Thursday, July 26, 2007

Steven Horwitz, Confused?

"A Mengerian understanding of the market process rejects the claim that an economy can be fruitfully understood through the use of simultaneous equations and equilibrium constructs… The Austrian approach rejects equilibrium theory as a description of actual economic events (although some Austrians would retain it as the never-achieved endpoint of economic activity) in favor of other theoretical and metaphorical devices." -- Steven Horwitz (2000: 8)
I don't know why Horwitz identifies "simultaneous equations" with equilibrium.

Consider this applet. I think one can characterize the underlying mathematics as a system of (countably infinite) simultaneous equations. Yet, one can hardly say that the interest in this mathematics lies in an equilibrium point, at least above a certain value of a parameter.

And that mathematics has economic applications. A Ricardian model can yield a logistic equation (Bhaduri). So can a cobweb cycle with an affine supply function and a quadratic demand function (Goodwin).

Advocates of the Austrian school should strive to write so one cannot read them as not being able to do math, instead of as simply choosing not to do math.

References
  • Bhaduri, Amit (1993). Unconventional Economic Essays: Selected Papers of Amit Bhaduri, Oxford University Press
  • Horwitz, Steven (2000). Microfoundations and Macroeconomics: An Austrian Perspective, Routledge
  • Goodwin, Richard M. (1990). Chaotic Economic Dynamics, Oxford University Press

11 comments:

Gabriel said...

I think there's an error here: "affine supply function and a quadratic supply function". Supply, supply?

Goodwin was hardcore. We should definitively have more endogenous cycle models in economics.

I'm not as sure as you are, though, that steady states are less interesting than the rest of the state space.

James said...

Where does Horwitz claim identify simultaneous equations with equilibrium? (Hint: "Economies cannot be fruitfully understood by A and B" is not the same as "A is identical to B".)

Ian said...

I can understand why Horwitz identifies simultaneous equations with equilibrium states: The fixed point of a system of differential equations is a system of simultaneous equations.

To model out-of-equilibrium processes requires either differential or difference equations, not simultaneous equations.

Robert Vienneau said...

Fixed. Thanks, Gabriel.

No, James. If my reading of Horwitz is hostile, it is not there. Horwitz is not using "and" in the strict sense of formal logic, as if he were a engineer designing digital electronics. It helps to read more of the conversation, if one wants to understand Horwitz. It would not surprise me if I've read more of the Austrian school than James has.

If I understand Ian, his point is to contrast "difference" and "simultaneous". Does the distinction rely on non-syntactical considerations? Is Horwitz's point invalidated if I just consider an application where state variables are subscripted by a one-dimensional space variable, instead of a time variable? (Would you be more willing to call difference equations a countably infinite set of simultaneous equations if, in the difference equations, the value of the state variables is a function of their value for both greater and lesser indices? I think this is called a non-causal system.) In any case, fixed points, that is, solutions in which the values of the state variables do not change with the value of the index, are picked out. This is an equilibrium, if you will.

I have the question mark in the title of this post deliberately. The mathematics of dynamical processes is about more than equilibria. But there are some aspects of economic processes I don't know can be formalized. For example, I don't know how to do interesting mathematical modeling without the law of one price, where variations in the price of a commodity arise because a Kirzner entreprener hasn't yet been alert enough to notice the opportunity for making pure economic profits. Maybe one could do something with abstract algebra, but I'm not at all sure formalization would have a point here.

Gabriel said...

I have little experience with these things but as far as I can see, one way to study dynamic systems (maybe the only way) is to look for steady states/equilibria, the local/global stability of those steady states and then do parameter sensitivity. And if you get chaotic behavior, you move on to that textbook. :-)

Since usually the dynamic system is the reduced form of a theoretic model, you really don't want chaos or fancy paths. -- Even something as simple as a saddle point is hard to interpret from a theory point of view.

With search models I think you can get trading at different prices, in the same period. But since one-sided search models use a probability distribution as a deux ex machina, I'm not sure it counts.

walt said...

Gabriel is right -- you can have a search model where different firms charge different prices. Under one plausible set-up, you in fact get that firms don't compete at all, and each firm charges the price they would in monopoly. (This is the Diamond paradox.)

Ian said...

> Does the distinction rely on non-syntactical considerations?

In the abstract, I'm not sure, and I agree there are often multiple mathematical representations of the same state-of-affairs. However, in the history of science differential and difference equations are normally employed to model dynamic processes, whereas simultaneous equations are normally employed to model stable states.

In economic theory, such as linear production theory, there is an absence of "adjustment laws" that govern the trajectory of the economic system through state space. Yet the economy is a dynamic system. Simultaneous equations and comparative statics in general misdirect our attention to very special cases that are never achieved in practice.

Gabriel said...

>> "In economic theory, such as linear production theory, there is an absence of "adjustment laws" that govern the trajectory of the economic system through state space."

How about the dynamic version of Leontief's I/O model? It's dynamic (discrete time), it's linear, it's multisectorial, it has steady states.

Robert Vienneau said...

A mathematical analysis of a dynamical system begins with an examination of equilibrium points and their stability. Further analysis might find limit cycles (i.e., fixed points of Poincare return maps). This is not yet chaos. I do not agree with Gabriel that finding chaos ends one's interest in a dynamical system.

I do agree with Gabriel that there are dynamic analyses in the literature of linear production models. I'll mention Goodwin, Pasinetti, and Rosser. I don't know that these models answer Ian's point.

I'm not sure if I made my point about simultaneous equations clear. Difference equations can be viewed as a system of simultaneous equations in which, in applications, the variables are indexed by time. Thus, being interested in (non-steady states of) dynamical systems does not preclude an interest in the mathematics of simultaneous equations.

I don't think probability distributions formalize entrepeneurship. In Kirzner's approach, the ability of entrepreneurs to discover opportunities for pure profit is not a function of how long they search. It's a matter of whether they are alert enough to notice facts that are right under their noses. My suggestion was to wonder if one could organize sets of some prices together in some sort of structures.

James said...

Robert,

Robert's high estimate of his own reading would probably be more interesting to his biographer, but it doesn't really relate to the original post or my initial comment. I asked him where Horwitz identifies simultaneous equations with equilibrium, but he neglected to provide a citation.

It's telling that Robert somehow missed my request for a citation in a post with but two sentences and then admonishes me to "read more."

Robert Vienneau said...

I made an error in my post. The Horwitz quote is from page 18, not page 8.