OPIM 952 – Market Equilibrium

Ralph Ahn

Today’s Lecture

A general introduction to market equilibria Walras-Cassel Model The Wald “corrections” The Arrow-Debreu Model “On the complexity of Equilibria”

The problem:

Individuals are endowed with “factors” (labor, capital, raw goods)

…and they demand produced goods. Firms demand factors and produce goods

with a fixed production technology. Does general equilibrium exist? What does that realistically tell us about

competitive markets?

Definition of equilibrium - in english The Walras-Cassel Model:

Where market demand is equal to market supply in all markets (factor or goods)

Maximizes utility for each consumer

Firms maximize proftis

Notation – Walras Cassel model Economy: H households, F firms, n produced

commodities, m primary factors

Produced commodities: Xh is a vector of commodities demanded by

household h Xf is a vector of commodities supplied by firm f p is the vector of commodity prices

Notation continued – Walras-Cassel Factors:

vh is a vector of factors supplied by household h vf is a vector of factors demanded by firm f w is the vector of factor prices

Technology (fixed proportions, same tech for all firms: bji = vj

f / xif is a unit output prod. coefficient

B is n x m matrix of unit-input coefficients

Elaboration of technology

bji = vjf / xi

f is a unit output prod. Coefficient Represents the amount of factor j necessary to

produce a unit of output I B’ is m x n matrix of unit-output coefficients

so that… v = B’x , where vector x is the supply of

produced goods and v is the demand for factors

Setting Equilibrium assumptions Perfect competition:

Entrepreneurs makes no positive profits or no losses, so…

Total revenue p’xf equals total cost w’vf for every firm f, or:

p’xf = w’vf

or as bji = vjf / xi

f, then the perfect comp. assumption implies

p = Bw

Objective of households (consumers) Maximize utility: Uh(xh, vh)

utility increases with consumption of produced commodities xh

Utility decreases with supply of factors vh

Given an announced set of prices (p, v), the hth household max. the following:

max Uh = Uh(xh, vh)

s.t. p’xh <= w’vh

(can’t consumer more than you have: household income comes in the form of selling factors w’vh)

Resulting Functions

A result of the budget constraint, we can derive a a set of output demand functions and factor supply functions of the following general form: xi = Di(p, w) for each commodity i = 1,..,n

vj = Fj(p, w) for each commodity j = 1,..,nasd

Market Clearing Functions

Supply must meet demand, more explicitly: Σh xi

h = Σh xif for each commodity i = 1, … ,n

Σh vih = Σh vi

f for each factor j = 1,…,m

Given our earlier notation, this can be rewritten as Di(p, w) = xi

Fj(p, w) = vj = Bj’ x

Bring it all together…

Perfect competition: pi = Biw Output market equil: Di(p, w) = xi

Factor market equil.: vj = Bj’ x Market factor supplies: Fj(p, w) = vj

Bring it all together…

Perfect competition: pi = Biw Output market equil: Di(p, w) = xi

Factor market equil.: vj = Bj’ x Market factor supplies: Fj(p, w) = vj

Bring it all together…

Caveat 1: These are supposed to be vectors Caveat 2: the output demand function is also a function

of w, and the factor demand function is also a function of p, so there is interaction between the diagrams which will cause the curves to shift around.

Equilibrium? The Walras proof Perfect competition: pi = Biw

Output market equil: Di(p, w) = xi

Factor market equil.: vj = Bj’ x

Market factor supplies: Fj(p, w) = vj

Unknowns:

Quan. of produced goods: xi

Quan. of factors: vj

Output Prices: pi

Factor Prices: wj

Equilibrium? The Walras proof Perfect competition: pi = Biw (n eq)

Output market equil: Di(p, w) = xi (n eq)

Factor market equil.: vj = Bj’ x (m eq)

Market factor supplies: Fj(p, w) = vj (m eq)

Unknowns:

Quan. of produced goods: xi (n unk)

Quan. of factors: vj (m unk)

Output Prices: pi (n unk)

Factor Prices: wj (m unk)

The proof:

There are 2n+2m equations and 2n+2m unknowns. This must be a solvable set of equations.

The problem: xy = 3 and x+y = 1, has no real solution

In context: have only one factory and 2 factors s.t. p = b1w1 + b2w2. Has no solution since there are numerous comb. of w1 and w2

that can yield the same output price.

Other problems: With the imposition of

the equalities v = B’x and p = Bw, prices may turn out to be negative

While negative prices and quantities are completely feasible, they mean nothing in this context.

Wald’s fixings

Idea of free goods. All factors are scarce and thus all have a price However, a factor is scarce only if there is more

demand of it than is available, but if supply far exceeds the demand , the factor should have zero price.

Real life examples: tap water, air, internet. Carl Menger “one does not know at the outset

which goods are free goods, so one should insert into the equality the poss. of an unused residual”

Wald’s system

The problem was, if introduce inequalities is existence of an equilibrium certain?

vj >= Bj’x In equil., for a particular factor j, either the quantity

supplies is equal to the quantity demanded (i.e v j= Bj’x) or the quantity of that factor supplied exceeds the demand for the factor vj> Bj’x, but then the corresponding return to factor j must be zero i.e. wj = 0

Primal and Dual Proof of Existence Primal: max p’x (output revenue)

s.t. v>=B’x ; x>=0 Dual: min w’v (factor paid)

s.t. p<=Bw ; w>=0 Other constraint was that we needed to make sure cost

of production (Bw) did not fall below output price, if it did then the company wouldn’t produce it, in other words x i = 0.

Duality Theorem of linear programming There exists a solution to the primal (x*) if

and only if there exists a solution to the dual (w*)

The maximized values of the objectives of the primal and dual are the same. (In our case, that means p’x* = w*v’, which is another way of imposing pure competition at equilibrium)

Duality Theorem of linear programming The langragian multipliers which satisfy the

primal problem is the solution vector to the dual (i.e. λ* = w*) and the multipliers which solve the second problem is the solution vector in the primal (i.e. μ* = x*). Implies that in equilibrium: w* [ v-B’x* ] = 0 x* [ Bw-p ] = 0

Complementary slackness conditions

Complementary slackness conditions These are important b/c they replace the

cond’s for equilibrium and allow for free goods: w* [ v-B’x* ] = 0

Either v = B’x* (factor market equil) or if there is an excess of a factor, vj > Bj’x* then wj* = 0 (the factor is free). Thus wj* > 0 only when vj = Bj’x* (i.e. a factor earns a positive return only if the market for that factor clears)

x* [ Bw-p ] = 0 Same as above except every produced good must equal

it’s cost of production.

Proof of existence

If we assume that v (factor supply) is inelastic i.e. quantity is fixed, then we only have to be concerned with output x*, factor return w*, and output price p*. The maximized values of the objectives of the

primal and dual are the same. i.e. p’x* = w*v’ assures us if we are given p* then w* and x* just follows.

More in depth proof online

Proof of existence (cont.) How do we know if p is p*?

Define equilibrium as a set of inequalities that must be met.

(1) Di (p*, w*) <= xi* for all i = 1, .., n (output market clearing)

(2) If Di (p*, w*) < xi*, then pi* = 0(rule of free goods) (3) vj >= Bj’ x* for all j = 1, .., m(factor market clearing) (4) If vj > Bj’ x*, then wj = 0(rule of free factors) (5) pi* <= Biw*(competition) (6) If pi* < Biw*, then xi = 0(rule of viability) the solution of the duality problem meets

many of these inequalities

Wald’s proof

Not only did Wald prove existence of an equilibrium, he also proved that it was unique.

A good rundown of the proof can be found here: http://cepa.newschool.edu/het/essays/get/

waldsys.htm

Arrow-Debreu Model Arrow-Debreu also proved the existence of

an competitive equilibrium. They defined the four conditions necessary

for a competitive equilibrium to be: Maximize profits for the company Maximize utility w/budget constraint (considered

stock rights and dividends) Prices had to be non-negative and not all zero The idea of a “free good”

Significance

They proved that markets achieve equilibrium by the price mechanism. “In any such situation, there is a price vector

pЄRm+ , the price equilibrium s.t. if xi(p) denotes

the allocation that optimizes ui (x) subject to px<=pei, then Σi xi(p) = Σi ei

In other words, if everyone just did their own thing, the market would clear on it’s own.

Produces an equilibrium w/o any communication between the parties. The market’s invisible hand?

Equilibrium under uncertainty They’ve also introduced the idea of “states” and

uncertainty for a market. Let there be S states of nature, n physically-

differentiated outputs, h households. The “commodity space” of agent h, xh is some subset of Rns

defined as xh = [x1h,…,xs

h ] where each xsh is a vector of

commodities received in state s. An economy-wide allocation X is defined as a vector [x1,…xH] where xh is the allocation to the hth household and xh is defined as before.

Agent h also has other state dependent variables similar to x’s vectors, price p, endowments eh=[e1

h,…,es

h], In addition ownerships shares θ (this was constant)

Consumer Preference

Then household h prefers commodity vector xh to another yh if exp. utility is greater i.e. Suppose agent h assigns a prob. πs to a particular

state s occurring xh>=yh if and only if ΣsЄS πsus

h(xsh)>= ΣsЄS πsus

h(ysh)

Lastly there are F firms that produce state-contingent production plans ys

f

Arrow-Debreu Equilibrium

An Arrow-Debreu equilibrium is a set of allocations (x*, y*) and a set of prices p* such that: (i) for every fЄF, yf* satisfies p*yf*>=p*yf for all yfЄYf

(firms maximize profits given state dependency) (ii) for every hЄH, xh* is maximal in the budget set

Bh = {xh Є Xh I p*xh <= p*eh + Σhθhfp*yf*}

(maximize consumption w/budget constraint)

(iii) ΣhЄH xh* = ΣfЄF yf* + ΣhЄH eh

(market clears, i.e. supply meets demand)

On the complexity of Equilibria Given the ei’s (supply) and some

representation of the ui’s, could we find the price equilibrium? “To date, there is no known poly-time algorithm for

it” NP-hardness is highly unlikely due to it being

guaranteed that an equilibrium exists. Although some very good approx. equil.

algorithms exists.

Indivisible goods

If the commodities are indivisible then an equilibrium may not exist. Possible approx. algorithms? (linear utilities) Market approx. clears: (1-Є) ΣieiΣixi<=Σiei

For all i, the utility is at least (1-Є) times the optimal solution subject to py<=pЄI

It is NP-hard to calculate or even approximate within any factor better than 1/3 the deficiency of a market with indivisible goods (even when there are only two agents, and even when it is known that equilibrium prices exist).

Polynomial time Algorithms

Given linear markets with a bounded number of divisible goods, there apparently is a poly-time algorithm for finding an equilibrium.

There is a poly-time algorithm for computing an Є-pareto curve in linear markets with indivisible commodities and a fixed number of agents.

With a bounded number of goods, there is a poly-time algorithm which, for any linear indivisible market for which a price equilibrium exists, and for any Є>0, finds an Є-approximate equilibrium.