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Uniqueness, Stability and Gross Substitutes
Arrow-Debreu Economy
Econ 2100
Fall 2014
Lecture 22, November 12
Outline
1
Uniquennes and Stability
2
Gross Substitute Property
3
Arrow-Debreu Economies:
1
2
3
4
State-Contingent Commodities
Markets and Prices
Budget Constraint
Arrow-Debreu Equilibrium
Lack of Uniqueness
Multiple Equilibria
8
Two goods and two consumers: u1 (x11 ; x21 ) = x11 18 (x21 ) and
8
u2 (x12 ; x22 ) = 81 (x12 ) + x22 , and ! 1 = (2; r ) and ! 2 = (r ; 2), r = 28=9
21=9 .
z1 (p1 ; 1):
Small changes in preferences or endowments do not change aggregate excess
demand much: multiplicity of equilibria is robust.
Since z1 ( ; 1) starts above zero and …nishes below zero, the picture also
suggests that the number of equilibria, if …nite, is odd.
This is a general result that follows from the Index Theorem.
Local Uniqueness
Even if there are multiple Walrasian equilibria, the previous example suggests
that each of these equilibria could be locally unique.
What does that mean? Equilibria are ‘isolated points’.
Local uniqueness: there is no other Walrasian equilibrium price vector ‘close’
to the orginal equilibrium price vector.
Formally, an equilibrium is not locally unique if its price vector p is the limit of
a sequence of other equilibrium prices.
In the example on the previous slide, there were three Walrasian equilibria and
each of them is locally unique.
Results
In many economies, competitive equilibria are locally unique.
Therefore, the set of Walrasian equilibria is …nite.
The formal statement says something like: the set of initial endowment such
that equilibria are locally unique has measure 1.
Equilbria are generically unique.
Local Uniqueness: Example
An example of local non-uniqueness would be if z1 (p1 ; 1) was equal to zero
over some interval of prices [p1 ; p1 ].
The …gure illustrates how this is extremely special.
Any small perturbation of z1 ( ; 1), for example because of a small change in
endowments, will move it a little and yield a …nite number of locally unique
equilibria.
Tâtonnement
How do equilbrium come to be? Is there a way for prices to converge
‘naturally’to their equilibrium value even if one starts with the wrong prices?
As much as we would like a simple yes answer, things are much more tricky.
Walras suggested the following price adjustment process called “tâtonnement”
(french for “trial and error”).
Tâtonnement
Agents meet in a public square and a “Walrasian" auctioneer calls out some
prices.
Afterwards, agents call out their demands at those prices.
The auctioneer sees demands and then calls out adjusted prices.
This process continues until prices are called so that demand equals supply.
If that is the case, the auctioneer stops, announces prices, and trade occurs.
Clearly, this is a very abstract way to think about price convergence.
Even in the stock market, trade actually takes place after prices are announced.
Second, consumers might not want to announce their true demands.
Tâtonnement Stability
A possible tâtonnement price adjustment rule is:
p(t + 1) = p(t) + z(p(t)) for small
> 0:
The only stationary points of this process are price vectors p at which
z(p ) = 0, i.e. Walrasian equilibrium prices.
An equilibrium price vector would be called locally stable if the price
adjustment rule converges to iy from any nearby starting prices.
If an equilibrium price vector is stable a small perturbation away from it moves
the economy away from equilibrium only for a short time as the auctioneer …nds
an equilibrium quickly.
An equilibrium price vector is globally stable if the price adjustment rule
converges to it from any initial prices.
Unfortunately, Walrasian tâtonnement can cycle without converging; Scarf
(1960) provides examples where both local and global stability failed.
The “convergence process” research agenda is now considered dead.
Gross Substitutes
We now consider a particular class of economies in which we can get
a¢ rmative answers to the uniquencess and stability questions.
Two goods are gross substitutes if an increase in the price of one good
increases the demand for the other.
A demand function satis…es the gross substitutes property if an increase in the
price of good k increases the demand for every other good l.
De…nition
A Walrasian demand function x (p) satis…es the gross substitutes property if,
whenever p and p 0 are such that pk0 > pk and pl0 = pl for all l 6= k, then
xl (p 0 ) > xl (p) for all l 6= k.
Remark
If Walrasian demand satisfyies gross substitutes for all consumers, then individual
and aggregate excess demand functions also satisfy the gross substitutes property.
Proof.
Homework
Gross Substitutes and Uniqueness
Theorem
If the aggregate excess demand function satis…es gross substitutes, the economy
has at most one Walrasian equilibrium
Proof.
We need to show that z(p) = 0 has at most one (normalized) solution.
By contradiction: p and p 0 are not linearly dependent and z(p) = z(p 0 ) = 0.
By homogeneity of degree zero, normalize the price vectors so that pl pl0 for
all l = 1; ::; L and pk = pk0 for some k.a
Move from p 0 to p in n 1 steps, increasing the prices of each of goods l 6= k
in turn.
At each step where a dimension of price increases strictly (and there must be
at least one such step), the aggregate demand for good k must strictly
increase, so that zk (p) > zk (p 0 ) = 0, yielding a contradiction.
a Set
p~l =
~l p l
l
p l0
,
pl
let ~l
0
l pl = pl .
arg maxl
l;
and then de…ne p~l
~l p.
Then p~~l = p~0 and
l
Gross Substitutes and Stability
When the gross substitutes property holds, Walrasian tatonnement converges
to the unique equilibrium.
Proposition
Suppose that the aggregate excess demand function z(p) satis…es gross substitutes
and that z(p ) = 0. Then for any p not collinear with p , p z(p) > 0.
Proof.
Homework (assume L = 2 for simplicity).
This Lemma is related to the weak axiom of revealed preference. Gross
substitutes implies that the weak axiom holds if one compares p , the unique
equilibrium price vector, to any other price vector p.
Theorem
Suppose that the aggregate excess demand function z(p) satis…es gross substitutes,
and that p is the Walrasian equilibrium price vector. Then the tatonnement
dp
adjustment process dy
= z(p(t)), with > 0, converges to p as t ! 1 for any
initial condition p(0).
Proof.
We show that the distance between p(t) and p (denoted D(p)) decreases
monotonically with time.
L
1X
2
(pl pl )
D(p) =
2
l =1
Then
dD(p(t))
dt
=
L
X
(pl
l =1
=
|{z}
dpl
pl )
=
dt
p
L
X
(pl
pl ) zl (p(t))
l =1
z(p)
0
by Walras’Law
The last inequality is strict unless p is proportional to p .
Now, because D(p(t)) is decreasing monotonically over time, it must
converge, either to zero or to some positive number.
In the former case, p(t) ! p , and we are done.
In the latter case, p(t) 9 p but dD(p(t))=dt ! 0.
This can happen only if p(t) becomes nearly proportional to p as t ! 1.
But then the relative prices of p(t) converge to those of p as t ! 1.
Gross Substitutes and Comparative Statics
A change that increases good k’s excess demand increases its equilibrium price.
Two Goods Comparative Statics
Set p2 = 1, and assume good 1 is a normal good for all consumers.
Increase the endowment of good 2. For any p1 , this increases aggregate
demand for good 1 (why?), and threfore increases aggregate excess demand.
The original curve is z1 (p1 ; 1; L), and the new one is z1 (p1 ; 1; H).
Because z1 (p1 ; 1; L) is continuous and crosses zero only once (remember that
equilibrium is unique), the new equilibrium must have a higher price for good 1.
This example is easy to formalize, and generalizes to more goods.
General Equilibrium Under Uncertainty
Main Objective
Extend general equilibirum theory to account for uncertainty.
We start by modeling uncertainty like we did in the subjective expected utility
world.
There are only two periods, today and tomorrow.
Consumption occurs after uncertainty is realized, after everyone knows the
realized state.
There is no consumption today, but this is only to keep notation simpler.
Notation
There are S possible states; a generic state is denoted s 2 f1; ::; Sg.
In state s = 1; ::; S, the agent consumes a vector of commodities
xs = (x1s ; ::; xLs ) 2 RL+
Now, however, the agent does not know which state she will be in, so she
makes plans for each of them. These plans are
x = (x1 ; ::; xS ) 2 RLS
+
A plan is a ‘state-contingent’good.
Plans are traded in markets, and preferences are over plans.
General Equilibrium Under Uncertainty
De…nitions
For each commodity l = 1; ::; L, and each state s = 1; ::; S, one unit of
state-contingent commodity ls is a title to receive one unit of good l if and
only if state s occurs.
A state-contingent commodity vector
x = (x1 ; :::; xs ; ::; xS ) = (x11 ; ::; xl 1 ; ::; xL1 ; ::; x1s ; ::; xLs ; ::; x1S ; ::; xLS ) 2 RLS
is a title to receive the commodity vector xs = (x1s ; ::; xLs ) 2 RL+ if and only if
state s occurs.
These are binding promises to obtain certain quantities of each of the goods in
a given state:
state-contingent commodities are ‘contracts’.
A commodity vector is also a random variable: it assigns a vector in RL to
each state of the world.
All State Contingent Commodities Are Traded
We assume that there is a market for each of these state contingent
commodities.
Preferences Over State Contingent Goods
Expected Utility Preferences Over State Contingent Commodities
RLS contains state-contingent commodities.
Agent i’s consumption set Xi
Agent i’s preferences %i over Xi have an expected utility representation.
For any xi and yi in Xi ,
xi %i yi
if and only if
S
X
si usi
(x1si ; ::; xLsi )
s =1
where
si
S
X
si usi
(y1si ; ::; yLsi )
s =1
is the probability consumer i assigns to state s.
Sometimes easier to write
xs ; ys 2 RL+ .
PS
s =1
si usi
(xsi )
Preferences can be state-dependent.
PS
s =1
si usi
(ysi ) where
These are ex-ante preferences; they describe what i thinks at time 0 about
what she might consume in the future.
i makes trade-o¤s between random variables; one more unit of good l in state s
versus one more unit of good l 0 in state s 0 ;
these trade-o¤s re‡ect the likelihood of di¤erent states and the shape of the
utility function in each state (combined in marginal rate of substitutions).
Utility functions and probability distributions di¤er across consumers.
The latter are ‘subjective’probabilities.
Initial Endowment
De…nition
The endowment of agent i is a state-contingent commodity bundle
! i = (! 1i ; :::; ! si ; ::; ! Si ) = (! 11i ; ::; ! L1i ; ::; ! 1Si ; ::; ! LSi ) 2 RLS
When state s occurs, agent i has endowment (! 1si ; ::; ! Lsi ) 2 RL .
A consumer can be rich in some states and poor in other states.
If ! ti > ! ti then the consumer i is richer in state t.
If ! ti = ! si for any t and s then the consumer wealth is constant.
We can also model the economy being richer or poorer by looking at the
aggregate endowment in each state.
If
P
i
! si >
P
P
i
! ti then the economy is richer in state s.
If i ! ti is constant for all t, we say there is no aggregate uncertainty since the
economy’s wealth is constant; the wealth distribution across consumers, however,
can change.
Production Plans for State Contingent Goods
Production plans are also state contingent.
De…nition
The production possibility set is represented by a set Yj
RLS for each …rm j.
Firms ‘produce’state-contingent commodities.
The …rm sells future production (at today’s prices) and makes a pro…t that is
distributed to consumers.
Shares in the …rms are not state contingent, but this is just to keep notation
simple.
Markets and Prices in an Arrow-Debreu Economy
In an Arrow-Debreu model each state-contingent commodity is traded.
Arrow-Debreu Markets
There are L S markets open at date 0 (when nobody knows the realized
state).
In these markets, a trade speci…es various amounts of goods to be delivered in
various states:
there is a price for each good in each state
p ls is the price agreed upon now for one unit of good l to be delivered in state s .
Agents trade promises to receive, or deliver, amounts of good l if and when
state s occurs.
In …nance lingo, these are “forward” markets where “futures” are traded.
Trades of these ‘contracts’are agreed upon now by all parties involved.
Even though decisions are made now, the ‘physical’exchange of goods only
happens after uncertainty is resolved.
Contracts are traded now, actual goods do not exist until the state of the
world is realized.
Budget Constraint
A consumer’s income is given by the current value of (i) the state-contingent
commodities she owns and (ii) her share of state-contingent pro…ts:
She sells future endowment.
She receives a share of pro…ts generated by …rms’sales of future production.
A consumer’s expenditure is the current expenditure in state-contingent
commodities.
Budget Constraint
Bi (p; ! i ) =
8
<
:
xi 2 X i :
S X
L
X
s =1 l =1
pls xlsi
S X
L
X
pls ! lsi +
s =1 l =1
The consumer cannot buy more than her income.
J
X
j =1
ji
S X
L
X
s =1 l =1
pls ylsj
9
=
;
Although physical exchanges are contingent on the realized state, payments
are not: they are made today.
Promises must be kept: Individuals cannot go bankrupt.
This system of payments and deliveries is well de…ned only if each consumer
knows which state occurred.
We call this symmetric information. It prevents situations where someone says
“This is state s” and someone else says “No, this is state t” since when that
Budget Constraint Notation
Budget Constraint Notation
Consumer i cannot spend more than she has:
8
overall expenditure
>
}|
{ z overall}|revenue {
z
>
>
>
expenditure
revenue from
>
>
in s
endowment in s
>
>
z }| {
z }| {
>
>
>
S X
L
S
L
J
<
X
X
X
X
Bi (p; ! i ) =
pls xlsi
pls ! lsi +
xi 2 X i :
>
>
s =1 l =1
s =1 l =1
j =1
>
>
>
>
>
>
>
>
>
:
=
=
where ps 2
RL+
8
<
:
8
<
:
xi 2 Xi :
S
X
ps xsi
s =1
xi 2 Xi : p xi
S
X
ps ! si +
s =1
p !i +
for each s, and p 2
J
X
j =1
RLS
+ .
J
X
j =1
9
=
ji (p yj )
;
ji
S
X
s =1
9
{>
>
>
j pro…ts >
>
in state s >
>
z }| {>
>
>
>
S X
L
=
X
pls ylsj
ji
>
>
s =1 l =1
>
>
>
>
>
>
>
>
>
;
z
ps ysj
j pro…ts
}|
9
=
;
Arrow-Debreu Economy
The Economy
An Arrow-Debreu economy is described by a set S of states of the world and:
for each agent i: a consumption set Xi RLS , a preference relation %i on Xi ,
and an endowment vector ! i 2 RLS
+ ;
for each …rm j: a production possibility set Yj RLS ; and
shares ji 0 denoting who owns what fraction of each …rm.
Individual characteristics (endowment, preference, and production) depend on
the realized state of the world.
Remark
Relative to the original general equilibirum model, this is nothing but a change of
labels.
There are L physical commodities, but SL commodities are traded. This model,
then, corresponds to the original GE setup with L replaced by SL.
For this to work one assumes that there exist competitive markets for all state
contingent commodities. We say this economy has complete markets.
Arrow-Debreu Equilibrium
Usual de…nition of competitive equilibrium: at the equilibirum prices individuals’
make optimal decisions and markets clear.
De…nition
An allocation (x ; y ) and a system of prices for the state contingent commodities
p constitute an Arrow-Debreu equilibrium if
1
Firm maximize pro…ts: for every j,:
p yj
p yj
for all yj 2 Yj
Consumer maximize preferences: 8
for every i
<
x i %i x i
for all xi 2 Bi (p ; ! i ) = xi 2 Xi : p
:
2
3
markets clear:
I
X
i =1
xi =
I
X
i =1
!i +
xi
p
!i +
J
X
j =1
J
X
yj
j =1
This is a special case of the general equilibrium model.
ji
p
yj
9
=
;
Problem Set 12
Due Monday 17 November
1
Consider an exchange economy with N consumers and L goods in which each consumer has a utility function ui : RL+ ! R that is continuous, strictly quasi-concave and strongly monotone, and an initial
endowment ! i which contains a positive amount of each good. Each consumer must pay a tax on the value of his or her gross consumption of each good, and these tax rates may vary across goods and
consumers. Thus if consumer i buys xij units of good j when the price for good j is pj , the tax she pays is tij pj xij . The tax rates ftij g are set exogenously by the tax authority; each consumer also receives a
lump-sum rebate Ri from the tax authority. In equilibrium, the total tax receipts must be rebated equally across the consumers in a lump-sum fashion.
An equilibrium
P with taxes is a price vector p , an allocation (x1 ; : : : ; xm ), and rebates (R1 ; : : : ; RN ) such that each consumer i is maximizing utility over the budget set at the bundle xi , all markets clear,
and Ri = N1 i ;j tij pj xij for each i = 1; : : : ; N.
Write down the budget set for a typical consumer in this economy.
2
Illustrate an equilibrium with taxes in an Edgeworth Box. Does the First Welfare Theorem hold for equilibria with taxes?
3
Show that an equilibrium with taxes exists. (This is hard; think about what is an equilibrium and what properties do consumers’demand functions satisfy in this economy?)
Now consider the case in which taxes are applied to net consumption (the di¤erence between consumption and initial endowment). Thus, consumer i net trades are xij ! ij units of good j and when the price for good j is
pj the tax she pays is tij pj (xij ! ij ).
4
Repeat the …rst two questions above for this case, and think about what can change in the third if anything.
1
2
8
1
2
3
4
2
Prove that if the two consumers have the same probabilities for the two states ( 11 = 12 ), then at an interior Arrow-Debreu equilibrium consumer 2 is fully insured (her consumption is the same in the two states).
Draw the corresponding Edgeworth box.
Prove that if the two consumers have di¤erent probabilities for the two states ( 11 6= 12 ), then at an interior Arrow-Debreu equilibrium consumer 2 is not fully insured. Draw the corresponding Edgeworth box. In
which state will she consume more (depending on the di¤erence in probabilities)? Also prove that consumer 1 does not gain from trading in this case.
Consider an economy with two consumers, a and b. There are two periods, today and tomorrow, and two possible states of nature tomorrow. Today there is one commodity, time (labor/leisure), denoted `.
Each consumer is endowed with one unit of ` today, and nothing tomorrow. Tomorrow there is a single consumption good (food) in each state. Each consumer’s utility function has the form
Ui (`i ; xi 1 ; xi 2 ) = 12 ln xi 1 + 12 ln xi 2 where `i denotes i’s consumption of leisure today, and xis denotes i’s consumption of food tomorrow if state s occurs, for s = 1; 2. There are three …rms. Consumer a
owns …rm 1, while consumer b owns …rms 2 and 3. The production sets of the …rms are:
Y1 = f( `; y1 ; y2 ) : y1 3`; y2 0; where ` 0g Y2 = f( `; y1 ; y2 ) : y1 0; y2 3`; where ` 0g Y3 = f( `; y1 ; y2 ) : y1 `; y2 `; where ` 0g
where ` denotes the amount of labor input used today, and ys denotes the amount of food produced tomorrow if state s occurs, for s = 1; 2. In particular, notice that …rm 1 produces only in state 1, while
…rm 2 produces only in state 2. Normalize the price of labor/leisure to be 1.
1
2
6
Find Walrasian demands of both consumers.
Let p2 = 1, write the aggregate excess demand function for the two goods, and use the aggreate excess demand of good 1 to …nd the three equilibria of this economy.
Prove that if in an exchange economy Walrasian demand satisfyies gross substitutes for all consumers, then individual and aggregate excess demand functions also satisfy gross substitutes.
Consider an Edgeworth box economy with a single consumption good, two states, and two consumers. Consumer i utility function has the expected utility form: Ui (x1i ; x2i ) = 1i ui (x1i ) + 2i ui (x2i ) where
xsi represents the amount of state s contingent consumption for individual i and si represents the probability individual i assigns to state s. Assume the two individuals each get half of the initial
endowment in each state; that is ! i = (! 1i ; ! 2i ) = 12 ! 1 ; 21 ! 2 for i = 1; 2. Finally, assume that individual 1 is risk-neutral (u1 ( ) is linear) while consumer 2 is risk-averse (u2 ( ) is strictly concave).
1
5
8
Consider a two goods two consumers economy where the utility functions are u1 (x11 ; x21 ) = x11 81 (x21 ) and u2 (x12 ; x22 ) = 18 (x12 ) + x22 , and the individual endowments are ! 1 = (2; r ) and
! 2 = (r ; 2), r = 28=9 21=9 . Find Walrasian demands of both consumers. Let p2 = 1, write the aggregate excess demand curve for the two goods, and …nd the three equilibriua of this economy.
Find the competitive equilibria in this economy.
Suppose …rm 3’s production set is instead Y~3 = f( `; y1 ; y2 ) : y1
3`; y2
3`; where `
0g . How does the answer to (a) change?
PS
Consider an economy with a single consumption good and S possible states of the world. There are I consumers, each with expected utility function Ui (xi ) = s =1 is ui (xis ) where is denotes i’s subjective
probability of state s, xis denotes i’s contingent consumption of the good in state s , and ui : R+ ! R is consumer i’s vNM index. Assume that ui is strictly increasing, continuous, di¤erentiable, and strictly
concave for each i. Consumer i’s initial endowment is ! i , and the aggregate endowment is !
0. Suppose that consumers have a common prior, where is = js > 0 for each i; j and for each state s.
1
2
3
Suppose there is no aggregate uncertainty, so that ! si = ! s 0 i for all states s; s 0 . (Thus the only uncertainty in this economy is idiosyncratic uncertainty involving individual endowments.) Show that every Pareto
optimal allocation involves full insurance. That is, show that in any Pareto optimal allocation (x1 ; : : : ; xm ), xis = xis 0 for all states s; s 0 and for each consumer i .
Does the result above hold if the consumers do not agree on the probabilities of the states? Explain.
Now go back to the common prior case, but assume there may be aggregate as well as idiosyncratic uncertainty, so it is possible that ! si 6= ! s 0 i for some states s; s 0 . Show that in any interior equilibrium allocation
(x1 ; : : : ; xm ), the variation in individual consumption across states is perfectly correlated with the variation in aggregate endowment across the states; that is, show that for each i : (xis xis 0 )(! s ! s 0 ) 0 for all s; s 0 .