Wednesday, April 25, 2012

Only Ever Equilibrium?

Roger Farmer has a somewhat puzzling guest post up at Noah Smith's place, arguing that economics is right to limit discussion to equilibrium:
An economic equilibrium, in the sense of Nash, is a situation where a group of decision makers takes a sequence of actions that is best, (in a well defined sense), on the assumption that every other decision maker in the group is acting in a similar fashion. In the context of a competitive economy with a large number of players, Nash equilibrium collapses to the notion of perfect competition.  The genius of the rational expectations revolution, largely engineered by Bob Lucas, was to apply that concept to macroeconomics by successfully persuading the profession to base our economic models on Chapter 7 of Debreu's Theory of Value... In Debreu's vision, a commodity is indexed by geographical location, by date and by the state of nature.  Once one applies Debreu's vision of general equilibrium theory to macroeconomics, disequilibrium becomes a misleading and irrelevant distraction. 
The use of equilibrium theory in economics has received a bad name for two reasons. 
First, many equilibrium environments are ones where the two welfare theorems of competitive equilibrium theory are true, or at least approximately true. That makes it difficult to think of them as realistic models of a depression, or of a financial collapse... Second, those macroeconomic models that have been studied most intensively, classical and new-Keynesian models, are ones where there is a unique equilibrium. Equilibrium, in this sense, is a mapping from a narrowly defined set of fundamentals to an outcome, where  an outcome is an observed temporal sequence of unemployment rates, prices, interest rates etc. Models with a unique equilibrium do not leave room for non-fundamental variables to influence outcomes... 
Multiple equilibrium models do not share these shortcomings... [But] a model with multiple equilibria is an incomplete model. It must be closed by adding an equation that explains the behavior of an agent when placed in an indeterminate environment. In my own work I have argued that this equation is a new fundamental that I call a belief function.
(Personally, I might just call it a convention.)
Some recent authors have argued that rational expectations must be rejected and replaced by a rule that describes how agents use the past to forecast the future. That approach has similarities to the use of a belief function to determine outcomes, and when added to a multiple equilibrium model of the kind I favor, it will play the same role as the belief function. The important difference of multiple equilibrium models, from the conventional approach to equilibrium theory, is that the belief function can coexist with the assumption of rational expectations. Agents using a rule of this kind, will not find that their predictions are refuted by observation. ...
So his point here is that in a model with multiple equilibria, there is no fundamental reason why the economy should occupy one rather than another. You need to specify agents' expectations independently, and once you do, whatever outcome they expect, they'll be correct. This allows for an economy to experience involuntary unemployment, for example, as expectations of high or low income lead to increased or curtailed expenditure, which results in expected income, whatever it was, being realized. This is the logic of the Samuelson Cross we teach in introductory macro. But it's not, says Farmer, a disequilibrium in any meaningful way:
If by disequilibrium, I am permitted to mean that the economy may deviate for a long time, perhaps permanently, from a social optimum; then I have no trouble with championing the cause. But that would be an abuse of the the term 'disequilibrium'. If one takes the more normal use of disequilibrium to mean agents trading at non-Walrasian prices, ... I do not think we should revisit that agenda. Just as in classical and new-Keynesian models where there is a unique equilibrium, the concept of disequilibrium in multiple equilibrium models is an irrelevant distraction.
I quote this at such length because it's interesting. But also because, to me at least, it's rather strange. There's nothing wrong with the multiple equilibrium approach he's describing here, which seems like a useful way of thinking about a number of important questions. But to rule out a priori any story in which people's expectations are not fulfilled rules out a lot of other useful ways about thinking about important questions.

At INET in Berlin, the great Axel Leijonhufvud gave a talk where he described the defining feature of a crisis as the existence of inconsistent contractual commitments, so that some of them would have to be voided or violated.
What is the nature of our predicament? The web of contracts has developed serious inconsistencies. All the promises cannot possibly be fulfilled. Insisting that they should be fulfilled will cause a collapse of very large portions of the web.
But Farmer is telling us that economists not only don't need to, but positively should not, attempt to understand crises in this sense. It's an "irrelevant distraction" to consider the case where people entered into contracts with inconsistent expectations, which will not all be capable of being fulfilled. Farmer can hardly be unfamiliar with these ideas; after all he edited Leijonhufvud's festschrift volume. So why is he being so dogmatic here?

I had an interesting conversation with Rajiv Sethi after Leijonhufvud's talk; he said he thought that the inability to consider cases where plans were not realized was a fundamental theoretical shortcoming of mainstream macro models. I don't disagree.

The thing about the equilibrium approach, as Farmer presents it, isn't just that it rules out the possibility of people being systematically wrong; it rules out the possibility that they disagree. This strikes me as a strong and importantly empirically false proposition. (Keynes suggested that the effectiveness of monetary policy depends on the existence of both optimists and pessimists in financial markets.) In Farmer's multiple equilibrium models, whatever outcome is set by convention, that's the outcome expected by everyone. This is certainly reasonable in some cases, like the multiple equilibria of driving on the left or the right side of the road. Indeed, I suspect that the fact that people are irrationally confident in these kinds of conventions, and expect them to hold even more consistently than they do, is one of the main things that stabilizes these kind of equilibria. But not everything in economics looks like that.

Here's Figure 1 from my Fisher dynamics paper with Arjun Jayadev:


See those upward slopes way over on the left? Between 1929 and 1933, household debt relative to GDP rose by abut 40 percent, and nonfinancial business debt relative to GDP nearly doubled. This is not, of course, because families and businesses were borrowing more in the Depression; on the contrary, they were paying down debt as fast as they could. But in the classic debt-deflation story, falling prices and output meant that incomes were falling even fast than debt, so leverage actually increased.

Roger Farmer, if I'm understanding him correctly, is saying that we must see this increase in debt-income ratios as an equilibrium phenomenon. He is saying that households and businesses taking out loans in 1928 must have known that their incomes were going to fall by half over the next five years, while their debt payments would stay unchanged, and chose to borrow anyway. He is saying not just that he believes that, but that as economists we should not consider any other view; we can rule out on methodological grounds the  possibility that the economic collapse of the early 1930s caught people by surprise. To Irving Fisher, to Keynes, to almost anyone, to me, the rise in debt ratios in the early 1930s looks like a pure disequilibrium phenomenon; people were trading at false prices, signing nominal contracts whose real terms would end up being quite different from what they expected. It's one of the most important stories in macroeconomics, but Farmer is saying that we should forbid ourselves from telling it. I don't get it.

What am I missing here?

27 comments:

  1. I have no patience with the "it's equilibrium or it's crap" school of thought, but I'm not sure that your last example is really as devastating as it should be. All that equilibrium requires is that people are doing as well as they could, conditional on what everyone else is doing. That doesn't have to be doing well, it just has to be doing better than the alternatives. And even a strategy which would typically do well, averaging over whatever noise the world presents, could happen to do poorly in retrospect. So all Farmer would need would be a story about why, given what people knew in 1928, loading up on debt then was better than the alternatives.

    As I said, I think telling such a story would contribute nothing at all to understanding the world, but one must be fair...

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  2. Well, you may be right. Let's think about this.

    The deflation of 1929-1933 meant that there was a sharp increase in ex post real interest rates for debts incurred in the late 1920s. So you'd expect household borrowing to fall, unless there was some other factor raising demand for loans. But the other factor we normally think of is income, and since that fell sharply in the early 1930s, that should have reduced household borrowing, not raised it. it seems to me that the only way you can tell an equilibrium story (in Farmer's sense) is if household utility just happened to be such that consumption in 1928 was much more desirable than consumption in 1932, by a much wider margin than for other four-year intervals. With a stable discount rate, I don't see how you can do it. Of course, you can always tell some story. Maybe we've greatly underestimated technological change in the 1930s and real incomes were actually rising.

    The alternative theory is that debt contracts in the 1920s were entered into in the expectation that recent income and price trends would continue, and that people then found themselves with real debt burdens larger than they had intended. I'm not sure I can prove that's a better tory, but I certainly don't see why it should be ruled out.

    The case of firms is even harder, because there we don't have the option of postulating some special utility function or unobserved welfare gains from technological change. Again, given the fall in incomes and prices in 1929-1933, realized returns on investments made in the late 1920s were very low, and ex post real interest rates were high. Yet business fixed investment peaked in 1929. It's very hard for me to see the coherent story in which that represented optimizing behavior given correct expectations. Do you see one?

    The other point is that the prediction that expectations are universally shared is in some ways even stronger than the prediction that they are rational. (This is one of Leijonhufvud's points about inflation -- that some of its biggest costs come from the fact that in conditions of high inflation expectations for future inflation will vary widely, so there's no longer any assurance that voluntary contracts are mutually beneficial.) But it's hard to detect heterogeneous expectations in aggregate data.

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    1. The return on equity in the U.S. has been six percentage points higher than the return on debt for as long as we have been collecting data. That excess risk premium is extremely hard to square with a conventional equilibrium model. It is not hard to reconcile with a multiple equilibrium model where any unemployment rate is an equilibrium.

      The Great Depression, from this perspective, was a shift from a state of euphoria, where prices were consistently rising and unemployment was falling, to an alternative equilibrium where the economy stagnated. The possibility that this outcome might occur was, and still is, incorporated into asset prices.

      The multiple equilibrium model can also explain the current slow recovery where more than 2 trillion dollars is held as liquid assets in U.S. corporations. Businesses are afraid to buy real assets whose prices could fall further. This paper

      http://rogerfarmer.com/NewWeb/PdfFiles/Farmer_Financial_Crises.pdf

      discusses the idea of a financial crisis as a shift from one equilibrium to another

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  3. JW
    Thank you for your thoughtful comments on my guest post at Noahopinion. Let me clarify my thoughts by distinguishing two different meanings of the term *equilibrium*. One refers to the equality of demand and supply. The other refers to the consistency of expectations with observations.

    Since at least the time of Hicks' classic book, *Value and Capital*, macroeconomists have modeled the world as a sequence of market meetings. Each period, households and firms enter the market carrying stocks of commodities and portfolios of financial assets, that have been accumulated from the past. During the period, they form expectations of the probabilities that any future sequence of events might occur. Give their commitments from the past, and their expectations of the future, agents trade with each other. This sequence of market meetings, closed with an assumption of how prices are set, is called a temporary equilibrium.

    There are two senses in which a temporary equilibrium model may be in *disequilibrium*. The first is that at each market meeting, demand may not equal supply for every commodity at market prices. The second, is that agents' beliefs of future prices may later turn out to have been incorrect. Bob Lucas argued that we should drop both senses of disequilibrium. In my view, he was right. But when combined with models of incomplete labor markets, as in my work http://www.rogerfarmer.com/ the twin equilibrium assumptions have somewhat different implications from the ones that most macroeconomists are familiar with.

    In the 1970's there was an active research agenda led by Jean-Paul Benassy and Jacques Dreze in Europe and Barro and Grossman, Bob Clower, Axel Leijonhufvud and Don Patinkin in the U.S., that sought to drop the market clearing assumption in each temporary market. That agenda was overtaken by the rational expectations revolution and I do not think that we should try to resuscitate it. If instead, one models carefully, the structure of the labor market, it is possible to write down sensible micro founded models in which involuntary unemployment may emerge as an equilibrium phenomenon. That is what I have been engaged in, in my recent books and papers.

    That brings me to the second sense of *disequilibrium* that was the point of your post. Should we model beliefs by assuming that the subjective probability distributions of the agents in the model coincide with realized distributions. That depends on the environment in which we place our agents. If we close our models with a *belief function*, the set of possible belief functions is not arbitrary. In a stationary environment, we would want the model to converge to a rational expectations equilibrium. But in a non-stationary environment, a situation of *uncertainty* in the sense of Frank Knight as opposed to *risk*, there is room for subjective expectations to differ from rational expectations. Indeed, it is not clear how anyone (including an outside observer) could ever know what rational expectations *are* if the sequence of probabilities that governs outcomes is arbitrary.

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    1. Roger,

      Thanks so much for the reply. Delighted you found the post worth responding to. You're certainly making an important argument here, that I am learning a lot from. But I'm afraid I'm still not entirely convinced.

      Point by point.

      On the economy as a series of market meetings, I think there's a rather large problem here that almost never gets acknowledged. As you present it here -- as it is almost always presented -- the implicit model is of a series of meetings happening sequentially in time. But once we introduce dated commodities, this interpretation is no longer possible. We cannot think of the meeting at time 1 setting prices and allocations of commodities available at time 1, the meeting at time 2 allocating commodities at time 2, and so on, because at each meeting all the commodities for all periods are priced and allocated. So we can think of the meetings as taking place in alternate worlds with different endowments, but we cannot logically think of one of them as lying in the future of another. This problem is normally "solved" in the most ad hoc way, by implicitly assuming that agents have perfect knowledge of all future contingencies except the particular one the modeler is interested in. I don't think it's an exaggeration to say that the normal way these problems is set up is to say, "Assume agents in the economy have rational expectations of all future states of the world. Now, something unexpected occurs." But, no. Intertemporal optimization and comparative statics do not mix. Or so it seems to me.

      (This is important to the specific issue here.)

      On research motivated by dropping the market-clearing assumption. I'm not familiar with this, I'm sorry to say, although I certainly admire the work of Leijonhufvud (actually he's sort of my intellectual hero), Clower and Patinkin. Will learn more, hopefully. Meantime, I'm happy to stipulate that we are not interested in disequilibrium in that sense and focus on the second sense that expectations are always correct, or that intentions are always realized.

      So on to that. I think I understand the logic of your approach. And it seems very promising and important. You are certainly right that self-validating expectations in a context of multiple equilibria are exceedingly important, and that many economic phenomena that seem to require some kind of disequilibrium can actually be understood in this light. For instance, I'd like to develop Keynes' -- and Marx's for that matter -- idea of the long-term interest rate as fundamentally unanchored, in terms of a self-stabilizing convention about the "normal" level of the long rate. Your approach could be very useful in showing how market interest rates can be "stuck" at a level too high for full employment, without any market frictions, thanks to the stabilizing speculation associated with well-anchored expectations. (The same kind of stabilizing speculation was important to the maintenance of the classical gold standard.) And I think you're right that the existence of involuntary unemployment can be rationalized in a multiple-equilibrium framework.

      That said, I think you run into the same problem as other intertemporal equilibrium models when it comes to transitions between equilibrium states. in 1929, unemployment is 3%. In 1933, it's 25%. I understand that 3% unemployment wi compatible with everyone expecting 3% unemployment forever, and 25% unemployment is compatible with everyone expecting 25% unemployment forever. But what about the transition? I don't see how you can tell a story of a shift from one of these states to the other without people's expectations being systematically mistaken, given that the existence of each equilibrium is specifically conditioned on the expectation that it will last forever.

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    2. (continued)


      You could say, as you sort of seem to, that expectations are of some probability distribution across states. But I don't think this help you. If the distribution is fixed prior to expectations, then we're really back in the world of a single unique equilibrium, it's just a mixed state rather than a pure state. And if the distribution depends on expectations, then you still have the problem of transitions between distributions/mixed states.


      Why does it matter? I think one big reason is that a natural place to end up from where you're starting is that it is only government actions that are unexpected. Logically, that doesn't make sense -- why can't rational agents anticipate the actions of the state? But in practice, as far as I can tell, people who talk about intertemporal equilibrium yet don't want to give up on historical time tend to end up saying that it's only policy changes that are unexpected and allow for transitions from one "intertemporal" equilibrium to another.

      i admit I don't fully understand your argument about the relationship between unemployment and the equity premium. (Which, as an aside, I don't agree with you on the empirics. There are quite lengthy periods historically when the equity premium is much lower than 6 points, even zero.) But if I am following you, I'm afraid your position seems logically contradictory. You are saying that unemployment is set in a multiple equilibrium framework, where the current state of employment can be maintained forever. But equity prices are being set in a unique (mixed) equilibrium, where there is a single probability distribution across states of the economy which is set by fundamentals.

      This contradiction is even clearer when you write

      The Great Depression, from this perspective, was a shift from a state of euphoria, where prices were consistently rising and unemployment was falling, to an alternative equilibrium where the economy stagnated. The possibility that this outcome might occur was, and still is, incorporated into asset prices.

      I find it very hard to read this passage except as an assertion that asset prices in the 1920s were both (1) set on the assumption that prices would consistently rise and (2) set on the assumption that prices might fall.

      You cannot say that there are multiple equilibria because expectations of state A are compatible with state A, and expectations of state B are compatible with state B, *and* that people expect the true mix of states A and B. It amounts to saying that there are multiple equilibria and a unique equilibrium. It's 1=2. I'm sorry, you just can't.

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    3. JW, on your first point, I don't see the problem. In one model (call it the Arrow-Debreu model), trade in time-contingent, state-contingent commodities takes place at time 0, before uncertainty is realized. In another model (call it the Radner model), at each point in time t there is both a spot market in date t commodities, and at least one market in state-contingent period t+1 commodities. Mas-Colell et al. tell me that the Radner model gives you the same equilibrium allocations as the Arrow-Debreu model, *if* agents have rational expectations about spot prices in the future. We can still interpret the Radner model as a series of meetings happening sequentially in time, and we can think of the t=2 meeting as lying in the future of the t=1 meeting.

      So Arrow-Debreu is not completely inconsistent with the idea of a series of market meetings, if we assume rational expectations. However, one could also consider the Radner model without RatEx, allowing for disequilibrium (in the inconsistent-plans sense, not the nonclearing-markets sense). Grandmont does this in 'Temporary General Equilibrium Theory' (1977). Agents have some 'expectation function' mapping observable signals into beliefs about the future (which could come from Bayes' Rule, from a possibly misspecified model of the world, etc.). This is not a million miles away from Farmer's approach. In models with a unique RatEx equilibrium, the expectation function is an alternative to rational expectations; in Farmer's models with multiple RatEx equilibria, the belief function comes in addition to RatEx. But I disagree with Farmer's statement that "In a stationary environment, we would want the model [with a belief function] to converge to a rational expectations equilibrium." Even if there were a unique RatEx equilibrium, I see no reason to expect that people ever learn their way to this equilibrium, let alone that they get there instantaneously.

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    4. Keshac,

      This isn't really my area and I may be missing something, but I don't understand the distinction between the two models you describe. If people are assumed to know the true probability distribution of all future outcomes, then it is logically impossible to speak of one equilibrium transitioning or evolving to another one. How is logically possible both to regard an equilibrium as being formed on the basis of true knowledge of the probability distribution of all future events, and then to speak of that distribution unexpectedly changing? If the change was possible, then by hypothesis, it was expected.

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    5. (Of course I also agree with you that the assumption is unrealistic but it seems to me we don't even get to the question of its realism because it's logically incoherent.)

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    6. JW
      Thanks once again for your comments. Here are a few more thoughts about how we should use the equilibrium concept in economic models.

      Start with an environment where there are no shocks to the economic fundamentals, (preferences, endowments or technology), and each of these fundamentals is represented by a constant sequence of numbers.

      To describe how agents would behave in this artificial world, assume that time proceeds in a sequence of market meetings and that at each meeting, agents trade goods and produce new commodities. Assume further that agents form beliefs about future prices, and given these beliefs, the market at each date is in equilibrium.

      *Equilibrium* here means that, at date t, demand equals supply for every date t commodity. This is a temporary equilibrium in the sense of Hicks. That is what happens in the models that I have constructed in my recent books and papers. But my work deviates from standard theory by incorporating incomplete labor markets. Labor is not traded in a spot market. It is traded in a search market. Because of search market frictions, I have shown here

      http://rogerfarmer.com/NewWeb/PdfFiles/fa-con-cra.pdf

      that ANY unemployment rate can be part of a temporary equilibrium.

      Now ask; how should we model beliefs about future prices in this stationary environment in which none of the fundamentals ever changes. Suppose we decide to stick with the perfect foresight assumption in which expectations are always correct. Then, in the models I work with, there are infinitely many stationary perfect foresight equilibria, one for every constant unemployment rate.

      But that is not all. In

      http://rogerfarmer.com/NewWeb/PdfFiles/Farmer_Financial_Crises.pdf

      I show that there are also infinitely many non-stationary perfect foresight equilibria where agents believe, correctly, that asset prices will keep increasing forever and where the unemployment rate gets closer and closer to a lower bound.

      We are not done yet. Now suppose that everybody in this economy reads the Wall Street Journal. Every period the WSJ gives an optimistic or a pessimistic signal and agents in the economy believe that this signal is accurate. This coordinating device can act as a signal, switching the economy from the non-stationary explosive growth path to one of the many stationary high unemployment paths. Using the terminology of Dave Cass and Karl Shell, I will call this shock, *extrinsic uncertainty*. (Shocks to the fundamentals are called *intrinsic*.) Since, in the non-stationary equilibrium, everyone understands that a crash can occur; assets that will fall in value as a consequence of a crash, triggered by extrinsic uncertainty, will earn a high premium.

      Is it reasonable to impose rational expectations in this environment? I confess to the sin of having thought mostly about models with a representative agent and in those worlds, it is not clear that one can distinguish a rational expectations equilibrium from a non-rational expectations equilibrium. But as Rajiv points out (see the post below) trade is inconsistent with common priors and one might take that as evidence against rational expectations. A more substantive issue is, that if either intrinsic or extrinsic shocks are non-stationary, no agent can ever know the true probability of a given event.

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    7. JW
      Thanks for your comments. Here are a few more thoughts about how we should use the equilibrium concept in economic models.

      Start with an environment where there are no shocks to the economic fundamentals, (preferences, endowments or technology), and each of these fundamentals is represented by a constant sequence of numbers.

      To describe how agents would behave in this artificial world, assume that time proceeds in a sequence of market meetings and that at each meeting, agents trade goods and produce new commodities. Assume further that agents form beliefs about future prices, and given these beliefs, the market at each date is in equilibrium.

      *Equilibrium* here means that, at date t, demand equals supply for every date t commodity. This is a temporary equilibrium in the sense of Hicks. That is what happens in the models that I have constructed in my recent books and papers. But my work deviates from standard theory by incorporating incomplete labor markets. Labor is not traded in a spot market. It is traded in a search market. Because of search market frictions, I have shown here

      http://rogerfarmer.com/NewWeb/PdfFiles/fa-con-cra.pdf

      that ANY unemployment rate can be part of a temporary equilibrium.

      Now ask; how should we model beliefs about future prices in this stationary environment in which none of the fundamentals ever changes. Suppose we decide to stick with the perfect foresight assumption in which expectations are always correct. Then, in the models I work with, there are infinitely many stationary perfect foresight equilibria, one for every constant unemployment rate.

      But that is not all. In

      http://rogerfarmer.com/NewWeb/PdfFiles/Farmer_Financial_Crises.pdf

      I show that there are also infinitely many non-stationary perfect foresight equilibria where agents believe, correctly, that asset prices will keep increasing forever and where the unemployment rate gets closer and closer to a lower bound.

      We are not done yet. Now suppose that everybody in this economy reads the Wall Street Journal. Every period the WSJ gives an optimistic or a pessimistic signal and agents in the economy believe that this signal is accurate. This signal can act as a coordinating device, switching the economy from the non-stationary explosive growth path to one of the many the stationary high unemployment paths. Using the terminology of Dave Cass and Karl Shell, I will call this shock, *extrinsic uncertainty*. (Shocks to the fundamentals are called *intrinsic*.) Since, in the non-stationary equilibrium, everyone understands that a crash can occur; assets that will fall in value as a consequence of a crash, triggered by extrinsic uncertainty, will earn a high premium.

      Is it reasonable to impose rational expectations in this environment? I confess to the sin of having thought mostly about models with a representative agent and in those worlds, it is not clear that one can distinguish a rational expectations equilibrium from a non-rational expectations equilibrium. But as Rajiv points out (see the post below) trade is inconsistent with common priors and one might take that as evidence against rational expectations. A more substantive issue is, that if either intrinsic or extrinsic shocks are non-stationary, no agent can ever know the true probability of a given event.

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    8. Roger,

      Thanks again for your replies. I'm a little reluctant to get into this further without having read your papers (which I will do, hopefully soon). But I have to say, while I don't dispute the validity or usefulness of what you're doing, I still think there's a danger of incoherence if we are not very careful about logical vs historical time.

      First of all, I have to point out that in your post at Noah's, you referred to an Arrow-Debreu world where there is a singe market for goods indexed over all periods and all states of nature. One CANNOT speak of equilibria in such a market as succeeding in each other chronologically. In an Arrow-Debreu world there is only ever a single equilibrium, you cannot speak of it changing, because allocation decisions for all periods have already been made.

      Now, in your comments here, you have separate markets in each period with trades only in that period. That's better, in the sense that it allows for the possibility of talking about different equilibria succeeding each other chronologically. But one has to be very clear about which assumptions are in use.

      This is the problem I see with your story as you're telling it here. (And again, I realize that blog comments are not the best medium for this and I need to read your articles.)

      Yes, it is possible to tell a story about multiple rational-expectations equilibria in the sense that, given the same fundamentals, many different states of the world are consistent with expectations of that state of the world. Expectations are rational in the sense that realized states are the same as expected states *in logical time*. But this is no longer true when we switch to chronological time and consider equilibria succeeding each other. If state A is consistent with expectations of state A obtaining forever, then when the transition to state B happens some expectations will turn out to have been false. OK, you say, then people expect some mix of states A and B. Well then, state A, considered alone, is no longer an equilibrium, since if it continues forever people's expectations of sometimes being in state B will be false. On the other hand, if there is a unique true mix of A and B that everyone expects, then you are back in a single equilibrium.

      The only way out of this problem is if people's choice of actions that leads to state A or B this period is unaffected by what they think the state will be next period. but that is assuming away the links between present and future that are the reason we are talking about expectations in the first place.

      If people in state A expect A to obtain forever, then their expectations will be false when A switches to B. On the other hand, if people expect A and B to alternate in some way, then A, taken alone, is not an equilibrium. This is not a problem when you think of A and B as logical alternatives but it is a (possibly insurmountable) problem when you think of A and B as succeeding each other in historical time.

      I think there is a rather profound problem here which is that life, and capitalism in particular, require us to make commitments based on future states which we simply cannot have rational beliefs about. From our point of view as social scientists, this means we have no choice but to ask how people actually have formed their beliefs historically.

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    9. Josh, I think that what you're missing here is that in models with indeterminacy, there are multiple paths for any given history that are *all* equilibria. Each of these paths is consistent with self-fulfilling beliefs. This indeterminacy persists over time, so the selection of a given path from time t to t+s does not pin down the continuation path after t+s. It could be argued that under such conditions coordination on an equilibrium path would be even less likely than in the case of a unique equilibrium, but the mere existence of indeterminacy doesn't involve any logical problems as far as I can see.

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    10. Well, if you think I am mistaken then I probably am.

      I do undertand the idea of multiple equilibria. But I still don't see how it solves the problem. At time t, people have expectations about the entire future. Either those expectations are realized, in which case we cannot speak of one equilibrium transitioning to another one, since a given equilibrium includes its whole future. Or those expectations are not realized, in which case they were not rational in the sense being used here.

      What am I not seeing?

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    11. You're right about this: strictly speaking, we cannot speak of one equilibrium transitioning to another since the transition itself is part of an equilibrium trajectory and is foreseen (as a possibility) before it occurs. What we can do is to compare different equilibrium trajectories: a steady growth path with low unemployment versus a path in which steady growth is interrupted and employment rises and persists. Both are equilibrium paths, so speaking of a transition from one equilibrium to another is imprecise and a bit misleading. You're right to point this out. But I don't think there's a logical fallacy in the underlying model.

      My objection to RE is on empirical grounds - it can't explain both AIG and Paulson unless they had identical beliefs and identical expected payoffs ex-ante, and would therefore have been happy to switch places. And I don't think we can understand the crisis without a theory that allows for AIG and Paulson to co-exist.

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    12. JW, I think the issue is Arrow-Debreu equilibria versus Hicksian temporary equilibria. Suppose that every day the WSJ (to continue Farmer's example) gives a positive or negative signal. If it gave a positive (negative) signal yesterday it will give a positive (negative) signal today with probability 0.9, and this is common knowledge. The structure of the economy is such that multiple equilibria are possible. Suppose the Arrow-Debreu equilibrium that actually obtains is the one in which everyone believes the WSJ. We stay in the same Arrow-Debreu equilibrium for all time - in this equilibrium people believe that if the economy is booming today, it will be booming tomorrow with 90% probability, etc. But the Hicksian 'temporary equilibrium' (the pattern of goods/assets trade on a particular market day) will randomly transition from high to low. So you are absolutely right that there is only one Arrow-Debreu equilibrium; however, we do transition between Hicksian temporary equilibria.

      "I think there is a rather profound problem here which is that life, and capitalism in particular, require us to make commitments based on future states which we simply cannot have rational beliefs about. From our point of view as social scientists, this means we have no choice but to ask how people actually have formed their beliefs historically." - I could not agree more with this statement.

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  4. RF: "In the context of a competitive economy with a large number of players, Nash equilibrium collapses to the notion of perfect competition."

    Sadly, no.

    I have a pretty good grasp of the microeconomics of industrial organization, and it constantly calls into question the competence -- even the sanity -- of macroeconomists. Everywhere about us, there are markets, which do not clear. There are many salient examples of industries, which remain in a persistent disequilibrium for decades on end, as they pursue increasing returns and technological advance. And, Nash equilibria for large numbers do not "collapse" to perfect competition -- never have, never will.

    Farmer's agenda, like the Lucas agenda, seems to be focused on models of economies, which could not possibly exist. That might not be a bad thing, as a matter of theory, but it is very, very bad if you do not understand that that is what you are doing. Perfect competition is a model of a market that could not possibly exist, but it is still a useful theory. Arrow-DeBreu is a model of an economy that could not possibly exist, but it is useful as theory.

    Lucas with a representative agent is just a fancy way of making an elementary error of composition. Perhaps, Farmer, in his sweetly subversive way, praises Lucas, while burying disequilibrium, as Antony praised Brutus, while burying Caesar. That would be my hope. My fear is that he simply doesn't know what he's talking about.

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    1. Bruce
      Here is a response I made to that same point on Noahpinion.

      @Keshav A number of responders have criticized me for confounding Nash equilibrium with Walrasian equilibrium and the point is well taken. You are correct to point to three different uses of the equilibrium concept and I do not want to disparage a research agenda that seeks to investigate, more closely, the connections between (1) (Arrow Debreu Equilibrium) and (2) (Nash Equilibrium).

      But I do not think that that agenda will help us to fix the problems with existing DSGE models in macro. Lucas pointed out that we cannot observe a market in disequilibrium. He was right. Indeed, in the sense in which the word is used in macroeconomics, we cannot observe a market. The best we can hope for is to observe sequences of trades and the prices at which those trades take place.

      How should we organize our observations? In my view, it is helpful to view agents as goal oriented decision makers who take actions to achieve what they perceive to be optimal outcomes. In a game theoretic formulation of an economy, we would need to specify the information structure of each agent, the timing of moves, the payoff structure of the game and the space of actions. That way of proceeding would explain data as a consistent set of plans; definition (2) in your taxonomy. (see the comment by Keshav on Noahpinion for my references to points (1) (2) and (3)).

      The general equilibrium approach of Debreu Chapter 7 ( your definition (1)), takes a short cut. No single agent sets prices; but prices are chosen such that quantities, prices and expectations of future prices are mutually consistent. In this rational expectations equilibrium environment, the data are explained as a sequence of choices that obey the equilibrium consistency requirements at all points in time and in all states of nature.

      Is that a useful way to seek to understand the world? That depends on two things. First, does it help us to understand data? Second, does it provide us with a guide to good policy? If one sticks to standard DSGE models with a unique equilibrium ( in the sense of (1) in (2)), a case can be made that the answer to both questions is no. But the same charge cannot be leveled at DSGS models with incomplete labor markets.

      Explaining persistent unemployment as the stubborn failure of prices to adjust to their equilibrium values will go only so far. At some point we should entertain the idea that persistent unemployment is itself an equilibrium phenomenon, and at that point, the idea of trading at 'false prices', in other words 'disequilibrium', becomes an impediment to our understanding.

      I should add that, by multiple equilibrium models, I do not restrict myself to dynamical systems where there is more than one steady state. Incomplete labor market models have a continuum of steady state equilibria as well as a continuum of non-stationary equilibria, where equilibrium here is in your sense (1). When closed with a specification of how agents form beliefs, these models are capable of understanding economic data and of guiding economic policy.

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    2. Roger, I just posted this reply on Mark Thoma's blog:

      I'm coming at this from a finance perspective where prices are generally flexible but heterogeneity is central: one needs to account for massive directional bets on both sides of the housing market, AIG betting on limited default and Paulson on a collapse for example.

      Hicks' notion of temporary equilibrium is very useful in understanding this because it allows for the clearing of spot markets (and any futures markets that exist) without requiring expectational consistency across individuals. In fact, temporary equilibrium is well-defined for any specification of expectations, including naive or adaptive expectations.

      The question is, what theory of expectations should we use to close the temporary equilibrium model. RE is one way to go but this would imply that both Paulson and AIG had the same beliefs about the housing market, which means that their ex-ante expected profits would be the same. From this perspective, Paulson just got lucky. Maybe so, but I think that allowing for heterogenous priors evolving under pressure of differential profitability is at the very least a promising direction for future research.

      I appreciate the enormous new insight that RE models with multiple equilibrium paths can bring to the understanding of the economy and am an admirer of your work as you know. But I don't think that models with mutually incompatible beliefs are an "irrelevant distraction" as your original post on Noah's blog seemed to suggest. Perhaps you were referring only to the non-market-clearing aspect of disequilibrium but a reader may have interpreted your remark to refer to all departures from RE as well.

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  5. The vast majority of trades in currency markets, commodity futures, stock options, and credit default swaps are zero sum speculative bets in which both parties are taking a price view (neither is hedging). These price views are inconsistent by definition, since they are taking opposite sides of the same contract. It is impossible for the subjective beliefs of both parties to such a contract to correspond to any objective probability distribution to which their actions give rise: at most one of them can have self-fulfilling expectations. One famous consequence of Aumann's (1976) Theorem is that there can be no speculative bets involving two parties, no matter how different their information may be, as long as they are updating based on a common prior. And once you allow for heterogeneous priors, individual plans will no longer be mutually consistent.

    The really interesting question to me is what determines the distribution of (mutually inconsistent) beliefs, and how this distribution evolves over time. To a first approximation, I think that the belief distribution evolves based on differential profitability: successful beliefs proliferate, regardless of whether those holding them were right or just fortunate. Woodford's reply to John Kay on the INET blog is worth reading. In particular:

    "The macroeconomics of the future, I believe, will still make use of general-equilibrium models in which the behavior of households and firms is derived from considerations of intertemporal optimality, but in which the optimization is relative to the evolving beliefs of those actors about the future, which need not perfectly coincide with the predictions of the economist’s model. It will therefore build upon the modeling advances of the past several decades, rather than declaring them to have been a mistaken detour. But it will have to go beyond conventional late-twentieth-century methodology as well, by making the formation and revision of expectations an object of analysis in its own right, rather than treating this as something that should already be uniquely determined once the other elements of an economic model (specifications of preferences, technology, market structure, and government policies) have been settled."

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    1. "The vast majority of trades in currency markets, commodity futures, stock options, and credit default swaps are zero sum speculative bets in which both parties are taking a price view (neither is hedging)."

      This sweeping generalization seems curiously unfounded to me, though I appreciate the gist of what you say about the implications of a diversity of views for rational expectations. (Didn't Farmer just say something similar?).

      First of all, what if most trades were hedged? So? If I do a hedge, it is because I recognize that the informational advantage I (think I) enjoy, which is leading me to bet, is narrow and isolated, while there are lots of things that will happen, which I will not be any better at anticipating than "the market" as a whole, or, maybe I do think I know better, but am not confident that I can stay solvent as long as the market can be "wrong". I recognize my own bounded rationality and uncertainty in general, and so I follow a rule. (But, everyone is guided by rules all the time in everything, for exactly those reasons.)

      Second of all, even in the absence of formal hedging, everyone has some portfolio diversification; most bets are made by agents handling other people's money; credit, re-hypothecation, etc. underlie much of the generation of funds to bet, etc. So, even without a formal, classic hedge structure, everyone's behavior in speculative markets is "hedged" in a general sense; everyone's beliefs are conditional on beliefs about everyone else's beliefs.

      Third, lots of bets are not actually taking a price view, at all. "Credit default swaps" trade an income stream against the possibility of a black swan event, and, if informed reporters are to be believed, have to do, customarily, with regulatory arbitrage. I'm probably garbling the interpretation terribly, but I get the feeling that the advantage of a buying a CDS to the fiduciary agent is relief from having to take a price view at all. Both sides of the trade may share the belief that the default will never take place, no?

      To me, the interesting issue wouldn't, usually, be the distribution of beliefs, or the evolution of the distribution of beliefs, per se, but rule-driven behavior in institutions for reconciling that diversity of beliefs. It is the rules, in observance and in the breach, that constrain the players, and it is the rules, which are being strategically revised in reform or eroded and corrupted in practice. A "credit default swap" isn't a fixed feature of nature, and, as a first-order approximation, I doubt that being right is nearly as profitable as being willing to do wrong.

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    2. Bruce, regarding the prevalence of speculation, think of AIG vesus Paulson, massive directional bets on opposite sides of the housing market. Or look at the volume of trading in oil futures, which swamps daily production. Any theory of expectations has to account for heterogenous beliefs of this kind and on this scale.

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    3. "Third, lots of bets are not actually taking a price view, at all. "Credit default swaps" trade an income stream against the possibility of a black swan event, and, if informed reporters are to be believed, have to do, customarily, with regulatory arbitrage."

      Bruce,

      I am not sure that is entirely true. CDS tend to involve collateral adjustments in response to changes in the market value of the underlying instrument that are marked-to-market frequently, if not daily. They don't usually work like an insurance policy, where the buyer pays a fixed stream of payments and only gets paid their losses in the event of a credit event. The seller also has to post collateral with the buyer, and as the credit quality of the underlying declines, they have to continue to post more collateral. Cash is paid regularly back-and-forth between the parties depending on changes in credit quality. That was what happened with AIGFP and Goldman (and Paulson) - AIGFP was not sitting pretty until a bunch of the MBS portfolio defaulted, they were getting bleed to death with collateral calls as the underlying MBS kept getting downgraded.

      CDS in actual use are a bet on the price of the credit quality of an asset - they are like treasury strips, only instead of stripping coupon from principle, they strip credit risk from the value of the asset. Both sides are making opposing bets on price of the credit quality of the underlying asset.

      With mark-to-market, everything becomes a bet on the continuously-adjusted market price.

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    4. I don't mean to be tedious, but I was focused exclusively on the curious claim in the first sentence of Rajiv's comment that most "bets" are unhedged. It seems curious to me for two reasons. First, because I do not think it true, and secondly, and most importantly, hedging would seem to be critical to the persistence of a diversity of views, so, why the qualification otherwise?.

      Look, in an imaginary world, where everyone made unhedged bets in complete and informationally efficient financial markets, all private information would be revealed, and everyone's expectations, informed by the Market process, would "collapse" into market price. Presumably, there could still be a diversity of interests pressing on market price from different directions, but, there would no longer be a diversity of expectations. The perfectly functioning markets would bring together the Hive Mind to coincide with the expectations of a Representative Agent, with complete knowledge and unbounded rationality.

      Here in the real world, everyone is always hedging, in a broad sense. Interests and expectations are bound up together in the ownership of assets, some "real" and some purely financial claims, and some financial claims tied to contingent claims on "real" assets. There are no pure two-way bets; everyone is making a three-way bet against the Market and one's other bets and ownership claims, where the persistence of a diversity of expectations depends particularly upon the view that the institution of the Market is falling short, failing. That's what a hedge is, after all: it is structuring a "bet" to take advantage of the extent to which the Market is failing to arrive at an informationally exhaustive balance of interests in price.

      To take up the example of Paulson "shorting" the mortgage-backed securities market illustrates the point. The sucker institution on the losing end of Paulson's trades was betting that the rules were being followed in the synthesizing and pricing of the securities, and the "market price" was "right", in the sense of informationally efficient, and therefore, could be relied upon in the routine bureaucratic business of putting money into large, diversified portfolios. Paulson was betting the Market was wrong; that the rules were not being followed -- which he knew, because he had investigated the matter and because he, himself, was cleverly inducing a bank to break the rules.

      So, it is not just that Paulson thinks "up" and someone else thinks, "down". It is that Paulson thinks the Market, qua institution, is broken, and someone else (a bureaucrat) is following rules premised on the institution functioning well.

      A theory of expectations, which encompasses a persistent diversity of views (persistent beyond their reconciliation in market price) must take into account the very particular circumstance of institutional failure, of rules being broken, of strategies of cheating.

      If Paulson had devised a conventional short, his "bets" would have immediately depressed the price of mortgage-backed securities "revealing" his information. But, he didn't. The vehicles he used had the effect of enabling the synthesizing of more bad securities -- actually worse securities. He was betting not only that the Market, qua institution, was broken, but in a way designed to take advantage of making that broken-ness apparent. Presumably, he was acting to take advantage of the Minsky Moment, when the Market returned to proper functioning, and the diversity of expectations again collapsed to an informationally efficient price, accepted in common by all.

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  6. Do you feel that new ideas in ambiguity aversion and other non-expected-utility decision theories have much to contribute here?

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    1. This isn't really my area -- I'm strictly macro. But personally, I am rather skeptical that the theory of economic agents as intertemporal optimizers with rational expectations can be replaced by a better theory of decisionmakers in general. My own view is that we need to accept that the question of how people make decisions does not have any general answer; it is historically specific and highly conventional.

      It would also be a good idea to distinguish more clearly between individuals/households and (capitalist) firms, rather than merging them as generic agents. Firms are subject to a process of selection whereby those that realize higher profits grow and those that realize lower profits shrink relatively and eventually disappear. There's no similar selective process causing individual human beings to maximize anything. Corollary to this, I think the idea of utility has no place in positive economics.

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  7. "we need to accept that the question of how people make decisions does not have any general answer; it is historically specific and highly conventional."

    So it would be impossible to write the "belief function"/algorithm that Mr. Farmer suggests is necessary to close the temporary-equilibrium model?

    Jumping a couple of concepts ahead, this makes me think of weather modeling. In that endeavor, what counts as an "exogenous shock." Nothing, really. So the attempt to model it via equilibria seems (obviously) quixotic at best...

    Probably not adding much to the high level of the discussion here, but for what it's worth.

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