The other day, I went looking for a Hallmark card to send to a friend of mine who turned 60 this year. There were several more or less facetious or risque cards on offer but none which said 'Congrats on getting ass-raped by all the snobs at the party to which you didn't invite me.' Thus I was obliged to send the fellow a text-message. If we lived in an Arrow-Debreu world, I would have been spared this type of menial labor. There would be a market for every possible type of Hallmark card including one in which my friend was depicted as ruefully applying a soothing balm to his posterior after his posh friends had utterly wrecked it. Furthermore, in the paradise imagined for us by the mathematical economists, there would be no need for Language or Science or Art or Education or Justice or Politics or anything else. A 'competitive equilibrium' would prevail such that everybody would instantaneously attain the highest level of felicity that Human Society could possibly procure for them without any extra effort whatsoever.
Ken Arrow said in his Nobel Prize speech-
From the time of Adam Smith’s Wealth of Nations in 1776, one recurrent theme of economic analysis has been the remarkable degree of coherence among the vast numbers of individual and seemingly separate decisions about the buying and selling of commodities.
Why stop there? Why not speak of the remarkable degree of coherence shown by producers and consumers of eggs? Why do all egg producers keep hens? How is it that everybody who buys eggs ends up cracking the shell of those eggs? Why do egg producers not keep cows instead of hens? Why do egg buyers not show a tender regard for the shells of their eggs?
I suppose the answer is mimetic effects. One guy starts keeping hens for eggs and, if he thrives, others copy him. You see your parents have a hard boiled egg for brekkie and copy them because you is a big boy- not a whiny little baby.
What distinguishes markets is that there is a price signal which goes up and down in response to changes in demand and supply. In other types of economic regimes, there may be a spoken or written order 'produce more eggs!', 'eat more eggs!' and this may have the same effect. This is a linguistic signal.
In everyday, normal experience, there is something of a balance between the amounts of goods and services that some individuals want to supply and the amounts that other, different individuals want to sell.
If prices are flexible- sure. But prices aren't flexible for various reasons. In everyday, normal experience, too many commuters want to use the trains and roads at peak hours. Not enough people want to lovingly suck your dick. Sad.
Would-be buyers ordinarily count correctly on being able to carry out their intentions, and would-be sellers do not ordinarily find themselves producing great amounts of goods that they cannot sell.
Unless they stick to writing poetry or painting pictures instead of getting a job delivering pizzas.
This experience of balance is indeed so widespread that it raises no intellectual disquiet among laymen; they take it so much for granted that they are not disposed to understand the mechanism by which it occurs. T he paradoxical result is that they have no idea of the system’s strength and are unwilling to trust it in any considerable departure from normal conditions. This reaction is most conspicuous in wartime situations with radical shifts in demand. It is taken for granted that these can be met only by price control, rationing, and direct allocation of resources.
Because otherwise military morale might collapse. If 'profiteering' appears to be going on, you might get a Mutiny or a Revolution. During war-time, we expect all sorts of things to be different. In particular, something like a 'principle of equi-marginal sacrifice' is entailed whereby the rich are seen to be making a bigger monetary sacrifice whereas the poor may be disproportionately at risk.
Yet there is no reason to believe that the same forces that work in peacetime would not produce a working system in time of war or other considerable shifts in demand. (There are undesirable consequences of a free market system, but sheer unworkability is not one of them.)
Nonsense! During war-time, you need to signal to the enemy that you will do whatever it takes to fuck it up. Indeed, you may do so even if you are going to abstain from conflict- e.g. our current policy w.r.t Ukraine. We are signalling that we can take a lot of economic pain from Russia provided we inflict yet more harm.
I do not want to overstate the case. The balancing of supply and demand is far from perfect. Most conspicuously, the history of the capitalist system has been marked by recurring periods in which the supply of available labor and of productive equipment available for the production of goods has been in excess of their utilization, sometimes, as in the 1930’s, by very considerable magnitudes.
& the history of Communism featured periods when very large numbers of peasants starved to death. The fault, in both cases, arose out of economic theories creating both irrational exuberance and then stupid shit.
Further, the relative balance of overall supply and demand in the postwar period in the United States and Europe is in good measure the result of deliberate governmental policies, not an automatic tendency of the market to balance. Nevertheless, when all due allowances are made, the coherence of individual economic decisions is remarkable. As incomes rise and demands shift, for example, from food to clothing and housing, the labor force and productive facilities follow suit.
Demand for food went up. But the income elasticity was low. As labor became more expensive, capital was substituted for labor and productivity went up in agriculture. However, this did not always happen. Sometimes migrants were brought in to do the agriculture or food was imported from poorer countries so that the reward for that type of labor fell.
Similarly, and even more surprising to the layman, there is a mutual interaction between shifts in technology and the allocation of the labor force. As technology improves exogenously, through innovations, the labor made redundant does not become permanently unemployed, but finds its place in the economy.
Or gets sent back across the border or is left to rot on meager Social Security benefits. More Labor may be employed but some laborers may now be unemployed while skilled people have come in from elsewhere.
It is truly amazing that the lessons of both theory and over a century of history are still so misunderstood. On the other hand, a growing accumulation of instruments of production raises real wages and in turn induces a rise in the prices of labor-intensive commodities relative to those which use little labor.
So an independent craftsman becomes an obsequious flunkey at a lower wage. Incidentally, giving beejays is a labor intensive service industry. Thankfully, Tardean mimetics- imitating superiors- causes people to find ways to stay off the game- for example by pretending to be Social Scientists.
All these phenomena show that by and large and in the long view of history, the economic system adjusts
just as the linguistic and social and political system adjusts or it doesn't the nation is conquered and enslaved.
with a considerable degree of smoothness and indeed of rationality to changes in the fundamental facts within which it operates. The problematic nature of economic coordination is most obvious in a free enterprise economy but might seem of lesser moment in a socialist or planned society.
Economic coordination is no more problematic than linguistic coordination in a free society. In a command economy it is an actual mathematical problem of great complexity which has to be solved or-what is more likely- fudged.
But a little reflection on the production and consumption decisions of such a society, at least in the modern world of complex production, shows that in the most basic aspects the problem of coordination is not removed by the transition to socialism or to any other form of planning.
This is a bizarre view. Coordination is spontaneous under free enterprise though it can always be improved by better public signals.
In the pure model of a free enterprise world, an individual, whether consumer or producer, is the locus both of interests or tastes and of information.
Whereas in an impure model, the individual is a focus of both vulgarity and ignorance
Each individual has his own desires,
Rather than those of his cat
which he is expected to pursue within the constraints imposed by the economic mechanism; but in addition he is supposed to have more information about himself or at least about a particular sphere of productive and consumptive activity than other individuals.
No. We make no such assumption. A consumer may be stupid and ignorant. A producer might be shit at his job. However, since consumers who poison themselves and producers who are shit get weeded out, for big enough markets we can think of consumers and producers as constrained by whatever ignorance and vulgarity prevails.
It might be that in an ideal socialist economy, all individuals will act in accord with some agreed ideas of the common good,
in which case they might choose the spontaneous order of free enterprise
though I personally find this concept neither realistic nor desirable, in that it denies the fact and value of individual diversity. But not even the most ideal socialist society will obviate the diversity of information about productive methods that must obtain simply because the acquisition of information is costly.
But dissemination of information need not be- witness the internet.
Hence, the need for coordination
Which is also needed to talk to people and to avoid tripping and falling down while walking
, for some means of seeing that plans of diverse agents have balanced totals, remains.
They don't need to have 'balanced totals'. That may be the aim, but it isn't the outcome.
How this coordination takes place has been a central preoccupation of economic theory since Adam Smith and received a reasonably clear answer in the 1870’s with the work of Jevons, Menger, and above all, Leon Walras: it was the fact that all agents in the economy faced the same set of prices that provided the common flow of information needed to coordinate the system.
But we don't face the 'same set of prices'. The point about the 'marginal revolution' was that 'limited arbitrage' was good enough. In other words so long as there is some interaction between sufficiently but not too heterogeneous agents everywhere then changes would occur in the 'right direction'
There was, so it was argued, a set of prices, one for each commodity which would equate supply and demand for all commodities; and if supply and demand were unequal anywhere, at least some prices would change, while none would change in the opposite case. Because of the last characteristics, the balancing of supply and demand under these conditions may be referred to as equilibrium in accordance with the usual use of that term in science and mathematics.
But any state of the economy which has no endogenous reason to change in the next time period is an equilibrium. There is a mathematical representation of it- though this may not be computable.
The adjective, “general,” refers to the argument that we cannot legitimately speak of equilibrium with respect to any one commodity; since supply and demand on any one market depends on the prices of other commodities, the overall equilibrium of the economy cannot be decomposed into separate equilibria for individual commodities
In which case Economic 'general equilibrium' is not different from the full equilibrium of the physical and biological and psychological and social and political and spiritual and fartological and linguistic and masturbatory equilibrium of everything under the Sun and in it.
Now even in the most strictly neoclassical version of price theory, it is not precisely true that prices alone are adequate information to the individual agents for the achievement of equilibrium, a point that will be developed later. One brand of criticism has put more stress on quantities themselves as signals, including no less an authority than the great Keynes [1936]; see especially the interpretation of Keynes by Leijonhufvud [1968, especially Chapter II].
But there are actual verbal signals in this case- everybody going 'the market is booming!' or 'the market is fucked!' as the case may be. If quantity can be a signal it is because you have a pretty sophisticated economy with trustworthy information aggregators and business analysts and Accountants and so on and so forth.
More recently the same argument has been advanced by Kornai [1971] from socialist experience. Nevertheless, while the criticisms are, in my judgment, not without some validity, they have not given rise to a genuine alternative model of detailed resource allocation. The fundamental question remains, how does an overall total quantity, say demand, as in the Keynesian model, get transformed into a set of signals and incentives for individual sellers?
Through language. Price and quantity and p/e and r.o.i may look a bit mathsy but they are in natural language.
If one shifts perspective from description to design of economies, it is not so hard to think of non-price coordinating mechanisms; we are in fact all familiar with rationing in one form or another. Here, the discussion of coordination shades off in that of efficiency. There has long been a view that the competitive price equilibrium is efficient or optimal in some sense that rationing is not.
You can start with rationing and then just let everybody swap stuff through some cheap and trustworthy network.
This sense and the exact statement of the optimality theorem were clarified by Pareto [1909, Chapter VI, sections 32-38] and, in the 1930’s by my teacher, Hotelling [1938] and by Bergson [1938]. An allocation of resources is Pareto efficient (or Pareto optimal) if there is no other feasible allocation which will make everyone better off (or, as more usually stated, make everyone at least as well off and at least one member better off).
This just means any mutually advantageous trade or swap or more complicated type of relationship featuring 'contingent' assets and liabilities or psychic goods- like friendship and lurve- improves things provided no third party is affected.
Simply increasing trustworthy methods for cheap interactions between larger and larger groups of people improve Pareto efficiency.
Then, by an argument that I shall sketch shortly, it was held that a competitive equilibrium necessarily yielded a Pareto efficient allocation of resources.
Any equilibrium would be Pareto efficient provided people could always swap stuff with others in a safe, cheap and reliable manner throughout the economy.
It was, of course, recognized, most explicitly perhaps by Bergson, that Pareto efficiency in no way implied distributive justice. An allocation of resources could be efficient in a Pareto sense and yet yield enormous riches to some and dire poverty to others.
But we can't know whether this is so. That homeless bum might be an ideal chum for that billionaire because they of some shared interest. The billionaire might decide to move the chap into this guest cottage and both of them now enjoy life much more with no third party being harmed. It may also be the case that God exists and Hell is where he sends rich fucks who don't help starving socioproctologists. Okay, I'm fat not starving but I could use a butler. And a chef. And a homeless bum who likes listening to me rant and rave against Amartya Sen when everybody knows it was Arrow who steered our homeboy wrong.
T HE HICKS-SAMUELSON MODEL OF GENERAL E QUILIBRIUM I will state more formally the model of general competitive equilibrium as it had been developed by about 1945, primarily through the detailed developments and syntheses of Hicks [1939] and Samuelson [1947]. Competitive analysis is founded on two basic principles: optimizing behavior on the part of individual agents in the presence of prices taken as given by them
But if everybody is a price-taker then no new good or service can enter the market (because no price exists for it and so the innovator has to do 'discovery' by setting a price). In addition to 'no innovation' there can be no arbitrage, no contingent assets- e.g. traded options, insurance policies- and no fiat money and no Financial Sector.
Thus the first 'basic principle' abolishes everything human about an economy. Still, the thing might describe the behavior of ants or low level robots. I'm kidding. Because of Djikstra concurrency or Razborov-Rudich or other such mathematical results coming out back then, Arrow should have known this basic principle was simply stupid.
and the setting of the prices so that, given this individual behavior, supply equals demand on each market.
Sadly, computing that price vector would take much longer than the life of the Universe. Again this was something known at the time Arrow said this.
The outcome of the competitive process is then to be evaluated in terms of Pareto efficiency and additional conditions on the resulting distribution of goods. The maximizing behavior of individuals has been well surveyed by Samuelson in his Nobel lecture [1971], and I will not go over that ground here. I just want to remind the listener of a few elementary points. The first is that the consumer’s choices are subject to a budget constraint.
Which is unknowable for the reason given above.
The consumer starts with the possession of some quantities of economically valuable goods, such as labor of particular types, land, or other possessions. Let us imagine there are n commodities altogether, and let &i be the amount of commodity i owned initially by individual h (this may well be zero for most commodities). If pi is the price of the ith commodity, then his total income available for expenditure is i pi &h
Hence nobody can ever know their own available total income. Did you know that you could use your smartphone to earn hundreds of dollars just by? ...dunno..some smart stuff I haven't figured out yet. If we knew that the price of a time machine is 100 billion, then we get extra information- viz. that time travel is possible. The assumption that the price vector is common knowledge means that everything that can be known is well defined because the tech associated with every possible type of knowledge would have a price tag.
Arrow tells us that the consumer is presumed to choose his most preferred bundle.
Which is reasonable.
The most usual interpretation of “most preferred” in this context is that there is a preference ordering over all possible bundles,
which is totally unreasonable. The fact that you ask a girl out means you prefer her to the others you know. It doesn't mean you know how you'd compare her to all the girls on the planet.
according to which, for every pair of bundles, one is preferred to the other or else the two are indifferent; and these pairwise judgments have the consistency property known to logicians as “transitivity;” thus, for example, if bundle A is preferred to bundle B and B to C, then A will be preferred to C. This “ordinalist” view of preferences was originally due to Pareto and to Irving Fisher, about 1900, and represented an evolution from the earlier “cardinalist” position, according to which a measurable satisfaction or “utility” was associated with each bundle, and the consumer chose that bundle which maximized utility within the budget set.
So, Econ had been getting stupider for over a century.
Obviously, a cardinal utility implies an ordinal preference but not vice versa ; and if the only operational meaning of utility is in the explanation of consumer choice, then clearly two utility functions which defined the same preference ordering are operationally indistinguishable.
But can't exist. The Universe won't last long enough for the thing to be computed.
The most preferred bundle then is a function, xh( (PI, . . ., pn) of all prices.
But prices are a function of preferences so the 'axiom of foundation' is being violated. So the thing can't yield 'ordered pairs' though it may be described as an algorithm such that there is a semantic model. But the thing would still be nonsense.
Notice that, from this viewpoint, all prices clearly enter into the determination of the demand for any one commodity.
And all preferences enter into the determination of the supply of one commodity.
For one thing, the rise in any one price clearly diminishes the residual income available for all other commodities. More specifically, however, the demands for some commodities are closely interrelated with others; thus, the demand for gasoline is perhaps more influenced by the use of automobiles and therefore by their price than it is by its own price.
Then some guy who prefers to drink whisky while driving his automobiles runs over you and suddenly your preference is for a wheel chair and crutches.
The interrelation of all demands is clearly displayed here. The characterization of consumer choice by optimization can, as we all know, be made more explicit. Let us recall Hicks’s definition of the marginal rate of substitution between two commodities for any individual. For any given bundle, (x7, . . ., x:), consider all bundles indifferent to it, i.e., neither preferred to it nor inferior to it. If we hold all but two commodity quantities constant, say Xk = xi (k # i, j) we can consider xi as a function of xj on this indifference surface
No we can't. A change in one commodity quantity could instantaneously change some or all the others in multiple ways. The function is not well defined.
...since the marginal rate of substitution for any pair of commodities is equal to the price ratio for all individuals, it is also true that the marginal rate of substitution for any two commodities is the same for all individuals.
assuming the two goods in question are equally satisfying or there is no 'income effect'- i.e. the change in relative prices does not have a wealth effect.
This suggests in turn that there is no possibility that two or any number of individuals can gain by trading with each other after achieving a competitive equilibrium.
Equilibrium means there is no further endogenous reason to change. However, trading isn't the only way mutual benefit is created.
The equality of the marginal rates of substitution means that a trade which would leave one individual on an indifference surface would do the same to the other. Hence, a competitive equilibrium satisfies the same kinds of conditions that are satisfied by a Pareto optimum.
Because of the stipulation that no further mutually advantageous interaction is possible. The mention of 'indifference surfaces' was otiose. It muddied the waters.
(It will be observed that the stated conditions for a consumer optimum and for a Pareto optimum are first-order conditions in the differential calculus. Hotelling, Hicks, and Samuelson also developed the second-order conditions which distinguish maxima from minima and showed that these had important implications.)
The opposite is the case. The stated condition for differential calculus- i.e. the existence of well defined functions- does not apply to Pareto optimum in any way whatsoever. This is because interaction can go from trade to lurve to something spiritual and of greater worth than life itself.
Evaluation of the performance of an economy with regard to distributive justice was far less studied, not surprisingly, since the deepest philosophical issues are at stake.
But the deepest philosophical issues are shit.
Arrow goes on to summarize the German tradition from Cassel to Von Neumann before explaining his own contribution
My own interest first centered on the relations between Pareto efficiency and competitive equilibrium. In particular, there was considerable discussion among economists in the late 1940’s about the inefficiencies resulting from rent control and different proposals for arriving at the efficiency benefits of a free market by one or another transition route.
Rent control redistributed endowments. It was part and parcel of other redistributive measures. However, it was also part of a deliberate restriction of freedoms of very sort so that 'equal sacrifice' on the part of civilians was demonstrated so as to maintain military morale. The only 'efficiency' of interest to society, was efficiency in achieving war aims- i.e. killing the enemy- and maintaining cohesion and morale to this end.
Part of the informal efficiency arguments hinged on the idea that under rent control people were buying the wrong kind of housing, say, excessively large apartments. It struck me that an individual bought only one kind of housing, not several.
Individuals may have 'bought' both a service- viz. accommodation- as well as an asset
The individual optima were at corners, and therefore one could not equate marginal rates of substitution by going over to a free market.
This would require a bit of 'mechanism design' so that accommodation, as a service, was disentangled from the underlying asset and then people were allowed to make mutually beneficial swaps through some transparent mechanism such that those with greater skill at swapping didn't end up controlling a lot of property.
The problem here is that what is or isn't an asset depends on contingencies- things which may or may not happen- and the instincts or unconscious valuation mechanisms of people at large. These could change quickly and unexpectedly without anything visible in Society changing. Thus one could never be sure one is disentangling the 'asset' from the 'service'.
Yet diagrammatic analysis of simple cases suggested to me that the traditional identification of competitive equilibrium and Pareto efficiency was correct but could not be proved by the local techniques of the differential calculus.
But that identification was not correct. Some fresh piece of information or even a contagious mood or a rumor might change behavior in any given market. Moreover, a market transaction might be replaced by a relationship or 'incomplete contract' because of increased uncertainty.
I soon realized that the theory of convex sets, and, in particular, the separation theorem, was the appropriate tool. Start with a Pareto-efficient allocation, and consider all logically possible allocations which would be preferred to it by everyone.
There are none. If an allocation is Pareto efficient there is no chain of mutually beneficial trades away from it. Arrow was thinking 'distributionally'- i.e. he thought who owns what matters. But Coase's theorem shows us that this needn't matter. If it is safe to make mutually advantageous trades, then ownership does not matter.
Of course, no such allocation can be feasible; otherwise the allocation we started with would not be Pareto efficient. Each such allocation is a statement of demand or supply of each commodity by each individual or firm.
But these statements would vary under different contingencies. Only by getting rid of Knightian uncertainty would there be a unique allocation. In other words, Arrow got rid of a fundamental fact about economic activity- viz it occurs in a world where nobody knows what is or isn't feasible at the aggregate level and thus the very notion of 'general equilibrium' is an oxymoron- so as to be able to do at the 'general' level that which was impossible to do at the local 'comparative statics' level. This is like saying 'we can't predict what will happen when two people meet but we can predict everything at the level of Society as a whole'.
Hence, by adding up over individuals and firms, with appropriate attention to signs, we can define the excess demand for each commodity.
No we can't. Nobody knows what will happen in every possible market and thus nobody can 'define' their excess demand for each and every commodity under any given price vector. In a given market there is a 'revealed preference' but the fact is some buyers are saying 'If only I could have been sure about such and such factor, I'd have bought more/less of the product'. The same is true of sellers. Arrow is pretending that people don't merely demand just as much as they want, they also know exactly how much of everything they would demand. But, in reality, we often buy an inferior product at a higher price because we don't know everything that is available and what price it sells for.
Let Z be the set of all excess demand vectors (z1 , . . ., zn ) generated this way. Since they are all infeasible, it must be true for each one that there is positive excess demand for at least one commodity.
We don't know this. It is possible that there is a workaround- e.g. two people sharing a cab or jointly reading a newspaper or making some other sort of mutual accommodation. Thus for Z to have a unique representation Arrow must assume that Society has reached a stage where no further 'cooperative' Pareto improvement is possible. In other words, his notion of 'competitive equilibrium' gains no traction till there is such perfect linguistic interchange and such perfect mutual knowledge of character and trustworthiness that economics is only concerned with things which are irreducibly 'rival'.
In the language of set theory, the set Z is disjoint from the non-positive orthant, i.e., the set of vectors (z1 , . . ., zn ) such that ti 5 0 for all i. The separation theorem for convex sets asserts that if two convex sets are disjoint, there is a hyperplane which separates them, so that one set is on one side and the other set on the other.
This may exist in 'the mind of God' but it is wholly inaccessible to us. Similarly we may say there exists some Latin incantation which would give me the power of levitation. We can't disprove this. We could even argue that a being must exist who can grant this power. Moreover, it is likely that some Latin incantation will flatter it sufficiently to actually do so. But the thing is improbable. The only people who might want to assert such a thing are likely to be charlatans whose stock in trade is Latin incantations of various sorts.
If theft exists or if taxes exist or if courts can order one person to hand over an asset to another person or, more generally, if transfers occur outside the market then 'convexity' is violated. This 'separation theorem' is useless. More generally, if 'incomplete contracts' can't be 'decomposed' into vectors of a suitable type, they lie outside the scope of Arrow's analysis. But this means almost all economic activity is outside its scope. Indeed, the thing may only be relevant to games- like Monopoly- played with children and drunken relatives on Christmas Day.
Taken altogether, it has been shown that if Z is a convex set, the Pareto efficient allocation can be achieved as a competitive equilibrium of the market, in the sense that prices and a suitable initial allocation of resources can be found such that each individual is achieving his satisfaction level at minimum cost, each firm is maximizing profits, and the markets are all in equilibrium in the generalized sense which permits corner equilibria.
In which case some agent or agents aren't making a 'trade-off' and thus, under some contingency, this really isn't an equilibrium at all. A motivation for a trade exists but hasn't yet been expressed because some information is lacking or some state of the world has not fructified. Non-action is an action waiting to happen. It does not represent contentment with the state of affairs.
The need to assume that Z is convex puts in sharper focus the convexity assumptions which had always implicitly underlain neoclassical theory.
It can do so explicitly and is useful enough because in the short run diminishing returns do set in. But it is foolish to think this heuristic device must be a metaphysical feature of the Economy at large. America, during the Second World War, went from a country with an army smaller than Portugal into a globe bestriding colossus altering the balance of power in Western Europe and the Far East. Who would have predicted that within four years American troops would be stationed in Berlin and Tokyo? Was there any 'competitive' trajectory by which this could be achieved?
The convexity of Z could be derived from the following two assumptions: (1) for each individual, the set of consumption vectors preferred to a given vector is convex; (2) for each firm, the set of technologically possible vectors is convex. The result states that, under suitable convexity conditions, a necessary condition for an allocation to be Pareto efficient is that it be realizable in the market as a competitive equilibrium.
Arrow must have bought presents for kids. Either he was a really shitty gift giver or else he violated this 'necessary condition'. Whenever you take pleasure in the pleasure others get from your gift you have made a Pareto improvement on a 'competitive equilibrium'.
A byproduct of the investigation was the proof of the converse theorem: a competitive equilibrium is always Pareto efficient, and this theorem is true without any convexity assumption. These results were embodied in Arrow [1951a]. But the idea that the theory of convex sets was the appropriate tool was clearly in the air. While I was working at Stanford, Gerard Debreu [1951] obtained very much the same results at the Cowles Commission for Research in Economics at Chicago.
To a guy who has a hammer everything looks like a nail. What this entailed was that, as Arrow says, ' It was necessary to require that no combination of activities as a whole permitted production without inputs.' Yet if there were 'corner solutions' or 'free goods' or prudential reserves then no economy would qualify as fitting Arrow's analysis. The wider problem is that for an Economy to exist there has to be a supply of various services- e.g. contract enforcement- and those services, by themselves, establish prices and valuations. Arrow says- The trouble is that the individual’s income also depends upon prices, and if the prices of those commodities which the individual owns originally fall to zero, his income falls to zero. When some prices and income are zero, however, the demand for the now-free goods may jump discontinuously. To illustrate, suppose an individual owned initially only one good, say, labor. So long as the price of that good was positive, he might retain some for his own use, but in any case could never consume more than he had initially. But when the price fell to zero, he could demand the same labor from others and in any amount he chooses. The existence of competitive equilibrium then does depend on assumptions which insure that for each individual there is at least one commodity he owns initially which is bound to have positive value.
In other words, competitive equilibrium can only exist if there is- at the very least- a Liberal 'night-watchman' state which secures the liberty of all. How would it finance itself? Taxes on property or incomes or expenditure or a combination of all three. But this gives the State an incentive to maintain property values and income levels and prevent 'under-consumption' crises. Arrow and Debreu may have thought they were showing there was something magical about competitive equilibria but the reverse was the case. The Second World War had shown that it took a lot of blood and treasure to keep that show on the road. The alternative was Nazi slave labor or Soviet Gulags.
Arrow had to abolish the fundamental fact of Social and Economic Life- viz Knightian uncertainty, our relative ignorance regarding not just what will happen but what can happen- in order to remain content with their magical discovery.
Suppose there is some uncertainty in production due, for example, to the weather. One type of weather will benefit one kind of producer and injure another, while another type will do the opposite. If we assume that individuals are averse to risk, there is room for a mutually profitable trade in insurance. Even apart from risk aversion, individuals and firms in planning for an uncertain future may want to make sure that their demands and outputs are mutually compatible. Thus, if there is uncertainty about the supply of grain, a miller may prefer to make future contracts for labor contingent on that uncertainty. We take from the theory of probability the concept of a state of the world, which is a description of the world so precise that it completely defines all initial holdings of goods and all technological possibilities. Uncertainty is not knowing which state will in fact hold.
This is risk, not Knightian uncertainty. Arrow accommodates risk in the following way-
Commodities in the ordinary sense are replaced by contingent commodities, promises to buy or sell a given commodity if and only if a certain state of the world occurs.
This is quite different from the type of 'incomplete contract' which in fact obtains such that there is a mutual adjustment over the lifetime of a business relationship precisely because both parties understand that it is impossible to specify all contingencies when contracting.
The market will then determine contingent prices.
But the market will fail unexpectedly because of mispricing. It is not safe to assume there will always be a riskless asset or that volatility surfaces are mathematically tractable. Warren Buffett is supposed to have called Arrow-Debreu securities 'weapons of mass financial destruction' a few years before the sub-prime crash.
Clearing of the markets means clearing of the contingent markets; the commitments made are sufficiently flexible so that they can always be satisfied. It should be noted that preference orderings over vectors of contingent commodities contain elements of judgment about the likelihoods of different states of the world as well as elements of taste in the ordinary sense. Other things being equal, one will invest less heavily in a demand contingent upon a state deemed unlikely.
The problem here is that there can be perverse 'income' effects or 'incentive incompatibility' in relevant markets- i.e. guys maximizing their commission doing deals which result in an asset bubble.
One can work out the implications of this model.
An entire industry may find it profitable to do so but it may also crash the entire financial system.
Clearly, the contingent commodities called for do not exist to the extent required, but the variety of securities available on modern markets serves as a partial substitute. In my own thinking, the model of general equilibrium under uncertainty is as much a normative ideal as an empirical description.
A 'normative ideal'! People who were wiped out by the crash may consider it a nightmare.
It is the way the actual world differs from the criteria of the model which suggests social policy to improve the efficiency with which risk-bearing is allocated. In fact, is is not a mere empirical accident that not all the contingent markets needed for efficiency exist but a necessary fact with deep implications for the workings and structure of economic institutions. Roughly speaking, information about particular events, even after they have occurred, is not spread evenly throughout the population. Two people cannot enter into a contract contingent on the occurrence of a certain event or state if only one of them in fact will know that the event has occurred. A particular example of this is sometimes known as “moral hazard” in the insurance and economic literature. The very existence of insurance will change individual behavior in the direction of less care in avoiding risks. The insurance policy that would be called for by an optimal allocation of risk bearing would only cover unavoidable risks and would distinguish their effects from those due to behavior of the individual.
It turned out that 'agent-principal hazard' affected Arrow-Debreu security markets. Furthermore, Fintech may have been a factor in turning dirty money derived from dirty energy, into dirty politics and now- as we see in Ukraine- a dirty war waged by oligarchs against innocent civilian populations.
Going forward, citizens of advanced economies understand that they will have to make sacrifices so as to guard against repugnant contingencies- e.g. financing a Russian dictator determined to stamp our democracy in a neighboring state. But, it is important to understand that this is the economic equivalent of war- we can't risk nuclear annihilation by intervening militarily. This means that, as happened during the Second World war, Governments will have to intervene to protect the working population and vulnerable sections of society while permitting the rich to invest in new, cleaner, types of energy and supply chains. Let the affluent take the capital gains while ensuring that ordinary people don't have to make a cruel choice between 'eating and heating'.
General competitive equilibrium above all teaches the extent to which a social allocation of resources can be achieved by independent private decisions coordinated through the market. We are assured indeed that not only can an allocation be achieved, but the result will be Pareto efficient.
No. Pareto efficiency can always be increased by cheaper and more reliable mechanisms for individual inter-change. I sell stuff I don't use anymore on Ebay and buy stuff you don't use anymore on Gumtree. However, non-commercial collaborative initiatives too increase Pareto efficiency. Over the lockdown, many people found their lives enriched by such friendships crossing continents and differences in age and culture.
But, as has been stressed, there is nothing in the process which guarantees that the distribution be just. Indeed, the theory teaches us that the final allocation will depend on the distribution of initial supplies and of ownership of firms. If we want to rely on the virtues of the market but also to achieve a more just distribution, the theory suggests the strategy of changing the initial distribution rather than interfering with the allocation process at some later stage.
But changing the initial distribution may provoke fears that more and more such arbitrary interventions occur. There may be a flight of capital and talent. A better approach may be to guarantee a social minimum and then use conventional fiscal and monetary means to finance this. There may even be a 'virtuous circle' such that guaranteed demand allows firms to exploit economies of scope and scale more efficiently. Price discrimination through branding and advertising may allow plentiful profit- some of which goes into R&D- while poorer people get good quality goods and services for a very affordable price.
Thus even under the assumptions most favorable to decentralization of decision-making, there is an irreducible need for a social or collective choice on distribution.
If the aim is a 'social minimum' worsening distribution may be a benign outcome. After all, efficiency may increase if 'control rights' are concentrated in the hands of those with greater expertise. There may be other ways to give indigenous populations a portion of National wealth without necessarily generating disincentive effects.
To be fair, Arrow was speaking 50 years ago when markets needed more freedom and a greater role in resource allocation. On the other hand, when we look at what oligarchs have done to Russia and the Ukraine we may well deplore the manner in which open markets in resources permitted the rise of sociopathic 'siloviki' strongmen for whom genocide is a management strategy.
In point of fact, there are a great many other situations in which the replacement of market by collective decision-making is necessary or at least desirable. In their different ways, both political scientists and economists have discussed the necessary role of the state. Among economists, these discussions have revolved around the concepts of externalities, increasing returns, and market failure; the clarification and application of these ideas have been among the major achievements of modern economic thought, but I have time now merely to recall them to you as helping to create the need for normative and descriptive analysis of collective decision-making. In the context of social choice, each individual may be assumed to have a preference ordering over all possible social states.
If they are omniscient- sure.
This ordering expresses not only his desire for his own consumption but also social attitudes, his views on justice in distribution or on benefits to others from collective decisions.
He would know which person, out of all the billions on this planet, would be the one he'd most enjoy sharing this pizza with. However, he'd also know which person on another continent he should split his desert with.
The ordinalist viewpoint forbids us from ascribing a definite quantitative expression to this preference, at least a quantitative expression which would have any interpersonal validity. Classical utilitarianism specifies that choices among alternative social states be judged in terms of their consequences for the members of the society; in the present terminology, this means in terms of the individual preference scales for social choices.
Those consequences are unknown and unknowable. I may think I'd be happier living in a Theocracy or a Communist utopia but I can't be sure. Moreover, it is likely that I am habituated to the way things are in the Society I live in. I would find it difficult to adjust to some wholly novel constitution.
This is obviously not a sufficient basis for choice in view of the diversity of individual preferences. It is implicit in classical utilitarianism and explicit in Bergson’s work that there is a second level at the individual judgments are aggregated into what might be termed a welfare judgment.
But it wouldn't be worth much time and effort making such judgments.
Thus the formation of welfare judgments is logically equivalent to what I will call a constitution.
Arrow noticed that his barber and the guy who watered his lawn were constantly pondering the relative merits of different constitutions and the manner in which they might be amended to meet different contingencies. Sadly, they'd get sacked because this ceaseless cogitation distracted them from doing the jobs they were paid to do.
Specifically, a constitution is a rule which associates to each possible set of individual preference orderings a social choice rule.
But we would always prefer not to have any such rule or, indeed, any such preference orderings. The thing is a waste of time.
A social choice rule, in turn, is a rule for selecting a socially preferred action out of any set of alternatives which may be feasible. So far, I would hold that the description of a constitution is a tautology, at least if we start from the view that social choice has to be based on the individual preference orderings.
But nobody has any such 'preference orderings'. I'd love to know which traded option I should buy so as to triple my money overnight. But I have no access to that information. I don't even know whether I'd prefer kiwi fruit to mango though I bought mango at Waitrose for my lunch. The truth is I've never eaten kiwi fruit.
The real question is what conditions are to be imposed on the constitution. One condition, which is already contained in Bergson’s work, is that for any given set of individual preferences, the social choice rule defined by them shall satisfy the technical conditions of an ordering, that is, that all possible alternative social states should be capable of being ranked and then the social choice from any particular set of alternatives should be the most preferred alternative, according to the ordering, in the available set. This is sometimes called the condition of Collective Rationality.
Which it would be irrational to have. Evolution ensured that our preferences aren't predictable because otherwise we'd be easy prey to a predator or a parasite. Some of our economic decisions are based on habit. But there are mimetic and 'discovery' processes going on as well.
A second condition, again in agreement with Bergson, is the Pareto principle; the social choice process shall never yield an outcome if there is another feasible alternative which everyone prefers according to his preference ordering.
Why not? It is easy enough to swap things around even if the waiter put the steak in front of me and the salad in front of you- contrary to our order. Pareto efficiency can take care of itself provided such swaps are cheap and reliable.
A third hardly controversial condition is that of Non-Dictatorship; the constitution shall not be such that there is an individual whose preferences automatically become those of society regardless of anyone else’s preferences.
But we may choose to let someone with superior information and public spirit make decisions for us. Winston Churchill was not a Dictator. He was a democratically chosen leader bound by the Law of the land. If you say 'cats are dogs' you are telling a stupid lie. From this stupid lie any other stupid lie can be deduced. This is the principle of explosion or ex falso quodlibet which has been known since the 12th century!
The fourth condition which I have suggested, that of the Independence of Irrelevant Alternatives, is more disputable, though I would argue that it has strong pragmatic justification : the social choice made from any set of alternatives will depend on only the orderings of individuals among alternatives in that set. To see what is at stake, suppose that a society has to make a choice among some alternatives and does so. After the decision is made, an alternative which has not previously been thought of is mentioned as a logical possibility, although it is still not feasible. The individuals can expand their preference orderings to place this new alternative in its place on their ranking; but should this preference information about an alternative which could not be chosen in any case affect the previous decision?
Yes. If we can be independent of Russia's dirty energy we should try to make this feasible as soon as possible. The alternative is that we end up paying for the killing of innocent Ukrainians.
Any form of voting certainly satisfies the condition of Independence of Irrelevant Alternatives;
No. Every form of voting would fail in this respect under some possible scenario. Suppose it is discovered that a candidate is a Martian. This revelation completely changes our preferences even if the candidate is disbarred or deported.
the preferences of voters as between candidates and non-candidates or as between non-candidates, are of course, never asked for or taken into account. It turns out (Arrow [1951b, 1963b]) that these four reasonable-sounding requirements are contradictory.
They were not reasonable. They were very silly. Nobody in their right mind would vote for this shit.
That is, if we devise any constitution, then it is always possible to find a set of individual orderings which will cause the constitution to violate one of these conditions. In one special form, this paradox is old. The method of majority voting is an appealing method of social choice. Like any other voting method, it satisfies Independence of Irrelevant Alternatives and certainly the Pareto principle and the condition on Non-Dictatorship. But as Condorcet pointed out as far back as [1785], majority voting may not lead to an ordering. More specifically, intransitivity is possible. Consider the following example. There are three alternatives X, y, and z, among which choice is to be made. One-third of the voters have the ranking ~,y, z; one-third, the ranking y, z, x; and one-third, the ranking z, X, y. Then a majority of the voters prefer x to y, a majority prefer y to z, and a majority prefer z to x. Unfortunately, this result is not due to a removable imperfection in the method of majority voting. The four conditions on social choice are mutually contradictory. The philosophical and distributive implications of the paradox of social choice are still not clear.
The implication was blindingly obvious. Social Choice theory was stooooopid.
Certainly, there is no simple way out. I hope that others will take this paradox as a challenge rather than as a discouraging barrier.
Arrow, like Samuelson, was a good man whose heart was in the right place. But a lot of what they wrote was bollocks. Today, more than ever, we are aware that Freedom is something people have to make sacrifices for. Our preferences must change in order to preserve the freedom to choose in a worthwhile manner at a future date. The war in Ukraine is an economic war of a global type. Government's must go on a war footing so as to preserve an 'equality of sacrifice' in the coming months and years.
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