Brad DeLong, Noah Smith, And Others On The Cambridge Capital Controversy

Brad DeLong and Noah Smith chat about the Cambridge Capital Controversy on their podcast, Hexapodia is the key insight. Noah says it was only mentioned in one of his classes on macroeconomics, but seems to say he was never formally taught it. Given his summary near the start of this discussion, he has obviously read something about it. Brad thinks the language used by the MIT economists in the 1960s was badly and inaccurately phrased and poorly suited to shed light. He says he had trouble figuring out why anybody would disagree that a single aggregate index could not be rigorous and theoretically justified. So this discussion, if I understand correctly, could be expected to fit well with the name of their podcast. But maybe not.

Brad distinguishes between the ownership of physical capital goods and the physical productivity of those goods. Do they anywhere distinguish between capital as the financial value of assets and capital as a heterogenous odd lot of means of production? Brad notes that 19th century economists brought up the (unconvincing) idea that those providing capital are incurring sacrifices by "waiting".

Noah knows that marginal productivity theory was developed as a theory of "just deserts", not that he accepts this idea. Aggregate models with a single index for the amount of capital are not helpful in figuring out how much business owners should be paid to be consistent with a flourishing society.

At one point, Noah says that somebody had a better index for capital, but he cannot recall who. Brad brings up Christopher Bliss and the slope of the production possibilities frontier for an intertemporal choice over, say, wheat today and wheat a week from now. I aggree Christopher Bliss' response to the CCC is an important marginalist response. But Noah should not have deferred so much at this time. He was trying to remember Edwin Burmeister's work.

I do not think Noah quite gets why the interest rate is generally not equal in equilibrium to the marginal product of capital. He brings up the question here of why did the participants in the CCC not also question an aggregate index for labor. Reswitching, capital reversing, the reverse substitution of labor, and so on can arise in models with heterogenous labor. Do either Brad or Noah distinguish between the price of the services of a capital good for, say, a year and the interest rate? Each kind of labor is measured in a homogeous unit, person-years. What is the analogy with capital supposed to be here?

At one point, Brad explains reswitching. Neither notes that in a comparison of long run positions, a higher wage can be associated with the adoption of a technique in which firms want to employ more labor to produce a given net output.

For me, what the English side showed is that prices of production do not follow the logic of supply and demand. Prices are not indices of relative scarcity. Marshall's principle of substitution does not characterize comparisons of (long-run) equilibrium. Classical political economy had a different approach to value and distribution, and that approach is logically consistent.

In trying to put what should have been the MIT side as strongly as possible, Brad describes the rate of profits "as a control variable" that provides a signal for how to allocate scarce resources. He thinks that by not acknowledging this role, the English side misses something important. Is Brad's position consistent with the above understanding of price theory?

I gather that this podcast is for a popular audience. It is not intended to be a comprehensive academic survey. So one should expect some gaps. They do not bring up Joan Robinson's distinction between historical and logical time or Post Keynesian's ideas on the difference between risk and uncertainty. To be fair, I do not discuss how long run positions can be reached very much myself.

Bill Mitchell now has a two part series on the CCC. Matias Vernengo points out he had an on-topic post in 2012. I find I had a bulleted summary in 2017. This example with three produced commodities is fairly comprehensive. Alexander Douglas has a 2018 Medium post offering an appreciation of Joan Robinson as a philosopher.

"When Economists Are Wrong"

In a blog associated with the Frankfurter Allegmeine, Gerald Braunberger criticizes the effects of Sraffian political economy on Italian policy in the 1970s. I rely on google translate and subject matter expertise to make some sense out of this. By the way, Bertram Schefold shows up in the comments. I would like to know more about the motivations behind this. Does Braunberger think the public is increasingly aware that mainstream economics is broken?

Before I disagree, I note Braunberger seems well-informed on some points. I know of grumbles about Garegnani's treatment of Sraffa's archives. And I have heard that the Trieste summer schools were torn between those who emphasized long period logic and Post Keynesians who emphasized uncertainty, money, and historical time. On another point, I do not see why Sraffians giving policy advice should care about whether their advice is consistent with the advice Ricardo had for Britain during and after the Napoleonic war. I would think Sraffians in Italy during the 1970s would be more interested in Marx's views, anyways.

I do not understand what non-Sraffian theory Braunberger thinks exists. All economists should (but do not) recognize no reason exists to think that a lower real wage, in a time of depression, will encourage firms to adopt more labor-intensive techniques and thereby increase employment. In parallel, higher real wages need not decrease employment through the adoption of less labor-intensive techniques. Supply and demand just is not a logically-consistent model, in which conclusions follow from assumptions, of wages and employment.

This does not mean that real wages can be increased willy-nilly, without any consequences. Income effects could be important. And one might want to worry about the possibility of a capital strike. I agree that the political slogan, "The wage as the independent variable" draws directly on Sraffa.

More than economics was involved in the going-ons in Italy in the 1970s. When activists are kidnapping and executing the prime minister and the government is imprisioning leftists without discrimination, firms, government, and unions are unlikely to come to a peaceful agreement about distribution.

Furthermore, Italy was not isolated from the wider world. Were the lira and the mark pegged to the dollar before Nixon ended Bretton Woods? The world-wide rise in oil prices was not the result of Sraffian policy advice. Wage-push inflation arose in many countries; it was not just an Italian problem. Sraffa's colleagues had worked out an explanation of stagflation long before the event. I would think this is an example of when (heterodox) economists are right. (A Tax-based Income Policy (TIP) is a policy idea I associate with American Post Keynesians, like Sidney Weintraub, that might have been worth trying in some countries in the 1970s.)

Some Positions Some Take On Sraffa's Book

This post lists some views on Production of Commodities by Means of Commodities: A Prelude to a Critique of Economic Theory.

• The quantity flows Sraffa takes as given are those observable in an actual economy at a given time, as with a snapshot (Roncaglia 1978).
• These quantity flows, on the contrary, are at the level of effectual demand (Garegnani 1990).
• These quantity flows are for an economy in a self-replacing state.
• The assumption of constant returns to scale is necessary for drawing any interesting conclusions from Sraffa's work (Samuelson 1990, Samuelson 2000).
• Market prices tend towards or orbit around Sraffa's prices of production in a process akin to gravitational attraction (Garegnani 1990).
• Sraffa's book is an investigation of logical consequences in a system of prices of production, akin to reasoning in geometry; no claims are put forth about tendencies or paths of market prices (Sinha 2012).
• Sraffa started, in the 1920s, in his research for his 1960 book from labor values and Marx's schemes of reproduction in Volume 2 of Capital (De Vivo 2003 and Gilibert 2003).
• Sraffa began, on the contrary, with a formalization of prices in terms of physical real cost; labor values are a corruption of this notion of real costs and Sraffa was not originally inspired by Marx in his economics (Gehrke and Kurz 2006).
• Sraffa showed that labor values are unnecessary and redundant for defining prices of production (Steedman 1981).
• Sraffa, on the contrary, vindicated Marx in his work (Porta 2012 and Bellofiore 2014).
• Sraffa's work cannot be set in historical time (Robinson 1985).
• Sraffa, for methodological reasons, rejected counterfactual reasoning and thus the marginal revolution (Sen 2003).

Some of the above statements are probably stated more strongly than the referenced scholars might endorse. I am also not at all sure those are the best references. They certainly are not the most up-to-date. It is clear at any rate that the Cambridge Capital Controversy was not solely about difficulties in aggregating capital and that Sraffa's approach to economics cannot be subsumed by general equilibrium theory (Hahn 1982).

References

• Bellofiore, Riccardo. 2014. The loneliness of the long distance thinker: Sraffa, Marx, and the critique of economic theory. In Bellofiore and Carter (2014).
• Bellofiore, Riccardo and Scott Carter. 2014. Towards a New Understanding of Sraffa: Insights from Archival Research. New York: Palgrave Macmillan.
• Bharadwaj, Krishna and Bertram Schefold (eds.). 1990. Essays on Piero Sraffa: Critical Perspectives on the Revival of Classical Theory. London: Unwin Hyman.
• de Vivo, Giancarlo. 2003. Sraffa's path to Production of Commodities by Means of Commodities. An interpretation. Contributions to Political Economy 22 (1): 1-25.
• Garegnani, Pierangelo. 1990. Classical versus Marginalist Analysis. In Bharadwaj and Schefold (1990).
• Gehrke, Christian & Heinz D. Kurz. 2006. Sraffa on von Bortkiewicz: Reconstructing the classical theory of value and distribution. History of Political Economy 38 (1): 91-149.
• Gilibert, Giorgio. 2003. The equations unveiled: Sraffa's price equations in the making. Contributions to Political Economy 22 (1): 27-40.
• Hahn, Frank H. 1982. The neo-Ricardians. Cambridge Journal of Economics 6: 352-374.
• Kurz, Heinz D. (ed.). 2000. Critical Essays on Piero Sraffa's Legacy in Economics. Cambridge: Cambridge University Press.
• Porta, Pier Luigi. 2012. Piero Sraffa's early views on classical political economy. Cambridge Journal of Economics 36: 1357-1383.
• Robinson, Joan. 1985. The theory of normal prices and the reconstruction of economic theory.
• Roncaglia, Alessandro. 1978. Sraffa and the Theory of Prices (trans. by J. A. Kregel). New York: John Wiley & Sons.
• Samuelson, Paul A. 1990. Revisionist findings on Sraffa. In Bharadwaj and Schefold (1990) and reprinted in Kurz (2000).
• Samuelson, Paul A. 2000. Sraffa's hits and misses. In Kurz 2000.
• Sen, Amartya. 2003. Sraffa, Wittgenstein, and Gramsci. Journal of Economic Literature 41: 1240-1255.
• Sinha, Ajit. 2012. Listen to Sraffa's silences: a new interpretation of Sraffa's Production of Commodities. Cambridge Journal of Economics 36 (6): 1323-1339.
• Steedman, Ian. 1981. Marx after Sraffa. London: Verso.

On Milana's Purported Solution To The Reswitching Paradox

Carlo Milana has posted a paper on arXiv. I was prepared to accept this paper's claims. Economists have developed price theory. Referring to Sraffian "paradoxes" and "perverse" switch points is a matter of speaking. There does not exist separate Sraffian and neoclassical versions of price theory. For a result to be "perverse", it need only contradict outdated neoclassical intuition. But it is as much a part of the mathematical economics as any other result. (It is another matter that much teaching in microeconomics is inconsistent with the mathematics.)

In equilibrium, the price of the services of each capital good in use is equal to the value of the marginal product of that good, with all prices discounted to the same moment in time. This discounting implies that the interest rate appears in a formal statement of these equations. These equalities are very different from the claim that the interest rate equals the marginal product of (financial) capital. In limited cases, one can prove something like the aggregate equality. No such thing as a marginal productivity theory of distribution, however, is restored. Milana might cite Hahn (1982) on this background.

But Milana goes further. He claims that reswitching is impossible or, at least, examples up to now are erroneous.

I think Milana's basic mistake is exposed in Salvadori and Steedman (1988). (I go through one of their examples here.) The Samuelson-Garegnani model is not a model of two-(produced) goods. The model contains as many capital goods as there are techniques. Potentially, there can be a continuum of capital goods in the model. As such, it is meaningless to require the price of the capital goods used in each technique that is cost-minimizing at a switch point to be equal to one another. That is analogous to requiring the price of a ton of iron be equal to the price of a ton of tin.

For some reason, Milana does not discuss examples of reswitching in flow-input, point output models, such as in Samuelson (1966). Nor does he acknowledge, as I read him, valid examples in which the same n commodities are produced in all techniques, and all commodities are basic in all techniques. (Does he say anything at all about the distinction between basic and non-basic commodities?) At a non-fluke or generic switch point, in such a framework, the two techniques that are cost-minimizing differ in a process in exactly one industry.

Milana should read and reference Bharadwaj (1970), as well as Bidard and Klimovsky (2004) on fake switches in models of joint production. Other Linear Programming formulations are available for considering the choice of technique. Vienneau (2005) presents one. What does Milana have to say about the direct method for analyzing the choice of technique in Kurz and Salvadori (1995)? I briefly provide a survey of different analysis in Vienneau (2017), as well as an algorithm for finding the cost-minimizing technique? Are all these approaches in error?

References

• Khrishna Bharadwaj. 1970. On the maximum number of switches between two production systems. Schweizerische Zeitschrift fur Volkswortschaft and Statistik (4): 401-428. Reprinted in Bharadwaj 1989. Themes in Value and Distribution: Classical Theory Reappraised, Unwin-Hyman.
• Christian Bidard and Edith Klimovsky. 2004. Switches and fake switches in methods of production. Cambridge Journal of Economics 28:89-97
• Frank Hahn. 1982. The neo-Ricardians Cambridge Journal of Economics 6:353-374.
• H. D. Kurz and N. Salvadori. 1995. Theory of Production: A Long-Period Analysis. Cambridge University Press.
• Carlo Milano. 2019. Solving the Reswitching Paradox in the Sraffian Theory of Capital. arXiv:1907.01189
• Neri Salvadori and Ian Steedman. 1988. No reswitching? No switching! Cambridge Journal of Economics 12: 481-486.
• Paul A. Samuelson. 1966. A Summing Up. The Quarterly Journal of Economics 80 (4): 568–583.
• Robert Vienneau. 2005. On labour demand and equilibria of the firm. The Manchester School 73(5): 612-619.
• Robert Vienneau. 2017. The choice of technique with multiple and complex interest rates. Review of Political Economy 29(3): 440-453.

On Hicks' Average Period of Production

Figure 1: APP Around Switch Points

1.0 Introduction

I take it that the Austrian theory of the business cycle builds on Austrian capital theory. The following two claims are central to Austrian capital theory:

• Given technology, profit maximizing firms adopt a more capital-intensive, more roundabout technique at a lower interest rate.
• The adoption of a more roundabout technique increases output per worker.

Originally, Eugen von Böhm-Bawerk proposed a physical measure of the average period of production, but economists of the Austrian school have been distancing themselves from this position for well over half a century. I have argued that the first claim fails, even in a framework without any scalar measure of capital-intensity or the average period of production.

Recently, Nicholas Cachanosky and Peter Lewin, in a series of articles, have championed J. R. Hicks' measure of the Average Period of Production (APP), as a justification of the first claim. They note that the APP, as defined here, is a function of the interest rate. Hence, it cannot fully support Böhm-Bawerk's theory. Saverio Fratini has argued this justification does not work, since the second claim above fails, when this APP is used as a measure of capital-intensity. Lewin and Cachanosky, in reply, argue that Fratini does not properly calculate the APP, since it should be forward looking and apply in disequilibria.

This post re-iterates Fratini's argument, with his example. I more closely follow Cachanosky and Lewin's approach, though.

2.0 Technology

Fratini considers a technology consisting of two techniques of production, Alpha and Beta (Table 1). Each technique requires three years of unassisted labor inputs, per bushel wheat produced at the end of the third year. Labor is advanced and paid at the end of the year. Labor is taken as numeraire. That is, the wage is assumed to be $1 per person-year. The price of a bushel wheat, p, is taken to be $12 dollars per bushel. As I hope becomes apparent, these assumptions generally characterize a disequilibrium.

Table 1: Inputs for Producing A Bushel Wheat

Year	Years Until Harvest	Technique
		Alpha	Beta
1	3	a3 = 1 Person-Yr.	b3 = 2 Person-Yrs.
2	2	a2 = 7 Person-Yrs.	b2 = 2 Person-Yrs.
3	1	a1 = 2 Person-Yrs.	b1 = 8 Person-Yrs.

The output per worker, in a stationary state, is determined by the chosen technique. Suppose the Alpha technique is adopted. In any given year, 10 person-years are employed per bushel wheat produced. Two person-years are being expended to produce each bushel of wheat harvested at the end of the year, seven person-years are being employed to produce each bushel of wheat available at the end of the next year, and one person-year is employed per bushel wheat harvested even a year later. That is, output per worker, under the alpha technique, y_α, is (1/10) bushels per person-year. Similarly, output per worker for the beta technique, y_β, is (1/12) bushels per person-year.

3.0 Net Present Value and the Choice of Technique

Suppose a wheat-producing firm faces a given annual interest rate, r. For convenience, define:

R = 1 + r

The discount factor, f, is defined to be:

f = 1/R = 1/(1 + r)

Consider a decision to choose a technique to adopt for next three years in producing wheat. Powers of the discount factor are used to evaluate the costs and revenues for each technique at the start of the given year. For example, the NPV of the alpha technique is:

NPV(α, f) = -a₃ f - a₂ f² + (p - a₁) f³

I have assumed that firms expect the given interest rate to remain unchanged for the decision period - a common convention. Revenues are positive, and costs (or outgoes) are negative.

Figure 2 graphs the difference between the NPV for the two techniques. A positive difference indicates that the alpha technique maximizes the NPV, while a negative difference arises when the beta technique is preferred. Which technique is chosen by cost-minimizing firm for each interest rate is shown. At switch points, firms are indifferent between the two techniques.

Figure 2: Difference in NPVs

Under the assumptions, NPV is always positive. (If the beta technique were adopted at an interest rate of zero, its NPV would be zero then.) If markets were competitive, the price of wheat would vary until the NPV was zero, given the interest rate. Fratini does indeed assume equilibrium and analyzes the choice of technique with backwards-looking calculations of costs, as Lewin and Cachanosky claim. But this makes no difference to his argument, so far.

4.0 The Average Period of Production

One might be interested in how NPV varies with the discount factor. The elasticity of the NPV, with respect to the discount factor, is a dimension-less number for assessing such sensitivity. Somewhat arbitrarily, I discount elasticity one period:

APP(α, f) = f [1/NPV(α, f)] [d NPV(α, f)/df]

Elasticity is the variation of NPV with variation of the discount factor, as a proportion of NPV.

APP(α, f) = [-a₃ f/NPV(α, f)] x 1

+ [- a₂ f²/NPV(α, f)] x 2

+ [(p - a₁) f³/NPV(α, f)] x 3

The APP for a technique, at a given discount factor, is the weighted average of the time indices, looking forward, for a given income stream. The weights are the proportion of the income stream received in each period. All income is discounted to the start of the first year.

So the elasticity of the NPV of an income stream, with respect to the discount factor can also be expressed as the average period of production.

Notice that the APP is not defined in equilibrium. The denominators in the above terms are zero, and the APP could be said to be infinite. If only costs are used in the above calculations (thus, no longer of a NPV), the APP is well-defined, at least in the flow-input, point output case. Fratini (2019) does this.

One could also express the APP as a function of the interest rate:

APP(α, r) = [-a₃ R²/NPV(α, r)] x 1

+ [- a₂ R/NPV(α, r)] x 2

+ [(p - a₁)/NPV(α, r)] x 3

where:

NPV(α, r) = -a₃ R² - a₂ R + (p - a₁)

I skip over some some algebraic manipulations above.

The above is not the definition of the APP in Fratini (2019), for example, in Equation 7. Where I have time indices of 1, 2, and 3, Fratini has indices of 3, 2, 1. I guess one can say that his definition of the APP is backwards-looking.

Fratini's argument still goes forward with Cachanosky and Lewin's (or Hicks') definition. One could present a mathematical proof that the APP is always increased around a switch point with a fall in the interest rate. But here I'll just graph it for the example. See Figure 1, at the top of this post. Around each switch point a lower interest rate is indeed associated with the adoption of a technique with a larger APP. But consider the switch point at an interest rate of 200 percent. The beta technique, adopted at a notionally lower interest rate, has a lower value of output per head.

4.0 Conclusion

The example illustrates that, around a switch point, a lower interest rate is associated with the adoption of a more roundabout technique, where roundaboutness is measured by Hicks' Average Period of Production. Incidentally, the example demonstrates that in a region where one technique is cost-minimizing the APP may decrease with the interest rate. But the adoption of a more roundabout technique can be associated with a decrease in output per worker. So much for Austrian capital theory and the Austrian theory of the business cycle.

Update (14 June 2019): Re-order numbers in table, as they are used in the calculations. References

• Nicholas Cachanosky and Peter Lewin. 2014. Roundaboutness is not a mysterious Concept: A financial application to capital-Theory. Review of Political Economy, 26(4): 648-665.
• Saverio M. Fratini. 2019. A note on re-switching, the average period of production and the Austrian business-cycle theory, Review of Austrian Economics.
• J. R. Hicks. 1946. Value and Capital: An inquiry into some fundamental principles of economic theory. Oxford University Press.
• Peter Lewin. 1999. Capital in Disequilibrium: The role of capital in a changing world. Routledge.
• Peter Lewin and Nicholas Cachanosky. 2019. Re-switching, the average period of production and the Austrian business-cycle theory: A comment on Fratini, Review of Austrian Economics.

"The Microeconomic Foundations of Aggregate Production Functions"

Figure 1: A Production Network

I here comment on Baqaee and Farhi (2018). I am still trying to absorb it. I suppose that it is nice that an economist at Harvard is revisiting the Cambridge Capital Controversy (CCC). (Where is Michael Mandler these days?)

My major criticism is they do not do what their title claims. That is, their supposed microeconomic foundations are still up in the air.

In many CCC examples, technology is specified in terms of fixed-coefficient production processes. Sometimes, more than one process is available for producing a specified commodity. This structure gives rise to a choice of technique. If one wanted, one could formulate a programming problem, in each sector, whose solution is a production function for that sector. This production function would not be differentiable everywhere.

Baqaee and Farhi, on the other hand, assume the existence of continuously-differentiable microeconomic production functions in each sector. In this paper, these production functions are specifically Constant Elasticity of Scale (CES) production functions. (As should be the case, their inputs and outputs are specified in physical units, not in price terms.)

I am willing to be convinced that this difference in starting points is a technical matter. Or that Baqaee and Farhi are making some progress towards a more complete framework that will include models of the production of commodities by means of commodities. I have some challenges, however.

There is a theorem whose status I am not sure of. It states that, given a continuously differentiable production function for a commodity that is basic, in the sense of Sraffa, for all techniques, reswitching is not possible. (Stephen Marglin was not the first to offer a proof of this theorem.) Thus, Baqaee and Farhi rule out, by assumption, many (most?) of the reswitching examples and much of the structure in the literature on the CCC. As they note, their assumptions do include an example from Paul Samuelson, in his 1966 "Summing Up" article. That example, had a flow-input, point-output structure, with no commodity basic in any technique.

Second, I gather Baqaee and Farhi think of themselves as starting with microeconomic data that is in principle empirically observable. These would be elasticities of substitution at points on factor demand curves that are chosen at an instance of time. Part of the point of the CCC is to question the existence of factor demand curves, including for intermediate inputs. In a comparison of long period positions, it is an incoherent thought experiment to vary one price at a time. On the other hand, as Han and Schefold have shown, empirical work can be based on given fixed coefficients processes. I think if Baqaee and Farhi were to take this point, they would have to rewrite a lot of their paper, including sections talking about the bias of technical change and macroeconomic elasticities of substitution between factors.

Baqaee and Farhi do have an interesting suggestion for visualizing a production network in logical time. (I've previously presented a less detailed approach from Bidard.) My diagram above is an attempt to expand on Baqaee and Farhi's approach. For each time period, four processes (a, b, c, and d) exist for producing one of two commodities from inputs of labor and those two commodities. The first two processes have the first good as output, and the second two processes produce the second good. The second commodity can be used for consumption, as well as a capital good in the production of either good. This is basically the technology for the examples in Vienneau (2005). The diagram could be simplified by not explicitly showing the demultiplexers and the summations.

References

• David Rezza Baqaee and Emmanuel Farhi (2018). The Microeconomic Foundations of Aggregate Production Functions. 26 November.
• Robert L. Vienneau (2005). On Labour Demand and Equilibria of the Firm. Manchester School 73(5): 612-619.

Catalog Of Neoclassical Responses To The Cambridge Capital Controversy

This is merely a list and, as usual, off the top of my head.

• Paul A. Samuelson (1966). A summing up. Quarterly Journal of Economics 80: 568-583.
• Mark Blaug (1975). The Cambridge revolution: Success of failure? A critical analysis of Cambridge theories of value and distribution. Institute of Economic Affairs.
• Joseph E. Stiglitz (1974). The Cambridge-Cambridge controversy on the theory of capital: A view from New Haven.
• Christopher J. Bliss (1975). Capital Theory and the Distribution of Income. Elsevier North-Holland.
• Avinash Dixit (1977). The accumulation of capital theory. Oxford Economic Papers 29: 1-29.
• Edwin Burmeister (1980). Capital Theory and Dynamics, Cambridge University Press.
• Frank Hahn (1982). The neo-Ricardians. Cambridge Journal of Economics 6: 353-374.
• Andreu Mas Colell (1989). Capital theory paradoxes: Anything goes. In Joan Robinson and Modern Economic Theory (ed. by G. R. Feiwel), Macmillan.
• Mario Ferretti (2004). The neo-Ricardian critique: An anniversary assessment.
• Gaetano Bloise and Pietro Reichlin (2005). An obtrusive remark on capital and comparative statics.
• Michael Mandler (). Sraffian economics (new developments). In New Palgrave, 2nd edition.

I could have cited many more references from Burmeister, Mandler, or Samuelson. I do not know if the last two were published as anything more than working papers. As far as I am concerned, most mainstream economists also ignore the neoclassical side of the CCC.

Inane Responses To The Cambridge Capital Controversy

I consider the following views, if unqualified and without caveats, just silly:

• The Cambridge Capital Controversy (CCC) was only attacking aggregate neoclassical theory.
• The CCC is just a General Equilibrium argument, and it has been subsumed by General Equilibrium Theory. (Citing Mas Colell (1989) here does not help.)
• The CCC does not have anything to say about partial, microeconomic models.
• Perverse results, such as reswitching and capital-reversing, only arise in the special case of Leontief production functions. If you adopt widely used forms for production functions, the perverse results go away.
• It is an empirical question whether non-perverse results follow from neoclassical assumptions. And nobody has ever found empirical examples of capital-reversing or reswitching.
• Mainstream economists have moved on since the 1960s, and their models these days are not susceptible to the Cambridge critique.

I would think that one could not get such ideas published in any respectable journal. On the other hand, Paul Romer did get his ignorance about Joan Robinson into the American Economic Review

References

• Andreu Mas-Colell (1989). Capital theory paradoxes: Anything goes. In Joan Robinson and Modern Economic Theory (ed. by George R. Feiwel), Macmillan.

A Sophisticated Neoclassical Response To Cambridge Capital Controversies

A Reproduction Scheme (Based on Biddard 1990)

1.0 Introduction

One way of reading the Cambridge Capital Controversy (CCC) is an internal exploration of and debate about neoclassical price theory¹. Both sides agreed to concentrate on the case of perfect competition, with no principal agent problems, no asymmetric information, etc. The Cambridge-Italian critics thought themselves to have demonstrated that neoclassical economists could not consistently with their theory claim that equilibrium prices were indices of relative scarcity. Such a claim is not well-founded in the theory, and economists should turn away from biotechnological determinism and turn toward developing price theories in which class power matters.

In this post, I want to outline the sophisticated neoclassical response, in the 1970s, to the Cambridge-Italian critics.

2.0 General Equilibrium Models of Intertemporal and Temporary Equilibrium

This neoclassical response asserted that price theory was best expressed in terms of General Equilibrium Theory (GET). Capital theory involves production over time. Models of intertemporal and temporary equilibrium have been developed in GET. And these models, it is claimed, are both logically consistent and unaffected by Cambridge-Italian criticism².

2.1 The Arrow-Debreu Model of Intertemporal Equilibrium

The Arrow-Debreu model is a model of disaggregated individuals interacting solely through a single centralized market in existence at the beginning of time. Commodities are distinguished by their physical characteristics, where they become available, when they become available, and the state of the world in which they become available. The givens in the Arrow-Debreu model consist of, roughly:

• Tastes: Each agent can choose the more preferred consumption plan, when presented with any pair of such plans. A consumption plan specifies what commodities the agent consumes at each point in space and time in each possible state. It also specifies what inputs that the agent controls are supplied as inputs into the firms at each point in space and time and for each state.
• Technology: What commodities can be produced for supply for consumption from each possible list of inputs is known by everybody.
• Endowments: A list of commodities available at the beginning of time is given. The specification of endowments includes who owns what. Likewise, the ownership (shares) of the given finite number of firms is known.

Equilibrium is achieved in some sort of no-time before the beginning of time. The centralized market opens, and the auctioneer informs all agents of the prices of all commodities. These commodities include, for example, a contract to buy an umbrella in New York City on 20 June 2015, given it is raining. The agents inform the auctioneer of which contracts they are willing to enter and offer at the given prices. The auctioneer adjusts prices, depending on mismatches between offers and demands, until all markets clear. No transactions are allowed to take place until equilibrium is achieved³.

In an equilibrium, the plans of all agents are pre-reconciled. For any commodity with a positive (spot or forward) price, the quantity supplied equals the quantity demanded. Some goods are not commodities in equilibrium; they have a price of zero. The supply of these goods exceeds the demand.

Once equilibrium has been established, all agents make their promised trades and carry out their plans throughout time.

2.2 The GET Model of Temporary Equilibrium

The idea that all plans are only made in one big market transaction at one instant of time is hard to swallow. This constraint is relaxed in models of temporary equilibrium, as developed by, for example, J. R. Hicks (1946).

In Hicks' model, a market opens at the start of each successive week. If I recall correctly, with a single exception, all markets are spot. That is, supplies and demands are contracted each Monday to be delivered or taken during the following week, but not for later weeks. If an agent plans, for example, to hire labor two weeks hence, they must agree on the wage on the Monday at the start of that week. No market exists in which a worker can be hired for a period of more than one week.

In this model, the plans of agents only need to be consistent in equilibrium for a single week. Supply and demand of both factors of production and commodities for consumption match in the spot market. But these supplies and demands may be based on plans that entail inconsistencies in future weeks. When agents see spot markets failing to clear, say, next week, they revise their plans. And, once again, these revisions and the haggling in the market are assumed to bring about instantaneous market clearing.

The single exception to the requirement that all markets be spot is a market for something like a bond or an annuity. Forward markets exist for all time periods in which one can offer to pay a unit of the numeraire commodity at the start of this week for delivery of a given quantity of the numeraire quantity at the start of, say, the next week. So a whole complex of interest rates exist for the numeraire. An individual's expectations include expectations of movements of future spot prices of all commodities. That is, each individual has expectations for not only future movements in, say, one week interest rates for the numeraire commodity but beliefs about own-rates of interest for all commodities. Nothing in the model brings about consistency among agents of these expectations and beliefs about the future.

This model continues to impose the requirement that no false trading occur. When the markets open on Monday, no transactions can take place until a market-clearing set of prices is found. Only after equilibrium has been achieved do commodities change hands. And production proceeds in each week during the period in which markets are closed.

These models impose very few restrictions on equilibrium, even with specific assumptions on expectations. Perhaps models of temporary equilibrium are better, within mainstream economics, than the Arrow-Debreu model of exploring the formation of expectations.

3.0 Acceptance of Cambridge-Italian Criticisms

These models of general equilibrium may be internally consistent. Both sides of the CCC, however, came to recognize they did not support the beliefs about causal properties still relied on to this day in mainstream applied theory. The faulty and unfounded idea is something like this: compare two equilibria, in which the exogenous (given) data is identical, except the quantity of some given endowment varies across the two cases. Then, if one abstracts from violations of the assumptions of pure competition and is ignorant of price theory, one might expect the price of that endowment to be higher in the case where it is more scarce. Similarly, such an ignoramus would expect the price of commodities produced more intensively with the more scarce endowment to be higher. Despite the short run nature of the models outlined above, such beliefs are unfounded in GET. Here is Christopher Bliss forthrightly acknowledging such:

"Even people who have made no study of economic theory are familiar with the idea that when something is more plentiful its price will be lower, and introductory courses on economic theory reinforce this common presumption with various examples. However, there is no support from the theory of general equilibrium for the proposition that an input to production will be cheaper in an economy where more of it is available." -- Christopher Bliss (1975).

3.1 The Meaning of Quantities, Prices, and Commodities in GET

One issue with GET is remaining clear on the meaning of prices, commodities, and endowments. As with Bliss, I have stated the claim about endowments, prices, and scarcity indices in a timeless, static equilibrium. However, the two disaggregated models outlined above are set in logical time. For example, consider the endowment of a natural resource, such as oil. Only the endowment at the beginning of time is taken as data; the quantities of oil available at the start of the second, third, etc., weeks are different commodities. And these quantities are found by solving the model; they are not taken as given. Likewise, a grade of gasoline produced from oil is a different commodity, depending on which week in which it is produced. Christian Bidard is clear about this distinction between commodities that may be physically identical, but available at different times:

"Intertemporal general equilibrium prices associated with a finite or infinite path of consumption and accumulation have no general properties: the reason being that goods called corn at date t, iron at date t, corn at date t + 1, iron at date t + 1 are formally considered as four distinct commodities of an atemporal economy." -- Christian Bidard (1990).

You might find in GET a statement about the price of one commodity and the quantity of one commodity that goes into the production of that commodity at some more-or-less distant time beforehand. But this is not a statement about the whole vector of prices of physically identical commodities distinguished by the time of their availability. In general, GET theory does not provide simple intuitive propositions about prices and quantities of multiple commodities.

3.2 Steady States

The possibility of a third model might be thought to provide a work-around. Consider the limit, as time increases without bound in the models, of relative quantities and relative prices all referring to that single point in time at infinity. This is a model of a steady-state⁴. Does this background help explain the following from Frank Hahn?:

"It is possible that the outputs produced in an Arrow-Debreu economy in the far distant future are independent of its initial endowments. That would mean that in such an economy the relative scarcities prevailing now would have no influence on the relative prices and rentals in the distant future. This should be enough to persuade the critics that the theory is not committed to a relative scarcity theory of distribution, though they seem to believe it is and that often motivates them in their attacks." -- Frank Hahn (1981).

Christopher Bidard also considers such a limiting process:

"The subscripts refer to the date, the horizontal arrow indicates production by means of inputs and labor and the data in italics are neoclassical theory calls the 'endowments' of the economy. The interesting feature of this scheme is that all inputs a_t, except for the very initial ones a₀, are obtained as the result of previous production. If we admit that the influence of the primitive inputs a₀ vanishes in the long run, we have a pure reproduction process (a mechanical endogenization of labour, identified with a given wage basket, would be useful for a comparison with von Neumann's theory, but the operation is not necessary here)." -- Christian Bidard (1990).

In a simple model of a steady state, one might as well drop time indices off state variables for relative quantities and relative prices. They are invariant over time in such a model. But even so, one cannot apply theorems of static equilibrium to such a model. The logic of steady states is not one of allocating scarce resources. Inputs into production, insofar as they are produced, are not exogenous givens. Rather, they are found as part of the model solution. And the kind of relationships that Bliss says above are without foundation in GET are equally unfounded in models of steady-states.

4.0 Does a Logically Consistent Vulgar Neoclassical Theory Exist?

I consider the beliefs that economics is solely about the allocation of scarce resources and that equilibrium prices are indices of relative scarcity to be vulgar neoclassical doctrines. Beliefs about properties of equilibrium, if you are interested in mathematically formalized Neoclassical models, should be logical conclusions derived from assumptions. And those assumptions should be on the primitives of the model. For example, one might have some assumptions about the tastes, production functions, or patterns of initial endowments.

Here Edwin Burmeister states that no such vulgar neoclassical theory is known to exist, albeit in the context of the analysis of an aggregate market for capital:

"Imposing some set of conditions on the technology ... should be sufficient to assure that real Wicksell effects are always negative. Such conditions would be of interest - especially if they could be empirically tested - since they would validate the qualitative conclusions derived from the one-good models often used in macroeconomics without any theoretical justification for ignoring aggregation problems. Moreover, Burmeister ... has proved that a negative real Wicksell effect is a necessary and sufficient condition for the existence of an index of capital ..., and a neoclassical aggregate production function defined across steady state equilibria such that (i) [consumption per head is a function of the index of capital per head], (ii) [the equilibrium interest rate is equal to the marginal product of the index of capital], and (iii) [This index of capital exhibits declining marginal returns]. Unfortunately, no set of such sufficient conditions is known, but the literature on capital aggregation suggests that they would impose severe restrictions on the technology." -- Edwin Burmeister (1987).

Economists who hold fast to vulgar neoclassical economics may present formal models. But often their mathematics is imprecise, disguising muddle and confusion. It is not a matter of having unrealistic assumpions; it is a matter of having assumptions that do not imply one's conclusions.

5.0 Conclusion

As far as I can tell, most economists do not learn the literature of their subject. Not only do most economists stay ignorant of the view of the English side of the CCC. They also remain ignorant of the most sophisticated neoclassical response. Thus, they ask to see irrelevant empirical evidence⁵ for the existence of reswitching, and do not acknowledge their applied stories⁶ are without foundation in rigorous neoclassical price theory.

Footnotes

1. For the purposes of this post, I bracket out the Sraffian reinterpretation of classical and Marxist economics, the Sraffian claim to have reconstructed a viable alternative price theory, and arguments over the compatibility of this theory with the economics of Keynes.
2. It is not clear that these models cannot be attacked by Sraffians. Do they contain or do they need to contain a market for income in general at each point of time, as in Walras's work?
3. Why this haggling over prices would ever approach equilibrium is unclear in theory.
4. The Turnpike Theorem asserts that intertemporal equilibrium paths starting from appropriately selected initial conditions will spend most of their time around a steady state, even if such a steady state is not the final destination of such a path at a final given final time at which the model ends. On the other hand, the Sonnenschein-Debreu-Mantel theorem suggests any dynamics is possible. Thus, intertemporal paths may have no tendency to approach such steady states, even if they have a local saddle-point stability.
5. I am not sure that all those who ask for empirical evidence of reswitching are clear what they are asking, even though I often use "rewitching" as a synecdoche myself. Anyways empirical evidence exists for Sraffa effects. If Sraffa effects were empirically unlikely, one would be faced with the (unmet) theoretical challenge of outlining a theory to explain this supposed unlikeliness.
6. For example, on the supposed decreasing employment effects, under perfect competition, of a minimum wage above the (what is that) equilibrium real wage.

References

• Christian Bidard (1990). From Arrow-Debreu to Sraffa, Political Economy: Studies in the Surplus Approach, V. 6: pp. 125-138.
• Christopher J. Bliss (1975). Capital Theory and the Distribution of Income, Amsterdam: North Holland Press.
• Edwin Burmeister (1980). Capital Theory and Dynamics, Cambridge: Cambridge University Press.
• Edwin Burmeister (1987). Wicksell Effects, The New Palgrave: A Dictionary of Economics (ed. by J. Eatwell, M. Milgate, and P. Newman).
• Frank Hahn (1981). General Equilibrium Theory, in The Crisis in Economic Theory (ed. by. D. Bell and I. Kristol), Basic Books.
• Frank Hahn (1982). The Neo-Ricardians, Cambridge Journal of Economics, V. 6: pp. 353-374.
• J. R. Hicks (1946). Value and Capital: An Inquiry into Some Fundamental Principles of Economic Theory, 2nd edition, Oxford: Oxford University Press.
• J. M. Grandmont (1977). Temporary General Equilibrium Theory, Econometrica, V. 45, N. 3: pp. 535-572.
• Fabio Petri (2004). General Equilibrium, Capital and Macroeconomics: A Key to Recent Controversies in Equilibrium Theory, Edward Elgar.

"Economics Textbooks - Decades of Scientific Fraud"

Lars Syll has written a post, titled "Economics Textbooks - Decades of Scientific Fraud". If you had not already read it, could you guess what it is about from the title?

I would expect likely guesses to be non-unique. It is not about:

• How the Cambridge Capital Controversy demonstrates that textbook teaching on labor markets and, for example,on the minimum wage is nonsense.
• The incoherence of textbook teaching on the justification for lack of tariffs by the theory of comparative advantage.
• The textbook misrepresentations of the theories of various economists, including John Maynard Keynes.

You can extend the above list at your leisure.

In many ways, economics seems to me to be an extraordinary subject. Good arguments have existed for decades for discarding most of mainstream teaching and practice. As far as I can see, the bulk of these arguments, including their very existence, are just ignored by most mainstream economists. I am willing to entertain demonstrations of the fallacy of theories taught in almost all mainstream textbooks for decades. I think my willingness to explore other demonstrations than those I have been previously aware of is partly due to my belief that most economists are socialized into willful ignorance.

I can see why some young mainstream economists may resist the notion that they have been taught, mostly, lies and nonsense. And so they may look in the research literature for arguments against the arguments and demonstrations that I accept, or at least try to explore. Since my favorite positions were established after long controversy, you can find neoclassical counter-arguments, of a sort. For example, one might cite, in response to the Cambridge Capital Controversy:

• Edwin Burmeister's championing of Champernowne's chain index measure of capital¹.
• Frank Hahn's advocacy, including in response to the Cambridge Capital Controversy, of General Equilibrium Theory².

In response to the application of the CCC to the theory of international trade, one might cite:

• Christopher Bliss's suggestion that the necessary existence of gains from trade follows from including an assumption that all produced goods, not just consumer goods, be traded internationally.
• Wilfred Ethier's claim that the endowment of capital be calculated in equilibrium prices in models of international trade.
• A suggestion that the theory of international trade be organized around comparisons of intertemporal equilibrium paths³.

One might think a conclusion is more justified from the weight of the evidence when multiple arguments reach that conclusion. So one might react to existence of such controversies in the research literature as allowing one to support mainstream teaching. However, one would be wrong in this attitude. If you look at these responses in some detail, you will find that the orthodox economists do not end up at the textbook position, but at some other point. But, as far as I can tell, neither side ends up being transitioned from the research literature to conventional teaching. Would you not be more confident in adopting some such conventional counter-argument to one of my favorite arguments if it were widely taught? Otherwise, should you not suspect yourself of adopting an idiosyncratic misinterpretation of the theory?

Footnotes

1. This chain index is endogenous, not exogenous, as needed for much of neoclassical theory. Furthermore, Burmeister accepts the validity of demonstrations of reswitching and capital-reversing.
2. A focus on intertemporal and temporary equilibria is a rather drastic change of theory from the traditional neoclassical focus on a comparison of long run equilibria. The latter comparisons seem to be to provide the (exploded) foundation for most mainstream policy advice.
3. Avinash Dixit (May 1981). The Export of Capital Theory, Journal of International Economics. V. 11, Iss. 2: pp. 279-294.

Frank Hahn (1925-2013)

Frank Hahn's 1982 article, "The Neo-Ricardians", in the Cambridge Journal of Economics is important in my understanding of Sraffa's economics, even though I think it was misdirected in many ways. Here are some posts where I have mentioned Hahn's work:

• Geoff Harcourt reminiscing about an oral debate with Hahn.
• Wondering if macroeconomists have yet met Hahn's challenge to make money matter in General Equilibrium Theory.
• Quoting Hahn on intellectual regression in macroeconomics.

Play It Cool, Daddy-O

1.0 Introduction "Is Von Neumann Square?" is one of my favorite titles for an article in economics¹. This post is about a case in which Von Neumann is more hep². Sraffa’s book presents a succession of models in which, after the second chapter, the system of price equations have one degree of freedom. This is usually taken to be a trade-off between wages and the rate of profits. Once the distribution of the surplus product is exogenously specified, prices are determined. Some, such as Michael Mandler and Paul Samuelson, have criticized Sraffian economics on the basis that this number of degrees of freedom is arbitrary. Cases can arise in which the system of price equations has either more or less than one degree of freedom. This post illustrates a case in which more than one degree of freedom exists. 2.0 The Example 2.1 Technology and Quantity Flows Consider a very simple economy in which laborers produce corn from seed corn on lands of definite types. Two types of land are available. Assume that this economy has 100 acres of land of each type available. Two Constant-Returns-to-Scale processes are known for producing corn. As shown in Table 1, each process requires inputs of a single type of land, as well as labor and seed corn. The technology is such that the order in which types of land will be rented can be read off directly from the technology. As I have previously pointed out, this is not a general property in long period models analyzing rent. I think this special case property, however, is not what drives the existence of possibly more than one degree of freedom.

Table 1: The Technology

	α Process	β Process
Labor	1 person-year	1 person-year
Type I Land	1 acre	0 acre
Type II Land	0 acre	1 acre
Corn	1/5 bushels	1/4 bushels
Outputs	1 bushel corn	1 bushel corn

Under the assumptions, anywhere from zero to 200 bushels of corn can be produced as gross output in this economy. Assume that the gross output of this economy is 100 bushels of corn. Then cost-minimizing firms will cultivate all of type I land, and all of type II land will lie fallow. 2.2 The Price System For stationary-state prices, no process can earn pure economic prices. This condition imposes the following inequalities:

(1/5)(1 + r) + ρ₁ + w ≥ 1

(1/4)(1 + r) + ρ₂ + w ≥ 1

• w is the wage (bushels per person-year), paid at the end of the year
• r is the rate of profits
• ρ1 is the rent (bushels per acre) on type I land, paid at the end of the year
• ρ2 is the rent (bushels per acre) on type II land, paid at the end of the year

An equality applies for any process in use. Land of a given type can be modeled, in an alternative specification of the technology, as jointly produced at the end of the period from the inputs of labor, seed corn, and that type of land³. As long as less than 200 bushels of corn are produced, at least one type of land will pay no rent:

ρ₁ ρ₂ = 0

2.2.1 First Special Case Consider what would happen if the gross output was infinitesimally less. Both types of land would be in excess supply. The rent on both would be zero:

ρ₁ = ρ₂ = 0

The solution in this case is:

0 ≤ r ≤ 4

w = (1/5)(4 - r)

Only type I land is cultivated. The number of processes in use is equal to the number of produced commodities, that is produced goods with a positive price. The system of price equations has one degree of freedom.

(1/5)(1 + r) + ρ₁ + w = 1

(1/4)(1 + r) + ρ₂ + w > 1

2.2.2 Second Special Case Consider, however, what would happen if the gross output was infinitesimally more. Both types of land would be cultivated. Type I land would not be able to produce all the output quantity needed for the requirements for use, and it would have a positive rent:

ρ₁ > 0

Type II land would be in excess supply, and it would have a rent of zero.

ρ₂ = 0

The price system becomes:

(1/5)(1 + r) + ρ₁ + w = 1

(1/4)(1 + r) + ρ₂ + w = 1

The solution is:

0 ≤ r ≤ 3

w = (1/4)(3 - r)

ρ₁ = (1/20)(1 + r)

Two produced commodities with positive prices exist: corn and the first type of land. And two processes are activated. 2.2.3 The Case With Two Degrees of Freedom But type II land does not need to be cultivated in the case under consideration. Thus, the costs of cultivating type II land can exceed the revenues, and the rent on Type I land is not determined by the price equations. Only the first equation in the system of equations for prices need obtain.

(1/5)(1 + r) + ρ₁ + w = 1

The second process is still characterized by an inequality:

(1/4)(1 + r) + ρ₂ + w ≥ 1

This system has the solution:

0 ≤ r ≤ 4

0 ≤ ρ₁ ≤ (1/20)(1 + r)

ρ₂ = 0

w = (1/5)(4 - 5ρ₁ - r)

Figure 1 illustrates one projection of this solution into two dimensions. The lines closer to the origin are drawn for a higher rent on the first type of land.

Figure 1: Variation in the Wage-Rate of Profits Frontier with Rent

3.0 Conclusions I am loath to argue that the extra degree of freedom in this example is negligible since it arises only for a knife-edge. If the quantity produced is a hair larger or a hair smaller, the input-output matrices for commodities with positive prices are square. But in a larger model, the quantity produced is a choice variable. I also don't see why Sraffian models must not have more than one degree of freedom. Footnotes ¹ If I’ve actually read this article, it must have been in a reprint in some collection. ² Some posts take me a while to write. I began this one the day after Arthur Laurents died. ³ I don't here show the derivation of rent from such a model. References

• Christian Bidard (1986) "Is Von Neumann Square?" Journal of Economics, V. 46: pp. 407-419.
• Michael Mandler (20xx) "Sraffian Economics (new developments)" New Palgrave, 2nd edition.

Kurz And Salvadori Peeved With Mark Blaug?

Usually when Heinz Kurz and Neri Salvadori want to explain some economist is mistaken, they confine themselves to saying something along the lines of certain propositions "cannot be sustained". Recently, I stumbled upon a 2010 paper in which they answer Mark Blaug. I find their tone sometimes striking:

"A careful scrutiny of [Blaug (2009)] shows that Blaug reiterates once again his previous criticisms, adds a few new ones, but does not enter into a serious discussion of the replies to his earlier efforts... Answering him in detail would necessitate repeating again our counter-arguments. We spare the readers this and ask them to consult our earlier replies to Blaug."

"Blaug has already been given the opportunity in this journal to answer his critics; see Blaug (2002). Apparently, he feels that his rejoinder was not effective. This is hardly surprising because Blaug did not attempt to counter the objections of his critics.

Scrutiny of his new effort reveals that the situation has not changed. Once again Blaug merely reiterates his previous criticisms, adds a few new ones, but neglects to answer his critics. He seems to feel that repeating his story often will render it credible."

"If Blaug was concerned with an historical reconstruction of the case under consideration, he needs to spend some time in Trinity College Library, Cambridge (UK), as we did, in order to study Sraffa's papers and library and find out when Sraffa had arrived at which results, and why. He would then see that his above speculation as well as many other statements he put forward concerning Sraffa's contributions are without foundation; they are pure fiction. Historians of economic thought ought to be aware of the usefulness of archival work."

"In order to give credibility to his (in itself rather strange) complaint that 'Sraffians' have not contributed to certain themes or fields in economics, Blaug re-labels some authors: in case X has/has not contributed to field Y, he or she is not/is a 'Sraffian'."

"In the context of a discussion of the problem of the gravitation of market prices to their 'natural' or normal levels, he contends that while Kurz and Salvadori point out 'that little is known about the dynamic behaviour of even simple linear production models; nevertheless, they express the hope that the problem will be "settled in the foreseeable future" (Kurz and Salvadori 1998[a], 20)' (229 n.20). The reader who checks the source mentioned will not find this statement. Has Blaug got the page wrong? No, in the entire book the reader won't find the statement quoted. Has Blaug perhaps confounded some of our books? Yes, he has, but things are worse still. The only passage we are aware of having written that can be related to Blaug's criticism is contained in a book published in 1995. After having pointed out the extreme complexity of the issue at hand ('gravitation') and the dependence of the results obtained on the specific conditions assumed, we conclude: 'It should then be clear that there is no fear that the issue of gravitation will be settled in the foreseeable future' (Kurz and Salvadori 1995, 20; emphasis added). Hence we say exactly the opposite of what Blaug contends we are saying. This is not only annoying but also raises doubts about the seriousness of the entire enterprise. What is the relevance of a critique that lacks the elementary rigor of not misrepresenting (let alone reversing) the view of the people criticised? Misconstruction is an error surely worse even than historically unfaithful reconstruction?"

"None of Blaug's criticisms stands up to close examination. He attributes views to us (and to other authors) we (they) never advocated. He contends that 'Sraffian' authors have not written about certain problems, while referring to writings which show precisely the opposite. He commits a number of elementary blunders and mistakes the mathematical form of an argument for its content. He variously contradicts himself in the paper. He puts forward bold statements that are contradicted by the facts."

I have commented before on the specific Mark Blaug paper Kurz and Salvadori are rejecting; on the history of Blaug's incomprehension of Sraffianism; and even on the Institute of Economic Affairs, a right-wing think tank sponsoring some of Blaug's work.

References

• Mark Blaug (1975) The Cambridge Revolution: Sccess or Failure? A Critical Analysis of Cambridge Theories of Value and Distribution, Institute of Economic Affairs
• Mark Blaug (1985) Economic Theory in Retrospect, Fourth Edition, Cambridge University Press
• Mark Blaug (1988) Economics Through the Looking Glass: The Distorted Perspective of the New Palgrave Dictionary of Economics, Institute of Economic Affairs
• Mark Blaug (1999) "Misunderstanding Classical Economics: The Sraffian Interpretation of the Surplus Approach", History of Political Economy, V. 31, N. 2: pp. 213-236.
• Mark Blaug (2002a) "Kurz and Salvadori on the Sraffian Interpretation of the Surplus Approach", History of Political Economy, V. 34, N. 1: pp. 237-240.
• Mark Blaug (2002b) "Misunderstanding Classical Economics: The Sraffian Interpretation of the Surplus Approach", in Competing Economic Theories: Essays in Memory of Giovanni Caravale (Edited by S. Nisticò and D. Tosato), Routledge
• Mark Blaug (2009) "The Trade-Off Between Rigor and Relevance: Sraffian Economics as a Case in Point", History of Political Economy, V. 41, N. 2: pp. 219-247.
• Pierangelo Garegnani (1987) "Misunderstanding Classical Economics? A Reply to Mark Blaug", History of Political Economy, V. 34, N. 1: pp. 241-254.
• Heinz D. Kurz and Neri Salvadori (2002) "Mark Blaug on the 'Sraffian Interpretation of the Surplus Approach'", History of Political Economy, V. 34, N. 1: pp. 225-236.
• Heinz D. Kurz and Neri Salvadori (2010) "In Favor of Rigor and Relevance. A Reply to Mark Blaug" (4 Feb).
• Carlo Panico (2002) "Misunderstanding the Sraffian Reading of the Classical Theory of Value and Distribution: A Note", in Competing Economic Theories: Essays in Memory of Giovanni Caravale (Edited by S. Nisticò and D. Tosato), Routledge

Still A Man Hears What He Wants To Hear And Disregards The Rest

Mark Blaug has another paper criticizing Sraffianism. Here's one quotation from it:

"One of the striking features of the Sraffian side of the debate, the victorious side, was their categorical refusal to throw light on the debate by empirical research, insisting along with Sraffa himself that an anomaly such as reswitching is a theoretical flaw, which can only be repaired by discarding the theory in which it occurs. This is a position that has been steadfastly maintained through a half century and has only recently been broken by two Sraffians, namely, Lynn Mainwaring and Ian Steedman (2000)... Despite diligent combing through the literature, I have been unable to find more than one or two pieces of empirical work inspired by the theoretical ideals of Sraffian economics." -- Mark Blaug, "The Trade-Off between Rigor and Relevance: Sraffian Economics as a Case in Point, History of Political Economy, V. 41, N. 2 (2009): 219-247

I still don't see how empirical work is necessary to demonstrate a logical error. But confining myself to work before Mainwaring and Steedman (2000) and work in English, I find more than two: Albin (1975), Prince and Rosser (1985), and Ozanne (1996). Asheim (2008) is based on work written up long ago.

Those works, though, are looking for empirical evidence of Sraffa effects. But a plethora of empirical work is somewhat consistent with Sraffianism. I refer to work following in Leontief's wake. Blaug even acknowledges the relevance of this tradition:

"I have inadverently slipped into the language of Leontief's input-output analysis, which of course is rooted in physiocracy and classical economics, but was later adapted by Leontief himself to the mode of analysis of G[eneral] E[quilibrium] T[heory]" -- Mark Blaug, ibid

I find tendentious the assignment of Leontief to General Equilibrium Theory.

A Neoclassical Response To The Cambridge Capital Controversy

1.0 Introduction
Around 1980, Edwin Burmeister could have justly thought that he was expressing the most prominent neoclassical response to the Cambridge Capital Controversy. He had championed David Champernowne's chain index as a defense of the aggregate neoclassical model, and continued to do so. Nowadays, though, mainstream economists make claims based on the aggregate model apparently in complete ignorance that they had ever been competently challenged:

"However, the damage had been done, and Cambridge, UK, 'declared victory': Levhari was wrong, Samuelson was wrong, Solow was wrong, MIT was wrong and therefore neoclassic economics was wrong. As a result there are some groups of economists who have abandoned neoclassical economics for their own refinements of classical economics. In the United States, on the other hand, mainstream economics goes on as if the controversy had never occurred." -- Edwin Burmeister (2000)

This post illustrates, by means of an example, elements of Burmeister's approach to the neoclassical aggregate model. It is exposition, with next to no criticism.

2.0 Technology
Consider a very simple economy in which a single consumption good, corn, is produced from inputs of labor, iron, and (seed) corn. All production processes in this example require a year to complete. Two production processes are known for producing corn, and two processes are known for producing iron. These processes require inputs to be available at the beginning of the year for each unit output produced and available at the end of the year. Each corn-producing process produces one bushel corn at the scale of operations shown in Table 1. Similarly, each iron-producing process produces one ton iron at the scale shown in Table 1.

Table 1: The CRS Technology

Inputs	Corn Industry		Iron Industry
	A	B	C	D
Labor (Person-Years):	2	3	1	2
Iron (Tons):	3/50	1/10	3/5	1/2
Corn (Bushels):	1/2	1/4	1/2	3/5
Output (Various):	1	1	1	1

Apparently, inputs of iron and corn can be traded off in producing corn outputs. The process that requires more iron also requires more labor. Inputs of iron and corn are also traded off in producing iron. But in iron production, the process requiring a greater quantity of iron input requires less labor.

A technique consists of a process for producing iron and a process for producing corn. Thus, there are four techniques in this example. They are defined in Table 2.

Table 2: Techniques and Processes

Technique	Processes
Alpha	A, C
Beta	A, D
Gamma	B, C
Delta	B, D

3.0 Quantity Flows
Suppose firms have adopted the alpha technique and they produce 20/43 bushels corn with process A and 3/43 tons iron with process C. One can see, from Table 1, that these firms will employ 40/43 person-years in the corn industry and 3/43 person-years in the iron industry - that is, a total of one person-year throughout the economy. Likewise, firms in the corn industry will purchase inputs of 6/215 tons iron, while firms in the iron industry will purchase inputs of 9/215 tons iron in the iron industry, for a total of 3/43 tons iron inputs throughout the economy. The produced iron at the end of the year exactly replaces the iron used as input, leaving a net output of 17/86 bushels corn. (Calculating corn inputs in the two industries is left as an exercise for the reader.)

Since these processes can be equally scaled up to any desired level, I have described a stationary economy on a per person-year basis. Table 3 shows the results of these calculations, as well as similar calculations for the gamma and delta technique. The beta technique is never cost-minimizing and is not shown in Table 3.

Table 3: Quantities Per Person-Year

Technique	Variable	Value
Alpha	Gross Outputs	(3/43 Tons, 20/43 Bushels)
	Capital Goods	(3/43 Tons, 23/86 Bushels)
	Net Output	17/86 Bushels Corn
Gamma	Gross Outputs	(1/13 Tons, 4/13 Bushels)
	Capital Goods	(1/13 Tons, 3/26 Bushels)
	Net Output	5/26 Bushels Corn
Delta	Gross Outputs	(1/17 Tons, 5/17 Bushels)
	Capital Goods	(1/17 Tons, 37/340 Bushels)
	Net Output	63/340 Bushels Corn

4.0 Prices
In a steady state, the same rate of profits is earned on all processes in use. Furthermore, charging that rate of profits on a process not eligible for use results in costs in that process exceeding the revenues. That is, we seek steady state prices corresponding with the cost-minimizing technique.

Suppose the alpha technique is cost minimizing. Prices for the iron-producing process (C) must satisfy the following equation:

(3/5 p_α + 1/2)(1 + r) + w_α = p_α,

where p_α is the price of iron, w_α is the wage, and r is the rate of profits. The wage is paid at the end of the year, and corn is taken as the numeraire (so the price of a bushel corn is unity). Likewise, prices for the corn producing process (A) satisfy the following equation:

(3/50 p_α + 1/2)(1 + r) + 2 w_α = 1

I have specified a system of two equations in three variables. The wage and price of iron can be found as a function of the third variable, that is, the rate of profits. Table 4 displays this solution, as well as the solutions for the corresponding systems of equations for the other three techniques.

Table : Solutions to Price Equations

Technique	Variable	Equation
Alpha	Wage	wα(r) = (27 r2 - 56 r + 17)/[2 (43 - 57 r)]
	Price of Iron	pα(r) = 25 (3 + r)/(43 - 57 r)
Beta	Wage	wβ(r) = (107 r2 - 286 r + 107)/[40 (14 - 11 r)]
	Price of Iron	pβ(r) = 5 (11 +r)/[2 (14 - 11 r)]
Gamma	Wage	wγ(r) = (2 r2 - 13 r + 5)/[2 (13 - 17 r)]
	Price of Iron	pγ(r) = 5(9 + 5 r)/[2 (13 - 17 r)]
Delta	Wage	wδ(r) = (13 r - 7)(r - 9)/[20 (17 - 13 r)]
	Price of Iron	pδ(r) = (33 + 13 r)/(17 - 13 r)

Figure 1 graphs the wage-rate of profits curves for each technique. The cost-minimizing technique corresponds to the curve on the outer envelope. The wage-rate of profits curves for the alpha, gamma, and delta technique comprise the wage-rate of profits frontier. Alpha is cost-minimizing at low rate of profits, delta is cost-minimizing at high rates, and gamma is cost-minimizing at intermediate rates. Notice that for each pair of techniques, the wage-rate of profits curves cross at most once in the first quadrant. There is no reswitching, either on or off the frontier, in this example.

Figure 1: Wage-Rate of Profits Frontier

5.0 Champernowne's Chain Index

The above analysis specifies for each rate of profits (or for each wage) which technique will be adopted by cost minimizing firms. At switch points, linear combinations of techniques are cost-minimizing. The above analysis also determines the price of each capital good (e.g. corn and iron) for each rate of profits, as well as the composition of capital goods used in each technique per person-year. Figure 2 can thus be drawn based on this analysis.

Figure 2: Value of Capital and the Rate of Profits

Figure 2 shows the effects of both real and price Wicksell effects. The two horizontal lines arise from switch points. At switch points the composition of capital goods varies with the technique, while the rate of profits and the prices of capital goods are fixed. In other words, "real" capital varies in some sense. So the horizontal lines show real Wicksell effects. The curved, non-horizontal, segments display price Wicksell effects. That is, at non-switching points, the composition of capital goods remains invariant, but the prices of capital goods vary. Consequently, the numeraire value of the basket of capital goods varies here also.

Champernowne's chain index (Figure 3) sums up real Wicksell effects alone. Price Wicksell effects are abstracted from. The value of capital goods at the rate of profits of zero is taken in Figure 3 from Figure 2. Horizontal lines are drawn in Figure 3 at the same rates of profits at which they appear in Figure 2. The horizontal lines are also the same length. Vertical lines are drawn between horizontal lines.

Figure 3: Chain Index Value of Capital and the Rate of Profits

Champernowne's chain index only makes sense of the neoclassical parable in this case because all steps in Figure 3 slope down to the right. In other words, for an infinitesimal variation of the rate of profits around a switch point, the capital intensity of the cost-minimizing technique at the lower rate of profits exceeds the capital intensity of the cost-minimizing technique at the higher rate of profits. That is, Burmeister's defense of the neoclassical parable only applies in cases in which real Wicksell effects happen to be always negative:

"It follows, then, that a negative real Wicksell effect is the appropriate concept of 'capital deepening' in a model with many heterogeneous capital goods... Imposing some set of conditions on the technology ... should be sufficient to assure that the real Wicksell effect is always negative. Such conditions would be of interest - especially if they could be empirically tested - since they would validate the qualitative conclusions derived from one-good models often used in macroeconomics without any theoretical justification... Unfortunately, no set of such sufficient conditions is known, but the literature on capital aggregation suggests that they would impose severe restrictions on the technology." -- Edwin Burmeister (1987)

6.0 A Pseudo-Production Function
I finally turn to the aggregate neoclassical production function used in the neoclassical parable:

Y = F(K, L),

where Y is net income, K is capital, and L is labor. Since Constant Returns to Scale are assumed, one can divide through by the labor input:

Y/L = F(K/L, 1)

Or:

y = f(k),

where y is net output per worker and k is capital per worker, in some sense. Figure 4 graphs this function for the example, where Champernowne's chain index is used to measure capital per worker. (If net output consisted of more than the numeraire good, a chain index would be used to measure output also.)

Figure 4: Pseudo-Production Function for the Example

Using this construction, the equilibrium condition that the rate of profits equal the marginal product of capital holds at switch points:

r = f ' (k)

This analysis has accepted that the value of capital goods (that is, the "quantity of capital") depends on the rate of profits. Recall, however, that this analysis only applies to examples in which real Wicksell effects happen to be always negative.

References

• Salvatore Baldone (1984). "From Surrogate to Pseudo Production Functions", Cambridge Journal of Economics, V. 8: 271-288
• Edwin Burmeister (1980) Capital Theory and Dynamics, Cambridge University Press
• Edwin Burmeister (1987) "Wicksell Effects", in The New Palgrave, (ed. by J. Eatwell, M. Milgate, and P. Newman), Macmillan
• Edwin Burmeister (2000). "The Capital Theory Controversy" in Critical Essays on Piero Sraffa's Legacy in Economics (ed. by H. D. Kurz), Cambridge University Press
• D. G. Champernowne (1953-1954). "The Production Function and the Theory of Capital: A Comment", Review of Economic Studies, V. 21: 112-135

Blaug Versus Sraffians

Apparently, Gavin Kennedy delivered a paper, "Adam Smith's Invisible Hand: From Metaphor to Myth", at the 40th Anniversary Conference of the History of Economic Thought (HET). Kennedy reports the conference was held in Edinburgh on 3-5 September. And he uses a report on Marg Blaug's keynote address to express irritation at Sraffians:

"Professor Tony Brewer, University of Bristol, took over the chair, for Professor Mark Blaug's keynote address, which did not stir up the opposition I had expected, but then I did not know the economics of most of the participants, and I was relieved to find out that there were no vocal 'Sraffians' among the audience (the obscurity of the Sraffian economics monologue defies summary and any explanation for why it excites, or once excited, the in-group enthusiasm of a small cell in Cambridge).

Mark Blaug's paper was on 'The Trade-off Between Rigor and Relevance: Straffian economics as a case in point', and what a demolition job it was too, summed up neatly in the title." -- Gavin Kennedy

I was unable to find any other references to Blaug's address on the web. I am aware, however, that Blaug has been, for a number of years, griping about the formalist revolution that occurred in economics after World War II (e.g., Blaug 2003). And that he groups Sraffa's Production of Commodities by Means of Commodities: Prelude to a Critique of Economic Theory with Debreu's Theory of Value: An Axiomatic Analysis of Economic Equilibrium as exemplars of formalism in economics.

Blaug has had something to say about Sraffians in the past. For example, he has written at least two pamphlets for the Institute of Economic Affairs. As I understand it, IEA is a right-wing think tank in Britain. Blaug (1975) is an attack on the Cambridge school, as it stood after the successes of the Cambridge Capital Controversies. In this attack, Blaug misunderstands Sraffa's mathematics in ways that he carries forward into his textbook. He also adopts the curious position that a demonstration, based on reswitching and capital-reversing, of the logical invalidity of neoclassical economics does not hold without the identification of empirical occurrences of the phenomena. Blaug (1988) is a review of The New Palgrave. Blaug claims that John Eatwell, Murray Milgate, and Peter Newman ("A Sraffian Trio") edited a "tendentious work":

"To have invited three Sraffians to edit a new Palgrave dictionary of economics is roughly equivalent to asking three atheists to edit an encyclopedia of Christianity" -- Mark Blaug (1988)

Since the incorrectness of most doctrines of orthodox economics are not recognized by many practitioners, editors of reference works have a problem. I welcome the recognition that belief in these doctrines are a matter of faith, although I am not sure Blaug is being fair to Christianity.

Sraffa had a revolutionary impact on how historians read Ricardo, in particular, and the Classical economists more generally. Recently, Blaug (1999) disputed the Sraffian interpretation of Classical economics. Kurz and Salvadori (2002), Blaug (2002), and Garegnani (2002) is a selection from the literature of responses and counter-responses to Blaug's article. The Sraffians seem to agree that Blaug disputes a straw person. In the Sraffian interpretation, the theory of value can be set out with rigorous mathematics. But the givens of the Classical theory of value are themselves explained within economics (in contrast to neoclassical General Equilibrium theory). Thus, the theory of value is only an element in an approach to a larger economics which investigates such issues as growth, development, population demographics, etc. Blaug failed to understand the instrumental role of the theory of value in the Sraffian interpretation. His criticism of the Sraffians for setting out the theory of value, in their understanding, without encompassing, for example, growth is simply misdirected.

Some Sraffians are competing with Blaug's Economic Theory in Retrospect (nth edition) in the market for textbooks on the history of economic thought. I gather that this market is shrinking, as mainstream economists purge the history of their field from the curriculum. Alessandro Roncaglia (2005) and Ernesto Screpanti & Stefano Zamagni (2005) are two textbooks from Sraffians.

• Mark Blaug (1975) The Cambridge Revolution: Success or Failure?, Institute of Economic Affairs
• Mark Blaug (1988) Economics Through the Looking Glass: The Distorted Perspective of The New Palgrave Dictionary of Economics, Institute of Economic Affairs
• Mark Blaug (1999) "Misunderstanding Classical Economics: The Sraffian Interpretation of the Surplus Approach", History of Political Economy, V. 31, N. 2: 213-236
• Mark Blaug (2002) "Kurz and Salvadori on the Sraffian Interpretation of the Surplus Approach", History of Political Economy, V. 34, N. 1: 237-240
• Mark Blaug (2003) "The Formalist Revolution of the 1950s", Journal of the History of Economic Thought, V. 25, N. 2: 145-156
• Pierangelo Garegnani (2002) "Misunderstanding Classical Economics? A Reply to Blaug", History of Political Economy, V. 34, N. 1: 241-254
• Heinz D. Kurz and Neri Salvadori (2002) "Mark Blaug on the 'Sraffian Interpretation of the Surplus Approach'", History of Political Economy, V. 34, N. 1: 225-236
• Alessandro Roncaglia (2005) The Wealth of Ideas: A History of Economic Thought, Cambridge University Press
• Ernesto Screpanti and Stefano Zamagni (2005) An Outline of the History of Economic Thought (Second edition)