Thursday, February 21, 2013

Inconsistencies In Keynes's General Theory And In General Equilibrium Theory?

1.0 Introduction

In this post, I raise some questions about how short-period and long-period equilibrium relate. I consider this issue in two frameworks:

  • Keynes's General Theory.
  • The Arrow-Debreu model of intertemporal equilibrium.

The issue in the first framework is at least partly a question of hermeneutics - can one read Keynes such that his theory is internally consistent? The issue in the second framework is, as I understand it, a question of mathematics.

2.0 Keynes's General Theory

It seems to me, Keynes, in his definition of long-period equilibrium simultaneously postulates that agents have common and divergent expectations. Can one find a consistent long-period equilibrium concept in the General Theory that includes both production and financial markets? I think of Post Keynesians, such as Joan Robinson, as having emphasized the extension of the General Theory to the long period. But I am not sure I can find a good resolution in the literature. I think my question is close to Jan Kregel's (1985) point.

Keynes defines long period equilibrium in Chapter 5, "Expectation as Determining Output and Employment":

"If we suppose a state of expectation to continue for a sufficient length of time for the effect on employment to have worked itself out so completely that there is, broadly speaking, no piece of employment going on which would not have taken place if the new state of expectation had always existed, the steady level of employment thus attained may be called the long-period employment1 corresponding to that state of expectation. It follows that, although expectation may change so frequently that the actual level of employment has never had time to reach the long-period employment corresponding to the existing state of expectation, nevertheless every state of expectation has its definite corresponding level of long-period employment...

...past expectations, which have not yet worked themselves out, are embodied in the to-day's capital equipment with reference to which the entrepreneur has to make to-day's decisions, and only influence his decisions in so far as they are so embodied. It follows, therefore..., to-day's employment can be correctly described as being governed by to-day's expectations taken in conjunction with to-day's capital equipment.

1 it is not necessary that the level of long-period employment should be constant, i.e. long-period conditions are not necessarily static. For example, a steady increase in wealth or population may constitute a part of the unchanging expectation. The only condition is that the existing expectations should have been foreseen sufficiently far ahead." -- John Maynard Keyes (1936): pp. 48-50.

Keynes analyzes financial markets in Chapter 12, "The State of Long-Term Expectation". It is in this chapter that he introduces his distinction between speculation and enterprise, as well as likening the markets for shares (stocks) and bonds to a beauty contest in which participants are not trying to pick the prettiest entrant, but rather the entrant who will be thought prettiest by common opinion, when all who are picking the entrant are looking at their choice from this perspective.

My claim is that the Chapter 5 definition requires common expectations, while the chapter 12 analysis presumes a diversity of expectations. In fact, according to Keynes, the momentary stability of financial markets depends on a balance of bulls and bears and, thus, divergent expectations.

3.0 General Equilibrium Theory

In the Arrow-Debreu model, an equilibrium can be considered a path through (logical) time. Some of those paths, in some simplified models, are steady states, in which each industry expands at the same rate of growth. One can read Von Neumann as setting forth the production side of such a model of a stationary state, if one so chooses.

In general, an equilibrium path in the Arrow-Debreu model is a (very) short-period equilibrium. The initial endowment of capital goods is taken as given, and expectations of the agents in the model are pre-reconciled. (Questions, perhaps unanswerable, exist about how such an equilibrium can be achieved.) One can ask about the limit behavior of each equilibrium path, as time increases without bound. Some of these paths might converge to stationary states, and some might diverge.

I associate the Turnpike Theorem with Paul Samuelson. I turn to either Dorfman, Samuelson, and Solow (1958) or, for example, Dixit (1976), when I want to read an exposition of this theorem. As I understand it, this theorem implies that stationary states, generically, have saddle point (in)stability in the Arrow-Debreu model.

I am also aware of the Sonnenschein-Mantel-Debreu theorem. As I understand it, this theorem implies almost any dynamics are possible in the Arrow-Debreu model. Some such dynamics, then, should be consistent with stationary states that are locally stable. Another dynamics in the model should be consistent with stationary states that are locally unstable. And some configuration of parameters should be consistent with multiple equilibrium mixing any combination of stable steady states, unstable steady states, and saddle-point steady states.

My understanding of the implications of the Turnpike and Sonnenschein-Mantel-Debreu theorems seems to be inconsistent. Where do I go wrong?

  • A. K. Dixit (1976). The Theory of Equilibrium Growth. Oxford University Press.
  • Robert Dorfman, Paul A. Samuelson and Robert M. Solow (1958). Linear Programming and Economic Analysis, Dover.
  • John Maynard Keynes (1936).The General Theory of Employment, Interest and Money. Harcourt, Brace and Company.
  • J. A. Kregel (1985). Hamlet without the Prince: Cambridge Macroeconomics without Money, American Economic Review. V. 75, No. 2 (May): pp. 133-139.
  • J. Von Neumann (1945-1946). A Model of General Economic Equilibrium, Review of Economic Studies. V. 13, No. 1: pp. 1-9.


BruceMcF said...

What Keynes dubs a long-period equilibrium seems to be a steady state in production consistent with the current state of effective demand. I don't see how a range if expectations as to whether or not the system is IN a steady state is incompatible with a steady state, provided the balance of opinion is that it is ~ and such a condition seems like it would be more robust against small fluctuations, in the same way that a balance of power between bulls and bears in financial markets yields a situation that is more robust in the face of small fluctuations than if everyone had identical expectations and everyone had to decide how to react to the small change. Given a thick and fairly even distribution between bulls and bears, with a few market participants near the margin between the two for any given price, a small fluctuation will flip the view of a small share of market participants resulting in a small correction in financial asset market values, while for most of the market they will remain bulls or bears, just with their view either strengthened or weakened a bit.

Imperfect information is a fundamental assumption of the General Information, and so in tackling the question we have to be careful that we take imperfect information seriously. Given imperfect information, the actions of producers are not just in reaction to expectations, but are also activities that uncover new information. A spread of expectations assures us of a range of producers activity ~ with respect to investment, research and development, inventory, marketing strategy, etc. ~ which will uncover a range of information. If the productive system is in fact in the near neighborhood of a steady state, a spectrum of actions exploring the neighborhood by a range of producers who expect that production in their industry is above, below or near to the long term sustainable level will reinforce the current steady state by confirming the underlying fact that they ARE in fact near to a long term sustainable level, given the present level of production of the other industries that they interact with.

The actions of a large group whom have identical expectations would be far less informative, and would not justify nearly as much confidence that the unanimous expectation is correct, even if that expectation is in fact correct; the actions of a large group whom have a range of expectations with the current situation being in the close neighborhood to a steady state being the balance of expectations would generate information that would justify far more confidence that the system is in the neighborhood of a steady state, if in fact it is.

Unlearningecon said...

I don't see how ch5 requires uniform expectations. It seems to me that a set of expectations can be wildly divergent between agents, yet still correspond to a particular long-run quantity of employment should they be fulfilled.

Robert Vienneau said...

Thanks for the comments.

I think of an input-output matrix as representing a dominant technique. After Bruce points it out, I have no problem with using such a matrix to represent a steady state in which not all producers are using the processes comprising that technique. Some are using old processes and earning less than profits than others. Some are exploring new processes that may later emerge to form the dominant technique. And some are exploring new processes that will turn out to be mistaken.

Since reading this post, I have skimmed Kregel (1976). I can accept his point that Keynes postulates long run expectations as being unaffected (for a time?) by disappointed short-run expectations.

It seems to me that Keynes' analysis of financial markets requires some short run expectations to become disappointed. I read Bruce and "Unlearning" as importing some aspects of this analysis to long-run expectations.

I find these responses interesting. Bruce, did you ever attend the Trieste summer school, where, apparently, disputes between Sraffians and Post Keynesians were never resolved.