Basic Themes

Frank Hahn’s significance for Cambridge economics extends in an important way beyond his published works. I will begin with a compact survey of his contributions to economic theory in general, and then turn to his central role in, as I see it, saving Cambridge economics from itself.

There are a few fundamental themes that run through Hahn’s body of work although, like any talented economist, he would be visited by a stray idea or provoked by something in the literature, and produce an article off the main track. The largest of those central themes is the formal theory of general equilibrium in a system of many decentralized markets. The great monument to this theme is General Competitive Analysis (1971, though no doubt years in the making) written with Kenneth Arrow. I will return soon to a closer look at the contents of this important book. It is interesting, however, that Hahn’s first stab at general equilibrium theory came in 1952-1953 with a short article in the Review of Economic Studies called ‘The General Equilibrium Theory of Money: A Comment’ (Hahn 1952-1953). Monetary economics was another long thread running through Hahn’s work, extending at least until he joined Benjamin Friedman in editing the Handbook of Monetary Economics (Friedman and Hahn 1990), to which he contributed the article on ‘Liquidity’ (Hahn 1990).

A third enduring interest was the broad question of the stability of equilibrium, more or less any kind of interesting and well-defined equilibrium. The main example, of course, is the stability of competitive general equilibrium as prices and quantities vary. Any serious approach to the issue requires a specification of the model economy’s behavior out of equilibrium, and this tends to take the analysis away from the tried and true. Many economists duck the issue, but Hahn did not, and in fact made important contributions to be discussed later. He also wrote extensively and originally on the stability of growth models, but here the question is usually about the behavior and possible divergence of equilibrium paths. To take another example, he and I together once explored an analogous but unrelated matter, mentioned but not analyzed by Keynes: whether perfect flexibility of wages might lead to ‘equilibrium’ paths that could fairly be described as pathological. (The answer was Yes.)

As significant examples of Hahn’s interests away from his main preoccupations, I will just mention two of his occasional topics. His 1951 PhD thesis at LSE, The Share of Wages in the National Income, was not published until 1972 (Hahn 1972a), though a 1951 article with the same title was among his first publications. The thesis was obviously much influenced by Nicholas Kaldor, his adviser; it makes use of the systematic difference between the marginal propensities to save from wage incomes and from profits (the remainder). But Hahn’s use of the assumption is more subtle (and both more Keynesian and more general equilibrium in character) than Kaldor’s. It is embedded in a more complete model of the economy, so that the level of output and the functional distribution of income are simultaneously determined. Two early papers, A Note on Profit and Uncertainty’ (Hahn 1947) and ‘Uncertainty and the Cobweb’ (Hahn 1955), which he thought to be one of his more important efforts, signal Hahn’s ongoing interest in questions related to uncertainty and expectations. He wrote others along the way, up to ‘Notes on Sequence Economies, Transaction Costs and Uncertainty’ (Arrow and Hahn 1999). But he never tried to develop a systematic treatment of uncertainty and expectations, and was highly critical of modern attempts to do so through the concept of ‘rational expectations’. The examples in this paragraph do not come close to exhausting the range of Hahn’s theoretical interests. What follows will be limited to the main themes of general equilibrium, money, and stability, which are closely enough related to be thought of as aspects of a single analytical project.

General Competitive Analysis is a difficult book, more difficult in its way than Gerard Debreu’s loftier Theory of Value (Debreu 1959), to which it pays tribute. Arrow-Hahn’s mathematics is much nitty-grittier than Debreu’s more abstract treatment, but this is because Arrow and Hahn are far more concerned to relate the mathematical arguments, even if only sketchily, to economic intuition. In the Preface, they face the question why anyone should be interested in so abstract and complicated a construction. They start with the conventional answer: Adam Smith had famously claimed that a decentralized market economy, motivated only by personal self-interest and guided by price signals, would lead not to chaos, as one might expect, but to a coherent outcome and, even more surprisingly, an outcome with many desirable properties. Smith’s proposition has become an article of faith for many untrained people. ‘The proposition having been put forward and very seriously entertained, it is important to know not only whether it is true, but also whether it could be true’ (Arrow and Hahn 1971: vii; italics in original). In other words, they go on, one needs to know if Smithian optimism will be overturned if there is imperfect rather than perfect competition, or if there are external economies, or apparent irrationalities like judging quality by price, or if some markets are missing, or in a monetary economy. ‘In attempting to answer the question “Could it be true?”, we learn a good deal about why it might not be true’ (ibid.). In later years, Hahn would often say that the main point of a book like that one is to show just how stringent the conditions are under which Smithian optimism has any claim to be believed.

Approximately the first third of General Competitive Analysis is given over to laying out in great detail the standard model of competitive equilibrium under favorable assumptions on consumer preferences and production possibilities, and then proving that this model has an equilibrium price vector (whose positive components correspond to goods with zero excess supply and goods with positive excess supply have prices equal to zero). In this treatment, commodities can be defined to have specific temporal and spatial dimensions. The treatment of uncertainty that more or less dissolves it by postulating the presence of a full set of state-contingent commodities is introduced but not explored in any detail. The next chapter (6) takes up some of the extensions or alternatives mentioned earlier: that prices might enter utility functions, that there are external economies or diseconomies in consumption or production, and that some markets are characterized by monopolistic competition. In each case, it is shown that, with appropriate modifications, an equilibrium price vector exists. In addition, there is an extended treatment of a two-period version of the sort of ‘temporary’ equilibrium introduced by John Hicks in Value and Capital (Hicks 1939): there are no futures markets, but there are certain storage possibilities. Once again, a short-period or temporary equilibrium is shown to exist. There is, however, no discussion of a monetary economy here; it is postponed to a later chapter on Keynesian economics.

The remainder of the book is taken up with special topics. There is a brief treatment of the case that some preferences are not convex (indifference surfaces have patches with the ‘wrong’ curvature). Chapter 8 on the core of a market economy is aimed at stating conditions under which, in a ‘large’ economy, an allocation in the core (an unblocked allocation) is approximately a competitive equilibrium. A longer chapter on possible uniqueness of competitive equilibrium is a change of pace, the focus falling on cases where all commodities are gross substitutes or almost effectively so. Otherwise, uniqueness is not to be expected at this level of generality. A chapter on ‘comparing equilibria’ (what most economists, following Samuelson, call ‘comparative statics’) concludes that, with only the standard assumptions, the theory of general equilibrium does not provide enough information to calculate how equilibrium prices and quantities respond to parametric changes. Since discovering at least the sign and perhaps the magnitudes of those responses is more or less the bread and butter of economics, the implication is that choice of special assumptions is a critical matter.

There follow three chapters on the stability of general equilibrium. The first two are fairly conventional: they accept the admittedly silly hypothesis that there is no trading out of equilibrium and explore the role of Walras’s ‘auctioneer’, especially with universal gross substitutability. It was already known from Herbert Scarf’s famous 1960 example that global instability cannot be excluded. The last of the three chapters allows trading out of equilibrium in a pure exchange economy, making use of earlier work by Hahn himself and Takashi Negishi. I will pass over it because, stability having been one of his major interests, I will soon describe Hahn’s special contributions to this subject.

The last chapter of the text, on ‘The Keynesian Model’, is unusual for a book on general equilibrium theory. Perhaps its presence reflects the fact that Hahn was already associated with the ins and outs of Cambridge economics, but perhaps not: both Arrow and Hahn were well aware of what was going on in the broader field of economics, well beyond the confines of formal general equilibrium theory. Nevertheless, the content of the chapter is quite precisely concentrated on recognizably general equilibrium issues.

First, they identify The General Theory as being concerned with short- period or temporary equilibrium, a topic already discussed in the form Hicks had given it. There are, however, at least two important differences: Keynes’s agents, unlike those treated earlier, enter the current short period with debts and perhaps other obligations left over from earlier periods; and the context in which they act is an explicitly monetary economy. The main concern of the chapter is to investigate whether earlier theorems on the existence of a temporary equilibrium can survive these extensions. The two main problems are to make allowance for bankruptcy possibilities introduced by the presence of previously incurred obligations, and to find reasonable assumptions that will make the price of money positive. On the whole these problems can be solved, and a temporary equilibrium shown to exist.

Arrow and Hahn understand that this is not what ‘Keynesian economics’ is meant to be about; they think of this chapter as merely laying some groundwork. They mention, but do not seriously pursue, some more typical questions concerned with target demand and effective demand, with the formation and function of expectations, and with the uses of wage flexibility. I will come back to the last of these.

Two of the three chapters about the stability of equilibrium are more or less in the standard Walrasian tradition: all trading is suspended while an ‘auctioneer’ tries to find an equilibrium price vector. They follow the formalization of Samuelson: each price moves instantaneously, increasing or decreasing as excess demand in its market is positive or negative, each excess demand taken as a function of all prices. Everyone had always realized that this was not how reality could work; since the auctioneer was ‘fictitious’, so was the whole process, and since trading takes place all the time, there must be trading at nonequilibrium prices.

The third of the three chapters, allowing for trading out of equilibrium but limited to a pure exchange economy, reflects important work in the 1960s by Hahn (1961, 1962), Negishi (1961), and the two together (Hahn and Negishi 1962). This line of work, though not large in volume, was an important advance in the theory of stability of general equilibrium, and counts as one of Hahn’s lasting contributions. The basic idea of the Hahn Process—as it was dubbed by Negishi—is straightforward economics. Trade takes place out of equilibrium, but it is assumed that markets are sufficiently orderly so that at any time, in any market, there may be excess suppliers (those who would like to sell but cannot find buyers at current prices) or excess demanders (just the opposite), but not both. Markets may not clear, but they are good enough not to have at the same time unsatisfied buyers and unsatisfied sellers.

If prices behave in the manner described by Samuelson, excess suppliers will see the prices of the things they want to sell falling, and vice versa for excess demanders. With some further simplifications and extensions, this is enough to create a general tendency for traders’ utilities to be falling, and this aggregative fact can be used to provide a proof that prices converge to an equilibrium. The story can be extended to include firms and production. But it is still unsatisfactory, especially in its account of—or its failure to provide an account of—the formation of expectations, and their consequences. Any sort of general theory of stability that includes a useful treatment of speculation would be a remarkable achievement. It may be too much to hope for. Where the theory stands now owes substantially to Hahn’s work. His later views are summed up in the chapter on ‘Stability’ he contributed to the Handbook of Mathematical Economics, Volume II (Hahn 1982a).

Beginning immediately after the publication of General Competitive Analysis and continuing until 1999, Hahn was sporadically engaged in following up another fault line in the classical theory of general equilibrium: that connected with transaction costs, missing markets, and what he called sequence economies. This train of thought began with ‘Equilibrium with Transaction Costs’ (Hahn 1971), even as General Competitive Analysis was being published, and continued with ‘On Transaction Costs, Inessential Sequence Economies and Money’ (Hahn 1973a), and ending only with ‘Notes on Sequence Economies, Transaction Costs and Uncertainty’ (Arrow and Hahn 1999). Evidently the notion that Arrow and Hahn were bemused Walrasians is not serious.

The 1973 paper works by comparing two almost-identical abstract economies: in both of them, goods are distinguished both by their physical qualities and their delivery date (one of a finite number of dates into the future). In the Debreu-type economy, all those markets are open at the beginning, a full set of equilibrium prices is somehow found, and households and firms make their purchase and sale decisions over all goods and all times. Households are constrained only by one intertemporal budget-balance condition, but there are vaguely formulated transaction costs. No further transactions need occur after the initial moment; future deliveries have been specified and payment is feasible and confidently expected. In the sequence economy, transactions occur period by period (though plans extend further) and households balance their budgets only period by period. Prices for a specific good may differ according to the date of the transaction. (Why should things happen this way? Perhaps because the claim of intertemporal budget feasibility over long intervals may not be credible to counterparties.) These assumptions entail that there is a meaningful sequential structure to household behavior. Hahn’s mature take on these questions can be found in a 1993 overview entitled ‘Sequence Economies and Incomplete Markets’ (Hahn 1993). Much the best criticism of general equilibrium theory comes from the best general equilibrium theorists.

With so abstract a set up, only abstract questions can be discussed. The paper is given over mainly to issues of existence of equilibrium, and whether an equilibrium of the Debreu economy is also an equilibrium of the sequence economy. In that case the latter equilibrium is, of course, Pareto efficient. If there are equilibria of the sequence economy that are not Debreu-equilibria, they are not efficient. An inessential sequence economy is one in which this does not occur. The paper introduces money to the sequence economy in a fairly general way, and then shows that inefficient equilibria are still possible. It is pretty clear that Hahn thinks of all this apparatus as merely laying the groundwork for a truly monetary version of general equilibrium theory, not the thing itself.

Another example of Hahn’s intention to get beyond simple Walras-Debreu assumptions without giving up rigor is ‘On Non-Walrasian Equilibria’ (Hahn 1978). It is interesting that he describes the particular motivation behind this paper as a wish to make precise sense of the notion that ‘the division of labor depends on the extent of the market’ (ibid.: 1). (The problem is that ‘the extent of the market’ is not well defined in the Walrasian setting.) But he also mentions the need to provide a general equilibrium setting for Keynesian economics.

In fact, the paper concentrates on the second of these objectives. It considers two ways of getting at the idea of Keynesian equilibria. The first is the approach (Dreze and Malinvaud-Younes) that holds prices fixed at nonmarket-clearing values in the short run and looks at possible outcomes when one side of the market is in some way rationed. Here, Hahn discusses mainly the existence of fixed-price equilibria that are orderly in the sense that only one side of any market has to be rationed. Such equilibria depend on the particular rationing scheme adopted for markets with either excess demand or excess supply. This rationing scheme is not a matter of individual choice.

Hahn prefers a different version of Keynesian underpinnings in which agents may, under some circumstances, have and understand that they have some market power: the constraint is not that prices are fixed, but that changing a price changes the amount that can be sold. Negishi (1960) had studied general equilibrium models with imperfect competition. Hahn’s angle is different: he wants to show that an economy that has a Walrasian equilibrium can also have non-Walrasian equilibria characterized by sellers’ conjectures about the demand curves they face. This is not really a satisfactory solution to the problem of finding a microeconomic basis for Keynesian economics because the model does not allow unconstrained agents (e.g. employers) to use to their own advantage the knowledge that counterparties (e.g. workers) are constrained. But it leads to an interesting discussion of the kinds of conjectures that are reasonable given the objective circumstances.

I will mention only two other research trails pursued by Frank Hahn, chosen because they intersect, one way or another, with my own. The first has to do with the theory of economic growth. The long survey article by Hahn and Matthews (1964) is everything that a survey article should be: comprehensive, clear, fair, and perceptive; it contains its own small but original contributions to the early stages of growth theory. Even earlier, with ‘The Stability of Growth Equilibrium’ (Hahn 1960), Hahn launched his own special project: he found that, under plausible assumptions, equilibrium paths in many good growth models need not converge to the standard sort of steady state. This work was continued in ‘Equilibrium Dynamics with Heterogeneous Capital Goods’ (Hahn 1966) and in ‘On Some Equilibrium Paths’ (Hahn 1973b).

The second is the book that he and I produced together: A Critical Essay on Modern Macroeconomic Theory (1995), written over a long period of years, one month per year, in Cambridge, Siena, and Helsinki. It was motivated by a shared distaste for the attempt of recent macroeconomic theory to lay ‘foundations’ by transferring the least credible assumptions of general equilibrium theory to a representative-agent context that only rendered them still less credible and even pernicious. We proceeded by following up a remark of Keynes: ‘[I]f competition between unemployed workers always led to a very great reduction of the money-wage, there would be a violent instability in the price-level’ (Keynes 1936 [1973]: 253) (and therefore in the real interest rate). In effect, we imposed perfect wage flexibility on a standard overlapping-generations model, so that the labor market always clears, and showed that many of the resulting trajectories could only be described as pathological. Unfortunately, we found no better way to demonstrate this conclusion than a prolonged application of brute force. It is no wonder that no one wanted to read this grim exercise. The fact that it was followed by some more palatable and, we thought, more acceptable macroeconomic modeling escaped notice. (For example, there is an attempt to explain wage rigidity as a response to piecemeal disturbance of traditional wage differentials, and there is an experiment with the incorporation of imperfect competition in a simple aggregative model.)

Quite a lot of Frank Hahn’s professional work stemmed from an acute aversion to nonsense. The nonsense in question could take several forms. One of them, certainly, was a lack of rigor in theory. This would account for his often- repeated remark that the point of General Competitive Analysis was to demonstrate, once and for all, how many very restrictive assumptions are required in order for classical general equilibrium theory to be true (including the First and Second Welfare Theorems). This may have been an afterthought: how could one know at the start of that massive enterprise where it would come out? The book may have begun with something more positive in mind. No matter: Hahn thought that the book, and several of his later articles as well, could stand as a warning against excessive claims that ‘the market’ could be relied upon to take care of this or that. The point is that he wanted such assertions to be rigorously refuted, not merely argued about.

Another form of nonsense consists of claims that have no real foundation in theory, but sound as if they did, especially with repetition. I think Hahn found this variety to be common in monetary and aggregative economics, where it might have the additional consequence of influencing policy. Something like that probably lay behind Money and Inflation (Hahn 1982b). Toward the end of his career, Hahn began to suspect that the sort of rigorous but realistically relevant theory that should be the economist’s goal might simply be too hard to be done properly in the traditional way. He thought that perhaps computer simulation would now be the preferred way to accumulate analytical understanding. Those who knew him well will realize how paradoxical this tentative conclusion must have seemed, rather like a lifelong maker of fine wooden boats seeing the future in fiberglass.