Feel the Power of Brouwer

8/14/2019 Feel the Power of Brouwer

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FEEL THE POWER OF BROUWER

RICHARD CRAIB

1. The Mathematics

Fixed Point Denition

Statement of Brouwers Fixed Point Theorem

Illustrating the Theorem

Visualizing the Theorem

Sketch of the Proof

Generalizing the Theorem

Believing the Generalization

Theorem and Proof History

2. Applications in Economics

The Fuss about Equilibrium

Theory of General Equilibrium and the Existence of Equilibria

Importance of General Equilibrium

3. References


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The Mathematics

Since Brouwers Fixed Point Theorem has been generalized and reformulated a number of times,

the applications of the theorem in the literature can be confusing. For example, a mathematician

might say that by Brouwers Fixed Point Theorem some result holds but it is unclear exactly which

formulation is being used. So I will start with a particular formulation, speak around it, and

formulate a generalization which is used in this paper to prove the Existence of General Equilibrium

in economics.

Fixed Point Denition

Source: Wolfram MathWorld

Statement of Brouwers Fixed Point Theorem

Source: Wolfram MathWorld

Illustrating the Theorem

The theorem is said to have originated by Brouwers observation that there is always a point on the

surface that appears motionless when one stirs a cup of coffee. However, as illustrations go this one

is poor because the center of turbulence moves when stirring and hence it is not necessarily the

xed point. Also it gives the impression that there is some point which cannot move but actually all

points are free to move as long as there is one xed at the end of the stir - after applying the

function.


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A better illustration, attributed to Brouwer, is to imagine two sheets of paper directly on top of each

other. Take the top sheet and crumple it up, then attening it out again above the other sheet such

that the edges dont meet - i.e. there is still room for creases in the on paper and non-zero distances

between the two. Brouwers Fixed Point Theorem (for n = 2) then states that there is at least one

point in the crumpled sheet of paper lying directly above its corresponding point on the at sheet.

Visualizing the Theorem

Consider the case where n = 1; the unit ball becomes just the closed interval from -1 to 1. We know

that this interval is homeomorphic to the closed interval from 0 to 1. The gure below shows some

continuous function f (the curved line) from [0,1] to [0,1] and compares it to the identity function

(the straight diagonal line) which sends y to y. The intuition now is obvious; if the curved line is

continuous we must see it intersect the diagonal line, and that intersection point, x, is a xed point

because f(x) = x .

Source: Wikipedia

Sketch of the Proof

As you will discover under Theorem and Proof History, Brouwer himself never did enjoy a good

proof by contradiction. There are constructive proofs but the mathematics is quite inaccessible. I

therefore just outline a proof of n = 2 given by Riley (2009). In essence you assume that there is a


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continuous function f with no xed point, and construct the line given in the diagram below. This

construction denes a continuous function F from the closed unit 2-ball the the circle of S1. This

function, F is a retraction. Considering your Fundamental Groups you can now construct an

identity map of the integers factoring through the trivial group which is clearly a contradiction.

Source: Wikipedia

Generalizing The Theorem

In the statement of Brouwers xed point theorem it is not necessary for the function to be bijective

or even surjective. There are only two properties of the function, f, in the theorem: it is continuous,

it has a xed point. Each of these properties are clearly invariant under homeomorphism. To the

topologist, therefore, the theorem is much stronger than it appears; Brouwers Fixed Point Theorem

applies to all sets homeomorphic to a closed unit ball not just the closed unit ball. For example,

Brouwers Fixed Point Theorem can be applied to non-unit closed balls, closed ovals, closed

squares, etc. We formalize this general case.

The General Brouwers Fixed Point Theorem

Every continuous function f from a convex compact subset K of a Euclidean space to K has a fixedpoint.

Believing The Generalization

By imposing convexity and compactness on the subset we can derive that K will be homeomorphic

to a the unit n-ball, and therefore Brouwers Fixed Point Theorem applies. I show this informally


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by demonstrating that the topological invariants of the unit-ball - closed, bounded, connected and

without holes - are captured by convexity and compactness.

Since K is convex, we clearly have K as path-connected (see gure below), and since path-

connected implies connected we have K connected. Also K is without holes because if there were a

hole (for example, between x and y in the gure below) we would violate convexity. Since K is

compact we can use Heine-Borel Theorem to get closure and boundedness. At this point, I trust

youre well-convinced of this generalization.

Source: Wikipedia

Theorem and Proof History

Brouwers Fixed Point Theorem is named after Luitzen Egbertus Jan Brouwer, a Dutch

mathematician who roamed the earth from 1881 to 1966. In 1886, when Brouwer was just 5 years

old, Henri Poincar proved a result that is equivalent to Brouwers xed point theorem. The case

for n = 3 was proved by Piers Bohl in 1904, and then by Brouwer in 1909. Jacques Hadamard

proved the general case in 1910, and then Brouwer came up with his own proof in 1912. A mean-

spirited person might say that Brouwer neither invented nor was the rst to prove his theorem, and

even if he had he would have had to done it before he was ve for it to be truly original.


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Brouwer raised objections to the proofs of his theorem including his own because they were indirect

and non-constructivist. Brouwer is considered the founder and strongest proponent of intuitionism

(it is exactly what it sounds like). As such, he was likely deeply unhappy that he could not nd a

constructivist proof to one of his most famous theorems. In 1966, Brouwer was struck [and

killed] by a vehicle while crossing the street in front of his house (van Stigt, 1990). Sadly, this was

just one year before techniques were discovered for a constructivist proof.


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Applications in Economics

In countless mathematics descriptions of markets in economics we nd applications of Brouwers

Fixed Point Theorem and its derivatives and generalizations - most notably Kakutanis (a 1982

survey by G. Debreu cites over 350 instances where Kakutani's xed point theorem is the main tool

for proving the existence of an economic equilibrium). Brouwers xed point Theorem is also a

key step in Nobel Prize winner John Nashs original proof of Nash equilibria (Mathematics in the

News, 2004), and in Nobel Prize winners Arrow and Debreus work on the theory of General

Equilibrium.

The Fuss About Equilibrium

In economics, markets are in equilibrium when supply equals demand. It turns out, it is a good

thing when markets are in equilibrium. Consider a case where for example, demand exceeds supply

in the market for iPhones. We observe that consumers are disgruntled by a situation like this; we

want a particular product at a particular price but there isnt enough to go around. The price

mechanism helps us avoid this problem. Holding iPhone supply constant the price mechanism

helps us solve this problem. If we assume. Apple likes money and consumers like money. Then if

Apple increases the price a bit, consumers as a whole will demand less iPhones (at that price

because they like money), and Apple will make more money on their nite supply of iPhones

(were holding supply constant). Apple will continue to adjust the price upwards (Apple likes

money), until supply equals demand and the market clears i.e. they sell all their iPhones to willing

buyers.

Theory of General Equilibrium and the Existence of Equilibria

The theory of General Equilibrium seeks to prove the equilibrium exists in the aggregate market

and that all prices are at equilibrium, in contrast to the iPhone example above which was just a


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discussion about one particular market. The aforementioned Nobel Memorial Prize Winners,

Kenneth Arrow and Gerard Debreu are the two key gures responsible for the modern theory.

The First Fundamental Theorem of Welfare Economics states that market equilibria are Pareto

Efcient - the situation where no trade (purchase or sale) is mutually benecial to both parties (a

good thing). The Second Fundamental Theorem of Welfare Economics is that every Pareto

Efcient allocation is supported by some set of prices. So the Second Fundamental Theorem

implies that just leaving the market to do its work (i.e. adjust prices as in the iPhone example) leads

to Pareto efciency.

However, neither of these theorems shows the existence of an equilibrium - they just assume it. But

there is an existence proof, and it involves Brouwers Fixed Point Theorem. I use excerpts from the

accessible and well organized proof given by Du, interspersed with my own comments.

Source: Du (2005)

Source: Du (2005)


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You should think of the goods above as extremely general - an example of a good could be pears

on Sunday in Central Par. Also notice that, in the net worth calculation above, we are assuming that

the consumer can sell all their endowed goods at the market price, p , a vector containing

information on the prices of all goods in the economy.

We allow our I consumers to with each other by selling their goods at the market price and buying

other goods. Clearly, there are some consumers who would want to do this because they are

unhappy with their current endowment. We dene a function - the demand function - for each

consumer to capture what each consumer wants at each income level and each price vector. The

demand function for each consumer would be . And denotes

the new bundle of goods demanded by consumer i.

Source: Du (2005)

Since we are looking at existence of equilibrium, Du gives the following denition to characterize

general equilibrium in our new notation.

Source: Du (2005)


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As weve discussed, the intuition with market equilibrium is that the price vector should converge

to a general equilibrium. We therefore are looking for some p , to satisfy the above equation. Du,

makes the problem even clearer below by dening aggregate excess demand.

Source: Du (2005)

Think about where we are at this point. A key part of nding p is going to involved understanding

each demand function f . It is no surprise, therefore, that at this point economists must come up with

some assumptions about f . Du gives these assumptions, and I talk about what they mean.

Source: Du (2005)

We see continuity pop up right away in (1), the rst hint that we might ever be able to use

Brouwers. But is it reasonable to have continuous demand function? According to Du its is

relatively harmless (Du, 2005), and I tend to agree: small changes in price and income should not

produce large changes in demand. Part (2) makes sense too; it says that multiplying both prices and

income by the same constant factor will not effect demand i.e. that consumers care about real prices


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not nominal prices. Part (3) says that the value of a new bundle bought by a consumer is the same

as his initial endowment value (net worth), and that the consumer uses all his income on

consumption. Part (4) says that if a good is free then everyone wants it - there is excess demand.

On the whole the restrictions are not too implausible or too restrictive (Du, 2005).

Du denes the following two lemmas for use in the upcoming proof:

Source: Du (2005)

Source: Du (2005)

Hmmm... What do we have here? First of all, why are we making the sum of all prices in the

economy equal to one? Well, remember part (2) of our assumption says that real prices and real

income is what matters so whatever prices this restricts us to we can just restrict our income

accordingly. What kind of set is this? It looks like a convex compact subset of Euclidean subspace,

does it not? Indeed, dear reader, it is. Maybe we can dene a map from it to itself? Surprise! This

is exactly how Du, starts his proof.


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Source: Du (2005)

Can you feel the power of Brouwer?

Source: Du (2005)

Boom! That just happened.

Source: Du (2005)


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Importance of General Equilibrium

The result is clearly extremely important for economics. It is at the center of economic ideas which

guide policy around the world. In a sense it is a proof the capitalism works. We have consumers

that are greedy, selsh and materialistic - as dened by their reasonable demand functions. Despite

all this, the result shows that there still exists a set of prices that leaves no desire unfullled and no

resource wasted (Du, 2005). I couldnt have put it better myself.


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Feel the Power of Brouwer

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