General equilibrium theory, or GET, is the metatheory on which all of mainstream economics rests; it remains very abstract, and it has been carefully studied by only a small number of economists. Invented by the French economist Léon Walras in 1899, GET was neglected for half a century as economists dealt with the intensifying business cycle, the emergence of central banks, and the Great Depression. The economists who began to take notice of it in the 1950s tended to be applied mathematicians, or, at least, economists with strongly analytical gifts.^{1} In the 1960s, a debate arose between John Maynard Keynes’s students at Cambridge University and mathematical economists at MIT. Known as the Cambridge capital controversy, the debate called into question the presumption that aggregation functions defined over microeconomic models could coherently yield a macroeconomic model of the economy.^{2}
Controversy and critique led economists to microfounded models in which micro and macroeconomic theories were, at least, presumptively consistent. The standard microfounded economic model, used by many central banks and presented to graduate students, is the dynamic stochastic general equilibrium model.^{3} It is available in two forms or flavors: the New Classical and the New Keynesian. Both sides see the economy tending toward a perfectly efficient outcome. They disagree only on how difficult it is to get there. New Classicals think it easy. They see economic laws operating in a vacuum, with little or no resistance from nonmarket actors. New Keynesians think it difficult. They see unavoidable rigidities in the real world—wages that do not fall fast enough—or factors internal to the market itself, such as the cost of changing price lists when prices change. For this reason, the New Keynesians often appeal to what they term short and long periods, times in which the economy’s general equilibrium is off and times in which it is on.
For all that, there is little disagreement about the general framework. Neither side doubts that markets move toward general equilibrium, if only because all neoclassical models assume that they do. The point is still more general: any model that invokes a market-clearing mechanism is committed to some sort of general equilibrium—an equilibrium in all markets simultaneously. This commitment is a matter of faith, and if ever questions are raised about GET, they remain within GET.^{4} Joan Robinson once said of economic models that they must abstract from the real economy. A map on a one-to-one scale is useless. She was right of course, but a map that does not accurately reflect the terrain is useless, no matter the scale. If the GET really does underpin the models that economists use to examine market economies, then surely we must be interested in how closely these models match the real world.
Groping and Auctions
In Éléments d’économie politique pure (Elements of Pure Economics), Walras introduced economists to the notion of tâtonnement:
What must we do in order to prove that the theoretical solution is identically [emphasis added] the solution worked out by the market? Our task is very simple: we need only show that the upward and downward movement of prices solve the system of equations of offer and demand by the process of groping [‘par tâtonnement’].^{5}
Two words stand out in this paragraph: “identically” and “simple.” Walras is not shy about making large claims for the theory that he is expounding: the theoretical solution to his system of equations is “identically the solution worked out by the market.” Did Walras have any proof of this? Not obviously. All we are told is that the “task is very simple.”
On the contrary, the task is not simple at all.
Within the literature of GET, background assumptions are sometimes encompassed by the figure of the Walrasian auctioneer.^{6} Imagine that all economic transactions take place at auction. The participants are highly numerate agents whose preferences are transparent and assigned exact prices. A is willing to pay $1 for a banana, but not one cent more; B, $275 million for a superyacht, but not one dollar more. The auctioneer collects the offers for all products at all their potential prices. These serve as inputs to Walras’s system of equations. G. L. S. Shackle sets out the underlying assumptions:
Equilibrium is … a means by which all persons in choosing their acts can be supposed to have equal and perfect relevant knowledge and equal freedom. Given a list of persons, besides himself, composing the society, and given for each of these persons a list of all the acts possible to that person, each person can be supposed to draw up a list of all the distinct combinations of acts, one or other of which combinations will constitute the circumstances surrounding his own act. For each of these combinations he can be supposed to specify the act that he himself would choose, in case he were assured that the combination in question would prevail. The conditional promises, one for each person, thus derived, can now be supposed to be treated as a system to be simultaneously solved.^{7}
That system of equations solved, the result is a general equilibrium at which the market is settled.
So much for the high-level view; but in the real world, one might think, an auction such as this would be asking a lot. Given an auctioneer recording prices, every market participant would be required to perform an enormous number of calculations and, what is more, generally without access to the thoughts of others. Walras’s auctioneer is obviously an imaginary figure, an aide to thought.^{8}
This Walras knew.
If general equilibrium is not reached by means of a simultaneous comparison of prices, plans, and intentions, convergence to general equilibrium happens anyway, Walras argued, through trial and error, a kind of controlled groping.
Thus, tâtonnement.
It is a word that has beguiled generations of economists. But it is curious that they have, those generations, left an important question unasked: how likely is it that tâtonnement could force an economy to converge to a point of general equilibrium?
The Test Case
It is almost always profitable to study questions about probabilities by considering something like a test case in which probabilities are perfectly random. If, in the real world, processes seem to depart from randomness, a test case allows the statistician to ask by what extent; and it allows the economist, much occupied by claims of tâtonnement—economic agents in a mutual grope—to ask whether all that groping might, by itself, explain a departure from randomness.
The test case admits of computation. It is possible to consider the number of transactions taking place in a market economy and then calculate the odds that a general equilibrium might be reached by chance. Under such a scheme, agents are undertaking various transactions randomly. They have no information relative to other agents, and the auctioneer posts no prices. Trial and error is useless. There are no trials, and hence no errors. The most comprehensive data that can be amassed are for the economy of the United States. The number of transactions per adult per month, together with the total number of adults, is laid out in Table 1.^{9}
Table 1.
Year | Adult population | Monthly transactions per adult |
---|---|---|
2000 | 223,468,159 | 46 |
2003 | 231,239,350 | 47 |
2006 | 239,661,291 | 52 |
2009 | 247,407,419 | 55 |
2012 | 254,333,400 | 57 |
2015 | 261,447,213 | 61 |
2016 | 263,745,653 | 62 |
If we assume absolutely no information on the part of various agents, we are faced with a combinatorial problem known as hats-in-a-ring. A number of men check their hats into a cloakroom. The cloakroom assistant is not paying attention and mixes them up. When the men come to collect their hats, the assistant gives hats back to the men at random. If a given agent is acting with zero information about the decisions of other agents, his transactions will be undertaken blindly, in the same way as the cloakroom assistant picks hats blindly and hands them back to the men.^{10} The solution to this problem is
$\mathrm{Pr}{\left(nmatches{hat}_{k}|E\right)}^{k}={\left(\frac{1}{\left(e*n!\right)}\right)}^{k}$,
where n is the number of agents, k is the number of hats or transactions, and e is the base of the natural logarithm. Results are presented in Table 2, along with a calculation of the probability of reaching an equilibrium if trials had been undertaken every month since the onset of the Big Bang—approximately 165,588,000,000 months.^{11}
Table 2.
Year | Probability of chance equilibrium | Probability of chance equilibrium occurring if attempted every month since the Big Bang |
---|---|---|
2000 | 10^{–81361711897} | 10^{–81361711885} |
2003 | 10^{–86182688876} | 10^{–86182688864} |
2006 | 10^{–99017447269} | 10^{–99017447257} |
2009 | 10^{–108303000000} | 10^{–108302999988} |
2012 | 10^{–115557196520} | 10^{–115557196508} |
2015 | 10^{–127316547417} | 10^{–127316547405} |
2016 | 10^{–130603480007} | 10^{–130603479995} |
These numbers are uninspiring.^{12} Finding a general equilibrium by chance is effectively impossible. I must not be misunderstood. The existence of a general equilibrium is not in doubt. A number of sophisticated mathematical economists have demonstrated this. But as much is true in the theory of ordinary differential equations, where any number of theorems guarantee that existence and uniqueness of solutions to various initial value problems. Finding the solution is another problem. And so is the problem of finding a point of general equilibrium. Whatever the answer, it cannot be by chance. These sorts of probabilities are in the region of miracles, not of science.
A Walk Back from Randomness
If general equilibrium is not reached randomly, just how is it reached? The auctioneer may be allowed to depart. If tâtonnement is left, it is no easy business to see how it prevails. No one would argue that economic agents have no information about the intentions and desires of other market actors. Yet every agent has extremely limited information.^{13} What does groping look like in a fixed-price retail outlet or in a situation that does not allow for haggling? These problems solved, what does it mean that the number of transactions seems to be increasing at a faster rate than the number of agents needed to process information?
That extra computing power—where is it coming from?
These questions are, if provocative, also vague. Still, Walras claimed that it was simple to transfer the model to a market economy. At the very least, in light of these calculations, that claim needs much more argument than anyone has provided. Equilibrium theories are broadly conceived in terms of long periods. In economic policy discussions, most of the debate centers on how far off this long period actually is. If this long period does, indeed, exist, and if an economist can demonstrate the probity of GET, this might open up a sensible empirical discussion of how far off it is. In this regard, consider these remarks, taken from an extremely popular textbook:
[M]arkets are usually a good way to organize economic activity… In any economic system, scarce resources have to be allocated among competing uses. Market economies harness the forces of supply and demand to serve that end. Supply and demand together determine the prices of the economy’s many different goods and services; prices in turn are the signals that guide the allocation of resources.^{14}
This statement might well be true. But it is not currently supported by economic theory. If GET cannot be shown to be an accurate reflection of existing market economies, then economics does not possess any theory that tells us that market forces automatically produce optimal outcomes, no matter how often mathematical economists demonstrate that optimal outcomes exist. If not really true, then perhaps the above statement is true in the more limited sense that alternative economic arrangements were put in place in the twentieth century that did not work well. This judgment rests on historical induction. Perhaps it will work better tomorrow.
What the facts suggest is that when agents are faced with relative pricing in a market economy, they are overloaded with information. Just possibly, we may not know how relative prices are set. That said, how do we ourselves behave when faced with an informational overload? Trying to price a good, I would emulate current market prices. That would be easy if the product I were selling was already available on the market. Introducing a novel good, I would try to find a similar item and base my price on it. Could such a process be modeled? It is not easy to see how. But it seems intuitively much closer to the truth than any appeals to imaginary auctions, auctioneers, or some barely coherent process of groping.
Scraping By
Economists trusting in the existence of GET have vested their confidence in a point that cannot be reached by chance; they have provided very little guidance on how it might be reached in any other way. Does this mean we just have to scrap economic theory? No. It does mean that economics would be a far more successful discipline if it studied aggregate behaviors that are easily observable in the wide range of economic statistics that we possess. Some economists have pointed in this direction, making reference to the latest talk in other disciplines about emergent properties.^{15} This buzzword signals a shifting in the sciences away from the methodological approaches of the atomists, who build up their picture of reality piece by piece, and toward a more Aristotelian approach, beginning from the basic principle that the whole is greater than the sum of its parts. A chaque jour suffit sa peine. This essay is meant only as a challenge to general equilibrium theory. To those championing the theory, a request: make a case for the departure from randomness that is intuitively plausible. Once done, if it can be done at all, discussion about many policy issues will become more sophisticated.
Until then, GET should be seen for what it is.^{16}