General Equilibrium, Overlapping
Generations Models,
and Optimal Growth Theory
By Truman F. Bewley
Harvard University Press, 2007
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Chapter 1: Why Study General Equilibrium?
The reader deserves a short explanation of what general equilibrium theory is and why
it might be interesting. The theory provides a summary description of economic interaction
in a society where people are free to pursue their own interests. The theory may be viewed
as an attempt to answer the question of whether trade arranged through markets could match
demands with supplies for millions of goods and services in an economically efficient way.
Notice that the question is could trade achieve an efficient outcome, not does it
do so. This second question is empirical, not theoretical.
The rough outlines of the general equilibrium model are easy to describe. Its participants
or agents are consumer-workers and firms, and it contains a description of the motives of
each of them and the constraints limiting their choices. The model's observable variables
are prices and the flows of goods and services to and from each of the agents. A state of
the economic system in the model is a specification of all of these flows. The flows occur
through central markets that are perfect in the sense that each commodity has only
one price, all buyers and sellers know this price, and transactions are costless. Firms
choose among technically feasible flows of inputs and outputs so as to maximize the flow
of profits. Consumers possess an initial allocation of goods and services, own the firms,
and have utility functions specifying their satisfaction from consumption. Consumers sell
their initial allocations and spend the proceeds together with the firms' profits on goods
and services so as to maximize utility. The economy is said to be in equilibrium when
all markets clear, that is, when the total demand for each commodity equals the
total supply. Since demands and supplies depend on prices, these are part of the
description of equilibrium. When time and uncertainty are included in the model, it allows
borrowing, lending, and insurance. The standard model of optimal growth is an
intertemporal general equilibrium model with no uncertainty, an infinite time horizon, and
just one consumer and one firm. This model becomes the well-known overlapping
generations model if the single consumer is replaced by a succession of consumers,
each of whom lives a finite number of periods. I treat the optimal growth and overlapping
generations models as special cases of the general equilibrium model.
The main conclusions of general equilibrium theory are that competitive equilibria exist
and are efficient in the sense that there is no other attain-able economic state that
would make some consumer better off and none worse off. The assertion that competitive
equilibria are efficient is called the first welfare theorem. A partial converse,
the second welfare theorem, is that any efficient state can he realized as a
general equilibrium occurring after lump-sum taxes and payments redistribute wealth among
consumers. In the growth model, the equilibrium state not only is efficient but also is
optimal in that it maximizes the single consumer's utility over all possible economic
states, where a state of the economy is a time path of consumption for the consumer and of
inputs and outputs for the firm. The main theorem applying to the growth model is that the
optimal state converges over time to an optimal stationary state, an assertion called the turnpike
theorem. The main theorem applying to the overlapping generations model is a version
of the second welfare theorem and asserts, roughly speaking, that efficient paths for the
economy can be achieved as equilibria with lump-sum taxes. If we imagine that the economy
has a government, then it has debt and uses the revenue from the taxes to pay for interest
on the debt. It is natural to think of the taxes as instruments of fiscal policy
and the interest rate as determined by monetary policy. With this interpretation, the
interest rate is not determined by equilibrium but is chosen by some public authority. The
theorem asserts, roughly speaking, that fiscal and monetary policy can achieve any
efficient outcome. Some of the possible efficient out-comes may have high economic growth,
and others may have low growth. Efficiency does not imply high economic growth. The rate
of growth is the outcome of choices made by a public authority, whether deliberately or
unconsciously.
One of the attractions of general equilibrium theory is that its conclusions are deduced
from simple assumptions about the basic constituents of the economic system, which are
consumers, firms, and markets. The theory is reductionist in that its main conclusions are
derived from premises about the behavior of the individuals in the system; most of the
theory requires no assumptions on collections of individuals. Within standard general
equilibrium theory, a need for nonreductionist assumptions arises only when an economy can
have several equilibria. When this is so, equilibrium is not determined by the
characteristics of consumers and firms; it is necessary to say that history or some
unmodeled process chooses it. Apart from the difficulty arising from multiple equilibria,
standard general equilibrium theory is built up entirely from microeconomic theory, which
deals with the behavior of individual firms, consumers, workers, and investors. Its main
assumptions are that firms maximize profits, that consumers make their sales and purchases
so as to maximize a preference ordering or a utility function subject to a budget
constraint, and that markets clear.
Economic equilibrium appears to have two properties that may explain some of the interest
in it. First of all, in achieving equilibrium, an economy solves an extraordinarily
complex computational problem — the calculation of equilibrium prices for each of
millions of commodities. Probably this problem would be too difficult for any computer to
solve, if it were given functions that expressed how the demands and supplies of all
commodities depended on prices. The second property is that the economy performs this
calculation without having to collect information about demand and supply functions;
consumers' and firms' purchases and sales automatically reveal the needed information.
Despite appearances, the general equilibrium model, strictly speaking, does not have the
two properties just listed. An economy's use of information and calculation of equilibrium
have to do with processes, whereas general equilibrium theory deals only with equilibrium
states. Because the theory does not describe a process that finds equilibrium, it does not
con-front the question of whether free trade could match demands and supplies in an
efficient way. It is understandable that this question is not addressed, because it is not
at all obvious how to model an economy's adjustment mechanisms.
Why, then, study general equilibrium theory? One reason is that the theory is the most
sensible way that has been found to reduce a confusingly complicated economic reality to a
structure that is simple enough to re-member, analyze, and interpret. The theory also
serves as a framework of thought for questions about how economies function as a whole. In
addition, the model can stimulate economic insights, where by insights I mean short
descriptions of mechanisms governing economic life. Although only careful empirical work
can substantiate such insights, they are sources of useful ideas. 1 describe a few
insights in this text, for instance, when I use the overlapping generations model to
discuss the impact of fiscal and monetary policy and of social security on the
steady-state capital stock. Another function of the theory is that its implications
inspire empirical work by suggesting phenomena that researchers might not otherwise have
thought to look for. For instance, the turnpike theorem suggests that per capita incomes
in regions with similar natural resources and technology should converge toward each other
as the regional economies grow. Robert Barro and Xavier Sala-i-Martin (1999, chap. 11)
have tested this convergence property extensively. Another advantage of general
equilibrium theory is that it focuses attention on the question of how to achieve economic
efficiency. The model serves as a reference point by describing what the world could be
like, but for various imperfections that prevent efficiency. Some of these imperfections
have to do with externalities, where an externality is any economic influence of one
economic agent on others distinct from the purchase and sale of commodities. Examples of
externalities are industrial pollution or the air-freshening effects of cultivating trees.
If there are externalities, equilibrium may not be efficient, and we can ask how public
policy could achieve efficiency. The first welfare theorem describes the ideal situation
we want the economy to imitate; we want it to achieve efficiency and to do so while
spontaneously computing the efficient state. Another use for the general equilibrium model
is as a basis for simulations of the whole economy that give useful estimates of its
evolution over time and of the impact of changes in taxes, technology, and resources. Such
simulations are studied in a subject called applied or computable general
equilibrium, and it has a huge literature.[1] An
obvious reason for learning general equilibrium theory is to understand the major debates
in economics, many of which are expressed in the language of the theory. Probably the
theory's most important use is to guide research by providing examples of conclusions that
could be drawn or might have to be modified as the underlying assumptions of the model are
made more realistic.
To understand general equilibrium theory, it is important to keep in mind what it is not
and what it should not be used for. First of all, it would surely be unwise to elaborate
the model in order to simulate an en-tire economy in detail with the hope of making
accurate predictions. Such simulations would require radical revision of the standard
general equilibrium model since it excludes many important aspects of reality, such as
externalities, imperfect markets, absence of certain markets, expectation formation,
increasing returns to scale, inflexible prices, and lack of market clearance. Although
many of these things are included in applied general equilibrium models, a model that
included all of them and represented an economy in detail would probably be so big and
complicated that no computer could handle it and economists could understand it no better
than the actual economy. Successful simulations use reasonably simple models to give rough
estimates.
Another caveat is that general equilibrium theory is not scientific in the sense that its
main implications are not empirically testable. This statement may seem surprising, and to
understand it, one must distinguish general equilibrium theory from the microeconomic
theory on which it is based. The main assertions of microeconomics are testable. For
instance, well-known work by Donald Brown and Rosa Matzkin (1996) tests the microeconomic
assumption that consumers maximize preference orderings subject to a budget constraint.
General equilibrium theory does contain testable assertions, but these require special
assumptions. The turnpike theorem is an example. It requires specific assumptions and has
been tested by Barro and Sala-i-Martin (1999), as was mentioned earlier. Another ex-ample
is the commonsense assertion that an increase in supply reduces price. The central
assumptions of general equilibrium theory do not imply this assertion, and it is easy to
construct a theoretical counterexample to it. The assertion is implied, however, by models
satisfying particular assumptions, and such models are testable. Werner Hildenbrand (1994)
has devised such a model and has tested it extensively. In contrast, the main assertions
of general equilibrium theory — the existence of equilibrium and the welfare theorems
— so obviously do not apply to reality in any strict sense that they are best thought
of as assertions about ideal models.
Consider the theorem that equilibria exist. First of all, it is important to understand
that this theorem does not imply that actual economies are in equilibrium. The theorem
cannot do so, because, like all theoretical statements, it is an assertion about a model.
It would be equally illogical to argue in reverse that equilibria exist in a general
equilibrium model because actual markets clear. The model and economic life are separate
entities. It does make sense to check the realism of the theorem's assumptions and its
assertion in order to see whether the theorem represents reality adequately. From this
point of view, the theorem is inadequate. Actual markets sometimes do not clear; there are
prolonged periods, namely recessions and depressions, when supply greatly exceeds demand
in very important markets, namely those for various types of labor. Also the special
assumptions required to prove the existence theorem are either unrealistic or difficult to
test. One special assumption is that consumers spend all their income, and another is that
total demand and supply for each commodity depend continuously on prices. The first
assumption is inaccurate; it is not always true that actual consumers spend all their
income, and that is one reason recessions occur. The other assumption may be impossible to
verify. It is best to treat the equilibrium existence theorem as a proposition applying
exclusively to economic models. Its significance is that it specifies conditions that may
be used to check whether a particular general equilibrium model has an equilibrium.
The first welfare theorem is not scientific for similar reasons. It asserts that
equilibrium states are efficient. Deviations from the conditions stated in the theorem
probably prevent it from ever applying to actual economies, even if they are in
equilibrium. Examples of deviations are the presence of monopoly power, imperfect markets,
lack of some markets, externalities, and taxes that are not lump sum. The efficiency of
general equilibrium should be thought of as an ideal that can be approximated in reality
by appropriate public policies designed to overcome obstacles to efficiency, such as those
just listed and lack of market clearance. For similar reasons, the second welfare theorem
is also best thought of as applying only to an ideal model.
A trap to be avoided, I believe, is to accept general equilibrium theory uncritically as
true. A healthier attitude is to think of the theory as tentative and to be modified as
knowledge accumulates about how actual economies function. As has just been explained,
only microeconomic theory should be thought testable; general equilibrium theory is a set
of useful tautologies derived from microeconomics. A great deal of evidence is
accumulating that the basic assertions of microeconomic theory should be adjusted. The
theory has, however, the advantage that it is simple and easy to use and remember. For
this reason, perhaps, many economists have long been reluctant to tamper with it. In the
nineteenth century, John Stuart Mill argued that the main propositions of economics are
valid because they are deduced from the self-evident premises of microeconomics.[2] More recently, Mil-ton Friedman (1953) argued that
microeconomic hypotheses should not be tested empirically, because what matters is whether
the theory's implications, not its assumptions, are valid. These seemingly opposite
arguments both discourage questioning of the standard microeconomic assumptions.
Currently, it is common for unusual microeconomic assumptions to be labeled as ad hoc, as
if only the standard assumptions were widely applicable and new ones were made up for
particular applications, though it may be that the new assumptions apply more generally
than do standard ones. We should keep an open mind and allow both microeconomics and
theories grounded on it to evolve with increased understanding.
This text, nevertheless, uses the standard assumptions. I describe the general equilibrium
model and present the propositions that form the basic structure of equilibrium theory,
namely, the equilibrium existence theorem, the two welfare theorems, the turnpike theorem,
application of these to the overlapping generations model, and various supporting
propositions. Because interesting economic ideas can be stimulated or illustrated through
simple examples, I emphasize their construction in both the text and problem sets.
This book is based on class notes for a one-semester course taught to third- and
fourth-year undergraduates. I teach the same material to first-year graduate students in
half a semester. More difficult sections — meant for a more advanced course —
often fall at the ends of chapters.[3]
Footnotes
[1] There follows an incomplete list of sources on applied general equilibrium, assembled with the help of Herbert Scarf. Books that discuss the subject as a whole are Shoven and Whalley (1992) and Ginzburgh and Keyzer (1997). Collections of papers that cover diverse applications are Scarf and Shoven (1984), Pigott and Whalley (1985), and Fossati and Wiegard (2002). Books on tax incidence and tax policy are Keller (1980) and Ballard, et al. (1985). Books on international trade policy are Srinivasan and Whalley (1986) and Kehoe and Kehoe (1995). Books on economic development are Dervis, et al. (1982) and Mercenier and Srinivasan (1994).
[2] See Mill (1836), Blaug (1980, 68), and Hausman (1992, 124-125).
[3] The advanced sections are 3.6, 3.7, 4.8, 4.9, 5.3, 6.5, 7.4, all of chapter 8, 9.6-9.9, and 10.7-10.14.