The Stability of Walrasian General Equilibrium with Decentralized Price Adjustment

Herbert Gintis and Antoine Mandel∗†

September 30, 2014

Abstract We study a completely decentralized Walrasian general equilibrium economy with trade occurring out of equilibrium. Prices are fully controlled by economic agents and updated according to an evolutionary learning process. This approach to price dynamics overcomes the failure of a centralized price adjustment process (tˆatonnement) to provide dynamical foundations for general equilibrium. In our setting, the law of one-price, price-taking behavior and market equilibrium are stochastically approximated long-run emergent properties of a complex dynamical system.

Journal of Economic Literature Classiﬁcations: C62—Existence and Stability Conditions of Equilibrium D51—Exchange and Production Economies D58—Computable and Other Applied General Equilibrium Economies

1 Introduction

Walras (1874) developed a general model of competitive market exchange, but provided only an informal argument for the existence of a market-clearing equilibrium for this model. Wald (1951[1936]) provided a proof of existence for a simpliﬁed version of Walras’ model. This proof was substantially generalized by Debreu (1952), Arrow and Debreu (1954), Gale (1955), Nikaido (1956), McKenzie (1959), Negishi (1960), and others.

∗The authors contributed equally to this research. Gintis: Santa Fe Institute; Mandel: Paris School of Economics, University of Paris 1 Panth´eon-Sorbonne. The authors acknowledgethe support of In- stitute for New Economic Thinking grant INO1200022. Antoine Mandel also acknowledgessupport from the EU-FP7 project SIMPOL. We dedicate this paper to Hirofumi Uzawa. He will be sorely missed. †Corresponding author, email:[email protected], tel/fax:+3314407 8271/8301

1 The stability of the Walrasian economy became a central research focus in the years following the existence proofs (Arrow and Hurwicz 1958, 1959, 1960; Ar- row, Block and Hurwicz 1959; Nikaido 1959; McKenzie 1960; Nikaido and Uzawa 1960). Following Walras’ tâtonnement process, these models assumed that there is no production or trade until equilibriumprices are attained, and out of equilibrium, there is a price profile shared by all agents (we call these public prices), the time rate of change of which is a function of excess demand. These efforts were largely unsuccessful (Fisher 1983). A novel approach to the dynamics of large-scale social systems, evolutionary game theory, was initiated by Maynard Smith and Price (1973), and adapted to dynamical systems theory in subsequent years (Taylor and Jonker 1978, Friedman 1991, Weibull 1995). The application of these models to economics involved the shift from genetic transmission to behavioral imitation as the mechanism for the replication of successful agents. Several authors have investigated market dynamics through this evolutionary lens. Vega-Redondo (1997) analyzes the convergence to the Walrasian outcome in a Cournot oligopoly where firms update quantities in an evolutionary fashion, Alós–Ferrer et al. (2000) provide an evolutionary model of Bertrand competition, Serrano and Volij (2008) study the stability of the Walrasian outcome in markets for indivisible goods, and a number of contributions, including Mandel and Botta 2009, 2011, investigate evolutionary dynamics in specific exchange economies. We here assume a decentralized price adjustment process in which each agent in each period has complete control of the relative prices at which he is willing to trade. We treat an exchange economy as the stage game of an evolutionary process in which each agent’s initial inventory is replenished in each period, agents enter into a series of bilateral trades, at the end of which each agent consumes his final inventory. There are no inter-period exchanges. An agent’s trade strategy consists of a vector of private prices for the goods he produces and the goods he consumes, such that, according to the individual’s private prices, a trade is acceptable if the value of goods received is at least as great as the value of the goods offered in exchange. This representation of the exchange process has strong commonalities with the strategic bargaining literature, and in particular Rubinstein and Wolinsky (1985,1990), Gale (1986) and Binmore and Herrero (1988). In our model, competition is effective because when there is excess supply in one period, a seller who failed to secure a transaction can gain by slightly under- cutting his competitors’ prices. Similarly, when there is excess demand, a buyer who failed to secure a transaction increases utility by slightly outbidding his competitors. When markets are in disequilibrium, then, there always exist agents who stand to gain by altering their prices. Under rather mild conditions on the trad-

2 ing process, market-clearing equilibria are the only Nash equilibria of our market system, and these equilibria are strict. Thus, if we assume that the strategies of traders are updated according to a monotone dynamic, the stability of equilibrium is guaranteed (Weibull 1995). The intuition as to why decentralized pricing entails a stable equilibrium while the centralized tâtonnement price adjustment process doesnotisthata smallchange in the private price vector for one agent has effects only for the few agents with whom he interacts, whereas a small change in publicprices impacts upon all agents in the economy and leads all buyers, respectively all sellers, to react in the same direction, producing significant macroeconomic effects. We illustrate this in Sec- tion 4. The paper is organized as follows. Section 2 introduces our model economy. Section 3 defines a class of bargaining games based on private prices in this economy for which we provide an example in Section 4. Section 5 proves the stability of equilibrium assuming that each agent’s set of consumption goods is given inde- pendent of relative prices, and Section 6 extends this result to an arbitrary exchange economy. Section 7 discusses in greater detail the necessary conditions for competition to entail a stable price adjustment process.

2 The Walrasian Economy

We consider an economy with a goods h = 1,...,l, and agents i = 1,...,m. Rl R Each agent i has consumption set X = +, a utility function ui: X → + and an initial endowment ei = (ei1,...,eil) ∈ X . We denote this economy by E(u, e), where e =(e1,...,em). We use the following concepts. • The set of allocations is X m. An allocation x ∈ X m of goods is feasible if

m m X xih ≤ X eih for h =1,...l. i=1 i=1

m We write A(e1,...,em) = A(e) ⊂ X for the set of feasible allocations. Rl • The set of prices is P ⊂ +. We assume strictly concave utility functions, so the demand of agent i is a mapping di : P × R+ → X that associates to a price p ∈ P and an income w ∈ R+ the unique utility-maximizing allocation:

di(p,w) = argmax{u(x) x ∈ X ,p · x ≤ w}.

3 • A feasible allocation x ∈ A(e) is an equilibrium allocation if there exists a price p such that for all i, xi = di(p,p · ei). The price p is then called an equilibrium price. We denote the set of equilibrium prices by Pequi(u, e). If Pequi(u, e) is ﬁnite, we assume it is a subset of P . • A feasible allocation x ∈ A(e) and a price p ∈P form a quasi-equilibrium (p, e) if for all yi ∈ X , ui(yi) > ui(xi) implies p · yi >p · xi. To ensure the existence of a quasi-equilibrium, we assume that utility functions satisfy the following standard conditions. The strict concavity assumption is not necessary for the existence of a quasi-equilibrium but implies demand mappings are single-valued, which will prove useful below.

Assumption 1 (Utility) For all i = 1,...,m, ui is continuous, strictly concave, and locally non-satiated.

To ensure that every quasi-equilibrium is an equilibrium allocation, it sufﬁces to assume that at a quasi-equilibrium, agents do not receive the minimal possible income (Hammond 1993, Florenzano 2005). This condition is satisﬁed when all initial endowments are in the interior of the consumption set, as well as in settings with corner endowments such as those investigatedin Scarf (1960) and Gintis (2007). Formally, the assumption can be stated as follows.

Assumption 2 (Income) For every quasi-equilibrium (p,x), and for every agent i, there exists an allocation yi such that p · xi >p · yi.

Assumptions 1 and 2 imply that the economy E(u, e) has at least one equilibrium (Florenzano 2005). We restrict attention to the generic case where the set Pequi of equilibria is ﬁnite (Balasco 2009).

3 Exchange Processes with Private Prices

Each agent i has a strategy consistingof a private price vector pi ∈P representing the prices at which he is willing to sell the goods he supplies to the market and the maximum prices he is willing to pay for the goods he demands. We associate with the economy E(u, e) the class of games G(u, e, ξ) where:

• Each agent i has a ﬁnite strategy set Pi ⊂P. The set of strategy proﬁles for the game is P = P1 × ... × Pm. • There is an exchange mechanism ξ : P→A(e) that associates with π ∈ P a feasible allocation ξ(π)=(ξ1(π),...,ξm(π)) ∈ A(e).

4 • The payoff of player i given strategy profile π is ui(ξi(π)). Many exchange processes satisfy these conditions, including double auctions and simultaneous, multilateral exchanges (Grandmont 1977, Bénassy 2005), as well as sequences of bilateral trades such as those considered by Gintis (2007, 2012). The following example illustrates how such a process can induce converging price dynamics in the Scarf economy, which is a well-known example of the instability of the tâtonnement process (Scarf 1960).

4 The Scarfeconomy

We consider the Scarf economy (Scarf 1960) with three goods (` = 3) and three types ofagents (m =3). Each agent1hasutility u1(x1,x2,x3) = min(x1/ω1, x2/ω2), x2 and initial endowment (ω1, 0, 0), each agent 2 has utility u2(x1,x2,x3) = min( /ω2, x3/ω3) and initial endowment (0, ω2, 0) and each agent 3 has utility u3(x1,x2,x3) = x3 x1 min( /ω3, /ω1) and initial endowment (0, 0, ω3). Let (ω1, ω2, ω3) = (10, 20, 400) so that equilibriumrelative prices, which must be proportionalto Pequi = (40, 20, 1), are extremely unequal (Anderson et al. 2004). We then consider 1000 agents of each type, each initially endowed with a private price vector. We assume that there is a smallest possible positive price, and each component of an agent’s price vector is drawn from the uniform distribution on the multiples of this smallest possible price in the interval (0, 1). Private prices evolve according to the followingdynamic, which we implement as a ﬁnite Markov process. At the beginning of each period, the inventory of each agent is reset to the value of its initial endowment. Each agent, in random order, is then designated a trade initiator and is paired with a randomly chosen responder, who can either accept or reject the proposed trade. Each agent is thus an initiator exactly once and responder on average once per period. The trade procedure is as follows. An initiator A offers a certain quantity of one good in exchange for a certain quantity of a second good. If the responder B has some of thesecond good,and ifthevalueof what B is offered in return exceeds the value of what B gives up, according to B’s private prices, then B agrees to trade. If B has less of the second good than the initiator wants, the trade is scaled down proportionally. The initiator’s acceptable trade ratios are given by his private prices. Which good he offers to trade for which other good is determined as follows. Let us call an agent’s endowment good his P-good, the additional good he consumes his C- good, and the good which he neither produces nor consumes his T-good. If the initiator A has his T-good in inventory, A offers to trade this for his C-good. If this offer is rejected, A offers to trade for his P-good. If the initiatorA does not have his

5 T-good but has his P-good, he offers this in trade for his C-good. If this is rejected, A offers to trade half his P-good for some of his T-good. If the initiator A had neither his T-goodnor his P-good, he offers his C-good in trade for his P-good, and if this fails A offers to trade for his T-good. In all cases, when a trade is carried out, the terms are dictated bythe initiatorA and the amountis the maximum compatible with the inventories of the initiator A and responder B. At the end of a trading period, agents consume from their updated inventories. Every tenth period is followed by a reproduction stage in which 5% of agents are randomly chosen either to copy a more successful agent or to be copied by a less successful agent, where success is measured by total undiscounted utility of consumption over the previous ten periods. Such an agent is chosen randomly and assigned a randomly chosen partner of the same type. The less successful of the pair then copies the private prices of the more successful. In addition, after the reproduction stage, each price of each agent is mutated with 1% probability, the new price either increasing or decreasing by 10%.

Figure 1: Convergence of Price to Equilibrium in a Three-good Scarf Economy with Pri- vate Prices

The results of a typical run are exhibited in Figure 1. Each of the curves in Figure 1 is given by (p∗ −p)/p∗ where p∗ is the equilibrium relative price. Because initial prices are generated by uniform distributions on the unit interval, the initial

6 mean price of good 3 is about 40 times its equilibrium value, and the price of good 2 is about 20 times its equilibrium value. Nevertheless, in sharp contrast to the oscillatory behavior of the tâtonnement process in this setting (see Scarf 1960), the system settles rapidly in a steady state where prices assume their equilibrium values, i.e we observe convergence to equilibrium. Similar convergence results have been obtained in economies with Leonti- eff preferences (see Gintis 2012) as well as in arbitrarily large economies with nested, arbitrarily heterogeneous, CES utility functions where fitness is measured by traded volumes rather than by utility (Gintis 2013). This set of results suggest that in a setting where agents use private prices as strategies, evolutionary dynamics based on imitation of success (see Schlag, 1998 and Benaim and Weibull, 2003) can provide dynamical foundations for general equilibrium in settings where the tâtonnement mechanism fails.

Figure 2: Scarf Price Dynamics with Public Price and Private Price Agents. Each of the curves is given by (p − p∗)/p∗, where p∗ is the equilibrium relative price and p is the average observed relative price. Equilibrium corresponds to zero on the y-axis. The top pane represents 10% public prices, and the bottom pane 40% public prices.

7 To show that public prices destabilize the Scarf economy, we reintroduce the tˆatonnement process, where a fraction f of the population uses public prices, and the auctioneer updates public prices in each period by aggregating the excess demands of the agents who use public prices. However, we retain the assumption that agents trade and consume in each period even though prices are not in equilibrium. Also, agents who use private prices can imitate the price structure of a public price agent who has been more successful over the past ten periods. Such an agent, however, remains a private price agent. Figure 2 shows the result of this run of the agent-based Scarf economy with 10% (top pane) and 40% (bottom pane) of agents using public prices. We see that even 10% public prices entails considerable initial price instability, but this wears off eventually. With 40% public price agents, there are recurrent price explosionsin which prices deviate manyfold from their equilibrium values. The auctioneer’s rate of adjustment of prices affects the period of price cycles but not their magnitude.

5 An Axiomatic Characterization of Stable Exchange Processes

l−1 Henceforth we assume P = K ×{1}, where K ⊂ R+ is a ﬁnite set of commodity prices with minimum pmin > 0 and maximum pmax > pmin. Good l is numeraire with price unity. We assume that P contains each of the ﬁnite number of equilibrium prices vectors, Pequi, of the economy. We will specify a broad class of exchange processes ξ for which the only Nash equilibria of the game G(u, e, ξ) are strict, and in which each agent’s strategy is an equilibriumprice. The analysis is simpler if we assume that each agent’s initial endowment does not include his consumption goods. We treat the general case in the next section.

Assumption 3 (Goods Partition) For each player i, let Ei be the (non-empty) set of i’s endowment goods; i.e., h ∈ Ei ⇔ eih > 0. There is a (non-empty) set Ci of i’s consumption goods disjoint from Ei, and i’s strategy set Pi ⊂ P are such that for any x, y ∈ X , a. If xih = yih for all h ∈ Ci, then ui(x) = ui(y). b. For any h ∈ Ci, let x = di(p,w) and y = argmax{u(z)|z ∈ X ,p · z ≤ w, zh = 0}. Then xh > 0 ⇒ u(y) < u(x). c. If pi ∈ Pi, then pih =0 for all h∈ / Ei ∪ Ci.

Condition (b) ensures that no two goods are perfect substitutes in consumption. Condition (c) ensures an agent’s strategy choice deals only with goods he either

8 consumes or produces. We then define the buyers of good h as Bh = {i|h ∈ Ci}, and the sellers of good h as Sh = {i|h ∈ Ei}. Note that for all goods h, Bh and Sh are nonempty and disjoint by construction. P Given a strategy profile π ∈ , let πmin = mini∈Sh πih and πmax = maxi∈Bh πih. We define the acceptable buyers of good h as Bh(π) = {i ∈ Bh|πih ≥ πmin}, the acceptable sellers as Sh(π) = {i ∈ Sh|πih ≤ πmax} and the feasible income of player i is wi(π) = P{πiheh|h ∈ Ei}. The set of price feasible allocations A(e, π) is then defined as follows.

Deﬁnition 1 An allocation x is in the set A(e, π) of price feasible allocations for strategy proﬁle π ∈ P if, for all players i:

1. xih > 0 ⇒ i ∈Bh(π) for all h ∈ Ci; 2. x ≤ e for all h; Pi∈Bh(π) ih Pi∈Sh(π) ih

3. xi ≤ di(πi,wi(π)).

In other words, a feasible allocation x, given endowment vector e, is price feasible for strategy profile π ∈ P if, for every good h and every player who consumes h, there is a seller of h with whom an exchange is possible, total demand for each good is less than or equal to total supply, no agent i is allocated by x more than i’s demand at prices πi and feasible income wi(π). We say a strategy profile π ∈ P is p-uniform for price p ∈ P if, for all agents i and for all h ∈ Ei ∪ Ci, πih = ph. Note that when a p-uniform strategy profile is attained, the market includes all agents, and each agent faces only his private budget constraint. Clearly, a necessary condition for market equilibrium with strategy profile π is that the market include all agents; i.e., for each good h, Bh(π) = Bh and Sh(π) = Sh. An exchange process ξ that ensures this must be such that an agent excluded from the market for a good h have an incentive to raise or lower his price ph so that there is a positive probability of securing an exchange. The following assumption, which is standard in the bargaining literature (Kalai and Smorodinsky 1975), captures this intuition.

Assumption 4 (Monotonicity) If π and τ are strategyprofiles such that A(e, π) ⊂ A(e, τ) then for all agents i we have ui(ξi(τ)) ≥ ui(ξi(π)). We say a p-uniform strategy profile π is an equilibrium price profile if p ∈ Pequi(u, e). An acceptable allocation mechanism must ensure that given an equilibrium price profile, the corresponding equilibrium allocation prevail. Thus we have:

9 Assumption 5 (Equilibrium) If π is a p-uniform equilibrium price profile, then for each agent i, we have ξi(π) = di(p,p · ei). If π is a non-equilibrium p-uniform strategy profile, by Walras’ Law there is excess demand for some good. An acceptable exchange processes must be competitive in the sense that in case of excess demand, at least one agent has an incentive to deviate from the common price. We assume: Assumption 6 (Competition) If π is a p-uniform strategy profile and there is a good h such that m m X dih(p,wi(π)) > X eih, i=1 i=1 then there is some agent i and a price q ∈ Pi such that ui(ξi(q,π−i)) > ui(ξi(π)). We discuss the conditions under which an exchange process satisfies this assumption in Section 7. Basically, an exchange process is competitivein thissense if it implements some form of priority for lowest bidding sellers and highest bidding buyers. In this case, when there is excess supply a seller who slightly undercuts his competitors increases his sales and hence his income while when there is excess demand, a buyer who slightly overbids his competitors overcomes rationing.

Proposition 1 Under Assumptions 3–6, a strategy proﬁle π is a strict Nash equilibrium of G(u, e, ξ) if and only if π is an equilibrium price proﬁle.

Proof: Suppose π is a p-uniform equilibrium price proﬁle. Suppose an agent i deviates from p to a price q ∈ Pi that does not decrease i’s payoff. By Assump- tion 5, wi(π) = p · ei. For h ∈ Ei, we have

• If qh >ph then i∈S / h(q,πi), so i can no longer sell good h.

• If qh

• If qh = ph then qheih = pheih

Hence either for all h ∈ Ei, qh = ph and wi(q,π−i) = wi(π), or wi(q,π−i) < wi(π). Thus wi(q,π−i) ≤ wi(π) (1) with equality if and only if qh = ph for all h ∈ Ei. Suppose for some h ∈ Ci, qh < ph. Then i∈ / Bh(q,π−i), which implies, by Assumption 3, ξih(q,π−i)=0. This implies, also by Assumption 3b, that ui(ξih(q,π−i)) < ui(di(p,p · ei)). Thus we must have qh ≥ ph for all h ∈ Ci.

10 Suppose qh >ph for some h ∈ Ci. Then by (1) and the strict concavity of ui, we have ui(di(q,wi(q,π−i))) < ui(di(p,wi(π))) so that

ui(ξi(q,π−i)) ≤ ui(di(q,wi(q,π−i))) < ui(di(p,wi(π))) = ui(ξi(π)).

It follows that qh = ph for all h ∈ Ci and all h ∈ Ei. For h∈ / Ei∪Ci, p, q ∈ Pi implies qih = pih = 0. Thus q = p, a contradiction. Thus for every equilibrium price profile p, the p-uniform strategy profile π is a strict Nash equilibrium For the converse, we must show that if π is not p-uniform for any p, or if π is p-uniform but p∈P / equi, then π is not a Nash equilibrium. If π is a p-uniform price profile where p is not an equilibrium price, then some good h is in excess demand, so Assumption 6 implies π is not a Nash equilibrium. Thus we can assume π is not a p-uniform price profile. If there is a good h and a player i ∈ Sh/Sh(π), choose q ∈ Pi such that qh = max{πih|i ∈ Bh} and qg = πig for all g ∈ Ei ∪ Ci with g 6= h. We obtain a strategy profile τ = (q,π−i) such that wi(τ) ≥ wi(π) while the other constraints in the definition of A(e, π) can only be relaxed. Therefore, A(e, π) ⊂ A(e, τ). According to Assumption4, π cannot be a strict Nash equilibrium. Similar arguments apply whenever there exists a good h and an agent i ∈ Bh/Bh(π). Thus we may assume Bh = Bh(π),Sh = Sh(π), and there are agents i and j and good h ∈ Sh ∪ Bh such that πih 6= πjh. Suppose πih < πjh. Then if i,j ∈ Sh, by setting τih = πjh and τkg = πkg for k 6= i or g 6= h, we have wi(τ) ≥ wi(π) and hence A(e, π) ⊂ A(e, τ) so that π can not be a strict Nash equilibrium according to assumption 4. The same holds true if i ∈ Sh and j ∈ Bh. Hence, we can now assume that for all i,j ∈ Sh,τih = πjh. Then if i,j ∈ Bh, by setting τjh = πih and τkg = πkg for k 6= i or g 6= h, the private budget constraint of agent j (condition 3 in the definition of A(e, π)) is relaxed and one then has A(e, π) ⊂ A(e, τ) so that π can not be a strict Nash equilibrium by Assumption 4. If i ∈ Bh and j ∈ Sh, it must be, given that Sh = Sh(π), that there exist k ∈ Bh such that πjh ≤ πkh and the preceding argument applies to i,k ∈ Bh such that πih <πkh. Thus a strategy profile π cannot be a strict Nash equilibrium of G(u, e, ξ) if it is not an equilibrium price profile. This ends the proof. A direct corollary of proposition1 is the asymptotic stabilityof equilibrium for the replicator dynamics. Proposition 2 Under assumptions (3) through (6), the only asymptotically stable strategy profiles for a monotonic dynamic in G(u, e, ξ) are those for which each agent uses an equilibrium price p ∈Pequi and agent i is allocated his equilibrium allocation di(p,p · ei).

11 Thisresult stronglycontrasts withthe lack of generic stabilityof the tˆatonnement. Both processes are driven by competition, the adaptation of prices to market condition. Yet, a fundamental difference is that in the tˆatonnement process, there is a single public price whose variations induce massive side effects whereas in our setting price changes are individual and, taken individually, only have a marginal impact on the economy.

6 Extension to a Generalized Economy

Proposition1 and 2 require that productionand consumption sets bedisjoint. In this section we drop this assumption but otherwise retain assumptions of the previous section, while changing some definitions. Given a strategy profile π ∈ P, we define:

• The set of buyers of good h is Bh(π) = {i|dih(πi) > eih};

• the set of sellers of good h is Sh(π) = {i|dih(πi) ≤ eih};

• potential buyers are those agents whose prices are above the lowest selling price; that is Bh(π) = {i|πih ≥ min{πjh|j 6= i,j ∈ Sh(π)};

• potential sellers are those agents whose prices are below the highest buying price, that is Sh(π) = {i|πih ≤ max{πjh|j 6= i,j ∈ Bh(π)}; S • The feasible income is now wi(π) = Ph{πiheh|i ∈ h(π)} The key change from the preceding section is that now a potential buyer is not necessarily a buyer, and a potential seller is not necessarily a seller.is not necessarily a buyer, and a potentialseller is not necessarily a seller. In buyer and a potential seller. Whether an agent ends up being a net buyer or seller for a given good depend on the complete demand proﬁle in the economy. The set of price feasible allocations A(e, π) is now redeﬁned as follows.

Deﬁnition 2 A feasibleallocation x isinthesetof price feasible allocations A(e, π) if

1. For all goods h and all i ∈ Bh(π), xih > eih ⇒ i ∈ Bh(π);

2. (x − e ) ≤ (e − x ) Pi∈Bh(π)∩Bh(π) ih ih Pj∈Sh(π) jh jh

3. xi ≤ di(πi,wi(π));

12 Assumptions 4, 5, and 6 sufﬁce to establish the counterpart of proposition 1, except for one complication: a single agent could now be both the sole buyer and sole seller of a good. Therefore, must to assume that at equilibrium there are at least one buyer and a distinct seller for every good. With this condition, we can prove of the following proposition.

Proposition 3 Under Assumptions4, 5 and 6, π ∈ Pm is a strict Nash equilibrium of G(u, e, ξ) if and only if there exists p ∈ Pequi(u, e) such that for all agents i, πi = p.

Proof: According to Assumption 5, if p-uniform strategy profile π is an equilibrium strategy profile, then for all i, ξi(π) = di(p,wi(π)), where wi(π) = p · ei. Assume then agent i deviates to a price q 6= p and let τ = (q,π−i). Suppose h ∈ Ei. If qh >ph, then i∈ / Sh(τ) so i can no longer sell good h. If qh Pi=1 eih, then assumption 6 implies π is not a (strict) Nash equilibrium. Thus henceforth we can assume that π is not a uniform price profile. Suppose for some good h, i ∈ Sh(π) but i∈ / Sh(π). Let qh = max{πih|j ∈ Bh(π),j 6= i} and qg = πig for all g 6= h, and let τ = (q,π−i) ∈ P. Then wi(τ) ≥ wi(π) and the other constraints in the definition of A(e, π) continue to hold. Therefore A(e, π) ⊂ A(e, τ). According to assumption 4, π cannot be a strict Nash equilibrium. Similar arguments apply whenever i ∈ Bh(π) but i∈ / Bh(π). Thus we may assume Bh(π) ⊂ Bh(π) and Sh(π) ⊂ Sh(π). There must then be agents i and j and good h such that πih 6= πjh. Suppose πih < πjh. If i, J ∈ Sh(π), let τ = π, except that τih = πjh. We then have wi(τ) ≥ wi(π), while the other constraints in the definition of A continue to hold. Therefore A(e, π) ⊂ A(e, τ) so that π can not be a strict Nash equilibrium according to Assumption 4. The same holds true if i ∈ Sh(π) and j ∈ Bh(π). Thus henceforth we can assume that for all i,j ∈ Sh,τih = πjh. Then if τ = π except that τjh = πih, the private budget constraint of agent j (condition 3 in the definition of A) is relaxed and one then has A(e, π) ⊂ A(e, τ) so π can not be a strict Nash equilibrium according to Assumption 4.

13 If i ∈ Bh(π) and j ∈ Sh(π), it must be, given that Sh(π) = Sh(π), that there exist k ∈ Bh such that πjh ≤ πkh and the preceding argument applies to i,k ∈ Bh(π) such that πih <πkh. Thus π cannot be a strict Nash equilibrium of G(u, e, ξ) if it is not an equilibrium price proﬁle in the Walrasian sense.

7 Characterizing the Competition Axiom

It remains to analyze Assumption 6, which posits a buyer’s incentive to increase price under conditions of excess demand. We first consider a basic example with two agents and two goods. Agent 1 has utility function u1(x1,x2) = x2, and initial endowment e1 = (1/4, 0). Agent 2 has Cobb-Douglas preferences u2(x1,x2) = x1x2 and initial endowment e2 = (1/4, 3/4). Good 2 is the numeraire and is price is fixed equal to one. It is straightforward to check that the only equilibrium is such that the price equals (1,1), agent 1 is allocated (0,1/4) and agent 2 is allocated (1/2,1/2). It is also clear that if both agents adopt (1,1) as private price, the only efficient price feasible allocation is the equilibrium allocation. Yet, if both agents adopt (p, 1) as private price, then the demands of agents 1 and 2, respectively, are d1(p) = (0, p/4) and d2(p)=(1/8 + 3/8p, p/8 + 3/8). Whenever p < 1, there is excess demand for good 1 and the only efficient allocation is (0, p/4) to agent 1 and (1/2, (3−p)/4) to agent 2. With respect to Assumption 6, agent 1, who is not rationed, would be worse off if he decreased his private price for good 1 and hence his income. Also increasing p unilaterally would price himself out of the market. Agent 2 cannot decrease his private price for good 1 withoutpricing himself out of the market. We might expect that “competition” would induces him to increase his price for good 1, but he has no incentive to do so as this would only decrease his income and hence his utility. The key issue is that agent 2 actually faces no competition on the good 1 market as there is no other buyer he could outbid by increasing his price. We consider a more competitive situation by doubling agent 2. That is, we consider an economy with three agents and two goods. Agent 1 continues to have utility function u1(x1,x2) = x2 and initial endowment e1 = (1/4, 0). Agents 2 and 3 have Cobb-Douglas preferences u2(x1,x2) = u3(x1,x2) = x1x2 and initial endowments e2 = e3 = (1/8, 3/8). It is straightforward to check that the only equilibrium has price (1,1), agent 1 is allocated (0,1/4) while agents 2 and 3 are allocated (1/4,1/4). It is also clear that if both agents adopt (1,1) as private price and the only efficient price feasible allocation is the equilibrium allocation. Now if each agent adopts (p, 1) as private price vector, then the demands respectively are d1(p) = (0, p/4) and d2(p) = d3(p)=(1/16 + 3/16p, p/16 + 3/16). Whenever

14 p < 1, there is excess demand for good 1. It seems natural to assume that the exchange process would then allocate its demand d1(p)= (0, p/4) to agent 1 who is not rationed. For agents 2 and 3, the allocation is efficient provided they are allocated no more than their demand 1/16 + 3/16p in good 1. It seems sensible to consider that the allocation is symmetric and hence that both agents are allocated (1/4, (3−p)/8) and so are rationed in good 1. As before agent 1 has no incentive to change his private price for good 1 and neither agent 2 nor agent 3 can further decrease his private price for good 1. However, if the exchange process implements a form of competition between buyers by fulfilling in priority the demand of the agent offering the highest price for the good, then agents 2 and 3 have an incentive to increase their private prices for good 1. Indeed assume that agent 2 increases his private price for good 1, so q >p. He will then be allocated his demand (1/16 + 3/16q, q/16 + 3/16) and be better off than at price p provided that (1/16 + 3/16q)(q/16 + 3/16) > (3−p)/32. It is straightforward to check that this equation holds for q = p (whenever p < 1) and hence by continuity in a neighborhood of p. In particular, there exists q>p such that agent 2 is better of adopting q as private price for good 1. This argument shows that necessary conditions for Assumption 6 to hold are potential and actual competition among buyers, which are respectively ensured via the presence of more thanone buyer for every goodand thepriority given to highest bidding buyers in the trading process. It turns out these two conditions are in fact sufficient. Since Edgeworth (1881), notably in (Debreu 1963), the seminal way to ensure competition is effective in a general equilibrium economy is to consider that the economy consists of sufficiently many replicates of a given set of primitive types. For our purposes, it suffices to assume that the economy E(u, e) is a two-fold replicate of some underlying simple economy.

Assumption 7 (Replicates) For every agent i there exists a distinct agent j such that ui = uj and ei = ej. Agents i and j are called replicates. A related requirement is that the exchange process be symmetric with respect to replicates using the same private price.

Assumption 8 (Symmetry) For every strategy profile π, if i and j are replicates and πi = πj, then ξi(π) = ξj(π). Finally, we ensure thatif there is excess demand for a good, the highest bidding seller of this good has priority access to the market, so that a buyer who deviates upwards from a uniform price profile has his demand fulfilled.

15 Assumption 9 (High Bidders Priority) Let π bea p-uniform strategyproﬁle, suppose i and j are replicates, and let τ be a strategy proﬁle that differs from π only in that τih >ph for h ∈ Ci. Then

ui(ξi(τ)) ≥ max {ui(xi)|xi ≤ ξi(π) + ξj (π) and τixi ≤ τiei}

That is, given that agent i gains priority over his replicate by increasing his price, i can effectively pick any allocation satisfying his private budget constraint in the pool formed by adding the his and his replicate’s former allocation. We then have Proposition 4 If Assumptions3, 4, 7, 8, and 9 hold, then Assumption6 holds. Proof: Let π be a p-uniform strategy proﬁle such that for some good h we have m m Pi=1 dih(p,wi(π)) > Pi=1 eih. We will prove that there exists a price vector q such that ui(ξi(q,π−i)) ≥ ui(ξi(π)). If there is an agent i such that for some h ∈ Ci, ξih(π)=0, the proof is straightforward according to assumption (3). Otherwise, let consider an agent i such that ξih(π) < dih(πi,wi(π)). It is clear, given condition 3 in the deﬁnition of A that ui(ξi(π)) < ui(dih(πi,wi(π)). Hence either

• the private budget constraint of agent i is not binding, so p · ξi(π)

0, ui(ξi(π) + tv) > ui(ξi(π)). If i’s budget constraint is not binding it is clear, given assumption 9, that i can still obtain ξi(π) if he shifts to a price vector q such that qh > ph for every h ∈ Ci, qh >= ph otherwise and q is sufficiently close to p. In the second case, where for all sufficiently small t > 0, ui(ξi(π) + tv) > ui(ξi(π)), denoting by j the replicate of i, who receives the same allocation according to assumption (8), we have for t > 0 sufficiently small: (ξi(π) + tv) ≤ ξi(π) + ξj (π). Then for any q such that q · ei qh >ph for every h ∈ Ci, and qh >= ph, let us set xi(q) = (ξi(π) + tv). We p · ei clearly have xi(q) ≤ ξi(π) + ξj(π) and q · xi(q) ≤ q · ei. This implies according to Assumption (9) that ξi(τ)) ≥ ui(xi(q)), where q is defined as in Assumption (9). Moreover, for q sufficiently close to p, we have using the continuity of the utility function that ui(xi(q)) > ui(ξi(π)). This ends the proof.

8 Conclusion

We have shown that the equilibria of a Walrasian market system in which there are no inter-period trades are the only Nash equilibria of an exchange game in which

16 the requirements of the exchange process are rather mild, and these Nash equilibria are strict. Assuming producers update their private price profiles periodically by adopting the strategies of more successful peers, we construct a multipopulation game in which Nash equilibria are strict and asymptotically stable in the replicator dynamic. Conversely, all stable equilibria of the replicator dynamic are strict Nash equilibria of the exchange process and hence Walrasian equilibria of the underlying economy. The major innovationof our model is the use of private prices, one set for each agent, in place of the standard assumption of a uniform public price faced by all agents, and the replacement of the tâtonnement process with a replicator dynamic. Considering prices as individual agent strategies is the only plausible assumption for a decentralized market system out of equilibrium, because there is no plausible way to define a common price system except in equilibrium. With private prices, each individual is free to alter his price profile at will, market conditions alone ensuring that something approximating a uniform system of prices will prevail in the long run. There are many general equilibrium models with private prices in the literature, based for the most part on strategic market games (Shapley and Shubik 1977, Sahi and Yao 1989, Giraud 2003) in which equilibrium prices are set on a market-by- market basis to equate supply and demand, and it is shown that under appropriate conditions the Nash equilibria of the model are Walrasian equilibria. These are equilibrium models, however, without known dynamical properties, and unlike our approach they depend on an extra-market mechanism to balance demand and supply. For even a small number of agents and goods, the equations of our dynamical system are too many and too complex to solve analytically or even to seek numeri- cal solutions. However, we can construct a discrete version of the systemas a finite Markov process. The link between stochastic Markov process models and deterministic replicator dynamics is well documented in the literature. Helbing (1996) shows, in a fairly general setting, that mean-field approximationsof stochasticpop- ulation processes based on imitation and mutation lead to the replicator dynamic. Moreover, Benaim and Weibull (2003) show that large population Markov process implementations of the stage game have approximately the same behavior as the deterministic dynamical system implementations based on the replicator dynamic. This allows us to study the behavior of the dynamical market economy for particular parameter values. For sufficiently large population size, the discrete Markov process captures the dynamics of the Walrasian economy extremely well with near certainty (Benaim and Weibull 2003). The dynamics of the Markov process model can be studied for various parameter values by computer simulation (Gintis 2007, 2012) , whereas

17 analytical solutions for the discrete system cannot be practically implemented, although they exist (Kemeny and Snell 1960, Gintis 2009). Macroeconomic models have been especially handicapped by the lack of a general stability model for competitive exchange. The proof of stability of course does not shed light on the fragility of equilibrium in the sense of its susceptibility to exogenous shocks and its reactions to endogenous stochasticity. These issues can be studied directly through Markov process simulations, and may allow future macroeconomists to develop analytical microfoundations for the control of exces- sive market volatility. Important topics for future modeling are

• Allow interperiod trades;

• Introduce multi-agent production (ﬁrms);

• Introduce capital goods and ﬁnancial markets.

• Allow for some public parameters (e.g., exchange rates) and introduce rational expectations concerning the adjustment of prices and quantities to these parameters.

We speculatethat noneof the changes will undermineour stabilityresults, although they willlikely increase the fragility of the market economyin thesenseofitsbeing more sensitive to random shocks.

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