Stability in classical and neoclassical theories*
Franklin Serrano
Introduction
In this paper, I want to make a brief comparative evaluation of the question of stability in classical (or Sraffian) and neoclassical (or marginalist) theories. The emphasis will be on the concepts of stability that are relevant for these different theories and on the implications of certain theoretical results. No new formal stability results will be presented.
The discussion is organised around three main questions that I believe summarise well the most common current reaction towards the Sraffian criticism, not only from the followers of neoclassical theory but also among economists who otherwise tend to be quite critical of the latter approach. The questions are:
1 Is a critique based on stability (and uniqueness) relevant, or are the ‘existence proofs’ all that matter?
2 What is the difference between the Sraffian critique and the difficulties with ‘income effects’, the seriousness of which are increasingly admitted by the neoclassicists?
3 Finally, if the Sraffians want to criticise neoclassical theory in what regards stability, what about the stability of their own classical long-period positions?
I start with a sketch of the Sraffian critique of neoclassical theories. I then move to discuss the neoclassical ‘existence proofs’. This is followed by a discussion of the stability problem in neoclassical theories. After that, the problem of the stability of classical long-period positions is taken up. A final section discusses some ambiguities concerning the question of ‘uniqueness’ in a classical framework.
The Sraffian critique
I think it is finally beginning to becoming clear that the critique of the neoclassical approach, of which Sraffa’s 1960 book was meant to be the prelude, applies to all versions of such theories, i.e. both to the older long period and the more recent intertemporal versions of general equilibrium theory.
In the case of the older long-period versions, one way of summarising the critique is by saying that, in the presence of heterogeneous capital, because of the dependence of relative prices upon distribution and because of the breakdown of the substitution principle, it seems impossible to guarantee the derivation of ‘well-behaved’ supply and demand functions for capital as a stock.
In case of the modern intertemporal versions, thanks to work by Garegnani (1990a, 2000) and Schefold (2000), we now see that it appears equally difficult, in the presence of heterogeneous capital, to guarantee the derivation of a ‘well-behaved’ excess demand function for new capital.
Thus, while in the older long-period versions the difficulties lie in the excess demand function for capital as a stock, in the modern intertemporal version the difficulties lie in the excess demand function for capital as a flow.
Moreover, in the long-period versions, difficulties with the excess demand function for capital will directly cause difficulties for the excess demand functions for all other factors.
In the intertemporal versions, the difficulties with the excess demand for new capital will disturb the demands for other factors by shifting the latter in such a way that the traditional presumed inverse relation between the quantity of factor and the price of its service will break down in a sequence of equilibria over time (Schefold 2000, Garegnani 1990a).
Existence proofs
I think it is uncontroversial to say that neoclassical general equilibrium theory, as an attempt to explain the workings of the market mechanism, is based on the idea that relative prices are explained by relative scarcity. Such scarcities, in an economy in which goods can be produced, can only be a consequence of the scarcities of the so-called factors of production.
That scarcity of the factors of production is, in its turn, explained by the interplay of the exogenous endowments and the operation of the substitution principle, both in consumption and in production, combined with factor price flexibility.
No ‘existence proof’ of the older long-period version of these theories is available, as it is now well known that would require the specification of a single quantity of the factor of production capital that would be independent of distribution.
It is true that the modern existence proofs, which concern the intertemporal versions of neoclassical theory, do away, albeit at a great methodological cost, with that requirement (Garegnani 1990a).
These mathematical ‘existence proofs’, however, are of such formal generality that they do not seem to be able to say anything about the validity of the specific neoclassical explanation of the workings of the market mechanism.
As is well known, such proofs are so ‘general’ that, for instance, strict convexity of production and consumption sets is not required for their validity. That, however, means that equilibria can be shown to exist even in a ‘model’ in which there is, for instance, fixed proportions in production and a fixed consumption basket. In such cases, there is simply no possibility of substitution in either production or consumption, yet what are still called neoclassical general equilibria can formally be shown to exist.
As an example of such curious ‘neoclassical general equilibria’, let us think of an economy that produces, with homogeneous labour and circulating capital only, a single good that is used both as consumption and capital good, using a single method of production.
In this economy, as there is neither substitution in consumption nor in production, it is clear that the factor demand functions for capital and labour will be completely inelastic.
Thus, depending on the technology and the relative size of the endowment of each of the two factors, we will normally have only two equilibria: one in which capital is abundant and the whole net product goes to the workers; and another one in which labour is abundant and the whole of the net product goes to profits.1
Of course, this model does not at all represent the neoclassical explanation of the market mechanism, as it lacks its main explanatory economic force, which is precisely the substitution principle.2
That example should suffice to show that existence proofs are too general to be used to evaluate the validity of the neoclassical explanation of the market mechanism, as the equilibria that can be shown to exist might have nothing to do with that explanation.3
It seems that it is precisely this confusion between the neoclassical explanation of the market mechanism and the mathematical properties of existence proofs that lies behind the modern tendency to consider as being also neoclassical general equilibria models based on a completely different analytical structure. The most obvious example perhaps is the Von Neumann growth model, which, with its subsistence wage, is clearly a special case of classical theory but is widely regarded as a neoclassical general equilibrium model.
We may understand that strange idea a bit better if we recall that, in the simple ‘fixed proportions’ model we have used as an example, when we take the case in which labour is abundant and further assume that a minimum subsistence wage for the workers is somehow included in the economy’s capital requirements and also that all the profit is invested, we immediately get a one-good version of the Von Neumann model. After that, we only need a further confusion, that between steady-state and long-period positions,4 to take the further step and claim, as Hahn (1982) has done, that even the analysis presented in Sraffa (1960) is also a special case of neoclassical general equilibrium theory.
Stability in neoclassical theories5
In order to discuss the question of stability, we should first distinguish between what is generally known as ‘Walrasian’ and ‘Marshallian’ (henceforth used without quotes) adjustment processes.6
The disequilibrium adjustment process is called Walrasian when prices increase whenever there is excess demand and accordingly fall in the presence of excess supply.
On the other hand, in the so-called Marshallian adjustment process, the quantities produced increase whenever the demand price is above the supply price and decrease when the demand price is below supply price.
It will be important for us also to distinguish between what is known as static stability and dynamic stability.7 Static stability conditions relate exclusively to the direction of the adjustment process, whereas dynamic stability conditions add information about the magnitude (often presented as the speed) of the adjustments.
As explained in a long footnote by Hicks (1965, p. 18, n. 2), static stability conditions are necessary but not sufficient conditions for dynamic stability.8 Take, for instance, a neoclassical partial equilibrium analysis of a market for a particular good, in which the supply schedule is inelastic and the demand schedule is positively sloped. Such (unique) equilibrium is statically unstable under the Walrasian adjustment, as excess supply (or demand) will lead to a fall in price and a further increase in excess supplies, making the economy move in the wrong direction away from the equilibrium. This means that this model is also going to be dynamically unstable for any magnitude or speed of adjustment.
However, assume now that the demand curve of our last example is negatively sloped. The equilibrium is clearly statically stable under the Walrasian rule, as adjustments go in the right direction. However, it is still logically possible (though perhaps not very plausible) that the magnitude of the adjustments is so large that there is overshooting, and the economy may move away from equilibrium in cycles of increasing amplitude.
Using these concepts, we can begin to evaluate the stability problem in neoclassical theories. First of all, I think it can be argued that the fundamental question when we want to discuss the relevance of the neoclassical explanation of the working of the market mechanism translates itself in terms of static rather than dynamic stability conditions. For, what we want to know at a high level of generality is if it is plausible that the economy subjected to the persistent forces envisaged by the theory will move in the direction of equilibrium, and, hence, if such equilibrium qualifies as a centre of gravitation of the economy, instead of being just the irrelevant ‘passing equality between supply and demand’ of which Marshall talked.9
On the other hand, it should be clear that, in the case of neoclassical general equilibrium theory, relying as it must on the condition of ‘equilibrium’ in the markets for the endowments of the factors of production, the relevant adjustment processes must be Walrasian, i.e. the sign of price changes following the sign of excess demands.10 In fact, Marshallian adjustment processes only make sense in the neoclassical approach when we are dealing with partial equilibrium in the markets for produced goods for a given set of ‘factor prices’.
Thus, in order to provide a plausible explanation of the working of the market mechanism, neoclassical general equilibrium theory must be shown to be statically stable for Walrasian adjustment rules.
Here, then, we see clearly the relevance of the Sraffian criticism summarised above. As is well known, the Walrasian static stability conditions depend crucially on the shape of the general equilibrium excess demand function for factors of production, which should be negatively sloped. As capital theoretic problems may involve a complete breakdown of the notion of substitution between factors of production, it is clear that these problems show that it is unlikely that we can trust neoclassical general equilibrium theory to be statically stable in the Walrasian sense. This, together with our earlier discussion that shows how little the existence proofs can say about the relevance of the neoclassical view of the operation of the market mechanism, is I think sufficient to refute Frank Hahn’s (1984) claim, according to which ‘funny shaped excess demands’ would not be a problem for neoclassical general equilibrium theory as ‘only discontinuities’ endanger the existence proofs.
We can at this point perhaps also try to answer an objection made by Ahmad (1998, p. 62), where he argues that we should not consider the Sraffian criticism as concerning stability, because the same model can be stable or unstable according to whether the adjustment process is Walrasian or Marshallian.
In order to demonstrate this argument, Ahmad (1998) draws a diagram in which a ‘perverse’, positively sloped demand for capital crosses a positively sloped supply of capital schedule, which happens somehow to be less steep than the demand schedule. In this case, the excess demand for capital is clearly positively sloped, implying static Walrasian instability. However, Ahmad argues that, if we think in terms of a Marshallian adjustment process, it is easy to see that the market is actually statically stable, as beyond the of equilibrium point demand price is below supply price, and before the equilibrium point the demand price lies above supply price.
However, our previous discussion allows us to see that Ahmad’s point is entirely based on what appears to be an inadequate application of the Marshallian rule to the market of a factor of production in which, according to the neoclassical theory, the factor endowment is given, and, hence, prices must follow the excess demand Walrasian rule.
Another common objection the Sraffian criticism can be found in Modigliani:
It seems to me that all the evidence suggests that the [factor price — F.S.] frontier is generally well behaved. The evidence suggests that the stock of capital and the flow of investments are functions of interest rates … the rental rate is an important element in the choice of capital intensity and in the right direction. So, I do not see why we should worry about the possibility of a perverse frontier, any more than about the possibility of concave utility functions or of global instability.
(1989, p. 580–1; our emphasis)
Leaving aside the empirical argument (for which, by the way, no reference is provided), Modigliani’s quote clearly equates the capital theoretical ‘paradoxes’ with the income effects that can lead to global instability even in a pure trade context.
Remembering the importance of static stability conditions, I think it is not difficult to see what is wrong with this increasingly common claim. For how could Modigliani know what is the ‘right direction’ if not by the application of the principle of substitution, which is supposed to be always valid? That principle is precisely what cannot be deduced theoretically when we have heterogeneous capital. The argument according to which the problems with general equilibrium income effects are empirical is necessarily based on the fact that there is no clear theoretical deduction of the sign and magnitude of such effects and, principally, on the hope that, if there is ‘enough substitutability’ in the economy, such effects will not prove too serious. That is so much so that one of the most common formal sets of conditions that ensures static stability is called precisely ‘gross substitution’.
Therefore, capital theoretic problems are much more serious than income effects as, when the very sign of the substitution effect can be anything, then it must be admitted that the forces contemplated by the theory (endowments, technology, preferences) cannot establish anything about what should be the ‘right direction’.
Modigliani mentioned ‘global instability’; therefore, we should perhaps include a word or two about the question of uniqueness and, hence, extend our discussion to global stability in neoclassical theories. In the case of these theories, of course, the same thing that prevents Walrasian static stability, i.e. ‘funny shaped’ factor excess demand functions will in all likelihood imply also a multiplicity of equilibria of which some at least will be (locally) unstable, namely those placed along the perversely shaped bits of those functions.
Moreover, in the case of multiple equilibria due to capital theoretic problems, neoclassical authors do not seem to give us any criteria or reason to argue that the economy might tend to one or another of such equilibria, adding to the lack of definite results of the analysis.
We can finish this section by looking at another argument of Hahn (1982) in which he concedes that the Sraffian criticism shows that at least ‘some (neoclassical — F.S.) adjustment stories are at risk’. Now, if we see that it is Walrasian static stability conditions that are crucial and that the only theoretical explanation the neoclassical theories present in favour of static stability is precisely the sign of the substitution effects, then I think it becomes clear that, as an explanation of the working of the market mechanism, neoclassical theory provides in fact a single ‘adjustment story’. This single story, which is the very principle of substitution upon which the whole neoclassical approach was originally based, is precisely the story that Hahn admits the possible ‘funny shaped excess demand’ for capital (or new capital) revealed by the Sraffians deprives of a consistent general theoretical foundation.
Stability in classical theories
We may now turn to the question of stability in what concerns the classical approach. In this case, of course, what matters is if the economy tends to gravitate towards its normal position determined according to the principle of the surplus, once the real wage, outputs and the technology in use are given. That, as is well known, involves market prices tending to normal prices and outputs adapting themselves to the set of effectual demands. In other words, the problem is to know if the normal position will be a centre of gravitation for the economy.
In terms of the concepts that we are using, it is clear that, in order for the normal position to qualify as a centre of gravitation, it must necessarily be statically stable, because this will show that the economy under fairly general conditions tends to move in the direction of the normal position.
On the other hand, as far as the type of adjustment process is concerned, it is important to notice the very different analytical structure of classical theory, in which there is no room for excess demand functions for factors of production, and distribution is determined by an exogenous level of the normal real wage. Therefore, the problem of stability in classical theory concerns an adjustment process in which there is production and in which the independent variables define a set of ‘supply prices’ (the prices of production). Thus, it is clear, in the standard terminology that we are using (which here shows itself as totally inadequate for classical theory), the relevant adjustment process should clearly be Marshallian rather than Walrasian. As the classics assumed or ‘took for granted’ (Garegnani 1997, p. 168), the reasonable argument that, whenever the quantity brought to the market was greater than the effectual demand the resulting market price would be lower than the normal price, and, conversely, whenever the quantity brought to market was lower than the effectual demand market price would end up being higher than the normal price, in fact it is true that in terms of Marshallian adjustment process that already points towards static stability conditions (see, however, the discussion below).
That is easily seen by the fact that, when, as is common in these studies, we take the empirical market price as a single theoretical demand price and think of the price of production as the supply price, it is clear that the classical notion of the effectual demand—natural price point implies that beyond the normal position the demand price will be lower than the supply price, and before the normal position the demand price is above the supply price, leading the economy to change output in the right direction.
Indeed, in a survey of a conference on the convergence to long-period positions, Caminati (1990) wrote that, with their notion of effectual demands, ‘the classical economists came close to assuming the stability of the prices of production’.
Caminati (1990) said ‘came close’ because, as he shows with the aid of a diagram, that does not prevent the logical possibility of dynamic instability as, even though the direction is right, the magnitude of the adjustment can be so great as to provoke cycles of constant or increasing amplitude (typical of cobwebs).
However, the classical postulate of given normal effectual demands and the associated idea that, when quantity brought to market is greater than effective demand, the market price falls below the normal price, however, is not sufficient to establish static Marshallian stability when we have more than two sectors11 or when the set of effectual demands may change endogenously during the process of gravitation.
Indeed, stimulated by a criticism made by Steedman, Garegnani (1990b, 1997) shows (in a paper that unfortunately does not seem to have attracted the attention that it deserves) how the classical analysis of gravitation must be modified to deal with more general cases.
First of all, with three or more sectors, there is indeed the possibility that, if we take into account that some goods are inputs for other goods, the market price of a product may be lower than its normal price, but still the market rate of profits in that branch moves above the normal rate of profits of the system, if the market prices of the inputs of that sector happens to be even lower relative to their own normal prices.
Garegnani shows in his paper that the gravitation process as a whole will go in the ‘correct’ direction, irrespective of the possible initial ‘perverse Steedman effects’ for some commodities.
Let us see why, first under the simplifying assumption that the set of effectual demands does not change during the gravitation process.
It is simpler to look, not at market prices, but directly at market rates of profits. Whatever happens initially to market prices of outputs and means of production outside the normal position, there will always be one (or more) sector(s) that will have the minimum market rate of profit of all. If (1) the sector with the minimum market rate reduces its final output, and (2) its relative market price is increased because of that, then this minimum market rate of profits will start rising.
When the minimum market rate of profit reaches the normal rate of profits, all the market prices will be equal to the normal prices, and the economy will have reached the normal position.
To prove this last proposition, Garegnani makes use of the fact that the sector with the lowest ratio of market to normal prices will have a market rate of profits below normal (which is obvious, as its costs cannot have fallen by more than its revenue) and that the sector with the highest ratio of market to normal prices will necessarily have a market rate of profits greater than the normal one (because of course its revenue has risen necessarily more than its costs).
These properties imply that all market rates of profit cannot be below the normal one at the same time. They also imply that all market rates of profit cannot be above the normal at the same time.
Thus, when the minimum market rate of profits reaches the level of the normal rate of profit, all other market rates of profit must be equal to it, and we have reached the normal position, for the only way other market rates of profit could be above that minimum rate was if all other market rates of profit were above the normal, but then, as the minimum is equal to normal (and no lower), that would contradict the above propositions that imply that, at anytime that market prices are different from normal at least one sector will have a rate of profit below normal.
Since, by focusing attention on the sector (or sectors) that at each point of the process of gravitation have the minimum market rate of profits, one gets around the ambiguity of sign of the relation between deviations between market and normal prices and the corresponding market and normal rate of profits, the question of gravitation is then reduced to the question of guaranteeing that, at any point in the process of gravitation, the sector that happens to have the minimum market profit rate at that stage will tend to reduce its output. If the answer to this question is affirmative, then, however strange the movements of market prices and quantities may initially be or became in any particular sector, they will be continuously being sent back in the correct direction, if not earlier, at least as soon as that sector becomes (or joins) the one(s) with the minimum market rate of profits.
But is the answer to that question affirmative? Will the sector with the minimum rate tend to reduce output?
Even taking all effectual demands of the economy as remaining constant during the process of gravitation, the answer cannot be the output will always be falling as, although the quantity brought to market is greater than effectual demand and market price is lower than natural price, we should not forget that quantity brought to market is not equal to current output but equal to output plus net decrease of inventories (stocks).
Thus, of course, it is possible that a market price lower than normal is met at least initially with the accumulation of inventories and thus with no immediate reduction in output. However, of course, largely unintended accumulation (or running down) of inventories is by its very nature a temporary (and very unprofitable) phenomenon, and thus it seems that we could in principle postulate that indeed there will be a definite and clear tendency for output to be reduced in the minimum rate of profit sector. Thus, while in this case dynamic Marshallian stability may still not obtain, the static Marshallian stability of the process is ensured.
However, even abstracting from the lags and cycles that are conceivable because of inventories and thus restricting the discussion to static stability conditions, Garegnani (1990b, 1997) points out that assuming that the set of effectual demands does not change during gravitation does not seem to be a very good assumption to make, particularly when dealing with the gravitation in markets for goods that are inputs of other goods.
In order to take these changes into account, Garegnani introduces the concept of market effectual demands for a sector, i.e. the demand for the product at a reference price that would yield the normal rate of profits in that sector, given the market prices of the other sectors. The reference price acts as an indicator of the relevant ‘supply price’ for that sector when the economy is not in the normal position.
Garegnani distinguishes market effectual demand (calculated at the reference price for that sector ) from normal effectual demand (calculated when normal prices rule everywhere and the economy is in the normal position) and argues that we must take this distinction into account when the during the gravitation process the effectual demands themselves may change endogenously the obvious example being the effectual demand for an input falling with the fall in the production of the goods that use it (to which we may if we want add the fall in the demand for consumption goods when less labour is employed due to reductions in output in many sectors). Thus the market effectual demand for a sector would be the relevant effectual demand (in the sense that it is the demand that gives normal profits) to look at during the process of gravitation .
Garegnani shows that the analysis of the behaviour of market prices should not be related to the difference between quantity brought to market and normal effectual demand, but should instead be deduced from the difference between quantity brought to market and market effectual demand that makes the market price deviate from the reference price.
Equipped with this concept of market effectual demand, we may then consider a further problem that may endanger gravitation. Assume that the sector with the minimum rate of profits tends to reduce its output. Will gravitation obtain? Now there is the possibility that the answer may be negative if the sector’s market effectual demand falls by more than its output does.
In that case, although output is falling, (market) effectual demand will fall by more, and, hence, market price will keep falling down rather than going up, as it is required for gravitation.
Garegnani points out that this can surely happen during gravitation, but he points out that this process cannot continue indefinitely unless the economy is ‘on its way to extinction’ (remembering he is considering only basic goods) and solves the problem formally, keeping maximum generality, by postulating that there is always a positive minimum to the market effectual demand of any good. If that minimum exists, then such cumulative effects will not persist, as, after market effectual demand reaches that minimum, output will necessarily fall faster than it.
We can see that Garegnani’s minimum market effectual demand will be easily met if we take final demand (or some component of it) as being given.
Thus, Garegnani shows that static Marshallian stability is preserved in a classical framework under very general assumptions, but admits that dynamic stability may not happen if inventories lead to cobweb phenomena.
In stark contrast to these rather positive and general results, we get a very negative impression from most surveys of the formal literature on gravitation models (including Caminati 1990), which do not take sufficiently into account these new results (Boggio (1998) does not mention Garegnani’s paper). This unjustified pessimism seems due to a number of reasons. First of all, most such studies tend to be concerned with ‘proving’ dynamic rather than static stability. Moreover, these studies seem to be unduly concerned with the so-called ‘cross-dual’ types of model, which tend to use Walrasian adjustment rules (or assume that Marshallian and Walrasian adjustments are at work simultaneously with the same speed). Finally (and perhaps more importantly), many of these ‘cross-dual’ models do not respect the logic of the point of effectual demand—price of production typical of the classical approach12 and thus cannot give much information concerning the conditions under which the properly classical gravitation process is stable or not.
Once these serious misunderstandings are cleared up, the difficulties of gravitation in classical theory are reduced to the possibility of cobweb type overshooting dynamic instability, which, although not impossible in some specific cases, anyway seem, as has been known for a quite a long time, to require extreme assumptions about repeated errors in expectations.13 This means that, in what concerns stability, the situation of classical theory is completely different from that of the neoclassical approach, as, in the former, the adjustment story tends to go in the ‘right direction’ under fairly general conditions.14
‘Uniqueness’ in classical theories
We will close this paper with some tentative remarks concerning the problem of ‘uniqueness’ and, hence, of whether the stability of a classical long-period position is global or just local. Here, the main difficulty is that we must distinguish between two current interpretations of what are the data or the independent variables of a classical long-period position. Although it is increasingly agreed that classical theory takes as given, in the determination of distribution and relative prices, (1) one distributive variable, (2) the set of normal effectual demands (or ‘requirements of use’) and (3) the technical conditions of production, there seem to be two very distinct interpretations of the meaning of the third element. On the one hand, we have the analysis of the old classical economists and also of Piero Sraffa, in his book Production of commodities, that always considered as the given technical conditions of production only the processes in use, i.e. actually employed in that particular situation. This means, in Sraffa’s book (‘at any rate in parts I and II’, (Sraffa 1960, preface), all processes actually in use, including the coexisting ones that give rise to rents and the obsolete ones that generate quasi-rents (in Sraffa’s particular sense).
In that case, the analysis of switches in the methods (or processes) of production — as when, for instance, a new technique is found to produce the same effectual demands (a potential change in the technical conditions of production) and is evaluated competitively using the criterion of ‘cheapness’ — has to be made on a one-by-one basis, following what has been called the ‘indirect method of choice of technique’. Moreover, a change in any of the other two sets of data (given distributive variable or effectual demands) will, in general, possibly cause some processes to be ‘superseded’ and thus provoke a change also in the previously given technical conditions of production.
In any case, if this is what is meant by the ‘given technical conditions of production’, it is clear that each long-period position is obviously ‘unique’ because it is assumed that the economy in that situation can both yield a surplus and viably meet (without overproduction) the set of given normal effectual demands. Thus, in this interpretation of what is a long-period position, no formal question of global versus local stability can arise.
There is, however, in the literature a second and increasingly common interpretation of the given ‘technical conditions of production’ in classical theories15 as referring to the ‘set of technical alternatives from which cost-minimizing producers can choose’ (Kurz and Salvadori 2000a). That different interpretation of the independent variables will lead to a different analysis of what is now called ‘choice of technique’, which has been called the ‘direct’ approach.16
In this latter approach, one can find in the mathematical literature the discussion that, under some circumstances of general joint production, the so-called ‘cost-minimizing system’ for a given set of normal effectual demands may not be unique.17
In this framework, the static Marshallian stability results (which in any case have not yet been fully extended to joint production systems)18 would definitely at most be local and thus, even if each possible long-period position is locally stable, the set of possible long-period positions can at most be quasi-globally stable.
Authors such as Kurz and Salvadori (2000b) maintain that the ‘direct’ approach is analytically ‘superior’. This ‘direct’ approach is of course the one taken by Von Neumann in his growth model, and it necessarily has to be the approach taken by neoclassical theory, which needs the whole set of available techniques to try to build the demand functions for the various factors of production.
In the context of the classical approach, we must note that whatever one thinks is the mathematical ‘superiority’, the approach of the ‘direct’ approach, such an approach does imply that producers have a very unrealistic and incredible vast amount of knowledge and sheer computing power.19 It is thus not clear why the simpler, and perhaps more sensible, approach of the classical economists and Sraffa should be considered obsolete.
Notes
* I wish to thank Cnpq/Brazil for continuing financial support, Professors P. Garegnani, F. Petri, B. Schefold, D. Tosato and F. Freitas for comments, discussions and/or correspondence in the research that led to this paper. Needless to say, none of them is responsible for the errors that remain.
1 For simplicity, I am leaving aside here the third possibility of an indeterminate situation in which the demand and endowment of both factors exactly coincide.
2 Professor Tosato has objected to my example, saying that it actually simply shows how neoclassical theory can always determine which factor is scarce. In reply, it can be said that, if we increase the number of factors of production in the example to say 768, the model will usually determine a single scarce factor, all the others becoming overabundant and, hence, acquiring zero price. This extended example would, I think, suffice to show how the lack of substitutability renders the neoclassical existence proofs quite incapable of explaining the scarcity of more than one factor of production.
3 That does not mean that existence proofs as such are without problems. We may mention that the occurrence of non-strict convexity in both consumption and production sets tends to imply that certain factors will be ‘free goods’ (usually n-1, see n. 3 above) and thus draw the attention precisely to the well-known and really artificial assumptions that are needed to prevent discontinuites in the excess demand functions for goods when some factor prices fall to zero.
4 On which see Petri (1999).
5 In this section, we are discussing both the long period and the intertemporal versions of neoclassical general equilibrium theory. The reader should notice that, as far as the problem of stability is concerned, the important difference is that, in the case of long-period versions, there is no need to assume that no transactions happen in disequilibrium, while the intertemporal version, because of the impermanence of the initial endowments of specific capital goods taken as given, requires such an extreme assumption (see Petri 1999).
6 Garegnani (1997, appendix), in a study of classical theory, uses the terms ‘first phase’ and ‘second phase’, respectively, for what we call, owing to common usage, Walrasian and Marshallian adjustment processes. Our terminology is obviously inacurate, not only because it is used to discuss classical theories (and authors) who, of course, could not possibly have anything to do with either Marshall or Walras, but also because we can find both Walrasian stability in Marshall and Marshallian stability in the work of Walras (and vice versa). For ease of communication, I have decided to stick to these received misnomers.
7 Samuelson (1947) introduced the concept and called it ‘true’ dynamic stability as he considered the ‘static’ conditions irrelevant. Here again the accepted terminology is really inadequate, as ‘static’ stability refers to a ‘dynamic’ problem, and also ‘static’ stability conditions are important to growth theories (which are obviously ‘dynamic’) theories also. I shall use it solely for ease of communication. Note, however, that Harrod, in his work on growth, refers to what we are calling here ‘static’ stability as ‘fundamental’ stability, as the latter was concerned only with the direction of adjustment and, hence, ‘independent of lags’ (for a preliminary assessment of the Harrodian ‘fundamental’ instability see Serrano (1996)).
8 Formal proof of this claim for general equilibrium can be found in Metzler (1945). As Hicks (1965) argues, Samuelson was not correct as far as global stability is concerned. Samuelson’s famous claim that static stability was neither necessary nor sufficient for ‘true’ dynamic stability is not formally incorrect, just very misleading. Of course, even if we have badly behaved excess demand functions, some equilibria may be stable, and thus it is true that the static stability conditions for the whole range of the excess demand functions may not be strictly necessary for the local dynamic stability of some of the possible multiple discrete equilibria.
9 Quoted by Kaldor (1934).
10 Cf. Garegnani (1997, p. 168, n. 2), where he says that Walrasian adjustment ‘generally is the only adjustment possible also in the markets for “factors of production” envisaged in modern theory’.
11 Note that Caminati’s (1990) discussion was framed in a two-sector framework.
12 A typical case (as Garegnani (1990b, 1997) pointed out) is the concern in the literature with what is curiously called ‘demand feedbacks’ by a number of authors. In this literature, the crucial assumption is that all quantities of means of production brought to the market in whatever situation the economy happens to be will be always immediately and necessarily used to increase production in the following period. It is easy to see that this creates what is in effect a ‘supply feedback’ (though in the literature these effects are referred to as ‘demand feedbacks’). When, for instance, there is initial excess of a commodity and the technology is ‘self-intensive’ (in the sense that the commodity is relatively more used in the production of itself than the others), the relative overproduction of that commodity will keep growing, making the lower than normal market price be associated with rising output and thus causing Marshallian instability. On the other hand, if the commodity in question is produced with a ‘hetero-intensive’ technology, the ‘supply feedback’ will make the relative production of other commodities (whose market price lies above the normal prices) increase, guaranteeing static Marshallian stability. In both cases, however, it is not clear what such arbitrary assumption about production has to do with the classical postulates concerning gravitation.
13 For the cobweb problem in neoclassical partial equilibrium with a Marshallian adjustment process see Kaldor (1934).
14 Note also that classical theory requires that market prices tend to gravitate either towards or just around normal prices, allowing also for permanent overshooting, as long as it remains contained within reasonable bounds.
15 And surprisingly, even for the writings of the old classical economists (though not for Sraffa’s book) as in Kurz and Salvadori (2000a). Here it seems that this attribution is connected with the idea that, in the case of rent (or joint production), for instance, ‘an element of choice of technique is always present’. It is of course correct that not only the coexisting, but all processes in use, must have been chosen at some point. However, the procedure actually adopted by the old classical economists (just like Sraffa) was always to start from a given system of production already in use and then analyse subsequent changes.
16 For the distinction between the direct and indirect approach, see Kurz and Salvadori (2000b).
17 Note, however, that there are those such as Schefold (1989) who have expressed doubts that such multiplicity is relevant, as it may require unrealistic assumptions about the nature of the available techniques.
18 Piccioni (1997), which explicitly follows the ‘indirect’ approach taken by the classics and Sraffa, shows that in such a framework, starting from any initial long-period position, joint production and even the simultaneous appearance of more than one possible ‘cost minimizing system’ will not create by themselves problems for gravitation, as, in each situation, only one of those systems will be chosen.
19 As convincingly demonstrated by the example, inspired by Kalecki’s rejection of the ‘direct’ approach as being unfeasible even for socialist central planners, given by Schefold (1997).
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