General Equilibrium theory and Finance

Robert Waldmann

I wrote a post below on how economic theory as presented to the public and economic theory are very different. In particular, even given the standard super strong assumptions, economic theory does *not* imply that Laissez faire is the best policy, does not imply that market outcomes are even Pareto efficient (a very weak statement anyway) does not imply that financial innovation can’t hurt everyone and certainly does not imply that financial markets can’t freeze up.

In this post I will write about general equilibrium theory [more on that after the jump] and assume symmetric information.

Furthermore I will talk about Pareto efficiency. [more after the jump].

I will assume that markets are incomplete. This is obvious. For markets to be complete, there must be a security (or portfolio) which pays positive value in any conceivable contingency and otherwise pays zero. It must be possible to buy in effect “an umbrella if it is raining” separately from “an umbrella if it is not raining”. No one has ever suggested that markets are approximately complete. To have any connection with reality, general equilibrium theory must concern economies with incomplete markets.

All that aside the conclusions of the literature is that

1) Generically (definition of generically after the jump) the market outcome is not constrained Pareto efficient. This means that the government can make everyone better off by restricting free market exchange on financial markets forbidding people to make mutually beneficial trades at a market clearing price.

2) If markets were complete the outcome would be Pareto efficient. However the introduction of a new security does not necessarily cause a Pareto improvement. In fact, for an open set of economies (definition after the jump) banning trading in a security causes a Pareto improvement. Generically, if markets are incomplete, there is a security whose introduction will make everyone worse off.

3) with incomplete markets sunspots (variables which have no effects on tastes or technology) can cause prices to change. Outcomes can occur with a sunspot that couldn’t possible occur without a sunspot. I don’t remember if this is generically true of just true for an open set of economies.

4) it is easy to write down a model in which the volume of trade is zero. In standard models agents who do not maximize welfare are introduced to cause trading volume to be greater than zero. It is possible to write a model in which there is some trading even if everyone is rational (for example due to life cycle saving and dis-saving or people hedging against risk in their labor income). No one writes such models as they give trading volume vastly lower than, say, current volume with markets that are “frozen” and have “seized up.”

Roughly none of the claims about general equilibrium with complete markets apply to general equilibrium with incomplete markets. The exceptions are not strange cases, rather the claims are true only if there are extraordinary coincidences.

General Equilibrium theory is extreme even as economic theory goes

I will therefore assume that agents have rational expectations — that is they know the probability of any conceivable event — and that they are price takers, that is there is perfect competition. I will assume that there are no non-pecuniary externalities like pollution (the only externality is that my demand affects the price that you have to pay too). I will also assume symmetric information although there is a large literature (founded by Robert Lucas) on general equilibrium with asymmetric information). Thus the assumptions are the very strong ones typically criticized by people who say that the support for laissez faire (libertarianism in English) from economic theory is irrelevant the real world.

Is it Pareto efficient is not an interesting question (as stressed I might add by Kenneth Arrow who proved the first welfare theorem and said it was no big deal).

This has little to do with efficiency in the normal sense of the word. An outcome is Pareto efficient unless it is possible to make someone better off without hurting anyone. Slavery was Pareto efficient if it helped the slave owners. The statement that an outcome is Pareto efficient tells us essentially nothing. In particular if we have two outcomes and one is Pareto efficient and the other isn’t that tells us nothing of interest. It certainly doesn’t at all mean that the Pareto efficient outcome is a Pareto improvement over the other outcome.

uh oh. I have to define what general equilibrium theorists mean when they say “an economy”

1) an economy includes a set of agents each of which has a utility function. The agents rationally maximize their expected utility.
2) Each agent has an initial endowment of goods and maybe abilities
3) There is a production possibilities set describing how goods and abilities can be used to produce goods.

In particular all the results mentioned above are for “fruit tree” economies in which no one works. Instead the production is of the form of fruit trees which produce goods by themselves. Typically the models have 2 periods (I think this assumption isn’t really restrictive). In period zero people start with endowments of fruit trees. They trade them. One can short sell a fruit tree.
In period 1 the trees produce the fruit. People exchange fruit. Then they eat it.

Clearly this is a very special case of general equilibrium and the fact that the standard results for a 1 period model (no fruit trees) don’t hold is especially striking.

Open: Actually this is an abstract concept. Given a set we can define a “topology” which is the set of open subsets of the set. The only rules are
1. the whole set is open
2. the empty set is open
3. The intersection of a finite number of open sets is open
4. The union of any set of open sets is open.

in general equilibrium theory “open” has a standard relatively narrow defintion. It always (almost always ?) means that if we take an economy in the set, there is an epsilon so low that if we change the endowments of goods initially owned by agents so that the largest over i and j change in the endowment of good j owned by agent i is less than epsilon the new economy with new endowments is in the set of economies.

So something is true of an open set of economies means I can take a little from this guy and give to that gal and it will still be true.

dense: a subset of a set is dense if for any point in the set, for any open set containing that point, there is an element of the subset in that set. For example rational numbers are a dense subset of real numbers.

Generically: Roughly something is true generically if it is true except in knife edge cases of extraordinary coincidences. Consider the set of pairs of real numbers. Generically the two numbers are not equal, only by an amazing coincidence would they happen to be equal.

Formally a statement is true generically if it is true for an open and dense subset of economies. In practice what is shown is if we have an economy for which it is true we can make little transfers of endowements so small that it is still true after the transfers. For any case in which it is not true for any epsilon, there are transfers where the biggest amount of any good given to or taken from anyone is less than epsilon such that after the transfers the statement becomes true.

Another way of saying it is that if we have a set and an open and dense subset, the complement of the subset (the points in the set which aren’t in the subset) has measure zero for any measure based on the topology of the set.