An unpleasant dilemma for contemporary general equilibrium theory

Abstract

By examining the contributions of two prominent contemporary neoclassical economists, i.e. Lucas and Hahn, the article attempts to shed light on the problematic relationship between neoclassical theory and observation. It is argued that this approach must face the unpleasant dilemma of having to choose between endowing general equilibrium theory with an explanatory role that is marred by its illegitimate notion of capital as a single factor of variable form (Lucas); or alternatively, to consistently treat each capital good as a distinct factor of production, with the bitter implication that the theory must simply renounce to have a correspondence with observation (Hahn).

Keywords:

• Lucas
• Hahn
• neo-Walrasian theory
• centre of gravitation
• capital theory

Acknowledgements

I wish to thank Profs. H. Kurz, R. Ciccone, F. Ravagnani, A. Vercelli, H. Trautwein, B. Schefold, A. Lazzarini and two anonymous referees for their comments to previous versions of this paper. Particular thanks are due to Prof. Fabio Petri for his invaluable help and support during the writing of this article.

Notes

1 The treatment of capital as analogous to that of labour or land also plays two additional crucial roles. First, it gives justification to the uniqueness and stability of the equilibrium, since it gives good reasons to believe, even in capital-goods producing economies, in the “correct” working of the factor substitution mechanisms, on which the negative slope of the factor demand curves is assumed to rest and, ultimately, the whole plausibility of the explanation of prices, distribution and employment in terms of an equilibrium between supply and demand forces. Second, once it is acknowledged that different techniques generally call for the employment of different kinds of capital goods, the “variable” form of capital justifies the sufficient substitutability among the different factors of production, and hence the sufficient elasticity of the factor demand curves. The latter is in turn essential to obtain plausible levels of the real wage rate and of the interest rate in equilibrium, and also to argue that, e.g. in the case of unemployment, wage reductions need not be implausibly drastic.

2 As Wicksell once put it “But it would clearly be meaningless – if not altogether inconceivable – to maintain that the amount of capital is already fixed before equilibrium between production and consumption has been achieved… a change in the relative exchange value of two commodities would give rise to a change in the value of capital… even if we conceive capital genetically, as being a certain quantity of labour and land accumulated in different years, a change in the value of commodities would also alter the conditions of their production and thus necessitate a larger or smaller change in the composition of capital” (Wicksell 1934, p. 202).

3 In any case, it should be noticed that it is the very inclusion of capital goods among the givens what forces the theory to deal with price-changes in the definition of the equilibrium. Given that the initial composition of the capital stock will considerably change from one period to the next along the equilibrium path, a fact that will generally entail changes in future equilibrium relative prices, this cannot be ignored by individuals when taking their economic decisions. On the other hand, the slowly changing data in the traditional versions of the theory make it legitimate to abstract from possible future price changes; it is therefore unnecessary to include the latter in the definition of equilibrium (cf. also footnote 7).

4 This explains why traditional neoclassical authors were forced to determine the composition of capital endogenously, and hence to measure the endowment of capital as a single factor, capable of changing form: as Petri (2003, p. 390) notices, it would have been clearly illegitimate “to assume equilibration on the market for produced goods and not to admit the variability of the relative amounts in existence of the several capital goods”.

5 As to the traditional notion of capital as a single factor, while neoclassical theorists openly admits that the total quantity of capital would be gradually altered by capital accumulation, its speed of variation is sufficiently slow so as to render it legitimate to consider the total endowment of capital as unchanging when studying the process of gravitation towards equilibrium, allowing its composition to be determined endogenously. Therefore, for instance, Knight (1931, pp. 208–9) writes that the “total supply of capital is ‘very large’ in comparison with possible variations in it, and that the opportunity for further investment is on a similar scale with the total… Manifestly no possible variation in the amount saved in a year could make enough of a variation in the total supply… The cumulative result would be detectable after a considerable number of years”.

6 To further grasp the relevance of the problem caused by the lack of persistence of the vectorial endowment of capital goods, consider an equilibrium situation and assume that, due to immigration, the real wage rate changes. This will presumably have repercussions in all sectors of the economy; it will thus take considerable time for prices to adapt to the changed costs, and for firms to adapt production so as to satisfy the new composition of final demand, which will generally change due to the changed prices of consumption goods. Along this adjustment process it can hardly be maintained that the several endowments of capital goods will not change (nor that their change is of minor importance).

7 In the temporary-equilibrium versions, also the datum relative to expectation functions lacks the necessary persistence since the way in which people form their expectations is generally influenced by all sort of accidental and transitory factors that will generally change during the adjustment process. Moreover, given that expectation functions can be influenced by a wide variety of factors – an in unpredictable ways –, their inclusion among the givens creates a serious problem of indeterminacy into the theory. Within traditional theory, on the other hand, these problems do not arise: the persistence of the data allows neglecting the possible changes in the theoretical position itself in the definition of the equilibrium. It becomes therefore unnecessary to include exogenous expectation functions among the determinants of equilibrium (or for that sake, to assume complete markets). We shall return to the problems caused by expectations on section III (§12).

8 As noted in footnote 1 in the traditional versions of the theory this problem does not arise.

9 As Muth (1961, p. 316) argued: individual's subjective expectations “tend to be distributed, for the same information set, about the prediction of the theory (or the ‘objective’ probability distributions of outcomes)”.

10 Cf. also in Lucas and Sargent (1981, p. 305), where the authors argue that the concept of equilibrium used by RATEX theorists “stemmed mainly from work by Arrow (1964) and Debreu (1959)”.

11 In Lucas's words: “One way to interpret a ‘contingent-claim’ equilibrium is as a description of an economy in which all state-contingent prices are determined in advance, in the clearing of a single grand futures market… Alternatively, one may… think of a contingent-claim equilibrium as being determined via a sequence of ‘spot’ markets, in which current prices are set given certain expectations about future prices” (Lucas 1980, p. 707). Furthermore, he assumes RATEX as “a principle to reconcile the price distributions implied by the market equilibrium with the distributions used by agents to form their own views of the future” (1980, p. 707). We may incidentally note that on the basis of this same citation Rodano (1984, p. 43) too has maintained that “Lucas’ idea [is] that an A–D [Arrow–Debreu] equilibrium can also be interpreted as being determined via a sequence of ‘spot’ markets, in current prices are set given certain expectations about future prices”, whereas Lucas notes, these “certain expectations” are RATEX. On this basis Rodano (1984, p. 25) argues that “Lucas and his followers maintain that the necessary and correct micro foundations [of macroeconomics] are to be found in modern General Equilibrium Theory”.

12 For the equivalence between the perfect-foresight path and the intertemporal path in contingent commodities, cf. e.g. Radner (1982, pp. 940–42) and Rodano (1984, section III). Although not essential for our purposes, which at this stage of the exposition only attempt to establish Lucas's alleged adoption of the notion of contingent-claim equilibrium, and more generally, of the neo-Walrasian method, as his so-called micro foundation, we may incidentally note that, beyond the formal aspect of the problem, Radner (1982, p. 942) and Rodano (1984, p. 42) have disputed that a perfect-foresight path – or in a stochastic environment the RATEX path – can be considered equivalent to the intertemporal path in contingent commodities. The reason is that the assumption of correct price expectations is contradictory with the assumption that the equilibrium must be found by the market (cf. also Petri 2009, pp. 14–15). More recently, the alleged equivalence between the perfect-foresight path and the intertemporal equilibrium path has been also convincingly questioned by Mandler (2002) and Fratini and Levrero (2011). We may also add that while Lucas occasionally argues that the assumption of RATEX implies that “agents are assumed to know the pertinent objective probability distributions” and that “This hypothesis is imposed by way of adhering to the tenets of equilibrium theory” (Lucas and Sargent 1981, p. 307), when one admits the possibility of contingent markets, modern general equilibrium theory does not require agents to have correct, common expectations about probabilities of future states; it can deal with any beliefs of individual agents about probabilities of future states of nature.

13 Cf. De Vroey (2001) for a detailed analysis of Lucas's (1972) business-cycle model.

14 As is probably well known, Friedman (1977) argues that the trade-off between labour unemployment and inflation depicted by the Phillips Curve can only hold in the short-run, since any attempt of the monetary authorities to decrease the unemployment rate by increasing the money supply will sooner or later make workers -- who are assumed to have adaptive expectations -- realize that the price level will eventually rise proportionally and, therefore, they will inevitably end up raising their demand for higher money wages so as to leave their real wage unaltered. There is a “normal” level of employment where the economy tends over sufficient time, and that is determined by the forces of supply and demand; hence, Friedman argues, a persistent level of inflation caused by monetary expansion will ultimately cause “perceptions [to] adjust to reality” and therefore “Ultimately, employment will be back at the level that prevailed before the assumed unanticipated acceleration in aggregate nominal demand” (Friedman 1977, p. 14).

15 As Lucas and Sargent (1981, p. 307) explain, the errors agents commit “are unavoidable given their limited information”.

16 Lucas extends the 1972 paper to model the production of capital goods and hence considers investment decisions in Lucas (1975; see §7).

17 In a subsequent contribution, Lucas (1988) also attempts to defend himself by arguing that, if the notion of RATEX equilibrium is extended over the infinite future, the economy would eventually reach a position of steady growth. Moreover, Lucas goes on to argue that it is this final equilibrium, and not necessarily the rational equilibrium path itself, the position the actual economy will be gravitating around: “What of economies that begin off the balanced path – surely the normal case? Cass showed – and this is exactly why the balanced path is interesting to us – that for any initial capital K(0) > 0, the optimal capital-consumption path (K(t), c(t)) will converge to the balanced path asymptotically. That is, the balanced path will be a good approximation to any actual path ‘most’ of the time” (Lucas 1988, p. 11). However, this claim seems to be devoid of justification: actual economies are far from being in a position of steady growth, and before the economy can reach this position, the data that determine the steady growth equilibrium path will have changed considerably.

18 However, as we shall see below, this justification is devoid of legitimacy because it presupposes the traditional notion of capital as a single factor.

19 At this juncture we may note, albeit largely implicitly, that individuals can only come to know the correct probability distribution after some adjustment process is implied by Lucas's (1980, p. 711) resort to an auctioneer-guided tâtonnement to justify how an RATEX equilibrium comes about. In fact, if the assumption of RATEX were to be taken at its words, individuals would correctly forecast equilibrium prices, and the respective quantities produced and sold in the market; no need would therefore arise to actually find the equilibrium values, as it is implied in the auctioneer-guided tâtonnement (on this point, cf. also Petri 2009, pp. 14–15; also Rodano 1984, p. 42; Radner 1982, p. 942).

20 Cf. however De Vroey (1998), Kirman (2003) and Vercelli (1991).

21 As noted in footnote 16, in the 1975 article Lucas extends the 1972 paper to a framework where the production of and the demand for capital goods (investment) is explicitly considered.

22 In the 1975 contribution, Lucas does not model the behaviour of the household sector explicitly. However, following the standard practice of Ramsey kind of models, he suggests that consumers’ demands can be derived from a representative consumer's well-behaved intertemporal-utility maximising problem. The assumption that the household sector behaves as if there was a single consumer who maximizes an infinite-horizon utility function is also found in e.g. Lucas (1988).

23 In this connection a point that seems to have been little noticed is worth stressing (cf. however Boianovsky 1998): it is Ramsey (1928, p. 556) himself, who, in his seminal contribution on the theory of economic growth, accepts that the very long run or secular equilibrium is useless to explain actual economic conditions since it may “never be reached”. The problem is surmounted by also determining an “equilibrium in the meantime”, i.e. at each point in the process of capital accumulation, on the basis of what he calls a “temporary” capital supply curve, which is simply the traditional given endowment of value capital, with the implication that the resulting equilibrium is a traditional long-period equilibrium.

24 We may note in this connection that, conveniently enough, the same hypothesis of a single capital good that justifies the persistence of the data of each RATEX equilibrium, and hence makes the identification of the latter with a traditional neoclassical equilibrium possible, also allows Lucas (1975, p. 1116) to derive well-behaved and sufficiently elastic factor demand curves for capital and labour. Indeed, the one-capital good model is sufficiently restrictive to exclude the possibility of reswitching and reverse capital deepening; phenomena that, as notably argued in Garegnani (1970), reveal that the adjustment processes traditionally envisaged by neoclassical authors to justify the tendency towards the position determined by the theory may not work in the expected direction, not only jeopardising the uniqueness of the equilibrium, but also its stability, both essential features to assign to equilibrium its traditional role as a centre of gravitation. Things would not be entirely different in this respect if, as Lucas occasionally does (cf. footnote 16), one were to argue that the centre of gravitation of the economy is a position of steady growth: as Schefold (2005) has recently pointed out in this connection (cf. also Burmeister 1980, pp. 124–26; Mc Kenzie 1986, p. 1337), the stability of the steady growth equilibrium would still require the absence of reswitching of techniques, a result that can be assured only if capital could be conceived as a single factor analogous to labour or land.

25 The claim that intertemporal equilibrium can be used as a benchmark will not be discussed here. For our purposes it is enough to note that this role of the notion of intertemporal equilibrium presumes that this equilibrium concept cannot indicate actual paths with sufficient approximation.

26 Cf. e.g. Cass and Shell (1983), in particular the appendix of that article for an example of the possibility of a self-fulfilling (sunspot) equilibrium that is not a randomization of the “real” equilibria.

27 In Hahn (1970, 1982a), there is a thorough assessment of the problems of stability in modern general equilibrium theory.

28 This however does not mean denying the influence of individuals’ expectations in economic variables. If necessary, this influence can be examined at a second level of approximation while studying the oscillations around the equilibrium path.

29 We may also note that the value notion of capital also considerably weakens the scope of a third objection occasionally raised by Hahn (1990, p. 237) to the RATEX school: Hahn argues that once complete future markets are accepted not to exist, ‘self-fulfilling’ beliefs, or possible expectational mistakes, can well cause Pareto inefficient outcomes. While a neoclassical author who, like Lucas, accepts that agents are likely to make mistakes would not disagree with Hahn on this issue, the traditional notions of equilibrium and of capital would give her good reasons to dismiss the relevance of this problem too: granted the notion of capital, the persistence of the data and the correct working of the factor substitution mechanisms would ensure that over sufficient time the action of the supply and demand forces would eventually assert itself fully, hence the possible inefficiencies (apart from those not surmountable by the market, e.g. externalities, etc.) will be gradually corrected in the light of experience, exerting a relatively minor negative effect on growth. Moreover, precisely because the tendency towards the full employment of resources would be constantly at work, this scholar would be on strong grounds to object to Hahn's claim that the absence of complete future markets “leaves vast scope for Government intervention” (1990, p. 246): she would argue, granted the tendency towards the full employment of resources, that the scope for Government intervention would actually be quite narrow.

30 In his well-known article “The neo-Ricardians” (1982b), Hahn (1982b, p. 370) “doubts” that one-capital good models are actually “useless” (1982b, p. 370), and then he adds, “we use simple models (e.g. macroeconomics) to gain insights of a certain kind. Simplification is never without cost and the cost is sometimes loss of rigour. It remains to be shown that the cost is too high in this instance”. Therefore, even Hahn seems to hesitate to give up the picture of economic growth derivable from one-goods models. This may explain why in his own work (cf. Hahn and Solow 1995, chapter 6), Hahn himself has had recourse to a standard neoclassical production function with labour and capital, the single factor, as arguments to model the behaviour of firms. This may also explain why Hahn has accepted in general terms the negative relationship between the rate of interest and investment decisions (cf. e.g. Hahn 1992, p. 11)

31 Incidentally, the appendix to this paper shows that Lucas and his school are not the only ones to attribute to modern general equilibrium theory an explanatory role by introducing traditional neoclassical reasonings that presume the old notion of capital as a single factor. This same attitude is also present in general equilibrium specialists like Arrow.

32 Cf. Petri (1999) and Garegnani (2003, appendix II) to find other statements by Hahn that further confirm the inability on the part of this author to understand the importance of the notion of capital as a single factor for neoclassical theory.

33 Unlike the composition of a given vectorial endowment of capital goods, which can be very quickly altered, the different kinds of labour (and land) need not be aggregated because their endowments can be safely assumed to change slowly relative to the speed of adjustment of disequilibrium prices towards their equilibrium values.

34 Hahn (1984, p. 10) has, for instance, argued that Arrow–Debreu theory “gives us the best base camp for sallies”, namely it must be the starting point of economic theorising. This may explain why even Hahn has eventually fallen in the temptation to endow modern general equilibrium theory with an explanatory role. Consider, for instance, the following sentence by Hahn: “The crudest empirical observations will convince one that there is no unique rate of profit to be observed in the economy. Do we conclude from that that competition is functioning badly? Answer: No. Consult any general equilibrium text” (Hahn 1975, p. 361). Note then how Hahn is here declaring that general equilibrium theory would have a positive role, namely it could explain why actually observed (own) rates of profits are not necessarily uniform in the different businesses, without this fact implying the absence of free competition.

35 In this respect, Petri (2004, chapter 5, appendix 5.A.2) has noticed that in his discussion of the problem of price taking (Arrow 1959), Arrow in fact does not seem to be very clear that traditional ways of reasoning are illegitimate when applied to modern versions of general equilibrium theory.