The Stability of WalrasianGeneral Equilibrium

Herbert Gintis

Santa Fe InstituteCentral European University

Institute for New Economic Thinking (INET)

November 2012

Existence of General Equilibrium

Gerard Debreu, 1921-2004Kenneth Arrow, 1921-

In the period 1952-1954, Kenneth Arrow and GerardDebreu showed that with plausible assumptions, there exists a set of equilibrium (market clearing) prices.

The Quest for Stability

The question of stability of the Walrasian economy was a central research focus in the years immediately following the existence proofs.

Arrow, Hurwicz, Uzawa et al. assumed that out of equilibrium, there is a set of disequilibrium prices, the time rate of change of prices being a function of aggregate excess demand.

So when a good is in excess demand, its price increases, and when it is in excess supply, its price decreases.

The problem is that this must happen in all markets simultaneously.

The Quest for Stability

But who changes the prices?It cannot be individual agents, because there is one price for

each good in the whole economy!Arrow et al. assumed that the price system was controlled by

a mythical “auctioneer” (commisaire-priseur in French)acting outside the economy to update prices in the current period on the basis of the current pattern of excess demand, using a process of “tâtonnement,” as was first suggested by Walras himself.

Walras’ Auctioneer

The auctioneer, before any buying and selling takes place,

1. “calls out” a set of prices, 2. asks firms and households say how much they want to

buy and sell at these prices,3. calculates the “excess demand” or “excess supply” for

each sector,4. adjusts the prices to bring the markets closer to

equilibrium,5. Then back to 1, until all markets are in equilibrium.6. Only then is production and trade permitted, at the

specified market-clearing prices.

The Quest for Stability

Even if this project had been successful, the result would have been of doubtful value, as the tâtonnement process is purely fanciful.

However, it was not successful.The quest for a general stability theorem was derailed by

Herbert Scarf''s (1960) simple examples of unstable Walrasian equilibria.

The Quest for Stability

General equilibrium theorists in the early 1970's harbored some expectation that plausible restrictions on the shape of the excess demand functions might entail stability, but Sonnenschein (1973), Mantel (1974, 1976), and Debreu (1974) showed that aggregate excess demand functions can have virtually any shape at all.

It follows that the tâtonnement process cannot generally be stable.

In fact, it turns out that tâtonnement is generically chaotic (Bala and Majumdar, 1992; Saari, 1995)

The Quest for Stability

Surveying the state of the art some quarter century after Arrow and Debreu's seminal existence theorems, Franklin Fisher (1983) concluded that little progress had been made towards a plausible model of Walrasian market dynamics.

The Quest for Stability

It is now more than another quarter century since Fisher's remarks, but it remains the case that the current literature offers us nothing systematic about the dynamics of decentralized competitive market economies.

Rethinking Macroeconomics My work returns to a study of the fully decentralized

Walrasian model, but this time with the understanding that the market economy is a complex, dynamic, nonlinear system that must be modeled using novel analytical tools.

The goal is a model of market dynamics that analytically specifies the conditions under which the system is robust.

The Stability of General EquilibriumAntoine Mandel (University of Paris, Sorbonne) and I

consider a completely decentralized market economy in which each agent produces a single good at the start of each period and enters an exchange process to acquire consumption goods.

In place of the common prices assumed in standard general equilibrium theory, we assume each agent is endowed with a set of personal prices on which the agent bases offers to trade and willingness to accept the offers of others.

The exchange mechanism is extremely generic, assuming only that agents exchange when both gain according to their respective personal prices.

The Stability of General EquilibriumThe competitivity assumption in the exchange

mechanism is that the lowest price offers are filled before higher price offers.

These assumptions define a game in which agent strategies are private price vectors and payoffs are the utility from consuming the goods obtained through the exchange process.

We show that a set of strategy profiles of this game is a Walrasian (market-clearing) equilibrium if and only if it a strict Nash equilibrium of the game.

Evolutionary DynamicsAn evolutionary dynamic begins with a stage game ,

which in our case is the Walrasian exchange process I just described.

We assume there are many players, each of which plays a particular strategy in the stage game in each period.

After each period, players who had high payoffs are copied by players with lower payoffs. This is called a replicator dynamic.

We study the long run dynamics of this evolutionary system.

The major theorem is: every market-clearing price system is a stable equilibrium of the evolutionary dynamic .

Markov Market DynamicsThe implementation of this model is difficult because there

are huge numbers of equations that can be solved neither analytically nor by numerical approximation.

However, the discrete version of the market dynamic using finite Markov processes is close to the replicator dynamic for large population size (Benaim and Weibull 2003) and various parameter values can be explored through computer simulation.

A Decentralized Market System with Individual Production

I have explored such Markov market economies in several publications (Gintis 2007, 2012). I find thatif we start with a random assignment of prices to each agent, the economy moves quickly to quasi-public prices,the latter being private prices with low relative standard error across agents, andin the long run, quasi-public prices move to general Walrasian quasi-equilibrium,which is a stationary distribution with near-market-clearing prices in almost all periods.

Private to Quasi-Public Prices

Quasi-Market Clearing

What Next?

We must add firms and inter-firm trade, as well as inter-period trade (inventories, money, wealth).

These moves will not change the stability properties of the system, which are very robust.

However, we can explore price bubbles in consumer durables based on expectations of rising prices across periods.

We can also explore the effects of shocks on bankruptcies and reduced production by extending our network analysis of firm interrelationships.

This can all be modeled using parameters from real economies.

Fragility vs. Stability

There is little doubt but that the above stability properties will extend to more complex decentralized market economies.

However a system can be stable, yet extremely robust or, by contrast, extremely fragile in reaction to shocks.

I find that in a fairly realistic model of a contemporary advanced economy, price bubbles occur rather frequently,although in the absence of a sophisticated financial sector, they do not produce large aggregate dislocations in labor and product markets.

Basic Assumptions

My more realistic agent-based model (The Economic Journal, 2007) assumes thatconsumers must engage in price searches in each period;workers have a subjective reservation wage that they use to determine whether to accept a job offer;firms know their production costs, but not their demand curves, and hence must experiment and learn.

There is a central bank and a tax-collecting authority, as well as a government sector that services unemployment insurance.

Basic Assumptions

Workers periodically search for alternative job opportunities;

firms maximize profits by experimentally varying their operating characteristics and copying the behavior of other firms that are more successful than themselves;

both prices and quantities respond to conditions of excess supply or demand;

all adjustment parameters are agent- and firm-specific, and evolve endogenously.

Adjustment ProcessesIn each period:For each firm inventory growth leads to lowering of price

a small amount, and excess demand leads to raising price a small amount.

average sector profits > 0 leads to a single firm entering the sector, and average profits < 0 leads to a single firm going bankrupt.

firms make limited searches for alternative employees, and workers make limited searches for alternative employers.

agents revise their consumption, production, employment, and trading strategies by sampling the population, and imitating the strategies of others who appear to be relatively successful.

all adjustment rates are endogenous

Adjustment ProcessesBecause all players (firms and workers) adjust their

behavior by imitating the successful, the economic dynamic is an evolutionary dynamicimitation leads to correlated errors, so the statistical independence of errors assumptions that plague traditional macroeconomic models are absent here: “fat tails” are the rule,and there are large excursions from equilibrium in the absence of macro-level shocks.

Main ResultsThe dynamical system satisfies the complex systems

counterpart to stability and uniqueness:excess supply in each sector;excess labor demand, as well as excess labor supply in

each period;labor demand differs from labor supply by only a few

percent;Prices are approximately equal to production costs;The wage rate in each sector is fairly stable, and wages

are approximately equal across sectors.There is a considerable level of fluctuation in price and

quantity series, even though there are no aggregate stochastic shocks to the system.

Price Stability with Excursions

Price Stability with Excursions

Excess Demand and Supply

Profits

Unemployment

Unemployment

Stability

Stability

•The vertical axis shows percentage efficiency.

percent

Stability

ConclusionSimple market exchange is robust to shocks, whereas

economies with sophisticated institutions can exhibit considerable fragility.

The fragility of sophisticated market competition exchange is based on endogenous random shocks and does not require exogenous shocks.

Agent-based simulation models provide insights into the dynamic performance of market economies.