Sraffa’s work was among the most important creations in the history of heterodox economics. His approach to relative prices and distribution, following from an alternative view of Ricardo’s economics, led to a unified system that allowed for explanation of relative prices movements and the determinants of the wage and profit rate without regards to marginalistic preferences; a feat in itself. However, a frequent critique that I have seen online, unfortunately by many Post-Keynesians too, is the claim that Sraffians rest upon a crude construction of temporary equilibrium in real, rather than historical time, and hence devoid of uncertainty of economic life. Marxists and Neo-Classical economists have also frequently leveled these criticisms, generally positing that Sraffa on reproduction was simply an example of either a special case of general equilibrium, or a special case constructed under ex nihilo reproduction(1). All of these criticisms generally from a line of thought that Sraffian cases of reproduction can only exist under equilibrium conditions, and some take these criticisms to the extreme, that of which Sraffian cases of reproduction may only be permitted to exist within special cases of equilibrium. These criticisms, however, have not been left alone by many Sraffians and broad tent Post-Keynesians. It’s important to note that Sraffian results are perfectly consistent under many generalized schemes of reproduction; and this blogpost is simply to briefly go over the responses to the equilibrium critiques by many heterodox and orthodox economists.
To not begin with Rosser Jr. on reswitching would be a crime, as his introduction of econophysics, particularly catastrophe theory, to Sraffian economics through bifurcations caused by reswitching was brilliant. Before I am to go onto the general approach used by Rosser Jr., it’s better to first explain what catastrophe theory is and its role in economics. Catastrophe theory, essentially, is a branch of bifurcation theory dealing with structural instabilities stemming from degenerate critical points of a function. The importance of this system, outside of pure mathematics, is quite vast, especially having to do with qualitative changes in the structural stability of an economic system stemming from a bifurcation or degradation of a critical point. We can follow the examples given in Mattuck’s seminal notes on ODEs, for the sake of expository clarity. To begin with, the question of the definition of “what” structural (in)stabilities are, is important. Structural instabilities have to do with the stability of a system’s critical points, i.e., if any variation in the parameters of the function do not cause a major geometrical change in the mathematical system, we can say that the system is structurally stable. A structural instability occurs when a change in the parameters of a function induces a major qualitative change in the geometrical structure of the mathematical system and its critical points. We can follow Mattuck’s notes for a rudimentary system where we may be able to describe structural stabilities with regards to changes in the parameters of a function (Or system of functions), from the 2 x 2 linear system:
x =ax+by
y = cx+dy
Which is said to be structurally stable if it happens to be a saddle, spiral, or a node. A saddle is essentially where, when using the quadratic formula, starts from a system where A and C have different signs in:
sqrt[B2-4AC]
This leads to the occurrence of a moving point originally tending towards (0,0) before tending towards infinity, essentially having to do with decaying or growing exponentials (If stable, there are no sines and cosines). Carrying on, we can continue with the above 2 x 2 linear system:
λ2-(a+d) λ+(ad-bc) =0
The eigenvalues of which being:
λ1, λ2=(a+d) +/- sqrt[(a+d)^2-4(ad-bc)]/2
Assuming that λ1 and λ2 are real (and distinct), Mattuck gives the possibilities with regards to the stability of the system:
λ1 > 0, λ2 > 0, an unstable node occurs
λ1 < 0, λ2 < 0, an asymptotically stable node occurs
λ1 > 0, λ2 < 0, an unstable saddle node occurs
What Mattuck shows, in essence, is that the roots (λ1, λ2) depend on a, b, c, and d. What this means is that a change in any of the parameters, even very slightly, e.g., a minor perturbation to the system’s parameters, a change would follow with the roots. This by itself is not interesting, but when considering special eigenvalues, a variation to the system’s parameters creates a major geometrical change. An example of this is provided below in the notes, e.g., imaginary eigenvalues, one zero eigenvalue, and a repeated real eigenvalue all have major implications for geometrical changes when varying parameters. In catastrophe theory, mathematicians deal with these major qualitative geometrical changes with regards to pure mathematical systems (ODEs) but also with physical systems (Uses in physics, particularly with states of water) and in economics, e.g., Puu’s work on the stability of an economic system(2). Rosser Jr,’s work was more or less a continuation of Puu’s work on disequilibrium in an economic system caused by a change in the parameters of an economic system, e.g., with the basic copied example above of a purely mathematical system.
Rosser Jr. (1983)(3) demonstrates that reswitching acts as a cause of discontinuous jumps along an optimal dynamic path. Reswitching then, acts as a cause of structural stability within an economic system traveling along a dynamic path, causing discontinuities that the system cannot merely “jump” over contra Bruno. These paradoxical zones can be seen as a demonstration of structural instabilities within Neo-Classical models which severely compromise the theoretical soundness of these models. Rosser Jr. provides a visual example of the reswitching model on page 185, demonstrating factor price movements with regards to the wage and profit rate cannot be treated as a smooth linear path, rather, a path that can generate complex movements and discontinuous jumps within the choice of technique. Roser Jr. observes that there are no discontinuities if k < minimum optimal steady state path k’; but a closer examination of the k-π relationship shows divergent pairs of π with regards to k’ and k’ increases beyond k’ min. The increasing divergence between π’s and the steady state k implies a larger discontinuous jump further away from the cusp point, the diagrammatical version being given on page 190. A variation of the rate of profit along the wage-profit frontier, for the sake of argument we can just assume that the variation will be smooth, the K/L ratio can vary in complex and disconnected ways. What’s important within these models is that Rosser Jr casts his results under a disequilibrium framework, in between steady states.
This means that one of the largest contributions to economic theory by Sraffians, particularly on choice of technique analysis (Which is in fact a constructive addition to economic theory, rather than being destructive as many economists seem to suggest, laying the foundation for technique analysis within joint product industries too), can be cast under disequilibrium conditions. This is in direct contrast to what many people, such as Burmeister, seem to suggest with regards to the validity of reswitching (Burmeister, however, does seem to have recanted his position on reswitching). Rosser Jr’s discontinuous capital-profit intensities simply demonstrates, in a more generalized manner, that the traditional marginal productivity theory of profit and wages, along with the interest rate dependent investment story, are simply too simplistic and cannot be thought of as valid in a real economy, especially when Rosser Jr’s model can cast upon capital theoretic paradoxes such as Bruno’s dynamic adjustment model. The wage, profit, and capital-per-worker value, can all have a leaping phenomena over intermediate values (Hence, non-linearity of the profit-capital relationship), or maintain a complex relationship with major discontinuous jumps, while the growth rate and consumption per worker reaches a final steady state. The importance of this cannot be understated, the economy in between steady states (The Hicksian traverse) can have multiple sources of instability through discontinuous jumps in the K/L and K/W ratio. Having been in-between steady states, the criticism of the Sraffian choice of technique analysis cannot hold, as it is in disequilibrium despite being on the traverse to a final steady state. It’s worth noting that this solution to reswitching’s usefulness in disequilibrium also resolves multiple issues debated within Post-Keynesian economics, e.g., the usefulness of the Sraffian analysis of technique under “radical uncertainty”.
Readers who have read parts of my older blog-posts should be aware of my position on uncertainty strongly echoicing that of George Shackle and Paul Davidson. While I have not fully recanted that position, I believe my previous analysis held many errors, I still maintain that uncertainty and analysis of decision making theory under uncertainty is integral to a proper functioning theory of choice of technique. Now, Joan Robinson’s critique(4) was primarily centred around the erroneous nature of a completely certain environment that Sraffa (1960) operated under for his single and joint product industries. Without an assumption of exogenously given “blueprints” and certainty about the viability of a production technique, the Sraffian analysis of choice of technique “missed its mark”, so to speak. Many Sraffians hold immense respect for Joan Robinson as a leading scholar in the Post-Keynesian tradition, albeit her questionable time as an ideologue for Maoist China, which has lead to this question being a hotly debated issue amongst Fundamentalist Post-Keynesians and Sraffians. A lot of recent work has been done by economists such as Fratini and Parrinello, but for the start of this section I’ll briefly talk about Rosser Jr’s solution to the Robinsonian critique. Rosser Jr, in his book From Catastrophe to Chaos: A General Theory of Economic Discontinuities, in chapter 8, extensively discusses the Sraffian approach to choice of technique analysis. Later in the chapter, a major point of discussion is the Robinsonian critique that Sraffian choice of technique analysis cannot hold under uncertain disequilibrium conditions. Aside from the given example above by Rosser Jr, that one cannot hold that reswitching can only occur in crude steady states rather than disequilibrium, he also provides a defense of the Sraffian analysis on a different ground. Rather than treating the blueprints and technical details of the technique itself, Rosser Jr. applies a technique cluster approach to reswitching of techniques. The Robinsonian critique is primarily centred around further heterogeneity of the choices of technique, i.e., techniques often have different composition dependent on its base, such as the differences between an iron based technique and a copper based technique.While the discontinuities may occur and the profitability of a certain technique relative to the profit rate being a complex function might hold theoretically, with transaction costs it makes little sense to suppose the major change in techniques with different basis’ would ever occur. Rather than adopting a more orthodox notion with a few finite techniques (This is not to say that reswitching isn’t possible under infinite technique conditions), what should be adopted is a conception of “technique clusters”. Technique clusters are techniques with similar technical similarities but are still heterogeneous enough to constitute a switch in the choice of technique and have the same implications for the marginalist theory of wages and profits. Rather than switching between completely different techniques, which is a possibility albeit not very likely, switching within technique clusters allows for a relatively “smooth” change while maintaining the complex non-linearities within the profit-capital relationship. Considering reswitching under more localized subgroups allows for a more microeconomic based analysis of reswitching while keeping the main analysis of Sraffa intact, which resolves the 2nd part of the Robinsonian critique. The final critique to be considered is that of radical uncertainty in an economic environment, to which reference to the aforementioned authors Fratini and Parrinello is in order.
Fratini(5) and Parrinello(6) both take the approach of treating prices and quantities as a stochastic variable, rather than treating them as something definitive and known with complete certainty. This analysis can incorporate the concept of gravitation into Post-Keynesian economics, rather than treating it as a deterministic process, prices and the concept of gravitation can hold as a stochastic concept which can still hold under random perturbations to the gravitation process (Parrinello also holds that the Sraffian choice of technique analysis can also hold under random perturbations, more on that later). Following Fratini, we can set up the probabilistic system as:
π_n,t=1/t (t^\sum_t=1 π_n,t)
The vector, again following Fratini, of average market prices (after t observations) is πt=[π1,t , π2,t , … , πN,t]. We can then put forth conceptual ideas with the random vector πt and its relationship to gravitation. Before continuing, it’s worth noting that this entire analysis is predicated on the non-exact-ness of the solutions to Sraffa’s equations, i.e., if we take these concepts as a stochastic rather than deterministic process, we lose the rigorous certain solution generated by a deterministic system; but retain the main components of Sraffa’s analysis on choice of technique and gravitation, after the initial deterministic price system has been solved for. The absolute value of the random vector πt subtracted by π is the (Euclidian, or non-curvature based gravitation path) distance between πt and π, in-line with the Euclidean addition postulates.
pr(|π1-π|) < ε
Is the probability that this distance is smaller than, a real number given to the equation. The conditions of natural prices, or prices of production, or the initial deterministic prices generated by Sraffa’s simultaneous equation method, to be the centre of gravitation for market prices would then be:
lim_t ->∞ pr(|π1-π|< ε =1
Which implies that market prices converge to the system’s “Sraffa prices” probabilistically, rather than being a certain deterministic process. Parrinello (1990) also suggests, additionally, near the end of the paper, that one can incorporate random perturbations into the Sraffian choice of technique analysis. Rather than treating the Sraffian choice of technique analysis as a cost minimizing technique under a steady state growth path, one can incorporate the effect of “seasonality” into the Sraffian choice of technique analysis as a statistical characteristic perturbing the Sraffian system. One can maintain that all of the usual components of the criterion for the choice of technique in the Sraffian system is correct, and hence the results from which (capital paradoxes) are valid, while incorporating some statistical characteristics of randomness into his analysis. Incorporating the effect of the weather, a random variable, particularly chaotic more than random, into the Sraffian choice of technique analysis would render it a more generalized system while maintaining the core components of Sraffa’s analysis. The distribution of random events in the Sraffian system can impact the choice of technique analysis, e.g., the weather, but it does not serve to invalidate the Sraffian analysis in a sufficient manner.
This short blog-post was to simply elucidate some of the Sraffian results that can hold in disequilibrium by copying and bringing together a few examples from various authors. I apologize for having to crudely format equations onto here, the equation function on blogger seemingly is terrible. I will attempt to resolve this, but for now, please bear with me.
With reference to my long break from this blog, I have had a lot of work recently to take care of, such as academics and research papers. I will resume a regular update schedule again, starting this week.
References:
1. ) Jefferies, W. (2019). Piero Sraffa’s physical price system and reproduction without production. Capital & Class, 44(1), 63–83. https://doi.org/10.1177/0309816819852771
2.) Puu, T. “Regional Modelling and Structural Stability.” Environment and Planning A: Economy and Space, vol. 11, no. 12, 1979, pp. 1431–1438., https://doi.org/10.1068/a111431.
3.) Barkley Rosser, J. (1983). Reswitching as a cusp catastrophe. Journal of Economic Theory, 31(1), 182–193. https://doi.org/10.1016/0022-0531(83)90029-7
4.) Robinson, Joan. “The Unimportance of Reswitching.” Quarterly Journal of Economics 89 (1975): 32-39.
5.) Fratini, Saverio M. and Naccarato, Alessia, The Gravitation of Market Prices as a Stochastic Process (November 2016). Metroeconomica, Vol. 67, Issue 4, pp. 698-716, 2016, Available at SSRN: https://ssrn.com/abstract=2847407 or http://dx.doi.org/10.1111/meca.12116
6.) Parrinello , Sergio. Some Reflections on Classical Equilibrium, Expectations, and Random Disturbances. Centro Sraffa, www.centrosraffa.org/pe/6,1-2/6,1-2.12.%20Parrinello.pdf