OPIM 952 – Market
Equilibrium
Ralph Ahn
Today’s Lecture
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A general introduction to market equilibria
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Walras-Cassel Model
The Wald “corrections”
The Arrow-Debreu Model
“On the complexity of Equilibria”
The problem:
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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
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The Walras-Cassel Model:
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Where market demand is equal to market supply
in all markets (factor or goods)
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Maximizes utility for each consumer
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Firms maximize proftis
Notation – Walras Cassel model
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Economy: H households, F firms, n produced
commodities, m primary factors
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Produced commodities:
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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
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Factors:
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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:
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bji = vjf / xif is a unit output prod. coefficient
B is n x m matrix of unit-input coefficients
Elaboration of technology
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bji = vjf / xif is a unit output prod. Coefficient
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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…
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v = B’x , where vector x is the supply of
produced goods and v is the demand for factors
Setting Equilibrium assumptions
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Perfect competition:
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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 / xif, then the perfect comp.
assumption implies
p = Bw
Objective of households (consumers)
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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)
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Resulting Functions
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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:
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xi = Di(p, w) for each commodity i = 1,..,n
vj = Fj(p, w) for each commodity j = 1,..,n
asd
Market Clearing Functions
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Supply must meet demand, more explicitly:
h
f
 Σh xi = Σh xi for each commodity i = 1, … ,n
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Σh vih = Σh vif for each factor j = 1,…,m
Given our earlier notation, this can be rewritten as
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Di(p, w) = xi
Fj(p, w) = vj = Bj’ x
Bring it all together…
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Perfect competition:
Output market equil:
Factor market equil.:
Market factor supplies:
pi = Biw
Di(p, w) = xi
vj = Bj’ x
Fj(p, w) = vj
Bring it all together…
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Perfect competition:
Output market equil:
Factor market equil.:
Market factor supplies:
pi = Biw
Di(p, w) = xi
vj = Bj’ x
Fj(p, w) = vj
Bring it all together…
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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:
 Output market equil:
 Factor market equil.:
 Market factor supplies:
Unknowns:
Quan. of produced goods:
Quan. of factors:
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Output Prices:
Factor Prices:
pi = Biw
Di(p, w) = xi
vj = Bj’ x
Fj(p, w) = vj
xi
vj
pi
wj
Equilibrium? The Walras proof
Perfect competition:
 Output market equil:
 Factor market equil.:
 Market factor supplies:
Unknowns:
Quan. of produced goods:
Quan. of factors:
pi = Biw
Di(p, w) = xi
vj = Bj’ x
Fj(p, w) = vj
(n eq)
(n eq)
(m eq)
(m eq)
Output Prices:
xi
vj
pi
Factor Prices:
wj
(n unk)
(m unk)
(n unk)
(m unk)
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The proof:
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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:
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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
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Idea of free goods.
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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
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The problem was, if introduce inequalities is
existence of an equilibrium certain?
vj >= Bj’x
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In equil., for a particular factor j, either the quantity
supplies is equal to the quantity demanded (i.e vj=
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
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Primal: max p’x (output revenue)
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s.t. v>=B’x ; x>=0
Dual: min w’v (factor paid)
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s.t. p<=Bw ; w>=0
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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 xi =
0.
Duality Theorem of linear programming
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There exists a solution to the primal (x*) if
and only if there exists a solution to the dual
(w*)
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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
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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:
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w* [ v-B’x* ] = 0
x* [ Bw-p ] = 0
Complementary slackness conditions
Complementary slackness conditions
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These are important b/c they replace the
cond’s for equilibrium and allow for free goods:
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w* [ v-B’x* ] = 0
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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
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Same as above except every produced good must equal
it’s cost of production.
Proof of existence
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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*.
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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.)
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How do we know if p is p*?
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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
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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:
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http://cepa.newschool.edu/het/essays/get/waldsys
.htm
Arrow-Debreu Model
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Arrow-Debreu also proved the existence of
an competitive equilibrium.
They defined the four conditions necessary
for a competitive equilibrium to be:
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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
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They proved that markets achieve equilibrium
by the price mechanism.
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“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
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They’ve also introduced the idea of “states” and
uncertainty for a market.
Let there be S states of nature, n physicallydifferentiated outputs, h households.
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The “commodity space” of agent h, xh is some subset of Rns
defined as xh = [x1h,…,xsh ] 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=[e1h,…,esh],
In addition ownerships shares θ (this was constant)
Consumer Preference
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Then household h prefers commodity vector xh
to another yh if exp. utility is greater i.e.
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Suppose agent h assigns a prob. πs to a particular
state s occurring
xh>=yh if and only if ΣsЄS πsush(xsh)>= ΣsЄS πsush(ysh)
Lastly there are F firms that produce statecontingent production plans ysf
Arrow-Debreu Equilibrium
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An Arrow-Debreu equilibrium is a set of
allocations (x*, y*) and a set of prices p* such
that:
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(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)
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(iii) ΣhЄH xh* = ΣfЄF yf* + ΣhЄH eh
(market clears, i.e. supply meets demand)
On the complexity of Equilibria
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Given the ei’s (supply) and some
representation of the ui’s, could we find the
price equilibrium?
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“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
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If the commodities are indivisible then an
equilibrium may not exist. Possible approx.
algorithms? (linear utilities)
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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
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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 Єpareto curve in linear markets with indivisible
commodities and a fixed number of agents.
With a bounded number of goods, there is a polytime algorithm which, for any linear indivisible
market for which a price equilibrium exists, and for
any Є>0, finds an Є-approximate equilibrium.