Decentralized Price Adjustment in 2 2 Replica Economies
|
|
- Jack Holmes
- 5 years ago
- Views:
Transcription
1 Decentralized Price Adjustment in 2 2 Replica Economies Maxwell Pak Queen s University August 2005 This version: March 2006 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 Rob [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-Redondo [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 R 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 : R 2 ++ R 2 +, where = {p 1, p 2 ) R 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 R-replica economy is the economy with 2R consumers in which R consumers are exact copies of consumer 1 of the underlying economy and the remaining R 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 R 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 R = I R, where a state i, p 1,..., p R ) Ξ R 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 Rule 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 R ) Ξ R 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 R will denote the set {1,..., R}. Trading within the current period can now be described in the following inductive manner. Let Ψ 0 = R, 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 R 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 R ) 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 R 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 Rule Define a best price correspondence B from Ξ R into as follows. For any ξ = i, p 1,..., p R ) Ξ R, let trades occur according to the trading rule described above. Then B is defined by Bξ) = { p r {p 1,..., p R } : ˆω ir i ˆω ir r 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 R is chosen according to some arbitrary initial distribution. At t = 1, 2, 3,...: Suppose ξ = i, p 1,..., p R ) 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 R Z ++ and ε 0, 1), the price adjustment rule, together with the trading rule, defines a Markov chain ξ ε on Ξ R. Let λ L be the Lebesgue measure on R. 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 R, let A k = R r=1 A kr = {p 1,..., p R ) : k, p 1,..., p R ) A}. Let 1 denote the indicator function. For any ξ = i, p 1,..., p R ) Ξ R, the transition probability from ξ to A is given by Prob ξ ε t A ξ ε t 1 = ξ ) = 1 2 R 1 ε) Bξ) Air r= {p 1,...,p R ) 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 R 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) 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 for all 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 R Z ++ and ε 0, 1). Then the unique invariant measure π ε for the chain ξ ε on Ξ R 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 R be the produce measure R r=1 µ L on R, and let µ 0 be the measure on {, {1}, {2}, {1, 2}} such that µ 0 {1}) = µ 0 {2}) = ) ) 1 m Rm ˆδε. 2 2 Let µ be the product measure µ 0 µ R on BΞ R ). Consider any A BΞ R ). Let A 1 = {p 1,..., p R ) : 1, p 1,..., p R ) A} and A 2 = {p 1,..., p R ) : 2, p 1,..., p R ) A} so that {1} A 1 ) {2} A 2 ) = A and {1} A 1 ) {2} A 2 ) =. For all ξ Ξ R, Pξ m A) = P ξ m {1} A 1) + Pξ m {2} A 2) ) ) 1 m Rm ) ) ˆδε 1 m Rm ˆδε µ R A 1 ) + µ R A 2 ) = µa). Since µ is non-trivial by construction, the conclusion follows from Theorem 3.1 with ρ ε = 1 µξ R )) 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-Redondo [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 R are suppressed to reduce clutter in the notation whenever their suppression does not affect the clarity of the meaning. Lemma 3.4 Fix any R 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) R and for all ξ Ξ\ˆΞ, P ξ ˆΞ) 1 2R 1 ε)r, π ε ˆΞ) 1 ε) R π ε 1 dξ) + ˆΞ Ξ\ˆΞ 2R 1 ε)r π ε dξ) = 1 ε) R π ε ˆΞ) + 1 2R 1 ε)r π ε Ξ\ˆΞ) > 0. Moreover, the above inequality yields π ε ˆΞ) 1 ε) R π ε ˆΞ) 1 2R 1 ε)r 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) 2R 1 ε)r as ε 0, 1 1 ε) R 13
14 The two lemmas in the Appendix, Lemma A.8 and Lemma A.10, show that the 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 R such that for all R > R the following holds. Let π ε be the limiting distribution of ξ ε on Ξ R 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 R 1 from Lemma A.9 and R 2 from Lemma A.10. Let R = max { R 1, R 2 } and fix R > R. 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 Remarks 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 The following gives definitions and notations that are used throughout the Appendix. For each i I and bundle ω R ++, let s i ω) be the price at which type i demands exactly ω when her endowment is ω. That is, x i s i ω), ω) = ω. We call s i ω) the supporting price for type i at ω. Let ˆp i = s i ω i ) denote the supporting price for i at i s initial endowment. Let T = {λˆp i + 1 λ)ˆp j : λ [0, 1]} be the set of prices that are convex combinations of ˆp i and ˆp j. Let T i = {λˆp j + 1 λ)p : λ [0, 1]}. Since there are only two goods, p T and T = T i T j. Finally, for any set A, let A denote the relative interior of A. Given the initial endowment ω i, type i prefers ˆp j over ˆp i since x i ˆp j, ω i ) i ω i = x i ˆp j, ω i ). Lemma A.1 below shows that more generally given any two prices in T, type i prefers the price vector that is closer to ˆp j. In particular, type i strictly prefers prices in T i \ {p } to p. Two goods economy also means that the excess demand functions of the two types are colinear. Lemma A.2 shows that, moreover, the directions of the excess demands are opposite if p T and the same if p T. An immediate consequence of this lemma is that some trade will occur if and only if p T. Lemma A.1 Take any i I and 1 > λ > λ > 0. Let p = λ ˆp j + 1 λ )ˆp i and p = λˆp j + 1 λ)ˆp i. Then x i p, ω i ) i x i p, ω i ). Proof. Since x i p, ω i ) i ω i = x i ˆp i, ω i ), ˆp i z i p, ω i ) > 0 by the weak axiom of revealed preference. We also have ˆp j z i p, ω i ) = 1 λ p 1 λ)ˆp i) z i p, ω i ) ) 1 λ = ˆp i z i p, ω i ) λ < 0. 16
17 Let η = λ λ > 0. Then p z i p, ω i ) = λ ˆp j + 1 λ ) )ˆp i zi p, ω i ) = λ + η)ˆp j + 1 λ η)ˆp i ) z i p, ω i ) = ηˆp j z i p, ω i ) ηˆp 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 ). Lemma A.2 Fix i I. For each p, there exists β R 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. Since p z j p, ω j ) = 0 = p z i p, ω i ), there exists β R 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 = λˆp j + 1 λ)p. ˆp 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. Next, ˆp j z j p, ω j ) > 0 since x j p, ω j ) j ω j = x j ˆp j, ω j ). Therefore, β < 0. Next, z j p, ω j ) = z i p, ω i ), and z j ˆp j, ω j ) < z i ˆp j, ω i ). Since z, ω 1, ω 2 ) is continuous, and zp, ω 1, ω 2 ) 0 for all p Ti, β < 1 as required. Now, suppose p T. Then either ˆp i = λp + 1 λ)ˆp j for some λ 0, 1), or ˆp j = λp + 1 λ)ˆp i for some λ 0, 1). Assume ˆp i = λp + 1 λ)ˆp j for some λ 0, 1). Then ˆp i z i p, ω i ) > 0 since x i p, ω i ) i ω i = x i ˆp i, ω i ). We have, ˆp i z j p, ω j ) = λp + 1 λ)ˆp j ) z j p, ω j ) = 1 λ)ˆp j z j p, ω j ) 17
18 > 0. Therefore, β > 0. Lastly, the case where ˆp j = λp + 1 λ)ˆp i for some λ 0, 1) is similar. For each i I, define f i : T i Z ++ by x i f i p, R), ω i R 1 ) R z jp, ω j ) = ω i R 1 R z jp, ω j ). Define T i : T i Z ++ 2 by { 1 T i p, R) = T i λp + 1 λ) 2 p + 1 ) } 2 f ip, R) : λ [0, 1]. For any p T i p, R), there exists λ [0, 1] such that p = λp + 1 λ)ˆp j. Similarly, define ˆf ) i : Ti by x i ˆfi p), ω i z j p, ω j ) = ω i z j p, ω j ), and define ˆT i : T i 2 by { 1 ˆT i p) = T i λp + 1 λ) 2 p + 1 ) } 2 ˆf i p) : λ [0, 1]. Lemma A.3 Fix any R Z ++, p T i, and p T i p, R) \ {p}. Then there exists β > 0 such that z i p, ω i R 1 ) R z jp, ω j ) = βz j p, ω j ). Proof. Since p z i p, ω i R 1 ) R z jp, ω j ) = 0 = p z j p, ω j ), there exists β R such that z i p, ω i R 1 ) R z jp, ω j ) = βz j p, ω j ). 18
19 To see that β > 0, first note that x j p, ω j ) j x j p, ω j ) by Lemma A.1. Thus, p z j p, ω j ) > 0. Let p = f i p, R). Then x i p, ω i R 1 ) R z jp, ω j ) i ω i R 1 R z jp, ω j ) = x i p, ω i R 1 ) R z jp, ω j ). Since p = λp + 1 λ)p for some λ 0, 1), p z i p, ω i R 1 ) R z jp, ω j ) = 1 p 1 λ)p ) z i p, ω i R 1 ) λ R z jp, ω j ) ) 1 λ = p z i p, ω i R 1 ) λ R z jp, ω j ) < 0. Therefore, β > 0. Lemma A.4 For any R Z ++, p = p 1, p 2 ) T i, and q = q1, q 2 ) T i p, R)\{p}, there exists γ > 0 such that q 1 q 2 = p1 p 2 + γz1 p, ω 1, ω 2 ). Proof. Since p T i, there exists β 0, 1) such that z jp, ω j ) = βz i p, ω i ). Also, since q T i p, R) \ {p}, there exists λ 0, 1) such that p = λq + 1 λ)ˆp j. ) 1 q zp, ω 1, ω 2 ) = p 1 λ)ˆp j ) 1 β)z i p, ω i ) λ ) 1 λ)1 β) = ˆp j z i p, ω i ) λ > 0 since ˆp j z i p, ω i ) < 0 as seen in the proof of Lemma A.2. Since q 1 q 2 z1 p, ω 1, ω 2 ) + z 2 p, ω 1, ω 2 ) > 0 = p1 p 2 z1 p, ω 1, ω 2 ) + z 2 p, ω 1, ω 2 ), q 1 q 2 z1 p, ω 1, ω 2 ) > p1 p 2 z1 p, ω 1, ω 2 ). 19
20 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 ). 20
21 Lemma A.5 Fix any p = p 1, p 2 ) and ω i R ++. Suppose p 1 = p 1 1, p2 1 ), p 2 = p 1 2, p2 2 ) is 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 p 2 z i p 1, ω i ) < 0 and p 1 z i p, ω i ) < 0. Hence, x i p 2, ω i ) i x i p 1, ω i ) i x i p, ω i ). Proof. Since 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, p 1 z i p, ω i ) < 0. Then p z i p 1, ω i ) > 0 by the weak axiom. Therefore, p 1 p 2 z1 i p 1, ω i ) + z 2 i p 1, ω i ) > 0 while p 1 1 p 2 1 z 1 i p 1, ω i ) + z 2 i p 1, ω i ) = 0. Thus, p 1 0 < p 2 p1 1 p 2 1 ) z 1 i p 1, ω i ) = γ 1 z 1 i p, ω i )z 1 i p 1, ω i ). Since γ 1 > 0, 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. Therefore, p 2 z i p 1, ω i ) < 0, and x i p 2, ω i ) i x i p 1, ω i ) i x i p, ω i ). 21
22 Lemma A.6 Fix any R Z ++. For any p \ T and p, there exists k I such that p Bk, p, p,..., p)). Proof. Suppose p T. Then, there exists λ 0, 1) such that p = λ ˆp i + 1 λ )ˆp j. Since p T, either ˆp i = λp+1 λ)ˆp j for some λ 0, 1], or ˆp j = λp+1 λ)ˆp i for some λ 0, 1]. Without loss of generality, assume ˆp i = λp + 1 λ)ˆp j for some λ 0, 1]. Then p = λ ˆp i + 1 λ )ˆp j = λ λp + 1 λ)ˆp j ) + 1 λ )ˆp j = λ λp + 1 λ λ)ˆp j. p z j p, ω j ) = 1 λ λ p 1 λ λ)ˆp j ) z j p, ω j ) 1 λ ) λ = λ ˆp j z j p, ω j ) λ < 0 since x j p, ω j ) j x j ˆp j, ω j ). Therefore, x j p, ω j ) j x j p, ω j ). Now, let ξ = i, p, p,..., p) Ξ R. Then Φ 1 = {2, 3, 4..., R}. Since x i p, ω i ) i ω i = x i ˆp i, ω i ), ˆp i z i p, ω i ) 0. Therefore, α [0, 1], ˆp i ω i α R ) R 1 z jp, ω j ) = ˆp i ω i αβ R ) R 1 z ip, ω i ) ˆp i ω i. for some β 0 So, ω i = x i ˆp i, ω i ) i ω i α R R 1 z jp, ω j ) for all α [0, 1]. Therefore, α 1 = arg max u i ω i α R ) R 1 z jp, ω j ) = 0. So, r 1, ˆω ir = ω i, and ω 2 jr = ω j. Let α 2 = arg max u i ωi αrz j p, ω j ) ). 22
23 Then ˆω i1 = ω i α 2 Rz j p, ω j ) i ω i = ˆω ir r 1. Therefore, p Bk, p, p,..., p)). Lastly, suppose p T. Then similar argument to above yields α 1 = α 2 = 0. Therefore, ˆω ir = ω i for all r R, and, trivially, p Bk, p, p,..., p)) for any k I. Lemma A.7 Fix any N δ). Then there exists R 1 such that for all R > R 1, the following holds. For any p T i \ N δ) and p T i p, R) \ {p}, let α = arg max u j ω j αrz i p, ω i R 1 )) R z jp, ω j ). Then ω j α Rz i p, ω i R 1 ) R z jp, ω j ) = x j p, ω j ). Proof. Let ζ j = max p T z j p, ω j ). Then ζ j 0, ). For each p T i \ N δ), let ĥ i p) = min p ˆT i p) z i p, ω i z j p, ω j )). Then ĥi ) is continuous by the theorem of the maximum, and ĥip) > 0 for all p T i \ N δ). Thus, ĥ i min p T i \N δ) ĥ i p) > 0. Consider any R > 1 and p T i \ N δ). Let h i p, R) = min p T i p,r) z i p, ω i R 1 ) R z jp, ω j ) > 0. Since h i, R) is continuous, h i R) min h ip, R) > 0. p T i \N δ) Moreover, h i R) ĥi > 0 as R. So, there exists R i and η i 0, ĥi) such that for all R > R i, h i R) > η i. Now, let R 1 > max { } R 1, R 2, ζ 1 η 2, ζ 2 η 1 and fix any R > R 1. Suppose p T i \ N δ) 23
24 and p T i p, R) \ {p}. By Lemma A.3, there exists β > 0 such that z i p, ω i R 1 ) R z jp, ω j ) = βz j p, ω j ). Therefore, for all α R, ω j αrz i p, ω i R 1 ) R z jp, ω j ) = ω j + αrβz j p, ω j ). Then arg max α R u j ω j αrz i p, ω i R 1 )) R z jp, ω j ) = 1 Rβ. Since βζ j βz j p, ω j ) = z i p, ω i R 1 ) R z jp, ω j ) η i, ) ) β η i ζj ζ j. Therefore, Rβ ηi η i ζ j = 1, and α = arg max u j ω j αrz i p, ω i R 1 )) R z jp, ω j ) = 1 Rβ as required. Lemma A.8 Fix any N δ). Then there exists R 1 such that for all R > R 1, the following holds. Suppose ξ = j, p, p,..., p) Ξ R is such that p T i \ N δ) and p T i p, R). Then p Bξ). Proof. Let R 1 be as in Lemma A.7 and fix R > R 1. Consider any ξ = j, p, p,..., p) satisfying the hypothesis. If p = ˆp j or p = p, then there is nothing to prove. So, assume p ˆp j and p T i p, R) \ {p}. Then there exists λ 0, 1) such that p = λp + 1 λ)ˆp j. In addition, since p T i \ {ˆp 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)ˆp j ) z j p, ω j ) = β 1 λ)ˆp j z j p, ω j ) 24
25 < 0. Therefore, x i p, ω i ) i x i p, ω i ), and Φ 1 = {2, 3, 4,..., R}. 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 R ) R 1 z ip, ω i ) = arg max = βr 1). R ) R u j ω j + α βr 1) z jp, ω j ) Therefore, ˆω jr = x j p, ω j ) for all r Φ 1, and ω 2 i = ω i + βr 1) z i p, ω i ) = ω i R 1 R R z jp, ω j ). Since Φ 2 = {1}, ˆω j1 = ω j α 2 Rz i p, ω i R 1 ) R z jp, ω j ), where α 2 = arg max u j ω j αrz i p, ω i R 1 )) R z jp, ω j ). Thus, ˆω j1 = x j p, ω j ) by Lemma A.7. Since x j p, ω j ) j x j p, ω j ) by Lemma A.1, p Bξ). Lemma A.9 Fix any N δ). Assume ε 0, 1 2) and ˆδ < δ. Then there exist N and R 1 Z ++ such that for all R > R 1 the following holds. Let A = { k, p,..., p ) Ξ R : k I and p N δ) }. For any ξ ˆΞ \ A, P ξ ξ ε τˆξ A) KεN, where K > 0 is a constant that does not depend on ξ. Proof. Fix any N δ), where δ > ˆδ, and let R 1 satisfy Lemma A.8. For each i I, p T i, and R Z ++, define ḡ i p, R) and ḡ i p) by T i p, R) N p, ˆδ) = {λp + 1 λ)ḡ i p, R) : λ 0, 1)}, and 25
26 ˆT i p) N p, ˆδ) = {λp + 1 λ)ḡ i p) : λ 0, 1)}. Let d i R) = min p T i \N δ) 1 p 2 p + 1 R)) 2ḡip, > 0, and choose ˆd i such that 0 < ˆd i < min p T i \N δ) 1 p 2 p ḡip)) Then there exists R 2 such that for all R > R 2, d i R) > ˆd i. Let R 1 > max {R 1, R 2 } and fix R > R 1. Let N > max S i p, R) = { ˆp1 p ˆd 2, ˆp 2 p ˆd 1 }. For all i I, let { 1 λḡ i p, R) + 1 λ) 2 p + 1 ) } 2ḡip, R) : λ [0, 1], and µ = min p T i \N δ) µ L S i p, R)) > 0. Now, take any ξ = k, p,..., p) Ξ R, 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, R) for all n. Since S i p n, R) T i p n, R) 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.4. 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 > ˆd 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.8. Consequently, Prob ξ ε t+1 { i, p, p n+1,..., p n+1 ) : p S i p n+1, R) } ξ ε t = i, p n+1, p n,..., p n ) ) 1 2 µε1 ε)r 1 ) 1 R µε. 2 26
27 Therefore, there exists a constant K > 0 that does not depend on ξ such that P ) ξ τˆξ A ξ t = ξ Kε N. Lastly, suppose ξ = k, p,..., p) Ξ R, where k I and p T. Then the result follows readily from the above argument and Lemma A.6. Lemma A.10 Fix any N δ) T and N Z ++. Assume ˆδ < δ. There exists R 2 such that for all R > R 2, the following holds. Suppose ξ = i, p 1, p 2,..., p N, p,..., p) Ξ R is such that p N δ) and p r N p, ˆδ) \ {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)}. ˆβ 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,ˆδ) z i p, ω i ) z j p, ω j ). Then ˆβ 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 {λˆp 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 R > R. R > max { ζ 1 N 2 δ β 1, ˆβ 1, ζ 2 N 2 δ β 2, ˆβ 2, N 1 β 1 β 1, N 1 β 2 β 2 } 27
28 Since there is nothing to prove if p N p i, ˆδ) N p j, ˆδ)), assume p N p i, ˆδ) N p j, ˆδ)) and p 1,..., p N N p, ˆδ) \ {p}. Let s = {p r {p 1,..., p R } : x j p r, ω j ) j x j p, ω j )}. Let ˆΨ 0 = R and ˆΦ 0 =. For each s = 1,..., s, define ˆΨ s and ˆΦ s inductively as follows. Let ˆΨ s = ˆΨ s 1 \ ˆΦ s 1. Let p {p r : r ˆΨ s } be such that x j p, ω j ) j x j p r, ω j ) for all r ˆΨ s, and let ˆΦ s = {r ˆΨ s : p r = p }. In particular, ˆΦ s+1 = {N + 1,..., R}. Case 1: Suppose p N p i, ˆδ). 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.5. 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.5. 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 = ˆΨ 1 and ω 1 j = ω j W j. Suppose for each stage s s, Ψ s = ˆΨ s and ω s j W j. Then Φ s = ˆΦ s, and Φ s N. Consider any r Φ s. If p r T, then a s = 0. If p r T, then p r T j since ˆδ < δ. So, α s = arg max = arg max = arg max u i ω i α R ) Φ s z jp r, ωj s ) u i ω i α ηr ) Φ s z jp r, ω j ) ) u i ω i + α β ηr Φ s z ip r, ω i ) for some η > β j for some β > 1 > ˆβ j 28
29 = Φs β ηr since Φ s β ηr N ˆβ j β j R ˆβ j β j N ) = ζ j N 2 δ β j, ˆβ 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 = ˆΨ s and Φ s = ˆΦ s, Ψ s+1 = ˆΨ s+1. Therefore, by induction, Ψ s = ˆΨ s, Φ s = ˆΦ s, and ω s j W j for all s = 1,..., s+1. In particular, Φ s+1 = {N + 1,..., R}. 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 ˆω ir for all r Φ s. On the other hand, α s+1 = arg max = arg max = arg max = R N βηr u i ω i α R ) R N z jp, ω s+1 j ) u i ω i α ηr ) R N z jp, ω j ) for some η > β j u i ω i + α βηr ) R N z ip, ω i ), where 1 β β j since R N βηr β j R N) β j R N) ηr β j R < 1. So, ˆω 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, ˆδ). 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 29
30 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 > ˆβ j. Thus, by similar reasoning as in Case 1, Ψ s = ˆΨ s, Φ s = ˆΦ s, and ω s j W j for all s = 1,..., s + 1. In particular, Φ s+1 = {N + 1,..., R}. α s+1 = arg max = arg max = arg max = 1 u i ω i α R ) R N z jp, ω s+1 j ) u i ω i α ηr ) R N z jp, ω j ) for some η < β i u i ω i + α βηr ) R N z ip, ω i ), where β < β i since R > N 1 β i β i > N 1 βη. Therefore, ω s+2 j = x j p, ω s+1 j ) and ˆω 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 ), 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.5. 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. 30
31 An induction argument yields that for all r Φ s, where s s + 2, ˆω ir = ω i. Therefore, for all p r such that p r N δ), p r Bξ). 31
32 References [1] Crockett, S., S. Spear, and S. Sunder. 2004). A Simple Decentralized Institution for Learning Competitive Equilibrium. Discussion Paper. [2] Foster, D. and P. Young. 1990). Stochastic Evolutionary Game Dynamics. Theoretical Population Biology, 38: [3] Hahn, F. 1982). Stability. Chap. 16 in Handbook of Mathematical Economics, vol. II, edited by K. Arrow and M. Intriligator. Amsterdam: North-Holland. [4] Kandori, M., G. Mailath, and R. Rob. 1993). Learning to Play Equilibria in Games with Stochastic Perturbations. Econometrica, 61: [5] Keisler, H. Jerome. 1996). Getting to a Competitive Equilibrium. Econometrica, 64: [6] Meyn, S. and R. Tweedie. 1996). Markov Chains and Stochastic Stability. NY: Springer. [7] Temzelides, T. 2001). A Note on Learning to Bid Walrasian in 2 2 Market Games. Discussion Paper. [8] Vega-Redondo, F. 1997). The Evolution of Walrasian Behavior. Econometrica, 65: [9] Walras, L. 1874). Elements d Economie Politique Pure. Lausanne: Corbaz. [Translated as: Elements of Pure Economics. Homewood, IL: Irwin, 1954.] [10] Young, P. 1993). The Evolution of Conventions. Econometrica, 61:
Decentralized Price Adjustment in 2 2 Replica. Economies
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
More informationDecentralized Price Adjustment in 2 2 Replica. Economies
Decentralized Price Adjustment in 2 2 eplica Economies Maxwell Pak Queen s University December 2009 Abstract This paper presents a model of price adjustment in replica economies with two consumer types
More informationUniqueness, Stability, and Gross Substitutes
Uniqueness, Stability, and Gross Substitutes Econ 2100 Fall 2017 Lecture 21, November 14 Outline 1 Uniquenness (in pictures) 2 Stability 3 Gross Substitute Property Uniqueness and Stability We have dealt
More informationKIER DISCUSSION PAPER SERIES
KIER DISCUSSION PAPER SERIES KYOTO INSTITUTE OF ECONOMIC RESEARCH Discussion Paper No.992 Intertemporal efficiency does not imply a common price forecast: a leading example Shurojit Chatterji, Atsushi
More informationTitle: The existence of equilibrium when excess demand obeys the weak axiom
Title: The existence of equilibrium when excess demand obeys the weak axiom Abstract: This paper gives a non-fixed point theoretic proof of equilibrium existence when the excess demand function of an exchange
More informationOn the Maximal Domain Theorem
On the Maximal Domain Theorem Yi-You Yang April 28, 2016 Abstract The maximal domain theorem by Gul and Stacchetti (J. Econ. Theory 87 (1999), 95-124) shows that for markets with indivisible objects and
More informationEconomics 201B Second Half. Lecture 12-4/22/10. Core is the most commonly used. The core is the set of all allocations such that no coalition (set of
Economics 201B Second Half Lecture 12-4/22/10 Justifying (or Undermining) the Price-Taking Assumption Many formulations: Core, Ostroy s No Surplus Condition, Bargaining Set, Shapley-Shubik Market Games
More informationThe B.E. Journal of Theoretical Economics
The B.E. Journal of Theoretical Economics Topics Volume 9, Issue 1 2009 Article 43 Simple Economies with Multiple Equilibria Theodore C. Bergstrom Ken-Ichi Shimomura Takehiko Yamato University of California,
More informationGeneral Equilibrium. General Equilibrium, Berardino. Cesi, MSc Tor Vergata
General Equilibrium Equilibrium in Consumption GE begins (1/3) 2-Individual/ 2-good Exchange economy (No production, no transaction costs, full information..) Endowment (Nature): e Private property/ NO
More informationIntroduction to General Equilibrium
Introduction to General Equilibrium Juan Manuel Puerta November 6, 2009 Introduction So far we discussed markets in isolation. We studied the quantities and welfare that results under different assumptions
More informationMISTAKES IN COOPERATION: the Stochastic Stability of Edgeworth s Recontracting. Roberto Serrano Oscar Volij. January 2003
MISTAKES IN COOPERATION: the Stochastic Stability of Edgeworth s Recontracting Roberto Serrano Oscar Volij January 2003 Abstract. In an exchange economy with a finite number of indivisible goods, we analyze
More informationQuestion 1. (p p) (x(p, w ) x(p, w)) 0. with strict inequality if x(p, w) x(p, w ).
University of California, Davis Date: August 24, 2017 Department of Economics Time: 5 hours Microeconomics Reading Time: 20 minutes PRELIMINARY EXAMINATION FOR THE Ph.D. DEGREE Please answer any three
More informationIntroduction to General Equilibrium: Framework.
Introduction to General Equilibrium: Framework. Economy: I consumers, i = 1,...I. J firms, j = 1,...J. L goods, l = 1,...L Initial Endowment of good l in the economy: ω l 0, l = 1,...L. Consumer i : preferences
More informationMathematical models in economy. Short descriptions
Chapter 1 Mathematical models in economy. Short descriptions 1.1 Arrow-Debreu model of an economy via Walras equilibrium problem. Let us consider first the so-called Arrow-Debreu model. The presentation
More informationCompetitive Equilibrium
Competitive Equilibrium Econ 2100 Fall 2017 Lecture 16, October 26 Outline 1 Pareto Effi ciency 2 The Core 3 Planner s Problem(s) 4 Competitive (Walrasian) Equilibrium Decentralized vs. Centralized Economic
More informationThe Consumer, the Firm, and an Economy
Andrew McLennan October 28, 2014 Economics 7250 Advanced Mathematical Techniques for Economics Second Semester 2014 Lecture 15 The Consumer, the Firm, and an Economy I. Introduction A. The material discussed
More informationSURPLUS SHARING WITH A TWO-STAGE MECHANISM. By Todd R. Kaplan and David Wettstein 1. Ben-Gurion University of the Negev, Israel. 1.
INTERNATIONAL ECONOMIC REVIEW Vol. 41, No. 2, May 2000 SURPLUS SHARING WITH A TWO-STAGE MECHANISM By Todd R. Kaplan and David Wettstein 1 Ben-Gurion University of the Negev, Israel In this article we consider
More informationMarket Equilibrium and the Core
Market Equilibrium and the Core Ram Singh Lecture 3-4 September 22/25, 2017 Ram Singh (DSE) Market Equilibrium September 22/25, 2017 1 / 19 Market Exchange: Basics Let us introduce price in our pure exchange
More informationDifferentiable Welfare Theorems Existence of a Competitive Equilibrium: Preliminaries
Differentiable Welfare Theorems Existence of a Competitive Equilibrium: Preliminaries Econ 2100 Fall 2017 Lecture 19, November 7 Outline 1 Welfare Theorems in the differentiable case. 2 Aggregate excess
More informationConvergence of Non-Normalized Iterative
Convergence of Non-Normalized Iterative Tâtonnement Mitri Kitti Helsinki University of Technology, Systems Analysis Laboratory, P.O. Box 1100, FIN-02015 HUT, Finland Abstract Global convergence conditions
More information1 Second Welfare Theorem
Econ 701B Fall 018 University of Pennsylvania Recitation : Second Welfare Theorem Xincheng Qiu (qiux@sas.upenn.edu) 1 Second Welfare Theorem Theorem 1. (Second Welfare Theorem) An economy E satisfies (A1)-(A4).
More informationCourse Handouts: Pages 1-20 ASSET PRICE BUBBLES AND SPECULATION. Jan Werner
Course Handouts: Pages 1-20 ASSET PRICE BUBBLES AND SPECULATION Jan Werner European University Institute May 2010 1 I. Price Bubbles: An Example Example I.1 Time is infinite; so dates are t = 0,1,2,...,.
More informationDiscussion Papers in Economics
Discussion Papers in Economics No. 10/11 A General Equilibrium Corporate Finance Theorem for Incomplete Markets: A Special Case By Pascal Stiefenhofer, University of York Department of Economics and Related
More informationDepartment of Agricultural Economics. PhD Qualifier Examination. May 2009
Department of Agricultural Economics PhD Qualifier Examination May 009 Instructions: The exam consists of six questions. You must answer all questions. If you need an assumption to complete a question,
More informationStochastic Stability of General Equilibrium. Antoine Mandel and Herbert Gintis
Stochastic Stability of General Equilibrium Antoine Mandel and Herbert Gintis June 1, 2012 1 Introduction The stability of Walrasian general equilibrium was widely studied in the years immediately following
More informationLecture 6: Communication Complexity of Auctions
Algorithmic Game Theory October 13, 2008 Lecture 6: Communication Complexity of Auctions Lecturer: Sébastien Lahaie Scribe: Rajat Dixit, Sébastien Lahaie In this lecture we examine the amount of communication
More informationFirst Welfare Theorem
First Welfare Theorem Econ 2100 Fall 2017 Lecture 17, October 31 Outline 1 First Welfare Theorem 2 Preliminaries to Second Welfare Theorem Past Definitions A feasible allocation (ˆx, ŷ) is Pareto optimal
More informationNear-Potential Games: Geometry and Dynamics
Near-Potential Games: Geometry and Dynamics Ozan Candogan, Asuman Ozdaglar and Pablo A. Parrilo January 29, 2012 Abstract Potential games are a special class of games for which many adaptive user dynamics
More informationMicroeconomics, Block I Part 2
Microeconomics, Block I Part 2 Piero Gottardi EUI Sept. 20, 2015 Piero Gottardi (EUI) Microeconomics, Block I Part 2 Sept. 20, 2015 1 / 48 Pure Exchange Economy H consumers with: preferences described
More informationImplementation of the Ordinal Shapley Value for a three-agent economy 1
Implementation of the Ordinal Shapley Value for a three-agent economy 1 David Pérez-Castrillo 2 Universitat Autònoma de Barcelona David Wettstein 3 Ben-Gurion University of the Negev April 2005 1 We gratefully
More informationA finite population ESS and a long run equilibrium in an n players coordination game
Mathematical Social Sciences 39 (000) 95 06 www.elsevier.nl/ locate/ econbase A finite population ESS and a long run equilibrium in an n players coordination game Yasuhito Tanaka* Faculty of Law, Chuo
More informationInformed Principal in Private-Value Environments
Informed Principal in Private-Value Environments Tymofiy Mylovanov Thomas Tröger University of Bonn June 21, 2008 1/28 Motivation 2/28 Motivation In most applications of mechanism design, the proposer
More informationRegularity of competitive equilibria in a production economy with externalities
Regularity of competitive equilibria in a production economy with externalities Elena del Mercato Vincenzo Platino Paris School of Economics - Université Paris 1 Panthéon Sorbonne QED-Jamboree Copenhagen,
More informationComputational procedure for a time-dependent Walrasian price equilibrium problem
Communications to SIMAI Congress, ISSN 1827-9015, Vol. 2 2007 DOI: 10.1685/CSC06159 Computational procedure for a time-dependent Walrasian price equilibrium problem M. B. DONATO 1 and M. MILASI 2 Department
More informationCan everyone benefit from innovation?
Can everyone benefit from innovation? Christopher P. Chambers and Takashi Hayashi June 16, 2017 Abstract We study a resource allocation problem with variable technologies, and ask if there is an allocation
More informationPositive Theory of Equilibrium: Existence, Uniqueness, and Stability
Chapter 7 Nathan Smooha Positive Theory of Equilibrium: Existence, Uniqueness, and Stability 7.1 Introduction Brouwer s Fixed Point Theorem. Let X be a non-empty, compact, and convex subset of R m. If
More informationPhD Qualifier Examination
PhD Qualifier Examination Department of Agricultural Economics July 26, 2013 Instructions The exam consists of six questions. You must answer all questions. If you need an assumption to complete a question,
More informationIn the Name of God. Sharif University of Technology. Microeconomics 1. Graduate School of Management and Economics. Dr. S.
In the Name of God Sharif University of Technology Graduate School of Management and Economics Microeconomics 1 44715 (1396-97 1 st term) - Group 1 Dr. S. Farshad Fatemi Chapter 10: Competitive Markets
More informationGeneral Equilibrium and Welfare
and Welfare Lectures 2 and 3, ECON 4240 Spring 2017 University of Oslo 24.01.2017 and 31.01.2017 1/37 Outline General equilibrium: look at many markets at the same time. Here all prices determined in the
More informationBayesian Persuasion Online Appendix
Bayesian Persuasion Online Appendix Emir Kamenica and Matthew Gentzkow University of Chicago June 2010 1 Persuasion mechanisms In this paper we study a particular game where Sender chooses a signal π whose
More informationRepresentation of TU games by coalition production economies
Working Papers Institute of Mathematical Economics 430 April 2010 Representation of TU games by coalition production economies Tomoki Inoue IMW Bielefeld University Postfach 100131 33501 Bielefeld Germany
More informationThe Bargaining Set of an Exchange Economy with Discrete. Resources
The Bargaining Set of an Exchange Economy with Discrete Resources Murat Yılmaz and Özgür Yılmaz November 17, 2016 Abstract A central notion for allocation problems when there are private endowments is
More informationThe Definition of Market Equilibrium The concept of market equilibrium, like the notion of equilibrium in just about every other context, is supposed to capture the idea of a state of the system in which
More informationExistence, Incentive Compatibility and Efficiency of the Rational Expectations Equilibrium
Existence, ncentive Compatibility and Efficiency of the Rational Expectations Equilibrium Yeneng Sun, Lei Wu and Nicholas C. Yannelis Abstract The rational expectations equilibrium (REE), as introduced
More informationNotes on General Equilibrium
Notes on General Equilibrium Alejandro Saporiti Alejandro Saporiti (Copyright) General Equilibrium 1 / 42 General equilibrium Reference: Jehle and Reny, Advanced Microeconomic Theory, 3rd ed., Pearson
More information1. Introduction. 2. A Simple Model
. Introduction In the last years, evolutionary-game theory has provided robust solutions to the problem of selection among multiple Nash equilibria in symmetric coordination games (Samuelson, 997). More
More informationANSWER KEY. University of California, Davis Date: June 22, 2015
ANSWER KEY University of California, Davis Date: June, 05 Department of Economics Time: 5 hours Microeconomic Theory Reading Time: 0 minutes PRELIMINARY EXAMINATION FOR THE Ph.D. DEGREE Please answer four
More informationThe Walrasian Model and Walrasian Equilibrium
The Walrasian Model and Walrasian Equilibrium 1.1 There are only two goods in the economy and there is no way to produce either good. There are n individuals, indexed by i = 1,..., n. Individual i owns
More informationLearning, Memory, and Inertia *
Learning, Memory, and Inertia * Carlos Alós Ferrer E mail: Carlos.Alos Ferrer@univie.ac.at Department of Economics, University of Vienna, Hohenstaufengasse 9, A 1010 Vienna, Austria This paper explores
More information1 Lattices and Tarski s Theorem
MS&E 336 Lecture 8: Supermodular games Ramesh Johari April 30, 2007 In this lecture, we develop the theory of supermodular games; key references are the papers of Topkis [7], Vives [8], and Milgrom and
More informationOn the Existence of Price Equilibrium in Economies with Excess Demand Functions
On the Existence of Price Equilibrium in Economies with Excess Demand Functions Guoqiang Tian Abstract This paper provides a price equilibrium existence theorem in economies where commodities may be indivisible
More informationSecond Welfare Theorem
Second Welfare Theorem Econ 2100 Fall 2015 Lecture 18, November 2 Outline 1 Second Welfare Theorem From Last Class We want to state a prove a theorem that says that any Pareto optimal allocation is (part
More informationThe Survival Assumption in Intertemporal Economy J.M. Bonnisseau and A. Jamin 1 November Abstract
The Survival Assumption in Intertemporal Economy J.M. Bonnisseau and A. Jamin 1 November 2009 Abstract In an economy with a non-convex production sector, we provide an assumption for general pricing rules
More information3. THE EXCHANGE ECONOMY
Essential Microeconomics -1-3. THE EXCHNGE ECONOMY Pareto efficient allocations 2 Edgewort box analysis 5 Market clearing prices 13 Walrasian Equilibrium 16 Equilibrium and Efficiency 22 First welfare
More informationAlfred Marshall s cardinal theory of value: the strong law of demand
Econ Theory Bull (2014) 2:65 76 DOI 10.1007/s40505-014-0029-5 RESEARCH ARTICLE Alfred Marshall s cardinal theory of value: the strong law of demand Donald J. Brown Caterina Calsamiglia Received: 29 November
More informationEC487 Advanced Microeconomics, Part I: Lecture 5
EC487 Advanced Microeconomics, Part I: Lecture 5 Leonardo Felli 32L.LG.04 27 October, 207 Pareto Efficient Allocation Recall the following result: Result An allocation x is Pareto-efficient if and only
More informationNoncooperative Games, Couplings Constraints, and Partial Effi ciency
Noncooperative Games, Couplings Constraints, and Partial Effi ciency Sjur Didrik Flåm University of Bergen, Norway Background Customary Nash equilibrium has no coupling constraints. Here: coupling constraints
More informationCore equivalence and welfare properties without divisible goods
Core equivalence and welfare properties without divisible goods Michael Florig Jorge Rivera Cayupi First version November 2001, this version May 2005 Abstract We study an economy where all goods entering
More informationThe Last Word on Giffen Goods?
The Last Word on Giffen Goods? John H. Nachbar February, 1996 Abstract Giffen goods have long been a minor embarrassment to courses in microeconomic theory. The standard approach has been to dismiss Giffen
More informationBROUWER S FIXED POINT THEOREM: THE WALRASIAN AUCTIONEER
BROUWER S FIXED POINT THEOREM: THE WALRASIAN AUCTIONEER SCARLETT LI Abstract. The focus of this paper is proving Brouwer s fixed point theorem, which primarily relies on the fixed point property of the
More informationUNIVERSITY OF VIENNA
WORKING PAPERS Carlos Alós-Ferrer Cournot versus Walras in Dynamic Oligopolies with Memory August 2001 Working Paper No: 0110 DEPARTMENT OF ECONOMICS UNIVERSITY OF VIENNA All our working papers are available
More informationEconomics 200A part 2 UCSD Fall quarter 2011 Prof. R. Starr Mr. Troy Kravitz1 FINAL EXAMINATION SUGGESTED ANSWERS
Economics 200A part 2 UCSD Fall quarter 2011 Prof. R. Starr Mr. Troy Kravitz1 FINAL EXAMINATION SUGGESTED ANSWERS This exam is take-home, open-book, open-notes. You may consult any published source (cite
More informationFixed Point Theorems
Fixed Point Theorems Definition: Let X be a set and let f : X X be a function that maps X into itself. (Such a function is often called an operator, a transformation, or a transform on X, and the notation
More informationThe Fundamental Welfare Theorems
The Fundamental Welfare Theorems The so-called Fundamental Welfare Theorems of Economics tell us about the relation between market equilibrium and Pareto efficiency. The First Welfare Theorem: Every Walrasian
More information18 U-Shaped Cost Curves and Concentrated Preferences
18 U-Shaped Cost Curves and Concentrated Preferences 18.1 U-Shaped Cost Curves and Concentrated Preferences In intermediate microeconomic theory, a firm s cost function is often described as U-shaped.
More informationImplementability, Walrasian Equilibria, and Efficient Matchings
Implementability, Walrasian Equilibria, and Efficient Matchings Piotr Dworczak and Anthony Lee Zhang Abstract In general screening problems, implementable allocation rules correspond exactly to Walrasian
More informationImplementation in Economies with Non-Convex Production Technologies Unknown to the Designer
Implementation in Economies with Non-Convex Production Technologies Unknown to the Designer Guoqiang TIAN Department of Economics Texas A&M University College Station, Texas 77843 and Institute for Advanced
More informationA Reversal of Rybczynski s Comparative Statics via Anything Goes *
A Reversal of Rybczynski s Comparative Statics via Anything Goes * Hugo F. Sonnenschein a and Marcus M. Opp b a Department of Economics, University of Chicago, 6 E. 59 th Street, Chicago, IL 60637 h-sonnenschein@uchicago.edu
More informationproblem. max Both k (0) and h (0) are given at time 0. (a) Write down the Hamilton-Jacobi-Bellman (HJB) Equation in the dynamic programming
1. Endogenous Growth with Human Capital Consider the following endogenous growth model with both physical capital (k (t)) and human capital (h (t)) in continuous time. The representative household solves
More informationShort correct answers are sufficient and get full credit. Including irrelevant (though correct) information in an answer will not increase the score.
Economics 200A Part 2 UCSD Fall 2012 Prof. R. Starr, Mr. Troy Kravitz Final Exam 1 Your Name: Please answer all questions. Each of the six questions marked with a big number counts equally. Designate your
More informationThe Existence of Equilibrium in. Infinite-Dimensional Spaces: Some Examples. John H. Boyd III
The Existence of Equilibrium in Infinite-Dimensional Spaces: Some Examples John H. Boyd III April 1989; Updated June 1995 Abstract. This paper presents some examples that clarify certain topological and
More informationLecture 1. History of general equilibrium theory
Lecture 1 History of general equilibrium theory Adam Smith: The Wealth of Nations, 1776 many heterogeneous individuals with diverging interests many voluntary but uncoordinated actions (trades) results
More informationImplementation of Marginal Cost Pricing Equilibrium Allocations with Transfers in Economies with Increasing Returns to Scale
Implementation of Marginal Cost Pricing Equilibrium Allocations with Transfers in Economies with Increasing Returns to Scale Guoqiang TIAN Department of Economics Texas A&M University College Station,
More informationPh.D. Preliminary Examination MICROECONOMIC THEORY Applied Economics Graduate Program May 2012
Ph.D. Preliminary Examination MICROECONOMIC THEORY Applied Economics Graduate Program May 2012 The time limit for this exam is 4 hours. It has four sections. Each section includes two questions. You are
More informationEconomics 501B Final Exam Fall 2017 Solutions
Economics 501B Final Exam Fall 2017 Solutions 1. For each of the following propositions, state whether the proposition is true or false. If true, provide a proof (or at least indicate how a proof could
More informationBoundary Behavior of Excess Demand Functions without the Strong Monotonicity Assumption
Boundary Behavior of Excess Demand Functions without the Strong Monotonicity Assumption Chiaki Hara April 5, 2004 Abstract We give a theorem on the existence of an equilibrium price vector for an excess
More informationEconomic Core, Fair Allocations, and Social Choice Theory
Chapter 9 Nathan Smooha Economic Core, Fair Allocations, and Social Choice Theory 9.1 Introduction In this chapter, we briefly discuss some topics in the framework of general equilibrium theory, namely
More informationVolume 29, Issue 3. Existence of competitive equilibrium in economies with multi-member households
Volume 29, Issue 3 Existence of competitive equilibrium in economies wit multi-member ouseolds Noriisa Sato Graduate Scool of Economics, Waseda University Abstract Tis paper focuses on te existence of
More informationBilateral Trading in Divisible Double Auctions
Bilateral Trading in Divisible Double Auctions Songzi Du Haoxiang Zhu March, 014 Preliminary and Incomplete. Comments Welcome Abstract We study bilateral trading between two bidders in a divisible double
More informationA Spatial Model of Perfect Competition
Working Paper 05-2014 A Spatial Model of Perfect Competition Dimitrios Xefteris and Nicholas Ziros Department of Economics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus Tel.: +357-22893700,
More informationPh.D. Preliminary Examination MICROECONOMIC THEORY Applied Economics Graduate Program June 2016
Ph.D. Preliminary Examination MICROECONOMIC THEORY Applied Economics Graduate Program June 2016 The time limit for this exam is four hours. The exam has four sections. Each section includes two questions.
More informationNear-Potential Games: Geometry and Dynamics
Near-Potential Games: Geometry and Dynamics Ozan Candogan, Asuman Ozdaglar and Pablo A. Parrilo September 6, 2011 Abstract Potential games are a special class of games for which many adaptive user dynamics
More informationCompetitive Equilibria in a Comonotone Market
Competitive Equilibria in a Comonotone Market 1/51 Competitive Equilibria in a Comonotone Market Ruodu Wang http://sas.uwaterloo.ca/ wang Department of Statistics and Actuarial Science University of Waterloo
More informationPrice and Capacity Competition
Price and Capacity Competition Daron Acemoglu, Kostas Bimpikis, and Asuman Ozdaglar October 9, 2007 Abstract We study the efficiency of oligopoly equilibria in a model where firms compete over capacities
More informationPERIODICITY AND CHAOS ON A MODIFIED SAMUELSON MODEL
PERIODICITY AND CHAOS ON A MODIFIED SAMUELSON MODEL Jose S. Cánovas Departamento de Matemática Aplicada y Estadística. Universidad Politécnica de Cartagena e-mail:jose.canovas@upct.es Manuel Ruiz Marín
More informationImplementation of marginal cost pricing equilibrium allocations with transfers in economies with increasing returns to scale
Rev. Econ. Design (2010) 14:163 184 DOI 10.1007/s10058-009-0088-5 ORIGINAL PAPER Implementation of marginal cost pricing equilibrium allocations with transfers in economies with increasing returns to scale
More informationEconomic Growth: Lecture 8, Overlapping Generations
14.452 Economic Growth: Lecture 8, Overlapping Generations Daron Acemoglu MIT November 20, 2018 Daron Acemoglu (MIT) Economic Growth Lecture 8 November 20, 2018 1 / 46 Growth with Overlapping Generations
More informationUnlinked Allocations in an Exchange Economy with One Good and One Bad
Unlinked llocations in an Exchange Economy with One Good and One ad Chiaki Hara Faculty of Economics and Politics, University of Cambridge Institute of Economic Research, Hitotsubashi University pril 16,
More informationThe Ohio State University Department of Economics. Homework Set Questions and Answers
The Ohio State University Department of Economics Econ. 805 Winter 00 Prof. James Peck Homework Set Questions and Answers. Consider the following pure exchange economy with two consumers and two goods.
More informationECONOMICS 001 Microeconomic Theory Summer Mid-semester Exam 2. There are two questions. Answer both. Marks are given in parentheses.
Microeconomic Theory Summer 206-7 Mid-semester Exam 2 There are two questions. Answer both. Marks are given in parentheses.. Consider the following 2 2 economy. The utility functions are: u (.) = x x 2
More information1 General Equilibrium
1 General Equilibrium 1.1 Pure Exchange Economy goods, consumers agent : preferences < or utility : R + R initial endowments, R + consumption bundle, =( 1 ) R + Definition 1 An allocation, =( 1 ) is feasible
More informationFoundations of Neoclassical Growth
Foundations of Neoclassical Growth Ömer Özak SMU Macroeconomics II Ömer Özak (SMU) Economic Growth Macroeconomics II 1 / 78 Preliminaries Introduction Foundations of Neoclassical Growth Solow model: constant
More informationA note on Herbert Gintis Emergence of a Price System from Decentralized Bilateral Exchange
Author manuscript, published in "The BE Journals in Theoretical Economics 9, 1 (2009) article 44" DOI : 10.2202/1935-1704.1560 A note on Herbert Gintis Emergence of a Price System from Decentralized Bilateral
More informationUNIVERSITY OF NOTTINGHAM. Discussion Papers in Economics CONSISTENT FIRM CHOICE AND THE THEORY OF SUPPLY
UNIVERSITY OF NOTTINGHAM Discussion Papers in Economics Discussion Paper No. 0/06 CONSISTENT FIRM CHOICE AND THE THEORY OF SUPPLY by Indraneel Dasgupta July 00 DP 0/06 ISSN 1360-438 UNIVERSITY OF NOTTINGHAM
More informationContracts under Asymmetric Information
Contracts under Asymmetric Information 1 I Aristotle, economy (oiko and nemo) and the idea of exchange values, subsequently adapted by Ricardo and Marx. Classical economists. An economy consists of a set
More informationThe Stability of Walrasian General Equilibrium under a Replicator Dynamic
The Stability of Walrasian General Equilibrium under a Replicator Dynamic Herbert Gintis and Antoine Mandel February 10, 2014 Abstract We prove the stability of equilibrium in a completely decentralized
More informationGeneral equilibrium with externalities and tradable licenses
General equilibrium with externalities and tradable licenses Carlos Hervés-Beloso RGEA. Universidad de Vigo. e-mail: cherves@uvigo.es Emma Moreno-García Universidad de Salamanca. e-mail: emmam@usal.es
More informationTrade Rules for Uncleared Markets with a Variable Population
Trade Rules for Uncleared Markets with a Variable Population İpek Gürsel Tapkı Sabancı University November 6, 2009 Preliminary and Incomplete Please Do Not Quote Abstract We analyze markets in which the
More informationCompetitive Market Mechanisms as Social Choice Procedures
Competitive Market Mechanisms as Social Choice Procedures Peter J. Hammond 1 Department of Economics, Stanford University, CA 94305-6072, U.S.A. 1 Introduction and outline 1.1 Markets and social choice
More informationThe Max-Convolution Approach to Equilibrium Models with Indivisibilities 1
The Max-Convolution Approach to Equilibrium Models with Indivisibilities 1 Ning Sun 2 and Zaifu Yang 3 Abstract: This paper studies a competitive market model for trading indivisible commodities. Commodities
More information