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 