As I’ve pointed out many times on this blog, equilibrium is an extremely important, but very problematic, concept in economic theory. What economists even mean when they talk about equilibrium is often unclear and how the concept relates to the real world as opposed to an imagined abstract world is even less clear. Nevertheless, almost all the propositions of economic theory that are used by economists in analyzing the world and in making either conditional or unconditional predictions about the world or in analyzing current or historical events are based on propositions of economic theory deduced from the theoretical analysis of equilibrium states,
Last year I wrote a paper for a conference marking the hundredth anniversary of Carl Menger’s death in 1921 and 150 years after his seminal work launching, along with Jevons and Walras, what eventually became neoclassical economic theory. Here is a link to that paper. Of late I have been revising the paper and I have now substantially rewritten (and I hope improved) one of the sections of the paper discussing Franklin Fisher’s important work on the stability of general equilibrium, which I have been puzzling over and writing about for several years, e.g., here and here, as well as chapter 17 of my book, Studies in the History of Monetary Theory: Controversies and Clarifications.
I’ve recently been revising that paper — one of a number of distractions that have prevented me from posting recently — and have substantially rewritten a couple sections of the paper, especially section 7 about Fisher’s treatment of the stability of general equilibrium. Because I’m not totally sure that I’ve properly characterized Fisher’s own proof of stability under a different set of assumptions than the standard treatments of stability, I’m posting my new version of the section in hopes of eliciting feedback from readers. Here’s the new version of section 7 (not yet included in the SSRN version).
Unsuccessful attempts to prove, under standard neoclassical assumptions, the stability of general equilibrium led Franklin Fisher (1983) to suggest an alternative approach to proving stability. Fisher based his approach on three assumptions: (1) trading occurs at disequilibrium prices (in contrast to the standard assumption that no trading takes place until a new equilibrium is found with prices being adjusted under a tatonnement process); (2) all unsatisfied transactors — either unsatisfied demanders or unsatisfied suppliers — in any disequilibrated market are all either on the demand side or on the supply side of that market; (3) the “no favorable surprises” (NFS) assumption previously advanced by Hahn (1978).
At the starting point of a disequilibrium process, some commodities would be in excess demand, some in excess supply, and, perhaps, some in equilibrium. Let Zi denote the excess demand for any commodity, i ranging between 1 and n; let commodities in excess demand be numbered from 1 to k, commodities initially in equilibrium numbered from k+1 to m, and commodities in excess supply numbered from m+1 to n. Thus, by assumption, no agent had an excess supply of commodities numbered from 1 to k, no agent had an excess demand for commodities numbered from m+1 to n, and no agent had an excess demand or excess supply for commodities numbered between k+1 and m.
Fisher argued that, with prices rising in markets with excess demand and falling in markets with excess supply, and not changing in markets with zero excess demand, the sequence of adjustments would converge on an equilibrium price vector. Prices would rise in markets with excess demand and fall in markets with excess supply, because unsatisfied demanders and suppliers would seek to execute their unsuccessful attempts by offering to pay more for commodities in excess demand, or accept less for commodities in excess supply, than currently posted prices. And insofar as those attempts were successful, arbitrage would cause all prices for commodities in excess demand to increase and all prices for commodities in excess supply to decrease.
Fisher then defined a function in which the actual utility of agents after trading would be subtracted from their expected utility before trading. For agents who succeed in executing planned purchases at the expected prices, the value of the function would be zero, but for agents unable to execute planned purchases at the expected prices, the value of the function would be positive, their realized utility being less than their expected utility, as agents with excess demands had to pay higher prices than they had expected and agents with excess supplies had to accept lower prices than expected. As prices of goods in excess demand rise while prices of goods in excess supply fall, the value of the function would fall until equilibrium was reached, thereby satisfying the stability condition for a Lyapunov function, thereby confirming the stability of the disequilibrium arbitrage proces.
It may well be true that an economy of rational agents who understand that there is disequilibrium and act arbitrage opportunities is driven toward equilibrium, but not if these agents continually perceive new previously unanticipated opportunities for further arbitrage. The appearance of such new and unexpected opportunities will generally disturb the system until they are absorbed.
Such opportunities can be of different kinds. The most obvious sort is the appearance of unforeseen technological developments – the unanticipated development of new products or processes. There are other sorts of new opportunities as well. An unanticipated change in tastes or the development of new uses for old products is one; the discovery of new sources of raw materials another. Further, efficiency improvements in firms are not restricted to technological developments. The discovery of a more efficidnt mode of internal organization or of a better way of marketing can also present a new opportunity.
Because a favorable surprise during the adjustment process following the displacement of a prior equilibrium would potentially violate the stability condition that a Lyapunov function be non-increasing, the NFS assumption is needed for a proof that arbitrage of price differences leads to convergence on a new equilibrium. It is not, of course, only favorable surprises that can cause instability, inasmuch as the Lyapunov function must be positive as well as being non-increasing, and a sufficiently large unfavorable surprise would violate the non-negativity condition.[1] While listing several possible causes of favorable surprises that might prevent convergence, Fisher considered the assumption plausible enough to justify accepting stability as a working hypothesis for applied microeconomics and macroeconomics.
However, the NFS assumption suffers from two problems deeper than Fisher acknowledged. First, it reckons only with equilibrating adjustments in current prices without considering that equilibrating adjustments are required in agents’ expectations of future prices on which their plans for current and future transactions depend. Unlike the market feedback on current prices in current markets conveyed by unsatisfied demanders and suppliers, inconsistencies in agents’ notional plans for future transactions convey no discernible feedback, in an economic setting of incomplete markets, on their expectations of future prices. Without such feedback on expectations, a plausible account of how expectations of future prices are equilibrated cannot — except under implausibly extreme assumptions — easily be articulated.[2] Nor can the existence of a temporary equilibrium of current prices in current markets, beset by agents’ inconsistent and conflicting expectations, be taken for granted under standard assumptions. And even if a temporary equilibrium exists, it cannot, under standard assumptions, be shown to be optimal. (Arrow and Hahn, 1971, 136-51).
Second, in Fisher’s account, price changes occur when transactors cannot execute their desired transactions at current prices, those price changes then creating arbitrage opportunities that induce further price changes. Fisher’s stability argument hinges on defining a Lyapunov function in which actual prices of goods in excess demand gradually rise to eliminate excess demands and actual prices of goods in excess supply gradually fall to eliminate those excess demands and supplies. But the argument works only if a price adjustment in one market caused by a previous excess demand or excess supply does not simultaneously create excess demands or supplies in markets not previously in disequilibrium, cause markets previously in excess demand to become markets in excess supply, or cause excess demands or excess supplies to increase rather than decrease.
To understand why, Fisher’s ad hoc assumptions do not guarantee that the Lyapunov function he defined will be continuously non-increasing, it will be helpful to refer to the famous Lipsey and Lancaster (1956) second-best theorem. According to their theorem, if one optimality condition in an economic model is unsatisfied because a relevant variable is constrained, the second-best solution, rather than satisfy the other unconstrained optimum conditions, involves revision of at least some of the unconstrained optimum conditions to take account of the constraint.
Contrast Fisher’s statement of the No Favorable Surprise assumption with how Lipsey and Lancaster (1956, 11) described the import of their theorem.
From this theorem there follows the important negative corollary that there is no a priori way to judge as between various situations in which some of the Paretian optimum conditions are fulfilled while others are not. Specifically, it is not true that a situation in which more, but not all, of the optimum conditions are fulfilled is necessarily, or is even likely to be, superior to a situation in which fewer are fulfilled. It follows, therefore, that in a situation in which there exist many constraints which prevent the fulfilment of the Paretian optimum conditions the removal of any one constraint may affect welfare or efficiency either by raising it, by lowering it, or by leaving it unchanged.
The general theorem of the second best states that if one of the Paretian optimum conditions cannot be fulfilled a second-best optimum situation is achieved only by departing from all other optimum conditions. It is important to note that in general, nothing can be said about the direction or the magnitude of the secondary departures from optimum conditions made necessary by the original non-fulfillment of one condition.
Although Lipsey and Lancaster were not referring to the adjustment process triggered by an adjustment process that follows a displacement from a prior equilibrium, nevertheless, their discussion implies that the stability of an adjustment process depends on the specific sequence of adjustments in that process, inasmuch as each successive price adjustment, aside from its immediate effect on the particular market in which the price adjusts, transmits feedback effects to related markets. A price adjustment in one market may increase, decrease, or leave unchanged, the efficiency of other markets, and the equilibrating tendency of a price adjustment in one market may be offset by indirect disequilibrating tendencies in other markets. When a price adjustment in one market indirectly reduces efficiency in other markets, the resulting price adjustments that follow may well trigger yet further indirect efficiency reductions.
Thus, in adjustment processes involving interrelated markets, price changes in one market can cause favorable surprises in other markets in which prices are not already at their general-equilibrium levels, by indirectly causing net increases in utility through feedback effects on related markets.
Consider a macroeconomic equilibrium satisfying all optimality conditions between marginal rates of substitution in production and consumption and relative prices. If that equilibrium is subjected to a macoreconomic disturbance affecting all, or most, individual markets, thereby changing all optimality conditions corresponding to the prior equilibrium, the new equilibrium will likely entail a different set of optimality conditions. While systemic optimality requires price adjustments to satisfy all the optimality conditions, actual price adjustments occur sequentially, in piecemeal fashion, with prices changing market by market or firm by firm, price changes occurring as agents perceive demand or cost changes. Those changes need not always induce equilibrating adjustments, nor is the arbitraging of price differences necessarily equilibrating when, under suboptimal conditions, prices have generally deviated from their equilibrium values.
Smithian invisible-hand theorems are of little relevance in explaining the transition to a new equilibrium following a macroeconomic disturbance, because, in this context, the invisible-hand theorem begs the relevant question by assuming that the equilibrium price vector has been found. When all markets are in disequilibrium, moving toward equilibrium in one market will have repercussions on other markets, and the simple story of how price adjustment in response to a disequilibrium restores equilibrium breaks down, because market conditions in every market depend on market conditions in every other market. So, unless all optimality conditions are satisfied simultaneously, there is no assurance that piecemeal adjustments will bring the system closer to an optimal, or even a second-best, state.
If my interpretation of the NFS assumption is correct, Fisher’s stability results may provide support for Leijonhufvud’s (1973) suggestion that there is a corridor of stability around an equilibrium time path within which, under normal circumstances, an economy will not be displaced too far from path, so that an economy, unless displaced outside that corridor, will revert, more or less on its own, to its equilibrium path.[3]
Leijonhufvud attributed such resilience to the holding of buffer stocks of inventories of goods, holdings of cash and the availability of credit lines enabling agents to operate normally despite disappointed expectations. If negative surprises persist, agents will be unable to add to, or draw from, inventories indefinitely, or to finance normal expenditures by borrowing or drawing down liquid assets. Once buffer stocks are exhausted, the stabilizing properties of the economy have been overwhelmed by the destabilizing tendencies, income-constrained agents cut expenditures, as implied by the Keynesian multiplier analysis, triggering a cumulative contraction, and rendering a spontaneous recovery without compensatory fiscal or monetary measures, impossible.
[1] It was therefore incorrect for Fisher (1983, 88) to assert: “we can hope to show that that the continued presence new opportunities is a necessary condition for instability — for continued change,” inasmuch as continued negative surprises can also cause continued — or at least prolonged — change.
[2] Fisher does recognize (pp. 88-89) that changes in expectations can be destabilizing. However, he considers only the possibility of exogenous events that cause expectations to change, but does not consider the possibility that expectations may change endogenously in a destabilizing fashion in the course of an adjustment process following a displacement from a prior equilibrium. See, however, his discussion (p. 91)
How is . . . an [“exogenous”] shock to be distinguished from the “endogenous” shock brought about by adjustment to the original shock? No Favorable Surprise may not be precisely what is wanted as an assumption in this area, but it is quite difficult to see exactly how to refine it.
A proof of stability under No Favorable Surprise, then, seems quite desirable for a number of related reasons. First, it is the strongest version of an assumption of No Favorable Exogenous Surprise (whatever that may mean precisely); hence, if stability does not hold under No Favorable Surprise it cannot be expected to hold under the more interesting weaker assumption.
[3] Presumably because the income and output are maximized at the equilibrium path, it is unlikely that an economy will overshoot the path unless entrepreneurial or policy error cause such overshooting which is presumably an unlikely occurrence, although Austrian business cycle theory and perhaps certain other monetary business cycle theories suggest that such overshooting is not or has not always been an uncommon event.
Uneasy Money 
Reblogged this on Calculus of Decay .
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To me, this line of research, which I think is quite worthwhile, points to the limits of using the general equilibrium metaphor. The metaphor is useful for spurring the imagination, but should not be used as a standard for evaluating the supposed shortcomings of real economies. Rather, evaluation should occur through comparing institutional arrangements that have really existed or are realistically close to those that have. A market economy allows for prices to act as signals to elicit and coordinate knowledge, not as well as in the imaginary construct of general equilibrium, but much better than in the realistic alternative of central economic planning. Economic coordination in a market economy, though bumbling and imperfect, is still good enough to produce amounts of wealth that would have seemed fantastical to our forbears.
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Kurt, Thans for your thoughtful comment. I certainly agree with your point about the superiority of a decentralized market economy to a centrally planned economy. And it was an important contribution by Hayek and others to explain why it would be dangerous to rely on central planning to organize and direct the bulk of economy activity in an economy. My point is not to challenge the general presumption in favor of a market economy. But the general equilibrium model has become the benchmark for macroeconomic modeling in a way that is much more than metaphorical, and I think that is a very unfortunate path for macroeconomics to have followed.
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David,
It has been interesting and frustrating watching you grapple over time with these questions around equilibrium. I get confused with your work when you appear to crossover from the abstract world to the real world. Sometimes it is difficult to know where you are. And in particular, when time is introduced, that is the analysis involves dynamics, clarity is lost even more.
It seems to me there are three general areas to be considered.
Firstly, there is the question of whether a particular model’s conclusions are a correct and logical extension of the model’s assumptions. A subsidiary question here is how does the introduction of time affect the outcome of models.
It seems to me that thinking about these questions is merely about agreeing that the model’s logic is sound. An infinite number of models can be devised. Change the assumptions and the logical outcome changes.
Fretting about which model is more useful than another has more to do with the second area which I discuss below. I can’t see the point of evaluating one model against the other without reference to their applicability to the real world.
Introducing time increases the complexity by an order and tends to confuse the analysis and stretch the analysis into delving with real world matters. It becomes increasingly difficult to discern where a discussion is situated, in the abstract or in reality.
Secondly, the next question concerns whether the model closely or not reflects the real world.
This area is essentially about whether the assumptions made reflect reality. It becomes a question of empirics, i.e. the model’s outcomes are tested against reality.
And thirdly, whether a model is useful for describing only microeconomic questions or whether it can be also or exclusively applied to macroeconomic questions. You always stray from the one area to the other.
You might remember we have discussed these sorts of issues before. As far as I can see, micro questions are distinct from macro questions. As I understand it, you believe that changes in relative prices can have an impact on the general level of employment, for instance, whereas I can’t see how this is possible.
So when I read your necessarily brief dissertations on these matters I get tangled up in all these issues. This may just be a function of limitations in my powers of conceptual analysis. (Of course, I would say not. 🙂 ) I think, at a minimum, you have to carefully distinguish which terrain you are traversing, the abstract world or the real world and then stick to it and make clear when you divert to the alternate direction.
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Thanks for your thoughtful comment, Henry. I’m glad that your still trying to figure out what I’m talking about. I think you are right to say that we want models that are coherent, useful, relevant and empirically realistic. It is a great challenge to come up with models that have all those qualities. I don’t think that current macro models fit the bill. Although I have high regard for Keynes’s contributions for which we can still learn a lot, I think there are too many issues with his model for us to rely on it. My work now is aimed mainly at criticizing the current standard macro models to the best of my ability. Unfortunately I am not competent to do more th,an make suggestions about possible alternatives, but not to actually develop one of my own. I do believe that it is important to try to gain an understanding of the interaction between micro and macro models and that the two categories have to be intergrated in some way. I don’t accept that there is no interaction between mind and body and I don’t accept that there is no interaction between micro and macroeconomic models. But explaining the nature of that interaction is very hard. Maybe a ne generation of economists will be able to do a better job than I or my predecessors have done, but they need to be persuaded that it is worth their time and effort to do so.
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David,
I have a copy of Fisher’s book but I have not read it yet. I have had a nose around some of the early chapters and it seems to me that it is also difficult to discern when he is talking in the abstract or in reality when dealing with the stability of equilibrium. He seems to wander. I might be wrong. Closer reading required.
I have outlined before my way of dealing with the dichotomy between micro and macro.
Micro models are structured such that they begin with the assumption that there is a fixed bundle of goods and resources to be allocated. Another way of saying this is to assume that income and the general level of output is fixed. That is, the general level of spending is fixed and what is under consideration is how this spending is allocated between competing uses. So, the aim of micro models is to understand how these fixed resources are allocated to various uses. The general level of unemployment cannot be addressed under these circumstances, only the distribution of employment over various uses. The abstract way to express this is to say the the production possibility frontier is singular and fixed in the output space.
And it needs to be said that where periods over time are in consideration, the decision to allocate resources from one period to another are decisions about the intertemporal allocation of resources. These are not decisions which alter the overall level of resource usage over time. The decisions relate to the allocation of a given bundle of resources over time. These are micro considerations. Consideration here is given to relative prices across time periods.
Whereas, macro models rely on the notion that income and the general level of output is not fixed, that is, why the general level of unemployment is a matter that can be validly examined in a macro model. The abstract way of considering this is to view the production possibility frontier as being mobile. This mobility is a function of the general level of spending. The general level of spending shifts the PPF in (or down) and out (or up) of the output space.
So micro models specify where the economy sits on a given PPF. Macro models specify where a PPF sits in the output space.
It seems to me that what is telling and important is to consider the kinds of decisions that economic actors (consumers and producers) make. To me this is the crux of the problem.
Consider a consumer. A consumer has under consideration the purchase of shoes from either of two brands. This is a micro decision. The consumer is making a decision largely based on the relative price of the two pairs of shoes. It will, in the end, influence how production is allocated at a narrow level.
Then consider a consumer who wants to buy a pair of shoes. The consumer is concerned whether he has sufficient income that will enable him to purchase the shoes. He also has to consider not only his current income but also future income, so his expectations (he doesn’t know with certainty what his future income might be) about future income are also relevant. This is a decision of a completely different nature compared to the choice between two pairs of shoes. And when circumstances arising from the general economic situation impinge on the matter, it will that consumers at large will be looking at their spending decisions in that light. These are macro related decisions.
Consider a producer. Say a producer has to make an investment decision. He has to decide which machine would best suit his purpose. He is making a micro related decision. He is choosing between alternatives based largely on the relative prices of the alternatives. This decision might have implications for the number of people he employs. But it will not have implications for the general level of employment.
Now take a producer who is considering making an investment decision. He wants to buy a new machine but is uncertain about the future path of general economic conditions. So his current level of output and his expectations about future demand for his goods are in consideration. The general economic situation will be such that the economic outlook is optimistic or pessimistic, producers generally looking to make this kind of investment decision will be looking to their current output and expectations about future output. These kinds of decisions have a macro impact. They will have an impact on the general level of employment.
I would suggest this is the kind of format that might be useful, even necessary, in the examination of the relationship between micro and macro considerations.
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Henry, Thanks for your lengthy reply. I’m afraid I don’t see any common ground between us given this paragraph, which expresses exactly the strict dualism between micro and macro that I reject.
“Micro models are structured such that they begin with the assumption that there is a fixed bundle of goods and resources to be allocated. Another way of saying this is to assume that income and the general level of output is fixed. That is, the general level of spending is fixed and what is under consideration is how this spending is allocated between competing uses. So, the aim of micro models is to understand how these fixed resources are allocated to various uses. The general level of unemployment cannot be addressed under these circumstances, only the distribution of employment over various uses. The abstract way to express this is to say the the production possibility frontier is singular and fixed in the output space.”
You write as if there is a Platonic ideal micro and a Platonic ideal macro. I am not an idealist; I’m a nominalist. Where is it written that micro assumes that there is a fixed bundle of goods and resources to be allocated?
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David,
“You write as if there is a Platonic ideal micro.”
There is.
It’s called the theory of perfect competition which has strict assumptions – given technology, given endowment of resources, perfect knowledge, perfect foresight, no barriers to exit and entry, there is a large number of buyers and sellers all of whom are price takers. Indifference curves are defined by a strict set of rules. There are also rules about production functions.
And when I was taught micro, the pure theory of competition relaxed the perfect knowledge and perfect foresight assumptions.
“I’m a nominalist.”
I’m not exactly sure what that means but it sounds like you ignore many of the assumptions of perfect competition and that rigour is subsidiary in the analysis. This the kind of analysis found in a textbook by the likes of Alchian and Allen. (From memory, this was your seminal economic text.)
Scarce resource allocation theories generally rely on optimization. This requires a fixed resource endowment and PPF and a budget constraint ( a fixed level of income).
“Where is it written that micro assumes that there is a fixed bundle of goods and resources to be allocated?”
I think the “bundles of goods” approach is found in Walras? Optimization makes no sense without the notion of a given set of resources.
The difficulty I have in reading your work is having a clear idea where you are coming from and where abstraction ends and reality begins and what the level of abstraction exactly is.
The formal models yield clarity and as the assumptions are relaxed, analysis can go anywhere and almost seems rudderless and even devolves into a confused morass where any position could be argued.
And if you want to talk about stability of equilibrium in a simple way I ask myself have consumers spent there allocated budgets and are producers achieving their expected stock levels. If either one or both of these don’t apply then there will be adjustments until they do apply. How long such a process takes in the real world, I would say could not be predictable.
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No there is not an ideal Platonic version of micro. Platonic ideals are a figment of his and it seems your imagination. There are many versions of micro, and it is up to everyone to pick out a version that he wants to use or to invent a new version of his own. The perfect competition model has its place, but few working economists nowadays use it as more than a convenient heuristic in doing actual applied microeconomic analysis. Perfect competition is akin to the no friction assumption in classical mechanics. It is a convenient way to get results for certain problems, but it is useless in getting results for a whole range of other problems.
Walras wrote over a century ago. I’m not overly impressed with the course of economic theory in the century since he wrote, but I would never argue that that there hasn’t been any progress. Optimization is another form of ideal analysis, which has its place, but practical microeconomists understand that optimization is a benchmark that is rarely attined in the real world that they analyze.
Abstraction is a tool that we use to simplify problems to make them tractable. The art of economic analysis is to figure out when and where to use abstraction. Good economists know where, when and how to use abstraction. And that artistic sense is the product of experience and practice in economics just as it is in doing science or any other intellectual discipline. All economists engage in this kind of selection process and they very often disagree about their selections are good or bad. You are free to have your own opinion about my selctions.
In the real world, people don’t really know what their budgets are. They can only guess about their budgets because their budgets depend on too many factors that they cannot know, control, or predict in advance. Your presumption that microeconomics is only applicable in the context of full information and full optimization is simply wrong, and I’m growing weary of trying to explain to you why your ideas about what microeconomics is and how it is practiced are completely misguided.
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David,
“Platonic ideals are a figment of his and it seems your imagination. ”
Isn’t that what pure abstraction is, a figment of someone’s imagination? Anyway, no big deal.
I pretty much agree with everything you say down to the last paragraph.
“In the real world, people don’t really know what their budgets are.”
Yes, pretty much. However, in the abstract world they can be totally defined.
“Your presumption that microeconomics is only applicable in the context of full information and full optimization is simply wrong…”
I didn’t quite say that.
Firstly, a discussion is facilitated if it is clear whether the discussion is in the abstract or it is in the real and if it is in the abstract what the underlying assumptions are.
Optimization only makes sense when decisions are made under defined opportunity/possibility and defined constraint. It doesn’t mean a discussion can’t be had if these aren’t defined but it is facilitated when possibility and constraint are specified otherwise “optimization” just becomes another word used in a discussion.
What I am essentially saying is that it is difficult to have a shared discussion if there is no common ground, i.e. if there is no clear formulation/specification, no matter how realistic or abstract it is, of the question in hand.
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Pose a specific question and I will try to answer if I can. If I can’t, I’ll tell you. Your abstract complaints about abstractions are getting us nowhere.
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