A Position On the Philosophy of Mathematics

A Slightly Economics Related Post

Recently this blog has been considered with some of the more practical considerations of growth and proper representatives models, along with the theory of distribution. I intend to continue with this, but for now, I believe it is worth considering a meta-theoretical note on formal mathematical theories in economics. 

The philosophy of mathematics is quite an extensive subject with varying interpretations, just as economics remains. Economics has concerned itself with the much more formalistic aspects of modeling for a great deal of time, particularly with the proper methodology of proofs for the existence of a mathematical phenomena and counter-examples to conjectures, e.g., the marginal productivity theory of remuneration of factors of production and the counter-example reswitching posited itself as. The reason for the incorporation of such topics into our economic understanding is due to the fact that economics has blurred the lines with pure mathematics as time has gone on, with the counter-revolutions starting in the 1960s in terms of the “Rational Expectations Hypothesis” and, e.g., the minor revolution led by Piero Sraffa at Cambridge in the 1960s. All of these have established a necessity for a meta-theory of economical proofs. The importance of a meta-theory for mathematics, and by the transitive property, economics, should not be understated in any attempt to further discourse on the formal aspects of economics. 

Mathematics itself is riddled with peculiarities, since the older Greco-Roman branch of mathematics established itself as an authority on the nature of mathematical objects as a real, but abstract, object. The existence of such fundamental objects in mathematics, e.g., the constant of π (Pi), are treated as real, but abstract objects that directly impact the functionality of the world. This position is generally called “Platonism” and is held as one of the oldest positions on the philosophy of mathematics. I do not find this view to be particularly useful with regards to the ability to justify economic proofs, if we treat mathematical objects as real bodies then the very obvious question arises of how we are to acquire knowledge of said objects. As mentioned above, these objects are held as abstracts, and hence the self evidence of these objects is not certain, which leads to dissent over the philosophy of certain mathematical objects. The question arises, however, of how individuals are able to recognize these objects for themselves. On a theoretical note, how do we acquire knowledge of these abstract objects? All justifications of such an abstract argument imply an element of mathematical anti-realism, the use of constructive intuition or by knowledge of empirical facts. The problem with such a position is, however, when introducing such individual dependent concepts into mathematics, the ability to verify the “independence” of Platonic objects becomes dubious, there is no premise to then state that mathematical objects exist independent of the observer.

    On the opposite end of such a spectrum is the concept of “formalism”; or that mathematics is fundamentally a game that evolves with societally given rules, and is nothing but the manipulation of alpha-numerical strings according to such a rule. This approach, pioneered by David Hilbert, essentially treated mathematics as a concept completely dependent on society and its own evolution. Assume, for the sake of argument, that as time evolves we forget everything we knew about mathematics, due to some catastrophe. Further, assume that we change a great deal in terms of our biological structure, so that our ability to manipulate said alpha-numerical strings changes. In such a situation, a formalist would hold that mathematics itself remains without any similarities to its older version prior to such a destructive, Schumpeter styled, evolution. Hilbert sought to reconstruct mathematics using his conception of mathematical objects as game-styled manipulation, using an axiomatic method. The problem with such an approach, however, was that its validity was dependent upon the validity of the all-encompassing nature of mathematics, that the axiomatic method was truly complete. One of the most notorious discoveries in the philosophy of mathematics was Gödel’s incompleteness theorems, stating that any system that was true was mathematically unprovable. I will refrain from discussing this, leaving it to a future post on the implications for economics, but in essence Gödel’s incompleteness theorem held that a true mathematical proposition, using the axiomatic method, cannot be proved within its own system, implying a fundamental element of uncertainty within our mathematical computational abilities. 

The two other positions on the philosophy of mathematics that I will briefly discuss is Logicism and Intuitionism/Constructivism. Logicism essentially purports itself as a theory that describes the fundamental nature of mathematics through logic. All mathematical statements are reducible to logical, true or false statements, and in that situation mathematics is a branch of logic. Such a theory sits well with most individuals, it explains to a satisfactory degree how we acquire knowledge of mathematics, and we find it almost completely obvious that logic is the fundamental piece of branches of mathematics such as geometry. Henri Poincaré (​​1902) among others, however, ruthlessly criticized logicism as a theory that adds little to our pure mathematical knowledge. Reducing everything to a few true or false statements essentially means math itself is tautologous, and that any acquisition of any further mathematical knowledge is simply equivalent to the knowledge known before. On a more theoretical point, Poincaré held that mathematics is the application of the pure intuition in a construction of the intuitive continuum, arithmetical and topological. Our knowledge of geometry comes from our pure intuition, our ability to construct mathematical objects using the pure intuition of time and space. The pure intuition of space manifests itself into geometry, we “construct” geometrical objects using our ability to form a topological continuum and analyze the intuitive relationships between lines, points, rays, and segments in Euclidean geometry1. The pure intuition of time acts as our ability to create a numerical continuum of real numbers, irrationals, and imaginary numbers. Poincaré took an essentially Kantian view on mathematics when criticizing the views of Dedekind, Russell, and Frege on the foundations of mathematics, utilizing a similar conception of the intuitive “synthetic a priori” and transcendental idealism. Our mathematical knowledge was all constructed through our use of intuition, and deductive proofs which may be reducible to true or false statements lack the ability to expand our knowledge of mathematics. Take the example of 1 + 1 = 2; Kant argued that such a statement was not true by its premise, it was not knowable a priori without intuition. The mind constructs a numerical continuum using the intuition and constructively finds the final result, the truth of 1+1 is not deductively reducible to a logical a priori statement due to the inability to extend one’s mind and intuitively construct and arithmetical continuum of numbers. The Brouwerian continuum hypothesis (Poincaré being a precursor to an intuitionistic-conventionalistic philosophy of mathematics) can formally be framed in terms of: 

Assume two predicates, A(α, x), α ranging over choice sequences (The intuitive idea of a set, a constructible set in essence, see Troelstra (1983)) and x over naturals. We can define extensionality as: 

Hence, the weak continuity principle is: 

    Where α and β range over choice sequences, m and x over naturals, and α’m being the initial segment of length m. This holds that we can assign a number to every choice sequence, using a tree-like conception of number-generators and free-choice sequences, given the initial segment of such a sequence, α. The importance of such given above implies that we can provide geometrical and arithmetical knowledge of objects using the continuum hypothesis, but recasting it under intuitionistic terms, as the non-constructive proof given by classical mathematics remains unacceptable. Providing a continuum allows for construction of mathematical objects using the mind, i.e., mathematical objects are mind dependent, and hence, constructions. This implies that the use of the intuition is necessary in mathematics, and hence, mathematics is not reducible to pure logico-mathematical statements, meaningless alpha-numerical strings, or abstract mind-independent objects. 

The proper interpretation of a meta-theory of proofs, seems to suggest in the direction of intuitionistic constructivism (To be distinguished from non-intuitionistic constructivism). In this line of thought, to prove an object is not to utilize the principle of the excluded middle, ¬(p∧¬p), cannot hold as a theory of proofs. The use of algorithmic constructive axiomatizations is the only valid manner to prove mathematical-economic propositions, formally. One peculiarity is the parallels to be found with Sraffa’s constructive, algorithmic method, and the BHK (Brouwer–Heyting–Kolmogorov) interpretation of constructive proofs. The manner of which Sraffa’s standard system is constructed is through a constructivist methodology of repeated algorithm like methods, along with the proof of the uniqueness of the standard system (Miyao 1977).  Such parallels, in terms of the constructive existence proof go quite deep, seemingly implying a conscious choice on the part of Sraffa’s proof style. This interpretation seems to be suggested by Velupillai (2008), and it is interesting to note that Sraffa’s time as an academic friend to Ludwig Wittgenstein in Cambridge, an (in)famous philosopher with a constructivist philosophy of mathematics (Rodych 2013) may provide itself as a link to Sraffa’s peculiar proof style. 

The purpose of this brief blogpost was to explore on a more meta-theoretical note, the foundations of mathematical-economics and the proper methodology for proofs in the aforementioned, along with a note on the mathematical foundations for Sraffa’s Production of Commodities by the Means of Commodities. I plan on exploring this more in depth later, but due to an unexpected sickness, I seem to lack the energy to write a sufficiently long blog-post. Old disclaimer applies with regards to equations, I apologize to have to screenshot equations from Google-Docs; but Blogger does not input it correctly otherwise.

Footnote:

1. Some hold that the intuitionistic methodology is incompatible with Non-Euclidean theories of geometry, found present in Einstein's Riemann Geometry based approach. Poincaré criticized this view through a conventionalist lense of geometry and the sciences, for further discussion with regards to the intuition of space and time and Non-Euclidean geometry, read Arledge (2014). 

References: 

Rodych, Victor. "Mathematical Sense: Wittgenstein’s Syntactical Structuralism". Wittgenstein and the Philosophy of Information: Proceedings of the 30th International Ludwig Wittgenstein-Symposium in Kirchberg, 2007, edited by Alois Pichler and Herbert Hrachovec, Berlin, Boston: De Gruyter, 2013, pp. 81-104. https://doi.org/10.1515/9783110328462.81

Velupillai, Kumaraswamy. (2008). Sraffa’s Economics in Non-Classical Mathematical Modes. 10.1057/9780230375338_15.

Miyao, Takahiro. “A Generalization of Sraffa’s Standard Commodity and Its Complete Characterization.” International Economic Review, vol. 18, no. 1, [Economics Department of the University of Pennsylvania, Wiley, Institute of Social and Economic Research, Osaka University], 1977, pp. 151–62, https://doi.org/10.2307/2525774.

Troelstra, A. S. “Analysing Choice Sequences.” Journal of Philosophical Logic 12, no. 2 (1983): 197–260. http://www.jstor.org/stable/30226270.

Poincaré, Henri, 1854-1912. Science and hypothesis. London, New York, Scott, 1905 (OCoLC)622773044

Arledge, Chris. (2014). Kant, non-Euclidean Geometry and A Priori Knowledge: A Reassessment.

    

A Note On The “Rigid” Interpretation of the Sraffian Supermultiplier

Since the development of Roy Harrod’s (1939)(1) growth model, there has been a great deal of interest from orthodox and heterodox economists on the topic of economic growth. Extension of the “Keynesian Hypothesis”, and other facets of growth taken to exist without any formal mathematical interpretation, were of great interest with regards to the new concepts Roy Harrod and Evsey Domar (1946)(2) independently developed. Orthodox economists, or what was left of it after Keynes’ devastating critiques of vulgar-classical economics (to be distinguished from the utilizable parts of classical economists, such as the circular flow/price determination method), formulated itself in “Neoclassical Synthesis Keynesianism”. Adopting the label Veblen originally pejoratively used, these Neoclassical Synthesis Keynesians, e.g., James Tobin, John Hicks, Paul Samuelson, and in particular, Robert Solow, attempted to demonstrate that Keynes’ results were consistent within a Neoclassical framework, aside from certain concepts out-right rejected for their inability to be formalized without violating prior assumptions, or simply merely neglected for presumed theoretical deficiency. Robert M. Solow, in particular, was primarily concerned with growth. Adopting the framework laid out by the Harrod-Domar model of economic growth, Solow (1956)(3) set out to project the role and determinants of economic growth into the long run. Rather than maintaining his “Keynesianism”, Solow set out to explain economic growth as constrained by labor force growth and predicated upon factor substitution and technological advancement. The “Solow-Swan” model remains the bedrock for most of New-Consensus macroeconomics, along with the recent work developed by the French economist Thomas Piketty, Capital in the Twenty-First Century(4) on inequality. Despite the slight changes made to the production-function through changes in the exogeneity of technology, the factor substitution based model of economic growth remains popular within the New Macroeconomic Consensus, the main idea of a lack of aggregate demand determining economic growth and the idea of growth being perfectly determined by other factors, e.g., endogenous technological progress functions and augmentation in the labor force population. Understandably, dissent has risen within most economists for the absolute lack of care for aggregate demand in determining growth, and the nature of these results projected into the long run. 

The Post-Keynesian economists and Marxist economists have provided the most satisfactory alternative to the traditional production function based growth model, that neglects aggregate demand as the key determining factor in economic growth. For the sake of the main topic of this short blog post, I’ll focus on the former rather than the latter, which has deficiencies of its own(5). The main contenders(6) for the replacement of the orthodox economic growth models within Post-Keynesian academia happen to be the Neo-Kaleckian models of growth, which attempt to project the paradox of costs and thrift into the long run, the Kaldor-Thirlwall models of growth which happen to be primarily centred around induced investment functions, the oldest Cambridge Kaldor-Pasinetti model of economic growth which attempts to explain the influence of aggregate demand on investment through endogenous shifts in the distribution of income, and the Sraffian Super-Multiplier model, a model of economic growth originally formulated by John Hicks(7) but culminated in Franklin Serrano (1995)’s “Sraffian” model which explained growth through induced demand and investment functions, a result from “autonomous demand”. For the sake of length, I will try my best to give a simple overview of the Sraffian “Second Keynesian Position” based critique of the rest of these growth models. To begin with, in Post-Keynesian academia it has been popular to primarily use the Neo-Kaleckian model of economic growth, centred around the role of investment in determining economic growth. I will briefly exposit this position of economic growth, and then proceed to lay out the “Sraffian” criticisms of the Neo-Kaleckian model of growth. 

A novel feature of this approach is that the rate of capacity utilization in the short and long run are both endogenous variables, in contradiction to the Pasinetti-Kaldor “Cambridge” model of growth, where changes in the rate of accumulation were accommodated by endogenous changes in the distribution of income. A basic model, without government malfeasance or balance of payments concerns, consists of solely,

The profit rate identity: 

r=(PY)(YYn)(Ynk)=Πuv

The saving function: 

Gs = sr

And, the accumulation function: 

gK = α + β(u - un) 

Where we can take α as the trend of the growth of sales, β taken as the parameter which represents the level at which entrepreneurs react to discrepancies from the actual rate of capacity utilization to the normal rate of capacity utilization. Using the familiar condition I = S, the equilibrium rate of capacity utilization is therefore: 

This, being the orthodox Neo-Kaleckian growth model, happens to be filled with a major oversight of the role of Harrodian instability within the model of economic growth, making it realistically impossible to approximate growth using that model if the actual rate of growth is to constantly diverge from the equilibrium rate of growth. As entrepreneurs,with adaptive expectations, revise their assessed growth rates as it diverges from the actual rate of growth in the economy. The constant reassessments in the expectations of the growth rate of entrepreneurs from an initial perturbation to the rate of capacity utilization ensures that there is a revision in the growth expectations from this initial perturbation. With constant reductions, from this new equilibrium growth rate e1 from the originally “reduced from” e, causes a further reduction in the growth expectations of the entrepreneurs, e1 falling to a new expected growth rate of e2, and due to the endogenous nature of capacity utilization in the orthodox Neo-Kaleckian models, a Myrdalian cycle of cumulative causation ensures major divergences from the rate given by the model and hence a deficiency in the predictive capability of said model. It also follows that the rate of capacity utilization, again, being an endogenous variable, can tend to 0 under these conditions of Harrodian instability. It may also follow that under these conditions, an initial increase in the rate of capacity utilization may trigger a similar cycle, the rate of capacity utilization tending to a finite or infinite rate of capacity utilization. This problem has notably plagued multiple growth models, and hence, the necessary revision of the Neo-Kaleckian models using the Sraffian Supermultiplier model. I will not go into detail, as I want to remain on topic for the title of this brief blog post, but in essence the amended Kaleckian model with a supermultiplier function still has the possibility of Harrodian instability if the perturbation is strong enough, something the Sraffian-Supermultiplier model does not suffer from. There have been multiple solutions given to the problem of dynamic Harrodian instability in Neo-Kaleckian growth models, and again, I will not be able to go over all of them. The first, more orthodox, or famous, one holds that there are  a range of values of capacity utilization, but again, this does not provide us with a proper answer as the possibility of a tendency to infinite or zero rate of capacity utilization is possible even with a varying range based rate of capacity utilization, for more discussion, see Cesaratto (2015)(8). The second, newer answer holds that the problem of Harrodian instability is solvable under ABM, agent based modeling(9). For, again, the sake of remaining on topic, I will not discuss this solution, but I do hold that this is a sufficient answer to the problem posed by Harrodian instability, but this may also fall under the work of Sraffian-Supermultipler models, i.e., what would result is a hybrid SSM-Neo Kaleckian model that Lídia Brochier seems to have promising work upon(10), assuming the issue of the rigid investment function is fixed. The other model under consideration, that I will critique much more briefly, is the Cambridge model of economic growth, the one formulated by Kaldor and Pasinetti. In these models, the issue of Harrodian instability is solved through offsets following from a change in accumulation endogenously in the distribution of income. Assuming that both only capitalists save, and that the Keynesian identity of I = S, the Cambridge equation follows: 

Gk = sπr

Where gk is the rate of accumulation, r is the profit rate, and sπ is the marginal propensity to save out of profits. The problem, from a Sraffian point of view, is quite simple. The manner of which we determine the relative prices of a set of n commodities is by taking the distribution of income as largely exogenous, i.e., either taking r or w as exogenous, we are able to get rid of ⅓ of the unknowns within the system and then solve for prices, without over or under determination of prices. If the identity above holds, then the distribution of income cannot be taken as exogenous and is a result of changes in the rate of accumulation. The return of circumventing Harrodian instability, from a Sraffian point of view, is not worth the cost of abandoning the fruitful, general, theory of relative prices/choice of technique analysis. Hence, it is necessary to abandon this “Cambridge-Keynesian” approach to growth, and look for a more dynamic theory of demand based economic growth, which can still keep the ‘Sraffian Hypothesis’. Finally, we can consider the Thirlwall-esque “Kaldorian” growth model, premised upon export-led growth. Being a generalized form of the Harrodian trade multiplier, the major source of stability within a export-led cumulative causation based BoP constrained growth model is the stabilizing effects of autonomous exports. Autonomous exports influence the value of the “Kaldorian multiplier” acting as a stabilizing effect for the Kaldorian export-led growth model. What I do not wish to do is critique the Kaldorian growth model, it remains a tremendously viable approach to describing variances in the rate of growth internationally, but rather, explain that the Sraffian supermultiplier is a generalized form of the Kaldorian export-led growth model. Rather than holding that the only source of autonomous demand arises from autonomous exports, it is more logical to hold that there are multiple “autonomous” sources of demand in a capitalist economy, e.g., credit financed consumption, government expenditure, etc.. The Kaldorian export led growth model is merely a specific model of the Sraffian-Supermultiplier approach to growth, pertaining primarily to the balance of payments constrained equilibrium growth rate. This thesis strongly resonates with the Kaldorian growth model proponent Professor J. S. L. McCombie(11), that the Harrodian trade multiplier can also be interpreted as the Hicksian Super-Multiplier. Considering my frequent references to the Sraffian Supermultipler Model, it’s only proper to discuss what the Sraffian-Supermultiplier is. As mentioned above, the SSM (Sraffian Supermultiplier Model) is a model of economic growth that follows from Hicks’s original Supermultiplier model, but retains many “Sraffian” elements, e.g., exogeneity of distribution. I will simply go over a basic model of the SSM, refraining from engaging in the more “specific” growth models which reference to, e.g., debt financed autonomous demand. An important property of this model is that the condition that investment generates savings, rather than savings generating investment, the famous ‘Keynesian hypothesis’ holds under the SSM. This implies that the nature of Say’s law in Neoclassical and Austrian models, predicated upon loanable funds, cannot be valid within the framework of these growth models, which is important to note for a demand-led growth regime extended to the long run. Furthermore, autonomous demand, not to be confused with exogenous demand, generates induced consumption through the multiplier, and generates induced investment through the accelerator process. The importance of the Hicks/Sraffian SM is that it rests upon a marriage of both the accelerator and multiplier principles, i.e., the economy is strongly dependent on aggregate demand as the source of capacity creation and the determinant of long run processes of capital accumulation. Finally, there exists the long run convergence of the rate of capacity utilization to an exogenous given rate of normal capacity utilization. This phenomena is called “long run effective demand”; a possibility is that, aside from the tendency based upon long run effective demand, is also dependent on societal customs, retaining the ‘Sraffian Hypothesis’ on distribution of income. We may introduce the basic model, now that the main important points of the SSM approach to growth has been highlighted: 

Y = C+I+G+(X−M)

C =C0 +Cy = C0 +c(1−t)Y

M = mY

I = hY

The output equation is quite standard, which happens to be the first equation above. Total consumption is defined as the total of autonomous consumption and induced consumption, and for the last 2 equations, investment and the import functions are linear functions of income at said time period. The investment function depends highly on expected demand, in this baseline model, one of the minor parting differences with the Neo-Kaleckian growth model. We may then derive that of effective demand, the totality is composed of autonomous and induced demand, and a further disaggregation results in that non-capacity creating autonomous demand is primarily composed of government spending, total exports, autonomous business expenditure, and autonomous consumption. Therefore, Z, autonomous demand, may be expressed identity wise as: 

Z = C0 + G + X + RD

Where X are total exports, G is total government expenditure, C0 is credit financed consumption, and RD is autonomous business expenditures. After taking the marginal propensity to save as s = 1 - c(1 - t), and then solving the equations given above for Y, total output: 

Where the growth rate of output happens to be: 

And we take v = K/YP. Now, coming to the investment function and its relationship to the main point of this blog-post, is the endogenous changes of the discrepancies in the utilization rate: 

h’=hγ(u−un)

Where, make note for the later discussion of Palley’s position on the SSM, γ > 0 is a reaction coefficient that is necessarily positive, because of the nature of aggregate demand in determining output-growth. Following from the above equations: 

Gi = gy+β1(u - un)

So, investment growth is dependent on output growth (given by the accelerator principle from autonomous demand, given above), and changes in capacity utilization rates. Expansion of productive capacity, being dependent on the level of effective demand at the time, we justify the equation given above. Now, we may finally find the long run position of the model: 

Gn = gi = gY = gK = gZ

This is important, specifically the second order of the equation above, gi = gy, which means that investment is fully induced by autonomous demand, which is a major difference from the autonomous Neo-Kaleckian investment function that allows for “profit-led growth”. This, importantly, implies that the dichotomy between profit and wage led growth cannot exist in a Sraffian Supermultiplier model, all growth is wage led as with a fully induced investment function, investment is dependent on the sources of autonomous demand, Z, given above. Due to the differing marginal propensities to save across income levels, i.e., those with lower incomes tend to spend more than those with higher incomes, most of effective demand is dependent on wage earners, not the profit share allocated to capitalists. With a semi-endogenous investment function, dependent on the level of effective demand at the particular time, most of economic growth is dependent on wage earners. This does then have implications for lower income stagnation, and another sufficient argument for the allowance of tax-cuts to improve the value of the Supermultiplier given above, dependent on the level of autonomous demand, determining the fully induced investment function given above when determining the long run position of the model. 

Finally, now arises the title of this blog-post. The reason I am discussing the notion of a lack of uncertainty within Sraffian supermultiplier models is that this remains one of the largest divisions between fundamentalist Post-Keynesians and Neo-Kaleckian Post-Keynesians with Sraffian Post-Keynesians. An economic environment devoid of uncertainty is devoid of little economic use, and cannot explain a majority of the dynamics found in capitalist economies. The problem is mainly with the treatment of the reaction coefficient,  γ, for the endogenous changes in the investment function. Some individuals, proponents of the SSM approach to growth and distribution(12) and those against the model(13) have a misfounded conception of the SSM assuming rational expectations within its investment function, thereby founding its investment function, the crowning achievement of the induced/autonomous distinction existent in making the SSM important for policy discussions, in an unrealistic environment which completely differs from the Fundamentalist Post-Keynesian and Neo-Kaleckian Post-Keynesian economic environments.  I wish to briefly go over this critique, and simply point out the ill-founded nature of attributing such to the SSM approach due to their shared confusion over the nature of expectations in the SSM, i.e., confusion of adaptive expectations and rational expectations. To begin with, individuals like T.I Palley seem to suggest, even supporting it due to having such a feature, that the Sraffian Supermultiplier model assumes rational expectations: 

“The combination of the Keynesian macro model and rational expectations (RE) therefore solves the problem of Harrodian instability which could arise from destabilizing revisions of expected demand growth. The Keynesian RE framing also provides a microeconomic justification for the assumption that expected demand growth equals autonomous demand growth (Bortis, 1997; DeJuan, 2017).”(14)

A very basic error that Palley makes here is the conflating of adaptive expectations with rational expectations. Adaptive expectations is perfectly consistent with a Keynesian uncertainty based approach to economic growth, as individual adaptive expectations, while assuming a stochastic learning process, on aggregate leads to complex dynamics over the ability for individuals with adaptive expectations to forecast the behavior of other agents; a phenomenon that strongly differs from rational expectations. Rational expectations is premised upon a learning mechanism that is either in a delayed-imperfect information setting or a perfect information setting, the expectational revisions that is within the concept of the rational expectations hypothesis assumes that individuals target an end and have an open path, which towards revision does not need to happen. Furthermore, the conception of path dependency does not exist in rational expectations either, a key assumption with agent modeling under adaptive expectations, i.e., history matters with expected trends(15) of demand or of government intervention. A dynamic system which displays path dependency by their components, e.g., heterogeneous agents under a trade setting similar to that of normal secondary markets, implies that the past positions of other agents, history, influences the decisions of the present. On the basis of this, individuals revise their expectations to fall in line with the socio-historical setting of the trading process. If the dynamics of a system are explosive, or an AR1 system has a unit root, then there will be permanent changes to the system based upon its past historical jump. Rational expectations lack the idea of hysteresis, and a subset of which being path dependency, something which the Sraffian Supermultiplier does not do. Key to the assumptions of long run effective demand with the Sraffian Supermultiplier is the idea that individuals revise their expectations to deal with a change in the historical settings of the economic system, nowhere is it assumed that individuals react spontaneously to deal with deviations of the capacity utilization rate from autonomous demand,  which is why the reaction coefficient γ takes on a value that can be in between 0 and 1. Rather than taking γ = 1, 1 > γ > 0, is a traditional setting for a capitalist economy, i.e., the reactions of entrepreneurs are not instantaneous but they do eventually reconcile the capacity utilization rate with the growth of autonomous demand, either downwards or upwards. The ability for individuals in a heterogeneous agent setting to revise their demand growth expectations with regards to the historical setting of previous economic decisions is a hallmark of adaptive expectations, not of rational expectations. The ability for the introduction of hysteresis is exceedingly important, as this “Sraffian” model can retain the expanded Minskyian hypothesis of financial instability and instability in the real sector while avoiding Harrodian instability or rational expectations. Palley’s contributions on the labor market to the Sraffian Supermultiplier are worth while to read, however, to treat them as a case of “rational expectations” is completely erroneous with regards to the Sraffian Supermultipler, contributing to further confusion within the broad tent Post-Keynesians over the proper model to realistically approximate differential country growth.

I apologize for having to use screenshots; that seems to be the most viable approach for posting equations, as I originally was writing on Google Docs, not Blogger. I'll try to figure out how to input equations onto Blogger without grossly distorting the formatting, but for now I most likely will have to use screenshots.

References:

1.  An Essay in Dynamic Theory. Author(s): R. F. Harrod. Source: The Economic Journal, Vol. 49, No. 193 (Mar., 1939), pp. 14-33.

2.  Domar, Evsey (1946). "Capital Expansion, Rate of Growth, and Employment". Econometrica. 14 (2): 137–147.

3.  Solow, Robert M. (February 1956). "A contribution to the theory of economic growth". Quarterly Journal of Economics. 70 (1): 65–94.

4.  Piketty, Thomas, 1971-. Capital in the Twenty-First Century. Cambridge Massachusetts :The Belknap Press of Harvard University Press, 2014.

5.  Hein, Eckhard (2016) : Post-Keynesian macroeconomics since the mid-1990s: Main developments, Working Paper, No. 75/2016, Hochschule für Wirtschaft und Recht Berlin, Institute for International Political Economy (IPE), Berlin

6.  Setterfield, Mark. “Handbook of Alternative Theories of Economic Growth.” (2011).

7.  Puu, Tönu. (2006). Short History of the Multiplier-Accelerator Model. 10.1007/3-540-32168-3_4.

8.  Cesaratto, Sergio. (2015). Neo-Kaleckian and Sraffian Controversies on the Theory of Accumulation. Review of Political Economy. 27. 10.1080/09538259.2015.1010708. 

9.  Emanuele Russo, 2021. "Harrodian instability in decentralized economies: an agent-based approach," Economia Politica: Journal of Analytical and Institutional Economics, Springer;Fondazione Edison, vol. 38(2), pages 539-567, July.

10.  An approach, after discussion with a few other individuals, seems to be a satisfactory approach to heterodox growth theory.

11.  Thirlwall, Anthony. “John McCombie’s Contribution to the Applied Economics of Growth in a Closed and Open Economy.” (2018).

12.  Palley, Thomas. (2018). The economics of the super‐multiplier: A comprehensive treatment with labor markets. Metroeconomica. 70. 10.1111/meca.12228.

13.  Michalis Nikiforos, 2018. "Some Comments on the Sraffian Supermultiplier Approach to Growth and Distribution," Economics Working Paper Archive wp_907, Levy Economics Institute.

14.  Palley, Thomas. (2018). The economics of the super‐multiplier: A comprehensive treatment with labor markets. Metroeconomica. 70. 10.1111/meca.12228. P. 4

15.  ​​Setterfield, Mark, Path Dependency (September 21, 2015). Available at SSRN: https://ssrn.com/abstract=2663719 or http://dx.doi.org/10.2139/ssrn.2663719