Motivation
Consumers
Excess Demand Function
Existence of Equilibrium
First Welfare Theorem
Introduction to General Equilibrium
Juan Manuel Puerta
November 6, 2009
Second Welfare Theorem
Motivation
Consumers
Excess Demand Function
Existence of Equilibrium
First Welfare Theorem
Second Welfare Theorem
Introduction
So far we discussed markets in isolation. We studied the
quantities and welfare that results under different assumptions on
market power.
The analysis of markets in isolation is referred as “partial
equilibrium”.
In contrast, “General Equilibrium” analysis, studies the
interactions of all the markets. In this case, all the prices are
variable and adjust. Equilibrium requires that all the markets
clear.
General equilibrium could be studied considering or not the
production of goods. We will study general equilibrium without
production and focusing on the exchange of goods.
Motivation
Consumers
Excess Demand Function
Existence of Equilibrium
First Welfare Theorem
Second Welfare Theorem
Some Definitions
There are n consumers in this economy.
A consumer is described by her utility function ui and her initial
endowment ωi = {ωi,1 , ..., ωi,k } for each of the k goods of this
economy.
The Consumption bundle of the consumers is given by
xi = {xi1 , ..., xik } and describes how much of each good is
consumed.
An Allocation is the collection of the n consumption bundles. In
other words an allocation is an n × k vector x = {x1 , ..., xn }
An allocation is feasible if the sum of consumption bundles is
less than the sum of endowments. That is,
n
n
X
X
xi ≤
ωi
(1)
i=1
i=1
In a 2 × 2-framework (i.e. 2 goods, 2 consumers), allocations can
be easily depicted with an Edgeworth box†.
Motivation
Consumers
Excess Demand Function
Existence of Equilibrium
First Welfare Theorem
Second Welfare Theorem
Some Definitions
Let mi denote the consumer’s income, i.e. the market value of
her endowment. mi = pωi
The consumer problem is given by:
max ui (xi )
xi
(2)
subject to mi = pxi
We studied that the solution to this problem is given by a
function 1 xi = xi (p, mi ). Note that unlike the standard case in the
consumer theory, mi = pωi is a function of prices!
1
Assuming strict convexity of preferences, the solution is unique
Motivation
Consumers
Excess Demand Function
Existence of Equilibrium
First Welfare Theorem
Second Welfare Theorem
Walrasian Equilibrium: Definition
Walrasian Equilibrium: The pair (x∗i , p∗ ) is a Walrasian
equilibrium if
n
n
X
X
x∗i (p∗ , mi ) ≤
ωi
i=1
(3)
i=1
that is, if the amounts demanded are less or equal than the
amount supplied for each good. No good is excess demand. Why
do we allow the possibility of excess supply? It could be the case
that a good is undesirable , so there is excess supply of it.
Motivation
Consumers
Excess Demand Function
Existence of Equilibrium
First Welfare Theorem
Second Welfare Theorem
Excess Demand Function: Properties
Under which conditions can we assure that there exist a p that
makes market to clear? In order to find an answer to this
question it is useful to define the excess demand function z(p)
n
P
z(p) = [xi (p, pωi ) − ωi ]
i=1
Properties of z(p)
1
2
3
Homogeneous of degree 0. From consumer theory, the demand
functions are homogeneous of degree 0 in prices, i.e.
xi (p, pωi ) = xi (kp, kpωi ). This property carries on to the excess
demand function (z(p)), defined as where we ignore the fact that
z(p) depends on ωi , as the initial endowments remain constant.
Continuity. If all xi are continuous, so is z(p)
Walras’ Law. For any price vector p, pz(p) = 0, that is, the value
of the excess demand is identically 0. Proof: The BC imply
pxi (p, pωi ) = pωi ). Sum across individuals and rearrange.
Motivation
Consumers
Excess Demand Function
Existence of Equilibrium
First Welfare Theorem
Second Welfare Theorem
Market Clearing. If demands equals supply in k-1 markets and
pk > 0, then demand equal supply in the k-th market.
Proof: Let zj denote the j-th element of the z(p) vector. Walras
P
P
law implies j pj zj (p) = 0. Rewrite as j,k pj zj (p) + pk zk (p) = 0.
If all but the k-th market clear, then pj zj (p) = 0 for j , k. Then
Walras’ Law implies that pk zk (p) = 0, i.e. zk (p) = 0 as pk > 0.
Free Goods. If p∗ is a Walrasian Equilibrium and z(p∗ ) ≤ 0, then
p∗j = 0. That is, if some good is in excess supply at the Walrasian
Equilibrium, it must be a free good.
If p∗ is a Walrasian Equilibrium, z(p∗ ) ≤ 0. Since prices are
P
non-negative, p∗ z(p∗ ) = k p∗k zk (p∗ ) ≤ 0. Then either the market
clears zk (p∗ ) = 0 or if zk (p∗ ) < 0, pk = 0. Otherwise
P
∗
k pk zk (p ) < 0 and the walras’ law does not hold.
Motivation
Consumers
Excess Demand Function
Existence of Equilibrium
First Welfare Theorem
Second Welfare Theorem
Definition: Desirability. We say a good i is desirable if pi = 0
implies zi (p) > 0
Idea: A good is desirable if when it is free, there is excess
demand of it!
It turns out that desirability allows us to find an important result,
equality between demand and supply.
Equality between demand and supply. If all goods are desirable
and p∗ is a price vector consistent with Walrasian equilibrium,
then z(p∗ ) = 0, that is, in equilibrium, supply equals demand in
every market.
Proof: Assume not. Then z(p∗ ) < 0. If that is the case then there
is some i for which pi = 0. Desirability then implies zi (p∗ ) > 0, a
contradiction.
Motivation
Consumers
Excess Demand Function
Existence of Equilibrium
First Welfare Theorem
Second Welfare Theorem
Summary
In equilibrium we will expect markets to clear or to show excess
supply.
If a market is in excess supply, then that means that the good has
to have a zero price in equilibrium.
If we assume that all goods are desirable in the sense that people
want to consume more than what is available at price zero, then
all markets should clear in equilibrium.
Motivation
Consumers
Excess Demand Function
Existence of Equilibrium
First Welfare Theorem
Second Welfare Theorem
Existence
Since z(p) is HD0, we can normalize prices and express
everything in terms of relative prices.
A convenient normalization of the absolute prices p̂i is given by
pi =
p̂i
k
P
p̂j
(4)
j=1
This implies that pi sum up to 1. So, let k be the number of goods
in this economy, we can restrict our attention to the following set
of normalized prices Sk−1 , the k-1 unit simplex
P
Sk−1 = {p ∈ Rk+ : k pk = 1}
Example: S1 and S2 †
Motivation
Consumers
Excess Demand Function
Existence of Equilibrium
First Welfare Theorem
Second Welfare Theorem
Brouwer’s Fixed Point Theorem
In order to prove existence of general equilibrium we need first a
couple of mathematical results.
Intermediate Value Theorem: Let f(x) be a continuous function
defined over [a,b]. Then, for every d between f(a) and f(b), there
exist a c such that d=f(c).
Brouwer’s Fixed Point Theorem Let f : Sk−1 → Sk−1 be a
continuous function from the k-1 unit simplex into itself, there is
some x in Sk−1 such that f(x) = x.
Proof for the one dimensional case. Let f : [0, 1] → [0, 1]. Let
g(x) = f (x) − x. Now, g(0) = f (0) − 0 ≥ 0 and g(1) = f (1) − 1 ≤ 0.
By the intermediate value theorem, there exist x such that
g(x) = 0. But this implies f(x)=x establishing the result.
graphical intuition. †
For a more general proof in the simplex see Starr (p.56-63)
(Beyond the scope of this class).
Motivation
Consumers
Excess Demand Function
Existence of Equilibrium
First Welfare Theorem
Second Welfare Theorem
Proof of existence I
Theorem
Existence of Walrasian Equilibria. If z : Sk−1 → Rk is a continuous
function that satisfies Walras’ Law, pz(p) ≡ 0, then there exist some
p∗ ∈ Sk−1 such that z(p∗ ) ≤ 0
Proof: Define a map g : Sk−1 → Sk−1 by
gi (p) =
pi + max{0, zi (p)}
for i=1,2,...,k
P
1 + kj=1 max{0, zj (p)}
(5)
Note that gi (p) is continuous as both z(p) and the max function are
continuous.Note also that g(p) is a point in the simplex since
P
gi (p) = 1.There is also a economic interpretation for this. The
relative price of goods in excess demand (zi > 0) is increased, so as to
Motivation
Consumers
Excess Demand Function
Existence of Equilibrium
First Welfare Theorem
Second Welfare Theorem
Proof of existence II
eliminate excess demand.Now, Brouwer’s fixed point theorem ensures
that there exists p∗ such that p∗ = g(p∗ ). That is
p∗i =
p∗i + max{0, zi (p∗ )}
for i=1,2,...,k
P
1 + kj=1 max{0, zj (p∗ )}
(6)
The last part of the proof requires that we show that the vector p∗
consistent (6) is associated with a Walrasian equilibrium. In order to
so, just rearrange the expression
p∗i + p∗i
k
X
j=1
max{0, zj (p∗ )}) = p∗i + max{0, zi (p∗ )}, for i=1,2,...,k
Motivation
Consumers
Excess Demand Function
Existence of Equilibrium
First Welfare Theorem
Second Welfare Theorem
Proof of existence III
Multiply these equations by zi (p∗ ) and sum up across goods we obtain
[
k
X
max{0, zj (p∗ )})]
j=1
By Walras’ Law
k
X
i=1
k
P
i=1
p∗i zi (p∗ ) =
k
X
zi (p∗ ) max{0, zi (p∗ )})
i=1
p∗i zi (p∗ ) = 0, so the expression is simplified to
k
X
zi (p∗ ) max{0, zi (p∗ )}) = 0
i=1
Which requires that zi (p∗ ) ≤ 0 for all i. This means that p∗ is the price
vector consistent with a walrasian equilibrium.
Motivation
Consumers
Excess Demand Function
Existence of Equilibrium
First Welfare Theorem
Second Welfare Theorem
Implications of the Theorem
The existence of general equilibrium could be proved under very
general conditions. We specifically had to assume just that:
1
2
Continuity of z(p): A sufficient condition for this is that
individual demand functions are continuous. This requires us to
assume strict convexity of preferences, then the demand
correspondence is single-valued (it is a function) and it is
continuous by the theorem of the maximum. But continuity of the
aggregate demand may still be achieved by aggregation of
non-continuous individual demand functions.
Walras’ Law: This property follows from the fact that individuals
are faced with some form of budget constraint.
There is a technical issue though. This is the assumption of
continuity may break down as pi → 0. As the price of something
that is actually desired goes to zero, excess demand may
explode. This point is overcome with a slightly more
complicated mathematical proof.
Motivation
Consumers
Excess Demand Function
Existence of Equilibrium
First Welfare Theorem
Second Welfare Theorem
Welfare Theorems
While it is important to establish the existence of Walrasian
equilibrium under broad circumstances, we have not been able to
say anything “normative” about this equilibrium? Is it good?
Could it be improved?
In order to answer these questions we have to ask what do we
mean by “good” and “improve”.
Definition (Pareto Efficiency)
A feasible allocation x is weakly Pareto efficient allocation if there is
no feasible allocation x0 such that all the agents strictly prefer x0 to x.
It is strongly Pareto efficient if all agents weakly prefer x0 to x and
there is at least one agent who strictly prefers x0 to x.
Motivation
Consumers
Excess Demand Function
Existence of Equilibrium
First Welfare Theorem
Second Welfare Theorem
Theorem (Pareto Efficiency)
If preferences are continuous and monotonic, then an allocation is
weakly Pareto efficient if, and only if, the allocation is strongly Pareto
Efficient. That is, the two definitions are equivalent.
Proof: Clearly SPE implies WPE (if you cannot improve 1 agent
without hurting other one, then you simply cannot improve all
the agents). The converse statement is not difficult to prove. If
you have 1 person that is strictly better off (xi ). You can take a
small amount (1 − θ)xi and redistribute it to all the other (n-1)
agents giving (1 − θ)xi /(n − 1) to each. Continuity implies that
we can choose θ close enough to 1 so that agent i still better off.
Monotonicity implies that all the rest are also better off, so that
the allocation is strongly pareto efficient.
Motivation
Consumers
Excess Demand Function
Existence of Equilibrium
First Welfare Theorem
Second Welfare Theorem
A pareto efficient allocation solves de maximization problem
max u1 (x1 )
{x1 ,x2 }
such that u2 (x2 ) ≥ ū
x1 + x2 = ω1 + ω2
Edgeworth box †
It should be clear that Pareto efficiency requires tangency of
indifference curves.
The locus of points of all the Pareto efficient allocations is the
contract curve
(7)
Motivation
Consumers
Excess Demand Function
Existence of Equilibrium
First Welfare Theorem
Second Welfare Theorem
Note that a walrasian equilibrium equalizes the price ratios to the
MRS of a consumer. Since all consumers face the same
walrasian prices, then this amounts to equalizing marginal rates
of substitution across individuals. Note that the solution to
problem (7) also involves equalization of marginal rates of
substitution.
Before we continue, it would be useful to define precisely what
we mean by Walrasian equilibrium. We will assume desirability
in order to simplify the argument to come.
Definition (Walrasian Equilibrium)
An allocation-price pair (x, p) is a Walrasian Equilibrium if the
allocation is feasible and each agent is maximizing his utility given
his budget set. That is,
1
Feasible:
n
P
i=1
2
xi =
n
P
ωi
i=1
Ut. Max: xi solves the UMP of agent i. That is, if x0i xi then
px0i > pωi
Motivation
Consumers
Excess Demand Function
Existence of Equilibrium
First Welfare Theorem
Second Welfare Theorem
Is there a correspondence between Walrasian equilibria and the
set of Pareto optima?
It turns out that there is. In the next two theorems will establish a
relationship. Roughly speaking,
1
2
The first welfare theorem says that every Walrasian equilibrium is
Pareto optimum.
The second welfare theorem proves the converse statement. For
every Pareto optimum, there exist a set of prices and
endownments such that it can be supported as a Walrasian
equilibrium.
Motivation
Consumers
Excess Demand Function
Existence of Equilibrium
First Welfare Theorem
Second Welfare Theorem
First Welfare Theorem
Theorem
First Theorem of Welfare Economics. If (x, p) is a Walrasian
equilibrium, then x is Pareto efficient.
Proof.
Assume not and let x0 be a feasible allocation that all agents prefer to
x. Then the definition of walrasian equilibrium implies that all agents
are maximizing utility, so since they prefer x0 to x, px0i > pωi . Sum up
P
P
across individuals to get, ni=1 px0i > ni=1 pωi . But this contradicts the
feasibility assumption. The contradiction establishes the result.
Motivation
Consumers
Excess Demand Function
Existence of Equilibrium
First Welfare Theorem
Second Welfare Theorem
Second Welfare Theorem I
Theorem
Second Theorem of Welfare Economics. Assume x∗ is a Pareto
Efficient allocation in which each agent holds a positive amount of
each good. Suppose that preferences are convex, continuous and
monotonic. Then x∗ is a Walrasian equilibrium for the initial
endowments ωi = x∗i for i = 1, 2, ..., n.
Proof: Let Pi = {xi ∈ Rk : xi i x∗i }. Pi is the set of bundles that agent
P
P
i prefers to x∗i . Define P = i Pi = {z : z = i xi , xi ∈ Pi }. Since each
Pi is convex by assumption and the sum of convex sets is convex.
Then P is convex. Before we go on we need to state a theorem that is
useful.
Motivation
Consumers
Excess Demand Function
Existence of Equilibrium
First Welfare Theorem
Second Welfare Theorem
Second Welfare Theorem II
Theorem (Separating Hyperplane)
If A and B are two nonempty, disjoint, convex sets in Rn , then there
exist a p , 0 such that px ≥ py for all x ∈ A and y ∈ B.
Let ω =
n
P
i=1
x∗i .Pareto optimality of allocation ω means that ω is not in
P. Since P is nonempty and convex and it is disjoint with the set that
includes only the element ω, we can use the separating hyperplane
theorem
pz ≥ p
n
P
i=1
x∗i for all z ∈ P
Motivation
Consumers
Excess Demand Function
Existence of Equilibrium
First Welfare Theorem
Second Welfare Theorem
Second Welfare Theorem III
In order to finalize the proof, we need to establish that, 1) p is a price
vector, that is, it is non-negative, 2) every agent is maximizing utility,
that is, if there is a bundle that they prefer to x∗i , it should be more
expensive than x∗i .
1. p ≥ 0: Let ei = (0, ..., 1, ..., 0) where the one is in the i-th position.
ω + ei ∈ P since it has one more unit of the good i, so everyone could
be made better off. But then, the separating hyperplane theorem
implies p(
ω + ei − ω) = pi ≥ 0. Since you can do this for any i, the
non-negativity of prices follows.
In order to establish that show that strictly preferred bundles are more
expensive we do it in 2 steps. First we show that they must be at least
as expensive and use that to prove the strict inequality.
2. if yj j x∗j then pyj > px∗j : We need to show if a particular agent j
prefers yj to xj , then yj is never cheaper. The trick is to redistribute the
extra good that j got among all the other consumers using strong
Motivation
Consumers
Excess Demand Function
Existence of Equilibrium
First Welfare Theorem
Second Welfare Theorem
Second Welfare Theorem IV
monotonicity and continuity. Then, we can use the separating
hyperplane theorem and we are done. Define,
zj = (1 − θ)yj
θ
yj , for i , j
zi = x∗i +
n−1
P
Now this new allocation i zi is in P as everyone is strictly better off
than before. Using the separating hyperplane theorem,
X
X
p
zi ≥ p
x∗i
i
i
But replacing the definitions of zi above,
p[(1 − θ)yj
X
i,j
zi x∗i +
X
i,j
X
θ
yj ] ≥ p[xj∗ +
x∗i ]
n−1
i,j
Motivation
Consumers
Excess Demand Function
Existence of Equilibrium
First Welfare Theorem
Second Welfare Theorem
Second Welfare Theorem V
p[(1 − θ)yj + θyj +
X
x∗i ] ≥ p[xj∗ +
i,j
X
i,j
∗
xi ]
pyj ≥ px∗j
Now the last step is to prove that this result actually holds with strict
inequality. Continuity of preferences implies that we can find θ
sufficiently close to 1 such that θyj j x∗j . By the argument just
presented, it is possible to show that
θpyj ≥ px∗j
(8)
Since,x∗j > 0, then px∗j > 0. So if the inequality pyj ≥ px∗j holds with
equality, then pyj = px∗j > 0 and θpyj < px∗j , which leads to a
contradiction with (8). Thus, if an allocation is preferred for an
individual j, then it must be more expensive than the original
allocation x∗j . With this condition it established that x∗ is a walrasian
equilibrium with vector price p concluding the proof.