## Rational Expectations, Efficient Markets, and Economic Welfare

I think Robert Waldmann is using the internet to carry on an argument among sometime roommates.

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 - p

^{2}]

The EMH still holds in equilibrium. Supply equals demand for the risky asset at its fundamental value of 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 - y^{2}/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... well known economists have done exactly that...