Rational Expectations, Efficient Markets, and Economic Welfare
6/17/09 9:07 PM
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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 http://delong.typepad.com/sdj/2009/06/rational-expectations-efficient-markets-and-economic-welfare.html
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Rational Expectations, Efficient Markets, and Economic Welfare
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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 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 - 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).
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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... rated 4.56 by you and 8 others [? ] You loved this post (
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Comments it would be preferable to ban gambling .... unless gambling was an inherent part of the process of allocating capital, experimenting with production, raising productivity etc.
Posted by: cheese | June 09, 2009 at 11:02 AM yes indeed the alleged math in my post was wrong (since corrected I think). Also I have deleted the words Andrei and Shleifer. Bit too late. I'd rather you delete them too as, well, its not totally nice to denounce something he wrote casually long ago (more than 10 years ago). Cheese: The statement about banning gambling was a statement about an economic model. The point of my post was to clarify a distinction in economic theory. I did not make any hint of any statement about the real world in my post. Posted by: robert waldmann | June 09, 2009 at 11:17 AM The problem with all of utility based models is that they assume people have the same risk tolerance which they don't. You can buy and sell risk. The other thing that you can do is to put all of the risky assets in a pool, and then have people buy shared of the pool rather than the assets themselves. Posted by: Twofish | June 09, 2009 at 01:03 PM The REH does not hold in an arbitrage-free asset pricing world for numeraire denominated assets, but does "hold" in such a world using the equivalent martingale measure in lieu of the traditional REH's econometric probability measure. Posted by: d4winds | June 09, 2009 at 01:57 PM > The problem with all of utility based models is > that they assume people have the same risk tolerance > which they don't. http://delong.typepad.com/sdj/2009/06/rational-expectations-efficient-markets-and-economic-welfare.html
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I would change that to "assumes they have the same philosophical concept of risk, which they don't". > You can buy and sell risk. That worked real well from 1998-2008. Posted by: Not Really | June 09, 2009 at 02:54 PM Wouldn't the agents want to update their beliefs about the probability measure as they start trading? Posted by: vkn | June 09, 2009 at 04:23 PM I would like to comment on this post as a recently retired investment management professional (trained I might add at the Haas School by folks like Hayne Leland and Barr Rosenberg, oh so many years ago.) I think as a practical matter the EMH should not be confused with the REH and in particular it is important to distinguish among its forms. The weak form, which effectively states that no mechanical trading rule using public information can create return in excess of the market return, is almost certainly a very good first approximation of capital market reality. What that is really saying, I think, is that with respect to markets that can be characterized as having mean outcomes with a positive variance, the current price is an unbiased estimate of the mean of the distribution, or if you will is a fair price for the probability weighted outcomes of different states of nature in an Arrow-Debreau economy. That most certainly does not mean that the price is correct in some epistemological sense. An example I use outside of capital markets is the establishment of odds by bookmakers. In 1987, a year of interest to financial economists is also of interest to baseball fans because of the unlikely outcome of the Minnesota Twins winning the world series. I can assure you that the odds makers in Vegas did not have the Twins as the favorites at any point in the season. Yet, I fail to see how this violates the EMH. The odds established were not systematically biased one way or the other, we took a draw from the distribution and we got the Twins. Only special information could have made you rich by betting on the Twins, (or of course just being a hopeful Twins fan taking a flyer.) With respect to the semi-strong form that has been too often violated in the markets over the course of a thirty year career for me to take it seriously. Most famous example, the tremendous discount from fair value that the original futures contract on the S&P 500 index traded at for its first year of existence. I and many other investment professionals made a lot of money for clients on that mispricing and after all, that is a simple cost of carry calculation that any competent MBA should be able to make. But institutional frictions exist that lead to mysterious outcomes such as that. The public policy problem I think is the misinterpretation of the EMH to meaning that markets are always "correct", whatever that means. But while this opens up a whole other line of argument, on the side of good, the EMH has steered a lot of inverstors into passive investment strategies (index funds) and that has helped numerous "regular people" get much better outcomes. Posted by: RPL | June 09, 2009 at 08:31 PM Dear Brad Thanks for the edits. vkn: Yes indeed, the agents in the model are really really really dumb much dumber than real world people. The point (if any) of the model is to show that, if we knew that the EMH is true, we wouldn't know much, because it is consistent with people being very very far from rational in a way that has extreme policy implications. RPL: I think you are making an important point -- the fact that it is very difficult to beat the market does not mean that the market price is close to the price there would be if everyone were rational. The point (noted by Shiller and separately Summers) is that "beat the market" is a statement about returns including especially the change in prices and an apparently fairly strong statement about changes in prices has only very weak implications for the level of prices. Basically the way present value works there is a huge difference between "a portfolio made of the risk free asset and the market portfolio has minimum risk for a given return" (EMH) which has strong implications for the level of prices and "a portfolio made of the risk free asset and the market portfolio has close to the minimum risk for a given return" which does not imply that asset prices are close to EMH asset prices.
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Posted by: robert waldmann | June 09, 2009 at 09:59 PM
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