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Monday, May 02, 2011

What did Say say ?

Jean Baptiste Say once wrote that there can't be a general glut -- that people sell things in order to buy other things so there can't be an outcome where everyone wants to sell more and no one wants to buy more.

As Brad DeLong notes from time to time, Say got over this error by 1826 noting that there are cases in which everyone wants more cash. There can't be excess supply of everything, but it can be that there is excess supply of everything except for pieces of paper bearing the pictures of some King of England (or whatever the hell they had on Franc notes in 1826, but the example was the bursting of the canal bubble in the UK).

Then Brad embraces a modified Says law which says there can only be a recession and more than frictional unemployment if there is excess demand for something sort of like money which might be cash, or M1 (cash in circulation plus demand deposits plus balances of checking accounts) or high quality bonds or in any case something which can't be produced freely by the private sector. The problem is that his case that economic theory has not advanced since 1826 is that it was much better in 1828 than in 1824 and much better in the UK and France in 1826 than in some US states in 2011.

It is well known that there can be undesirable fluctuations in the volume of trade in economies without any financial assets at all so no money, no bonds, no stock. I don't claim these models have anything to do with the recent recession. I am talking about math not social science. I just claim that there is no metaphysical necessity for there to be money (or anything like it) to have a recession

Before the jump, I will just list the many models with recessions without money.

1) The Diamond search model. In applications it is always assumed that the search function has constant returns to scale. In the original paper it had increasing returns to scale and there could be no trade even though there were gains from trade. There was no money in the model.

2. The Dreze expectational equilibrium model. The standard Walrasian assumption is that people can buy or sell as much as they want at current prices. There can be no rational expectations equilibrium in which this expectation is irrational. This is, I think, a very fair translation of Say's hypothesis into modern language. It is also obviously a tautology. One can imagine a model in which the non-Walrasian auctioneer calls out prices and rations (limits on how much good j agent i can sell or limits on how much good k agent l can buy). Then agents maximize within their feasible sets (which aren't budget sets). Non-Walrasion Nash Equilibria of this game occurs when the amount of each good offered for sale is equal to the amount demanded for purchase and all achieve maximum feasible utility. Walrasian equilibria are Dreze equilibria, but there are many others which are often Pareto worse than any Walrasian equilibrium.

3. Imperfect competition with strategic complements. This can give multiple Pareto ranked equilibria.

4. incomplete markets (OK here there are financial assets but all asset markets clear). There can be a continuum of equilibria some of which are Pareto worse than others.

5. A Romer 86 model with endogenous labor supply can have a continuum of equilibria all but one of which are Pareto inefficient (sorry had to or my co-author would be angry).

All of these add something new which was not known to Say or Mill in 1830.

After the jump I try to explain the first two

A Dreze example. Consider a model with two goods apples and oranges and agents whose happiness is the product of the fourth root of the number of apples they eat and the fourth root of the number oforanges they eat U = (apples*oranges)^0.25. There is no production. Agents of type A start out with 40 apples. Agents of type O start out with 40 oranges. There are equal (and very large) of agents of each type.

This is, I think, the simplest model of tastes and technology.

Now there is a special agent called the non-Walrasian auctioneer. She calls out a price P apples for an orange (so oranges are the numeraire good -- pardon my French).
But she also may tell each and every agent that they are rationed. So She might say
"agent i can buy no more than 3 apples" or "agent j can sell no more than two oranges." Oh my how multidimensional. The strategy space of the non Walrasian auctioneer isn't just relative prices (in this case one relative price). It has number of goods times number of agents more variables. This means that the problem is N-dimensional and N is a very large number. This huge number of dimensions makes the mathematics no more difficult. Theorists work in N dimensions and it matters if N is finite but a quintillion is just as simple for them as 2.

As mentioned above, agents maximize their utility choosing a consumption bundle in their feasible set which is not a budget set (it is the intersection of their budget set and the arbitrary rationing sets imposed on them by the non-Walrasian auctioneer).

The non-Walrasian auctioneers sole aim is to make sure that, given these constrained choices, total apples demanded = total apples and total oranges demanded = total oranges so non-Walrasian demand (also called effective demand) equals non-Walrasian supply.

An example might make this comprehensible (and pigs might fly)

One possible Nash equilibrium of the game is the Walrasian equilibrium -- the price is one apple for one orange, there is no rationing and everyone ends up consuming 20 apples and 20 oranges. They each have U = 400^0.25

There are an infinite number of equilibria which are trivially different from the Walrasian equilibrium -- the non-Walrasian auctioneer might tell people endowed with oranges that they can't buy oranges (so what they don't want to) or that they can't sell more than 21 oranges (so what they want to sell 20). Here trade, consumption and welfare are Walrasian and, without listening in on the non-Walrasian auctioneer we can't tell that the world isn't Walrasian.

But there are many other Nash equlibria. For example, the price is two apples for an orange and orange sellers are restricted to sell no more than 10 oranges each (or equivalently to buy no more than 20 apples each). The result is that apple sellers freely choose to sell 20 apples each and end up with 20 apples and 10 oranges -- the price and marginal utility of an additional orange are twice the price and marginal utility of an additional apple, so they have optimized subject to their budget constraint (and no other constraint). The orange sellers each buy 20 apples for 10 oranges. They want to buy more but they can't. There is an equilibrium with less trade than the Walrasian equilibrium.

Or the non-Walrasian auctioneer can be nasty and tell half of the orange owners that they can't buy or sell at all and leave the others free. The unconstrained orange sellers will sell 20 oranges for 40 apples and end up with 40 apples and 20 oranges. Demand equals supply.

Let's say the non-Walrasian auctioneer chooses which orange owners to ban from the market at random. The expected welfare of an orange seller is
0.5*0+0.5*(800^0.25) = 0.5*(2^0.25)(400^0.25) < 400^0.25.
The expected welfare of an apple seller is (200^0.25 < 400^0.25

There is a Nash equilibrium with lower trade than the Walrasian in which everyone has lower expected utility that in the Walrasian equilibrium.

This is a one period model with no financial assets whatsoever.

There isn't a general glut, because apple sellers don't want to sell more apples. But there is a glut of oranges and no shortage of apples (apple sellers don't want to sell fewer apples either).

The paradox notes by Say is that if we assume that people choose what to supply and demand assuming that they can buy and sell as much as they want at market prices, and they have rational expectations, then they can buy and sell as much as they want at market prices. It is, in fact, a tautology. There is nothing there. There is no need to introduce money or bonds or anything else to demonstrate that Say was wrong when he wrote in 1803 that it must be so, that no other possibility is conceivable.

The Diamond Search Model is very well known (see the latest Nobel Memorial Prize citation) so I don't think this is really needed, but it is also very simple.

there are two types of agents A and B. Agents of type A are endowed with one unit of good a and want to consume good b. So far the simplest model of a market possible. But agents must find each other in order to barter (see no money). First agents decide whether to search or not. Search is tiring. The probability that a given agent of type A finds an agent of type B and trades is the lesser of one and a constant C times the number of agents of type B who are searching. Also vice versa. Utility is 0 minus a small cost epsilon which must be paid if one searches plus one if one finds a match and trades.

The model as described has a Nash equilibrium in which no one searches. If one player searches the chance of finding a trading partner is zero, so the highest possible utility is 0 obtained by not searching. For some C and epsilon there is also an equilibrium in which everyone searches and has expected utility greater than 0.

There are no financial assets in the model. It is not an obscure model.

The key assumption which has been changed in all current search models is that the matching function [ matches = C*(number of type A searching)*(number of type B searching)] has increasing returns to scale. If the matching function has constant returns to scale, then there is a unique equilibrium. But neither Say nor Mill nor DeLong nor (as far as I know) Rowe ever wrote anything about returns to scale of the matching function.

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