The article, written especially for the 100th issue of Cuadernos de Economía, will try to concentrate the reader´s attention on one aspect of another piece written four years ago, in the 88th issue of the same journal and dedicated to the Great Recession under the title «Una Vision (Semi)Heterodoxa de la Crisis, 2009». The unorthodox part of it had to do with what, once upon a time, was called Disequilibrium Macroeconomics, and more specifically, with the idea of the Neoclassical Corridor, also known as the «stability corridor», a notion related directly to the topic of Effective Demand Failures, and indirectly to fixed prices.
The idea of the corridor marks the exact moment of the turning point in the hegemony of Disequilibrium Macroeconomics, an approach that can be said to have its remote origin in chapter 13 of the 2nd edition (1965) of Patinkin’s famous book of 1956, its actual development through several works of Barro, Grossman, Clower, and Leijonhufvud (see references) in the late sixties or the early seventies, and, after an overlap with that part of Monetary Theory that was trying to introduce fiat money into the General Equilibrium Model, begun to ebb down at the reported point in 1973,only to abruptly disappear at the end of the seventies, in full swing of its popularity, marked by the widespread use in the teaching of Macro of both by Barro and Grossman’s 1976 book and by Malinvaud’s of 1977. In spite of the fact that some up-to-date dynamic macromodels in the Keynesian tradition use fixed prices with great academic success, one can safely say that Disequilibrium Macroeconomics, and especially the Neoclassical Corridor, are by now really «sleeping beauties».
The purpose of what follows is to describe the filter that put these beauties to sleep, discover its intellectual flaws, if any, and shed some light on what was really hidden in the beautifulness which can thereof wake up our senses to what could be used as a guide to the understanding of the present crisis, to the way to be followed out of it and, most importantly, to the nature of an economic system. It is a paper on ideas, but it is not a piece on the history of economic doctrines. It is, rather, a call to take into account some intuitions coming out from complex systems (see W. Brian Arthur 1994 and 1997 and Kirman 2011). These intuitions which not only are behind the evolution of ideas in general, showing their unpredictability and the possibility of the return of some of them, but more specifically, are the offspring of a clear understanding of the notion of Effective Demand Failures, ultimately leading to the current attempts to change the pace of Eeconomics.
I will proceed as follows. In the next section, I will present quite briefly some simple ways of understanding the evolution of ideas and its application for the so-called Disequilibrium Macroeconomics, its origins and the filter that put it to sleep. In the third section, the main part of this paper, I will make use of a very old note of mine (never published before) in order to present the notion of the Neoclassical Corridor as a case study especially important, not only to illustrate the evolution of ideas, but to understand it properly and underline its potential, now that the filter has been shown not as effective as thought for forty years. In the fourth section, the just above-mentioned potential is analyzed as useful for diagnosis and prognosis of the present economic situation.
The final section summarizes and offers some final comments.
2. Evolution of ideas. From vision withoit technique to technique without hindsight
I learned Macroeconomics at the end of the sixties and the beginning of the seventies of the last century. The program dedicated two semesters to the topic. In the first one, I read with awe two great books by Hicks (Mr. Keynes and the Classics, 1937, and Value and Capital, 1939, with its appendix dedicated to general equilibrium) and the locus classicus of the neoclassical synthesis represented by the already mentioned Patinkin’s Money, Interest and Prices, especially the (at the time) almost esoteric chapter 13). In the second semester, I was exposed to the first writings on Disequilibrium Macroeconomics and became a real fan of an approach which did not make a sharp distinction between the not very well defined Macro and the apparently indestructible and towering General Equilibrium Model of Arrow and Debreu (1954) and followers. The challenge was, indeed, to build up microfoundations to bridge the gap between Macro and Micro.
All that led to the writing of a thesis that tried to build models of growth where money played a role that was just as important as the idea of an exchange structure that eliminated the auctioneer. Money should be not neutral, monetary policy having therefore a role to play, and the Phillips Curve was not necessarily negatively sloped. In this dissertation, the then-recent contributions of Clower and Leijonhufvud were a must, together with the first writings of H. Grossman, with or without Barro (see references). But the task was not that easy, given that another development started almost simultaneously. Questions of expectations, aggregation of assets and aggregation of agents had far from obvious answers, and the efforts devoted to introducing money into the axiomatic approach to general equilibrium (as the one by Hahn’s in 1965) were not that successful at the time, in spite of the efforts made by people trying to think about temporary equilibrium, distribution effects and alternative ways of thinking about interest rates depending on the aggregation of assets (see M. C. Gallastegui and J. Urrutia 1988)
The intellectual climate was fascinating and its continuation was something the profession should have promoted. However, this was not the case. The general -equilibrium theorists continued with their task of completing the model, a task that took at least until the middle eighties, and their work became the obvious reference for excellence.
It was not the right moment for challenging its prominence, or asking for a new model that could introduce money, take into account differences between individuals (so as to take into consideration distribution effects), contemplate the effects of alternative aggregation of assets (curiously out of the picture) and use temporary equilibrium, corresponding to the above mentioned period analysis, so as to discuss expectations meaningfully. This task would have been formidable enough, and we can all understand the rather opportunistic behavior of almost all macroeconomists away from static models and towards dynamic ones, which, by abstracting away many traits of reality, could «easily» introduce rational expectations. The last gasps of life of Disequilibrium Macroeconomics took place in the two books by Barro and Grossman (1976) and Malinvaud (1977), while simultaneously the so -called DSGE (Disequilibrium Stochastic General Equilibrium) models began to rule the roost.
The so-called stagflation of the late seventies definitively killed off the disequilibrium approach, and now, after five years of recession, the DSGE model is facing the same kind of danger. It seems ironic that some of the ideas of the disequilibrium approach («vision without technique», as Backhouse and Boianovsky call it in their 1988 working paper) could now come to the rescue of DSGE macroeconomics which might be characterized by «technique without hindsight». That is, however, the contention of the present paper. But it is not only my idea. It seems that Leijonhufvud can help me at this point. In the 21st of November, 2009, he wrote a long post in VOX («Stabilities and instabilities in the macroeconomy»), from which I abstract the main paragraphs containing the ideas I want to illustrate.
Around the turn of the century …macroeconomists came to a «brackish» compromise known as the New Neoclassical Synthesis. The New Keynesians adopted the dynamic stochastic general equilibrium (DSGE) framework pioneered by the New Classicals while the latter accepted the market «frictions» and capital market «imperfections» long insisted upon by the former.
This New Synthesis, like the Old Synthesis of fifty years ago, postulates that the economy behaves like a stable general equilibrium system whose equilibrating properties are somewhat hampered by frictions. Economists of this persuasion are now struggling to explain that what has just happened is actually logically possible. But the recent crisis will not fit.
The syntheses, Old and New, I believe, are wrong. They stem from a fundamental misunderstanding of the nature of a market economy. Further technical innovations in economic modeling will not bring real progress as long as «stability-with-frictions» remains the ruling paradigm. The genuine instabilities of the modern economy have to be faced.
The economy is an adaptive dynamical system. It possesses the self-regulating, «equilibrating» properties that we usually refer to as «market mechanisms». But these mechanisms do not always suffice to ensure the coordination of activities in the complex system. Almost forty years ago, I proposed the «corridor hypothesis». The hypothesis suggested that the economy might show the desirable «classical» adjustment properties within some «corridor» around a hypothetical equilibrium path but that its self-regulating capabilities would be impaired in the «Keynesian» regions outside the corridor. For large displacements from equilibrium, therefore, the market system might not be able recover unless aided by stabilization policy.
The original argument for the corridor concerned the conditions under which to expect significant deviation-amplifying multiplier effects and might not be all that persuasive by itself. It is the case, however, that all other known complex dynamical systems, whether human-made or occurring in nature, are known to have the property that their homeostatic capabilities are bounded. It is extremely unlikely that the economy would be different in this regard.
It is reasonable to believe, therefore, that the state-space of the system -in addition to regions with good equilibrating properties- has regions where deviation-amplifying processes have impaired these properties…
3. Efective demand failures: The Neoclassical corridor
The purpose of Leijonhufvud’s 1973 paper was to stimulate the reallocation of intellectual resources towards the difficult, important, and unresolved questions which gravitate around the notion of effective demand failures, and which are in urgent need of theoretical modeling and empirical testing. Grossman’s 1974 comment can be understood as stating, by implication,that the questions raised by Leijonhufvud one year earlier were not unresolved or, at any rate, were not new or important and that, therefore, they were in no need of special theoretical modeling.
Here, it will be argued that Grossman’s misgivings were caused by his concentrating on what I will call the Clower-Leijonhufvud first thought (CL1), instead of looking for the new suggestions provided by Leijonhufvud, which constitute the Clower-Leijonhufvud second thought (CL2)1. However, since these new suggestions were neither brought up clearly by Leijonhufvud nor thoroughly exploited by him at that time, or later, for that matter, there seems to be room for additional analysis, taking advantage of what we have learned in the last forty years. In the process of proceeding to this additional analysis, some (admittedly incomplete) hints for the correct modeling of those ideas, new at the time, will be offered.
Theory versus Analysis2 According to Leijonhufvud, the central issue of Macro-theory is whether or not the market sectors of an economy behave as a «self regulating system» -an expression that sounds understandable now that we are familiar with complex systems, but was not so at the time.
The social correlate of this central issue was, and is, the «coordination of economic activities», where full coordination has to be understood as clearance of every market and has nothing to do with optimality of allocation. The problem is, then, whether there exists automatic tendencies towards full coordination, and how strong they are. In Grossman’s 1974 compact statement, the central issue concerns the «nature of the market -clearing process» (p.1).
A superficial reading of the previous statement of the central issue, especially in Grossman’s compact translation, may lead one to think that the problem is not new, and that it has been adequately treated by the literature on General Equilibrium (G.E.) and the stability of G.E. allocations3. However, the standard G.E. model (Arrow-Debreu, 1954) and most4 of the available exercises on the stability of G.E configurations are analytical constructions and what Leijonhufvud is arguing for are theoretical constructions. Now, the problem of analysis is how to fit reality into it, i.e., to what extent is one particular feature of reality well represented by the analytical features of the model and, therefore, to what extent can the analytical model yield insights into that particular feature of reality. The problem of theory, on the other hand, is how to model this particular feature of reality, i.e., how to construct an analytical apparatus in terms of the theoretical features of reality. As stated, the distinction is not very clear, but it can be illuminated by an example important for our present purposes.
Consider the particular feature of reality that we call monetary exchange. The now forty-years-old contributions to the literature on the integration of monetary and value theory can be classified as analytical or theoretical, according to whether they try to fit money into the standard G.E. model or whether they try to introduce choice-theoretic elements into the theory of money. The motto of analytical contributions is: «A model of the economy as satisfying as Debreu’s, but which can accommodate money» (Hahn 1971, p. 418). The motto of theoretical constructions might very well be: «A model of the economy as satisfying as Keynes’, but which can accommodate choice-theoretic elements»5. The distinction illustrated here may just represent two different attitudes towards a problem, and both approaches (analytical and theoretical) might very well converge, but in fact, they have not, as far as I know. In any case, Leijonhufvud’s quest for theory seemed to imply that a reallocation of intellectual resources ought to have taken place. We should have moved from G.E. analysis to the Theory of Coordination of Activities and from stability analysis to a Theory of Exchange capable of explaining, in economic terms, how exchange is actually carried6. The Theory of Exchange would supposedly find sets of conditions for the emergence of well-defined transactions structure, and the Theory of Coordination of Activities then tell us under what conditions would a particular transactions structure lead to full coordination7. It will be argued in the sequel (and it has to be argued,because it is not clear in Leijonhufvud’s original 1973 paper) that much of what he has to offer is related to the theoretical necessity of incorporating the idea of transactions structure. This is, in my opinion, what Grossman fails to see in his comment of 1974.
CL1, its generalizations, and the essence of CL2
The just described quest for theory is basic to Clower (1965) and Leijonhufvud (1968) in their reappraisal of Keynes’ theory. Clower (1969), starting from the theoretical idea of a monetary economy in which there are as many spot markets as non-money commodities (n), which are visited sequentially, and no futures markets, builds an analytical partial equilibrium model without an auctioneer and without stocks. At a non-equilibrium price vector, «false» trading will take place and will lead to the «labor sales-constrained demand» typical of the so-called dual decision hypothesis. The essential reason for this lies in that, in a monetary economy, each market reveals only the excess demand for that commodity, without revealing at the same time excess demands for all other commodities. This construction is sufficient to isolate the crucial choice-theoretic foundation of Keynes’ theory. If we now add, following Leijonhufvud (1968) that, in a monetary economy, there are reasons for laborers to engage in a search behavior before adjusting wage rates, we have that once the price vector is displaced from equilibrium by any shock, quantity adjustments take place and multiplier mechanisms are triggered immediately. This constitutes CL1, and from here on, when we mention quantity adjustments, we refer only to those here described.
We can now move into CL2, but before we do it, let us begin by noting that the dual decision hypothesis can be generalized into a system of general «sales-constrained demands» and «purchase-constrained supplies», together with the agents search behavior necessary to justify the fixity of prices. In this Hicks’ fixed price method, it is analytically sound to let a tatônnement in quantities to run its course until it reaches what Benassy (1974) called a K-equilibrium, which is obviously Pareto inefficient8. It is only then that, according to the method of what, at the time, was called period analysis (Grandmont and Laroque 1976), prices are allowed to adjust in response to certain pressures which take into account the interaction among markets implicit in the definition of effective, or sales-constraineddemands, (see Grossman (1971))9. Now we can move to CL2. One aspect- of CL2, CL2A, is the removal of a theoretically unsound feature of CL1, namely the absence of stocks: «modern economies maintain, in normal times, an enormous, and elaborate system of physical and financial buffer stocks», asserts Leijonhufvud (1973), p.38). An easy, although roundabout, way of capturing the theoretical necessity of incorporating the existence of buffer stocks is to reflect upon two of the different roles realized saleshave been made to play in CL1, or realized transactions in its generalizations or modifications10. First, under Say’s Principle and in a world without stocks, realized sales are equal to current income, and the latter is the relevant constraint on purchases. If this constraint is binding, we say we have aggregate demand deficiency. In the real stock-flow world, however, current income may exceed realized sales by an amount dependent on the volume of buffer stocks which include the stock of trade credit sustainable by the volume of physical and monetary stocks.
Second, it is only in a monetary economy without stocks, with markets visited sequentially and with money being the only counterpart of any transaction, that the relevant constraint on the demand signals a household may emit in one market are his realized sales in other markets. If this constraint is binding, we say that effective demand has failed.
It is not as much deficiency of aggregate demand, but a communication failure that we want to focus on. In a transactions structure with n(n-1)/2 trading posts (direct barter), or in a sequence of simultaneous bilateral trades, or in a system of a single «market» with an auctioneer, realized sales do not effectively constraint the emission of demand signals. In a monetary economy with buffer stocks, demand signals are not totally limited by actual or expected realized sales in other markets.
Even though including stocks and trade credit is, as we have just seen, one aspect of CL2, another important aspect of it, CL2B, can be detected by rhetorically asking whether the distinction between aggregate demand deficiencies and effective demand failures is a meaningful one. It will be meaningful if, and only if, i) aggregate demand is not deficient, but there are effective demand failures, and/or ii) there are not effective demand failures, but aggregate demand is deficient. It is rather clear that in a monetary economy such as the one visualized by CL1, even if stocks and trade credit are appended to it, neither i) nor ii) hold. It follows that, for the distinction between aggregate demand deficiency and effective demand failure to make some sense, it must be connected to some idea other than the existence of buffer stocks. It might be fruitful to connect to it a distinction between alternative transaction structures or trading arrangements. For instance, under a direct barter arrangement (see Veendorp 1971), G.E. allocations, assumed to exist, might be impossible to reach (deficiency) and if so they will not be reached (failure). Under a sequence of simultaneous bilateral trades without money (see Ostroy 1973), G.E. allocations, assumed to exist, could be reached (not deficiency) but would not be reached (failure).
But, by adding money and a monetary authority to this latter trading arrangement, G.E. allocations will be reached if they exist. With a single «market» and an auctioneer (Walrasian transactions structure), G.E. allocations will be reached if they exist.
These examples suggest that the reason why certain trading arrangements do not lead to communication failures is that, under them, individuals are not bound by pure flow constraints, but can generate trade credit. Hence, it is quite safe to state that the essence of CL2 is an appeal to the explicit consideration of a) the possibility of «fooling» the budget constraint by means of the generation of trade credit to an extent supposedly dependent on the volume of accumulated stocks (CL2A), and of b) the underlying trading arrangement (CL2B).
The two aspects of CL2 are logically separable. The second one is how traders meet, regardless of whether money is used or not. This is the aspect of transactions structure proper and, in this respect, whether we have direct barter, a sequence of simultaneous bilateral trades, or a Walrasian transactions structure is indeed very relevant. This aspect is now easily recognized as posing a problem of networks. The first aspect is whether, given the transactions structure, the conventional budget constraint can be «fooled». In this respect, the existence of money and/or trade credit is the crucial question. Although logically separable, the two aspects are linked by the fact that the volume of stocks is probably not independent of the transactions structure11. Whether or not Leijonhufvud had both aspects in mind is a moot question.
Certainly, the explicit wording of Leijonhufvud in 1973 focuses primarily on CL2A, but if the little logical exercise here performed is correct, he can be said to imply both CL2A and CL2B. Indeed, some of his assertions not understood by Grossman (1974) cannot indeed be understood without taking the into account the second aspect.
Speaking generally, Grossman asserts in his 1974 paper that «… the substance of Leijonhufvud’s discussion is deficient on a number of points and… much of what Leijonhufvud has to say seems to be seriously misleading…» It will be argued now that, with a correct grasp of both aspects of CL2, it will follow that the substance of Leijonhufvud’s discussion is not deficient, and that much of what he has to say hammers the nail right on the head. We will discuss in turn the two points raised by Grossman in his 1974 comment.
Significance of the monetization of exchange
Since this point is actually related to CL1, it can be commented upon very briefly. Grossman agrees with Leijonhufvud, and with everybody else, for that matter, in that the kind of exchange structure visualized by CL1 is crucial for the theory of effective demands, as opposed to notional demands. He, however, argues that Leijonhufvud’s argument is unconvincing because
«in his barter example, the economy has only two goods (labor and commodities) and one market (and) obviously with a single market, market interactions, including those which generate effective demands, are nonexistent. In his monetary example, the economy has three goods (labor, commodities and money) and two markets. However, it is not clear from his example that the fact that the third good serves as a medium of exchange is crucial».
It is difficult to admit that Leijonhufvud’s argument is unconvincing, since the only thing he is doing at this point is to explain Clower’s paradigm, which lies at the very heart of CL1. It was indeed from this paradigm that it became clear that an exchange structure consisting of markets visited sequentially and money as the unique medium of exchange, could generate effective demands different from notional ones.
In any case, the model put forward by Grossman «in order to bring out the significance of the monetization of exchange in a more effective and transparent way», is as interesting as other work by himself fdone alone (1971), or in collaboration with Barro (1971), or as the 1974 paper by Benassy that he refers to. As it was said above, these are powerful generalizations of CL1,and can, in fact, be considered as the analytical completion of CL1 Leijonhufvud is asking for in order to explain why the monetary system in question does not home in towards a fully coordinated state.
However, by using CL2 properly understood, Leijonhufvud’s examples can be interpreted in an alternative way, which I take to be more attuned to his purposes in his paper of 1971. It can be said that in the barter example, there are two «markets», and that every good is a medium of exchange. The transactions structure can then be interpreted as a Walrasian one, which clearly creates no coordination failures. On the other hand, in the monetary example, there are two markets visited sequentially and a unique medium of exchange. The transactions structure is that associated with CL1, and, without stocks and trade credit, it will create coordination failures.
This suggests that the interesting thing is not whether the number of markets is sufficient to generate market interactions, but rather, which kind of transactions structure is assumed to hold and what kind of network is formed by the agents. Even in the barter example, with only one «market» and two goods, there are possible alternative transactions structures: how are transactors going to meet, and according to which rules are they going to transact if the transactions structure is, e.g., one of bilateral trading?
Alternative transactions structures are associated with different patterns of information flows and, presumably, with different behaviors of stocks and this is what Leijonhufvud is pointing out rather clearly. In particular, he is inquiring into the possible effects on coordination failures that the introduction of stocks and trade credit in any transactions structures, and specifically in the transactions structure implicit in CL1, might have. I now turn to this
Relation between price and quantity adjustments
According to Leijonhufvud, the introduction of buffer stocks into the exchange structure visualized in CL1 generates a Neoclassical Corridor fully surrounded by a Keynesian zone. Within the corridor, any deviation from a fully coordinated path generates deviation counteracting feedback mechanisms, and the larger the displacement the stronger they are. Outside the corridor, any deviation from a fully coordinated path triggers deviation-amplifying feed-back mechanisms due to effective demand failures12. It is obvious that, under CL1 and the exchange structure compatible with it, the width of the corridor is literally zero, and that, therefore, any deviation from a fully coordinated path will generate immediately quantity adjustments. The reasons for that are, once again, the exchange structure visualized in CL1 and the search procedure consistent with this exchange structure.
The gist of CL2 is that, under the exchange structure of CL1, supplemented with buffer stocks and trade credit, the width of the corridor can be positive. Within the corridor, we have price adjustments. Outside the corridor, we have quantity adjustments, which, through a tatônnement in quantities, will lead to a K-equilibrium. Prices will supposedly adjust later in response to the appropriate pressures as discussed above. Since Leijonhufvud abstracts from this latter type of price adjustment, he can be said to consider price and quantity adjustments always as alternatives, in spite of his explicit wording.
Grossman claims that this is flatly wrong. He is not denying that, for given displacement from the fully coordinated path, the smaller is the coefficient of price adjustment λ (0 < λ <∞), the stronger are the pressures which produce quantity adjustments13.
What he is asserting is that, for given λ, the greater the displacement, the greater are the pressures for quantity adjustments, but the greater are also the pressures for price adjustments. He thinks, furthermore, that this deprives the corridor of its analytical basis.
The two issues are, however, separable. Even assuming that price adjustments would take place after a displacement, and in the ad- hoc fashion of stability analysis14, CL2 implies that, for a given λ, the greater the displacement, the more likely quantity adjustments are; and that makes the corridor an analytically meaningful notion. To see this, consider the classical analytical problem proposed by Grossman in his 1974 comment of 1974. For a given λ, the greater is the decline in the money supply, M, the greater is the price adjustment per period, and the greater are the pressures for quantity adjustments.
In this respect, Grossman is correct. Under CL2, however, individuals are not bound by pure flow constraints, but hold stocks of every commodity and can generate trade credit up to a limit presumably dependent on the volume of stocks they hold. Therefore, the volume of stocks determines the number of periods over which individuals can still make their notional demands effective and, therefore, allow only for price adjustments.
Now, the larger the displacement, the smaller this number of periods, and, therefore, the larger the probability that eventually, effective demands diverge from their notional counterparts. Consequently, the likelier are quantity adjustments are. It has to be noted that the argument just made does not imply that price and quantity adjustments are necessarily alternatives. What I think it implies is that it is only when the displacement from a fully coordinated path is very substantial that we are allowed to use Hicks’ fixed price method. For it is only in such a case that effective demand failures will eventually occur, and that, given the exchange structure considered, a search procedure consistent with it will set in, justifying the consideration of prices as fixed.
It can thus be seen that CL2 has a methodological (if you like) implication which works against the method of analysis used by CL1 and its generalizations. In particular, the work of Grossman, Benassy and Grandmont and Laroque, referred to above as generalizations of CL1, seem to be relevant only for fixed price vectors which differ widely from the equilibrium one, due to the substantial size of the original displacement. In consequence,these pieces of analysis can hardly be considered relevant to explain or describe the workings of the system under the small displacements characterizing normal times. It also follows, it seems to me, that the Veendorp’s result  on local stability reported by Grossman in  is not relevant to the issue.
After all this being said, Grossman’s confessed inability to imaging «conditions under which some prices may show no tendency to change although desires to sell and to buy do not coincide in the respective markets» is indeed paradoxical. If these conditions are not apparent to him, he should not have used the fixed price method, as he did in his 1971 paper or in his paper with Barro That same yea. Of course, his use of it was legitimate, because the exchange structure of CL1, together with the search behavior consistent with it, provides an obvious example of the conditions under consideration. In fact, Leijonhufvud devoted his famous 1968 book to the examination of these very conditions15.
Finally, in the argument made up to show the meaningfulness of the notion of the corridor, use was only made of the first aspect of CL2.
However, the second aspect of it is necessary to understand that all «prices might be at their right (G.E.) levels, but amounts transacted differ persistently from the desired rates of sale and purchase in some markets». It is certainly true that, under certain transactions structures, G.E. configurations will not be reached even if prices are at their right levels (see Ostroy 1973 and Veendorp 1970). This is not true, however, in the exchange structure of CL1 Leijonhufvud is referring to at this point and, therefore, Grossman is justified in objecting to the validity of the statement.
Difficulties in the modeling of the new ideas
The most important implication of what has been said so far is, perhaps, that in order to describe and explain the short-run workings of a market system in normal times, we need a formal model which allows simultaneously for both, price and quantity adjustments. This has not been done by the analytical generalizations of CL1. I submit that if and when it is done, it will have to take into account CL2 as explained here.
This is easy to say, but the actual incorporation of CL2 into such a formal model might be difficult. A very basic difficulty is that we cannot hope (without making use of some not very standard ideas about complex systems) to fully integrate the Theory of Exchange and the Theory of Coordination of Activities. This impossibility is reflected in the usual procedure of specifying a transactions technology set, which is actually a way of skirting the question of which is, in fact, the transactions structure (pace, e.g., Hahn 1971 )16. That is, it is difficult to imagine how the passage of time in the model will generate successive transactions structures, each of them making for a different theoretical model of Coordination of Activities. It might be thought that incorporating the idea of technological progress to the transactions technology set of the previous footnote could solve the difficulty. It does not, because the idea of technological progress is an analytical construction which covers our theoretical ignorance: why, at a certain moment, does a new method of transactions become available? How does society know it may be more efficient without trying it? Why should society try it? What we can hope to have is a Theory of Exchange which isolates conditions for the emergence of a particular transactions or exchange structure. Be it the economic fact of transactions costs or the political fact of a monetary authority, these conditions have to be incorporated into the model so as to have a Theory of the Coordination of Activities under the particular transactions or exchange structure consistent with these conditions17. Even if all this had been accomplished for, say, the monetary exchange structure visualized in CL1, we still face the problem of how to model the budget constraint. The value of purchases at current prices can exceed the amount of outside money the individual holds by an amount limited by the size of the individual’s stocks of money and other physical goods. The model has to explain not only that, under the exchange structure considered, people will hold stocks, but also how the size of these stocks is determined and what amount of trade credit will this particular size support.
Even assuming that this can be done in a meaningful way, a host of other problems appears. Suppose M declines. In the first day of the recursive process, each individual forms his notional demands, taking into account the new kind of budget constraint at the old equilibrium pricevector. Each individual visits sequentially each market. Some individuals will be rationed in, say, the first market visited. Each individual will now form his effective demands taking into account the new kind of budget constraint at the old equilibrium price vector and the fact that he or she has been rationed in the first market visited. And this goes for any market and all individuals.
But, now we are not allowed to let this tatônnement in quantities run its course until a K-equilibrium is reached because, at the end of the day, somebody, according to some policy, is going to partially adjust the prices. We can safely assume that some agent with temporary monopoly power in disequilibrium modifies prices as part of a rational search procedure, but this search procedure cannot be described as it was in the case of quantity adjustments without price adjustments.
Whatever story we devise to describe it, it has to be consistent with the transactions structure envisioned. If we now look at the following day of the recursive process, I do not think it will be reasonable to let individuals form their effective demands taking into account, besides the new kind of budget constraint at the new non-equilibrium price vector, expectations of future transactions based on past actual transactions. And this because yesterday’s prices are not today’s prices, and the information provided by the latter has already been incorporated in the budget constraint. Thus, there seems to be no justification for the use of static expectations as in the generalizations of CL1. Even worse, there seems to be no basis for any kind of rational expectations.
For instance, individuals might think that the new price vector is the equilibrium one, and will accordingly express their notional demands, which take into account the new kind of budget constraint and nothing else. Since it has been assumed that the new price vector is a non-equilibrium one, some individual will be rationed in, say, the first market visited. A new set of effective demands will be formed as in the first day of the recursive process. At the end of the second day, some agent will move the price according to the rational search procedure. The next day, individuals might very well be at a loss concerning how to form their effective demands. No surprise, then, that we (economists) feel at a loss concerning the modeling of their behavior.
These difficulties reported here are challenges, but they do not constitute an impossibility theorem. Assume, then, that they have been overcome and that the resulting recursive process can be shown to have a fixed point we call a P-Q equilibrium.
What are the likely implications of this new choice-theoretic apparatus for those macro constructions which, like the consumption function (as in Clower’s 1965) or the accelerator (as in Grossman’s 1972 paper), had received such a nice choice-theoretic foundation under CL1? My optimistic conjecture is that what was true under CL1 will still be true, because quantity adjustments continue to occur, but that the new choice-theoretic foundation will yield more specific implications. This is the main reason underlying Leijonhufvud’s urge for formal modeling.
We could, finally, inquire into the efficiency properties of the P-Q- equilibrium. If it is identical with G.E., it will be Pareto-efficient. If it is not identical with G.E., the efficiency of the P-Q equilibrium could be analyzed, according to the criterion proposed by Arrow and Hahn (1971) and used by Benassy (1974) in his analysis of K-equilibria. I do not think, however, we want to do that, irrespective of the transactions structure considered, because the alternative efficiency criterion refers to a bilateral barter trading arrangement. We could try to develop alternative efficiency criteria, suited to the particular transactions structure in question, but what CL2 suggests is that it is perhaps time to lay down the importance of static efficiency criteria, and to devise new concepts relating to the coordinative performance of the economic system.
Since I have shown that, for a given λ, the greater the volume of stocks the larger is width the corridor, it follows that the greater the stocks, theless likely is the occurrence of unpleasant unemployment situations following a shock of random magnitude. That is, the greater the volume of stocks, the greater the «safety» of the system18. Now, as I have already said, the volume of stocks is probably dependent on the transactions structure. Therefore, it appears as if we could begin the analysis of the relative «safety» of alternative transactions structures.
It is hoped that the ideas here discussed show the importance and far-reaching implications (even, or especially, for Macro- theory) of CL2. We should perhaps talk less about the micro-foundations of Macro- theory, and more about the macro-stimulus to Micro-theory. This was appropriate to say in the seventies, and continues to be appropriate forty years later.
4. Recessions under the forgotten lense
In order to wrap up this paper, it is quite convenient to show that the ideas underlying the development of the notion of Effective Demand Failures are useful to the understanding of different observed crisis observed. The focus will be now placed on the two most important in the last forty years. The 35 -years -old oil crisis, and the Great Recession we are still in, at least in Europe.
The oil crisis
This is the appropriate place to remember the oil crisis of 73/74 and 79/80, and the feeling of bafflement about what the cause was of the increase in the price of oil, and what was to be done. Together with a colleague of mine we wrote a paper in Economies et Societés in 1983, the spirit of which was indeed tinted by ideas of Disequilibrium Macroeconomics. To begin with, we wanted to think helped by the idea of the fixity of prices. As we already knew, and at the time, had not been yet forgotten, after any shock, it is possible that prices do not balance the situation immediately, allowing for quantity adjustments into the workings of multipliers and accelerators. We justified this possibility on the basis of the Neoclassical Corridor or, as it is now known, the «stability corridor». We did focused on the width of this corridor and the determinants of this width. The size of Monetary and Fiscal policies, relative to the magnitude of the shock, were indeed the main determinants. Restrictive macropolicies lead to a narrower corridor. After the OPEC-engineered shock, it was rather obvious that we were poorer, we had a smaller productive capacity, and our potential growth was indeed smaller.
At the time, we submitted the suggestion that among the determinants of the width of the corridor, we had to introduce the notion of Credit Limit, a notion not at all standard at the time. While as long as this limit was not reached, the corridor’s width will be large enough, but the situation changes immediately once we approach this limit and quantity adjustments begin to be felt. We did mention that it was likely that the Monetary and Fiscal policies might not be generous enough because their size was calibrated according to the old growth potential, but we did not however continue the argument, underlying the fact that the Credit Limit depended on the level of leverage of the banking system -something we have finally learned. The difference from the present situation is precisely that the banking system now is indeed exhausted and incapable of continue to give credit. The robustness of the financial system in general is, therefore, crucial. One of the main fiascoes of the Spanish economy in the recent years has been the misinformation about this robustness.
The Great Recession
Let us now turn to the notion of the Neoclassical Ccorridor, because I think it is appropriate to understand what has come to be known as the Great Recession, and discuss the ways out of it.
This notion, as we have seen, is not easy to grasp in all its richness with the tools of neoclassical economics; this being the reason for the little success it has attained, as Laidler (2008) said in his laudatio of Leijonhufvud in the homage UCLA offered him.
However, this «sleeping beauty» has been visited by various princes (such as, e.g. Kehoe and Levine (2008) in the context of debt-constrained markets), and the notion of the corridor has been used in a way which turns out to be useful now. In 2007, three Japanese authors (A.Dohtani, T. Inaba and H. Osaka, wrote an article on neoclassical growth in which the notion is used in terms that are more recognizable now. They complicate the standard model, introducing a consumption function depending on permanent income.
This turns out to be sufficient for the model to show the same and unique stationary state shown by the standard model, but in which there is a stability corridor around the growth path. Within the corridor, the neoclassical vision is validated, because every path converges to the unique equilibrium one, and outside it, every path diverges from the neoclassical long -run equilibrium.
Let us now see how this now awakened beauty may help to understand what is happening during the Great Recession. We are used to recognizing the failures of effective demand when an unemployed worker cannot make his demand effective, because he lacks income. We also know that a savings decision is not automatically translated into an effective demand of future goods. And, there is an additional way of showing a failure of effective demand. This happens when the financial system is in such a disarray that makes it impossible for an entrepreneur to credibly show his present demand of factors of production due to the impossibility of supplying now future goods. In a 2002 interview, conducted by Snowdon, Leijonhufuvd says quite expressively that this third way of effective demand failure makes it impossible to credibly say the following:
«I have this investment project that will pay off in the future, and I want to trade that prospect for the factors of production necessary today to produce those future goods»
And he continues: «And that’s where we end up if the financial system is totally clogged up with bad loans. That has been and still is the Japanese situation».
If the problem was the conventional Keynesian one (of consumers being cash-constrained), then there is a rationale for public works. But that was never the Japanese problem. Their problem was that they did not move directly to clean up the banking system after the collapse of the real estate and stock market bubble”
This was the situation in Japan twenty years ago, and it might not be mere chance that some of the notions of disequilibrium macroeconomics have been preserved in Japan. That situation is not very different from what it is happening nowadays in the Great Recession, except that the real state crisis is now happening in the USA. In situations of this nature, is not enough to increase public spending but, because for this move to be effective, it is necessary to have a clean financial system capable of serving as a conveyor belt of this public spending. But the financial system is crowded with bad loans. If the government and the banking system are in connivance, the situation looks as rather dark indeed. That happened in the USA, with Lehman Brothers, and in the UK, with Northern Rock, and also with the apparently sound situation of banks in Spain.
Quite recently (2009), Leijonhufvud concluded in a column of Vox that:
Fiscal stimulus will not have much effect as long as the financial system is deleveraging. Even if that problem were to be more or less solved, the government deficit would have to offset both the decline in industry investment and the rise in household saving -a gap that is rising as the recession deepens. Here, too, the public is skeptical and prone to conclude that a program that only slows or stops the decline but fails to «jump start» the economy must have been a waste of tax payers’ money. The most effective composition of such a program is also a problem
5. Final comments
In hope that I have made clear that the dynamics of any intellectual field is complex, and as such, it may generate unexpected patterns of not necessarily optimal behavior. This is what has happened in Economics in the last forty years, which have witnessed new ways of thinking in macro beyond the dynamic stochastic general equilibrium models, even though a new paradigm has not emerged. There is a large variety of approaches which shake the foundations of mainstream Economics. From Behavioral Economics (which combins results in Experimental Economics, Experimental Psychology and Neuronomics), which casts doubts on the canonical notion of rationality already under suspicion coming from Bounded Rationality ideas, to those new ideas about uncertainty that are so challenging and humiliating for the forecasters, and continuing through to Agent-based Models, Networks Analysis and the dynamics of Complex Systems so carefully cultivated in the Santa Fe institute for more than twenty years now, and which allow us to distinguish between erklärung (to explain) and verstehen (to understand). What deserves to be underlined now however is that a particular, albeit rudimentary, complex system had been there for forty years, and even had its days of glory, which were anyway soon to be forgotten. As has been shown in this paper, I am referring here to the disequilibrium approach to macroeconomics, and more specifically to the idea of a corridor, which has been thoroughly explored above.
As has been emphasized, had we not forgotten this clever idea, we would have not tried to understand the Great Recession in terms of DSGE models, stressing rational expectations in order to make room for the inefficacy of conventional policy measures, and we could have been able to understand four years ago that the system was working outside the Neoclassical Corridor, and move to a way of thinking that made room for complex systems and their possible unexpected results.
If we would had been able to follow this way of reasoning, we would have immediately realized that the more leveraged the productive system is, the narrower is the Neoclassical Corridor, so that we could have come to avoid completely useless policies by taking into account the size of both the disequilibria and the parameters of the leverage, as well as the value losses in the balance sheets. Then we could have produced a correct diagnosis and addressed the need to globally coordinate efforts. We might have even been able to explain that to socialize the losses of the financial system was better than to let it collapse, because if so happened, the corridor would have become a simple line without any width.
But, as I have stressed from the beginning, the waking up of a «sleeping beauty» has not been the only purpose of the present paper.
I also wanted to illustrate the so-called path dependency in the social construction of ideas. And here, the figure of Leionhufvud has played a double role.
To begin with, he was the author who was capable of really departing from the general equilibrium framework, demanding new microfoundations, which, as I think I have shown, could have avoided quite a heavy burden of erroneous policies, but then, he was also an early whistle blower on the dangers of not leaving the well-trodden path and not turning towards what we could now call Econophysics Regarding the first issue, it is still notable that he has been ignored, that Barro explicitly rejected the disequilibrium approach, and even wished he had not written the joint book with Grossman, and even more strange that the very entry on the Neoclassical Corridor has been erased from Google on the Internet. I am not heading towards any strange explanation, even though the reading of Backhouse and Boianovsky (1988) offers material for this kind of story. I have only wanted to illustrate the path dependency already emphasized by Brian Arthur (1994 and 1997).
As a precursor of really changing the pace and turning to Econophysics, Leijonhufvud is also a case study, since he himself tried to organize the profession around Computational Economics, as a way dealing with certain problems which were difficult to model, but «easy» to tackle on the computer. In 1991, he organized the Center for Computable Economics at UCLA, and he was its director until 1997. His iniciative was not very much appreciated among his colleagues, and he moved to the University de Trento inthe Computable and Experimental Economics Laboratory (CEEL) of whichhe could continue his involvement in this branch of Econophysics
1. For Clower´s ideas, see three references offered.
2. The distinction is basic to Leijonhufvud 1973 and appears explicitly or implicitly in the following passages: p. 29 at the beginning of last paragraph; p. 29, in the last paragraph of fn.1; p. 31, first complete sentence; p. 3. 4th and 5th lines; p. 35, in the last paragraph of fn. 3; p. 35, first paragraph of fn. 2; p. 38, the last three lines of the second paragraph under ii)
3. That stability analysis is widely off the mark, even in analytical terms (to be defined more precisely below), has been implicitly recognized by all recent contributions dealing with markets without perfect information. For a survey, see Rothschilde 1973.
4. The 1962 work by Hahn and Negishi on the stability of a non–tatonnement stability mechanism could be considered as theoretical in the sense that it incorporates the theoretical feature of lack of recontracting. See also Arrow and Hahn 1971, ch.13.
5. One of the first examples of the first approach is Hahn 1971. Clower’s 1965 and 1969 and Leijonhufvud’s 1968 works are the original examples of the alternative approach.
6. For a terse exposition, see Hirshleifer 1973.
7. The distinction between analysis and theory is not very different from Hahn’s 1971 distinction between what he calls, without taxonomic purposes, abstract and realistic models,and it should replace the obsolete distinction between Micro-theory and Macro-theory. On the other hand, the distinction under discussion has nothing to do with the usual ones between unrealism and realism, or irrelevancy and relevancy, since those terms are contingent on the problem at hand. Take Ostroy’s 1973 model of a sequence of simultaneous bilateral (between two individuals) trades. It is a theoretical model, in the precise sense that it incorporates a specific transactions structure. The model is quite relevant for the understanding ofthe existence and nature of monetary exchange. However, the specific transactions structure assumed seems unrealistic to me, abstracting from whether it is more realistic than other transactions structures implicitly used by other models. On the other hand, take the standard G.E, Debreu 1959 model as the paradigm of analytical models. It is certainly quite irrelevant to the understanding of how society transacts. However, it might be a realistic representation of a future, and perhaps utopian, transactions structure (see Kurz and Wilson 1973).
8. See Barro and Grossman 1971, Benassy 1974, and Grossman 1971. At the so-called K-equilibrium,markets are cleared, but people are not doing what they would like to do at current prices. This runs counter to the standard definition of equilibrium. This explains why the work referred to in the text is generally associated with disequilibrium analysis.
9. The dual decision hypothesis can also be modified for ad- hoc purposes. Urrutia 1978 forces Clower’s idea into a sequential structure of non –overlapping markets in order to analyze the coexistence of inflation and unemployment in a growing monetary economy.
10. There is a third role of realized sales, the one they play as expected income. In generalized versions of the dual decision hypothesis, the aim of which is to prove the existence of a K-equilibrium as a fixed point of recursive process (Benassy 1974), it is assumed that expected transactions at t are equal to perceived constraints on transactions at t-1. When the aim of the generalizations of the dual decision hypothesis is not the proof of existence, as it is the case of Barro and Grossman 1971 and Grossman 1971 , the recursive process can be collapsed into an instantaneous tatônement in quantities allowing for expected transactions to be equal to currently perceived constraints.
In both cases, perceived constraints are intimately related to realized transactions, and a crucial point is its specific relation.
11. One of the implications of the fact that CL2A and CL2B are logically separable is that, contrary to what was thought under CL1, money is not necessarily related to any particular transactions structure. The following terminology will be used: transactions structure or trading arrangements refer to CL2B, while exchange structure refers to CL2A and CL2B.
12. His wording is, of course, quite careful and he never rules out the possibility of both kinds of adjustment taking place simultaneously (Leijonhufvud 1973, p. 32), but he does not make any use of this implicit qualification.
13. Although Grossman devotes some space to showing that this is so, this has nothing to do with the issue at stake. See footnote 1 in p. 37 of Leijonhufvud 1973.
14. The two assumptions are not equivalent since even if it is true that the price adjustment as a proportion of the excess demand created by the displacement is the assumed behavior of the auctioneer, it is not necessarily the behavior of a price setter in disequilibrium. See footnote 3.
15. It goes without saying that much of the book has to be revised in accordance with CL2. CL2 is indeed very sweeping. Cf. footnote 3 in pages 34-35 of Leijonhufvud 1973.
16. This is reminiscent of the usual practice in. G.E. analysis of specifying a social production set not derived from the individual production sets. The analogy would be totally correct only if it were the case that we did not know how individual producers do what they do.
17. Note the care with which Hahn 1971 strives to incorporate into the transaction set what he thinks are conditions for the existence of money.
18. A great volume of stocks is normally taken as a sign of lack of efficiency, and here it is argued that it provides “safety”. The two things are not contradictory and suggest that these might be a trade-off between efficiency and «safety».
Arrow, K. and Debreu, G. (1954): “Existence of an Equilibrium for a Competitive Economy”, Econometrica , 22, pp. 265-290
Arrow, K. J. and Hahn, P.H (1971): General Competitive Analysis, San Francisco, Holden-Day Inc
Arthur, Brian W. ( 1994): Increasing Returns and Path Dependence in the Economy,University of Michigan Press, Ann Arbor
………………….. (1997): The Economy as an Evolving Complex System II, edited with Steven Durlauf and David Lane, Addison-Wesley, Reading, MA, Series in the Sciences of Complexity
Backhouse, R. and M. Boianovsky (1988): Disequilibrium Macroeconomics: an episode in the transformation of modern macroeconomics, working paper????? Barro, R. J. and Grossman, H. I. (1971): A Disequilibrium Model of Income and Employment, American Economic Review,61, pp.82-93 Barro, R. J. and H. I. Grossman (1976): Money, Employment and Inflation. Cambridge: Cambridge University Press
Benassy, J.P. (1974 ): “Neokeynesian Disequilibrium Theory in a Monetary Econonomy”, TheReview of Economic Studies, 42, pp. 503-523
Clower, R.W. (1965),The Keynesian Counter-Revolution: A Theoretical Appraisal in F.H.Hahn and F.P.R.Brechling (eds), The Theory of Interest Rates , London: Macmillan ………… (1967) “A Reconsideration of the Microfoundations of Monetary Theory”, Western Economic Journal,6, pp.1-9 ………….(1971) Theoretical Foundations of Monetary Policy, in Clayton,G., J.C. Gilbert and R. Sedgwick (eds) Monetary Theory and Monetary Policy in 1970 ‘ s, Oxford: Oxford University Press
Debreu, G. ( 1959): Theory of Value,. An Axiomatic aAnalysis of Economic Equilibrium, New York, Wiley & Sons Dohtani,A,T.Inaba and H.Osaka(2007 ):“Corridor Stability of the Neoclassical Steady State” in Time and Space in Economics, Springer Verlag, Japan Gallastegui,M. C. and J. Urrutia (1983): “Crisis, teoría económica y margen de maniobra,”Econiomies et Societés,nº 25 pp
Gallastegui, M. C. and J. Urrutia (1988): “Problemas de agregación: una sugerencia para complicar aun más la teoría de la evaluación de la política economica, ” in La Herencia de Keynes, Madrid: Alianza Universidad
Grandmont, J.M. and Laroque, G. (1976): “On Keynesian Temporary Equilibria,” The Review of Economic Studies, 43, pp. 53-67
Grossman, H. I. (1971): Money,Interest and Prices in Market Disequilibrium, Journal of Political Economy,79,pp. 943-961 …………..: (1972): A Choice-Theoretic Model of an Income Investment Accelerator, The American Economic Review,62, pp. 630-641 …………….(1974): “Effective Demand Failures: A Comment,”, Sweedish Journal of Economics ,pp. 358-366
Hahn, F. H. (1965): “On Some Problems in Proving the Existence of Equilibrium in a Monetary Economy”, in The Theory of Interest Rates ,in F.H.Hahn and F.P.R.Brechling (eds) London: Macmillan …………(1971): “Equilibrium with Transaction Costs”, Econometrica » vol. 39, nº 3
…………(1974) On the Notion of Equilibrium in Economics,Inaugural Lecture, Cambridge: Cambridge University 1973 Hahn, F. B. and Nagishi, T. (1962 ): “A Theorem on Non-Tatonnement Stability”,Econometrica 30, pp.463-9
Hirshleifer, J. (1973) “Exchange Theory: The Missing Chapter,” Western Economic Journal , 11, pp.129-146
Hicks, J. R. (1937): “Mr. Keynes and the Classics, A Suggested Interpretation, Econometrica, Volume 5, nº 2, pp.147-159” ……………… (1939): Value and Capital, Oxford; Clarendon Press
Kehoe, T.J J. and D. Levine (2008): “Bankruptcy and collateral in debt constrained markets”, in Roger E.H. Farmer (ed.) Macroeconomics in the Small and the Large, Edward Elgar Publishing Inc, pp. 99-114
Kirman Alan (2011): Complex Economics. Individual and collective rationality, Routledge, New York Kurz, K. and Wilson, R. (1974): The Structure of Trade, Economic Inquiry, 12, nº 4, pp. 493–516 Laidler, D. (2008):“Axel Leijonfhufvud and the quest for micro-foundations: some reflections”in Roger E.H. Farmer (ed.) Macroeconomics in the Small and the Large, Edward Elgar Publishing, Inc. pp 1-22
Leijonhufvud, A. (1968): On Keynesian Economics and the Economics of Keynes , London: Oxford University Press
……………… ……. (1973) “Effective Demand Failures,” Sweedish Journal of Economics, 75,nº 1,pp. 28-48
Leijonhufvud, A. (2009): http://www.voxeu.org/article/no-ordinary-recession-there-much-fearbeyond-fear-itself
Malinvaud, E. (1977): The Theory of Unemployment Reconsidered, Oxford: Basil Blackwell
Ostroy,J.M.(1973):” The informational of Monetary Exchange”,The American Economic Review,vol.63.nº4,pp.567-610 P Patinking, D. (1965): Money, Interest and Prices. An iIntegration of Monetary and Value Theory, 2nd edition, New York, Harper & Row Publishers
Rothschilde, M. (1973): Models of Market Organization with Imperfect Information: A Survey, Journal of Political Economy, 81, nº 6, pp 1283-1308 Snowdon, B. (2002): “Outside the Mainstream: An Interview with Axel Leijonhufvud, Macroeconomic Dynamics , February 2004, pp
117-145.Cambridge University Press
Urrutia, J. (1978) “Crecimiento monetario, Inflación y Desempleo,” Cuadernos de economía, vol. 6, nº 15,pp. 97-124
………………(2009): “Una visión (Semi)Heteroxa de la crisis. Un simulacro de formalización parcial”, Cuadernos de Economía,vol.32,nº 88,pp.13-41 Veendorp, E. C. H. (1970): “ General Equilibrium Theory for a Barter Economy”, Western Economic Journal, No. 8, pp.1 – 23 …………………(1975): Stable Spillovers among Substitutes, The Review of Economic Stud