Saturday, September 5, 2009

Completeness, expectations, and efficiency (amv)

I remember one of the many enlightening conversations I have enjoyed with fg. Some weeks ago, we again confronted our respective fields of research, our ways of thinking about economic issues. His major fields of research, if I get them right, are finance and monetary macroeconomics, and he is especially interested in the link between the two. I am concerned with intertemporal general equilibrium theory, with decentralized growth, theories of investment and capital structure, and market process theory (issues related to the link between intertemporal and dynamic equilibria and global stability). In our discussion, a major topic came up: the significance of ‘completeness’ for real-world market performance and market efficiency.

‘Completeness’ relates to the completeness of markets. If the economy is complete, there is a set of not necessarily open (futures) markets with elements for each kind of dated good and for all possible events. In an Arrow-Debreu economy, we choose an initial point in time before consumption and production can take place, an allow all agents to express for a given arbitrary price vector their respective excess demands (positive or negative) on markets for all these goods (zero excess demand is feasible; most are zero in disequilibrium; all are zero in equilibrium). Thus, there is an excess demand correspondence for each agent, the domain being the price vector for all goods at each and every point in time and for all events or states of the world imaginable (uncertainty of Mother Nature's choice; all feasible states imaginable; price vector is public knowledge). Now imagine time to be infinite. Think of the set of possible states or events to become ever finer over time (from the perspective of the initial period). The number of feasible states or events may again be infinite. It is thus clear that ‘completeness’ asks for an infinite set of markets, each for every dated event (which is logically only possible at zero transaction cost, since otherwise resources would be exhausted).

Such an assumption is what economists call ‘highly restrictive’. If an intertemporal equilibrium is possible iff the equilibrium price vector is formulated in the light of a (countably) infinite number of markets, there is little hope that intertemporal general equilibrium analysis relates to anything in the real-world. Completeness is therefore a problem. Fg and I agree on this.

Let us turn to market efficiency, which is no more than just an equilibrium definition (full-information general equilibrium imposed - yet not explained - by a no-arbitrage condition). But the concept of market efficiency is also used to relate general equilibrium theory to data and the real world. Chicago's dictum to argue 'as if' the real economy is at least close to its (temporary) equilibrium is alive and kicking in the market efficiency hypothesis: as an empirical hypothesis it says that at any point in real time asset prices are close approximations of their full-information values (full-information relaxed for weaker types of efficiency).

The significance of completeness for market efficiency is obvious: If a general equilibrium analytically rests on an unbounded set of markets, how can we hope for real-world asset prices to reflect their fair values? Full information is simply too much to be handled given only countable number of goods, markets, and prices. However, to assume that the economy is approached by general equilibrium analysis (remember Chicago’s ‘as if’) is to assume that somehow there is some substitute for the completeness of markets, an assumption that makes the use of reduced-form models more plausible. This question we discussed.

Whereas I stressed that there is no such substitute, fg’s position was that there may be one. To understand his argument, you have to know that it is possible to replicate the optimal allocation of the Arrow-Debreu economy, which is a static model by its very nature, in a dynamic setting by formulating a sequence economy (we call such an inessential sequence an Arrow-equilibrium). To do so, we have to introduce what came to be known as Arrow securities (you may guess why Arrow earned a Nobel prize). Arrow securities are one-period contingent claims (each a claim to payoffs attributed to dated events). Again, Arrow securities have to be complete, that is, there must be one for each event of the succeeding period. Nevertheless, the total number of markets that open over any time horizon is very much reduced compared to an equivalent horizon in the Arrow-Debreu economy.

This is so, because Arrow securities (the capital market) are cost-reducing technologies, concentrating all intertemporal transfers on two single markets (for future claims produced in the present and the future claims of the immediate past maturing at present.) Thus, instead of a variety of intertemporal commodity markets, we have fewer security markets. The only other markets that have to open over time are spot goods markets and maturing Arrow securities can be turned into present consumption or saving (purchase of new Arrow securities).

Fg’s argument rest on the following insight: Of course, Arrow securities are analytical constructs. For each future good, you have to buy a complete bundle of Arrow securities, each for an event imaginable for that good in the succeeding period (since prices are real-valued you can imagine how many events there may be). And here is fg’s major point: You don’t need a complete set of Arrow securities. All you need are two kinds of securities: stocks and bonds (and possible derivates therefrom). Together, they allow resembling all allocations possible with a complete set of Arrow securities.

And fg is right! I did not get his point at the time of our discussion but I agree today. However, my point may be true nevertheless. For my critique of using reduced-form models, local stability analysis, etc, rests on a problem not solved by the Arrow equilibrium, but which is rather accelerated by it. For: the reduction of necessary open markets by introducing Arrow securities comes at a cost. Outside an initial period with complete and open multi-lateral markets, and with some kind of super-auctioneer, there seems to be no way to construct the equivalent Arrow allocation other than by imposing on each agent complete foresight of equilibrium prices for all dated events still to come (rational expectations). The concept of rational expectations is more than just an empirical description of human beings, that they are forward-looking, avoiding systematic errors by learning, etc. ‘Rational expectation’ is an equilibrium definition. Since there is little hope for the stochastic cousin of perfect foresight to be a matter of the real world, there is also little reason to assume that the economy is close to its equilibrium, to assume market efficiency (since market efficiency means full-information and the relevant information set contains future equilibrium values).

Thus, even though the burden of ‘completeness’ can be reasonably reduced to acceptable levels, this is accompanied by an increasing burden in form of the informational requirements we have to impose on economic agents way beyond acceptable levels (see again Arrow). Like the Arrow-Debreu economy, the Arrow-sequence economy collapses the future into the present. In the case of the former, this is done by a complete set of futures markets in the intitial period. In the case of the latter, this is done by imposing on agents the complete knowledge of the future yet to come.