## General Equilibrium Theory by Popular Demand

Robert Waldmann

I’m not kidding. Someone in some thread said that he or she thought it would be great if I could give a simple intuitive explanation of Geanakoplos and Polemarchakis (1985). Also I would get an interesting perspective on the crisis if I could fly to the moon.

I will try after the jump. The G&P result is that, if markets are incomplete, unless there is an amazing coincidence, there exists a regulation of financial markets which makes everyone better off than laissez faire.

This does not mean we should decide to regulate. The fact that such a regulation exists, doesn’t mean that it will be implemented if we just convince people that laissez faire is not optimal. It doesn’t even mean that economists can figure out a Pareto improving regulation and the problem is that those rotten politicians won’t implement it. Thus the result has only one practical application. If someone says that econmic theory or common sense tell us that free interactions of rational people must be in the interests of those rational people — that person has just said that he doesn’t know what he is talking about. This is not a judgment call. There is a mathematical proof.

Before the jump, I will just notice that there are many reasons why free market outcomes are presumably not Pareto efficient. The list of sufficient conditions in the most general description of economies with Pareto efficient market outcomes is very long, and for each sufficient condition for optimality it is easy to construct examples where, if the condition doesn’t hold, regulation can be good for everyone.

The requirements are rational expectations, well defined property rights, price taking behavior (which is really stronger than saying no agent has market power), no nonpecuniary externalities which implies, among other things, that people have no sense of pity and don’t mind knowing that others are starving, symmetric information and, finally, complete markets.

After the jump I will consider only the implications of incomplete markets as the assumption of complete markets — that for every distinguishable state of the world there is an asset which pays a positive return in that state and only in that state — is so absurdly false that no one claims it is a good approximation.

OK so the problem is to explain how rational choice by price taking agents with well defined property rights and symmetric information whose happiness depends only on their consumption leisure and whose actions don’t cause externalities except via prices (from now on RCBPTETC)can make all these agents worse off than they would be if the rational choices were constrained by a regulator who knows know more than they do and who can’t introduce new assets (like social insurance) which weren’t available to the private agents (from now on a RWKNETC).

The problem is that the proof really is based on an argument like “we have N inequalities and more than N variables so, generically, we can find variables such that all N inequalities are satisfied.” This is not intuitive. I was asked for examples, but I won’t get to them in this post. I will try to give examples in the next post.

I will define my terms (search for EOD to skip this part)

1. Price Taking: Agents decide what to do given prices, including prices of financial assets and their exactly accurate forecasts of future prices as a function of the state of nature. They assume that they can buy or sell any amount at current prices and ignore any effect that their buying and selling might have on prices. The bolded passage is necessary for Pareto efficiency in a one period model without uncertainty (a model without financial markets). It is also key to proving that equilibrium with financial markets is almost certainly inefficient. Oh note this is describing free markets. If goods are rationed by the regulator, agents know that they are.

2. Symmetric information: agents may not know much but they have the same information. This, along with rationality means that they have the same accurate belief about the probability that something will happen. With asymmetric information, insurance markets which are good for everyone can fail to exist due to the adverse selection death spiral. Forcing all people to participate in such markets can be good for everyone.

3. Well defined property rights: IIRC An assumption in all proofs that the market outcome is inefficient is that agents have an endowment and can choose to consume exactly that endowment neither buying nor selling. This is related to externalities. If anyone is free to pollute the air, then the good “clean air” doesn’t belong to anyone in particular. If fish in the sea belong to no one in particular, there will be overfishing — that is all people including fisherman might be helped by restrictions on catches.

4. Externalities include envy, shame at one’s unearned good fortune and pity. If people mind knowing that other people are starving, a world without starvation is a public good. I help you if I give food to the starving. Even if I care as much about you as I do about the starving person, if I care more about myself, then we can all be made happier by taxing each other to end starvation (and the formerly starving would be much happier). The sort of externality that is assumed not to exist in proofs of the Pareto efficiency of the market outcome is any interest in anything except my own consumption and leisure. Note not that people aren’t selfless but that we are completely totally absolutely selfish.

Becker showed that the results can be generalized if people are divided into non overlapping sets (called “families”) where people only care about other people in their own set and someone in each set cares enough that if you take from someone else in the set and give to them, they will choose to give it back. This is still very strong. He assumed that families don’t overlap (wonder if Becker explained that to his in-laws).

5. The regulator doesn’t know anything that the private agents don’t know. This seems clear. If the regulator knows better, then she can force people to do things which are in their interests but they don’t know it. This result will interest no one who observes the actual senate actually legislating.

6. This corresponds to the “constrained” in Geanokoplos and Polemarchakis title. The point is that, if there are incomplete markets, except for an amazing coincidence, making the markets complete *and* lump sum taxes and transfers can make everyone better off. It makes the challenge of finding a Pareto improvement harder in an important way as one way in which public intervention is widely believed to have made us better off is by introducing new kinds of insurance such as unemployment insurance. That’s too easy for G and P.

7. Pareto efficient: Look this is a very weak claim. It doesn’t mean efficient in any normal sense of the word. A Pareto improvement (making everyone better off) is interesting. The result that no Pareto improvement is possible (Pareto efficiency) isn’t interesting at all.

8: except for amazing coincidences. This is an effort at an Egnlish translation for “generically” which means “for an open and dense set of economies.” The claim is be ” if for some set of tastes technology and endowments the market outcome(s) isn’t (aren’t) constrained Pareto efficient, then there is a change in endowments so tiny that, after the change in endowments, the new economy has a market outcome which isn’t constrained Pareto efficient (or many outcomes which aren’t).” That means the set of economies with constrained Pareto inefficient outcomes is open. Also if there is an economy with a constrained Pareto efficient outcome, there is a tiny change (as tiny as you want) in endowments so that with the new endowments the outcome of the new economy is constrained Pareto inefficient (that’s the dense part).

EOD

OK the idea of the G&P result. First the model is a general equilibrium market with financial assets. The idea is that there are 2 periods, period 0 and period 1. IN period 0 agents trade financial assets which are all in zero net supply. The state the world will be in in period 1 is not known but everyone knows the probability of any possible state. Assets have payouts which depend on the state of the world. After the uncertainty is resolved (that is in period 1) agents buy and sell goods on ordinary spot markets, then they consume, then the universe ends. General equilibrium theorists claim that theis simple structure isn’t really as restrictive as it seems.

The point is that trades in the financial assets in period 0 will generally affect the prices in period 1 except in the case of amazing coincidences. One such amazing coincidence if if everyone has identical homothetic preferences so aggregate excess demand is a function of the aggregate endowment and relative prices (so demand can be represented as demand by a representative consumer). In this case only transfering wealth from one person to another (what financial assets do) can have no effect on spot prices in period 1.

So generically decisions made in period 0 will affect spot market prices in period 1. This is a pecuniary externality.

Now if we just look at one state of the world in period 1, there will be no way to make everyone better off by messing around with trade on the spot market in period 1. By then the economy has become a one period Walrasian economy with a Pareto efficient market outcome.

If the regulator messes with trade in financial assets in period 0 then some people iwll be richer and some poorer in some realizations (say one called s) of the state of the world in period 1. If these people have different tastes or if the distribution of wealth affects demand (as in some goods are luxuries) then this will affect spot prices in those states. The change in spot prices will help people who are selling goods whose relative price goes up and hurt people whose are buying those goods. That effect, the change in spot prices due to changes in trade in financial assets in period 1, will move the state s outcome from one Pareto efficient outcome to another. It will help some people and hurt others.

As always, it might increase the happiness of the people it helps by a tiny number of utils and hurt the people it hurts by a huge number of utils. That is Pareto efficient is a very weak result. To be more exact, choose one good and call it the numeraire. the market outcome in state s will maximize the weighted sum of utility of all the agents with weights equal to the inverse of that agents marginal utility of consumption of the numeraire good in state s.

This means in each state a weighted sum of utility will be maximized but the weights will, in general, be different for different states (with low weight in state s on the utility of an agent who is poor in state s)

The key point is that we know that the indirect effects on utility of messing with financial markets through spot prices will be of different signs for different agents in each state, but we don’t know anything about the effect on expected utility. Each agent knows that she will be helped some times and hurt some times, but knows nothing about the probability weighted average effect on utility.

OK so to get to the result that messing with what people buy and sell on the financial market (making them hold different amounts of financial assets than they want to) can make them all better off it is necessary to have 2 things.

1. The relative weights on different agents are different in different states. If there are complete markets, then the relative weights will always be the same (the weights are constants all multiplied by the same state specific factor for each agent). So the market outcome is Pareto efficient.

2. There have to be enough assets that the arbitrary vectors of changes in welfare do to messing around with portfolios are numerous enough that a linear combination of such changes adds up to a vector with all positive elements — that is a Pareto improvement.

The theorem requires an assumption about the number of assets compared to the number of different types of agents. Dimensions of messing with portfolios will be something like number of assets times number of agents minus 1 (because the assets are in zero net supply). There are as many inequalities to satisfy to get a Pareto improvement as there are agents.

Except for amazing coincidences, you can satisfy N inequalities if you have N unknowns so basically it is needed that the numbers of dimensions of meddling with portfolios is greater than the number of different types of agents.