zk(p

))2, (5.11)

and now sum over all k in Case 2, obtaining

(1 )

kCase2

pkZk(p) =

kCase2

k(Zk(p))2. (5.12)

Walras Law says

0 =Nk=1

pkZk(p) =

kCase1

pkZk(p) +

kCase2

pkZk(p). (5.13)

But for k Case 1, pkZk(p) = 0, and so

0 = kCase1

pkZk(p). (5.14)

Therefore, kCase2

pkZk(p) = 0. (5.15)

Hence, from (5.11) we have

0 = (1 )

kCase2

pkZk(p) =

kCase2

k(Zk(p))2. (5.16)

Using Walras Law, we established that the left-hand side equals 0, but the

right-hand side can be zero only if Zk(p) = 0 for all k such that pk > 0 (k

in Case 2). Thus, p

is an equilibrium. This concludes the proof.QED

The demonstration here is striking; it displays the essential economic and

mathematical elements of the proof of the existence of general equilibrium.

These are the use of a fixed-point theorem, of Walras Law, and of the

continuity of excess demand. If the economy fulfills continuity and Walras

Law, then we expect it to have a general equilibrium. The mathematics that

assures us of this result will be a fixed-point theorem. Most of the rest of

this book is devoted to developing, from more fundamental economic and

mathematical concepts, the tools and properties demonstrated here.

5.1 Bibliographic note

The treatment in this chapter parallels Arrow-Hahn (1971) chapter 2.

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Exercises

5.1 Consider the following example of supply and demand relations be-

tween two markets. There are two goods, denoted 1 and 2, with

prices p1 and p2, supply functions S1(p1, p2) and S2(p1, p2), and de-

mand functions D1(p1, p2) and D2(p1, p2). These are specified by the

expressions

S1(p1, p2) = 3p1; D1(p1, p2) = 8 4p2 p1;p2 2

and

S2(p1, p2) = 5p2; D2(p1, p2) = 24 6p1 p2;p1 4.

The market for good 1 is said to be in equilibrium at prices (po1, po2)

where S1(po1, p

o2) = D1(p

o1, p

o2). The market for good 2 is said to

be in equilibrium at prices (p

1, p

2) where S2(p

1, p

2) = D2(p

1, p

2).Demonstrate that each market has an equilibrium when the others

price is fixed. Show that, nevertheless, no pair of prices exists for

the two markets at which they are both in equilibrium. Does this

supply-demand system provide a counterexample to Theorem 5.2,

the existence of general equilibrium prices?

5.2 Recall the intermediate value theorem:

Intermediate Value Theorem: Let [a, b] be a closed interval in R and

let h be a continuous real valued function on [a, b] so that h(a) < h(b).

Then for any real k so that h(a) < k < h(b) there is x [a, b] so that

h(x) = k.

Consider a two-commodity economy with an excess demand function

Z(p). The price space is the unit simplex in R2. Let Z(p) be con-

tinuous, bounded, fulfill Walras Law as an equality (p Z(p) = 0).

The notation (0,1) indicates the price vector with the price of good

1 equal to 0; (1,0) indicates the price vector with the price of good

1 equal to 1. Assume Z1(0, 1) > 0, Z1(1, 0) < 0, Z2(0, 1) < 0,

Z2(1, 0) > 0. Using the Intermediate Value Theorem, and without

using the Brouwer fixed point theorem, show that the economy has

a general equilibrium. That is, show that there is a price vector p

so that Z1(p) = (Z1(p

), Z2(p)) = (0, 0).

5.3 Walras Law can be stated as p Z(p) = 0, where Z(p) is the N-

dimensional excess demand function.

The Walras Law is sometimes interpreted as saying that if all mar-

kets but one clear, then the remaining market must clear as well.

Demonstrate this result, assuming pn > 0 for all n = 1, 2, . . . , N .

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Exercises 9

5.4 The price space in a general equilibrium model is typically described

as the unit simplex in RN. What is the economic significance of

this choice? Would it apply equally well to a model with money?

5.5 Define P as the unit simplex in RN. The Brouwer Fixed PointTheorem can be stated as

Let f : P P, f continuous. Then there is p in P so that f(p) =

p.

This theorem is used to prove the existence of general competitive

equilibrium. To allow for a more general price space, we might wish

to allow the price space to be RN+ (the closed nonnegative quadrant

of RN) instead of P. Would the Brouwer Fixed Point Theorem

apply equally well to RN+ ? Would a continuous mapping from RN+

into RN+ generally have a fixed point?

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