# The REH vs the EMH

Robert Waldmann

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. It is, therefore, not a statement about prices only but about behavior generally. As used it definitely amounts to much more the assumption that observable aggregates have the values they would have if everyone had rational expectations. It is common for microeconometric models to be estimated the assumption of rational expectations. Clearly a statement only about aggregates does not have implications for micro data. This use of the phrase “rational expectations” to refer to individual behavior not aggregates is common and, as far as I know, uncontroversial.

The assumption of ratinal expectations also has important theoretical implications. For example, the 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 it is safe to say that the fist welfare theorem has a well established place in economic thought. The assumption that it is a matter of no relevance is not easily reconciled with the history of the profession.

In the post below, I assumed that this point is plainly obvious. Now I think an extremely elementary proof might be useful. The proof after the jump.

update: totally wrong math corrected.

there are 2 time periods t = 1 and t = 2.
In the model there are 2 assets. 1 is a risk free asset which is the numeraire. one unit of risk free asset gives one unit of consumption good in period 2.

There is also a coin which is flipped. It comes up heads in period 2 with probability 0.5.

The economy is populated by a continuum of agents indexed by i which goes from 0 to one, who maximize the sum of the log of their consumption in period 2. They have identical preferences and endowements. Each owns one unit of the risk free asset.

It is possible for them to bet on the coin. for a price p one can get an asset which pays 1 unit of consumption good if the coin comes up heads.

Rational expectations implies that agents know that the probability the coin comes up heads is 0.5.

If everyone has rational expectations, then the market will clear with p = 0.5 for each t. 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 beliefs about the probability that the coin will come up heads is i 1 if i>0.5 and 0 if i
The market clearing price is 0.5. at p = 0.5 half of the agents will buy 2 units each of the risky asset from each of the other half of the agents.

The EMH still holds. p = 0.5.

The outcome is somewhat different. In period 2 half of the agents consume 2 and half consume 0. The outcome is no longer Pareto efficient. Each agent has expected welfare equal to negative infinity.

now correct analysis of my original model.

Now assume that Assume that agent i believes that the probability that the coin will come up heads is i. The outcome is somewhere in between. The market clearing price is still 0.5 so the efficient markets hypothesis still holds. However, agent i will have consumption 2-2i if the coin comes up tails and 2i if it comes up heads. Since the agents, except for agent i = 0.5, are making mistakes, their true objective actual expected welfare is lower than it would be if they were rational. All but i=0.5 think that they think they are doing better than just playing safe but they are all doing worse. mr or ms 0.5 plays safe, invests all in the safe asset.

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. Alternatively the state could impose a 100% tax in period 2 and distribute the receipts equally.

I think it is safe to say that there is a difference of interest to economists between a model in which the optimal policy is laissez faire and a model in which the optimal policy is confiscation and equal distribution of all wealth.

update 2: snark deleted.