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)

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.

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

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.

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