Walrasian Equilibrium G2

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Walrasian Equilibrium

Having developed a model of the economic environment and characterizedecient allocations in that environment, we turn next to the question of market-mediated mechanisms for allocating resources in the economy. We will focus inparticular on a competitive, private-ownership economy in which resources areallocated via voluntary exchange among agents, each of whom is a negligiblepart of the economy, in response to commonly observed market prices.

Denition: A price systemfor the economy is a vector p 2 R`+ (so that pk

is the price of good k .)

Given a price vector, the value of a consumption plan is given by p x =

P`i=1pixi: Similarly, the value of a production plany is justp y=PJi=1piyi:Our assumption of private ownership means that all rms are owned by

agents in the economy. To keep our analysis simple, we will assume that theshares of the rms owned by each agent are xed, which implies that the nancialside of the economy is trivial. We letji denote the share of rm j owned by

agent i: Hence, we have ji 2 [0; 1] andPMi=1 ji = 1: With this assumption,

the economy is now fully described by

=

(Xi; i; !i)

Mi=1 ; (Yj)

Jj=1 ;

ji

j=1;:::;Ji=1;:::;M

:

Maximizing Behavior

Producers in the model are assumed to act to maximize prot j(p) =maxyj2Yj

p yj : The rationale for this assumption is the interests of the owners of

the rm, who are the residual claimants to the rms prots. As long as someowner has non-satiated preferences, they will prefer more income to less, andwill wish for the rm to act to maximize prots. We let j(p) be the set ofsolutions to rm j s optimization, so that

j(p) = arg maxyj2Yj

p yj :

j(p)is rm j s supply correspondence.

Consumers in the model act to maximize utility subject to budget con-straints. The income or wealth of consumeri for given prices p is

wi(p) = p !i+JXj=1

jij(p) :

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Agent is budget constraint is then given by

i(p) = fx 2 Xi j p x wi(p)g :

Agent is demand is then

xi 2 i(p) = fx 2 i(p) j 8x0 2 i(p) , x i x

0g :

Denition: AWalrasianor competitive equilibrium for consists of(x;y; p)satisfying

1. (x;y) 2 F()

2. p is a price system

3. For allj; yj 2 j(p)

4. For alli; x

i 2 i(p

) :

The set of Walrasian equilibrium allocations for is denoted W E() :

To analyze the Walrasian equilibrium, it will be convenient to let

! =Xi

!i (aggregate endowment)

(p) =Xi

i(p) (aggregate demand)

(p) =Xi

i(p) (aggregate supply)

z (p) = (p) (p) ! (excess demand).

Properties of the Excess Demand Correspondence

1. z (p) = z (tp)for all t >0 (z is homogeneous of degree zero).

(Prove this as an exercise.)

The degree zero homogeneity of excess demand means that only relativeprices matter in determining equilibrium. Hence, since the absolute pricelevel is indeterminate, we are justied in normalizing prices. There arevarious ways of doing this. One way is to select anumeraire good (good1, for example) and simply declare that the price of this good is 1. Theprice of good k 6= 1 is then given in terms of how many units of good 1exchange for one unit of goodk. A second way of normalizing prices is torequire that they have norm 1. A third way (which we will adopt here)is to require that the prices sum to 1. This normalization puts the priceson the unit simplex

=

x 2 R`+ j x= 1

whereT = [1; :::; 1] :

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2. The economysatises Walras Law if, for every p 2 ; p z (p) = 0:

Proposition 1: If for all i= 1;:::;M; i is locally non-satiated, then satises Walras Law.

Proof: Under the non-satiation assumption, we know that p i(p) =wi(p) since if it were not, it would be possible for agent i to purchase astrictly positive amount of some good yielding higher utility, contradict-ing the assumption that i(p) was the most preferred aordable bundle.Next,

p z (p) = p

24 MXi=1

(i(p) !i) JXj=1

j(p)

35=M

Xi=1p i(p) M

Xi=1p !i J

Xj=1p j(p)=MXi=1

p i(p) MXi=1

p !i JXj=1

MXi=1

jij(p)

=MXi=1

p i(p) MXi=1

24p !i+ JXj=1

jij(p)

35=MXi=1

[p i(p) wi(p)] = 0:

Notice that we have not said anything so far about the existence of equi-librium: Walras law holds (under the non-satiation assumption) as an

identity for all prices, by virtue of the budget constraints being satisedwith equality for all consumers.

Existence of Equilibrium

To show the existence of equilibrium in the general Arrow-Debreu model,we will need the following technical result (which we will apply without proof).

Kakutanis Fixed-Point Theorem: Let : X X be a correspondencefrom the compact, convex metric space X to itself. If is non-empty-valued,compact- and convex-valued, and upper-hemi-continuous, the there exists a pointx 2 (x) :

To apply this result, we will construct a correspondence with the requiredproperties to apply the theorem, and the resulting xed point will then con-stitute an equilibrium. To construct the correspondence, we note rst thatthe excess demand correspondence z (p) is upper-hemi-continuous. This fol-lows from the maximum theorem (which implies that individual excess demand

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correspondences are uhc) and standard aggregation results for uhc correspon-dences. We now also add the assumption that for alli, i is convex. With this

assumption,z (p)will also be convex-valued. Next, dene the correspondence

B (x) =

p 2 j p x= max

q2q x for x 2 R`:

Problem: To apply the Kakutani theorem, we need the excess demand cor-

respondence to be compact-valued. If some consumer is globally non-satiated,then z (p) will not be compact valued for any p 2 @: In this case, however,we know that such a price cant be an equilibrium. If, on the other hand,preferences are such that somepk = 0 is consistent with equilibrium, then z willbe compact-valued at such prices. Hence, we make the following modication.

For K R` with 0 2 int K; and K compact and convex, dene

zK(p) = z (p)\K: IfKis suciently large, then, at any equilibriumpricep; z (p) R`_ will be such thatz (p

) K:

Theorem 1: There exists p 2 such that z (p) \ R`

6=:Proof: The proof has 3 steps.

1. zK : Kis uhc, compact-, convex-valued, and satises Walras Law:

p zK(p) 0

for all p 2 : Also, and Kare compact and convex.

2. With B : K dened as above, B is uhc (by the maximum theo-

rem), compact- and convex-valued. Compact-valuedness follows from thecompactness of; while convex-valuedness follows from the fact that if

p1; p2 2 B (x) ;thenp1 x= p2 xsop x= [p1+ (1 )p2]x= maxq2

qx:

3. Dene a new correspondence' : K K by

' (p; z) = [B (z) ; zK(p)] :

Clearly, the correspondence ' has the properties required to apply theKakutani theorem. Hence, we know there exists (p; z) 2 ' (p; z) :Let us show that this is in fact an equilibrium price and net trade.

Sincep 2 B (z) ; p maximizes the value ofz; so that

p z p z 0for all p 2

(where the last inequality follows from Walras Law). Lettingp = ei (theith unit vector) for i = 1;:::;`; we nd that zi 0 for all i; so z

0:Sincez 2 z (p) ; it follows that z is an equilibrium net trade.

Not nally that z 2 R`

cannot be unbounded without violating bound-edness from below of consumption sets.

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The example is a simple pure exchange economy in which there are 3 agentswho trade 3 goods. We specify person 1s utility function as

u1(x1; x2; x3) =

b3

x21+

1

x22

for b >3, and let !1 = [1; 0; 0]. The preferences and endowments for agents 2and 3 are then obtained by cyclically permuting the indices on the goods andprices. Thus, for example, agent 2 will have the same utility function as agent1, except that x1 would be replaced by x2, x2 byx3, and x3 byx1. Similarlyagent 3 would have the same utility function, with x3 replacingx1,x1 replacingx2, and x2 replacing x3. Agent 2s endowment is then!2 = [0; 1; 0], and agent3s is !2 = [0; 0; 1].

With these specications we calculate demand functions in the usual way:equating agent 1s MRS to the ratio or the prices yields the rst-order condition

bx2x1

3=

p1p2

:

Solving this for x2 in terms ofx1;substituting into the budget constraint andsolving the resulting equation for x1 then yields the demand function

x1 = bp

2=31

bp2=31 +p

2=32

:

Back-substituting into the equation for x2 then yields

x2= p1

bp2=31 p

1=32 +p2

:

One obtains the demand functions for the other agents similarly.Now, make a change of variable i = p

1=3i and substitute into the demand

functions you have obtained. By Walras Law, it suce to consider only thedemands (and excess demands) for goods 1 and 2. Making this substitution,we obtain excess demand functions

z1 = b21b21+

22

+ 33

b231+ 31

1

z2 = 31

b212+ 32

+ b22

b22+ 23

1:

From this pair of equations, it is apparent that 1 = 2 = 3 = 1 is anequilibrium since bb+1+

1b+1 1 = 0.

To show that this equilibrium is unique, consider the case ofb = 3. For thiscase it is convenient to renormalize prices so that 1 = 1: Market clearing thenrequires that

22

1 + 323 33

+ 333 = 0

322+ 23 3

232

23

32 = 0:

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Evaluating this expression at 1 = 2 = 1 then yields

Dz = " b3(b+1)2 2b(b+1)2b+3(b+1)2

b3(b+1)2

# :The characteristic polynomial for this matrix is

ch () =

"

b 3

(b + 1)2

#2+

2b (b + 3)

(b + 1)4

which reduces to

ch () = 2 2(b 3)

(b + 1)2 + 3

b2 + 3

(b + 1)4

:

Applying the quadratic formula to this yields roots

= 1

(b + 1)2

h(b 3)

p2 (b2 + 3)

i:

Clearly, the roots are complex conjugates with positive real parts as longas b > 3, so the Walras tatonnement is completely unstable at the uniquecompetitive equilibrium for this economy.

Geometrically, the dynamic system corresponding to the Walras taton-nement for this economy spirals away from the competitive equilibrium, andapproaches a limit cycle, as illustrated in Figure 2.

Research over the past decade into the problem of tatonnement has shownthat it is always possible to construct a tatonnement procedure i.e. to specifya dierential equation of the form

_p= H[z (p)]

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which converges to the competitive equilibrium for any given economy . Un-fortunately, the specication of the appropriate function H requires knowledge

of all agents preferences, at least up to the second derivatives of the utilityfunctions. One such approach is based on a global version of Newtons Methodfor nding the zeros of a function. This approach, rst derived by Smale, isbased on the dierential equation

Dpz (p) _p= z (p)

which can be shown to converge to some competitive equilibrium for almostany starting value ofp in the unit simplex. Note, however, that because theJacobian matrixDpz (p)will generally be non-diagonal, this procedure not onlyrequires that we have information about the derivatives of the excess demand, italso requires some mechanism for coordinating the rates at which prices adjustin any given market with the rates of price adjustment in other markets. It

is dicult to think of obvious economic institutions or mechanisms capable ofimplementing this kind of tatonnement procedure. Thus, while economistsgenerally believe that the concept of a competitive equilibrium does manifestitself in the real world of commerce, we dont know how it gets implemented.

One alternative to the idea of tatonnement processes is to postulate thateconomic equilibria, unlike those of physical systems, must be learned by theagents in the system. In physical systems, equilibria occur as natural restpoints in dynamic processes based on xed laws of motion of the system. Eco-nomic equilibrium, on the other hand, involves not only a physical rest pointcondition (market-clearing), but also a psychological condition (satisfaction ofneeds or wants) interacting with an articial construct (prices) derived fromthe psychological condition. Indeed, it is easy to nd market-clearing allo-cations: any bully can do it very eectively. It is less easy to nd Paretooptimal allocations, although well-dened systems of property rights togetherwith enforcement mechanisms for ensuring that contracts are honored makes im-plementing Pareto improving trades possible, which can lead under very simplesearch procedures to optimal allocations. But, as we already know, not everyPareto optimum is a competitive equilibrium. Thus, it seems likely that if theconcept of the competitive equilibrium is to be useful in economic analysis, weneed a mechanism for explaining how agents in the economy might learn whatthe competitive prices are.

Recent work by Gode and Sunder took a very dierent approach to theproblem of implementing the competitive equilibrium. In their approach, they

analyzed a partial equilibrium situation involving a single market with manyinteracting agents, based on the standard supply and demand experiments pi-oneered by Vernon Smith and Charlie Plott. In the experimental version ofthis market, one group of agents play the role of buyers, the other the role ofsellers. Buyers can purchase one unit of the good, and this one unit is wortha given reservation price to them. Hence, if they buy the good for a price at

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or below their reservation value, they earn a prot. Sellers can each sell up toone unit of the good. If they sell their unit, they incur a givenproduction cost.

Hence, if they sell at a price at or above their cost, they make a prot. It iswell established in the literature on experimental markets by now that humantraders in this environment eventually end up trading the competitive amountof the good at prices the closely approximate the competitive equilibrium (i.e.,transactions take place according to the price and aggregate quantity speciedby the intersection of the supply and demand schedules for the experiment). Wenote, however, that agents in this experiment generally require several roundsof trading before they learn what the relevant equilibrium prices are, so that thedata generated in such experiments exhibits a convergence of prices and quanti-ties to the predicated competitive equilibrium prices and quantities, rather thanan abrupt and direct implementation of the equilibrium.

Gode and Sunder asked whether this process of nding the right prices and

allocations was one requiring very sophisticated learning, or whether it couldbe implemented with zero intelligence search procedures. They proceeded toreplicate the basic experimental setup using computerized robots. The robottraders in their model generated simple random bids (if they were buyers) oroers (if they were sellers) with the only restriction on behavior being that nobid or oer should make an agent worse o. Thus, buyers were restricted tobid below their oer price, while sellers were restricted to bid above their costs.In simulations of the model, Gode and Sunder found that while prices dontconverge to the competitive equilibrium prices (as they do with human subjects),the infra-marginal prices (i.e. the prices of the last observed transactions) alwaysoccur at or near the CE price, while the eciency of the market is in excessof 90% of the maximum (which occurs when the quantity of the good tradedis the CE quantity). These results tell us that the double auction mechanismof the classic supply and demand experiment will implement the competitiveequilibrium under very mild conditions on agents behavior.

The zero intelligence trading result does not, however, answer the question ofwhether the competitive paradigm can be implemented easily in environmentswhere many agents trade many goods.

Follow-on work to the Gode and Sunder work by Gode, Spear and Sundershowed that, at least in the context of a 2 agent, 2 good exchange economy,simple random search easily nds Pareto optimal equilibria. (Notes on thisresearch can be found on the course web-site in PDF format.) The randomsearch process does not, however, nd the competitive equilibrium. The reasonfor this is self-evident. The random search process generates a uniform set of

random trajectories from the initial endowment to the contract curve. Theending allocations are, therefore, uniformly distributed on the contract curveabout the average trajectory generated by the search procedure.

More recent research by Gode, Spear and Sunder has focused on the ques-tion of how much additional intelligence is required of agents in the zero-

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intelligence exchange environment in order to nd the competitive equilibrium.The answer turns on the issue of whether agents can price the optimal allocation

they nd, in the sense of learning the (common) normalized utility gradient atthe optimal allocation.

An -intelligent Implementation of Competitive Equilibrium

The economic model is one in which a nite number of agents trade a nitenumber of goods and services in a pure exchange environment. By way ofnotation, we index agents as i = 1; : : : ;M < 1 and goods as j = 1;:::;` ; then we repeat our construction ofthe near Pareto optimal allocation, but with the added set of constraints that

pt

xt+1i !i

> ti+

where is small and positive, for any i such that ti < 0: These constraintsguarantee that any agent who was providing subsidies at stagetwill be providingsmaller subsidies at stage t + 1:

Note that in passing from one stage to the next, we must always carry alongthe subsidization constraints, even if we move to a new allocation in which anagent who was providing a subsidy in the previous stage is receiving a subsidyin the current stage. If we forget the past subsidization constraint, then thealgorithm could go back to an allocation in which this agent was again makinglosses, possibly larger than in the previous stage. Hence, at each stage t; thedata required for each agent is [bxti; i ; p], where(i ; p)were the price and lossin the last stage at which agenti incurred a loss. We illustrate this in Figure3, for the 2-by-2 economy. Here, at some earlier allocation marked A, agent 1incurred a loss, so subsequent generations of optimal allocations must lie abovethe upper line parallel to the tangent line at A. Similarly, at a dierent stage,agent B incurred a loss, so subsequent allocations must lie below the lower lineparallel to the tangent line at B. This restricts the set of potential reallocationsthat the ZI mechanism can draw from to those in the grey shaded area. In theabsence of the loss constraints, the set of potential reallocations would be thefull lens-shaped region between the pair of indierence curves containing theendowment point.

Since only the CE allocation has no agent making a loss, the manner in whichwe increment the loss constraints implies that the set of potential allocationsfrom the ZI algorithm selects must decrease in size from one stage to the next,which implies in turn that the ZI algorithm must converge to the -competitiveequilibrium.

This algorithm improves signicantly on the tatonnement results, to the ex-tent that it requires that agents only be able to price Pareto optimal allocations,a process which requires information only about the common normalized utilitygradient at the optimal point, rather than information about both rst- andsecond-derivatives of all agents utility functions. It also provides a more re-alistic foundation for actually implementing the competitive equilibrium than

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does the ctitious price-adjusting auctioneer, based on standard bargaining the-ory. In this framework, once agents learn that they are in fact subsidizing other

agents, they may use the threat of refusing to trade with agents who are benet-ting from this subsidization to extract concessions (in the form of trades whichreduce the degree of subsidization). As we will see in the following section,this threat is made quite credible by the fact that in a large economy, it isalways possible for subsets of agents (trading among themselves) to Pareto im-prove on any non-CE allocation. Finally, the implication that the competitiveequilibrium must be learned also explains the recurrence of the idea throughouteconomic history of competitive prices as normal prices around which actualmarket prices uctuate.

Welfare Properties of Walrasian Equilibrium

While our results on learning and competitive equilibrium suggest that mar-kets may not attain equilibrium quite as quickly as economists had typicallyassumed, they do provide robust support for the notion of the competitive equi-librium as an economic attractor in the sense that realistic trading processeswill tend toward the competitive equilibrium given sucient time for learningto take place. Thus, we are justied in asking what properties the competitiveequilibrium exhibits, and the most important of these are its welfare properties.

Denition: The set ofindividually rationalfeasible allocations inis denedby

IR () = f(x;y) 2 F() j 8i; xii ! ig :

Proposition 2: If satises0 2 Yjforj = 1;:::;J, thenW E() IR () :

Proof: Exercise.

Theorem 2: (First Welfare Theorem) If satises i is locally non-satiated for i= 1;:::;M then W E() P O () :

Proof: Let (x;y) 2 W E() and suppose (x;y) =2 P O () : Thenthere exists an allocation (~x; ~y) 2 F()which Pareto improves on (x;y) : Inparticular, we have

8i ~xiix

i

with strict preference for some i. Let agents be indexed such that ~x1 1 x

1:Then, by local non-satiation, it must be that p ~x1 > p

x1, since otherwisewe would have p ~x1 p

x1 = w1(p), contradicting the assumption that

x1 2 1(p).

Next, for any i >1, it must be that p ~xi p xi : If this is not the case

for some i, then p ~xi < p xi ; and we can choose > 0 such that for everyx 2 B(~xi) ; p x < p xi : By local non-satiation, there exists x 2 B(~xi)such that x i x

i which again contradicts the assumption that x

i 2 i(p) :

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Combining these results, we have

MXi=1

p ~xi >MXi=1

p xi :

Now, by prot maximization, we know that for all j, p yj p ~yj which

implies that

pMXi=1

xi =p

24 MXi=1

!i+JXj=1

yj

35 p24 MXi=1

!i+JXj=1

~yj

35= p MXi=1

~xi:

But this implies thatM

Xi=1p xi

M

Xi=1p ~xi

which is a contradiction.

Note that if we combine Proposition 2 and the First Welfare Theorem, wehave W E() IR () \ P O () :

You should be able to construct a counter-example in a pure exchangeenvironment to the claim that the local non-satiation assumption is un-necessary in the proof above.

Denition: A quasi-equilibrium with transfer payments is an (M+ J+ 1)-

tuple[(x

;y

) ; p

] satisfying

1. (x;y) 2 F()

2. p is a price system

3. 8j = 1;:::;J yj 2 j(p)

4. 8i= 1;:::;M x i x

i )p x p xi :

The transfer payment to consumer i is then Ti(p) = p xi wi(p) :

Theorem 3: (Second Welfare Theorem) Let satisfy

1. For some i= 1;:::;M i is locally non-satiated

2. For all i, fx 2 Xi j x i xgis convex for all x 2 Xi

3. For all j; Yj is convex

4. Y

R`+

:

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If (x;y)is a Pareto optimal allocation in ;then there exists a price systemp such that [(x;y) ; p] is a quasi-equilibrium with transfer payments.

Proof: Without loss of generality, we assume that assumption 1 is satisedfor agent 1. Let(x;y) 2 P O () : Dene

G= fx j x 1 x

1g +MXi=2

fxj x i x

i g JXj=1

Yj :

1. (a) Claim: ! =2 G: If!2 G then there exists (~x; ~y) such that forallj = 1;:::;J ~yj 2 Yj ; and

!=MXi=1

!i =MXi=1

~xi JXj=1

~yj

i.e. (~x; ~y) 2 F() and ~x1 1 x

1 and ~xi i x

i for i = 2;:::;M.

But this contradicts the assumed optimality of(x;y) :(b) By assumptions 2 and 3, the setG is convex. By the separating

hyperplane theorem (Minkowskis theorem), then, and the factthat! =2 G; there exists p 2 R`; p 6= 0 such that

p ! p v

for allv 2 G: If we use the denition ofGto write this expressionout in full, it states that

MXi=1

p !i MXi=1

p ~xi JXj=1

p ~yj

for(P~xi P~yi) 2 G:(c) Claim: p is a price system. We need to show thatp 2 R`+:Suppose for some good k (w.l.o.g. k= 1) we havep1 < 0: Thenby assumption 4, e1 2 Y; where e1 = [1; 0; :::; 0] and > 0:By the argument in b), for any x such that for all i = 1;:::;M;xi2 Xi; x1 1 x1 and xi i x

i fori = 2;:::;Mwe have that

p !MXi=1

p ~xi p e1 =

MXi=1

p ~xi+ p

1

for any >0: But this is impossible for suciently large valuesof:

(d) Claim: Firms maximize prots. Let fxq1g1

q=1 be a sequence

such that for all q, xq1 2 X1 and x

q1 1 x

1; andxq1 ! x

1: (Notethat the existence of such a sequence is guaranteed by assumption1.) Lety 2 Y: Then by the argument in b),

p ! p xq1+MXi=2

p xi p y:

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Taking limits, we have

p ! p " MXi=1

xi y#

:

Now, consider producer j = 1 (say). By feasibility, we knowthat

p ! = p

24 MXi=1

xi JXj=1

yj

35 p

2

4MXi=1

xi y1 JXj=2

yj

3

5for any y1 2 Y1: We conclude from this thatp y1 p

y1

for any y1 2 Y1; i.e. producer 1 is maximizing prots.

(e) As an exercise, you can use a similar argument to establish thateach consumer must be minimizing her expenditures at the *allocation relative to any allocation in G:

Corollary 3: Under the assumptions of Theorem 3 and the additionalassumption that for all i; iis a continuous preference ordering, if in the quasi-equilibrium [(x;y) ; p] the followingminimum wealth condition is satised

8i p xi >inf fp xj x 2 Xig

then [(x;y) ; p] can be implemented as a Walrasian equilibrium after thesuitable transfer payments have been made.

Proof: Exercise.

In Figure 4, we illustrate a counter-example (know as the Arrow exceptionalcase) to the corollary when the minimum wealth condition is not met.

In the diagram, the Pareto set has a limit as x11 = 0; and x

12 > 0: Fur-thermore, the supporting prices at this allocation have p2 = 0: It should beclear from the way person 2s indierence curves are in the Edgeworth box dia-gram that at these prices, person 2 will demand all of the good 2, not just x12:

Hence, for this case, the * allocation cannot be implemented as a competitiveequilibrium. Note, however, that if person 2 has strictly positive wealth at oneof the Pareto optimal allocations, then the corollary will apply and we will beable to implement the allocation as a Walrasian equilibrium.

Exercise: Consider the following 2 2 exchange economy.

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There are two commodities available for trade. We denote a typical con-sumption vector by

x=

x1x2

and assume that there is one unit of each commodity available to allocate amongagents.

The two agent types are denoted by A and B, and are completely charac-terized by their preferences and endowments. Type A agents preferences aregiven by the utility function

uA xA= min xA1; 2xA2 wherexA denotes agent As consumption bundle of the two goods. Agent Bspreferences are given by

uB

xB

= min

2xB1; xB2

Endowments are denoted by!A for type A agents, and by !B for type B agents.

1. Find the Pareto set and illustrate it in the Edgeworth box.

2. Consider the case where

!A = 1 "" !B =

"1 "

Show that for this case, there is a unique competitive equilibrium

with strictly positive prices. Find the equilibrium prices.

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Show that this equilibrium is unstable under the standard Walrasiantatonnement process.

Are there other competitive equilibria for this case? If so, what arethey?

What equilibrium is selected in this case by the "-intelligent learningalgorithm (from the Crockett, Spear and Sunder paper)?

3. Now consider the case where

!A =

"1 "

!B =

1 "

"

Again, show that there is a unique competitive equilibrium havingstrictly positive prices.

Show that this is the only equilibrium for this case.

Show that this equilibrium is stable under the Walrasian tatonnementprocess.

Show that the "-intelligent learning algorithm also converges to thisequilibrium.

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Walrasian Equilibrium G2