 Market Selections
 It is conventional to assume that traders in asset markets have rational expectations about asset returns, and choose savings rates and portfolios as if they maximize expected utility using these beliefs. As the hypothesis that traders are expected utility maximizers places few restrictions on behavior in the absence of the rational expectations hypothesis, much attention has been focused on the validity of assuming rationality (correctness) of expectations. Although traders
would certainly prefer to hold accurate rather than inaccurate beliefs, an explanation of how traders come to correctly forecast endogenous equilibrium rates of return is lacking.
Various approaches have been devised to provide the missing foundation for the rational expectations hypothesis. One approach posits that correct beliefs can be learned. In other words, rational expectations are stable steady states of learning dynamics; see [2] and [5] for more discussion. The learning approach is not completely satisfactory. Achieving rational expectations through learning requires too much prior knowledge, to the point where the requirements for learning essentially assume the conclusion. An alternative approach based on evolutionary forces operating through wealth dynamics has arisen in recent years. This approach actually has a long tradition. Both Alchian [1] and Friedman [7] argued in the early 1950’s that evolutionary forces would eventually result in behavior consistent with correct maximization. Although their argument is plausible, until recently there was no careful analysis of the market dynamics that would supposedly select for expected utility maximizers, and, within the class of expected utility maximizers, select for those with rational expectations.
In [3] we analyzed an economy with repeated markets for risky, one period assets. We
showed that if savings rates are equal across traders, then wealth dynamics do not necessarily
lead to traders acting as if they maximize expected utility using rational expectations. Expected
utility maximization may fail to emerge because the market selects for traders whose investment
portfolios generate higher expected growth rates of their share of wealth. Although our primary
focus was not on the link between portfolio rules and expectations, we showed that a trader with
logarithmic utility and correct expectations maximizes the expected growth rate of wealth share
and so dominates the market.
However, traders with correct expectations and non-logarithmic
utility need not maximize the expected growth rate of wealth share and so can be driven out of
the market even by traders with incorrect expectations.
Our analysis addressed the possible emergence of expected utility maximization. The
question of belief selection among expected utility maximizers was examined by Sandroni [9],
who analyzed an economy with infinitely lived risky assets. He showed that if markets are
dynamically complete, and some assumptions are made on returns, then when savings rates are
endogenous, and all traders are expected utility maximizers with a common discount factor, only
traders with rational expectations survive.
This occurs because when markets are complete,
traders can place bets on any disagreement about the probability of states and traders with
correct expectations will win the bets. So in Sandroni’s world the market selects for those
traders whose expectations are correct. This analysis differs from [3] in that savings behaviors are
endogenous, and the exogenous parameters are the primitives involved in describing preferences,
such as payoff functions and discount factors.
In this paper we explore more completely when selection occurs, when it does not occur
and why. We return to the asset structure of [3], so first we show that with our assets, market
completeness and some ancillary assumptions imply selection for rational expectations. Second,
we show that dynamic completeness of markets is necessary to guarantee selection for rational
expectations. In economies with incomplete markets, a trader who is overly optimistic about the
return on some asset in some state can choose to save enough to more than overcome the poor
asset allocation decision that his incorrect expectations create. This result is even more striking
than it seems, because when traders’ beliefs are heterogeneous, some market incompleteness
is inevitable.
For with heterogeneous beliefs, market completeness implies that traders can
bet on any differences in beliefs. This amounts to opening a new set of markets every time a
new trader with different beliefs enters the economy. In the context of the model, the relevant
state space contains the union of the supports of each player’s beliefs. Thus adding a new
trader can require expanding the state space, and therefore adding new markets. We conclude
that, within the evolutionary framework, the conditions required to ensure market selection for
rational expectations are too strong to be useful. In general there are no market impediments
to long-run heterogeneous beliefs.
 |
