Showing posts with label Robert Waldmann. Show all posts
Showing posts with label Robert Waldmann. Show all posts

Tuesday, June 9, 2009

people bet on their beliefs, and because their beliefs are wrong utility-subtracting risk enters the world

TO BE NOTED:

"
Rational Expectations, Efficient Markets, and Economic Welfare

I think Robert Waldmann is using the internet to carry on an argument with our ex-roommate Andrei Shleifer. I do, however, think that I need to rework Robert's example.

Robert Waldmann:

The REH vs the EMH: I think I should explain a claim I made in the post below. I assert that the efficient markets hypothesis (EMH) does not imply the rational expecations hypothesis (REH). The EMH states that asset prices are the same as they would be if everyone had rational expectations. The strong form EMH adds the assumption that everyone has complete information. The semi-strong form, like the REH has implications only for expected values conditional on public information. The EMH makes no statement about individual portfolios. It is absolutely not assumed or implied that each investor has an efficient portfolio.

In contrast, the rational expectations hypothesis says that the expected value of expectational errors conditional on public information is zero.... As used it definitely amounts to much more the assumption that observable aggregates have the values they would have if everyone had rational expectations.... This use of the phrase "rational expectations" to refer to individual behavior not aggregates is common and, as far as I know, uncontroversial....

[T]he first welfare theorem requires the assumption of rational expectations. It is absolutely not sufficient for aggregates to be the same as they would be if people had rational expectations.... I think an extremely elementary proof might be useful...


Me:

  • There are 2 time periods t = 1 and t = 2.
  • There is a coin which is flipped. It comes up heads in period 2 with probability 0.5.
  • There are 2 assets:
    • A is a risk-free asset, the numeraire: One unit of risk-free asset gives one unit of consumption good in period 2.
    • B is a risky asset that gives one unit of consumption goods in period 2 if the coin comes up heads. It sells for an equilibrium price p.
  • There are a continuum of agents indexed by i over [0.5-z,0.5+z] who maximize their expected value of the log of their consumption in period 2.
  • Each agent is endowed with one unit of the risk-free asset.

Rational expectations implies that agents know that the probability the coin comes up heads is 0.5. If everyone has rational expecations, then the market will clear with a period-1 price p = 0.5. Each risk averse agent will find it optimal to invest 0 in the risky asset. there is 0 net supply of the risky asset. Markets clear. This outcome is Pareto efficient and maximizes total utility. The EMH therefore is satisfied if the price of the risky asset is 0.5.

Now relax the assumption of rational expectations. Assume that agent i believes that the probability that the coin will come up heads is i.

The market clearing price is 0.5. At p = 0.5 agent i's net demand for asset B is:

x(i) = [i - p]/[p - p2]

The EMH still holds. p = 0.5.

The welfare outcome, however, is different. In period 2 agent i's consumption is either 1+x(i) or 1-x(i). The outcome is no longer ex-ante Pareto efficient: people bet on their beliefs, and because their beliefs are wrong utility-subtracting risk enters the world. The expected utility of agent i is:

(1/2)ln(1+x(i)) + (1/2)ln(1-x(i))

(1/2)ln(1+4[i-0.5]) + (1/2)ln(1-4[i-0.5])

If z is small so that we can approximate ln(1+y) by y - y2/2, then the approximate expected utility of agent i is:

E(U(i)) = -8[i - 0.5]2.

In the model with rational expectations, the optimal policy is laissez faire.

In the model with efficient markets but without rational expectations it would be preferable to ban gambling--to impose a 100% tax on net trading profits, and redistribute the proceeds (if any).


Robert again:

I think it is safe to say that... [the] difference...between a model in which the optimal policy is laissez faire and a model in which the optimal policy is confiscation and equal [re]distribution of all [trading profits is of interest to economists]. I do not see how it is possible for anyone who can understand the model above to conflate rational expectations and efficient markets. Oddly, however, at least two well known economists have done exactly that. One is named Andrei Shleifer.

Like many economists he has decided to call "The Efficient Markets Hypothesis", "Rational Expectations."

TO BE NOTED: From Angry Bear:

"Shrill

Robert Waldmann

I usually try to be semi polite. I especially don't usually deliberately write rude things about smart economists. However, here goes. Evidently Tyler Cowen* wrote

In a strict rational expectations model, we might expect some people to overtrust others and one view of rational expectations is that investors’ errors will cancel one another out in each market period. Another view of rational expectations is that investors’ errors will cancel one another out over longer stretches of time but that the aggregate weight of the forecasts in any particular period can be quite biased owing to common entrepreneurial misunderstandings of observed recent history. In the latter case, entrepreneurial errors magnify one another rather than cancel one another out. That is one simple way to account for a widespread financial crisis without doing violence to the rational expectations assumption or denying the mathematical elegance of the law of large numbers.


After the jump an argument which might be of some interest (added as an update) and a rant.

* Update: spelling error corrected.



Update: I just noticed something odd about the paragraph by Cowen. Like many economists he has decided to call "The Efficient Markets Hypothesis", "Rational Expectations." So before his redefinition (which I think must be absolutely condemned as I do below) he wrote "one view of rational expectations is that investors’ errors will cancel one another out in each market period." If by "errors" he means avoidable errors, then that would be the efficient markets hypothesis. However, it would not amount to rational expectations.

When the rational expectations assumption is used, it is used to mean "the things we care about have the same values they would have if everyone were rational." In particular, inferences about welfare which some people actually take seriously, are derived from models including the rational expectations hypothesis.

Then somehow, when it is tested it changes to a quite different hypothesis. "aggregate variables which we can measure have the same values they would have if everyone were rational." So, in finance, it becomes a statement about asset prices. However, for the use of the assumption in contributions to the policy debate to be defensible, one would need a model in which welfare is what it would be if everyone were rational. I am fairly confident that you can't do this unless you assume people are risk neutral, that is make an assumption which is overwhelmingly rejected by the data.

If aggregates act as if people are rational and they make irrational mistakes which cancel out then they will bear more risk than they would if they were rational. Welfare will be lower. The amount of irrational risk bearing can be influenced by policy. The optimal policy is laissez faire *if* people are rational. If they are not rational yet aggregates behave as if they were rational, then the optimal policy will not be laissez faire.

Before moving on to discuss Cowen's effort to dodge data by redefining terms,
I note that it does *not* become a statement about asset prices and trading volume for the simple reason that no one can write down a model with the rational expectations hypothesis which isn't overwhelmingly rejected by the data. I believe that there are simply no models of trading volume including the rational expectations assumption in the literature. For more than 20 years agents who trade with no motive described in the model have been present in (as far as I know) all models of market microstructure. Now it may be hinted that there might be some rational reason for noise traders to trade as they do. However, no one (as far as I know and I am ignorant) has presented such a model, because the volume of trading that could be rationalized is a tiny tiny tiny fraction of observed trading volume.

My original rant is below. 5 minutes after posting, I stand by it, but I think the objection above is actually of some potential interest, while the shrillness below is, of course, something that has been said and written many many times.



Cowen has chosen to redefine the rational expectations hypothesis. He has also defined it so that it is meaningless, unfalsifiable, and not a hypothesis.

His intellectual accomplishment can be reproduced in other fields. If I redefine "The Ptolomaic model" to mean "The hypothesis that the earth orbits the sun" then I can save it from its recent difficulties.

I think he could have made his point more clearly if he decided to redefine "the rational expectations hypothesis" to be the hypothesis that 2+2=4. Oh and while he's at it he could define "the law of large numbers" to mean large numbers are larger than small numbers."

His use of the phrase "law of large numbers" shows that he is absolutely unwilling to consider the actual statement of any actual theorem. Oh and that he doesn't know the difference between mathematics and science.

I think a refutation of Cowen's argument which is just as valid as his argument is "I am Tyler Cowen and I retract abjure and reject my argument". Technically, I am not Tyeler Cowen, but if he can redefine the rational expectations hypothesis as he pleases then why can't I redefine "Tyler Cowen" to mean "Robert Waldmann."