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.