76 years ago, Arrow & Debreu wrote the paper which claimed to prove the existence of a competitive equilibrium for an integrated model of production, exchange and consumption
The main results of this paper are two theorems stating very general conditions under which a competitive equilibrium will exist. Loosely speaking, the first theorem asserts that if every individual has initially some positive quantity of every commodity available for sale,
in other words, everybody is an arbitrageur participating in every market
then a competitive equilibrium will exist.
under various assumptions. In particular, there will be no speculative demand. But if so, why are there arbitragers? What is their incentive? As a matter of fact, arbitragers can be infected by 'animals spirits' and drive up all asset prices before there is a crash and a bit shake out. If you assume there are no animal spirits and no speculative demand, why would there be any arbitragers?
The second theorem asserts the existence of competitive equilibrium if there are some types of labor with the following two properties: (1) each individual can supply some positive amount of at least one such type of labor;
I suppose arbitrage is a type of labor. Guys who do that for a living tend to wear Rolex watches and drive sports cars.
and (2) each such type of labor has a positive usefulness in the production of desired commodities.
I suppose 'market makers' are useful if they don't get infected by irrational optimism or pessimism. However, that isn't what Arrow-Debreu mean. They are saying marginal physical product is never zero or negative. This is unrealistic. The supply of coal miners does not immediately disappear the moment all the coal has been mined. Moreover hiring more and more cooks means that they get in each other's way and so the output of the kitchen falls.
The conditions of the second theorem, particularly, may be expected to be satisfied in a wide variety of actual situations,
Not unless labor is highly mobile between occupations and industries with positive returns can scale up quickly and costlessly. These certainly weren't the conditions faced by people over the previous four decades.
though not, for example, if there is insufficient substitutability in the structure of production
Which was certainly the case for all extant economies. Arrow-Debreu were describing the economy of Fairy-land.
Worse, they were saying something which could not be a set (because the 'intension' was epistemic and thus not well defined. This is the intensional fallacy) was actually a set.
For each production unit j, there is a set Y j of possible production plans.
If there is such a set, then all future technological and scientific advances are already known today which is why there is a set Yj known to each firm. What is more every member of the firm already knows the date of their death and the date of the birth of their great-great-grand daughter which is why they can order their grave-stone today and also arrange for a birthday card to be sent to their distant descendants.
The truth is there is no set Yj. There may be a set of production plans currently thought to be feasible by such and such person. But such plans don't matter. After all, every enterprise does have a plan. Sadly, it keeps having to be revised because sales didn't meet the target or there were unexpected supply problems or the CEO had a mental breakdown.
If there are no sets, there are no subsets and no vectors or matrices or topology of any kind. Convexity and continuity and so forth can't arise. Mathematics has nothing to work with.
Arrow/Debreu have a bizarre notion of diminishing returns (which arises because factors of production, with given technology, can't be substituted for each other. Thus, if there is no more coal to be found, hiring more coal miners won't increase coal production). They say- 'The existence of factors private to the firm is the standard justification in economic theory for diminishing returns to scale'. This is nonsense. Coal productivity in the UK fell because, with existing technology, access to good quality coal seams was decreasing. Britain would have to transfer coal miners into other occupations. The General Strike of 1926 could not magically raise the productivity of coal mines. True, rationalization- maybe even nationalization- could tap financial, management, and other economies of scope and scale. It could fund research into productivity enhancing technology. But it could not prevent, in the short to medium term for the Coal industry, diminishing returns to Labor, and hence much lower wages than had been paid during the War. What is true is that if a particular firm earns higher profit, we say there is a rent or return accruing on something non-fungible owned by the firm- e.g. the big, big, brain of the Manager.
Assumption 1.a. implies non-increasing returns to scale- in other words, what is doing the heavy lifting is good old fashioned diminishing returns. Talk of 'convexity' is misleading because there is no underlying set.
1.b. says that one cannot have an aggregate production possibility vector with a positive component unless at least one component is negative. I.e., it is impossible to have any output unless there is some input. This rules out discoveries & windfalls or inventions which have the same effect as windfalls. Yet Arrow-Debreu had already witnessed just such discoveries. Enormous energy could be gained by splitting the atom.
1.c. asserts the impossibility of two production possibility vectors which exactly cancel each other, in the sense that the outputs of one are exactly the inputs of the other.
This means you can't have autarkic islands or a wholly isolated subsistence sector.
The simplest justification for 1.c. is to note that some type of labor is necessary for any production activity, while labor cannot be produced by production-units.
Because a farm which produces all the food and woolen yarn and fuel the family living on it needs isn't itself that family. But it is a family farm. Labor produced it and it produced that Labor in the sense of permitting the laborers to marry and have kids who, in turn, would provide labor.
Arrow/Debreu mention Koopmans 'impossibility of the land of Cockaigne'- i.e. one where there is no scarcity. It isn't impossible at all. The snake in that Eden is either Malthusian overpopulation or invasion or immigration.
Other assumptions are tantamount to
1) no bliss point (non-satiation)
2) no speculative demand or hedging
3) negligible income effects- i.e. high substitutability for all goods. The last two taken together are what makes General Equilibrium 'anything goes'- i.e. there is no uniqueness or categoricity or even 'naturality' re. spotting the optimum.
The problem here is that if all these hold, why would agents be arbitragers? Essentially what Arrow-Debreu are saying is 'if everybody was a market maker without a motive to be a market maker then, if they behave exactly as we say they will behave, we conclude by misusing mathematics that there will be a competitive equilibrium'. Why not simply say 'if everybody did just as I want them to, then I would get what I want. Euclid proved this long ago.' ?
The second half of 1V.a. asserts in effect that every individual could consume out of his initial stock in some feasible way and still have a positive amount of each commodity available for trading in the market.
In a recent paper, 'Thermo-economist' Jing Chen has pointed out that (Arrow/Debreu) 'assumes that every individual possesses an initial stock that satisfies all his current and future consumption. In effect, Arrow Debreu model indicates that equilibrium only exists in a system (where) no one needs to do anything
This assumption is clearly unrealistic. However, the necessity of this assumption or some parallel one for the validity of the existence theorem points up an important principle; to have equilibrium, it is necessary that each individual possess some asset or be capable of supplying some labor service which commands a positive price at equilibrium.
In other words, the fact that poor people who are old or sick or disabled have nothing to supply must not stop us from assuming this is not the case. Otherwise, equilibrium would mean that a portion of the population starves to death. After all, that's what happens to animals. Ethologists have no qualms in explaining why animal populations will fluctuate greatly. Economists, however, should have some qualms about engaging in mathsy masturbation in order to say something nice, but wholly untrue, about the free market system.
I suppose you could say 'Arrow Debreu can easily put in compulsory collective insurance'- i.e. a 'Social Minimum'. Why didn't they do so? The answer is that it would screw up the Math. Essentially, the scheme has to be 'backstopped' by arbitrary redistribution or periodic entitlement failure. But both arise outside the market from a Public Sector. In other words, you can't model the thing in a pure market economy. Even if you do persist, you won't have convexity and fixed points. This is actually quite interesting. Essentially, there are going to be holes in the decision space and 'topological dictators'. Also, there will be retractions of a 'holographic' type- i.e. instead of a 'fixed point' reflective equilibrium, there will be information available at the boundary for disequilibrium dynamics. Why settle for boring old equilibrium theory- which requires you to make dismal assumptions about diminishing returns- when you could be getting rich as an arbitrageur of far from equilibrium chrematistics?
Jing Chen, who uses advanced math to teach 'quants', quotes Kaldor (1985)
Returning to prevailing theory, as far as I understand it, the axioms of equilibrium theory were originally chosen in order to secure the desired result, in other words, the assumptions required for proving the existence of a unique and possibly stable general equilibrium. But its authors were motivated by the belief that they were only laying the foundations of an explanation of how a market economy works, an initial stage of the analysis which is in the nature of "scaffolding": it has to be erected before the permanent building can be built, but will be removed step by step as the permanent building nears completion. However, since Walras first wrote down his system of equations over 100 years ago, progress has definitely been backwards not forwards in the sense that the present set of axioms are far more restrictive than those of the original Walrasian model. The ship is no nearer to the shore, but considerably farther off, though in a logical, mathematical sense, the present system of derived tautologies is enormously superior to Walras' s original effort.
Arrow-Debreu securities, though 'weapons of financial mass destruction', are useful enough precisely because they raise volatility and thus liquidity as well as 'shake outs' and hence 'Creative Destruction'.
Still, the true argument for competitive markets has to do with mimetic effects raising general purpose productivity and creating co-evolved mechanisms (e.g. futures markets) which tame Knightian Uncertainty. That's what the Chinese realized back in the Eighties. Sadly, Sen-tentious Indians will continue to stick their heads, like ostriches, into the quicksand of Arrow-Debreu and the belief that Knightian Uncertainty does not exist. We know all possible future states of the world and our Pundits can pick from them in the same way that they pick food items from a restaurant menu.
In a game, the pay-off to each player depends upon the strategies chosen by all, but the domain from which strategies are to be chosen is given to each player independently of the strategies chosen by other players.
In life, Tardean mimetics obtains. Your choice of strategy is determined by what guys smarter and more successful than you are doing. True, you may fail to emulate them. Still, it is 'regret minimizing' to try. It is one thing to try and fail to get a good job. To never have tried, is to live with the knowledge that you yourself think you are shit. In the old days in India people would describe themselves as B.A (fail). This showed they had self-belief and wanted to rise in the world. Maybe they weren't good at academics. Perhaps they would be good at business. What was important was that they wanted to better themselves.
An abstract economy, then, may be characterized as a generalization of a game in which the choice of an action by one agent affects both the pay-off and the domain of actions of other agents.
This is not an economy. It is also not a competitive economy where all agents are price takers and hence no choice of action by one agent has any material effect on any other.
The need for this generalization in the development of an abstract model of the economic system arises from the special position of the consumer.
Consumers are predictable. It is Investment which is volatile.
His "actions" can be regarded as alternative consumption vectors; but these are restricted by the budget restraint that the cost of the goods chosen at current prices not exceed his income.
You can borrow. There is a market for that.
But the prices and possibly some or all of the components of his income are determined by choices made by other agents.
Prices for consumers are likely to be stable for all sorts of reasons. It is investment and speculative demand where 'impredicativity' becomes a problem and equilibrium becomes far to seek.
Why was Arrow so utterly crap at Econ? The answer is that he didn't study Econ. He studied Math. But he wasn't a great mathematician. Koopmans & Kantorovich were first class mathematicians but they too didn't get that the Economy is about mimetics on the one hand and 'market making' arbitrage on the other. I have analyzed Arrow's Nobel lecture elsewhere and thus will end this blog post by pointing to a final absurdity in the Arrow-Debreu paper
Assumption VII plays a key role in the following proof. Part (a) simply asserts that no labor service, at least of those included in 6, can be produced by a production unit.
It is very strange that an American (who knew the great role played by American industry and the American army in turning agricultural workers from the old world into a highly skilled industrial class) should say that enterprises don't themselves create the types of labor they need.
Stranger yet, is what follows-
A case where VII might not hold is an economic system with fixed technological coefficients where production requiring a given type of labor also requires, directly or indirectly, some complementary factors. It is easy to see intuitively in this case how an equilibrium may be impossible. Given the amount of complementary resources initially available, there will be a maximum to the quantity of labor that can be employed in the sense that no further increase in the labor force will increase the output of any commodity. Now, as is well known, the supply of labor may vary either way as real wages vary and broadly speaking is rather inelastic with respect to real wages.
Arrow-Debreu forget that they made each labor contract independent of every other. What happens at the margin does not matter because of the assumption they made.
In particular, as real wages tend to zero,
the marginal wage. You may still get some takers- e.g. unpaid 'interns'.
the supply will not necessarily become zero; on the contrary, as real incomes decrease, the necessity of satisfying more and more pressing needs may even work in the direction of increasing the willingness to work despite the increasingly less favorable terms offered.
That occurs after existing contracts have expired.
It is, therefore, quite possible that for any positive level of real wages, the supply of labor will exceed the maximum employable and hence a fortiori the demand by firms.
but they have individual contracts in their model. What is impossible is to have competitive markets which 'clear' because of the 'iron law of wages', Trade Unions, efficiency wage arguments etc. There is too much hysteresis. Ergodicity would be far to seek.
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