Decentralized Price Adjustment in 2 2 Replica. Economies

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1 Decentralized Price Adjustment in 2 2 eplica Economies Maxwell Pak Queen s University This version: April 2009 Abstract This paper presents a model of price adjustment in replica economies with two consumer types and two goods. The model provides a trading rule that allows out-of-equilibrium trading and a decentralized price adjustment rule that features learning through noisy imitation. It is shown that for all sufficiently large economies, the process of experimentation and imitation favors adjustment of prices in the direction of the excess demand. When the experimentation probability is small, the price adjustment process mostly follows a tâtonnementlike dynamics, and the limiting distribution is concentrated around the Walrasian equilibrium. Price adjustment, tâtonnement, exchange economy, stochastic stabil- Keywords: ity. JEL Classification: C7, D51, D83. This paper has benefitted greatly from numerous discussions with Chris Shannon. I am also deeply indebted to Steve Evans for his guidance on the continuous state space Markov chain literature. Department of Economics, Queen s University. Kingston, Ontario K7L 3N6, Canada. Phone: Fax: pakm@qed.econ.queensu.ca. 1

2 1 Introduction Walrasian equilibrium defines equilibrium price vector in an exchange economy as the prices at which all markets clear when all consumers are maximizing their preferences given these prices. The intuitive appeal of this definition makes Walrasian equilibrium the natural equilibrium concept not only for an exchange economy but also for many other economic models. Not surprisingly, then, there have been many attempts at modeling the dynamic process through which a Walrasian equilibrium may arise. A well-known example is tâtonnement dynamics, first proposed in 1874 by Walras [9], which assumes that the price of a good adjusts in the direction of its excess demand. 1 Tâtonnement dynamics captures the seemingly correct intuition that for an economy to reach its equilibrium, the price of a good should rise when its demand exceeds its supply, and fall when its supply exceeds its demand. Also well known about tâtonnement dynamics, however, are its many shortcomings as a model of price adjustment. First, its stability is not guaranteed in economies with more than two goods. Second, it leaves the motivation of the price-setting agent unmodeled. Tâtonnement models typically assume the existence of an exogenous agent, commonly called the Walrasian auctioneer, who learns the demand of all the market agents at given prices and adjusts the price of each good according to the sign of its excess demand. However, assuming that the auctioneer is exogenous and not modeling why she would want to adjust prices in this particular manner is a significant omission in models in which all agents are assumed to maximize their own preferences. Third, it does not specify how trading occurs when the economy is not in equilibrium. Typically, the models assume that prices first adjust toward their limit, and that there is no actual trading until the equilibrium prices have been reached. Without out-of-equilibrium trades, however, there is no incentive for agents to reveal their 1 For general discussion of tâtonnement dynamics, see Hahn [3]. 2

3 demand. Thus, tâtonnement dynamics fails to be a satisfactory model of price adjustment even in settings where its stability can be assured. This paper addresses this issue by modeling a decentralized, endogenous price adjustment process that is tâtonnementlike. In particular, it provides a trading rule and a price adjustment rule in replica economies with two consumer types and two goods. The trading rule is constructed to allow trades to occur out of equilibrium. The price adjustment rule assumes that the two consumer types set prices in different periods and adjust prices through a learning through noisy imitation rule, in which the prices that were most successful in the previous period are adopted with high probability but random experiments are also taken with low but strictly positive probability. 2 It is shown that for all sufficiently large economies, the noisy imitation rule favors adjustment of prices in the direction of the excess demand. As a result, when the experimentation probability is small, the price adjustment process mostly follows a tâtonnement-like dynamics and leads eventually to a Walrasian equilibrium. More precisely, this paper shows that for any fixed experimentation probability, the distribution of the prices converges to some limiting distribution. Following the standard approach in evolutionary game theory, the limit of the limiting distributions, as the experimentation probability decreases to zero, is then considered. The main result shows that for any small neighborhood of the equilibrium price vector, the limit of the limiting distributions is concentrated inside the neighborhood for all sufficiently large economies. These results, however, are derived in the restrictive setting of a two-consumers, two-goods, replica economy with a unique equilibrium. The two-consumers, twogoods economy with a unique equilibrium provides the simplest setting where tâtonnement dynamics is globally stable. Because the price adjustment dynamics incorporates learning through imitation, the existence of other agents from whom a given agent 2 See Foster and Young [2], Kandori, Mailath, and ob [4], and Young [10] for pioneering examples of evolutionary models that feature persistent random experimentation. 3

4 can learn is required. Thus, a replica economy in which there are many identical agents of each consumer type provides a natural setting for the model. Ultimately, like other attempts at providing a foundation for tâtonnement dynamics, this paper addresses some issues while leaving others unresolved. For example, Keisler [5] models a price adjustment process that approximates tâtonnement dynamics and also features out-of-equilibrium trading and decentralized price setting. Keisler assumes that a large, finite set of agents takes turns trading with a market maker and shows that if at each period the market maker adjusts the price vector in the direction opposite to the changes in her inventory, the price vector approaches a Walrasian equilibrium under suitable conditions. While Keisler s model resolves many deficiencies of tâtonnement dynamics without resorting to the restrictive setting of this paper, it leaves unmodeled the motivation of the market maker to adjust the prices in the specified manner. In contrast, the focus in this paper is on providing a model, albeit in a simple setting, in which price setters adjust prices because it is in their interest to do so. 3 Studying a price adjustment process through an evolutionary game theory approach is not new. In a partial equilibrium context, Vega-edondo [8] shows that learning through noisy imitation leads to the competitive equilibrium price in Cournot competition with identical firms. In addition, Temzelides [7] applies noisy imitation to the market game of Shapley and Shubik in 2 2 replica economies and shows that it leads to the Walrasian equilibrium. While this paper shares the same replica economy setting of Temzelides [7], the papers differ in that the market game requires the existence of an auctioneer who collects the bids and use them to determine the market clearing prices. Methodologically, this paper also departs from the existing literature on learning 3 More recently, Crockett, Spear, and Sunder [1] provides a decentralized process that leads to a competitive equilibrium in a general setting. Agents in their model use a random search to first get near a Pareto optimal allocation and then use modified random searches to move towards a competitive equilibrium. However, the modified random searches require some agents to accept allocations that make them strictly worse off, and the motivation for accepting such allocation is left unmodeled. 4

5 through noisy imitation. The existing literature is limited to finite state space models since it relies on the use of the tree-surgery technique to characterize a limiting distribution. However, as prices get closer to the equilibrium price vector, the excess demand approaches zero, and the set of prices that can be successfully adopted through experimentation and imitation becomes arbitrarily small. As a result, considering only a finite set of prices, however large, leads to unnecessary complications. Consequently, this paper foregoes the tree-surgery technique and applies a method that is applicable to general state space models. The remaining pages are organized as follows. Section 2 presents the trading rule and the price adjustment rule considered in this paper. Section 3 provides the main result, and Section 4 gives a brief conclusion. The lemmas and their proofs that support the main result are given in the Appendix. 2 The Price Adjustment Process Since a replica economy is an economy in which there are many copies of the consumers of some underlying economy, the description of the model begins with the specification of the underlying economy. The underlying economy is a pure exchange economy consisting of two consumers and two goods. The set of consumers is denoted by I = {1, 2}. For each i I, let ω i 2 ++ denote consumer i s initial endowment, and let i denote consumer i s preference, which is assumed to be continuous, strongly monotone, and strictly convex. Let u i ) denote the continuous utility function representing i. Consumer i s demand function is denoted by x i : , where = {p 1, p 2 ) 2 ++ : p 1 + p 2 = 1}. 4 Let z i : p, ω i ) x i p, ω i ) ω i denote consumer i s excess demand function, and let z : p, ω 1, ω 2 ) z 1 p, ω 1 ) + z 2 p, ω 2 ) denote the market excess demand function. A Walrasian equilibrium price vector of the 4 The assumptions on the preferences imply that x i, ) is a continuous function. 5

6 underlying economy is a price vector p that satisfies zp, ω 1, ω 2 ) = 0. It is assumed that the Walrasian equilibrium is unique and satisfies x i p, ω i ) ω i for all i I. The -replica economy is the economy with 2 consumers in which consumers are exact copies of consumer 1 of the underlying economy and the remaining consumers are exact copies of consumer 2. That is, the consumers in the underlying economy are interpreted as consumer types so that I now denotes the set of consumer types. For each type i I, there are consumers with the identical preference i and the identical initial endowment ω i. These consumers are called type i replicas, and r-th replica of type i is denoted ir. A replica economy is related to the underlying economy in that price vector p together with each replica ir consuming x i p, ω i ) is also a Walrasian equilibrium in the replica economy. The following two subsections present the price adjustment model considered here. In the model, each replica ir starts with the same endowment ω i in every period. In the beginning of each period, a consumer type is chosen randomly as the price setter. After the prices have been set, trades occur according to the trading rule specified in Subsection 2.1. After all the trades have been completed, consumptions occur and the new period begins. In the next period, each replica again receives her endowment, a new price-setter type is chosen randomly, and the prices are set according to the learning rule specified in Subsection 2.2. As seen below, these two rules together imply that the evolution of prices can be modeled as a Markov chain on the state space Ξ = I, where a state i, p 1,..., p ) Ξ has the interpretation that type i consumer is the price setter and that replica ir has set price vector p r. 5 5 In the discrete time model developed here, only one consumer type is chosen as the price setter in each period. This assumption may seem more plausible if the discrete time model is thought of as being embedded in a continuous time model in which price adjustments occur at random times. Suppose each consumer type sets prices independently of each other and that the waiting time between the price adjustments has exponential distribution. Assume further that each type reacts first to the other type s price changes before attempting to set its own prices. Then since the probability of two adjustments occurring at any given time is zero, watching this continuous time process only at random times in which a price adjustment occurs is effectively equivalent to the original discrete time setup. 6

7 2.1 The Trading ule The trading rule assumes that replicas of the price-taker type trade with replicas of the price-setter type in sequential stages, starting with the price setters with the most favorable prices and ending with the price setters with the least favorable prices. The price takers are active traders in that they choose the order of their trading partners and set the desired net trade vector. The price setters are passive in that they only trade when asked to trade by a price taker and are required to trade in an amount proportional to the net trade vector desired by the price takers. More precisely, suppose a state i, p 1,..., p ) Ξ has been realized at the beginning of the current period. In particular, let j i denote the price-taker type. In the following, Ψ s denotes the set of price setters that the price takers have not yet traded with as of the beginning of stage s, and Φ s denotes the price takers most favorable trading partners among those in Ψ s. Each price setter trades only once, and the result of her trade is denoted ˆω ir. Since all price takers have the same preference and endowment and face the same set of prices, they are assumed to behave identically. The commodity vector each type j replica has at the beginning of stage s is denoted by ωj s, and the final result of her trading in the current period is denoted by ˆω j, without using subscripts to distinguish among replicas. Finally, in this paper any reference to j I will always mean j i, and will denote the set {1,..., }. Trading within the current period can now be described in the following inductive manner. Let Ψ 0 =, Φ 0 =, and ω 1 j = ω j. At stage s, let Ψ s = Ψ s 1 \ Φ s 1. Assume Ψ s. Let p {p r : r Ψ s } be such that x j p, ω s j ) j x j p r, ω s j ) for all r Ψs, and let Φ s = {r Ψ s : p r = p}. The total net trade desired by each type j replica from the type i replicas in Φ s is given by z j p, ωj s ). Since type j replicas are indifferent among their 7

8 trading partners in Φ s, each type j replica is assumed to desire 1 Φ s z jp, ω s j ) from each replica in Φ s. Thus, each type i replica in Φ s receives a total order of Φ s z jp, ωj s ) as the desired net trade from type j replicas. After an order is received, each type i replica in Φ s gives α s z j p, ωj s ) to each type j replica, where α s = arg max u i ω i α ) Φ s z jp, ωj s ). In particular, it is assumed that price setters are allowed to partially fill the orders they receive as long as they trade in the amount proportional to the trade vector desired by the price takers. The result of the trading in s-th stage is given by: ω s+1 j = ω s j + α s z j p, ω s j ), and r Φ s, ˆω ir = ω i α s Φ s z jp, ωj s ). The trading proceeds to stage s + 1 if Ψ s+1 = Ψ s \ Φ s. Otherwise, all trades have been completed and ˆω j = ω s+1 j. 2.2 The Price Adjustment ule Define a best price correspondence B from Ξ into as follows. For any ξ = i, p 1,..., p ) Ξ, let trades occur according to the trading rule described above. Then B is defined by Bξ) = { p r {p 1,..., p } : ˆω ir i ˆω ir r }. Thus, Bξ) is the set of prices that were most successful for type i. Let ˆδ > 0 and for all p = p 1, p 2 ), let N p, ˆδ) = {q 1, q 2 ) : q 1 p 1 < ˆδ} be the ˆδ-neighborhood of p. The price adjustment process is governed by the following learning through noisy 8

9 imitation rule: At t = 0: A state in Ξ is chosen according to some arbitrary initial distribution. At t = 1, 2, 3,...: Suppose ξ = i, p 1,..., p ) is the state chosen at period t 1. Then a new state is chosen at period t in the following way. 1. First, k I is chosen with uniform probability. 2. If k = i, then each replica ir independently chooses a price vector in either of two ways. With probability 1 ε > 0, replica ir imitates. That is, replica ir chooses an element of Bξ) with uniform probability. With probability ε > 0, replica ir experiments. That is, replica ir chooses an element of N p r, ˆδ) with uniform probability. 3. If k = j, then each replica jr adopts the previous period s prices by setting p r. 6 Formally, for any Z ++ and ε 0, 1), the price adjustment rule, together with the trading rule, defines a Markov chain ξ ε on Ξ. Let λ L be the Lebesgue measure on. Define measure µ L on subsets of by µ L C) = λ L {p 1 : p 1, 1 p 1 ) C}). For any Borel subset A of Ξ, let A k = r=1 A kr = {p 1,..., p ) : k, p 1,..., p ) A}. Let 1 denote the indicator function. For any ξ = i, p 1,..., p ) Ξ, the transition probability from ξ to A is given by Prob ξ ε t A ξ ε t 1 = ξ ) = ε) Bξ) Air r= {p 1,...,p ) A j }. Bξ) ) + εµ L N p r, ˆδ) A ir ) 6 This specification implicitly assumes that consumers only remember the immediate past. If the price setters in the current period had not been the price setters in the previous period, they would have no information about which prices had been successful. It is assumed that under this scenario they simply adopt the previous period s prices. 9

10 3 Limiting Distribution This section characterizes the long-run behavior of the price adjustment dynamics. As a starting point, Subsection 3.1 shows that, for any fixed experimentation probability, the price adjustment dynamics is stochastically stable. That is, starting from any arbitrary initial distribution, the dynamics eventually settles down to the same limiting distribution given by the unique invariant distribution. However, instead of deriving the limiting distribution explicitly, this paper derives the limit of the limiting distributions as the experimentation probability goes to zero. This limit is viewed as an approximation of the limiting distribution when the experimentation probability is small. Subsection 3.2 shows that as the experimentation probability goes to zero, the limiting distribution becomes concentrated around the states corresponding to the Walrasian equilibrium price vector. 3.1 Existence of the Limiting Distribution Theorem 3.2 below shows that for each ε > 0, the Markov chain ξ ε corresponding to the price adjustment dynamics has a limiting distribution and gives an upper bound on the rate of convergence. Although establishing the existence of a limiting distribution in general state space models is often delicate, the existence can be readily shown in models with persistent randomness by appealing to a special case of Doeblin s Theorem. 7 In the following, let X n and BX) be the Borel σ-field of X. Let ζ be a Markov chain on state space X. For any A BX) and x X, P m x A) denotes the m-step transition probability Probζ t+m A ζ t = x) and P x A) denotes P 1 x A). Theorem 3.1 Doeblin). Let ζ be a Markov chain on state space X. Suppose there exist m > 0 and a non-trivial measure ν m on BX) such that P m x A) ν m A) for all 7 For further discussion of Doeblin s condition and the uniform ergodicity of general state space chains, see Theorems and , and the related discussion in Meyn and Tweedie [6]. 10

11 x X and A BX). Then the unique invariant measure λ for ζ exists. Moreover, sup x P t x ) λ ) 1 ν m X)) t m. Theorem 3.2. Fix any Z ++ and ε 0, 1). Then the unique invariant measure π ε for the chain ξ ε on Ξ exists. Moreover, there exists a constant ρ ε < 1 such that sup ξ P t ξ ) πε ) ρ t ε. Proof. Let m > 1 + 2ˆδ. Let µ be the produce measure r=1 µ L on, and let µ 0 be the measure on {, {1}, {2}, {1, 2}} such that µ 0 {1}) = µ 0 {2}) = ) ) 1 m m ˆδε. 2 2 Let µ be the product measure µ 0 µ on BΞ ). Consider any A BΞ ). Let A 1 = {p 1,..., p ) : 1, p 1,..., p ) A} and A 2 = {p 1,..., p ) : 2, p 1,..., p ) A} so that {1} A 1 ) {2} A 2 ) = A and {1} A 1 ) {2} A 2 ) =. For all ξ Ξ, Pξ m A) = P ξ m {1} A 1) + Pξ m {2} A 2) ) ) 1 m m ) ) ˆδε 1 m m ˆδε µ A 1 ) + µ A 2 ) = µa). Since µ is non-trivial by construction, the conclusion follows from Theorem 3.1 with ρ ε = 1 µξ )) 1 m. As the above argument makes clear, the existence of a limiting distribution for the chain ξ ε is not a deep result. As in finite state space evolutionary models with persistent randomness, it is essentially the consequence of the irreducibility generated by allowing random experimentation. The more interesting result is the characterization of the limiting distribution, which is given next. 11

12 3.2 Characterization of the Limiting Distribution Since finding the exact expression for the limiting distribution π ε is difficult, this paper characterizes π ε by deriving the limit of π ε as ε 0. Meyn and Tweedie [6] gives a useful characterization of an invariant measure that simplifies this derivation. A simple version of their theorem is stated below as Theorem In the theorem and elsewhere, given chain ζ on X and A BX), let τ A = inf {t 1 : ζ t A} denote the hitting time of the set A. Theorem 3.3 Meyn and Tweedie). Under the assumptions of Theorem 3.1, the unique invariant measure λ for ζ satisfies the following. For any B BX) such that λb) > 0 and A BX), λa) = B λdx)e x [ τb t=1 1 {ζt A} ]. Meyn and Tweedie s theorem states that for any fixed set B of positive measure, the measure λ places on A is determined by how often the chain visits A before returning to B. Lemma 3.4 and Theorem 3.5 below exploit this return time characterization. Following Vega-edondo [8], let the states in which all the replicas set the same prices be called monomorphic states. Consider the expected number of times the price adjustment dynamics, starting from a monomorphic state, will visit non-monomorphic states before returning to the set of monomorphic states. Consider also the expected number of times the dynamics, starting now from a nonmonomorphic state, will visit monomorphic states before returning to the set of non-monomorphic states. When the experimentation probability is small, the probability of replicas taking an imitation step is greater than the probability of replicas 8 For the statement of this theorem in its full generality, see Theorem of Meyn and Tweedie [6]. In particular, Theorem only requires ζ to be recurrent, which is weaker than the hypothesis stated in Theorem 3.3. Furthermore, Theorem requires B to satisfy ψb) > 0, where ψ is the maximal irreducibility measure for ζ. However, since ψ and λ are equivalent measures, this simpler statement of the theorem is used. 12

13 taking a random experimentation step. Since imitation steps lead to a monomorphic state, the expected number of visits to monomorphic states is greater than the expected number of visits to non-monomorphic states. Therefore, according to the return time characterization, the limiting distribution π ε puts greater measure on the set of monomorphic states. In the limit, as the experimentation probability goes to zero, full measure is placed on the set of monomorphic states. This is formally stated and shown as Lemma 3.4 below. In the remainder of the paper, superscripts denoting are suppressed to reduce clutter in the notation whenever their suppression does not affect the clarity of the meaning. Lemma 3.4. Fix any Z ++. Let ˆΞ = {i, p,..., p) : i I and p }. Then for all ε 0, 1), π ε ˆΞ) > 0. Moreover, π ε ˆΞ) 1 as ε 0. Proof. Fix any ε 0, 1). Since π ε Ξ) > 0 and τ Ξ = 1 P ξ -a.s. for all ξ Ξ, π ε ˆΞ) = Ξ P ξ ˆΞ)π ε dξ) = P ξ ˆΞ)π ε dξ) + P ξ ˆΞ)π ε dξ) ˆΞ Ξ\ˆΞ by Theorem 3.3. Since for all ξ ˆΞ, P ξ ˆΞ) 1 ε) and for all ξ Ξ\ˆΞ, P ξ ˆΞ) ε), π ε ˆΞ) 1 ε) π ε 1 dξ) + ˆΞ Ξ\ˆΞ 2 1 ε) π ε dξ) = 1 ε) π ε ˆΞ) ε) π ε Ξ\ˆΞ) > 0. Moreover, the above inequality yields π ε ˆΞ) 1 ε) π ε ˆΞ) ε) π ε Ξ\ˆΞ). Since π ε ˆΞ) 1 as ε 0. π ε ˆΞ) 1 π ε ˆΞ) = πε 1 ˆΞ) π ε Ξ\ˆΞ) 2 1 ε) as ε 0, 1 1 ε) The two lemmas in the Appendix, Lemma A.6 and Lemma A.10, show that the 13

14 price adjustment dynamics favors adjustment in the direction of the excess demand for all economies with sufficiently large number of replicas. As a result, for any small neighborhood of the equilibrium price vector, the maximum number of experimentation steps needed to transition from the states outside the neighborhood into the neighborhood is smaller than the minimum number of experimentation steps needed to transition from the states inside the neighborhood to the outside. Therefore, the return time characterization yields that, as the experimentation probability goes to zero, the limiting distribution becomes concentrated around the monomorphic states corresponding to the equilibrium price vector. This is formally stated and shown below as Theorem 3.5. In the following, let N δ) = N p, δ) denote the δ-neighborhood of p. Theorem 3.5. Fix N δ) and assume ˆδ < δ. There exists such that for all > the following holds. Let π ε be the limiting distribution of ξ ε on Ξ and let A = {i, p,..., p) : i I and p N δ)}. Then π ε A) 1 as ε 0, Proof. Let B = ˆΞ\A. If B =, then the theorem follows from Lemma 3.4. So, assume B. Take N and 1 from Lemma A.9 and 2 from Lemma A.10. Let = max { 1, 2 } and fix >. Take any ε 0, 1 2). Since π ε ˆΞ) > 0, Theorem 3.3 yields π ε A) = = P ξ ξ ε A)π ε dξ) τˆξ ˆΞ P ξ ξ ε A)π ε dξ) + τˆξ A B P ξ ξ ε τˆξ A)π ε dξ). By Lemma A.9, for all ξ B, P ξ ξ ε τˆξ A) Kε N, where K > 0 is a constant. 14

15 For all ξ A, P ξ ˆΞ \ A) = 0. Then by Lemma A.10, for all ξ A, P ξ ξ ε A) P τˆξ ξ A) + Prob ξt+2 ε A, ξt+1 ε ˆΞ ) ξt ε = ξ εn+1 ) 1 ε N+1. Therefore, π ε A) 1 ε N+1 ) π ε dξ) + Kε N π ε dξ) A B = 1 ε N+1) π ε A) + Kε N π ε B). Since π ε A) 1 π ε A) = πε A) π ε B) KεN as ε 0, εn+1 π ε A) 1 as ε 0. 4 Concluding emarks The decentralized price adjustment model presented in this paper resolves some of the difficulties in interpreting tâtonnement dynamics. First, the model specifies outof-equilibrium trading so that a fictional time scale in which prices adjust without trade is not needed. Second, the price adjustment rule is decentralized and endogenous so that the model does not require an exogenous agent whose motivation for adjusting prices is unmodeled. These results, however, are derived in the restrictive setting of 2 2 replica economy, and to what extent the results can be generalized is an open question. 15

16 A Appendix By supporting price for i at ω i, we mean a price vector at which type i will demand exactly ω i when her endowment is ω i. Let p i denote the supporting price for i at her initial endowment ω i so that x i p i, ω i ) = ω i. Let T = {λ p i + 1 λ) p j : λ [0, 1]} be the set of prices that are convex combinations of p i and p j. For any set A, let A denote the relative interior of A. Since x i p j, ω i ) i ω i = x i p i, ω i ), type i prefers p j over p i when her endowment is ω i. The following lemma shows that more generally, given any two prices in T, type i prefers the price that is closer to p j. Lemma A.1. Take any p = λ p j + 1 λ ) p i and p = λ p j + 1 λ) p i, where 1 > λ > λ > 0. Then x i p, ω i ) i x i p, ω i ). Proof. Since x i p, ω i ) i ω i = x i p i, ω i ), p i z i p, ω i ) > 0 by the weak axiom of revealed preference. This implies p j z i p, ω i ) = 1 λ p 1 λ) p i) z i p, ω i ) ) 1 λ = p i z i p, ω i ) λ < 0. Letting η = λ λ > 0, we have p z i p, ω i ) = λ p j + 1 λ ) ) p i zi p, ω i ) = λ + η) p j + 1 λ η) p i ) z i p, ω i ) = [η p j η p i ) + λ p j + 1 λ) p i )] z i p, ω i ) = η p j z i p, ω i ) η p i z i p, ω i ) + p z i p, ω i ) < 0. Therefore, x i p, ω i ) i x i p, ω i ). Let T i = {λ p j + 1 λ)p : λ [0, 1]} be the set of prices that are convex com- 16

17 bination of the equilibrium price and the supporting price for j at her initial endowment. Since p z i p, ω i ) = 0 = p z j p, ω j ) by Walras Law, z i p, ω i ) and z j p, ω j ) are always colinear. Lemma A.2 below shows that, in addition, if p T i, then the excess demand vectors of the two types are in the opposite direction, with the magnitude of i s excess demand exceeding that that of j s. However, if p T then the excess demand vectors are in the same direction. Since p T, T = T i T j. Therefore, an immediate consequence of this lemma is that a trade will occur at the initial endowment allocation and price p if and only p T. Lemma A.2. For all p \{ p i }, there exists β such that z j p, ω j ) = βz i p, ω i ). Furthermore, if p T i then β 1, 0), and if p T then β > 0. Proof. If p = p j, z j p, ω j ) = 0 = βz i p, ω i ) with β = 0. If p \ { p j, p i }, then z j p, ω j ) 0 z i p, ω i ). Moreover, p z j p, ω j ) = 0 = p z i p, ω i ) by Walras Law. Therefore, z 1 j p, ω j) z 2 j p, ω j) = p 2 p 1 = z1 i p, ω j) z 2 j p, ω j). So, for any p \ { p i }, there exists β such that z j p, ω j ) = βz i p, ω i ). Suppose p T i. Then there exists λ 0, 1) such that p = λ p j + 1 λ)p. p j z i p, ω i ) = 1 λ p 1 λ)p ) z i p, ω i ) ) 1 λ = p z i p, ω i ) λ < 0 since x i p, ω i ) i x i p, ω i ) by Lemma A.1. But, β [ p j z i p, ω i )] = p j z j p, ω j ) > 0 since x j p, ω j ) j ω j = x j p j, ω j ). Therefore, β < 0. Next, z i p j, ω i ) > 0 = z j p j, ω j ) while z i p, ω i ) = z j p, ω j ). Since the excess demand functions are continuous functions of prices and the equilibrium is unique, z i p, ω i ) > z j p, ω j ) for all p = λ p j + 1 λ)p. Therefore, β < 1 as claimed. Now, suppose p T. Then either p i = λp + 1 λ) p j for some λ 0, 1), or 17

18 p j = λp + 1 λ) p i for some λ 0, 1). Assume p i = λp + 1 λ) p j for some λ 0, 1). Then p i z j p, ω j ) = λp + 1 λ) p j ) z j p, ω j ) = 1 λ) p j z j p, ω j ) > 0 since x j p, ω j ) j ω j = x j p j, ω j ). Also, p i z i p, ω i ) > 0 since x i p, ω i ) i ω i = x i p i, ω i ). Therefore, β > 0. Lastly, the case where p j = λp + 1 λ) p i for some λ 0, 1) is similar. Using the above two lemmas, we can now show that it takes only one experimentation to successfully transition away from a monomorphic state if the replicas are setting a price p in \ T. In particular, consider a monomorphic state i, p,..., p). Lemma A.3 below shows that if p T, then for some appropriate k I, p Bk, p, p,..., p). This implies that it takes only a single experimentation step to transition from i, p,..., p) to k, p,..., p ) via k, p, p,...p). Lemma A.3. Fix any replica population size Z ++. For all p \ T and p, there exists ξ = k, p, p,...p) Ξ such that p Bξ). Proof. Since p T, either p i = λp + 1 λ) p j for some λ 0, 1], or p j = λp + 1 λ) p i for some λ 0, 1]. Without loss of generality, assume p i = λp + 1 λ) p j for some λ 0, 1]. First, suppose p T so that p = λ p i + 1 λ ) p j for some λ 0, 1). Then p = λ λp + 1 λ) p j ) + 1 λ ) p j = λ λp + 1 λ λ) p j. So, p z j p, ω j ) = 1 λ λ p 1 λ λ) p j ) z j p, ω j ) 1 λ ) λ = λ p j z j p, ω j ) λ < 0 since x j p, ω j ) j ω j = x j p j, ω j ). 18

19 Therefore, x j p, ω j ) j x j p, ω j ). Let ξ = i, p, p,..., p) Ξ. Since x j p, ω j ) j x j p, ω j ), j-replicas will want to trade first with i-replicas setting price p. That is, Φ 1 = {2, 3, 4..., }, with each replica ir, r Φ 1, receiving a total trade order of 1 z jp, ω j ). However, since the desired trades of the two types are in the opposite direction, no actual trade will occur in the first stage. To see this, note that since x i p, ω i ) i ω i = x i p i, ω i ), p i z i p, ω i ) 0. Using Lemma A.2, we obtain p i ω i α ) 1 z jp, ω j ) = p i ω i αβ ) 1 z ip, ω i ) p i ω i for all α [0, 1]. for some β 0 Meaning, ω i i ω i α 1 z jp, ω j ) for all α [0, 1]. Therefore, α 1 = arg max u i ω i α ) 1 z jp, ω j ) = 0 so that ω ir = ω i for all r Φ 1 and ω 2 jr = ω j for all r. In the second stage, we have j-replicas wanting to trade with replica i1. Since p is assumed to be in T, some trade will actually occur, leaving i1 better off than replicas ir, r Φ 1. To see this, note that Lemma A.2 yields α 2 = arg max u i ωi αz j p, ω j ) ) = arg max u i ωi + αβz i p, ω j ) ) for some β > 0 { } = min 1 β, 1 by continuity of u i ). Since a 2 > 0, we have ω i1 = ω i + α 2 βz i p, ω j ) i ω i = ω ir r 1. Therefore, Bi, p, p,..., p) = {p }. Second, suppose p T. Then similar argument to above yields that no actual trade will occur under p or p. Therefore, ω ir = ω i for all r, and Bi, p, p,..., p) = {p, p}. 19

20 For each p T i and the size of the replica population,, let f i p, ) be the supporting price for i at ω i 1 z jp, ω j ). That is, define f i : T i Z ++ by x i f i p, ), ω i 1 ) z jp, ω j ) = ω i 1 z jp, ω j ). To see the relevance of f i p, ), consider a state ξ in which 1 many j-replicas have set price p while a single j-replica has set price p, where x i p, ω i ) i x i p, ω i ). If p Ti, then trade will occur since the excess demands of the two types are in the opposite direction. In fact, as shown below in the proof of Lemma A.6, the result of the first round trade for type i will be ωi 1 = ω i 1 z jp, ω j ). Thus, f i p, ) is the price that supports i s endowment after the first round of trading. Now, let T i p, ) be the set of convex combination of p and 1 2 p f ip, ) that are also in T i. That is, define T i : T i Z ++ 2 by { 1 T i p, ) = T i λp + 1 λ) 2 p + 1 ) } 2 f ip, ) : λ [0, 1] = T i { λp + 1 λ) f i p, ) : λ [ 1 2, 1]}. The proof of Lemma A.6 also shows that if p T then f i p, ) p so that the relative interior of { λp + 1 λ) 1 2 p f ip, ) ) : λ [0, 1] } is not empty, which then implies that Ti p, ). Moreover, since T ip, ) T i, if p Ti p, ), trade will occur at p when each type s endowment is its initial endowment. Lemma A.4 below goes one step further and shows the excess demands of the two types are in the opposite direction even when type i has endowment ω i 1 z jp, ω j ) and type j has endowment ω j. Therefore, trade will also occur in the second round in state ξ if p Ti p, ). Lemma A.4. For all p Ti, Z ++, and p Ti p, ), there exists β > 0 such that z i p, ω i 1 ) z jp, ω j ) = βz j p, ω j ). Proof. Since p z i p, ω i 1 z jp, ω j ) ) = 0 = p z j p, ω j ), there exists β 20

21 such that z i p, ω i 1 z jp, ω j ) ) = βz j p, ω j ). In the following, we show that β > 0. First, x i p, ω i 1 ) z jp, ω j ) i ω i 1 z jp, ω j ) = x i f i p, ), ω i 1 ) z jp, ω j ) Since p = λp + 1 λ)f i p, ) for some λ 1 2, 1), p z i p, ω i 1 ) z jp, ω j ) = 1 p 1 λ)f i p, ) ) z i p, ω i 1 ) λ z jp, ω j ) ) 1 λ = f i p, ) z i p, ω i 1 ) λ z jp, ω j ) < 0 by the weak axiom. Second, p Ti ; so, by Lemma A.2, there exists γ 0, 1) such that ω i 1 z jp, ω j ) = ω i + 1 γz ip, ω i ) = 1 1 ) ) 1 γ ω i + γ x i p, ω i ). Since p ω i 1 z jp, ω j ) ) = p ω i, x i p, ω i ) i ω i 1 z jp, ω j ) = x i f i p, ), ω i 1 ) z jp, ω j ). Appealing to the weak axiom again, we obtain 0 < f i p, ) x i p, ω i ) = f i p, ) x i p, ω i ) = 1 1 γ ω i 1 )) z jp, ω j ) 1 1 ) 1 γ ω i ) f i p, ) z i p, ω i ). ) ) γ x i p, ω i ) 21

22 By Lemma A.2, there is β 0, 1) such that p z j p, ω j ) = p β )z i p, ω i ) = 1 p 1 λ)f i p, ) ) β )z i p, ω i ) λ ) 1 λ = λ β f i p, ) z i p, ω i ) > 0. Since p z i p, ω i 1 z jp, ω j ) ) < 0 while p z j p, ω j ) > 0, we have β > 0 as desired. Lemma A.5. Fix any N δ). Then there exists 1 such that for all > 1, the following holds. For any p T i \ N δ) and p T i p, ) \ {p}, let α = arg max u j ω j αz i p, ω i 1 )) z jp, ω j ). Then ω j α z i p, ω i 1 ) z jp, ω j ) = x j p, ω j ). Proof. Suppose p T i \ N δ) and p T i p, ) \ {p}. By Lemma A.4, there exists β > 0 such that z i p, ω i 1 ) z jp, ω j ) = βz j p, ω j ). Then ω j αz i p, ω i 1 ) z jp, ω j ) = ω j + αβz j p, ω j ). Therefore, arg max u j ω j αz i p, ω i 1 )) { } 1 z jp, ω j ) = min β, 1. It remains to show that β 1 for all sufficiently large. 22

23 Price vector f i p, ) was defined earlier as the supporting price for i at ω i 1 z jp, ω j ). Similarly, let f i p) be the supporting price for i at ω i z j p, ω j ). That is, let f i : T i be defined by x i fi p), ω i z j p, ω j ) ) = ω i z j p, ω j ). Also, analogously to T i p, ), define T i : T i 2 by T i p) = T i { λp + 1 λ) 1 2 p f i p) ) : λ [0, 1] } = T i { λp + 1 λ) f i p) : λ [ 1 2, 1]}. Since f i p) = p if and only if p = p, T i p) for all p T i \ {p }. For each p T i \ N δ), let h i p) = min p T i p) zi p, ω i z j p, ω j ) ). Since p p and f i p) T i p) by construction, z i p, ω i z j p, ω j )) > 0 for all p T i p). Since z i, ω i z j p, ω j )) is continuous and T i p) is compact, h i p) > 0. Moreover, h i ) is itself continuous by the theorem of the maximum; therefore, we have h i min p T i \N δ) h i p) > 0. Next, for each p T i \ N δ) and Z ++, let h i p, ) = min p T i p,) z i p, ω i 1 ) z jp, ω j ). Since h i, ) is continuous and strictly positive on T i \ N δ), we have h i ) min h ip, ) > 0. p T i \N δ) Moreover, h i ) h i > 0 as. So, there exists i and η i > 0 such that for 23

24 all > i, h i ) > η i. Let ζ j = max p T z j p, ω j ). Then ζ j 0, ). Now, let 1 > max and fix any > 1. Since { } 1, 2, ζ 1 η 2, ζ 2 η 1 βζ j βz j p, ω j ) = z i p, ω i 1 ) z jp, ω j ) η i, ) ) β η i ζj ζ j. Therefore, β ηi η i ζ j = 1, and α = arg max u j ω j αz i p, ω i 1 )) z jp, ω j ) = 1 β as desired. Lemma A.6. Fix any N δ). Then there exists 1 such that for all > 1, the following holds. Suppose ξ = j, p, p,..., p) Ξ is such that p T i \ N δ) and p T i p, ). Then p Bξ). Proof. Let 1 be as in Lemma A.5 and fix > 1. Consider any ξ = j, p, p,..., p) satisfying the hypothesis. If p = p j or p = p, then there is nothing to prove. So, assume p p j and p T i p, ) \ {p}. Then there exists λ 0, 1) such that p = λp + 1 λ) p j. In addition, since p T i \ { p j }, there exists β > 0 such that z i p, ω i ) = β z j p, ω j ) by Lemma A.2. Thus, p z i p, ω i ) = β λp + 1 λ) p j ) z j p, ω j ) = β 1 λ) p j z j p, ω j ) < 0. Therefore, x i p, ω i ) i x i p, ω i ), and Φ 1 = {2, 3, 4,..., }. 24

25 Since p T i, there exists β 0, 1) such that z jp, ω j ) = βz i p, ω i ). Thus, arg max u j ω j α ) 1 z ip, ω i ) = arg max = β 1). ) u j ω j + α β 1) z jp, ω j ) Therefore, ω jr = x j p, ω j ) for all r Φ 1, and ω 2 i = ω i + β 1) z i p, ω i ) = ω i 1 z jp, ω j ). Since Φ 2 = {1}, ω j1 = ω j α 2 z i p, ω i 1 ) z jp, ω j ), where α 2 = arg max u j ω j αz i p, ω i 1 )) z jp, ω j ). Thus, ω j1 = x j p, ω j ) by Lemma A.5. Since x j p, ω j ) j x j p, ω j ) by Lemma A.1, p Bξ). The next two lemmas relate our price adjustment dynamics to the tâtonement dynamics. The first lemma, Lemma A.7 shows that if p T i and q p is in T i p, ), then q can be obtained from p by moving in the direction of the excess demand function, zp, ω i, ω j ). The second lemma, Lemma A.8 shows that, given two price vectors that are obtained from some p by moving in the direction opposite to type i s excess demand, z i p, ω i ), type i would prefer the price vector that is further away from p. Lemma A.7. For all p = p 1, p 2 ) Ti, Z ++, and q = q 1, q 2 ) Ti p, ), there exists γ > 0 such that q 1 q 2 = p1 p 2 + γz1 p, ω 1, ω 2 ). Proof. Since p Ti, Lemma A.2 implies that there exists β 0, 1) such that 25

26 z j p, ω j ) = βz i p, ω i ). Therefore, q 1 z 1 p, ω 1, ω 2 ) + q 2 z 2 p, ω 1, ω 2 ) = q zp, ω 1, ω 2 ) = λp + 1 λ)f i p, )) z i p, ω i ) + z j p, ω j )) = 1 λ)1 β)f i p, ) z i p, ω i ) > 0 as shown in the proof of Lemma A.4 while p 1 z 1 p, ω 1, ω 2 ) + p 2 z 2 p, ω 1, ω 2 ) = p zp, ω 1, ω 2 ) = 0. Therefore, q 1 q 2 z1 p, ω 1, ω 2 ) > p1 p 2 z1 p, ω 1, ω 2 ). So, whether z 1 p, ω 1, ω 2 ) > 0 or z 1 p, ω 1, ω 2 ) < 0, there exists γ > 0 such that q 1 q 2 = p1 p 2 + γz1 p, ω 1, ω 2 ). Lemma A.8. Fix any p = p 1, p 2 ) and ω i ++. Suppose p 1 = p 1 1, p2 1 ), p 2 = p 1 2, p2 2 ) are such that p 1 1 p 2 1 = p1 p 2 γ 1z 1 i p, ω i ) and p 1 2 p 2 2 = p1 p 2 γ 2z 1 i p, ω i ) for some γ 2 > γ 1 > 0. Then x i p 2, ω i ) i x i p 1, ω i ) i x i p, ω i ). Proof. Using p z i p, ω i ) = 0, we obtain p 1 1 p 2 1 z 1 i p, ω i ) + z 2 i p, ω i ) = ) p 1 p 2 γ 1zi 1 p, ω i ) zi 1 p, ω i ) + zi 2 p, ω i ) = γ 1 zi 1 p, ω i )zi 1 p, ω i ) < 0. 26

27 Thus, p 1 z i p, ω i ) < 0, which implies x i p 1, ω i ) i x i p, ω i ). Next, p 1 z i p, ω i ) < 0 also implies p z i p 1, ω i ) > 0 by the weak axiom. So, p 1 p 2 z1 i p 1, ω i ) + zi 2 p 1, ω i ) > 0 = p1 1 zi 1 p 1, ω i ) + zi 2 p 1, ω i ). p 2 1 Subtracting yields 0 < ) p 1 p 2 p1 1 p 2 zi 1 p 1, ω i ) = γ 1 zi 1 p, ω i )zi 1 p 1, ω i ). 1 Since γ 1 > 0, this means that z 1 i p, ω i)z 1 i p 1, ω i ) > 0. p 1 2 p 2 2 z 1 i p 1, ω i ) + z 2 i p 1, ω i ) = = = ) p 1 p 2 γ 2zi 1 p, ω i ) zi 1 p 1, ω i ) + zi 2 p 1, ω i ) ) p 1 p 2 γ 1 + γ 2 γ 1 ))zi 1 p, ω i ) zi 1 p 1, ω i ) + zi 2 p 1, ω i ) ) p 1 1 γ 2 γ 1 )zi 1 p, ω i ) zi 1 p 1, ω i ) + zi 2 p 1, ω i ) p 2 1 = γ 2 γ 1 )zi 1 p, ω i )zi 1 p 1, ω i ) < 0 by above. Therefore, p 2 z i p 1, ω i ) < 0, which implies x i p 2, ω i ) i x i p 1, ω i ). Lemma A.9. Fix any N δ). Assume ε 0, 1 2) and δ < δ. Then there exist N and 1 Z ++ such that for all > 1 the following holds. Let A = { k, p,..., p ) Ξ : k I and p N δ) }. For any ξ Ξ \ A, P ξ ξ ε τ Ξ A) KεN, where K > 0 is a constant that does not depend on ξ. Proof. Fix any N δ), where δ > δ, and let 1 satisfy Lemma A.6. For each i I, p T i, and Z ++, define ḡ i p, ) and ḡ i p) by T i p, ) N p, δ) = {λp + 1 λ)ḡ i p, ) : λ 0, 1)}, and T i p) N p, δ) = {λp + 1 λ)ḡ i p) : λ 0, 1)}. 27

28 Let d i ) = min p T i \N δ) 1 p 2 p + 1 )) 2ḡip, > 0, and choose d i such that 0 < d i < min p T i \N δ) 1 p 2 p ḡip)) Then there exists 2 such that for all > 2, d i ) > d i. Let 1 > max { 1, 2 } and fix > { 1. Let N > max p1 p, d p 2 p }. For all i I, let 2 d 1 S i p, ) = { 1 λḡ i p, ) + 1 λ) 2 p + 1 ) } 2ḡip, ) : λ [0, 1], and µ = min p T i \N δ) µ L S i p, )) > 0. Now, take any ξ = k, p,..., p) Ξ, where k I and p T i \ N δ). Consider any sequence of prices p 1, p 2,..., p N with p 1 = p and p n+1 S i p n, ) for all n. Since S i p n, ) T i p n, ) for all n, there exists γ n > 0 such that p 1 n+1 p 2 n+1 = p1 n p 2 + γ n z 1 p n, ω 1, ω 2 ) n by Lemma A.7. Since tâtonnement dynamics is globally stable in the underlying economy, p n+1 p < p n p for all n. Moreover, p n+1 p n > d i for all p n N δ). Therefore, there exists m N such that p m N δ) and p n N δ) for all n < m. For any n < m, p n+1 Bi, p n+1, p n,..., p n )) by Lemma A.6. Consequently, Prob ξ ε t+1 { i, p, p n+1,..., p n+1 ) : p S i p n+1, ) } ξ ε t = i, p n+1, p n,..., p n ) ) 1 2 µε1 ε) 1 ) 1 µε. 2 28

29 Therefore, there exists a constant K > 0 that does not depend on ξ such that P ξ τ Ξ A ξ t = ξ ) Kε N. Lastly, suppose ξ = k, p,..., p) Ξ, where k I and p T. Then the result follows readily from the above argument and Lemma A.3. Lemma A.10. Fix any N δ) T and N Z ++. Assume δ < δ. There exists 2 such that for all > 2, the following holds. Suppose ξ = i, p 1, p 2,..., p N, p,..., p) Ξ is such that p N δ) and p r N p, δ) \ {p} for each r = 1,..., N. Then p r Bξ) for any p r such that p r N δ). Proof. Let p i T j For each j I, let and p j T i be such that N δ) = {λ p i + 1 λ) p j : λ 0, 1)}. β j = inf p T i N δ) z j p, ω i ) z i p, ω j ) and β j = sup p T j N p i, δ) z i p, ω i ) z j p, ω j ). Then β j 0, 1) and β j 0, 1). Let β j 1, 1 β j ). Since x j p j, ω j ) j ω j for all j, there exists δ > 0 such that, for each j I, W j = {ω j : ω j ω j < δ and p j ω j p j ω j } has the following property. For any ω j, ω j W j and p, p {λ p i + 1 λ) p j : λ [0, 1]}, if x j p, ω j ) j x j p, ω j ), then x j p, ω j ) j x j p, ω j ). Moreover, by choosing δ small enough, W j can be made such that for all p T and ω j W j, z j p, ω j ) = ηz j p, ω j ) for some η β j, β i ). For each j I, let ζ j = max { 1, sup p T, ω j W j z j p, ω j ) } <. Let and fix >. > max { ζ1 N 2 δ β 1, β 1, ζ 2 N 2 δ β 2, β 2, N 1 β 1 β 1, N 1 β 2 β 2 } 29

30 Since there is nothing to prove if p N p i, δ) N p j, δ)), assume p N p i, δ) N p j, δ)) and p 1,..., p N N p, δ) \ {p}. Let s = {p r {p 1,..., p } : x j p r, ω j ) j x j p, ω j )}. Let Ψ 0 = and Φ 0 =. For each s = 1,..., s, define Ψ s and Φ s inductively as follows. Let Ψ s = Ψ s 1 \ Φ s 1. Let p {p r : r Ψ s } be such that x j p, ω j ) j x j p r, ω j ) for all r Ψ s, and let Φ s = {r Ψ s : p r = p }. In particular, Φ s+1 = {N + 1,..., }. Case 1: Suppose p N p i, δ). By Lemma A.2, z j p, ω j ) = βz i p, ω i ) for some β > 1. Consider any p r such that p1 r p 2 r Since p 1 r p 2 r = p1 p 2 γ rβ 1)z 1 i p, ω i ), = p1 p 2 + γ r z 1 p, ω 1, ω 2 ) for some γ r > 0. x i p r, ω i ) i x i p, ω i ) by Lemma A.8. Then p z j p r, ω j ) = βp z i p r, ω i ) < 0, and, therefore, x j p, ω j ) j x j p r, ω j ). Next, consider any p r such that p1 r p 2 r γ r z 1 p, ω 1, ω 2 ) for some γ r > 0. Since = p1 p 2 p 1 r p 2 r = p1 p 2 γ r 1 1 ) zj 1 p, ω j ), β x j p r, ω j ) j x j p, ω j ) by Lemma A.8. Therefore, x j p r, ω j ) j x j p, ω j ) if and only if p1 r p 2 r = p1 p 2 γ r z 1 p, ω 1, ω 2 ) for some γ r > 0. In particular, since tâtonnement dynamics is globally stable in the underlying economy, x j p r, ω j ) j p r N δ). x j p, ω j ) if Now, note that Ψ 1 = Ψ 1 and ω 1 j = ω j W j. Suppose for each stage s s, Ψ s = Ψ s and ω s j W j. Then Φ s = Φ s, and Φ s N. Consider any r Φ s. If 30

31 p r T, then a s = 0. If p r T, then p r Tj since δ < δ. So, α s = arg max = arg max = arg max = Φs β η u i ω i α ) Φ s z jp r, ωj s ) u i ω i α η ) Φ s z jp r, ω j ) ) u i ω i + α β η Φ s z ip r, ω i ) for some η > β j for some β > 1 > β j since Φ s β η N β j β j β j β j N ) = ζ j N 2 δ β j, β j δ ζ j N < 1. Therefore, ω s+1 j ωj s δ ζ j N z jp r, ωj s ) δ N. So, in either case, ω s+1 j W j. In addition, since Ψ s = Ψ s and Φ s = Φ s, Ψ s+1 = Ψ s+1. Therefore, by induction, Ψ s = Ψ s, Φ s = Φ s, and ω s j W j for all s = 1,..., s+1. In particular, Φ s+1 = {N + 1,..., }. Take any r Φ s, where s s. Since x j p r, ω s j ) j x j p, ω s j ), p z jp r, ω s j ) > 0. Therefore, p ω i α s z j p r, ω s j ) ) p ω i. Thus, x i p, ω i ) i ω ir for all r Φ s. On the other hand, α s+1 = arg max = arg max = arg max = N βη u i ω i α ) N z jp, ω s+1 j ) u i ω i α η ) N z jp, ω j ) for some η > β j u i ω i + α βη ) N z ip, ω i ), where 1 β β j 31

32 since N βη β j N) β j N) η β j < 1. So, ω ir = x i p, ω i ) for all r Φ s+1. Therefore, for all p r such that p r N δ), p r Bξ). Case 2: Suppose p N p j, δ). By Lemma A.2, z j p, ω j ) = βz i p, ω i ) for some β < 1. By similar reasoning as in Case 1, x j p r, ω j ) j x j p, ω j ) if and only if p1 r p 2 r = p1 p 2 + γ r z 1 p, ω 1, ω 2 ) for some γ r > 0. In particular, by the stability of the tâtonnement dynamics, if x j p r, ω j ) j x j p, ω j ), then p r N δ), and, therefore, z j p r, ω j ) = β z i p r, ω i ) for some β > β j. Thus, by similar reasoning as in Case 1, Ψ s = Ψ s, Φ s = Φ s, and ω s j W j for all s = 1,..., s + 1. In particular, Φ s+1 = {N + 1,..., }. α s+1 = arg max = arg max = arg max = 1 u i ω i α ) N z jp, ω s+1 j ) u i ω i α η ) N z jp, ω j ) for some η < β i u i ω i + α βη ) N z ip, ω i ), where β < β i since > N 1 β i β i > N 1 βη. Therefore, ω s+2 j = x j p, ω s+1 j ) and ω ir i ω i for all r Φ s+1. Now, consider any r Φ s+2. p r z j p r, ω s+2 j ) = 0 = p r z i p r, ω i ). Since x j p r, ω s+2 j ) ω s+2 j j = x j p, ω s+2 j ), 32

33 p z j p r, ω s+2 j ) > 0. On the other hand, since p 1 r p 2 r = p1 p 2 γ rz 1 p, ω 1, ω 2 ) = p1 p 2 γ r1 β)z 1 i p, ω i ), p r z i p, ω i ) < 0 by Lemma A.8. Therefore, p z i p r, ω i ) > 0. Therefore, there exists β > 0 such that z j p r, ω s+2 j ) = β z i p r, ω i ). So, for any K > 0, arg max ) u i ω i αkz j p r, ω s+2 j ) = 0. An induction argument yields that for all r Φ s, where s s + 2, ω ir = ω i. Therefore, for all p r such that p r N δ), p r Bξ). 33

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