Archive for the 'expectations' Category

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.

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My Paper “The Fisher Effect and the Financial Crisis of 2008” Is Now Available

Back in 2009 or 2010, I became intrigued by what seemed to me to be a consistent correlation between the tendency of the stock market to rise on news of monetary easing and potentially inflationary news. I suspected that there might be such a correlation because of my work on the Great Depression inspired by Earl Thompson, from whom I first learned about a monetary theory of the Great Depression very different from Friedman’s monetary theory expounded in his Monetary History of the United States. Thompson’s theory focused on disturbances in the gold market associated with the demonetization of gold during World War I and the attempt to restore the gold standard in the 1920s, which, by increasing the world demand for gold, was the direct cause of the deflation that led to the Great Depression.

I later came to discover that Ralph Hawtrey had already propounded Thompson’s theory in the 1920s almost a decade before the Great Depression started, and my friend and fellow student of Thompson, Ron Batchelder made a similar discovery about Gustave Cassel. Our shared recognition that Thompson’s seemingly original theory of the Great Depression had been anticipated by Hawtrey and Cassel led us to collaborate on our paper about Hawtrey and Cassel. As I began to see parallels between the financial fragility of the 1920s and the financial fragility that followed the housing bubble, I began to suspect that deflationary tendencies were also critical to the financial crisis of 2008.

So I began following daily fluctuations in the principal market estimate of expected inflation: the breakeven TIPS spread. I pretty quickly became persuaded that the correlation was powerful and meaningful, and I then collected data about TIPS spreads from 2003, when the Treasury began offering TIPS securities, to see if the correlation between expected inflation and asset prices had been present 2003 or was a more recent phenomenon.

My hunch was that the correlation would not be observed under normal macroeconomic conditions, because it is only when the expected yield from holding money approaches or exceeds the yield from holding real assets that an increase in expected inflation, by reducing the expected yield from holding money, would induce people to switch from holding money to holding assets, thereby driving up the value of assets.

And that’s what the data showed; the correlation between expected inflation and asset prices only emerged after in 2008 in the period after a recession started at the end of 2007, even before the start of the financial crisis exactly 10 years in September 2008. When I wrote up the paper and posted it (“The Fisher Effect Under Deflationary Expectations“), Scott Sumner, who had encouraged me to write up the results after I told him about my results, wrote a blogpost about the paper. Paul Krugman picked up on Scott’s post and wrote about it on his blog, generating a lot of interest in the paper.

Although I was confident that the data showed a strong correlation between inflation and stock prices after 2008, I was less confident that I had done the econometrics right, so I didn’t try to publish the original 2011 version of the paper. With Scott’s encouragement, I have continued to collected more data as time passed, confirming that the correlation remained even after the start of a recovery while short-term interest rates remained at or near the zero lower bound. The Mercatus Center whose Program on Monetary Policy is directed by Scott has just released the new version of the paper as a Working Paper. The paper can also be downloaded from SSRN.

Aside from longer time span covered, the new version of the paper has refined and extended the theoretical account for when and why a correlation between expected inflation and asset prices is likely be observed and when and why it is unlikely to be observed. I have also done some additional econometric testing beyond the basic ordinary least square (OLS) regression estimates originally presented, and explained why I think it is unlikely that more sophisticated econometric techniques such as an error-correction model would generate more reliable results than those generated by simple OLS regrissions. Perhaps in further work, I will attempt to actually construct an explicit error-correction model and compare the results using OLS and an error-correction model.

Here is the abstract of the new version of the paper.

This paper uses the Fisher equation relating the nominal interest rate to the real interest rate and
expected inflation to provide a deeper explanation of the financial crisis of 2008 and the subsequent recovery than attributing it to the bursting of the housing-price bubble. The paper interprets the Fisher equation as an equilibrium condition in which expected returns from holding real assets and cash are equalized. When inflation expectations decline, the return to holding cash rises relative to holding real assets. If nominal interest rates are above the zero lower bound, equilibrium is easily restored by adjustments in nominal interest rates and asset prices. But at the zero lower bound, nominal interest rates cannot fall, forcing the entire adjustment onto falling asset prices, thereby raising the expected real return from holding assets. Such an adjustment seems to have triggered the financial crisis of 2008, when the Federal Reserve delayed reducing nominal interest rates out of a misplaced fear of inflation in the summer of 2008 when the economy was already contracting rapidly. Using stock market price data and inflation-adjusted US Treasury securities data, the paper finds that, unlike the 2003–2007 period, when stock prices were uncorrelated with expected inflation, from 2008 through at least 2016, stock prices have been consistently and positively correlated with expected inflation.

Hayek, Radner and Rational-Expectations Equilibrium

In revising my paper on Hayek and Three Equilibrium Concepts, I have made some substantial changes to the last section which I originally posted last June. So I thought I would post my new updated version of the last section. The new version of the paper has not been submitted yet to a journal; I will give a talk about it at the colloquium on Economic Institutions and Market Processes at the NYU economics department next Monday. Depending on the reaction I get at the Colloquium and from some other people I will send the paper to, I may, or may not, post the new version on SSRN and submit to a journal.

In this section, I want to focus on a particular kind of intertemporal equilibrium: rational-expectations equilibrium. It is noteworthy that in his discussions of intertemporal equilibrium, Roy Radner assigns a  meaning to the term “rational-expectations equilibrium” very different from the one normally associated with that term. Radner describes a rational-expectations equilibrium as the equilibrium that results when some agents can make inferences about the beliefs of other agents when observed prices differ from the prices that the agents had expected. Agents attribute the differences between observed and expected prices to the superior information held by better-informed agents. As they assimilate the information that must have caused observed prices to deviate from their expectations, agents revise their own expectations accordingly, which, in turn, leads to further revisions in plans, expectations and outcomes.

There is a somewhat famous historical episode of inferring otherwise unknown or even secret information from publicly available data about prices. In 1954, one very rational agent, Armen Alchian, was able to identify which chemicals were being used in making the newly developed hydrogen bomb by looking for companies whose stock prices had risen too rapidly to be otherwise explained. Alchian, who spent almost his entire career at UCLA while moonlighting at the nearby Rand Corporation, wrote a paper at Rand listing the chemicals used in making the hydrogen bomb. When news of his unpublished paper reached officials at the Defense Department – the Rand Corporation (from whose files Daniel Ellsberg took the Pentagon Papers) having been started as a think tank with funding by the Department of Defense to do research on behalf of the U.S. military – the paper was confiscated from Alchian’s office at Rand and destroyed. (See Newhard’s paper for an account of the episode and a reconstruction of Alchian’s event study.)

But Radner also showed that the ability of some agents to infer the information on which other agents are causing prices to differ from the prices that had been expected does not necessarily lead to an equilibrium. The process of revising expectations in light of observed prices may not converge on a shared set of expectations of future prices based on common knowledge. Radner’s result reinforces Hayek’s insight, upon which I remarked above, that although expectations are equilibrating variables there is no economic mechanism that tends to bring expectations toward their equilibrium values. There is no feedback mechanism, corresponding to the normal mechanism for adjusting market prices in response to perceived excess demands or supplies, that operates on price expectations. The heavy lifting of bringing expectations into correspondence with what the future holds must be done by the agents themselves; the magic of the market goes only so far.

Although Radner’s conception of rational expectations differs from the more commonly used meaning of the term, his conception helps us understand the limitations of the conventional “rational expectations” assumption in modern macroeconomics, which is that the price expectations formed by the agents populating a model should be consistent with what the model itself predicts that those future prices will be. In this very restricted sense, I believe rational expectations is an important property of any model. If one assumes that the outcome expected by agents in a model is the equilibrium predicted by the model, then, under those expectations, the solution of the model ought to be the equilibrium of the model. If the solution of the model is somehow different from what agents in the model expect, then there is something really wrong with the model.

What kind of crazy model would have the property that correct expectations turn out not to be self-fulfilling? A model in which correct expectations are not self-fulfilling is a nonsensical model. But there is a huge difference between saying (a) that a model should have the property that correct expectations are self-fulfilling and saying (b) that the agents populating the model understand how the model works and, based know their knowledge of the model, form expectations of the equilibrium predicted by the model.

Rational expectations in the first sense is a minimal consistency property of an economic model; rational expectations in the latter sense is an empirical assertion about the real world. You can make such an assumption if you want, but you can’t credibly claim that it is a property of the real world. Whether it is a property of the real world is a matter of fact, not a methodological imperative. But the current sacrosanct status of rational expectations in modern macroeconomics has been achieved largely through methodological tyrannizing.

In his 1937 paper, Hayek was very clear that correct expectations are logically implied by the concept of an equilibrium of plans extending through time. But correct expectations are not a necessary, or even descriptively valid, characteristic of reality. Hayek also conceded that we don’t even have an explanation in theory of how correct expectations come into existence. He merely alluded to the empirical observation – perhaps not the most faithful description of empirical reality in 1937 – that there is an observed general tendency for markets to move toward equilibrium, implying that, over time, expectations somehow do tend to become more accurate.

It is worth pointing out that when the idea of rational expectations was introduced by John Muth (1961), he did so in the context of partial-equilibrium models in which the rational expectation in the model was the rational expectation of the equilibrium price in a particular market. The motivation for Muth to introduce the idea of a rational expectation was the cobweb-cycle model in which producers base current decisions about how much to produce for the following period on the currently observed price. But with a one-period time lag between production decisions and realized output, as is the case in agricultural markets in which the initial application of inputs does not result in output until a subsequent time period, it is easy to generate an alternating sequence of boom and bust, with current high prices inducing increased output in the following period, driving prices down, thereby inducing low output and high prices in the next period and so on.

Muth argued that rational producers would not respond to price signals in a way that led to consistently mistaken expectations, but would base their price expectations on more realistic expectations of what future prices would turn out to be. In his microeconomic work on rational expectations, Muth showed that the rational-expectation assumption was a better predictor of observed prices than the assumption of static expectations underlying the traditional cobweb-cycle model. So Muth’s rational-expectations assumption was based on a realistic conjecture of how real-world agents would actually form expectations. In that sense, Muth’s assumption was consistent with Hayek’s conjecture that there is an empirical tendency for markets to move toward equilibrium.

So, while Muth’s introduction of the rational-expectations hypothesis was an empirically progressive theoretical innovation, extending rational-expectations into the domain of macroeconomics has not been empirically progressive, rational-expectations models having consistently failed to generate better predictions than macro-models using other expectational assumptions. Instead, a rational-expectations axiom has been imposed as part of a spurious methodological demand that all macroeconomic models be “micro-founded.” But the deeper point – one that Hayek understood better than perhaps anyone else — is that there is a difference in kind between forming rational expectations about a single market price and forming rational expectations about the vector of n prices on the basis of which agents are choosing or revising their optimal intertemporal consumption and production plans.

It is one thing to assume that agents have some expert knowledge about the course of future prices in the particular markets in which they participate regularly; it is another thing entirely to assume that they have knowledge sufficient to forecast the course of all future prices and in particular to understand the subtle interactions between prices in one market and the apparently unrelated prices in another market. It is those subtle interactions that allow the kinds of informational inferences that, based on differences between expected and realized prices of the sort contemplated by Alchian and Radner, can sometimes be made. The former kind of knowledge is knowledge that expert traders might be expected to have; the latter kind of knowledge is knowledge that would be possessed by no one but a nearly omniscient central planner, whose existence was shown by Hayek to be a practical impossibility.

The key — but far from the only — error of the rational-expectations methodology that rules modern macroeconomics is that rational expectations somehow cause or bring about an intertemporal equilibrium. It is certainly a fact that people try very hard to use all the information available to them to predict what the future has in store, and any new bit of information not previously possessed will be rapidly assessed and assimilated and will inform a possibly revised set of expectations of the future. But there is no reason to think that this ongoing process of information gathering and processing and evaluation leads people to formulate correct expectations of the future or of future prices. Indeed, Radner proved that, even under strong assumptions, there is no necessity that the outcome of a process of information revision based on the observed differences between observed and expected prices leads to an equilibrium.

So it cannot be rational expectations that leads to equilibrium, On the contrary, rational expectations are a property of equilibrium. To speak of a “rational-expectations equilibrium” is to speak about a truism. There can be no rational expectations in the macroeconomic except in an equilibrium state, because correct expectations, as Hayek showed, is a defining characteristic of equilibrium. Outside of equilibrium, expectations cannot be rational. Failure to grasp that point is what led Morgenstern astray in thinking that Holmes-Moriarty story demonstrated the nonsensical nature of equilibrium. It simply demonstrated that Holmes and Moriarity were playing a non-repeated game in which an equilibrium did not exist.

To think about rational expectations as if it somehow results in equilibrium is nothing but a category error, akin to thinking about a triangle being caused by having angles whose angles add up to 180 degrees. The 180-degree sum of the angles of a triangle don’t cause the triangle; it is a property of the triangle.

Standard macroeconomic models are typically so highly aggregated that the extreme nature of the rational-expectations assumption is effectively suppressed. To treat all output as a single good (which involves treating the single output as both a consumption good and a productive asset generating a flow of productive services) effectively imposes the assumption that the only relative price that can ever change is the wage, so that all but one future relative prices are known in advance. That assumption effectively assumes away the problem of incorrect expectations except for two variables: the future price level and the future productivity of labor (owing to the productivity shocks so beloved of Real Business Cycle theorists).

Having eliminated all complexity from their models, modern macroeconomists, purporting to solve micro-founded macromodels, simply assume that there are just a couple of variables about which agents have to form their rational expectations. The radical simplification of the expectational requirements for achieving a supposedly micro-founded equilibrium belies the claim to have achieved anything of the sort. Whether the micro-foundational pretense affected — with apparently sincere methodological fervor — by modern macroeconomics is merely self-delusional or a deliberate hoax perpetrated on a generation of unsuspecting students is an interesting distinction, but a distinction lacking any practical significance.

Four score years since Hayek explained how challenging the notion of intertemporal equilibrium really is and the difficulties inherent in explaining any empirical tendency toward intertempral equilibrium, modern macroeconomics has succeeded in assuming all those difficulties out of existence. Many macroeconomists feel rather proud of what modern macroeconomics has achieved. I am not quite as impressed as they are.

 

Hayek and Rational Expectations

In this, my final, installment on Hayek and intertemporal equilibrium, I want to focus on a particular kind of intertemporal equilibrium: rational-expectations equilibrium. In his discussions of intertemporal equilibrium, Roy Radner assigns a meaning to the term “rational-expectations equilibrium” very different from the meaning normally associated with that term. Radner describes a rational-expectations equilibrium as the equilibrium that results when some agents are able to make inferences about the beliefs held by other agents when observed prices differ from what they had expected prices to be. Agents attribute the differences between observed and expected prices to information held by agents better informed than themselves, and revise their own expectations accordingly in light of the information that would have justified the observed prices.

In the early 1950s, one very rational agent, Armen Alchian, was able to figure out what chemicals were being used in making the newly developed hydrogen bomb by identifying companies whose stock prices had risen too rapidly to be explained otherwise. Alchian, who spent almost his entire career at UCLA while also moonlighting at the nearby Rand Corporation, wrote a paper for Rand in which he listed the chemicals used in making the hydrogen bomb. When people at the Defense Department heard about the paper – the Rand Corporation was started as a think tank largely funded by the Department of Defense to do research that the Defense Department was interested in – they went to Alchian, confiscated and destroyed the paper. Joseph Newhard recently wrote a paper about this episode in the Journal of Corporate Finance. Here’s the abstract:

At RAND in 1954, Armen A. Alchian conducted the world’s first event study to infer the fuel material used in the manufacturing of the newly-developed hydrogen bomb. Successfully identifying lithium as the fusion fuel using only publicly available financial data, the paper was seen as a threat to national security and was immediately confiscated and destroyed. The bomb’s construction being secret at the time but having since been partially declassified, the nuclear tests of the early 1950s provide an opportunity to observe market efficiency through the dissemination of private information as it becomes public. I replicate Alchian’s event study of capital market reactions to the Operation Castle series of nuclear detonations in the Marshall Islands, beginning with the Bravo shot on March 1, 1954 at Bikini Atoll which remains the largest nuclear detonation in US history, confirming Alchian’s results. The Operation Castle tests pioneered the use of lithium deuteride dry fuel which paved the way for the development of high yield nuclear weapons deliverable by aircraft. I find significant upward movement in the price of Lithium Corp. relative to the other corporations and to DJIA in March 1954; within three weeks of Castle Bravo the stock was up 48% before settling down to a monthly return of 28% despite secrecy, scientific uncertainty, and public confusion surrounding the test; the company saw a return of 461% for the year.

Radner also showed that the ability of some agents to infer the information on which other agents are causing prices to differ from the prices that had been expected does not necessarily lead to an equilibrium. The process of revising expectations in light of observed prices may not converge on a shared set of expectations of the future based on commonly shared knowledge.

So rather than pursue Radner’s conception of rational expectations, I will focus here on the conventional understanding of “rational expectations” in modern macroeconomics, which is that the price expectations formed by the agents in a model should be consistent with what the model itself predicts that those future prices will be. In this very restricted sense, I believe rational expectations is a very important property that any model ought to have. It simply says that a model ought to have the property that if one assumes that the agents in a model expect the equilibrium predicted by the model, then, given those expectations, the solution of the model will turn out to be the equilibrium of the model. This property is a consistency and coherence property that any model, regardless of its substantive predictions, ought to have. If a model lacks this property, there is something wrong with the model.

But there is a huge difference between saying that a model should have the property that correct expectations are self-fulfilling and saying that agents are in fact capable of predicting the equilibrium of the model. Assuming the former does not entail the latter. What kind of crazy model would have the property that correct expectations are not self-fulfilling? I mean think about: a model in which correct expectations are not self-fulfilling is a nonsense model.

But demanding that a model not spout out jibberish is very different from insisting that the agents in the model necessarily have the capacity to predict what the equilibrium of the model will be. Rational expectations in the first sense is a minimal consistency property of an economic model; rational expectations in the latter sense is an empirical assertion about the real world. You can make such an assumption if you want, but you can’t claim that it is a property of the real world. Whether it is a property of the real world is a matter of fact, not a matter of methodological fiat. But methodological fiat is what rational expectations has become in macroeconomics.

In his 1937 paper on intertemporal equilibrium, Hayek was very clear that correct expectations are logically implied by the concept of an equilibrium of plans extending through time. But correct expectations are not a necessary, or even descriptively valid, characteristic of reality. Hayek also conceded that we don’t even have an explanation in theory of how correct expectations come into existence. He merely alluded to the empirical observation – perhaps not the most accurate description of empirical reality in 1937 – that there is an observed general tendency for markets to move toward equilibrium, implying that over time expectations do tend to become more accurate.

It is worth pointing out that when the idea of rational expectations was introduced by John Muth in the early 1960s, he did so in the context of partial-equilibrium models in which the rational expectation in the model was the rational expectation of the equilibrium price in a paraticular market. The motivation for Muth to introduce the idea of a rational expectation was idea of a cobweb cycle in which producers simply assume that the current price will remain at whatever level currently prevails. If there is a time lag between production, as in agricultural markets between the initial application of inputs and the final yield of output, it is easy to generate an alternating sequence of boom and bust, with current high prices inducing increased output in the following period, driving prices down, thereby inducing low output and high prices in the next period and so on.

Muth argued that rational producers would not respond to price signals in a way that led to consistently mistaken expectations, but would base their price expectations on more realistic expectations of what future prices would turn out to be. In his microeconomic work on rational expectations, Muth showed that the rational-expectation assumption was a better predictor of observed prices than the assumption of static expectations underlying the traditional cobweb-cycle model. So Muth’s rational-expectations assumption was based on a realistic conjecture of how real-world agents would actually form expectations. In that sense, Muth’s assumption was consistent with Hayek’s conjecture that there is an empirical tendency for markets to move toward equilibrium.

So while Muth’s introduction of the rational-expectations hypothesis was an empirically progressive theoretical innovation, extending rational-expectations into the domain of macroeconomics has not been empirically progressive, rational expectations models having consistently failed to generate better predictions than macro-models using other expectational assumptions. Instead, a rational-expectations axiom has been imposed as part of a spurious methodological demand that all macroeconomic models be “micro-founded.” But the deeper point – a point that Hayek understood better than perhaps anyone else — is that there is a huge difference in kind between forming rational expectations about a single market price and forming rational expectations about the vector of n prices on the basis of which agents are choosing or revising their optimal intertemporal consumption and production plans.

It is one thing to assume that agents have some expert knowledge about the course of future prices in the particular markets in which they participate regularly; it is another thing entirely to assume that they have knowledge sufficient to forecast the course of all future prices and in particular to understand the subtle interactions between prices in one market and the apparently unrelated prices in another market. The former kind of knowledge is knowledge that expert traders might be expected to have; the latter kind of knowledge is knowledge that would be possessed by no one but a nearly omniscient central planner, whose existence was shown by Hayek to be a practical impossibility.

Standard macroeconomic models are typically so highly aggregated that the extreme nature of the rational-expectations assumption is effectively suppressed. To treat all output as a single good (which involves treating the single output as both a consumption good and a productive asset generating a flow of productive services) effectively imposes the assumption that the only relative price that can ever change is the wage, so that all but one future relative prices are known in advance. That assumption effectively assumes away the problem of incorrect expectations except for two variables: the future price level and the future productivity of labor (owing to the productivity shocks so beloved of Real Business Cycle theorists). Having eliminated all complexity from their models, modern macroeconomists, purporting to solve micro-founded macromodels, simply assume that there is but one or at most two variables about which agents have to form their rational expectations.

Four score years since Hayek explained how challenging the notion of intertemporal equilibrium really is and the difficulties inherent in explaining any empirical tendency toward intertempral equilibrium, modern macroeconomics has succeeded in assuming all those difficulties out of existence. Many macroeconomists feel rather proud of what modern macroeconomics has achieved. I am not quite as impressed as they are.

Hayek and Temporary Equilibrium

In my three previous posts (here, here, and here) about intertemporal equilibrium, I have been emphasizing that the defining characteristic of an intertemporal equilibrium is that agents all share the same expectations of future prices – or at least the same expectations of those future prices on which they are basing their optimizing plans – over their planning horizons. At a given moment at which agents share the same expectations of future prices, the optimizing plans of the agents are consistent, because none of the agents would have any reason to change his optimal plan as long as price expectations do not change, or are not disappointed as a result of prices turning out to be different from what they had been expected to be.

The failure of expected prices to be fulfilled would therefore signify that the information available to agents in forming their expectations and choosing optimal plans conditional on their expectations had been superseded by newly obtained information. The arrival of new information can thus be viewed as a cause of disequilibrium as can any difference in information among agents. The relationship between information and equilibrium can be expressed as follows: differences in information or differences in how agents interpret information leads to disequilibrium, because those differences lead agents to form differing expectations of future prices.

Now the natural way to generalize the intertemporal equilibrium model is to allow for agents to have different expectations of future prices reflecting their differences in how they acquire, or in how they process, information. But if agents have different information, so that their expectations of future prices are not the same, the plans on which agents construct their subjectively optimal plans will be inconsistent and incapable of implementation without at least some revisions. But this generalization seems incompatible with the equilibrium of optimal plans, prices and price expectations described by Roy Radner, which I have identified as an updated version of Hayek’s concept of intertemporal equilibrium.

The question that I want to explore in this post is how to reconcile the absence of equilibrium of optimal plans, prices, and price expectations, with the intuitive notion of market clearing that we use to analyze asset markets and markets for current delivery. If markets for current delivery and for existing assets are in equilibrium in the sense that prices are adjusting in those markets to equate demand and supply in those markets, how can we understand the idea that  the optimizing plans that agents are seeking to implement are mutually inconsistent?

The classic attempt to explain this intermediate situation which partially is and partially is not an equilibrium, was made by J. R. Hicks in 1939 in Value and Capital when he coined the term “temporary equilibrium” to describe a situation in which current prices are adjusting to equilibrate supply and demand in current markets even though agents are basing their choices of optimal plans to implement over time on different expectations of what prices will be in the future. The divergence of the price expectations on the basis of which agents choose their optimal plans makes it inevitable that some or all of those expectations won’t be realized, and that some, or all, of those agents won’t be able to implement the optimal plans that they have chosen, without at least some revisions.

In Hayek’s early works on business-cycle theory, he argued that the correct approach to the analysis of business cycles must be analyzed as a deviation by the economy from its equilibrium path. The problem that he acknowledged with this approach was that the tools of equilibrium analysis could be used to analyze the nature of the equilibrium path of an economy, but could not easily be deployed to analyze how an economy performs once it deviates from its equilibrium path. Moreover, cyclical deviations from an equilibrium path tend not to be immediately self-correcting, but rather seem to be cumulative. Hayek attributed the tendency toward cumulative deviations from equilibrium to the lagged effects of monetary expansion which cause cumulative distortions in the capital structure of the economy that lead at first to an investment-driven expansion of output, income and employment and then later to cumulative contractions in output, income, and employment. But Hayek’s monetary analysis was never really integrated with the equilibrium analysis that he regarded as the essential foundation for a theory of business cycles, so the monetary analysis of the cycle remained largely distinct from, if not inconsistent with, the equilibrium analysis.

I would suggest that for Hayek the Hicksian temporary-equilibrium construct would have been the appropriate theoretical framework within which to formulate a monetary analysis consistent with equilibrium analysis. Although there are hints in the last part of The Pure Theory of Capital that Hayek was thinking along these lines, I don’t believe that he got very far, and he certainly gave no indication that he saw in the Hicksian method the analytical tool with which to weave the two threads of his analysis.

I will now try to explain how the temporary-equilibrium method makes it possible to understand  the conditions for a cumulative monetary disequilibrium. I make no attempt to outline a specifically Austrian or Hayekian theory of monetary disequilibrium, but perhaps others will find it worthwhile to do so.

As I mentioned in my previous post, agents understand that their price expectations may not be realized, and that their plans may have to be revised. Agents also recognize that, given the uncertainty underlying all expectations and plans, not all debt instruments (IOUs) are equally reliable. The general understanding that debt – promises to make future payments — must be evaluated and assessed makes it profitable for some agents to specialize in in debt assessment. Such specialists are known as financial intermediaries. And, as I also mentioned previously, the existence of financial intermediaries cannot be rationalized in the ADM model, because, all contracts being made in period zero, there can be no doubt that the equilibrium exchanges planned in period zero will be executed whenever and exactly as scheduled, so that everyone’s promise to pay in time zero is equally good and reliable.

For our purposes, a particular kind of financial intermediary — banks — are of primary interest. The role of a bank is to assess the quality of the IOUs offered by non-banks, and select from the IOUs offered to them those that are sufficiently reliable to be accepted by the bank. Once a prospective borrower’s IOU is accepted, the bank exchanges its own IOU for the non-bank’s IOU. No non-bank would accept a non-bank’s IOU, at least not on terms as favorable as those on which the bank offers in accepting an IOU. In return for the non-bank IOU, the bank credits the borrower with a corresponding amount of its own IOUs, which, because the bank promises to redeem its IOUs for the numeraire commodity on demand, is generally accepted at face value.

Thus, bank debt functions as a medium of exchange even as it enables non-bank agents to make current expenditures they could not have made otherwise if they can demonstrate to the bank that they are sufficiently likely to repay the loan in the future at agreed upon terms. Such borrowing and repayments are presumably similar to the borrowing and repayments that would occur in the ADM model unmediated by any financial intermediary. In assessing whether a prospective borrower will repay a loan, the bank makes two kinds of assessments. First, does the borrower have sufficient income-earning capacity to generate enough future income to make the promised repayments that the borrower would be committing himself to make? Second, should the borrower’s future income, for whatever reason, turn out to be insufficient to finance the promised repayments, does the borrower have collateral that would allow the bank to secure repayment from the collateral offered as security? In making both kinds of assessments the bank has to form an expectation about the future — the future income of the borrower and the future value of the collateral.

In a temporary-equilibrium context, the expectations of future prices held by agents are not the same, so the expectations of future prices of at least some agents will not be accurate, and some agents won’tbe able to execute their plans as intended. Agents that can’t execute their plans as intended are vulnerable if they have incurred future obligations based on their expectations of future prices that exceed their repayment capacity given the future prices that are actually realized. If they have sufficient wealth — i.e., if they have asset holdings of sufficient value — they may still be able to repay their obligations. However, in the process they may have to sell assets or reduce their own purchases, thereby reducing the income earned by other agents. Selling assets under pressure of obligations coming due is almost always associated with selling those assets at a significant loss, which is precisely why it usually preferable to finance current expenditure by borrowing funds and making repayments on a fixed schedule than to finance the expenditure by the sale of assets.

Now, in adjusting their plans when they observe that their price expectations are disappointed, agents may respond in two different ways. One type of adjustment is to increase sales or decrease purchases of particular goods and services that they had previously been planning to purchase or sell; such marginal adjustments do not fundamentally alter what agents are doing and are unlikely to seriously affect other agents. But it is also possible that disappointed expectations will cause some agents to conclude that their previous plans are no longer sustainable under the conditions in which they unexpectedly find themselves, so that they must scrap their old plans replacing them with completely new plans instead. In the latter case, the abandonment of plans that are no longer viable given disappointed expectations may cause other agents to conclude that the plans that they had expected to implement are no longer profitable and must be scrapped.

When agents whose price expectations have been disappointed respond with marginal adjustments in their existing plans rather than scrapping them and replacing them with new ones, a temporary equilibrium with disappointed expectations may still exist and that equilibrium may be reached through appropriate price adjustments in the markets for current delivery despite the divergent expectations of future prices held by agents. Operation of the price mechanism may still be able to achieve a reconciliation of revised but sub-optimal plans. The sub-optimal temporary equilibrium will be inferior to the allocation that would have resulted had agents all held correct expectations of future prices. Nevertheless, given a history of incorrect price expectations and misallocations of capital assets, labor, and other factors of production, a sub-optimal temporary equilibrium may be the best feasible outcome.

But here’s the problem. There is no guarantee that, when prices turn out to be very different from what they were expected to be, the excess demands of agents will adjust smoothly to changes in current prices. A plan that was optimal based on the expectation that the price of widgets would be $500 a unit may well be untenable at a price of $120 a unit. When realized prices are very different from what they had been expected to be, those price changes can lead to discontinuous adjustments, violating a basic assumption — the continuity of excess demand functions — necessary to prove the existence of an equilibrium. Once output prices reach some minimum threshold, the best response for some firms may be to shut down, the excess demand for the product produced by the firm becoming discontinuous at the that threshold price. The firms shutting down operations may be unable to repay loans they had obligated themselves to repay based on their disappointed price expectations. If ownership shares in firms forced to cease production are held by households that have predicated their consumption plans on prior borrowing and current repayment obligations, the ability of those households to fulfill their obligations may be compromised once those firms stop paying out the expected profit streams. Banks holding debts incurred by firms or households that borrowers cannot service may find that their own net worth is reduced sufficiently to make the banks’ own debt unreliable, potentially causing a breakdown in the payment system. Such effects are entirely consistent with a temporary-equilibrium model if actual prices turn out to be very different from what agents had expected and upon which they had constructed their future consumption and production plans.

Sufficiently large differences between expected and actual prices in a given period may result in discontinuities in excess demand functions once prices reach critical thresholds, thereby violating the standard continuity assumptions on which the existence of general equilibrium depends under the fixed-point theorems that are the lynchpin of modern existence proofs. C. J. Bliss made such an argument in a 1983 paper (“Consistent Temporary Equilibrium” in the volume Modern Macroeconomic Theory edited by  J. P. Fitoussi) in which he also suggested, as I did above, that the divergence of individual expectations implies that agents will not typically regard the debt issued by other agents as homogeneous. Bliss therefore posited the existence of a “Financier” who would subject the borrowing plans of prospective borrowers to an evaluation process to determine if the plan underlying the prospective loan sought by a borrower was likely to generate sufficient cash flow to enable the borrower to repay the loan. The role of the Financier is to ensure that the plans that firms choose are based on roughly similar expectations of future prices so that firms will not wind up acting on price expectations that must inevitably be disappointed.

I am unsure how to understand the function that Bliss’s Financier is supposed to perform. Presumably the Financier is meant as a kind of idealized companion to the Walrasian auctioneer rather than as a representation of an actual institution, but the resemblance between what the Financier is supposed to do and what bankers actually do is close enough to make it unclear to me why Bliss chose an obviously fictitious character to weed out business plans based on implausible price expectations rather than have the role filled by more realistic characters that do what their real-world counterparts are supposed to do. Perhaps Bliss’s implicit assumption is that real-world bankers do not constrain the expectations of prospective borrowers sufficiently to suggest that their evaluation of borrowers would increase the likelihood that a temporary equilibrium actually exists so that only an idealized central authority could impose sufficient consistency on the price expectations to make the existence of a temporary equilibrium likely.

But from the perspective of positive macroeconomic and business-cycle theory, explicitly introducing banks that simultaneously provide an economy with a medium of exchange – either based on convertibility into a real commodity or into a fiat base money issued by the monetary authority – while intermediating between ultimate borrowers and ultimate lenders seems to be a promising way of modeling a dynamic economy that sometimes may — and sometimes may not — function at or near a temporary equilibrium.

We observe economies operating in the real world that sometimes appear to be functioning, from a macroeconomic perspective, reasonably well with reasonably high employment, increasing per capita output and income, and reasonable price stability. At other times, these economies do not function well at all, with high unemployment and negative growth, sometimes with high rates of inflation or with deflation. Sometimes, these economies are beset with financial crises in which there is a general crisis of solvency, and even apparently solvent firms are unable to borrow. A macroeconomic model should be able to account in some way for the diversity of observed macroeconomic experience. The temporary equilibrium paradigm seems to offer a theoretical framework capable of accounting for this diversity of experience and for explaining at least in a very general way what accounts for the difference in outcomes: the degree of congruence between the price expectations of agents. When expectations are reasonably consistent, the economy is able to function at or near a temporary equilibrium which is likely to exist. When expectations are highly divergent, a temporary equilibrium may not exist, and even if it does, the economy may not be able to find its way toward the equilibrium. Price adjustments in current markets may be incapable of restoring equilibrium inasmuch as expectations of future prices must also adjust to equilibrate the economy, there being no market mechanism by which equilibrium price expectations can be adjusted or restored.

This, I think, is the insight underlying Axel Leijonhufvud’s idea of a corridor within which an economy tends to stay close to an equilibrium path. However if the economy drifts or is shocked away from its equilibrium time path, the stabilizing forces that tend to keep an economy within the corridor cease to operate at all or operate only weakly, so that the tendency for the economy to revert back to its equilibrium time path is either absent or disappointingly weak.

The temporary-equilibrium method, it seems to me, might have been a path that Hayek could have successfully taken in pursuing the goal he had set for himself early in his career: to reconcile equilibrium-analysis with a theory of business cycles. Why he ultimately chose not to take this path is a question that, for now at least, I will leave to others to try to answer.

Roy Radner and the Equilibrium of Plans, Prices and Price Expectations

In this post I want to discuss Roy Radner’s treatment of an equilibrium of plans, prices, and price expectations (EPPPE) and its relationship to Hayek’s conception of intertemporal equilibrium, of which Radner’s treatment is a technically more sophisticated version. Although I seen no evidence that Radner was directly influenced by Hayek’s work, I consider Radner’s conception of EPPPE to be a version of Hayek’s conception of intertemporal equilibrium, because it captures essential properties of Hayek’s conception of intertemporal equilibrium as a situation in which agents independently formulate their own optimizing plans based on the prices that they actually observe – their common knowledge – and on the future prices that they expect to observe over the course of their planning horizons. While currently observed prices are common knowledge – not necessarily a factual description of economic reality but not an entirely unreasonable simplifying assumption – the prices that individual agents expect to observe in the future are subjective knowledge based on whatever common or private knowledge individuals may have and whatever methods they may be using to form their expectations of the prices that will be observed in the future. An intertemporal equilibrium refers to a set of decentralized plans that are both a) optimal from the standpoint of every agent’s own objectives given their common knowledge of current prices and their subjective expectations of future prices and b) mutually consistent.

If an agent has chosen an optimal plan given current and expected future prices, that plan will not be changed unless the agent acquires new information that renders the existing plan sub-optimal relative to the new information. Otherwise, there would be no reason for the agent to deviate from an optimal plan. The new information that could cause an agent to change a formerly optimal plan would either affect the preferences of the agent, the technology available to the agent, or would somehow be reflected in current prices or in expected future prices. But it seems improbable that there could be a change in preferences or technology would not also be reflected in current or expected future prices. So absent a change in current or expected future prices, there would seem to be almost no likelihood that an agent would deviate from a plan that was optimal given current prices and the future prices expected by the agent.

The mutual consistency of the optimizing plans of independent agents therefore turns out to be equivalent to the condition that all agents observe the same current prices – their common knowledge – and have exactly the same forecasts of the future prices upon which they have relied in choosing their optimal plans. Even should their forecasts of future prices turn out to be wrong, at the moment before their forecasts of future prices were changed or disproved by observation, their plans were still mutually consistent relative to the information on which their plans had been chosen. The failure of the equilibrium to be maintained could be attributed to a change in information that meant that the formerly optimal plans were no longer optimal given the newly acquired information. But until the new information became available, the mutual consistency of optimal plans at that (fleeting) moment signified an equilibrium state. Thus, the defining characteristic of an intertemporal equilibrium in which current prices are common knowledge is that all agents share the same expectations of the future prices on which their optimal plans have been based.

There are fundamental differences between the Arrow-Debreu-McKenzie (ADM) equilibrium and the EPPPE. One difference worth mentioning is that, under the standard assumptions of the ADM model, the equilibrium is Pareto-optimal, and any Pareto-optimum allocation, by a suitable redistribution of initial endowments, could be achieved as a general equilibrium (two welfare theorems). These results do not generally hold for EPPPE, because, in contrast to the ADM model, it is possible for agents in EPPPE to acquire additional information over time, not only passively, but by investing resources in the production of information. Investing resources in the production of information can cause inefficiency in two ways: first, by creating non-convexities (owing to start-up costs in information gathering activities) that are inconsistent with the uniform competitive prices characteristic of the ADM equilibrium, and second, by creating incentives to devote resources to produce information whose value is derived from profits in trading with less well-informed agents. The latter source of inefficiency was discovered by Jack Hirshleifer in his classic 1971 paper, which I have written about in several previous posts (here, here, here, and here).

But the important feature of Radner’s EPPPE that I want to emphasize here — and what radically distinguishes it from the ADM equilibrium — is its fragility. Unlike the ADM equilibrium which is established once and forever at time zero of a model in which all production and consumption starts in period one, the EPPPE, even if it ever exists, is momentary, and is subject to unraveling whenever there is a change in the underlying information upon which current prices and expected future prices depend, and upon which agents, in choosing their optimal plans, rely. Time is not just, as it is in the ADM model, an appendage to the EPPPE, and, as a result, EPPPE can account for many phenomena, practices, and institutions that are left out of the ADM model.

The two differences that are most relevant in this context are the existence of stock markets in which shares of firms are traded based on expectations of the future net income streams associated with those firms, and the existence of a medium of exchange supplied by private financial intermediaries known as banks. In the ADM model in which all transactions are executed in time zero, in advance of all the actual consumption and production activities determined by those transactions, there would be no reason to hold, or to supply, a medium of exchange. The ADM equilibrium allows for agents to borrow or lend at equilibrium interest rates to optimize the time profiles of their consumption relative to their endowments and the time profiles of their earnings. Since all such transactions are consummated in time zero, and since, through some undefined process, the complete solvency and the integrity of all parties to all transactions is ascertained in time zero, the probability of a default on any loan contracted at time zero is zero. As a result, each agent faces a single intertemporal budget constraint at time zero over all periods from 1 to n. Walras’s Law therefore holds across all time periods for this intertemporal budget constraint, each agent transacting at the same prices in each period as every other agent does.

Once an equilibrium price vector is established in time zero, each agent knows that his optimal plan based on that price vector (which is the common knowledge of all agents) will be executed over time exactly as determined in time zero. There is no reason for any exchange of ownership shares in firms, the future income streams from each firm being known in advance.

The ADM equilibrium is a model of an economic process very different from Radner’s EPPPE, because in EPPPE, agents have no reason to assume that their current plans, even if they are momentarily both optimal and mutually consistent with the plans of all other agents, will remain optimal and consistent with the plans of all other agents. New information can arrive or be produced that will necessitate a revision in plans. Because even equilibrium plans are subject to revision, agents must take into account the solvency and credit worthiness of counterparties with whom they enter into transactions. The potentially imperfect credit-worthiness of at least some agents enables certain financial intermediaries (aka banks) to provide a service by offering to exchange their debt, which is widely considered to be more credit-worthy than the debt of ordinary agents, to agents seeking to borrow to finance purchases of either consumption or investment goods. Many agents seeking to borrow therefore prefer exchanging their debt for bank debt, bank debt being acceptable by other agents at face value. In addition, because the acquisition of new information is possible, there is a reason for agents to engage in speculative trades of commodities or assets. Such assets include ownership shares of firms, and agents may revise their valuations of those firms as they revise their expectations about future prices and their expectations about the revised plans of those firms in response to newly acquired information.

I will discuss the special role of banks at greater length in my next post on temporary equilibrium. But for now, I just want to underscore a key point: in the EPPE, unless all agents have the same expectations of future prices, Walras’s Law need not hold. The proof that Walras’s holds depends on the assumption that individual plans to buy and sell are based on the assumption that every agent buys or sells each commodity at the same price that every other transactor buys  or sells that commodity. But in the intertemporal context, in which only current, not future prices, are observed, plans for current and future prices are made based on expectations about future prices. If agents don’t share the same expectations about future prices, agents making plans for future purchases based on overly optimistic expectations about the prices at which they will be able to sell, may make commitments to buy in the future (or commitment to repay loans to finance purchases in the present) that they will be unable to discharge. Reneging on commitments to buy in the future or to repay obligations incurred in the present may rule out the existence of even a temporary equilibrium in the future.

Finally, let me add a word about Radner’s terminology. In his 1987 entry on “Uncertainty and General Equilibrium” for the New Palgrave Dictionary of Economics, (Here is a link to the revised version on line), Radner writes:

A trader’s expectations concern both future environmental events and future prices. Regarding expectations about future environmental events, there is no conceptual problem. According to the Expected Utility Hypothesis, each trader is characterized by a subjective probability measure on the set of complete histories of the environment. Since, by definition, the evolution of the environment is exogenous, a trader’s conditional probability of a future event, given the information to date, is well defined.

It is not so obvious how to proceed with regard to trader’s expectations about future prices. I shall contrast two possible approaches. In the first, which I shall call the perfect foresight approach, let us assume that the behaviour of traders is such as to determine, for each complete history of the environment, a unique corresponding sequence of price system[s]. . .

Thus, the perfect foresight approach implies that, in equilibrium, traders have common price expectation functions. These price expectation functions indicate, for each date-event pair, what the equilibrium price system would be in the corresponding market at that date event pair. . . . [I]t follows that, in equilibrium the traders would have strategies (plans) such that if these strategies were carried out, the markets would be cleared at each date-event pair. Call such plans consistent. A set of common price expectations and corresponding consistent plans is called an equilibrium of plans, prices, and price expectations.

My only problem with Radner’s formulation here is that he is defining his equilibrium concept in terms of the intrinsic capacity of the traders to predict prices rather the simple fact that traders form correct expectations. For purposes of the formal definition of EPPE, it is irrelevant whether traders predictions of future prices are correct because they are endowed with the correct model of the economy or because they are all lucky and randomly have happened simultaneously to form the same expectations of future prices. Radner also formulates an alternative version of his perfect-foresight approach in which agents don’t all share the same information. In such cases, it becomes possible for traders to make inferences about the environment by observing prices differ from what they had expected.

The situation in which traders enter the market with different non-price information presents an opportunity for agents to learn about the environment from prices, since current prices reflect, in a possibly complicated manner, the non-price information signals received by the various agents. To take an extreme example, the “inside information” of a trader in a securities market may lead him to bid up the price to a level higher than it otherwise would have been. . . . [A]n astute market observer might be able to infer that an insider has obtained some favourable information, just by careful observation of the price movement.

The ability to infer non-price information from otherwise inexplicable movements in prices leads Radner to define a concept of rational expectations equilibrium.

[E]conomic agents have the opportunity to revise their individual models in the light of observations and published data. Hence, there is a feedback from the true relationship to the individual models. An equilibrium of this system, in which the individual models are identical with the true model, is called a rational expectations equilibrium. This concept of equilibrium is more subtle, of course, that the ordinary concept of equilibrium of supply and demand. In a rational expectations equilibrium, not only are prices determined so as to equate supply and demand, but individual economic agents correctly perceive the true relationship between the non-price information received by the market participants and the resulting equilibrium market prices.

Though this discussion is very interesting from several theoretical angles, as an explanation of what is entailed by an economic equilibrium, it misses the key point, which is the one that Hayek identified in his 1928 and (especially) 1937 articles mentioned in my previous posts. An equilibrium corresponds to a situation in which all agents have identical expectations of the future prices upon which they are making optimal plans given the commonly observed current prices and the expected future prices. If all agents are indeed formulating optimal plans based on the information that they have at that moment, their plans will be mutually consistent and will be executable simultaneously without revision as long as the state of their knowledge at that instant does not change. How it happened that they arrived at identical expectations — by luck chance or supernatural powers of foresight — is irrelevant to that definition of equilibrium. Radner does acknowledge that, under the perfect-foresight approach, he is endowing economic agents with a wildly unrealistic powers of imagination and computational capacity, but from his exposition, I am unable to decide whether he grasped the subtle but crucial point about the irrelevance of an assumption about the capacities of agents to the definition of EPPPE.

Although it is capable of describing a richer set of institutions and behavior than is the Arrow-Debreu model, the perfect-foresight approach is contrary to the spirit of much of competitive market theory in that it postulates that individual traders must be able to forecast, in some sense, the equilibrium prices that will prevail in the future under all alternative states of the environment. . . .[T]his approach . . . seems to require of the traders a capacity for imagination and computation far beyond what is realistic. . . .

These last considerations lead us in a different direction, which I shall call the bounded rationality approach. . . . An example of the bounded-rationality approach is the theory of temporary equilibrium.

By eschewing any claims about the rationality of the agents or their computational powers, one can simply talk about whether agents do or do not have identical expectations of future prices and what the implications of those assumptions are. When expectations do agree, there is at least a momentary equilibrium of plans, prices and price expectations. When they don’t agree, the question becomes whether even a temporary equilibrium exists and what kind of dynamic process is implied by the divergence of expectations. That it seems to me would be a fruitful way forward for macroeconomics to follow. In my next post, I will discuss some of the characteristics and implications of a temporary-equilibrium approach to macroeconomics.

 

Correct Foresight, Perfect Foresight, and Intertemporal Equilibrium

In my previous post, I discussed Hayek’s path-breaking insight into the meaning of intertemporal equilibrium. His breakthrough was to see that an equilibrium can be understood not as a stationary state in which nothing changes, but as a state in which decentralized plans are both optimal from the point of view of the individuals formulating the plans and mutually consistent, so that the individually optimal plans, at least potentially, could be simultaneously executed. In the simple one-period model, the plans of individuals extending over a single-period time horizon are constrained by the necessary equality for each agent between the value of all planned purchases and the value of all planned sales in that period. A single-period or stationary equilibrium, if it exists, is characterized by a set of prices such that the optimal plans corresponding to that set of prices such that total amount demanded for each product equals the total amount supplied for each product. Thus, an equilibrium price vector has the property that every individual is choosing optimally based on the choice criteria and the constraints governing the decisions for each individual and that those individually optimal choices are mutually consistent, that mutual consistency being manifested in the equality of the total amount demanded and the total amount supplied of each product in that single period.

The problem posed by the concept of intertemporal equilibrium is how to generalize the single-period notion of an equilibrium as a vector of all the observed prices of goods and services actually traded in that single period into a multi-period concept in which the prices on which optimal choices depend include both the actual prices of goods traded in the current period as well as the prices of goods and services that agents plan to buy or sell only in some future time period. In an intertemporal context, the prices on the basis of which optimal plans are chosen cannot be just those prices at which transactions are being executed in the current period; the relevant set of prices must also include those prices at which transactions already being planned in the current period will be executed. Because even choices about transactions today may depend on the prices at which future transactions will take place, future prices can affect not only future demands and supplies they can also affect current demands and supplies.

But because prices in future periods are typically not observable by individuals in the present, it is not observed — but expected — future prices on the basis of which individual agents are making the optimal choices reflected in their intertemporal plans. And insofar as optimal plans depend on expected future prices, those optimal plans can be mutually consistent only if they are based on the same expected future prices, because if their choices are based on different expected future prices, then it is not possible that all expectations are realized. If the expectations of at least one agent, and probably of many agents, will be disappointed, implying that the plans of at least one and probably of many agents will not be optimized and will have to be revised.

The recognition that the mutual consistency of optimal plans requires individuals to accurately foresee the future prices upon which their optimal choices are based suggested that individual agents must be endowed with remarkable capacities to foresee the future. To assume that all individual agents would be endowed with the extraordinary ability to foresee correctly all the future prices relevant to their optimal choices about their intertemporal plans seemed an exceedingly unrealistic assumption on which to premise an economic model.

This dismissive attitude toward the concept of intertemporal equilibrium and the seemingly related assumption of “perfect foresight” necessary for an intertemporal equilibrium to exist was stridently expressed by Oskar Morgenstern in his famous 1935 article “Perfect Foresight and Economic Equilibrium.”

The impossibly high claims which are attributed to the intellectual efficiency of the economic subject immediately indicate that there are included in this equilibrium system not ordinary men, but rather, at least to one another, exactly equal demi-gods, in case the claim of complete foresight is fulfilled. If this is the case, there is, of course, nothing more to be done. If “full” or “perfect” foresight is to provide the basis of the theory of equilibrium in the strictly specified sense, and in the meaning obviously intended by the economic authors, then, a completely meaningless assumption is being considered. If limitations are introduced in such a way that the perfection of foresight is not reached, then these limitations are to be stated very precisely. They would have to be so narrowly drawn that the fundamental aim of producing ostensibly full rationality of the system by means of high, de facto unlimited, foresight, would be lost. For the theoretical economist, there is no way out of this dilemma. ln this discussion, “full” and “perfect” foresight are not only used synonymously, but both are employed, moreover, in the essentialIy more exact sense of limitlessness. This expression would have to be preferred because with the words “perfect” or “imperfect”, there arise superficial valuations which play no role here at all.

Morgenstern then went on to make an even more powerful attack on the idea of perfect foresight: that the idea is itself self-contradictory. Interestingly, he did so by positing an example that would figure in Morgenstern’s later development of game theory with his collaborator John von Neumann (and, as we now know, with his research assistant who in fact was his mathematical guide and mentor, Abraham Wald, fcredited as a co-author of The Theory of Games and Economic Behavior).

Sherlock Holmes, pursued by his opponent, Moriarity, leaves London for Dover. The train stops at a station on the way, and he alights there rather than traveling on to Dover. He has seen Moriarity at the railway station, recognizes that he is very clever and expects that Moriarity will take a faster special train in order to catch him in Dover. Holmes’ anticipation turns out to be correct. But what if Moriarity had been still more clever, had estimated Holmes’ mental abilities better and had foreseen his actions accordingly? Then, obviously, he would have traveled to the intermediate station. Holmes, again, would have had to calculate that, and he himself would have decided to go on to Dover. Whereupon, Moriarity would again have “reacted” differently. Because of so much thinking they might not have been able to act at all or the intellectually weaker of the two would have surrendered to the other in the Victoria Station, since the whole flight would have become unnecessary. Examples of this kind can be drawn from everywhere. However, chess, strategy, etc. presuppose expert knowledge, which encumbers the example unnecessarily.

One may be easily convinced that here lies an insoluble paradox. And the situation is not improved, but, rather, greatly aggravated if we assume that more than two individuals-as, for example, is the case with exchange-are brought together into a position, which would correspond to the one brought forward here. Always, there is exhibited an endless chain of reciprocally conjectural reactions and counter-reactions. This chain can never be broken by an act of knowledge but always only through an arbitrary act-a resolution. This resolution, again, would have to be foreseen by the two or more persons concerned. The paradox still remains no matter how one attempts to twist or turn things around. Unlimited foresight and economic equilibrium are thus irreconcilable with one another. But can equilibrium really take place with a faulty, heterogeneous foresight, however, it may be disposed? This is the question which arises at once when an answer is sought. One can even say this: has foresight been truly introduced at all into the consideration of equilibrium, or, rather, does not the theorem of equilibrium generally stand in no proven connection with the assumptions about foresight, so that a false assumption is being considered?

As Carlo Zappia has shown, it was probably Morgenstern’s attack on the notion of intertemporal equilibrium and perfect foresight that led Hayek to his classic restatement of the idea in his 1937 paper “Economics and Knowledge.” The point that Hayek clarified in his 1937 version, but had not been clear in his earlier expositions of the concept, is that correct foresight is not an assumption from which the existence of an intertemporal equilibrium can be causally deduced; there is no assertion that a state of equilibrium is the result of correct foresight. Rather, correct foresight is the characteristic that defines what is meant when the term “intertemporal equilibrium” is used in economic theory. Morgenstern’s conceptual error was to mistake a tautological statement about what would have to be true if an intertemporal equilibrium were to obtain for a causal statement about what conditions would bring an intertemporal equilibrium into existence.

The idea of correct foresight does not attribute any special powers to the economic agents who might under hypothetical circumstances possess correct expectations of future prices. The term is not meant to be a description of an actual state of affairs, but a description of what would have to be true for a state of affairs to be an equilibrium state of affairs.

As an aside, I would simply mention that many years ago when I met Hayek and had the opportunity to ask him about his 1937 paper and his role in developing the concept of intertemporal equilibrium, he brought my attention to his 1928 paper in which he first described an intertemporal equilibrium as state of affairs in which agents had correct expectations about future prices. My recollection of that conversation is unfortunately rather vague, but I do remember that he expressed some regret for not having had the paper translated into English, which would have established his priority in articulating the intertemporal equilibrium concept. My recollection is that the reason he gave for not having had the paper translated into English was that there was something about the paper about which he felt dissatisfied, but I can no longer remember what it was that he said he was dissatisfied with. However, I would now be inclined to conjecture that he was dissatisfied with not having disambiguated, as he did in the 1937 paper, between correct foresight as a defining characteristic of what intertemporal equilibrium means versus perfect foresight as the cause that brings intertemporal equilibruim into existence.

It is also interesting to note that the subsequent development of game theory in which Morgenstern played a not insubstantial role, shows that under a probabilistic interpretation of the interaction between Holmes and Moriarity, there could be an optimal mixed strategy that would provide an equilibrium solution of repeated Holmes-Moriarity interactions. But if the interaction is treated as a single non-repeatable event with no mixed strategy available to either party, the correct interpretation of the interaction is certainly that there is no equilibrium solution to the interaction. If there is no equilibrium solution, then it is precisely the absence of an equilibrium solution that implies the impossibility of correct foresight, correct foresight and the existence of an equilibrium being logically equivalent concepts.

A Primer on Equilibrium

After my latest post about rational expectations, Henry from Australia, one of my most prolific commenters, has been engaging me in a conversation about what assumptions are made – or need to be made – for an economic model to have a solution and for that solution to be characterized as an equilibrium, and in particular, a general equilibrium. Equilibrium in economics is not always a clearly defined concept, and it can have a number of different meanings depending on the properties of a given model. But the usual understanding is that the agents in the model (as consumers or producers) are trying to do as well for themselves as they can, given the endowments of resources, skills and technology at their disposal and given their preferences. The conversation was triggered by my assertion that rational expectations must be “compatible with the equilibrium of the model in which those expectations are embedded.”

That was the key insight of John Muth in his paper introducing the rational-expectations assumption into economic modelling. So in any model in which the current and future actions of individuals depend on their expectations of the future, the model cannot arrive at an equilibrium unless those expectations are consistent with the equilibrium of the model. If the expectations of agents are incompatible or inconsistent with the equilibrium of the model, then, since the actions taken or plans made by agents are based on those expectations, the model cannot have an equilibrium solution.

Now Henry thinks that this reasoning is circular. My argument would be circular if I defined an equilibrium to be the same thing as correct expectations. But I am not so defining an equilibrium. I am saying that the correctness of expectations by all agents implies 1) that their expectations are mutually consistent, and 2) that, having made plans, based on their expectations, which, by assumption, agents felt were the best set of choices available to them given those expectations, if the expectations of the agents are realized, then they would not regret the decisions and the choices that they made. Each agent would be as well off as he could have made himself, given his perceived opportunities when the decision were made. That the correctness of expectations implies equilibrium is the consequence of assuming that agents are trying to optimize their decision-making process, given their available and expected opportunities. If all expected opportunities are correctly foreseen, then all decisions will have been the optimal decisions under the circumstances. But nothing has been said that requires all expectations to be correct, or even that it is possible for all expectations to be correct. If an equilibrium does not exist, and just because you can write down an economic model, it does not mean that a solution to the model exists, then the sweet spot where all expectations are consistent and compatible is just a blissful fantasy. So a logical precondition to showing that rational expectations are even possible is to prove that an equilibrium exists. There is nothing circular about the argument.

Now the key to proving the existence of a general equilibrium is to show that the general equilibrium model implies the existence of what mathematicians call a fixed point. A fixed point is said to exist when there is a mapping – a rule or a function – that takes every point in a convex compact set of points and assigns that point to another point in the same set. A convex, compact set has two important properties: 1) the line connecting any two points in the set is entirely contained within the boundaries of the set, and 2) there are no gaps between any two points in set. The set of points in a circle or a rectangle is a convex compact set; the set of points contained in the Star of David is not a convex set. Any two points in the circle will be connected by a line that lies completely within the circle; the points at adjacent edges of a Star of David will be connected by a line that lies entirely outside the Star of David.

If you think of the set of all possible price vectors for an economy, those vectors – each containing a price for each good or service in the economy – could be mapped onto itself in the following way. Given all the equations describing the behavior of each agent in the economy, the quantity demanded and supplied of each good could be calculated, giving us the excess demand (the difference between amount demand and supplied) for each good. Then the price of every good in excess demand would be raised, the price of every good in negative excess demand would be reduced, and the price of every good with zero excess demand would be held constant. To ensure that the mapping was taking a point from a given convex set onto itself, all prices could be normalized so that they would have the property that the sum of all the individual prices would always equal 1. The fixed point theorem ensures that for a mapping from one convex compact set onto itself there must be at least one fixed point, i.e., at least one point in the set that gets mapped onto itself. The price vector corresponding to that point is an equilibrium, because, given how our mapping rule was defined, a point would be mapped onto itself if and only if all excess demands are zero, so that no prices changed. Every fixed point – and there may be one or more fixed points – corresponds to an equilibrium price vector and every equilibrium price vector is associated with a fixed point.

Before going on, I ought to make an important observation that is often ignored. The mathematical proof of the existence of an equilibrium doesn’t prove that the economy operates at an equilibrium, or even that the equilibrium could be identified under the mapping rule described (which is a kind of formalization of the Walrasian tatonnement process). The mapping rule doesn’t guarantee that you would ever discover a fixed point in any finite amount of iterations. Walras thought the price adjustment rule of raising the prices of goods in excess demand and reducing prices of goods in excess supply would converge on the equilibrium price vector. But the conditions under which you can prove that the naïve price-adjustment rule converges to an equilibrium price vector turn out to be very restrictive, so even though we can prove that the competitive model has an equilibrium solution – in other words the behavioral, structural and technological assumptions of the model are coherent, meaning that the model has a solution, the model has no assumptions about how prices are actually determined that would prove that the equilibrium is ever reached. In fact, the problem is even more daunting than the previous sentence suggest, because even Walrasian tatonnement imposes an incredibly powerful restriction, namely that no trading is allowed at non-equilibrium prices. In practice there are almost never recontracting provisions allowing traders to revise the terms of their trades once it becomes clear that the prices at which trades were made were not equilibrium prices.

I now want to show how price expectations fit into all of this, because the original general equilibrium models were either one-period models or formal intertemporal models that were reduced to single-period models by assuming that all trading for future delivery was undertaken in the first period by long-lived agents who would eventually carry out the transactions that were contracted in period 1 for subsequent consumption and production. Time was preserved in a purely formal, technical way, but all economic decision-making was actually concluded in the first period. But even though the early general-equilibrium models did not encompass expectations, one of the extraordinary precursors of modern economics, Augustin Cournot, who was way too advanced for his contemporaries even to comprehend, much less make any use of, what he was saying, had incorporated the idea of expectations into the solution of his famous economic model of oligopolistic price setting.

The key to oligopolistic pricing is that each oligopolist must take into account not just consumer demand for his product, and his own production costs; he must consider as well what actions will be taken by his rivals. This is not a problem for a competitive producer (a price-taker) or a pure monopolist. The price-taker simply compares the price at which he can sell as much as he wants with his production costs and decides how much it is worthwhile to produce by comparing his marginal cost to price ,and increases output until the marginal cost rises to match the price at which he can sell. The pure monopolist, if he knows, as is assumed in such exercises, or thinks he knows the shape of the customer demand curve, selects the price and quantity combination on the demand curve that maximizes total profit (corresponding to the equality of marginal revenue and marginal cost). In oligopolistic situations, each producer must take into account how much his rivals will sell, or what prices they will set.

It was by positing such a situation and finding an analytic solution, that Cournot made a stunning intellectual breakthrough. In the simple duopoly case, Cournot posited that if the duopolists had identical costs, then each could find his optimal price conditional on the output chosen by the other. This is a simple profit-maximization problem for each duopolist, given a demand curve for the combined output of both (assumed to be identical, so that a single price must obtain for the output of both) a cost curve and the output of the other duopolist. Thus, for each duopolist there is a reaction curve showing his optimal output given the output of the other. See the accompanying figure.cournot

If one duopolist produces zero, the optimal output for the other is the monopoly output. Depending on what the level of marginal cost is, there is some output by either of the duopolists that is sufficient to make it unprofitable for the other duopolist to produce anything. That level of output corresponds to the competitive output where price just equals marginal cost. So the slope of the two reaction functions corresponds to the ratio of the monopoly output to the competitive output, which, with constant marginal cost is 2:1. Given identical costs, the two reaction curves are symmetric and the optimal output for each, given the expected output of the other, corresponds to the intersection of the two reaction curves, at which both duopolists produce the same quantity. The combined output of the two duopolists will be greater than the monopoly output, but less than the competitive output at which price equals marginal cost. With constant marginal cost, it turns out that each duopolist produces one-third of the competitive output. In the general case with n oligoplists, the ratio of the combined output of all n firms to the competitive output equals n/(n+1).

Cournot’s solution corresponds to a fixed point where the equilibrium of the model implies that both duopolists have correct expectations of the output of the other. Given the assumptions of the model, if the duopolists both expect the other to produce an output equal to one-third of the competitive output, their expectations will be consistent and will be realized. If either one expects the other to produce a different output, the outcome will not be an equilibrium, and each duopolist will regret his output decision, because the price at which he can sell his output will differ from the price that he had expected. In the Cournot case, you could define a mapping of a vector of the quantities that each duopolist had expected the other to produce and the corresponding planned output of each duopolist. An equilibrium corresponds to a case in which both duopolists expected the output planned by the other. If either duopolist expected a different output from what the other planned, the outcome would not be an equilibrium.

We can now recognize that Cournot’s solution anticipated John Nash’s concept of an equilibrium strategy in which player chooses a strategy that is optimal given his expectation of what the other player’s strategy will be. A Nash equilibrium corresponds to a fixed point in which each player chooses an optimal strategy based on the correct expectation of what the other player’s strategy will be. There may be more than one Nash equilibrium in many games. For example, rather than base their decisions on an expectation of the quantity choice of the other duopolist, the two duopolists could base their decisions on an expectation of what price the other duopolist would set. In the constant-cost case, this choice of strategies would lead to the competitive output because both duopolists would conclude that the optimal strategy of the other duopolist would be to charge a price just sufficient to cover his marginal cost. This was the alternative oligopoly model suggested by another French economist J. L. F. Bertrand. Of course there is a lot more to be said about how oligopolists strategize than just these two models, and the conditions under which one or the other model is the more appropriate. I just want to observe that assumptions about expectations are crucial to how we analyze market equilibrium, and that the importance of these assumptions for understanding market behavior has been recognized for a very long time.

But from a macroeconomic perspective, the important point is that expected prices become the critical equilibrating variable in the theory of general equilibrium and in macroeconomics in general. Single-period models of equilibrium, including general-equilibrium models that are formally intertemporal, but in which all trades are executed in the initial period at known prices in a complete array of markets determining all future economic activity, are completely sterile and useless for macroeconomics except as a stepping stone to analyzing the implications of imperfect forecasts of future prices. If we want to think about general equilibrium in a useful macroeconomic context, we have to think about a general-equilibrium system in which agents make plans about consumption and production over time based on only the vaguest conjectures about what future conditions will be like when the various interconnected stages of their plans will be executed.

Unlike the full Arrow-Debreu system of complete markets, a general-equilibrium system with incomplete markets cannot be equilibrated, even in principle, by price adjustments in the incomplete set of present markets. Equilibration depends on the consistency of expected prices with equilibrium. If equilibrium is characterized by a fixed point, the fixed point must be mapping of a set of vectors of current prices and expected prices on to itself. That means that expected future prices are as much equilibrating variables as current market prices. But expected future prices exist only in the minds of the agents, they are not directly subject to change by market forces in the way that prices in actual markets are. If the equilibrating tendencies of market prices in a system of complete markets are very far from completely effective, the equilibrating tendencies of expected future prices may not only be non-existent, but may even be potentially disequilibrating rather than equilibrating.

The problem of price expectations in an intertemporal general-equilibrium system is central to the understanding of macroeconomics. Hayek, who was the father of intertemporal equilibrium theory, which he was the first to outline in a 1928 paper in German, and who explained the problem with unsurpassed clarity in his 1937 paper “Economics and Knowledge,” unfortunately did not seem to acknowledge its radical consequences for macroeconomic theory, and the potential ineffectiveness of self-equilibrating market forces. My quarrel with rational expectations as a strategy of macroeconomic analysis is its implicit assumption, lacking any analytical support, that prices and price expectations somehow always adjust to equilibrium values. In certain contexts, when there is no apparent basis to question whether a particular market is functioning efficiently, rational expectations may be a reasonable working assumption for modelling observed behavior. However, when there is reason to question whether a given market is operating efficiently or whether an entire economy is operating close to its potential, to insist on principle that the rational-expectations assumption must be made, to assume, in other words, that actual and expected prices adjust rapidly to their equilibrium values allowing an economy to operate at or near its optimal growth path, is simply, as I have often said, an exercise in circular reasoning and question begging.

Making Sense of Rational Expectations

Almost two months ago I wrote a provocatively titled post about rational expectations, in which I argued against the idea that it is useful to make the rational-expectations assumption in developing a theory of business cycles. The title of the post was probably what led to the start of a thread about my post on the econjobrumors blog, the tenor of which  can be divined from the contribution of one commenter: “Who on earth is Glasner?” But, aside from the attention I received on econjobrumors, I also elicited a response from Scott Sumner

David Glasner has a post criticizing the rational expectations modeling assumption in economics:

What this means is that expectations can be rational only when everyone has identical expectations. If people have divergent expectations, then the expectations of at least some people will necessarily be disappointed — the expectations of both people with differing expectations cannot be simultaneously realized — and those individuals whose expectations have been disappointed will have to revise their plans. But that means that the expectations of those people who were correct were also not rational, because the prices that they expected were not equilibrium prices. So unless all agents have the same expectations about the future, the expectations of no one are rational. Rational expectations are a fixed point, and that fixed point cannot be attained unless everyone shares those expectations.

Beyond that little problem, Mason raises the further problem that, in a rational-expectations equilibrium, it makes no sense to speak of a shock, because the only possible meaning of “shock” in the context of a full intertemporal (aka rational-expectations) equilibrium is a failure of expectations to be realized. But if expectations are not realized, expectations were not rational.

I see two mistakes here. Not everyone must have identical expectations in a world of rational expectations. Now it’s true that there are ratex models where people are simply assumed to have identical expectations, such as representative agent models, but that modeling assumption has nothing to do with rational expectations, per se.

In fact, the rational expectations hypothesis suggests that people form optimal forecasts based on all publicly available information. One of the most famous rational expectations models was Robert Lucas’s model of monetary misperceptions, where people observed local conditions before national data was available. In that model, each agent sees different local prices, and thus forms different expectations about aggregate demand at the national level.

It is true that not all expectations must be identical in a world of rational expectations. The question is whether those expectations are compatible with the equilibrium of the model in which those expectations are embedded. If any of those expectations are incompatible with the equilibrium of the model, then, if agents’ decision are based on their expectations, the model will not arrive at an equilibrium solution. Lucas’s monetary misperception model was a clever effort to tweak the rational-expectations assumption just enough to allow for a temporary disequilibrium. But the attempt was a failure, because Lucas could only generate a one-period deviation from equilibrium, which was too little for the model to pose as a plausible account of a business cycle. That provided Kydland and Prescott the idea to discard Lucas’s monetary misperceptions idea and write their paper on real business cycles without adulterating the rational expectations assumption.

Here’s what Muth said about the rational expectations assumption in the paper in which he introduced “rational expectations” as a modeling strategy.

In order to explain these phenomena, I should like to suggest that expectations, since they are informed predictions of future events, are essentially the same as the predictions of the relevant economic theory. At the risk of confusing this purely descriptive hypothesis with a pronouncement as to what firms ought to do, we call such expectations “rational.”

The hypothesis can be rephrased a little more precisely as follows: that expectations of firms (or, more generally, the subjective probability distribution of outcomes) tend to be distributed, for the same information set, about the prediction of the theory (or the “objective” probability distributions of outcomes).

The hypothesis asserts three things: (1) Information is scarce, and the economic system generally does not waste it. (2) The way expectations are formed depends specifically on the structure of the relevant system describing the economy. (3) A “public prediction,” in the sense of Grunberg and Modigliani, will have no substantial effect on the operation of the economic system (unless it is based on inside information).

It does not assert that the scratch work of entrepreneurs resembles the system of equations in any way; nor does it state that predictions of entrepreneurs are perfect or that their expectations are all the same. For purposes of analysis, we shall use a specialized form of the hypothesis. In particular, we assume: 1. The random disturbances are normally distributed. 2. Certainty equivalents exist for the variables to be predicted. 3. The equations of the system, including the expectations formulas, are linear. These assumptions are not quite so strong as may appear at first because any one of them virtually implies the other two.

It seems to me that Muth was confused about what the rational-expectations assumption entails. He asserts that the expectations of entrepreneurs — and presumably that applies to other economic agents as well insofar as their decisions are influenced by their expectations of the future – should be assumed to be exactly what the relevant economic model predicts the expected outcomes to be. If so, I don’t see how it can be maintained that expectations could diverge from each other. If what entrepreneurs produce next period depends on the price they expect next period, then how is it possible that the total supply produced next period is independent of the distribution of expectations as long as the errors are normally distributed and the mean of the distribution corresponds to the equilibrium of the model? This could only be true if the output produced by each entrepreneur was a linear function of the expected price and all entrepreneurs had identical marginal costs or if the distribution of marginal costs was uncorrelated with the distribution of expectations. The linearity assumption is hardly compelling unless you assume that the system is in equilibrium and all changes are small. But making that assumption is just another form of question begging.

It’s also wrong to say:

But if expectations are not realized, expectations were not rational.

Scott is right. What I said was wrong. What I ought to have said is: “But if expectations (being divergent) could not have been realized, those expectations were not rational.”

Suppose I am watching the game of roulette. I form the expectation that the ball will not land on one of the two green squares. Now suppose it does. Was my expectation rational? I’d say yes—there was only a 2/38 chance of the ball landing on a green square. It’s true that I lacked perfect foresight, but my expectation was rational, given what I knew at the time.

I don’t think that Scott’s response is compelling, because you can’t judge the rationality of an expectation in isolation, it has to be judged in a broader context. If you are forming your expectation about where the ball will fall in a game of roulette, the rationality of that expectation can only be evaluated in the context of how much you should be willing to bet that the ball will fall on one of the two green squares and that requires knowledge of what the payoff would be if the ball did fall on one of those two squares. And that would mean that someone else is involved in the game and would be taking an opposite position. The rationality of expectations could only be judged in the context of what everyone participating in the game was expecting and what the payoffs and penalties were for each participant.

In 2006, it might have been rational to forecast that housing prices would not crash. If you lived in many countries, your forecast would have been correct. If you happened to live in Ireland or the US, your forecast would have been incorrect. But it might well have been a rational forecast in all countries.

The rationality of a forecast can’t be assessed in isolation. A forecast is rational if it is consistent with other forecasts, so that it, along with the other forecasts, could potentially be realized. As a commenter on Scott’s blog observed, a rational expectation is an expectation that, at the time the forecast is made, is consistent with the relevant model. The forecast of housing prices may turn out to be incorrect, but the forecast might still have been rational when it was made if the forecast of prices was consistent with what the relevant model would have predicted. The failure of the forecast to be realized could mean either that forecast was not consistent with the model, or that between the time of the forecast and the time of its realization, new information,  not available at the time of the forecast, came to light and changed the the prediction of the relevant model.

The need for context in assessing the rationality of expectations was wonderfully described by Thomas Schelling in his classic analysis of cooperative games.

One may or may not agree with any particular hypothesis as to how a bargainer’s expectations are formed either in the bargaining process or before it and either by the bargaining itself or by other forces. But it does seem clear that the outcome of a bargaining process is to be described most immediately, most straightforwardly, and most empirically, in terms of some phenomenon of stable and convergent expectations. Whether one agrees explicitly to a bargain, or agrees tacitly, or accepts by default, he must if he has his wits about him, expect that he could do no better and recognize that the other party must reciprocate the feeling. Thus, the fact of an outcome, which is simply a coordinated choice, should be analytically characterized by the notion of convergent expectations.

The intuitive formulation, or even a careful formulation in psychological terms, of what it is that a rational player expects in relation to another rational player in the “pure” bargaining game, poses a problem in sheer scientific description. Both players, being rational, must recognize that the only kind of “rational” expectation they can have is a fully shared expectation of an outcome. It is not quite accurate – as a description of a psychological phenomenon – to say that one expects the second to concede something; the second’s readiness to concede or to accept is only an expression of what he expects the first to accept or to concede, which in turn is what he expects the first to expect the second to expect the first to expect, and so on. To avoid an “ad infinitum” in the description process, we have to say that both sense a shared expectation of an outcome; one’s expectation is a belief that both identify the outcome as being indicated by the situation, hence as virtually inevitable. Both players, in effect, accept a common authority – the power of the game to dictate its own solution through their intellectual capacity to perceive it – and what they “expect” is that they both perceive the same solution.

Viewed in this way, the intellectual process of arriving at “rational expectations” in the full-communication “pure” bargaining game is virtually identical with the intellectual process of arriving at a coordinated choice in the tacit game. The actual solutions might be different because the game contexts might be different, with different suggestive details; but the intellectual nature of the two solutions seems virtually identical since both depend on an agreement that is reached by tacit consent. This is true because the explicit agreement that is reached in the full communication game corresponds to the a prioir expectations that were reached (or in theory could have been reached) jointly but independently by the two players before the bargaining started. And it is a tacit “agreement” in the sense that both can hold confident rational expectation only if both are aware that both accept the indicated solution in advance as the outcome that they both know they both expect.

So I agree that rational expectations can simply mean that agents are forming expectations about the future incorporating as best as they can all the knowledge available to them. This is a weak common sense interpretation of rational expectations that I think is what Scott Sumner has in mind when he uses the term “rational expectations.” But in the context of formal modelling, rational expectations has a more restrictive meaning, which is that given all the information available, the expectations of all agents in the model must correspond to what the model itself predicts given that information. Even though Muth himself and others have tried to avoid the inference that all agents must have expectations that match the solution of the model, given the information underlying the model, the assumptions under which agents could hold divergent expectations are, in their own way, just as restrictive as the assumption that agents hold convergent expectations.

In a way, the disconnect between a common-sense understanding of what “rational expectations” means and what “rational expectations” means in the context of formal macroeconomic models is analogous to the disconnect between what “competition” means in normal discourse and what “competition” (and especially “perfect competition”) means in the context of formal microeconomic models. Much of the rivalrous behavior between competitors that we think of as being essential aspects of competition and the competitive process is simply ruled out by the formal assumption of perfect competition.

Rational Expectations, or, The Road to Incoherence

J. W. Mason left a very nice comment on my recent post about Paul Romer’s now-famous essay on macroeconomics, a comment now embedded in his interesting and insightful blog post on the Romer essay. As a wrote in my reply to Mason’s comment, I really liked the way he framed his point about rational expectations and intertemporal equilibrium. Sometimes when you see a familiar idea expressed in a particular way, the novelty of the expression, even though it’s not substantively different from other ways of expressing the idea, triggers a new insight. And that’s what I think happened in my own mind as I read Mason’s comment. Here’s what he wrote:

David Glasner’s interesting comment on Romer makes in passing a point that’s bugged me for years — that you can’t talk about transitions from one intertemporal equilibrium to another, there’s only the one. Or equivalently, you can’t have a model with rational expectations and then talk about what happens if there’s a “shock.” To say there is a shock in one period, is just to say that expectations in the previous period were wrong. Glasner:

the Lucas Critique applies even to micro-founded models, those models being strictly valid only in equilibrium settings and being unable to predict the adjustment of economies in the transition between equilibrium states. All models are subject to the Lucas Critique.

So the further point that I would make, after reading Mason’s comment, is just this. For an intertemporal equilibrium to exist, there must be a complete set of markets for all future periods and contingent states of the world, or, alternatively, there must be correct expectations shared by all agents about all future prices and the probability that each contingent future state of the world will be realized. By the way, If you think about it for a moment, the notion that probabilities can be assigned to every contingent future state of the world is mind-bogglingly unrealistic, because the number of contingent states must rapidly become uncountable, because every single contingency itself gives rise to further potential contingencies, and so on and on and on. But forget about that little complication. What intertemporal equilibrium requires is that all expectations of all individuals be in agreement – or at least not be inconsistent, some agents possibly having an incomplete set of expectations about future prices and future states of the world. If individuals differ in their expectations, so that their planned future purchases and sales are based on what they expect future prices to be when the time comes for those transactions to be carried out, then individuals will not be able to execute their plans as intended when at least one of them finds that actual prices are different from what they had been expected to be.

What this means is that expectations can be rational only when everyone has identical expectations. If people have divergent expectations, then the expectations of at least some people will necessarily be disappointed — the expectations of both people with differing expectations cannot be simultaneously realized — and those individuals whose expectations have been disappointed will have to revise their plans. But that means that the expectations of those people who were correct were also not rational, because the prices that they expected were not equilibrium prices. So unless all agents have the same expectations about the future, the expectations of no one are rational. Rational expectations are a fixed point, and that fixed point cannot be attained unless everyone shares those expectations.

Beyond that little problem, Mason raises the further problem that, in a rational-expectations equilibrium, it makes no sense to speak of a shock, because the only possible meaning of “shock” in the context of a full intertemporal (aka rational-expectations) equilibrium is a failure of expectations to be realized. But if expectations are not realized, expectations were not rational. So the whole New Classical modeling strategy of identifying shocks  to a system in rational-expectations equilibrium, and “predicting” the responses to these shocks as if they had been anticipated is self-contradictory and incoherent.


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.

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