Hi Gil, We're as one on the first issue, so I'll focus on the second. Firstly, on the issue of who's more obsessed with equilibrium, again no argument: Sraffians are as obsessed as neoclassicals, and that's something I addressed very directly in my 1998 ROPE paper. But I see that more as a failing of the Sraffians than of Sraffa, if you get my drift. The technical point re stability is therefore something that is a criticism of both neoclassicals and Sraffians (and therefore of Walras, but not of Sraffa!): it is provably the case that (a) the properties of substitutable production functions are irrelevant in the vicinity of equilibrium and (b) the equilibrium is unstable if it describes a growing economy. Put the two together and the implications are: (1) CES production functions don't save neoclassical economics from Sraffa's static equilibrium critique. If one takes a CES production function and one does a Taylor series expansion of it at that point, then the coefficients of all the nonlinear components (x^2, x^3,... x^n) are zero. The CES is thus a Leontief in equilibrium, and that applies to any other nonlinear/substitutable factors production. This is a simple consequence of the definition of a Taylor series. Take any function (say sin[x] to use a scalar example). Expand it using a Taylor series to get [0 + x - x^3/3! + x^5/5! -...]. Define 0 as your equilibrium and consider a point a very small distance away from zero--say 0.1. Then sin[x] is very close to [0 + 0.1 - 0.1^3/6 + 0.1^5/120]. As you can see, the first nonlinear term in that approximation is 0.00017, about 1/6000th the size of the linear component. If we are considering the stability of the function six[x] within the vicinity of zero (meaning in this case an x substantially less than 1), then we are justified in ignoring the nonlinear terms. (2) Jorgenson's papers from the 1960s (plus those of McManus; see references below [the best summary of this is given by Blatt]) show that a Leontief technology with spot market pricing [the usual 'market clearing' neoclassical assumption] will have either an unstable price or quantity vector. The quickest explanation of this is that it's a consequence of the Perron-Frobenius theorem on the dominant eigenvalue of a positive semi-definite matrix: since this is greater than zero, and since the matrix determines one vector (say quantity) while its inverse determines the other (price), then either the matrix or its inverse will have a dominant eigenvalue greater than one. Therefore, unless the economy starts precisely in equilibrium (and remains without disturbance for all time), it will rapidly diverge from equilibrium. Of course, once it gets sufficiently distant from equilibrium, then the nonlinear terms in any CES production function can come into play (in my sin[x] example, once you get to about 0.5, 1+x becomes an increasingly poor approximation for sin[x]: 0+x=.5, sin[0.5]=.479) and stop the system blasting off into infinity (whereas a Leontief technology would lead to negative prices, etc.). Bringing this back to Sraffa's critique of neoclassical GE, and the response of neoclassicals that it's irrelevant because it doesn't consider factor substitutability, this means that their defence is only valid *if they drop the presumption of equilibrium*. But if you drop equilibrium, then by definition marginal product won't determine anything, since all the neoclassical optima only apply exactly in equilibrium. Neoclassicals would also have to drop static methods and learn about differential equations, etc--something none of them show any real inclination to do. When you ask "unstable with respect to what perturbations", a puff of wind caused by a passing comet's effect on the earth's path around the sun will do. Unless we are willing to model the economy as if it is a completely deterministic closed system, then any model we build of it will have to take into account that the system will be perturbed from equilibrium by something, even if it begins in equilibrium. Cheers, Steve >Secondly, though the Leontief IO matrix is a "restrictive assumption", it > >is also the linear component of any more general nonlinear production > >technology (it's just the first term [Jacobian] in a Taylor series > >expansion). Since the linear component of any nonlinear system dominates > >the nonlinear in the proximity of equilibrium, what applies to a Leontief > >IO system will apply to any more general technology in the same region. > >The mathematical truth in this, which I endorse, nevertheless does not >rebut my point that in all cases but the limit of CES production, to which >the Leontief system corresponds, substitution among factors (and thus >potential "marginalist" considerations) arise, perhaps with indefinitely >large elasticities of substitution, even "in the proximity of >equilibrium"---indeed, *particularly* "in the proximity of equilibrium," >because this is exactly the region in which marginalist comparative static >analysis applies. > > >Since neoclassical economics is obsessed with equilibrium, the defence that > >a more general production system applies in general is NO DEFENCE. > >Again, I question whether the neoclassical system can legitimately >considered any more "obsessed" with equilibrium than the Sraffian system. >The latter presupposes that the law of one price obtains universally, and >implicitly, that the set of prices thus rendered are not simply transitory >in the context of its theoretical representation of capitalism (otherwise >why study it?). Even if you don't choose to call this representation an >"equilibrium," it is formally equivalent to it. > > >This is an elementary point from differential/difference equation > >mathematics, but it is something that few economists appreciate. It also > >leads to a result most Sraffians aren't aware of--that the equilibrium of a > >Leontief production system with spot prices is unstable. This result was > >derived back in the 1960s, but again most economists of all persuasions > >don't seem to appreciate its import. > >I don't understand this point--"unstable" with respect to what perturbations? > >Gil > > > >Cheers, > >Steve > >At 05:26 AM 7/02/2002 Thursday, you wrote: > >>Who has advanced our understanding of capitalism since Marx? I think it is > >>impossible to answer this question briefly. When I try, I start with a > >>short list of "the usual suspects" (though my list might differ from that > >>of others on OPE-L, perhaps.....), and then find myself immediately > >>branching out to recognize the related efforts of many others who have > >>contributed small but important insights. Like Gary, I put Sraffa > >>relatively high up on my list, but not exactly for the reasons he gives. > >>Here's why: > >> > >>Gary writes: > >> > >> >I would have to say Sraffa (surprise!), mainly for showing that the > >> classical > >> >P.E. tradition running through Ricardo to Marx is robust: i.e. its > >>fundamental > >> >insights regarding value and distribution hold up when the problematic > >> labor > >> >value analysis is discarded. He also pointed the way to a critique > of the > >> >marginalist theory that displaced classical and Marxian P.E. > >> > >>To me, the validity of the latter statement depends strongly on what one > >>means by "marginalist" theory. > >>Speaking broadly, I would say that Sraffa's "critique" of marginalist > >>theory is only achieved by arbitrarily dismissing or obscuring realistic > >>conditions under which "marginal" considerations *might* plausibly arise. > >>Three examples: > >> > >>1) Capitalism universally features markets for financial capital and labor > >>power in addition to markets for commercially produced commodities, but the > >>former are not incorporated into formal Sraffian analysis. Thus the > >>"un-marginalist" Sraffian "result" about the relation between wage and > >>profit rates arises in part from begging the question of how these prices > >>(and they are indeed prices, in addition to representing claims on total > >>product) are determined in their respective markets. > >> > >>2) The standard Sraffian model presumes that Leontief (fixed-coefficient, > >>constant returns to scale) production conditions obtain universally. > >>Whether this condition reasonably describes actual capitalist economies is > >>a matter of debate, but in analytical terms it is extremely restrictive. > >>In particular, Leontief can be understood as the limiting case of a more > >>general constant elasticity of substitution production function, and in > >>every instance of this function *except* the limit as the elasticity > >>parameter approaches infinity, factors are substitutable at the margin, and > >>thus "marginalist" considerations would presumptively apply. > >> > >>3) The standard Sraffian model also imposes rather stringent market > >>conditions, without providing any formal grounds for understanding when and > >>why they might obtain. In particular, it is assumed that the "law of one > >>price" holds throughout, even in the (otherwise unanalyzed!) markets for > >>capital and labor power. If you were to ask what sort of economic behavior > >>would make this strong condition tenable, the only answer I know of would > >>be arbitrage: market participants taking advantage of non-cost-based price > >>differentials by shifting their choices of quantities demanded and > >>supplied. But arbitrage is a form of optimizing behavior, and what's more > >>it necessarily involves optimizing *at the margin*--otherwise price > >>differentials would not be driven to zero, as the Sraffian model presumes. > >>So if the model implicitly presumes marginalist optimizing behavior here, > >>on what legitimate grounds does it categorically rule this behavior out in > >>other plausible contexts? > >> > >>[Related thought experiment: Marx's analytical "base case," employed > >>subsequent to V. I Ch. 5 of Capital, presumes that commodity prices are > >>proportional to their respective values. Sraffian analysis replaces this > >>basic scenario with one in which the law of one price obtains universally, > >>but commodities typically do not exchange at their respective values. As > >>Marx notes in the last footnote of Ch. 5, neither representation describes > >>actual capitalist market outcomes. On what grounds *intrinsic to the > >>theory* could one justify invoking the Sraffian in preference to the > >>Marxian base case? ] > >> > >>Moreover, and more importantly, "marginalist" analysis, in the sense of > >>taking derivatives of production, cost, or utility functions, is not the > >>essence of *neoclassical* theory as it has developed in the 20th century. > >>Which leads me to the next point: > >> > >> >The contribution was both positive and critical: he gave us a reason to > >> stick > >> >with classical P.E. (the theory holds up) and a reason to ditch > >> neoclassical > >> >theory (it DOESN'T hold up). > >> > >>I don't see this claim at all, since by the second welfare theorem, any > >>market outcome generated by the Sraffian model could be supported as a > >>solution of a neoclassical general equilibrium market model, and moreover > >>one with Leontief production for which no derivatives are taken anywhere > >>(so no "marginalist" conditions arise, in that limited sense). Thus any > >>valid *positive* outcome emerging from the Sraffian model (as opposed to > >>the "results" that emerge from simply failing to incorporate certain > >>real-world markets or implicit behavioral assumptions) can also be > >>demonstrated with the appropriately specified neoclassical general > >>equilibrium model, with the important difference that in the latter the > >>grounds for restrictive conditions such as the "law of one price" emerge > >>from explicitly stated underlying conditions (conditions, by the way, that > >>are never ruled out by Marx or Sraffa). For example, as Mas-Colell has > >>demonstrated, one can readily generate "reswitching" phenomena in a > >>standard neoclassical GE model. > >> > >>In light of this, I wonder how Sraffian analysis can be used to demonstrate > >>that neoclassical theory "doesn't hold up," or for that matter that > >>classical theory "holds up" any better. > >> > >>Gil > > References: Blatt, J.M., (1983). Dynamic Economic Systems, ME Sharpe, Armonk. Jorgenson, D.W., (1960). 'A dual stability theorem', Econometrica 28: 892-899. Jorgenson, D.W., (1961). 'Stability of a dynamic input-output system', Review of Economic Studies, 28: 105-116. Jorgenson, D.W., (1963). 'Stability of a dynamic input-output system: a reply', Review of Economic Studies, 30: 148-149. McManus, M., (1963). 'Notes on Jorgenson's model', Review of Economic Studies, 30: 141-147.
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