The Stability of Walrasian General Equilibrium with Decentralized Price Adjustment

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 (tâtonnement) 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 Panthéon-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 possi- ble 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 set- tings 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).

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