Archive for the 'Sonnenschein-Mantel-Debreu theorem' Category

Lucas and Sargent on Optimization and Equilibrium in Macroeconomics

In a famous contribution to a conference sponsored by the Federal Reserve Bank of Boston, Robert Lucas and Thomas Sargent (1978) harshly attacked Keynes and Keynesian macroeconomics for shortcomings both theoretical and econometric. The econometric criticisms, drawing on the famous Lucas Critique (Lucas 1976), were focused on technical identification issues and on the dependence of estimated regression coefficients of econometric models on agents’ expectations conditional on the macroeconomic policies actually in effect, rendering those econometric models an unreliable basis for policymaking. But Lucas and Sargent reserved their harshest criticism for abandoning what they called the classical postulates.

Economists prior to the 1930s did not recognize a need for a special branch of economics, with its own special postulates, designed to explain the business cycle. Keynes founded that subdiscipline, called macroeconomics, because he thought that it was impossible to explain the characteristics of business cycles within the discipline imposed by classical economic theory, a discipline imposed by its insistence on . . . two postulates (a) that markets . . . clear, and (b) that agents . . . act in their own self-interest [optimize]. The outstanding fact that seemed impossible to reconcile with these two postulates was the length and severity of business depressions and the large scale unemployment which they entailed. . . . After freeing himself of the straight-jacket (or discipline) imposed by the classical postulates, Keynes described a model in which rules of thumb, such as the consumption function and liquidity preference schedule, took the place of decision functions that a classical economist would insist be derived from the theory of choice. And rather than require that wages and prices be determined by the postulate that markets clear — which for the labor market seemed patently contradicted by the severity of business depressions — Keynes took as an unexamined postulate that money wages are “sticky,” meaning that they are set at a level or by a process that could be taken as uninfluenced by the macroeconomic forces he proposed to analyze[1]. . . .

In recent years, the meaning of the term “equilibrium” has undergone such dramatic development that a theorist of the 1930s would not recognize it. It is now routine to describe an economy following a multivariate stochastic process as being “in equilibrium,” by which is meant nothing more than that at each point in time, postulates (a) and (b) above are satisfied. This development, which stemmed mainly from work by K. J. Arrow and G. Debreu, implies that simply to look at any economic time series and conclude that it is a “disequilibrium phenomenon” is a meaningless observation. Indeed, a more likely conjecture, on the basis of recent work by Hugo Sonnenschein, is that the general hypothesis that a collection of time series describes an economy in competitive equilibrium is without content. (pp. 58-59)

Lucas and Sargent maintain that ‘classical” (by which they obviously mean “neoclassical”) economics is based on the twin postulates of (a) market clearing and (b) optimization. But optimization is a postulate about individual conduct or decision making under ideal conditions in which individuals can choose costlessly among alternatives that they can rank. Market clearing is not a postulate about individuals, it is the outcome of a process that neoclassical theory did not, and has not, described in any detail.

Instead of describing the process by which markets clear, neoclassical economic theory provides a set of not too realistic stories about how markets might clear, of which the two best-known stories are the Walrasian auctioneer/tâtonnement story, widely regarded as merely heuristic, if not fantastical, and the clearly heuristic and not-well-developed Marshallian partial-equilibrium story of a “long-run” equilibrium price for each good correctly anticipated by market participants corresponding to the long-run cost of production. However, the cost of production on which the Marhsallian long-run equilibrium price depends itself presumes that a general equilibrium of all other input and output prices has been reached, so it is not an alternative to, but must be subsumed under, the Walrasian general equilibrium paradigm.

Thus, in invoking the neoclassical postulates of market-clearing and optimization, Lucas and Sargent unwittingly, or perhaps wittingly, begged the question how market clearing, which requires that the plans of individual optimizing agents to buy and sell reconciled in such a way that each agent can carry out his/her/their plan as intended, comes about. Rather than explain how market clearing is achieved, they simply assert – and rather loudly – that we must postulate that market clearing is achieved, and thereby submit to the virtuous discipline of equilibrium.

Because they could provide neither empirical evidence that equilibrium is continuously achieved nor a plausible explanation of the process whereby it might, or could be, achieved, Lucas and Sargent try to normalize their insistence that equilibrium is an obligatory postulate that must be accepted by economists by calling it “routine to describe an economy following a multivariate stochastic process as being ‘in equilibrium,’ by which is meant nothing more than that at each point in time, postulates (a) and (b) above are satisfied,” as if the routine adoption of any theoretical or methodological assumption becomes ipso facto justified once adopted routinely. That justification was unacceptable to Lucas and Sargent when made on behalf of “sticky wages” or Keynesian “rules of thumb, but somehow became compelling when invoked on behalf of perpetual “equilibrium” and neoclassical discipline.

Using the authority of Arrow and Debreu to support the normalcy of the assumption that equilibrium is a necessary and continuous property of reality, Lucas and Sargent maintained that it is “meaningless” to conclude that any economic time series is a disequilibrium phenomenon. A proposition ismeaningless if and only if neither the proposition nor its negation is true. So, in effect, Lucas and Sargent are asserting that it is nonsensical to say that an economic time either reflects or does not reflect an equilibrium, but that it is, nevertheless, methodologically obligatory to for any economic model to make that nonsensical assumption.

It is curious that, in making such an outlandish claim, Lucas and Sargent would seek to invoke the authority of Arrow and Debreu. Leave aside the fact that Arrow (1959) himself identified the lack of a theory of disequilibrium pricing as an explanatory gap in neoclassical general-equilibrium theory. But if equilibrium is a necessary and continuous property of reality, why did Arrow and Debreu, not to mention Wald and McKenzie, devoted so much time and prodigious intellectual effort to proving that an equilibrium solution to a system of equations exists. If, as Lucas and Sargent assert (nonsensically), it makes no sense to entertain the possibility that an economy is, or could be, in a disequilibrium state, why did Wald, Arrow, Debreu and McKenzie bother to prove that the only possible state of the world actually exists?

Having invoked the authority of Arrow and Debreu, Lucas and Sargent next invoke the seminal contribution of Sonnenschein (1973), though without mentioning the similar and almost simultaneous contributions of Mantel (1974) and Debreu (1974), to argue that it is empirically empty to argue that any collection of economic time series is either in equilibrium or out of equilibrium. This property has subsequently been described as an “Anything Goes Theorem” (Mas-Colell, Whinston, and Green, 1995).

Presumably, Lucas and Sargent believe the empirically empty hypothesis that a collection of economic time series is, or, alternatively is not, in equilibrium is an argument supporting the methodological imperative of maintaining the assumption that the economy absolutely and necessarily is in a continuous state of equilibrium. But what Sonnenschein (and Mantel and Debreu) showed was that even if the excess demands of all individual agents are continuous, are homogeneous of degree zero, and even if Walras’s Law is satisfied, aggregating the excess demands of all agents would not necessarily cause the aggregate excess demand functions to behave in such a way that a unique or a stable equilibrium. But if we have no good argument to explain why a unique or at least a stable neoclassical general-economic equilibrium exists, on what methodological ground is it possible to insist that no deviation from the admittedly empirically empty and meaningless postulate of necessary and continuous equilibrium may be tolerated by conscientious economic theorists? Or that the gatekeepers of reputable neoclassical economics must enforce appropriate standards of professional practice?

As Franklin Fisher (1989) showed, inability to prove that there is a stable equilibrium leaves neoclassical economics unmoored, because the bread and butter of neoclassical price theory (microeconomics), comparative statics exercises, is conditional on the assumption that there is at least one stable general equilibrium solution for a competitive economy.

But it’s not correct to say that general equilibrium theory in its Arrow-Debreu-McKenzie version is empirically empty. Indeed, it has some very strong implications. There is no money, no banks, no stock market, and no missing markets; there is no advertising, no unsold inventories, no search, no private information, and no price discrimination. There are no surprises and there are no regrets, no mistakes and no learning. I could go on, but you get the idea. As a theory of reality, the ADM general-equilibrium model is simply preposterous. And, yet, this is the model of economic reality on the basis of which Lucas and Sargent proposed to build a useful and relevant theory of macroeconomic fluctuations. OMG!

Lucas, in various writings, has actually disclaimed any interest in providing an explanation of reality, insisting that his only aim is to devise mathematical models capable of accounting for the observed values of the relevant time series of macroeconomic variables. In Lucas’s conception of science, the only criterion for scientific knowledge is the capacity of a theory – an algorithm for generating numerical values to be measured against observed time series – to generate predicted values approximating the observed values of the time series. The only constraint on the algorithm is Lucas’s methodological preference that the algorithm be derived from what he conceives to be an acceptable microfounded version of neoclassical theory: a set of predictions corresponding to the solution of a dynamic optimization problem for a “representative agent.”

In advancing his conception of the role of science, Lucas has reverted to the approach of ancient astronomers who, for methodological reasons of their own, believed that the celestial bodies revolved around the earth in circular orbits. To ensure that their predictions matched the time series of the observed celestial positions of the planets, ancient astronomers, following Ptolemy, relied on epicycles or second-order circular movements of planets while traversing their circular orbits around the earth to account for their observed motions.

Kepler and later Galileo conceived of the solar system in a radically different way from the ancients, placing the sun, not the earth, at the fixed center of the solar system and proposing that the orbits of the planets were elliptical, not circular. For a long time, however, the actual time series of geocentric predictions outperformed the new heliocentric predictions. But even before the heliocentric predictions started to outperform the geocentric predictions, the greater simplicity and greater realism of the heliocentric theory attracted an increasing number of followers, forcing methodological supporters of the geocentric theory to take active measures to suppress the heliocentric theory.

I hold no particular attachment to the pre-Lucasian versions of macroeconomic theory, whether Keynesian, Monetarist, or heterodox. Macroeconomic theory required a grounding in an explicit intertemporal setting that had been lacking in most earlier theories. But the ruthless enforcement, based on a preposterous methodological imperative, lacking scientific or philosophical justification, of formal intertemporal optimization models as the only acceptable form of macroeconomic theorizing has sidetracked macroeconomics from a more relevant inquiry into the nature and causes of intertemporal coordination failures that Keynes, along with many some of his predecessors and contemporaries, had initiated.

Just as the dispute about whether planetary motion is geocentric or heliocentric was a dispute about what the world is like, not just about the capacity of models to generate accurate predictions of time series variables, current macroeconomic disputes are real disputes about what the world is like and whether aggregate economic fluctuations are the result of optimizing equilibrium choices by economic agents or about coordination failures that cause economic agents to be surprised and disappointed and rendered unable to carry out their plans in the manner in which they had hoped and expected to be able to do. It’s long past time for this dispute about reality to be joined openly with the seriousness that it deserves, instead of being suppressed by a spurious pseudo-scientific methodology.

HT: Arash Molavi Vasséi, Brian Albrecht, and Chris Edmonds

[1] Lucas and Sargent are guilty of at least two misrepresentations in this paragraph. First, Keynes did not “found” macroeconomics, though he certainly influenced its development decisively. Keynes used the term “macroeconomics,” and his work, though crucial, explicitly drew upon earlier work by Marshall, Wicksell, Fisher, Pigou, Hawtrey, and Robertson, among others. See Laidler (1999). Second, having explicitly denied and argued at length that his results did not depend on the assumption of sticky wages, Keynes certainly never introduced the assumption of sticky wages himself. See Leijonhufvud (1968)

Filling the Arrow Explanatory Gap

The following (with some minor revisions) is a Twitter thread I posted yesterday. Unfortunately, because it was my first attempt at threading the thread wound up being split into three sub-threads and rather than try to reconnect them all, I will just post the complete thread here as a blogpost.

1. Here’s an outline of an unwritten paper developing some ideas from my paper “Hayek Hicks Radner and Four Equilibrium Concepts” (see here for an earlier ungated version) and some from previous blog posts, in particular Phillips Curve Musings

2. Standard supply-demand analysis is a form of partial-equilibrium (PE) analysis, which means that it is contingent on a ceteris paribus (CP) assumption, an assumption largely incompatible with realistic dynamic macroeconomic analysis.

3. Macroeconomic analysis is necessarily situated a in general-equilibrium (GE) context that precludes any CP assumption, because there are no variables that are held constant in GE analysis.

4. In the General Theory, Keynes criticized the argument based on supply-demand analysis that cutting nominal wages would cure unemployment. Instead, despite his Marshallian training (upbringing) in PE analysis, Keynes argued that PE (AKA supply-demand) analysis is unsuited for understanding the problem of aggregate (involuntary) unemployment.

5. The comparative-statics method described by Samuelson in the Foundations of Econ Analysis formalized PE analysis under the maintained assumption that a unique GE obtains and deriving a “meaningful theorem” from the 1st- and 2nd-order conditions for a local optimum.

6. PE analysis, as formalized by Samuelson, is conditioned on the assumption that GE obtains. It is focused on the effect of changing a single parameter in a single market small enough for the effects on other markets of the parameter change to be made negligible.

7. Thus, PE analysis, the essence of micro-economics is predicated on the macrofoundation that all, but one, markets are in equilibrium.

8. Samuelson’s meaningful theorems were a misnomer reflecting mid-20th-century operationalism. They can now be understood as empirically refutable propositions implied by theorems augmented with a CP assumption that interactions b/w markets are small enough to be neglected.

9. If a PE model is appropriately specified, and if the market under consideration is small or only minimally related to other markets, then differences between predictions and observations will be statistically insignificant.

10. So PE analysis uses comparative-statics to compare two alternative general equilibria that differ only in respect of a small parameter change.

11. The difference allows an inference about the causal effect of a small change in that parameter, but says nothing about how an economy would actually adjust to a parameter change.

12. PE analysis is conditioned on the CP assumption that the analyzed market and the parameter change are small enough to allow any interaction between the parameter change and markets other than the market under consideration to be disregarded.

13. However, the process whereby one equilibrium transitions to another is left undetermined; the difference between the two equilibria with and without the parameter change is computed but no account of an adjustment process leading from one equilibrium to the other is provided.

14. Hence, the term “comparative statics.”

15. The only suggestion of an adjustment process is an assumption that the price-adjustment in any market is an increasing function of excess demand in the market.

16. In his seminal account of GE, Walras posited the device of an auctioneer who announces prices–one for each market–computes desired purchases and sales at those prices, and sets, under an adjustment algorithm, new prices at which desired purchases and sales are recomputed.

17. The process continues until a set of equilibrium prices is found at which excess demands in all markets are zero. In Walras’s heuristic account of what he called the tatonnement process, trading is allowed only after the equilibrium price vector is found by the auctioneer.

18. Walras and his successors assumed, but did not prove, that, if an equilibrium price vector exists, the tatonnement process would eventually, through trial and error, converge on that price vector.

19. However, contributions by Sonnenschein, Mantel and Debreu (hereinafter referred to as the SMD Theorem) show that no price-adjustment rule necessarily converges on a unique equilibrium price vector even if one exists.

20. The possibility that there are multiple equilibria with distinct equilibrium price vectors may or may not be worth explicit attention, but for purposes of this discussion, I confine myself to the case in which a unique equilibrium exists.

21. The SMD Theorem underscores the lack of any explanatory account of a mechanism whereby changes in market prices, responding to excess demands or supplies, guide a decentralized system of competitive markets toward an equilibrium state, even if a unique equilibrium exists.

22. The Walrasian tatonnement process has been replaced by the Arrow-Debreu-McKenzie (ADM) model in an economy of infinite duration consisting of an infinite number of generations of agents with given resources and technology.

23. The equilibrium of the model involves all agents populating the economy over all time periods meeting before trading starts, and, based on initial endowments and common knowledge, making plans given an announced equilibrium price vector for all time in all markets.

24. Uncertainty is accommodated by the mechanism of contingent trading in alternative states of the world. Given assumptions about technology and preferences, the ADM equilibrium determines the set prices for all contingent states of the world in all time periods.

25. Given equilibrium prices, all agents enter into optimal transactions in advance, conditioned on those prices. Time unfolds according to the equilibrium set of plans and associated transactions agreed upon at the outset and executed without fail over the course of time.

26. At the ADM equilibrium price vector all agents can execute their chosen optimal transactions at those prices in all markets (certain or contingent) in all time periods. In other words, at that price vector, excess demands in all markets with positive prices are zero.

27. The ADM model makes no pretense of identifying a process that discovers the equilibrium price vector. All that can be said about that price vector is that if it exists and trading occurs at equilibrium prices, then excess demands will be zero if prices are positive.

28. Arrow himself drew attention to the gap in the ADM model, writing in 1959:

29. In addition to the explanatory gap identified by Arrow, another shortcoming of the ADM model was discussed by Radner: the dependence of the ADM model on a complete set of forward and state-contingent markets at time zero when equilibrium prices are determined.

30. Not only is the complete-market assumption a backdoor reintroduction of perfect foresight, it excludes many features of the greatest interest in modern market economies: the existence of money, stock markets, and money-crating commercial banks.

31. Radner showed that for full equilibrium to obtain, not only must excess demands in current markets be zero, but whenever current markets and current prices for future delivery are missing, agents must correctly expect those future prices.

32. But there is no plausible account of an equilibrating mechanism whereby price expectations become consistent with GE. Although PE analysis suggests that price adjustments do clear markets, no analogous analysis explains how future price expectations are equilibrated.

33. But if both price expectations and actual prices must be equilibrated for GE to obtain, the notion that “market-clearing” price adjustments are sufficient to achieve macroeconomic “equilibrium” is untenable.

34. Nevertheless, the idea that individual price expectations are rational (correct), so that, except for random shocks, continuous equilibrium is maintained, became the bedrock for New Classical macroeconomics and its New Keynesian and real-business cycle offshoots.

35. Macroeconomic theory has become a theory of dynamic intertemporal optimization subject to stochastic disturbances and market frictions that prevent or delay optimal adjustment to the disturbances, potentially allowing scope for countercyclical monetary or fiscal policies.

36. Given incomplete markets, the assumption of nearly continuous intertemporal equilibrium implies that agents correctly foresee future prices except when random shocks occur, whereupon agents revise expectations in line with the new information communicated by the shocks.
37. Modern macroeconomics replaced the Walrasian auctioneer with agents able to forecast the time path of all prices indefinitely into the future, except for intermittent unforeseen shocks that require agents to optimally their revise previous forecasts.
38. When new information or random events, requiring revision of previous expectations, occur, the new information becomes common knowledge and is processed and interpreted in the same way by all agents. Agents with rational expectations always share the same expectations.
39. So in modern macro, Arrow’s explanatory gap is filled by assuming that all agents, given their common knowledge, correctly anticipate current and future equilibrium prices subject to unpredictable forecast errors that change their expectations of future prices to change.
40. Equilibrium prices aren’t determined by an economic process or idealized market interactions of Walrasian tatonnement. Equilibrium prices are anticipated by agents, except after random changes in common knowledge. Semi-omniscient agents replace the Walrasian auctioneer.
41. Modern macro assumes that agents’ common knowledge enables them to form expectations that, until superseded by new knowledge, will be validated. The assumption is wrong, and the mistake is deeper than just the unrealism of perfect competition singled out by Arrow.
42. Assuming perfect competition, like assuming zero friction in physics, may be a reasonable simplification for some problems in economics, because the simplification renders an otherwise intractable problem tractable.
43. But to assume that agents’ common knowledge enables them to forecast future prices correctly transforms a model of decentralized decision-making into a model of central planning with each agent possessing the knowledge only possessed by an omniscient central planner.
44. The rational-expectations assumption fills Arrow’s explanatory gap, but in a deeply unsatisfactory way. A better approach to filling the gap would be to acknowledge that agents have private knowledge (and theories) that they rely on in forming their expectations.
45. Agents’ expectations are – at least potentially, if not inevitably – inconsistent. Because expectations differ, it’s the expectations of market specialists, who are better-informed than non-specialists, that determine the prices at which most transactions occur.
46. Because price expectations differ even among specialists, prices, even in competitive markets, need not be uniform, so that observed price differences reflect expectational differences among specialists.
47. When market specialists have similar expectations about future prices, current prices will converge on the common expectation, with arbitrage tending to force transactions prices to converge toward notwithstanding the existence of expectational differences.
48. However, the knowledge advantage of market specialists over non-specialists is largely limited to their knowledge of the workings of, at most, a small number of related markets.
49. The perspective of specialists whose expectations govern the actual transactions prices in most markets is almost always a PE perspective from which potentially relevant developments in other markets and in macroeconomic conditions are largely excluded.
50. The interrelationships between markets that, according to the SMD theorem, preclude any price-adjustment algorithm, from converging on the equilibrium price vector may also preclude market specialists from converging, even roughly, on the equilibrium price vector.
51. A strict equilibrium approach to business cycles, either real-business cycle or New Keynesian, requires outlandish assumptions about agents’ common knowledge and their capacity to anticipate the future prices upon which optimal production and consumption plans are based.
52. It is hard to imagine how, without those outlandish assumptions, the theoretical superstructure of real-business cycle theory, New Keynesian theory, or any other version of New Classical economics founded on the rational-expectations postulate can be salvaged.
53. The dominance of an untenable macroeconomic paradigm has tragically led modern macroeconomics into a theoretical dead end.

Phillips Curve Musings

There’s a lot of talk about the Phillips Curve these days; people wonder why, with the unemployment rate reaching historically low levels, nominal and real wages have increased minimally with inflation remaining securely between 1.5 and 2%. The Phillips Curve, for those untutored in basic macroeconomics, depicts a relationship between inflation and unemployment. The original empirical Philips Curve relationship showed that high rates of unemployment were associated with low or negative rates of wage inflation while low rates of unemployment were associated with high rates of wage inflation. This empirical relationship suggested a causal theory that the rate of wage increase tends to rise when unemployment is low and tends to fall when unemployment is high, a causal theory that seems to follow from a simple supply-demand model in which wages rise when there is an excess demand for labor (unemployment is low) and wages fall when there is an excess supply of labor (unemployment is high).

Viewed in this light, low unemployment, signifying a tight labor market, signals that inflation is likely to rise, providing a rationale for monetary policy to be tightened to prevent inflation from rising at it normally does when unemployment is low. Seeming to accept that rationale, the Fed has gradually raised interest rates for the past two years or so. But the increase in interest rates has now slowed the expansion of employment and decline in unemployment to historic lows. Nor has the improving employment situation resulted in any increase in price inflation and at most a minimal increase in the rate of increase in wages.

In a couple of previous posts about sticky wages (here and here), I’ve questioned whether the simple supply-demand model of the labor market motivating the standard interpretation of the Phillips Curve is a useful way to think about wage adjustment and inflation-employment dynamics. I’ve offered a few reasons why the supply-demand model, though applicable in some situations, is not useful for understanding how wages adjust.

The particular reason that I want to focus on here is Keynes’s argument in chapter 19 of the General Theory (though I express it in terms different from his) that supply-demand analysis can’t explain how wages and employment are determined. The upshot of his argument I believe is that supply demand-analysis only works in a partial-equilibrium setting in which feedback effects from the price changes in the market under consideration don’t affect equilibrium prices in other markets, so that the position of the supply and demand curves in the market of interest can be assumed stable even as price and quantity in that market adjust from one equilibrium to another (the comparative-statics method).

Because the labor market, affecting almost every other market, is not a small part of the economy, partial-equilibrium analysis is unsuitable for understanding that market, the normal stability assumption being untenable if we attempt to trace the adjustment from one labor-market equilibrium to another after an exogenous disturbance. In the supply-demand paradigm, unemployment is a measure of the disequilibrium in the labor market, a disequilibrium that could – at least in principle — be eliminated by a wage reduction sufficient to equate the quantity of labor services supplied with the amount demanded. Viewed from this supply-demand perspective, the failure of the wage to fall to a supposed equilibrium level is attributable to some sort of endogenous stickiness or some external impediment (minimum wage legislation or union intransigence) in wage adjustment that prevents the normal equilibrating free-market adjustment mechanism. But the habitual resort to supply-demand analysis by economists, reinforced and rewarded by years of training and professionalization, is actually misleading when applied in an inappropriate context.

So Keynes was right to challenge this view of a potentially equilibrating market mechanism that is somehow stymied from behaving in the manner described in the textbook version of supply-demand analysis. Instead, Keynes argued that the level of employment is determined by the level of spending and income at an exogenously given wage level, an approach that seems to be deeply at odds with idea that price adjustments are an essential part of the process whereby a complex economic system arrives at, or at least tends to move toward, an equilibrium.

One of the main motivations for a search for microfoundations in the decades after the General Theory was published was to be able to articulate a convincing microeconomic rationale for persistent unemployment that was not eliminated by the usual tendency of market prices to adjust to eliminate excess supplies of any commodity or service. But Keynes was right to question whether there is any automatic market mechanism that adjusts nominal or real wages in a manner even remotely analogous to the adjustment of prices in organized commodity or stock exchanges – the sort of markets that serve as exemplars of automatic price adjustments in response to excess demands or supplies.

Keynes was also correct to argue that, even if there was a mechanism causing automatic wage adjustments in response to unemployment, the labor market, accounting for roughly 60 percent of total income, is so large that any change in wages necessarily affects all other markets, causing system-wide repercussions that might well offset any employment-increasing tendency of the prior wage adjustment.

But what I want to suggest in this post is that Keynes’s criticism of the supply-demand paradigm is relevant to any general-equilibrium system in the following sense: if a general-equilibrium system is considered from an initial non-equilibrium position, does the system have any tendency to move toward equilibrium? And to make the analysis relatively tractable, assume that the system is such that a unique equilibrium exists. Before proceeding, I also want to note that I am not arguing that traditional supply-demand analysis is necessarily flawed; I am just emphasizing that traditional supply-demand analysis is predicated on a macroeconomic foundation: that all markets but the one under consideration are in, or are in the neighborhood of, equilibrium. It is only because the system as a whole is in the neighborhood of equilibrium, that the microeconomic forces on which traditional supply-demand analysis relies appear to be so powerful and so stabilizing.

However, if our focus is a general-equilibrium system, microeconomic supply-demand analysis of a single market in isolation provides no basis on which to argue that the system as a whole has a self-correcting tendency toward equilibrium. To make such an argument is to commit a fallacy of composition. The tendency of any single market toward equilibrium is premised on an assumption that all markets but the one under analysis are already at, or in the neighborhood of, equilibrium. But when the system as a whole is in a disequilibrium state, the method of partial equilibrium analysis is misplaced; partial-equilibrium analysis provides no ground – no micro-foundation — for an argument that the adjustment of market prices in response to excess demands and excess supplies will ever – much less rapidly — guide the entire system back to an equilibrium state.

The lack of automatic market forces that return a system not in the neighborhood — for purposes of this discussion “neighborhood” is left undefined – of equilibrium back to equilibrium is implied by the Sonnenschein-Mantel-Debreu Theorem, which shows that, even if a unique general equilibrium exists, there may be no rule or algorithm for increasing (decreasing) prices in markets with excess demands (supplies) by which the general-equilibrium price vector would be discovered in a finite number of steps.

The theorem holds even under a Walrasian tatonnement mechanism in which no trading at disequilibrium prices is allowed. The reason is that the interactions between individual markets may be so complicated that a price-adjustment rule will not eliminate all excess demands, because even if a price adjustment reduces excess demand in one market, that price adjustment may cause offsetting disturbances in one or more other markets. So, unless the equilibrium price vector is somehow hit upon by accident, no rule or algorithm for price adjustment based on the excess demand in each market will necessarily lead to discovery of the equilibrium price vector.

The Sonnenschein Mantel Debreu Theorem reinforces the insight of Kenneth Arrow in an important 1959 paper “Toward a Theory of Price Adjustment,” which posed the question: how does the theory of perfect competition account for the determination of the equilibrium price at which all agents can buy or sell as much as they want to at the equilibrium (“market-clearing”) price? As Arrow observed, “there exists a logical gap in the usual formulations of the theory of perfectly competitive economy, namely, that there is no place for a rational decision with respect to prices as there is with respect to quantities.”

Prices in perfect competition are taken as parameters by all agents in the model, and optimization by agents consists in choosing optimal quantities. The equilibrium solution allows the mutually consistent optimization by all agents at the equilibrium price vector. This is true for the general-equilibrium system as a whole, and for partial equilibrium in every market. Not only is there no positive theory of price adjustment within the competitive general-equilibrium model, as pointed out by Arrow, but the Sonnenschein-Mantel-Debreu Theorem shows that there’s no guarantee that even the notional tatonnement method of price adjustment can ensure that a unique equilibrium price vector will be discovered.

While acknowledging his inability to fill the gap, Arrow suggested that, because perfect competition and price taking are properties of general equilibrium, there are inevitably pockets of market power, in non-equilibrium states, so that some transactors in non-equilibrium states, are price searchers rather than price takers who therefore choose both an optimal quantity and an optimal price. I have no problem with Arrow’s insight as far as it goes, but it still doesn’t really solve his problem, because he couldn’t explain, even intuitively, how a disequilibrium system with some agents possessing market power (either as sellers or buyers) transitions into an equilibrium system in which all agents are price-takers who can execute their planned optimal purchases and sales at the parametric prices.

One of the few helpful, but, as far as I can tell, totally overlooked, contributions of the rational-expectations revolution was to solve (in a very narrow sense) the problem that Arrow identified and puzzled over, although Hayek, Lindahl and Myrdal, in their original independent formulations of the concept of intertemporal equilibrium, had already provided the key to the solution. Hayek, Lindahl, and Myrdal showed that an intertemporal equilibrium is possible only insofar as agents form expectations of future prices that are so similar to each other that, if future prices turn out as expected, the agents would be able to execute their planned sales and purchases as expected.

But if agents have different expectations about the future price(s) of some commodity(ies), and if their plans for future purchases and sales are conditioned on those expectations, then when the expectations of at least some agents are inevitably disappointed, those agents will necessarily have to abandon (or revise) the plans that their previously formulated plans.

What led to Arrow’s confusion about how equilibrium prices are arrived at was the habit of thinking that market prices are determined by way of a Walrasian tatonnement process (supposedly mimicking the haggling over price by traders). So the notion that a mythical market auctioneer, who first calls out prices at random (prix cries au hasard), and then, based on the tallied market excess demands and supplies, adjusts those prices until all markets “clear,” is untenable, because continual trading at disequilibrium prices keeps changing the solution of the general-equilibrium system. An actual system with trading at non-equilibrium prices may therefore be moving away from, rather converging on, an equilibrium state.

Here is where the rational-expectations hypothesis comes in. The rational-expectations assumption posits that revisions of previously formulated plans are never necessary, because all agents actually do correctly anticipate the equilibrium price vector in advance. That is indeed a remarkable assumption to make; it is an assumption that all agents in the model have the capacity to anticipate, insofar as their future plans to buy and sell require them to anticipate, the equilibrium prices that will prevail for the products and services that they plan to purchase or sell. Of course, in a general-equilibrium system, all prices being determined simultaneously, the equilibrium prices for some future prices cannot generally be forecast in isolation from the equilibrium prices for all other products. So, in effect, the rational-expectations hypothesis supposes that each agent in the model is an omniscient central planner able to solve an entire general-equilibrium system for all future prices!

But let us not be overly nitpicky about details. So forget about false trading, and forget about the Sonnenschein-Mantel-Debreu theorem. Instead, just assume that, at time t, agents form rational expectations of the future equilibrium price vector in period (t+1). If agents at time t form rational expectations of the equilibrium price vector in period (t+1), then they may well assume that the equilibrium price vector in period t is equal to the expected price vector in period (t+1).

Now, the expected price vector in period (t+1) may or may not be an equilibrium price vector in period t. If it is an equilibrium price vector in period t as well as in period (t+1), then all is right with the world, and everyone will succeed in buying and selling as much of each commodity as he or she desires. If not, prices may or may not adjust in response to that disequilibrium, and expectations may or may not change accordingly.

Thus, instead of positing a mythical auctioneer in a contrived tatonnement process as the mechanism whereby prices are determined for currently executed transactions, the rational-expectations hypothesis posits expected future prices as the basis for the prices at which current transactions are executed, providing a straightforward solution to Arrow’s problem. The prices at which agents are willing to purchase or sell correspond to their expectations of prices in the future. If they find trading partners with similar expectations of future prices, they will reach agreement and execute transactions at those prices. If they don’t find traders with similar expectations, they will either be unable to transact, or will revise their price expectations, or they will assume that current market conditions are abnormal and then decide whether to transact at prices different from those they had expected.

When current prices are more favorable than expected, agents will want to buy or sell more than they would have if current prices were equal to their expectations for the future. If current prices are less favorable than they expect future prices to be, they will not transact at all or will seek to buy or sell less than they would have bought or sold if current prices had equaled expected future prices. The dichotomy between observed current prices, dictated by current demands and supplies, and expected future prices is unrealistic; all current transactions are made with an eye to expected future prices and to their opportunities to postpone current transactions until the future, or to advance future transactions into the present.

If current prices for similar commodities are not uniform in all current transactions, a circumstance that Arrow attributes to the existence of varying degrees of market power across imperfectly competitive suppliers, price dispersion may actually be caused, not by market power, but by dispersion in the expectations of future prices held by agents. Sellers expecting future prices to rise will be less willing to sell at relatively low prices now than are suppliers with pessimistic expectations about future prices. Equilibrium occurs when all transactors share the same expectations of future prices and expected future prices correspond to equilibrium prices in the current period.

Of course, that isn’t the only possible equilibrium situation. There may be situations in which a future event that will change a subset of prices can be anticipated. If the anticipation of the future event affects not only expected future prices, it must also and necessarily affect current prices insofar as current supplies can be carried into the future from the present or current purchases can be postponed until the future or future consumption shifted into the present.

The practical upshot of these somewhat disjointed reflections is, I think,primarily to reinforce skepticism that the traditional Phillips Curve supposition that low and falling unemployment necessarily presages an increase in inflation. Wages are not primarily governed by the current state of the labor market, whatever the labor market might even mean in macroeconomic context.

Expectations rule! And the rational-expectations revolution to the contrary notwithstanding, we have no good theory of how expectations are actually formed and there is certainly no reason to assume that, as a general matter, all agents share the same set of expectations.

The current fairly benign state of the economy reflects the absence of any serious disappointment of price expectations. If an economy is operating not very far from an equilibrium, although expectations are not the same, they likely are not very different. They will only be very different after the unexpected strikes. When that happens, borrowers and traders who had taken positions based on overly optimistic expectations find themselves unable to meet their obligations. It is only then that we will see whether the economy is really as strong and resilient as it now seems.

Expecting the unexpected is hard to do, but you can be sure that, sooner or later, the unexpected is going to happen.

About Me

David Glasner
Washington, DC

I am an economist in the Washington DC area. My research and writing has been mostly on monetary economics and policy and the history of economics. In my book Free Banking and Monetary Reform, I argued for a non-Monetarist non-Keynesian approach to monetary policy, based on a theory of a competitive supply of money. Over the years, I have become increasingly impressed by the similarities between my approach and that of R. G. Hawtrey and hope to bring Hawtrey’s unduly neglected contributions to the attention of a wider audience.

My new book Studies in the History of Monetary Theory: Controversies and Clarifications has been published by Palgrave Macmillan

Follow me on Twitter @david_glasner


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