Download Pure Exchange Economy

Pure Exchange Economy
What is an economy?
An economy consists of technologies, tastes, endowments.
→ Firms produce goods using inputs (technology of
production)
→ Individuals consume goods and have preferences (utility
function)
→ Endowments are the resources available in the economy
(example: time, talent, goods)
For simplicity we analyze the pure exchange economy.
The economy consists of consumers only described by their
preferences (utility function).
They own some goods as initial endowment.
The agents exchange these goods between themselves.
Their goal is to maximize their utility.
Question of interest of the theory of general equilibrium:
 How are goods allocated among agents?
 What is the outcome of such process?
 Is it desirable?
 What are the mechanisms achieving the desirable
outcome?
1 Agents and goods
There is a large number, n, of consumers in the economy
indexed by i , i = 1, …, n.
They have preferences described by the utility function
u i  
There are k different goods in the economy.
Each consumer has an initial endowment of the k
commodities
i
Each consumer behaves competitively: prices are taken as
given.
Question
How are goods allocated among agents?
Notations and vocabulary
j
We denote by x i the amount of good j that individual i
holds.
The consumption bundle of individual i is
xi  ( xi1 , xi2 ,..., xik )
An allocation x  ( x1 , x2 ,..., xn ) is a collection of n
consumption bundles describing what each agent holds.
A feasible allocation is one that is physically possible
In the pure exchange economy: i xi  i i
(equality means that the allocation uses up all goods).
n
n
The case of two agents with two goods is very convenient
to represent allocations, preferences and endowments.
→ Edgeworth box
Example
2 individuals: i = 1,2
Total amount of good 1:  1  11   21 ; width of the
Edgeworth box
Total amount of good 1:  2  12  22 ; height of the
Edgeworth box
Preferences:

 

u1 x11 , x12 , u 2 x12 , x22 ,
Figure 1
Every feasible allocation of the two goods between the two
agents is represented by a point in the box.
The point x11 , x12  indicates how much of good 1 and 2
individual 1 holds.
At the same time, the point x12 , x22    1  x11 ,  2  x12 
indicates how much of good 1 and 2 individual 2 holds.
2 Walrasian equilibrium
There is a vector of market prices
(one price for each good)
p  ( p1 , p 2 ,..., pk )
Each consumer takes the prices as given.
He/she chooses the most preferred bundle from his/her
consumption set. The problem is
max ui xi 
xi
subject
to
px i  p i
Spending must be lower or equal to the income.
If preferences are monotonic, the budget constraint is
binding: pxi  pi
The solution of this program leads to the demand function:
xi ( p, pi ).
The equilibrium price vector is the one that clears all
markets.
We define a Walrasian equilibrium to be a pair ( p * , x * )
such that:


i xi p* , p*i  i i
n
n
p * is a Walrasian equilibrium if there is no positive excess
demand for any good.
3 Graphical analysis
It is very simple to describe the equilibrium in the 2 agents,
2 goods case (Edgeworth box)
Figure 2
Every agent maximizes his utility on his budget line and the
demands are compatible with the total supplies available.
At equilibrium the two indifference curves are tangent: that
is, the utility maximisation requires that each individual’s
marginal rate of substitution equals the common price
ratio.
4 Existence of Walrasian equilibria
Recall that the demand functions are homogenous of degree
zero in prices
xi  p, pi   xi p, pi  for all
 > 0.
The aggregate excess demand is
z( p)  i [ xi ( p, pi )  i ]
n
It is homogenous of degree zero in prices as it is the sum of
homogenous of degree zero functions.
It is continuous as long as it is the sum of continuous
functions.
→It must satisfy the Walras’ Law.
Walras’ Law. For any price vector p, we have pz ( p)  0 ;
i.e. the value of the excess demand is identically zero.
Proof: multiply the aggregate excess demand by p and
simplify:
pz( p)  pi [ xi ( p, pi )  i ]  i [ pxi ( p, pi )  pi ]  0
n
n
since xi  p, pi  satisfies the budget constraint which is
binding: pxi  pi .
The Walras law says that if each individual satisfies his/her
budget constraint, so that the value of his/her excess
demand is zero, then the sum of the excess demands must
be zero.
Combining the Walras law with the definition of
equilibrium, we obtain the following Propositions:
Market clearing: If demand equals supply in k-1 markets,
and pk  0 , then demand must equal supply in the kth
market.
Proof: If not, the Walras’ s Law would be violated.
*
Free goods: if p is a Walrasian equilibrium and
*
then p j  0 .
→If some good is in excess supply at Walrasian
equilibrium it must be a free good.
z j ( p* )  0 ,
Proof: Since p* is a Walrasian equilibrium, z ( p )  0 .
k
Since prices are nonnegative, p * z( p * )  i1 pi* zi ( p * )  0 . If
*
*
and p j  0 we would have p* z ( p* ) < 0,
contradicting Walras’ Law.
z j ( p* )  0
Suppose that all goods are desirable in the following sense:
Desirability: If pi  0 , then z ( p )  0 for i = 1, …, k.
If some price is zero, the aggregate excess demand for that
good is strictly positive.
Equality of demand and supply: If all goods are desirable
*
and p is a Walrasian equilibrium, then z ( p * )  0 .
Proof: Assume z i ( p * )  0 , then by the free good Proposition,
p * = 0. But then, by the desirability assumption, z i ( p * )  0 a
contradiction.
To sum-up:
All we require for an equilibrium is that there is no excess
demand for any good.
If a good is in excess supply, its price must be zero.
5 Existence of an equilibrium
The aggregate demand function is homogenous of degree
zero.
→We can normalize prices.
→ express demands in terms of relative prices.
We replace each absolute price pi by a normalized price
pi 

pi
k
j 1
pj
A consequence of this normalization:

k
j 1
pj 1.
Therefore, we can restrict our attention to the k-1
dimensional unit simplex


S k 1  p   k :  j 1 p j  1
k
Figure 3 about here.
The proof uses the Brouwer fixed-point theorem.
Brouwer fixed-point theorem. If f : S k 1  S k 1 is a
continuous function from the unit simplex to itself, there
exists some x in S k 1 such that x  f (x) .
Existence of Walrasian equilibria. If z : S k 1   k is a
continuous function that satisfies Walras’ Law, pz ( p )  0 ,
*
k 1
then there exists some p in S such that z ( p * )  0 .
Proof: Exercise.
The theorem of existence is very general.
All that is needed is the excess demand function to be
continuous and satisfy Walras’ Law.
Even though at the individual level demands function
display discontinuities, at the aggregate level the demand
may be continuous.
The continuous assumption at the aggregate level is a weak
requirement.
Example
Assume that
  x 
u ( x , x )  x  x 
u1 ( x11 , x12 )  x11
2
1
2
2
2
a
1 b
2
2 1a
1
2 1b
2
, with 0<a<1 and 1  1,0
, with 0<b<1 and 2  0,1
Demand functions are
x11 ( p1 , p2 , m1 ) 
am1
p1 , where
is the income of agent 1.
m1  1 p1  0  p2  p1
x12 ( p1 , p2 , m2 ) 
bm2
p1 , where
m2  0  p1  1 p2  p2 is the income of agent 2.
The equilibrium price is such that Demand=Supply
By Walras’ Law, we only need to find the system of prices
that clear the market of good 1.
x11 ( p1 , p2 , m1 )  x12 ( p1 , p2 , m2 ) 
am1 bm2 aP1 bP2
bP



 a  2 1
p1
p1
p1
p1
p1
p2 1  a

p1
b
Only the relative price is determined at equilibrium.
6 The first theorem of welfare economics
The Walrasian equilibrium theory is very interesting from a
positive perspective as long as we believe in the
behavioural assumptions on which the model is based.
Even if sometimes, the assumptions are not especially
plausible, the Walrasian equilibrium theory is interesting
for its normative content.
Definition of Pareto efficiency:
1) A feasible allocation x is a weakly Pareto efficient
allocation if there is no feasible allocation x’ such that
all agents strictly prefer x’ to x.
2) A feasible allocation x is a strongly Pareto efficient
allocation if there is no feasible allocation x’ such that
all agents weakly prefer x’ to x and some agent strictly
prefers x’ to x.
A strongly Pareto efficient allocation is weakly Pareto
efficient. But the reverse is, in general not true.
Equivalence of strongly and weakly Pareto efficiency:
Suppose that preferences are continuous and monotonic.
Then, an allocation is weakly Pareto efficient if and only if
it is strongly Pareto efficient.
Proof:
a) strongly Pareto efficient implies weakly Pareto efficient.
If an allocation is strongly Pareto efficient, then it is weakly
Pareto efficient: if you can’t make one person better off
without hurting someone else, you can’t make everyone
better off.
b) weakly Pareto efficient implies strongly Pareto efficient.
Suppose that it is possible to make some particular agent i
better off without hurting anyone. We must find a way to
make everyone better off. To do this, scale back i’s
consumption bundle by a small amount and redistribute the
good equally to the others.
Specifically,, replace i’s consumption x i by x i and
replace each other agent j’s consumption bundle by
x j  (1   ) xi /( n  1) . By continuity of preferences, it is
possible to choose  close enough to one so that agent i is
still better off. By monotonicity, the other are all better off.
In general ‘Pareto efficient’ is said to mean ‘weakly Pareto
efficient’.
Pareto efficiency is a weak normative concept.
An allocation where an agent gets everything and the
others nothing is Pareto efficient (if agents are not
satiated).
Pareto efficient allocations can be depicted in the
Edgeworth box.
Pareto efficient allocations are found by fixing one agent’s
utility at a given level and maximizing the other agent’s
utility.
Formally the problem to solve is
max u1x1 
x1
subject
to
u 2 ( x2 )  u 2
x1  x 2  1   2
This problem can be solved analytically using the
Lagrangian method.
In the Edgeworth box, we simply find the point on one
agent’s indifference curve where the other agent reaches the
highest utility.
Figure 5 about here.
Pareto efficient points are characterized by a tangency
condition: the marginal rate of substitution must be the
same for each agent.
The set of Pareto efficient points, the Pareto set, is the
locus of tangencies drawn in the Edgeworth box.
This is known as the contract curve.
It gives the set of efficient ‘contract’ or allocation.
In fact comparing Figure 2 and 5, we see there is a one to
one correspondence between the set of Walrasian equilibria
and the set of Pareto efficient allocations.
The Walrasian equilibrium satisfies the first order condition
for utility maximisation that the marginal rate of
substitution between the two goods. Since agents face the
same price at a Walrasian equilibrium, all agents must have
the same marginal rates of substitution.
If we pick an arbitrary Pareto efficient allocation, the
marginal rate of substitution are equal across two agents.
Thus, there exists a system of price that equals to this
common value, i.e. which sustains a Walrasian equilibrium.
Definition of a Walrasian equilibrium. An allocationprice pair (x, p) is a Walrasian equilibrium if 1) the
allocation is feasible, and 2) each agent is making an
optimal choice from his budget set.
In equations:
n
n
1) i xi  i i
2) if
x i'
is preferred by agent i to x i , then
pxi'  p i
This definition is equivalent to the original definition of
Walrasian equilibrium as long as the desirability
assumption is satisfied.
First Theorem of Welfare Economics. If (x, p) is a
Walrasian equilibrium, then x is Pareto efficient.
Proof: Suppose not and let x’ be a feasible allocation that
all agents prefer to x. Then, by property 2 of the definition
of Walrasian equilibrium, we have
pxi'  p i , for i=1, …, n
.
Summing over i=1, …., n, and using the fact that x’ is
feasible we have
n
n
n
pi 1i  pi 1 xi'  pi 1 xi which is a contradiction.
This says that if the behavioral assumptions of the model
are satisfied then, the market equilibrium is efficient. A
market equilibrium is not necessarily ‘optimal’ in the
ethical sense since it can be ‘unfair’. The outcome depends
on the initial endowment.
To choose among the efficient allocation further ethical
criterion are needed.
→ concept of welfare function (see end of the chapter).
7 The Second Welfare Theorem
Second Theorem of Welfare Economics (Version 1).
Suppose 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 endowment
 i  x * for i=1, …, n.
Proof: Exercise.
Second Theorem of Welfare Economics (Version 2).
Suppose x is a Pareto efficient allocation and that
preferences are non-satiated. Suppose further that a
competitive equilibrium exists from the initial endowment
  x and let it be given by (p’, x’). Then, (p’, x*) is a
competitive equilibrium.
*
*
i
*
Proof: Since x i is in the consumer i’s budget set by
'
*
x
construction, i is preferred or indifferent to x i . Since
x*
'
is Pareto efficient, this implies that x i* is indifferent to x i .
'
*
Thus, x i is optimal, so is x i . Hence, (p’, x*) is a
competitive equilibrium.
8 Pareto efficiency and calculus
We have shown that if a competitive equilibrium exists
from a Pareto efficient allocation, then that Pareto efficient
allocation is itself a competitive equilibrium.
In this section we derive the first order conditions that
characterize market equilibria and Pareto efficiency.
Then we compare these two sets of conditions.
Calculus characterization of equilibrium. If (x*, p*) is a
market equilibrium with each consumer holding a positive
amount of every good, then there exists a set of numbers
(1 , 2 ,..., n ) such that:
Du i ( x * )  i p * ,
i=1, …, n.
Proof : if we have a market equilibrium, then each agent is
maximised o his budget set, and these are just the first
order conditions for such utility maximisation. The i ’s
are the agents’ marginal utilities of income.
Calculus characterization of Pareto efficiency. A feasible
allocation x* is Pareto efficient if and only if x* solves the
following n maximization problems for i=1,…,n:
max ui xi 
such that
 x   g = 1, …, k.
u x   u x  , j  i
n
i 1
j
*
j
g
i
g
j
j
Proof :Suppose x* solves all maximisation problems but x*
is not Pareto efficient. This means that there is some
allocation x’ where everyone is better off. But then, x*
could not solve any of the problems, a contradiction.
Conversely, suppose x* is Pareto efficient, but it does not
solve one of the problems. Instead, let x’ solve that
particular problem. Then x’ makes one of the agent better
off without hurting any of the other agents, which
contradicts the assumption that x* is Pareto efficient.
Let us solve one maximisation problem.
Let q g for g=1, …, k be the Kuhn-Tucker multipliers for
the resource constraints, and a j j  i be the multipliers for
the utility constraints:
L  ui xi   g 1 q g
k



g
g
*
x



a
u
(
x

i
i
j
j
j )  u j (x j )
i 1
j 1
n
n

The first order conditions lead to
 
u i xi*
 q g  0 , g= 1, …, k.
g
xi
aj
 q
u i x *j
x
g
j
g
 0, j i,
g= 1, …, k.
These conditions imply that the relative values of the q are
independent of the choice of i:
 
 
u i xi* / xig
qg
 h
*
h
u i xi / xi
q
, g= 1, …, k,
ji
, h= 1, …, k.
Since x* is given, q g / q h must be independent of the
problem we solve. In fact, when we maximise agent i’s
utility, it is as if we are arbitrarily setting agent i’s KuhnTucker multiplier to one: ai  1.
Using the First Welfare theorem, we get interpretations of
the weights ( ai ) and ( q g ).
If x* is a market equilibrium:
Du i ( x * )  i p * , i=1, …, n.
All market equilibria are Pareto efficient and satisfy
ai Du i ( x * )  q , i=1, …, n.
From above, we can choose p*=q and ai  1/  .
i
That is, the multipliers on the resources constraints are just
the competitive prices, and the multipliers on the utilities
are the reciprocals of their marginal utilities of income.
Rearranging the above conditions, one gets:
 
 
p *g q g
ui xi* / xig
 *  h , g= 1, …, k,
*
h
ui xi / xi
ph q
ji
, h= 1, …, k.
Each Pareto efficient allocation must satisfy the condition
that the marginal rates of substitution between each pair of
goods is the same for every agent.
This value corresponds to the ratio of the competitive
prices.
Intuition: If two agents had different marginal rates of
substitution, they could arrange a small trade that would
make them better off, contradicting the Pareto efficiency.
It is useful to note that the first order conditions for a Pareto
efficient allocation are the same as the ones maximizing a
weighted sum of utilities:
max i 1 ai ui xi 
n
such that

g
g
x


g = 1, …, k.
i
i 1
n
This gives
ai Du i ( x * )  q i=1, …, n.
The weights (a1 , a2 ,..., an ) are called ‘welfare weights’.
As they vary, we can trace out the set of Pareto efficient
allocations.
9 Welfare maximization
The criterion of Pareto efficiency is normative but not very
specific. In particular it is not concerned about distribution
of welfare.
A way to solve this problem is to assume the existence of a
social welfare function.
This function aggregates the utility of all agents to come up
with a social utility.
We can interpret it as a social decision maker’s preferences
about how to trade-off the utilities of individuals.
Specifically, we have
W : n  
so that W (u1 , u 2 ,..., u n ) is the social utility function.
We assume that W( ) is increasing in each of its arguments.
The society chooses an allocation x
max W u1 x1 , u 2 x2 ,..., u n xn 
such that
*
solution of

g
g
x


g = 1, …, k.
i
i 1
n
Question:
How do the allocations that maximize this welfare function
compare to the Pareto efficient allocations?
*
Welfare maximization and Pareto efficiency. If x
maximizes a social welfare function, then x * is Pareto
efficient.
*
Proof: If x were not Pareto efficient, then there would be
some feasible allocation x’ such that u i ( xi' )  u i ( xi* ) for
i=1,…,n.
But then, W u1 x1' , u 2 x2' ,..., u n xn'  > W u1 x1*' , u 2 x2* ,..., u n xn*'  .
The property of Pareto efficient allocation and welfare
maxima are the same:
- Welfare maxima satisfy the same first order conditions
as Pareto efficient allocation.
- Under convexity assumption, every welfare maxima is
a competitive equilibrium: every welfare maxima is a
competitive equilibrium for some distribution of
endowments.
Welfare maximum are Pareto efficient. Is the converse
true?
Pareto efficiency and Welfare maximization. Let x * be a
*
Pareto efficient allocation with x >>0 for i=1,…,n. Let the
utility functions u i be concave, continuous, and monotonic.
Then, there is some choice of welfare weights a such that
*
x maximizes  ai ui ( xi ) subject to the resource constraints.
*
*
Furthermore, the weights are such that ai  1/  where *i is
the ith agent’s marginal utility of income; that is, if mi is the
value of agent i’s endowment at the equilibrium prices p*,
then
*
i
*
i
vi ( p * , mi )
 
vi ( p* , mi ) is the indirect utility
,
where
mi
*
i
function.
Proof: Exercise.
Summary
1) Competitive equilibria are always Pareto efficient;
2) Pareto efficient allocations are competitive equilibria
under convexity assumptions and endowments
redistribution;
3) Welfare maxima are always Pareto efficient;
4) Pareto efficient allocations are welfare maxima under
convexity assumptions for some choice of welfare
weights.
A competitive market leads to efficient allocations, but
this says nothing about distribution.
The choice of distribution of income is the same as the
choice of a reallocation of endowments, and this in turn is
equivalent to choosing a particular welfare function.