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Models: What Are They Good For?

  • Science
09/19/2002Gene Callahan

>Many large brokerage houses, as well as small boutiques, rely heavily upon mathematical models in their day-to-day trading. Clearly, there is a market for this kind of modeling, yet Austrian economists have generally held the view that the neoclassical, mathematical approach to economics is a strictly limited approach.

How can we reconcile the market success of mathematical models of trading with the Austrian skepticism toward theoretical models expressed in math? Let's examine this question.

The Limits of Models

Mises explained the fundamental gulf between economics and mathematics in Human Action:

"Logic and mathematics deal with an ideal system of thought. The relations and implications of their system are coexistent and interdependent. We may say as well that they are synchronous or that they are out of time. A perfect mind could grasp them all in one thought…. Within such a system the notions of anteriority and consequence are metaphorical only. They do not refer to the system, but to our action in grasping it. The system itself implies neither the category of time nor that of causality. There is functional correspondence between elements, but there is neither cause nor effect. What distinguishes… the praxeological system from the logical system is precisely that it implies the categories both of time and of causality" (1998: V.1).

Let us take a famous mathematical creation, the Pythagorean theorem, as an example of what Mises is talking about. As is well known to school children everywhere, the theorem says that there is an immutable relationship between the three sides of a right triangle, where the sum of the squares of each of the shorter legs equals the square of the longer leg (a2 + b2 = c2). Mises is saying that this type of relationship has a fundamentally different nature than do the relationships implied by human action.

None of the legs of a triangle causes any of the other legs to be a certain length. Neither Pythagoras's equation nor any of the infinite number of triangles that it describes has any temporal relationship to each other. Once we clearly conceive the concept "right triangle," then the universe of right triangles, along with the relationship of their sides and all other geometric facts about them, emerge as the aspects of a completely timeless, ideal form. Although our limited minds must approach these aspects piecemeal, their existence is simultaneous with the very notion "right triangle," and none of these aspects are prior to or stand in a casual relationship to any other aspect of the ideal form.

Human action is different. Just as the idea of a right triangle implies the Pythagorean theorem, the idea of human action implies "before" and "after," "cause" and "effect." We cannot make sense of human plans unless we understand that there is a past which, for the human actor, provides the soil in which the seeds of action might be sown, that there is a present during which the sowing will transpire, and that there is a future in which the actor hopes to reap the fruit of his action. Similarly, we must see that the actor hopes that his action will be the cause of a desired effect, or he would not act.

The Neoclassical Overreach

Viewing the economy as if it were a mathematical form is coherent if it is seen, for instance, as the study of a limiting state—equilibrium—that the real economy may have some tendency to gravitate toward. Equilibrium theorizing is the construction of an abstract economy in which economic activity proceeds strictly according to an interrelated system of equations. As such, it may or may not yield an increased understanding of certain aspects of an actual economy.

But when taken as more than that--as, for instance, a description of the real economy, a normative ideal to which the real economy must measure up, or as a causative force generating economic actions--it creates confusion. It eliminates from view real human choices, the very phenomena that differentiate economic activity from all else going on in experience.

Let us look at an example. In Steven Landsburg's microeconomics textbook, Price Theory, he says:

"It is important to distinguish causes from effects. For an individual demander or supplier, the price is taken as a given and determines the quantity demanded or supplied. For the market as a whole, the demand and supply curves determine both price and quantity simultaneously" (1999: 18).

Landsburg is saying we must not think of prices as being determined by the actions of individuals--individuals simply take prices as a given. Instead, it is the abstract mathematical notions of supply and demand curves that simultaneously determine what occurs in the market.

We can agree with Landsburg that it is important to distinguish causes from effects. At the same time, we must contend that there is a real confusion in Landsburg's presentation of cause and effect in economics. If Landsburg wants to construct a mathematical model in which individual market participants have no influence, but instead are moved by equations describing intersecting curves, he is engaged in a coherent enterprise, the value of which he would have to demonstrate by drawing our attention to whatever insight it provides.

But in doing so, it is incoherent to explain how we arrived at the equilibrium point by invoking producers' or consumers' decisions to raise or lower their bids and asks, as Landsburg does. By hypothesis, such explanations have already been excluded as causative factors in the model, so to bring them back into play makes the whole enterprise self-contradictory.

In the real economy, as Landsburg acknowledges, prices and quantities change as the result of human action. Where in a real economy can a new price come from if not a human deciding to bid or ask above or below the current market price? It is the striving of individuals to better their circumstances, in the face of an uncertain future, that drives the market process.

Landsburg is forced into his odd posture because, on the one hand, he wishes to use mathematics as the basis for economics. On the other hand, he wants his economics to describe the real world. His goals are at odds with each other, because mathematical equations cannot take into account creative human decisions based on the categories of cause and effect, before and after.

What they describe is a world of timeless correlations from which causation is absent. Human intentions play no part in the model, as the model assumes all humans know everything relevant to their situation and can only accept it as a given. Faced with the prospect of acknowledging the limits of his model, Landsburg opts for eliminating human action from the economy, but then slips back into considering human action when giving a verbal explanation of the model.

The fact that supply and demand curves can give us a rough picture of market behavior is an effect of human action, and certainly not the cause of it. No one acts with the goal of bringing supply and demand into balance. (Well, no one except Fischer Black, as we'll see later.) People act in the market in order to profit, in the broadest sense of the word: they exchange because they feel they will be better off after the exchange than they were beforehand. That their search for profit tends to bring supply and demand into balance is a by-product of their actual goals. As Hayek said: "…the modern theory of competitive equilibrium assumes the situation to exist which a true explanation ought to account for as the effect of the competitive process" (1948: 94).

In other words, mathematical economics simply assumes that prices do bring supply and demand into balance, without asking how we could have found the prices that did so. Rather than a market process that consists of a groping for "correct" prices, in equilibrium models, everyone knows the price to trade at before trading can begin.

The neoclassical mistake is not the mere use of such models. Seen for what they are, they may be useful. Instead, the mistake is the notion that equilibrium constructs are good descriptions of the actual market, or normative ideals to which we should compare the real market. In fact, they are mental constructs that abstract out the most crucial part of economics--human action--from the world, and treat the world of economics as if it consisted only of quantities of goods and services.

So What Is It Good For?

But that leaves us with a puzzle: Just what are all of the investment banks, trading companies, and so on doing spending so much money on modeling? Are they just nuts?

From an Austrian perspective, it would be strange to find large groups of businessmen who were wasting resources on a useless activity. So are Austrians wrong about mathematics and economics? I don't think so, because the Austrian arguments are sound as far as I can see, and what Peter Boettke calls the "precise irrelevance" of much of modern economics is good evidence that the arguments are sound.*

My own experience at the equity trading company led me to think about that question a good deal. What I finally decided was this: Mathematical equations can be useful for modeling the result of people following through on previously made plans, for capturing "equilibrium-like" phases of markets.

Analogously, we could say that once a player in a basketball game decides to shoot jumper, we could use an equation that, based on the initial force vector that the shooter chooses to apply to the ball, then predicts the progress of the shot. Such an equation will be of little use, however, in predicting whether the player will change his mind and pass instead.

Similarly, the relative price of the two stocks in a merger may move in line with the predictions of a mathematical model most of the time. But if traders find out something that alters their perception of the merger, the relative price of the stocks may differ greatly from the model's prediction.

If rumors emerge indicating that the merger might fall through, the relative price of the seller might plunge. Traders have to employ their entrepreneurial judgment in an attempt to grasp how other market participants will react to this news. Once this evaluation is completed, a new risk factor for deal failure can be fed into the model and it may again function reasonably well

Such a model cannot capture the change of perception in the market, which is the beginning of the creation of a new plan. That is the moment of human choice, as the plan must aim for one goal while setting aside others, and choose some means to achieve that goal while rejecting others. Mathematical economics models the equilibrium-like phases of markets, when no plans are being created or revised.

Fischer Black and CAPM

Now, as I mentioned, no human has ever acted with the goal of bringing supply and demand into balance… except Fischer Black. For those of you who don't know of him, he was one of the two creators of the famous Black-Scholes option pricing model, and he also has contributed to other areas of finance and economics, especially the Capital Asset Pricing Model [CAPM], one of the most famous models in finance. I take up Black here because his work is so well known in this area.

I make him the exception to the rule because it seems he really tried to live his life with the goal of bringing about a general equilibrium, or what Austrians might call the evenly rotating economy. (See Mehrling, 2000.)

Perry Mehrling  (ibid: 5-6) says:

"From [Black's] point of view, economic growth appears as a process of increasing sectoral differentiation and increasing temporal roundaboutness, a process with no apparent end in sight. What we observe as accumulation of capital, physical and human, is just the form that the process takes."

So, again, we see that the actual concrete choices people make, the individual items of physical and human capital they choose to accumulate, are seen as the happenstance "forms" of a "process," in the same way that concrete decisions to buy and sell are seen as the outcome not of human choice but of mathematical curves. Since Black wanted to create mathematical models that captured economic events, he had to take such a view--as we saw above, mathematics inherently can't deal with human action.

In Black's model, all security prices are explained by considering only the rate of interest and the price of risk. Of course, any model must simplify like that, as it's not possible to have a model in which the input is "everything." Such a model would have to be as complex as the entire universe!

Whatever parameters a model includes, however, as more people employ it, those parameters not included will tend to move away from equilibrium pricing. Let's say that I know of some firm with a great R&D department. Well, having a great R&D department is not a parameter in CAPM, so it is not taken into account by those trading using CAPM. As more people employ CAPM, that great R&D department will become more and more underpriced, and the profits I can make by buying the firm will increase. No such model could "achieve" equilibrium, as that would result in a state where no one is profiting. But it is precisely the desire to profit that drives human action.

The Evenly Rotating Economy

What Black was hoping to achieve was an economy without true economic profit--all investors would receive simply the market rate of return. Mises pointed out the absurdity of an economy without profit when he discussed the evenly rotating economy--roughly what mainstream economists call general equilibrium.

The evenly rotating economy is an economy where the price and quantity of all goods is in balance. There are no price fluctuations, as everyone already knows what everyone else wants to buy and sell, and how much. Such an "economy" is an endless cycle of the same events being repeated. The same number of babies is born each year, and that number exactly equals the number of people dying. The same goods are manufactured each year and demanded in the exact same quantities. No harvest ever fails, no business ever goes bankrupt, no new products are ever introduced, and no person's tastes ever change.

Such a world could not possibly exist, but it can be helpful to create the image of such a world for use as a mental tool. By introducing a single change into our mental construction, we can isolate what the effects of that particular change would be, apart from the welter of complicating data that exists in our real world.

Why isn't such a world possible? For one thing, we no longer have any motive force driving the market process. Why watch the stock market when it never goes anywhere and everyone achieves the same return? Why search for a better price for a good you want if you already know that everyone offers the same price?

David Friedman (1996: 9-10) presents a nice analogy to the stock market by discussing supermarket checkout lines. If we were in an equilibrium-always world, all checkout lines would be equally fast, and there would be no benefit to assessing their length--everyone would pick one at random. But if that were the case, there would be no driving force tending to equalize the speed of the lines, and they would become wildly imbalanced at times!

In the real world, the weather and animal populations change, resources are depleted and discovered, tastes change, learning occurs, and change is omnipresent. In the real world, entrepreneurs hoping to profit from such changes are the force driving prices toward equilibrium.

There is a further conceptual problem with a world where everyone traded stocks using the same model. Whenever the model said "sell," everyone in the market would attempt to sell. Who would they be selling to? Exchange is driven by differences in opinion as to the value of various goods, and a "market" where everyone has the same valuation of all goods is nonsensical, as there is no exchange in such a market.

The Use of Mathematical Models in the Social Sciences

As I mentioned, the above considerations do not lead me to believe that it is useless to examine the mathematical aspects of markets. Political philosopher Terry Nardin calls statistical studies in the social sciences "disguised descriptions" of customs, traditions, and practices. A particular model may work fine, as long as there are no major new alterations of those circumstances. (How large is "major"? The answer depends on how far off can you afford to have your model be!)

Karen Vaughn makes what I take to be the same point:

"In the first case [of action without a significant alteration in the social context], we try to explain a set of choices and their consequences within an established culture and an established market--within a given set of institutions. Because there are established institutional parameters, we can make informed theoretical predictions about the outcome of any action. In the second case [where new practices are being formulated], we are asking questions about the process of market creation and institutional change brought about by the discovery of new knowledge or the perception of previously unimagined opportunities. With changing institutional parameters brought about by discovery or changing perceptions, we can predict very little even in principle, since we cannot know in advance what is going to be learned or perceived [i.e., what new practice will be adopted]."

Employing Models

So, if you're going to employ models as an aid to investing, what rules of thumb might the above considerations suggest?

First of all, speculating in any market involves entrepreneurship, whether done employing mathematical models or not. As Mises (1998: XIV.7) formulates the notion, "The term entrepreneur as used by catallactic theory means: acting man exclusively seen from the aspect of the uncertainty inherent in every action. In using this term one must never forget that every action is embedded in the flux of time and therefore involves a speculation." Mises notes that:

"Like every acting man, the entrepreneur is always a speculator. He deals with the uncertain conditions of the future. His success or failure depends on the correctness of his anticipation of uncertain events. If he fails in his understanding of things to come, he is doomed. The only source from which an entrepreneur's profits stem is his ability to anticipate better than other people the future demand of the consumers. If everybody is correct in anticipating the future state of the market of a certain commodity, its price and the prices of the complementary factors of production concerned would already today be adjusted to this future state. Neither profit nor loss can emerge for those embarking upon this line of business" (ibid.: XV.8).

Someone using mathematical models for investing is betting that his models are going to be better able to anticipate future market conditions than are the efforts of others to do the same.

Here are some principles of using financial models that emerge from these considerations:

  • You must have reason to think that you've discovered an existing pattern before others have. Therefore, if "everyone" knows about a model, forget about it. The disappearance of the "Dogs of the Dow" effect once it became widely known is an example of this. The technique consisted of buying the 10 Dow stocks with the highest dividend yield and holding them for at least a year. After it became popularly known, it underperformed the market in 1998 and 1999, had a mediocre return in 2000, and lost money in 2001.      
  • You must constantly watch for the pattern dissipating, because eventually it will. Either market conditions will change, or others will start to perceive the outlines of your model. The profits you are earning are always temporary. You must watch the results of the model carefully, to see when its applicability begins to fade.      
  • A corollary to the above: Be prepared to innovate and find new patterns if you want to keep playing. Never stop trying to develop, or trying to find from others who have developed, new models, even while the old ones are still making profits.      
  • Finally, never trust anyone--including yourself--who says he has a universal system for trading securities. Such a system implies that we will arrive at the evenly rotating economy, or at least a financial-market-wide equilibrium. As we saw above, it also implies all market participants attempting to buy or sell at the same time.


* That does not mean I think economic reasoning must by empirically tested -- I agree with Say, Senior, Mises, Robbins, Knight, Rothbard, Hoppe, and others that fundamental economic reasoning is a priori. I merely would add that mistakes in a priori reasoning are possible, and that empirical results widely divergent from what one's a priori reasoning indicates should occur do not disprove one's theory, but they are a sign to check one's theory carefully. The converse, that events are proceeding as one's theorizing leads one to suspect they should, similarly does not prove one's theory, but it can increase one's confidence in it.

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Friedman, D. (1996) Hidden Order: The Economics of Everyday Life, New York: Harper Collins.

Hayek, F.A. (1948) "The Meaning of Competition" in Individualism and Economic Order, Chicago, Ill.: The University of Chicago Press: 92-106.

Hoppe, H. H. (1995) Economic Science and the Austrian Method, Auburn, Ala.: Ludwig von Mises Institute.

Landsburg, S. E. (1999) Price Theory & Applications, Fourth Edition, Cincinnati, Ohio: South-Western College Publishing.

Mehrling, P. (2000) "Understanding Fischer Black," working paper presented to the NYU Austrian Colloquium.

Mises, Ludwig von ([1949] 1998) Human Action, Scholar’s Edition, Auburn, Ala.: Ludwig von Mises Institute.

Nardin, T. (2001) "Oakeshott's Philosophy of the Social Sciences," Presentation to the First Annual Michael Oakeshott Association Conference, London School of Economics.

Vaughn, K. I. (1982) "Subjectivism, Predictability, and Creativity: Comment on Buchanan," in Kirzner, Israel, ed., Method, Process, and Austrian Economics: Essays in Honor of Ludwig von Mises, Lexington, Mass.: D.C. Heath and Company, pp. 21-29.

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