The Stability of Walrasian
General Equilibrium
Herbert Gintis
Santa Fe Institute
Central European University
Institute for New Economic Thinking (INET)
November 2012
Existence of General Equilibrium
In the period 1952-1954,
Kenneth Arrow and Gerard
Debreu showed that with
plausible assumptions, there
exists a set of equilibrium
(market clearing) prices.
Gerard Debreu, 1921-2004
Kenneth Arrow, 1921-
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 Equilibrium
Antoine 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 Equilibrium
The 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 Dynamics
An 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 Dynamics
The 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 that
if 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, and
in the long run, quasi-public prices move to general
Walrasian quasi-equilibrium,
which is a stationary distribution with near-marketclearing 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 interperiod 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 that
consumers 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 Processes
In 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 Processes
Because all players (firms and workers) adjust their
behavior by imitating the successful,
the economic dynamic is an evolutionary dynamic
imitation 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 Results
The 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
percent
Stability
•The vertical axis shows percentage efficiency.
Stability
Conclusion
Simple 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.