Sunday, May 17, 2009

Reflections On "Sraffian Economics (New Developments)"

Michael Mandler has an article, "Sraffian Economics (New Developments)" in the latest edition of The New Palgrave Dictionary of Economics. I have been trying to read this. (Paul Samuelson's article, "Sraffian Economics", in the original New Palgrave is also heavy going.)

I have previously read Mandler as an anti-Sraffian willing to take the views he opposes seriously. I wonder if he is more positive now. Perhaps he feels that, although Sraffians are mistaken in theory, their mistakes are worthwhile to explore.

That is all subjective on my part, of course. Mandler is explicit on the issues of the indeterminateness of equilibrium and of tâtonnement stability. An indeterminate equilibrium is not merely a case of multiple equilibria. Rather, a continuum of equilibria arise. Perturbations of an equilibrium along this continuum would not set up stable or unstable forces driving the economy back towards or away from the original equilibrium. Rather the economy would just be in another equilibrium. The tâtonnement is a particular kind of exchange process that arises before the beginning of time in the Arrow-Debreu model of intertemporal equilibrium. Mandler argues that Sraffa has failed to demonstrate indeterminateness, and that issues of tâtonnement instability are not essentially connected to Sraffa's model of production; they arise from elements of utility-maximization.

Mandler has certainly been engaged by Sraffians (or vice versa) on exactly these issues. But I'm not sure that I agree that Mandler has picked out the essential points of Sraffa's book. The distribution of income is indeterminate in Sraffa's open model. I do not read Sraffa as claiming this property would still obtain if he closed his model by appending a specification of utility-maximizing consumers, including intertemporally. Rather, I take Sraffa as offering an open model demonstrating non-neoclassical theories of value and distribution can be constructed. If one insists on a closed mathematical model (for example), an empirical issue arises. I think Sraffa did not insist that his model be closed, at least, with elements of a model at the same level of abstractness and generality.

While tâtonnement (in)stability is interesting, I take Sraffian analysis to point towards stability isses elsewhere in, say, the Arrow-Debreu model. One can construct models of spot prices corresponding to the forward prices in the Arrow-Debreu model. These spot prices have their own dynamics that would arise even if spot markets always cleared instantaneously over time. Sraffa's model of production supports an exploration of limit points of this dynamics.

I have constructed examples with bifurcations, pointing to possibilities of complex dynamics in models of temporary or momentary equilibrium. (I don't claim to have a good grasp of the distinction, if any.) One can also show, through an analysis of structural stability, that many of the stories applied economists like to tell are without logical foundation.

Variations in the supply of labor can be modeled by perturbing a parameter in utility functions. An increased supply of labor is modeled by an increased desire for consumption, as opposed to leisure. Nevertheless, the corresponding equilibrium associated with an increased supply of labor, all other parameters held constant, might have a higher wage. The increased supply of labor need not drive the equilibrium wage down.

Likewise, variations in the supply of savings can be modeled by perturbations in a parameter describing intertemporal utility-maximizing. And greater savings can be associated, all other parameters held constant, with a higher equilibrium interest rate.

Relating the structural (in)stability of equilibrium limit points to the dynamics of temporary or momentary equilibria is a challenge to me. I am not sure whether interesting bifurcations are tied to capital-theoretic "paradoxes" such as reswitching and capital-reversing. I think it may depend on details of the model. In one reswitching example, I have found that whether the normal or "perverse" switch is associated with bifurcations depends on whether intertemporal maximizing representative agents are also modeled as choosing between leisure and consumption. Whether the latter choice is included or not seems to flip the result. But perhaps in some model where one has fixed the modeling choice, the existence of interesting dynamic behavior, in some sense, may be tied to the existence of perverse switches.

I may never resolve these theoretical issues to my own satisfaction.

4 comments:

BruceMcF said...

But the reason for being interested in Sraffa's model is that it does not require closure by a false model of consumer behavior ... inquiring into the properties of a system that involves a Sraffian system closed with a utility theoretic model is like asking whether pigs could fly if they had wings, and were not afraid of heights.

Robert Vienneau said...

I agree that combining an interest in Sraffa with explaining existing economies in the real world would lead one to be uninterested in closing the model with utility theory.

But I am willing to pursue questions of abstract math in generating an internal critique of orthodox economics.

YouNotSneaky! said...

This is a really good exposition, an interesting post and a fair treatment of the issue. What happened to the completely unwarranted "so much for ....." conclusions?

In particular I think your description of Mandler's intention:

"I have previously read ... as an ... willing to take the views he opposes seriously. I wonder if he is more positive now. Perhaps he feels that, although ... are mistaken in theory, their mistakes are worthwhile to explore."

is as much a description of some of your antagonistic readers as maybe a subconscious advice to yourself, mirror style.

BruceMcF - without some kind of closure, the Sraffian model can pretty much predict anything you want it to. Any kind of counter example to anything you want it to. Empirically it is useless. At least if we close the model with intertemporal maximization we pin down the rate of profits (through the rate of impatience) and then we can start talking about whether all the paradoxes that the model allows for are likely to occur at or below those kinds of rates.

An alternative would be a well specified model of bargaining or social power. Closest we got to that is a pretty ad hoc Philips curve. Maybe some kind of Sraffa-Goodwin hybrid? I think you're still gonna get that anything whatsoever can happen.

On the other side, Mandler's perfectly correct that the possibility of multiple equilibria has nothing to do with the production side as these are driven by the presence of wealth effects in the utility function. I think I've said this like five times already, every time Steve Keen came up at least. But multiple equilibria are not really a problem as long as there is a finite number of them.

What Mandler HAS argued somewhere is that, combined with the demand-side/marginalist/neoclassical Sonnenschein-Mantel-Debreu theorem, the anomalies on the production side can get you an infinite - hence locally indeterminate - number of equilibria (which is a problem, which of course does not exist in a non-closed model like Sraffa's, by construction).

Discussing Mandler's work (on Wikipedia) is how, IIRC, first met Robert and probably the reason I still read this blog.

BruceMcF said...

"BruceMcF - without some kind of closure, the Sraffian model can pretty much predict anything you want it to."

Without a closure it is not a functional system ... rather than being many-to-one or one-to-one, it is one-to-many.

But if social outcomes are overdetermined, that implies that outcomes within institutional subsystems cannot be both completely determined and functional ... they must either be open in the sense of being in part determined by outcomes of other institutional subsystems, or open precisely in the sense of being one-to-many systems, imposing boundaries upon the range of outcomes rather than selecting the final outcome within the range.

If a system of production requires prices that are consistent with system reproduction, and that implies a range of price sets rather than a price set, then the shape of that space of viable price sets is more useful information about the economic system than a specification of a price set or model of price sets arrived at by closing the system with a false model of human behavior.

(There is, of course, the separate line of inquiry that uses the Sraffian model as a forensic tool for examining the hypothetical neoclassical economy and better understanding how it differs from the economy within which we live.)

"An alternative would be a well specified model of bargaining or social power."

It seems likely that "well-specified" is a framing that ensures, for some institutional subsystems at least, that none of the models will be valid. That would be a quite useful framing for rescuing an refined and known to be invalid model from an invidious distinction wrt cruder models that are not known to be invalid.