I did something weird (for me) last year: I wrote an economics paper. This was weird, because by profession I’m an astrophysicist. It was also unproductive, since I could not persuade editors at any of several economics journals that an astrophysicist might have anything useful to say about asset pricing. So after three submission attempts (none of which even got to a referee) I gave up. The thesis of the paper — that the notion of “value” of an asset, which underlies the Efficient Market Hypothesis, has no meaning, because asset markets have no stable price equilibria — evidently only persuaded the editors of these journals that they were dealing with a crackpot.

Nonetheless, I believe that there are valuable ideas in the paper. It’s a bit technical (although anyone with a little training in differential equations ought to be able to read it), but the main ideas can be described reasonably clearly without resorting to math. At least, that’s what I’m going to attempt to do here. After all, it’s in the finest tradition of crackpottery that we publish our stuff ourselves…

**The Efficient Market Hypothesis**

The foundational idea in the theory of asset prices called the “Efficient Market Hypothesis” (EMH). It’s mostly associated with the name of Eugene Fama, who wrote the basic papers on the subject in the mid-1960s. The idea that underlies the EMH is that assets (securities, bonds, futures, etc.) have prices that are statistical fluctuations about their underlying “values”. The prices are transient, but the values are not. “Value”, in this view, is the outcome of the balance between supply and demand for each asset in its respective market. Price is merely a noisy representation of value, the noise coming about in consequence of day-to-day randomness in the decisions of market players. An asset’s “value” may be measured, in this view, by averaging the price over some time that is longer than the typical scale of these noisy fluctuations.

Got that? Something subtle has already happened here — Asset “Value” in effect has *two* definitions:

- Theoretically, asset value is the result of a “tatonnement”, an auction-like meeting and equilibration of supply and demand;
- Empiricaly, asset value is the “expectation value” (i.e. the mean) of asset price, over some timescale.

It is a *hypothesis* that these two definitions identify the same thing. That hypothesis is an important (though unstated) background assumption of the EMH.

The EMH itself may now be stated: *Asset value is a function of all information relevant to the asset, and has no other dependencies. The market process factors all information (earnings, prevailing interest rates, splits, etc.) available at any time into the asset value at that time, and that value can only change in consequence of an information “shock”, wherein arrival of new information changes the asset’s value*.

There are some qualifications and shadings of this statement (“Weak” versus “Semi-Strong” versus “Strong” forms of the EMH) which need not detain us here. The point to keep in mind is that according to the EMH, all changes in asset prices reflect one of two effects: either noisy fluctuations about a steady mean (the “value”) or a secular adjustment of that underlying mean due to the advent of new information.

**Is The EMH True?**

On the one hand, the EMH has received much empirical confirmation, in the sense that statistical analyses of price time series in various markets have failed to turn up the sort of evidence for the kind of correlations that would constitute a departure from the EMH. In this sense, the EMH is widely considered to be empirically confirmed.

On the other hand, note that this empirical confirmation really only has to do with the second, empirical notion of asset “value” above. It says that if you average out the statistical noise in asset price series over some shortish time scale, no correlations are discernible in these “smoothed” time series. It says nothing about the relation that those smoothed, average “values” bear to information available to market players.

In fact, on its face, the EMH basically says something that is clearly at variance with what we know about markets: It says that there is no such thing as an asset bubble. Bubbles don’t exist, because if they did exist they would represent a third price-moving effect besides noise and information shock. Bubbles, by definition, are non-fluctuating, secular price motion whose source is irrationality, not new information. Supporters of the EMH deny that such things are possible, and in effect claim that (for example) the wild ride that the stock market took between 2003 and 2008, so far from constituting the inflating and popping of a bubble, in fact correctly represented at all times the underlying value of the assets in question.^{1}

How, then, can a hypothesis that appears to have received such strong empirical confirmation have implications that are, on their face, so absurd?

**Suppy And Demand For Assets**

I’ve already telegraphed the answer above: the trouble lies not with the empirical, operational definition of asset value above, which in fact appears to have correlation properties predicted by the EMH. The trouble lies instead with *the idea of value as the equilibrium of supply and demand*.

This is a big claim, which may be the reason the paper was not taken seriously. Supply and demand equilibrium is the foundational idea of all economics, after all. It’s so built-in to practically all economic thinking that questioning its relevance must strike professionally-trained economists the way denying relativistic gravitation strikes a professionally-trained physicist. Nonetheless, I believe that in constructing the intellectual infrastructure underlying the EMH (and consequently all modern portfolio theory, among other things), economists helped themselves improperly to the notion of “equilibrium”, which they imported into the theory of asset markets from the theory of commodities markets. The bulk of the paper comprises a mathematical argument for the thesis that the notion of supply/demand equilibrium, while appropriate for commodities markets, is in fact a grotesquely bad model for how asset markets actually function. In effect, people do not buy stocks in the same way, or for the same reasons, that they buy eggs, and the difference requires a profound adjustment in the view of what asset “value” really means.

Capital assets differ substantially from goods and services in their economic properties. In the first place, capital assets are *scarce*. Their “production” is not related to “factors of production” in the standard way that is assumed in normal commodity market models, and their supply at a given price is not determined by considerations of marginal production costs.

Moreover, on the demand side, it is clear that buyers’ incentives are quite different with respect to capital assets than they are with respect to commodities — simply put, we don’t buy stock for the same reasons we buy potatoes, or haircuts. Capital assets have a *store of value* function that is non-existent for goods and services. That is to say, assets are partly valued for their ability to appreciate, or at least to not decline in price (relative to other assets).

These differences mean that asset markets function in a radically different manner from commodities markets, because *asset supply and demand functions have different structure from their commodity counterparts.* In particular, the demand for an asset is a function not only of its price P, but also of its price rate-of-change dP/dt. An expensive asset that appreciates rapidly is in higher demand than a cheap asset whose value is declining (think shares of Google versus those of GM) — and complementarily, there are more people prepared to unload the cheap, depreciating asset than are willing to sell the expensive appreciating asset, so the supply of the cheaper asset is *higher* than that of the more expensive one in this case. This common example yields precisely the reverse of the universal rule of commodities markets, wherein cheaper commodities are always in higher demand, and lower supply, than their more expensive substitutes.

It turns out that this basic structure of the supply and demand functions for assets undermines the essential property of price equilibrium that is required for equilibrium price to serve as a notion of “value”: such price equilibria are *unstable*.

**Equilibrium Stability**

Why are equilibrium prices so important in economics? The reason is that detailed market dynamics are so very complicated, and so poorly-understood, that it is not possible to model them with any fidelity whatever. Instead, supply/demand equilibria furnish a proxy for those dynamics. They allow one to say, in effect,

“OK, the dynamics of the price are a mess, but we know that these equilibria exist, so the detailed dynamics will presumably not allow prices to wander very far away from their equilibrium points. So we’ll interpret the equilibria as the true underlying asset values, and treat the price excursions from those points as small random fluctuations. When new information comes along, we’ll model it as a motion of the equilibrium points (the values), and just assume that the prices will obediently come along for the ride.”

That’s all well and good, but in order for this to be a useful way to proceed, the equilibrium points must have an important technical property: they must be stable. This means that the detailed dynamics must be such that it leads prices *towards* the equilibrium points, or at least never leads them very far away. This is a highly non-trivial requirement! The sciences are littered with dynamical systems possessed of unstable equilibria, in which the state of the system is actually *repelled* by the formal equilibrium points. In such systems, the state may gyrate wildly about the phase space, never settling down in any particular neighborhood, but rather zooming hither and yon under its own internal dynamics, and studiously avoiding the equilibria. Even if you start such a system arbitrarily close to an equilibrium configuration, it leaves that configuration at (usually) an exponentially-growing rate.

Clearly, if the supply/demand equilibrium points of an asset market were unstable, they would certainly not supply suitable proxies for representing the action of the market dynamics. The equilibria might in fact move with new information, but that motion would not impress itself dominantly upon the motion of the prices themselves. The notion of “value” would turn out to be vacuous. This would be a catastrophe for asset market theories based on the EMH. So, do we know whether this situation is encountered or avoided in asset markets?

It turns out that a few decades ago, economists studying *commodity* market dynamics did worry about equilibrium stability — the subject even goes back as far as Samuelson’s 1941 opus. The conclusion seem to be that given reasonably general conditions, commodity market price equilibria are, in fact, stable, and so the requirement of supply/demand equilibrium is actually a useful model for such markets. This is very reassuring, and confirms the intuitions we form from our experiences of actual working commodity markets.

So far as I can tell, however, nobody has taken the trouble to re-examine the question of equilibrium stability for the very different case of asset markets, where the supply and demand functions are functions not only of price, but also of price rate-of-change, and where supply is characterized by strict scarcity. The paper I’m telling you about appears to be the first in this regard.

**Asset Market Price Equilibria Are Unstable**

That’s the conclusion of the paper. Assume very simple and general constraints on the dependence of the supply and demand functions on price and on price rate-of-change, and try to construct model markets satisfying those constraints: In every analytically tractable market that I was able to construct, price equilibria exist, but not only are they unstable, they are *catastrophically* unstable, from nearly every direction that the system can approach those points.^{2} As proxies for the dynamics, the equilibrium points are worthless. The prices just go careening around price space under their own power, executing complicated motions through parts of the space that aren’t even in the same ZIP code as the supply/demand equilibria.

On reflection, this is not a terribly surprising conclusion. Local stability is a very special condition, which requires considerable effort to arrange and to verify even in classical markets. Despite the obvious attractiveness of the notion that one can simply use an equilibrium price as a proxy for complicated and poorly -understood dynamical laws, it was always risky to assume that one could summarily take equilibrium stability as read in asset markets, without attempting to verify the plausibility of this assumption.

If we accept these conclusions, we must face up to the fact that “asset value” is a metaphysically empty concept. *There is no value, only price*. The operational definition of value — the expectation value of a price time series in the absence of new and relevant information — is still available, but it is not possible to attach it to the conception of equilibrium price in a competitive market model. Whatever empirical regularities those expectation values may exhibit, they must be explained in terms of the market bouncing around price space under the power of its own dynamics, as much as reacting to the shocks of new information.

In this light, the EMH’s view of value as a hidden variable, incorporating all or most market information, changing only in response to changes in such information, and imposing observational consequences on price time series, acquires a somewhat theological tinge. One may as well speak of an asset’s soul as of its “value”.

^{1} For a critique of the absurdity of the EMH’s view of bubbles, see George Cooper’s brilliant and eminently readable “The Origin of Financial Crises” (Random House, 2008).

^{2} Note that even one unstable direction is a catastrophe, since any tiny perturbation of the price configuration in that direction gets amplified and runs away. For true stability, an equilibrium point must be stable in *all* directions.

December 4, 2010 at 9:51 pm |

That’s right, but isn’t it sort of obvious that buying something to consume it is different from buying it to invest in it?

It seems to me that the equilibria in a pure-investment market would be repelling, ie absolutely never stable.

December 5, 2010 at 9:29 am |

Hi Bengt.

You’re right. It does seem kind of obvious. Nonetheless, the assumptions buried in the notion of price equilibrium were never really re-examined in light of this observation, so far as I can see.

Intuitively, I also agree that it is unlikely that any stable equilibria can exist in an asset market. It’s far too easy to get runaway behavior — a set of appreciating assets create new demand in virtue of their appreciation, thereby increasing their appreciation rate, and similarly for depreciating assets. Classic bubble behavior appears to be built-in to asset markets, just due to their demand/supply structure.

However I was unable to prove the impossibility of stable equilibria rigorously. What I was able to do was to show that in every case of analytically simple and un-contrived model market that I could think of, no stable equilibrium ever arises. A proof of the general impossibility of stable equilibria would certainly be very interesting.