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Journal of Economic Theory 147 (2012) 400–406
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Introduction to general equilibrium
Yves Balasko a,∗ , John Geanakoplos b,c
a Department of Economics and Related Studies, University of York, UK
b Yale University, New Haven, CT, United States
c Santa Fe Institute, Santa Fe, NM, United States
Received 18 September 2011; final version received 3 October 2011; accepted 20 January 2012
Available online 30 January 2012
Abstract
This introduces the symposium on general equilibrium.
© 2012 Elsevier Inc. All rights reserved.
JEL classification: C62; C63; D04; D51; D52; D53; D58; D61; E32; E44; E58
Keywords: Default; Efficiency; Equilibrium; Existence; Incomplete markets; Leverage; Overlapping-generations model;
Public goods; Regular equilibria; Sunspot equilibria
1. A short and incomplete perspective
A goal in this symposium is to offer a snapshot of some of the best research currently going
on in the theory of general equilibrium. The most active research themes in this field can be
traced back to a rather small number of influential sources. The following is neither a detailed
listing nor an exhaustive account of the research in general equilibrium theory during the past
one hundred years, a goal that would be far beyond the scope of this short essay. The role of
our short and necessarily incomplete historical references is to offer some perspective on the
research papers in the symposium. The book and papers we have highlighted are: (1) The 1951
articles of Arrow [2] and Debreu [18], in which equilibrium is shown to be Pareto efficient – the
first theorem of welfare economics; (2) The 1954 articles by Arrow and Debreu [5] and McKenzie [32], where the existence of an equilibrium is proved by a fixed-point argument; (3) Debreu’s
monograph [19], which applies to the general equilibrium model of Walras [37] the axiomatic
* Corresponding author.
E-mail addresses: yves.balasko@york.ac.uk (Y. Balasko), john.geanakoplos@yale.edu (J. Geanakoplos).
0022-0531/$ – see front matter © 2012 Elsevier Inc. All rights reserved.
doi:10.1016/j.jet.2012.01.022
Y. Balasko, J. Geanakoplos / Journal of Economic Theory 147 (2012) 400–406
401
approach advocated in the treatise of Nicolas Bourbaki [13] – see in particular the part on “Mode
d’emploi de ce traité;” (4) The formulation in Arrow [3,4] of a two-period model with uncertainty in the second period with the explicit introduction of assets to enable the transfer of wealth
through time and states of nature. Arrow shows the equivalence between the two-period model
with complete securities and spot commodity markets and the contingent-goods model in Chapter 7 of Debreu [19]. (5) The overlapping-generations paper of Samuelson [33] with the discovery
that existence of equilibrium with a strictly positive value of money and even its efficiency are
not incompatible with monetary policies that are not balanced; (6) The infinite horizon paper by
Malinvaud [31] that contains the observation that the first theorem of welfare economics does
not hold true in infinite horizon models and Shell [34] who establishes that the sole source of
inefficiency in the overlapping-generations model of Samuelson [33] is the open time-horizon.
Malinvaud [31] also contains the proof that prices tending to zero at infinity is a sufficient condition for the efficiency of equilibrium allocations, a result subsequently improved by Cass [14] and
by Balasko and Shell [11] into a complete characterization of efficiency in growth models and
overlapping-generations models respectively; (7) The smooth economies paper of Debreu [20]
showing that generically there are only a finite number of locally unique equilibria around each
of which differentiable comparative statics is well-defined; (8) The Cass and Shell [17] sunspot
paper that contains the first formulation of a sunspot model and the proof of the existence of
sunspot equilibria under restricted market participation. (9) The existence and indeterminacy of
equilibrium motivated by the analysis of the various sources of sunspot equilibrium established
for a two period model with nominal and incomplete asset payoffs by Cass [15,16], the working paper versions of these articles having been published almost simultaneously in 1984. The
existence and indeterminacy properties were extended to the general case by Balasko and Cass
[9] and Geanakoplos and Mas-Colell [27]; (10) The paper by Geanakoplos and Polemarchakis
[28] showing that, for generic economies with at least two agents and incomplete asset markets,
equilibrium is not only Pareto inefficient but also constrained Pareto inefficient; (11) The articles by Dubey, Geanakoplos and Shubik [22] (circulated in 1988) and by Geanakoplos [25] and
Geanakoplos and Zame [29] that introduced default and punishment and default and collateral
respectively into the general equilibrium model with incomplete asset markets.
Cross-fertilization between these different lines of research has been particularly fruitful.
There is rarely a theory paper nowadays that does not combine at least two of the above research
lines. It is hardly surprising to see that the rich literature on default, incomplete markets and efficiency is represented in the current symposium by no less than three papers, namely the papers
by Araujo, Kubler and Schommer [1], by Fostel and Geanakoplos [24] and by Bottazi, Luque and
Pascoa [12]. Henriksen and Spear [30] work with incomplete markets and efficiency, Balasko [6]
deals with sunspot equilibria and regular economies, and Villanacci and Zenginobuz [36] deal
with existence and efficiency of equilibrium.
2. Existence and efficiency
Villanacci and Zenginobuz [36] is a direct descendant of the Arrow–Debreu and McKenzie
existence papers. It deals with a general equilibrium model with one public good and a collection of private goods. The public good is produced by a firm that uses the private goods as inputs.
Villanacci and Zenginobuz use the concept of subscription equilibrium introduced for that setup
by Malinvaud. Previous results developed in simple versions of the model have suggested that
a certain neutrality property would be satisfied by subscription equilibria in these simple models. Roughly speaking, the level of public good provision at equilibrium does not depend on
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individual endowments of agents who contribute to the production of the publish good. More
specifically, as long as the set of contributors remains the same, a tax-financed increase of government spending on the public good reduces voluntary private contributions to the public good
by an equal amount. That neutrality property has been established using very restrictive assumptions, including the lack of general equilibrium effects. Villanacci and Zenginobuz show that the
neutrality property does not hold true once we have a genuine general equilibrium environment.
The contrary would have been surprising, but the main strength of the result is that it invalidates
any hasty conclusion that might be drawn from partial equilibrium or other unrealistic models. In
fact, the authors establish that, for a generic set of economies, every subscription equilibrium can
be Pareto improved by suitable government interventions. This paper is unquestionably in the
lineage of the Arrow–Debreu and McKenzie papers, but also exploits techniques of differential
topology that are the hallmark of much of recent research in the theory of general equilibrium.
What is true of the model with one public good and many private goods is likely to be also true
of other simple models and their properties, whether these models deal with macroeconomics or
finance. This will have to be kept in mind when reading the conclusions of some of the other
papers in the symposium.
3. Overlapping-generations and efficiency
Henriksen and Spear explore efficiency properties of equilibrium allocations in a standard
stochastic exchange overlapping-generations model with three-period lived agents and a productive asset but no informational frictions. That paper is not only an offspring of the two papers
by Malinvaud and Samuelson [31,33], it combines earlier work of Spear on rational expectations
equilibria in the overlapping-generations model [35] and the huge literature on incomplete asset
markets. In addition, the paper crucially exploits the characterization of efficiency given by Cass
for growth models in [14] and for the overlapping-generations model by Balasko and Shell [11].
Their main conclusion is that markets are sequentially incomplete in their model and, as a
consequence, risk is not allocated efficiently at competitive equilibria. They then show that a
financial reform which replaces the positive net supply asset with a set of insurance contracts
in zero net supply generates a Pareto improvement on the competitive equilibrium allocation.
They also show by way of numerical simulations that tax and transfer schemes can also generate
Pareto improvements.
The paper by Henriksen and Spear [30] is an important improvement on Demange [21], where
it is shown that if markets are sequentially complete then the stationary competitive equilibrium
in a model in which agents trade only a single good and live more than two periods are Pareto
optimal.
By showing that risk is not allocated efficiently at equilibrium when markets are not sequentially complete, Henriksen and Spear provide further evidences that systemic risk as identified in
2008 financial market meltdown can be the consequence of fundamental market incompleteness.
4. Collateral, regulation and leverage
Three papers come under this heading. This number illustrates the dynamism of some of the
research lines that have evolved from the study of Arrow’s asset model into the study of general
equilibrium models with incomplete markets combined with the explicit introduction of default
and promises backed by collateral. The stakes are high because these models aim at nothing less
than understanding the most recent and current developments of the world economy and finance,
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from the subprime crisis of 2007 to the sovereign default crisis of some eurozone countries in
2011.
4.1. Durable goods as collateral and their regulation
The model considered by Araujo, Kubler and Schommer [1] follows Geanakoplos [25,26] and
Geanakoplos and Zame [29]. The model has two time periods, with uncertainty over the states
of the world in the second period. There are two commodities, one perishable and the second
durable. The durable good serves as collateral. A financial asset in this model is characterized
by its state-contingent promises in the second period and by its collateral requirement. As in
Geanakoplos [25], financial contracts consist of the durable collateral and a promise. The actual payoff of the asset will be the minimum of its promise and the value of the collateral. In
equilibrium every such contract is priced, but not every such contract will be positively traded.
Evaluated at equilibrium prices, each contract determines a loan to value ratio (the ratio of the
value of the promise divided by the value of the collateral) or equivalently a margin (1 minus
the loan to value). Clearly contracts with low margins will have a greater likelihood of defaulting. One question is whether contracts with margins low enough to entail default in some future
states will be positively traded. Araujo, Kubler and Schommer [1] show that default can occur at
equilibrium.
Another question is whether the government can make everyone better off by prohibiting
contracts with margins below some threshold. More precisely, the paper investigates how the
regulation of margin requirements can affect the equilibrium allocation and how regulation can
make all or some of the consumers better off. This question is very topical since the asset with
the lowest collateral-requirement can be interpreted as a subprime loan.
Similar exercises have been done for all kinds of models in economic theory. They are still
the bread and butter of cost-benefit analysis. The huge literature on the theory of the second-best
also warns us of the difficulty if not the impossibility of reaching clear cut answers in general
models. The model considered here is no exception. After the negative result that the conditions
for making every consumer better off through margin regulation are so restrictive that there is
little chance that they can be satisfied in the real world, the authors consider a series of numerical
examples. They then show that the rich and the poor gain at the expense of the middle-class when
only subprime loans can be traded and markets for prime loans are closed. These examples also
show that only subprime loans are traded when the borrower owns almost no collateralizable
goods. In the opposite direction, one example shows that subprime loans are not traded when
borrowers own substantial amounts of the durable good.
4.2. Securities as collateral and repo markets
Following the pioneering work of Duffie [23], Bottazzi, Luque and Páscoa [12] develop a
model where loans are backed by securities instead of durable collateral of the kind considered
in the previous paper. In models where assets are backed by durable collateral like housing, the
available quantity of collateral caps the amount of securities (mortgage) that can be issued. (See,
for example, the Geanakoplos and Zame’s paper [29].)
Collateral used in repo markets are securities and they are not an argument of preferences or
utility functions. The collateral is then fully recyclable. It can be lent and sold by the counterparty it has been lent to. Shorting and issuing of securities, which are formally identical in the
traditional asset models, are now treated differently. Shorting occurs once the securities have been
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issued. The possibility of re-hypothecation in this model is far more important than the actual nature of the collateral as re-hypothecation drives the potential leverage of positions. The definition
of equilibrium excludes the possibility of default (failure to return money) and fails (failure to
return a security). The authors show that equilibrium exists under exogenously determined limited re-hypothecation. Two cases are modeled: (1) regulatory limits on dealers’ leverage and (2)
segregation of haircuts.
4.3. The pro-cyclical pattern of leverage
Numerous data show that the pattern followed by leverage is pro-cyclical: in good times
leverage is rising, and after bad news leverage falls. Why? A recent literature starting with
Geanakoplos [26] explains how an increase in volatility reduces leverage. Without a theory that
explains why bad news increases volatility, the explanation of the pro-cyclical pattern of leverage
is incomplete. The paper by Fostel and Geanakoplos [24] fills in this gap by suggesting a reason
why bad news is more often than not associated with higher future volatility.
This paper considers a multi-period model in which projects can be built in period 0 that pay
dividends in the last period. In intermediate periods information is revealed about how productive
the projects will be. Fostel and Geanakoplos find that, all else equal, the market gives a higher
price to assets or projects for which bad news (if it occurs) dribbles out, because those assets can
be leveraged more. Thus those are the assets that will get built. But for those assets, any piece of
bad news creates the possibility of more bad news, which means more uncertainty. So according
to the paper we should see uncertainty rise after bad news for the majority of projects actually
in production, and furthermore, the majority of projects we see should go wrong slowly when
things go bad. Remember that the crisis of 2007–2009 developed very slowly, with every bank
dribbling out information about losses a few billion dollars at a time, each time creating more
uncertainty.
The paper by Araujo, Kubler and Schommer in this symposium gives a two-period example
of an asset which is used as collateral in two different actively traded contracts. The paper by
Fostel and Geanakoplos presents for the first time a three period model in which an asset is
endogenously traded simultaneously at different margin requirements in equilibrium.
5. Bifurcations to sunspot equilibria
Nonsunspot equilibria in the Cass–Shell sunspot model [17] do not depend on the level of
restrictions to market participation faced by the consumers because the nonsunspot equilibria
coincide with the equilibria of the associated certainty economy. In addition, if consumers face no
restrictions in the sunspot model, the only equilibria of that model are the nonsunspot equilibria;
see the sunspot immunity theorem in Cass and Shell [17]. If the certainty economy features more
than one equilibrium and states of nature are equiprobable, a simple continuity argument shows
that there is some threshold level for restrictions in market participation beyond which sunspot
equilibria exist since the fully constrained sunspot economy features a number of equilibria that
is equal to the number of equilibria of the certainty economy raised to the power of the number
of states of nature. See Balasko [7]. The total number of equilibria is then much larger than
the number of nonsunspot equilibria, a number that is equal to the number of equilibria of the
certainty economy, which proves the existence of sunspot equilibria in such cases; see Balasko,
Cass, and Shell [10].
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405
The only way for a nonsunspot equilibrium to bifurcate into a sunspot equilibrium when market participation is varied—for example because more agents are prevented from participating
in the asset market—is therefore for a stable nonsunspot equilibrium of the sunspot economy
without restrictions in market participation to become unstable once restrictions in market participation are imposed on the economy.
The main contribution of Balasko [6] is the proof that this phenomenon exists or, in other
words, that stable nonsunspot equilibria may become unstable. The overall picture, however, is
far from simple. First, it is necessary to define an adjustment dynamics that treats symmetrically
the various states of nature, an essential characteristic of the sunspot model especially when
states of nature are equiprobable. This is done with the help of the variation on Walras tatonnement considered in Balasko [8], a dynamic process where goods are treated symmetrically and
adjustment speeds endogenously determined. The nonsunspot equilibria associated with stable
certainty equilibria are then stable for that dynamics in the two extreme and polar cases, namely
the fully constrained and the fully unconstrained sunspot economies.
Is it then possible to observe loss of stability between the two extremes, i.e., for intermediary
levels of restrictions in market participation? Stable equilibria with small net trade vectors are
shown to be immune to the level of restrictions in market participation in the sense that they
remain stable even if the speed of convergence to the equilibrium may vary up to some level. But
the speed does not vary sufficiently to become equal to zero and for the equilibria to bifurcate into
instability. This immunity property fails to be true for the certainty equilibria that feature large
net trade vectors. The corresponding nonsunspot equilibria may then be unstable for intermediary
levels of restriction in market participation. A consequence of the stability of the stable certainty
equilibria in the two extreme cases of fully constrained and unconstrained participation is the
lack of a general rule about the appearance of instability as a function of the level of restrictions
in market participation.
6. Conclusion
A widely held view among the general public is that the Great Depression and the current
financial and economic crisis triggered by the subprime debacle are evidence that economics
is not a science in the same sense that mathematics or mechanics are sciences. The research
reported in this symposium is proof of the contrary. It illustrates how mathematical models of
the economic and financial worlds developed from first principles before the crisis can help us
understand a world in crisis and how we got there. Regulators call for the monitoring of systemic
risk. This can only mean the study of general equilibrium.
Economics is a difficult science with many still unchartered territories. It is through more
research of the kind reported in the articles published in this symposium that our understanding
of the operations of modern economies will be able to improve.
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