Greg Mankiw Should Try To Make A Honest Living

A Discussion On The Best (or Worst) Of Mankiw

Peter Bofinger has expanded a series of tweets to point out some stuff that is just wrong in Mankiw's introductory textbook. The video above is a virtual panel discussion in which Mankiw graciously pretends to respond to Bofinger. Rüdiger Bachmann and Anna Reisch also participate. I do not know the host, Thomas Fricke. Questions from the audience are fielded towards the end. I concentrate on Sascha Buetzer (1:07:45) below. Other questions are from Janina Urban (1:18:41) and Thomas Kopp (1:21:29). I am probably missing something.

I wonder whether Anna Reisch knows about Adolph Lowe's political economics. I know of this through his 1965 book On Economic Knowledge. I gather Lowe thought it was the task of economists to say whether a given end state is internally consistent and to explain how it could be reached.

I want to point out some hypocrisy from Mankiw. He does not even bother arguing that Bofinger has pointed out confusion and nonsense in his textbook. He says that he sees his job as presenting the consensus of mainstream economics, not his own theories. He tries to minimize the imposition of his own idiosyncrasies. Now Buetzer is, if I hear correctly, the senior advisor to the German director to the International Monetary Fund (IMF). Buetzer offers the difficult proposition that "textbooks should strive to be factually correct." And one of his points is that, "all modern empirical evidence ... point to ... there is no equity-efficiency tradeoff from moderate levels of redistribution, but rather the opposite." This is the "mainstream in mainstream institutions". Does Mankiw say he will then update his textbook in the next edition to reflect the mainstream view? Of course not. He starts presenting his own idiosyncratic reasons for rejecting empirical evidence.

Economists should strive not to teach falsehoods and nonsense and not to promote the teaching of falsehoods and nonsense. Maybe Mankiw is correct that if he discarded from his textbooks stuff that is, at best, just wrong, his textbooks would not sell as well. That is no justification for retaining balderdash. Although Mankiw may disagree, he is not entitled to an income from textbooks.

On The Empirical Verification Of The Cambridge Capital Controversy

1.0 Introduction

My consistent position is that Sraffa and his followers, besides recovering an alternate approach to value and distribution found in classical economics and Marx, demonstrated the logical invalidity of marginalist economics. Empirical results are irrelevant to questions of logical validity.

Wage curves, as constructed from input-output matrices, are rational functions with the numerator and denominator both being some high order polynomial functions. I would have liked to see some more wobbles in those constructed empirically and more examples of reswitching and capital-reversing. Nevertheless, the finding that frontiers are close to linear functions, with only a few switch points, is not consistent with an emphasis on widespread marginal adjustments. It is more consistent with Marx's theory of value and Joan Robinson's understanding of technical change, in which the question of the choice of technique at a given moment in time is, at most, a secondary concern. Schefold's recent work (Schefold 2013, Schefold 2016, Götz and Schefold 2020) with random matrices is of interest here for trying to explain the empirical posts.

I have written about empirical results before. In this post I concentrate on Zambelli (2018) as the most recent, most extensive empirical examination of input-output matrices. See also the comments on Zambelli's work in Götz and Schefold (2020).

2.0 Progress in Empirical Research Work

Increased computer power and more complete consistent national income and product accounts (NIPAs) has supported empirical research. If I recall correctly, Ochoa (1987) looks at wage curves as based on input-output matrices from different times. He looks for pairs that intersect more than onec.

In looking at such a pair, however, many more wage curves are available. One can construct input-output matrices, with one process for each industry, where the processes are not all from one matrix but combine processes among industries from the different matrices. Han and Schefold (2006) take this approach.

But this is not all. One need not limit oneself with processes from pairs of wage curves. One should look at the full range of techniques, where the process for each industry might be from any input-output matrix in your database. Zambelli (2018), in following this approach, uses an algorithm that he and his colleagues cleverly constructed to select the wage curves on the frontier, thereby keeping the combinatorial explosion in this approach somewhat under control.

Ideally, one would like internationally consistent classifications of industries in make and use tables and Leontief input-output matrices that include joint production. If the latter is not available, which it usually not, one needs consistent approximations for single-production. Since make and use tables, and the resulting Leontief input-output tables are typically price data, one needs price indices for industries or commodities. At what level of aggregation do some industries only appear in some tables? Many more questions arise here that are probably beyond me.

3.0 'Perverse' Phenomena

What supposedly 'perverse' phenomena should one look for in techniques formed out of empirical input-output matrices? I suggest instances of the reswitching of techniques, capital reversing, the reverse substition of labor, and the recurrence of processes in individual industries would be of interest. Reswitching on the frontier is sufficient, but not necessary for the occurrence of positive real Wicksell effects. I, like many others, define capital reversing (also known as reverse capital deepening) as equivalent to positive real Wicksell effects. Zambelli (2018), on the other hand, defines capital reversing to arise with positive real or price Wicksell effects.

I tend, in pointing out the invalidity of marginalist economics, to de-emphasize any concern with the direction of price Wicksell effects. As I understand it, the direction of price Wicksell effects is dependent on the selection of the numeraire. Also, I am aware of Burmeister's championing of Champernowne's chain index for capital. On the other hand, Baldone (1984) suggest this defense of mainstream economics fails. Fratini (2010) has an example with a continuous variation of techniques along the wage frontier and in which negative price Wicksell effects swamp positive real Wicksell effects, which I guess is a propos here.

4.0 Wage Frontiers and Aggregate Production Functions

I have been talking about wage frontiers and wage curves above. One can construct the aggregate production 'function', given the analysis of the choice of technique. In this analysis, one takes net output as of a given physical composition. It is convenient to take net output as the numeraire. The composition of capital goods varies at switch points, and their prices vary between switch points. At one point, though, Zambelli considers variations in the composition of capital goods between switch points, as I understand it. I relegate an explanation of what he is doing here to an appendix.

5.0 Conclusion

Zambelli (2018) is impressive empirical work. The failure of so-called neoclassical theory in 60 percent of the cases examined, as I understand it results, from a concentration on price Wicksell effects, which would not disconcert, for example, Burmeister. I also have difficulties with how Zambelli relates the aggregate production function to a problem of minimizing the value of aggregate capital.

Appendix: The Construction of a Microeconomic Production Function

I illustrate the construction of a production funcition as the solution of a maximization problem. A more general presentation would start with netput vectors and assume convexity. I briefly glanced at the appendix to chapter VI in Pasinetti (1977) in writing this.

For concreteness, suppose the managers of a firm have given quantities, x₁, x₂, and x₃, of three resources and know of four fixed-coefficient processes for producing a single commodity. The coefficicients of production for these four processes are:

(a_.j)^T = (a_{1, j}, a_{2, j}, a_{3, j}), j = 1, 2, 3, 4.

Let q_i, i = 1, 2, 3, 4, be the decision variables denoting how much output is produced with each process. Consider the linear following linear program (LP). Maximize output y:

y = q₁ + q₂ + q₃ + q₄

such that:

a_{1, 1} q₁ + a_{1, 2} q₂ + a_{1, 3} q₃ + a_{1, 4} q₄ ≤ x₁

a_{2, 1} q₁ + a_{2, 2} q₂ + a_{2, 3} q₃ + a_{2, 4} q₄ ≤ x₂

a_{3, 1} q₁ + a_{3, 2} q₂ + a_{3, 3} q₃ + a_{3, 4} q₄ ≤ x₃

q_i ≥ 0, i = 1, 2, 3, 4 = 1, 2, 3, 4.

The constraints express the condition that no more of a resource (also known as a factor of production) can be used than is given. Every process must be operated at a non-negative level. Let f express the solution of this LP as a function of factors of production:

y = f(x₁, x₂, x₃)

This is a discrete version of the production function for a given commodity. It has properties commonly assumed in marginalist economics. It exhibits constant returns to scale (CRS) and non-increasing marginal products. If one wanted to construct a production function differentiable everywhere, one could assume an uncountably infinite set of production processes.

I might as well write down the dual problem. It is to choose factor prices w₁, w₂, w₃ to minimize:

w₁ x₁ + p₂ x₂ + p₃ x₃

such that:

a_{1, 1} w₁ + a_{2, 1} w₂ + a_{3, 1} w₃ ≥ 1

a_{1, 2} w₁ + a_{2, 2} w₂ + a_{3, 2} w₃ ≥ 1

a_{1, 3} w₁ + a_{2, 3} w₂ + a_{3, 3} w₃ ≥ 1

a_{1, 4} w₁ + a_{2, 4} w₂ + a_{3, 4} w₃ ≥ 1

w₁ ≥ 0, w₂ ≥ 0, w₃ ≥ 0

For a solution of these two LPs, the values of their objective functions are equal. Factor prices are such that output is completely distributed among the owners of the resources whose services are used in producing the given commodity. If a constraint in the dual is met with inequality, the corresponding decision variable in the primal LP is set to zero. That process is not operated. If a constraint in the primal LP is met with an inequality, that resource is in excess supply and its price is zero. Even though you see no derivatives above, this is an exposition of an aspect of the theory of marginal productivity.

All the parameters and variables in the primal LP are in physical units (for example, bushels, tons, person-years). It does not make much sense to me in an aggregate production function, with output and arguments in price terms, to maximize the value of output for a given value of capital or to minimize the value of capital for a given value of output. Nevertheless, that is what Zambelli does in Section 5.3 of his paper. I suppose he wanted to present a comprehensive empirical exploration of aggregate neoclassical theory, taking its illogic as given.

References

• Baldone, Salvatore. 1984. From surrogate to pseudo production functions. Cambridge Journal of Economics 8: 271-288.
• Burmeister, E. 1980. Capital Theory and Dynamics. Cambridge: Cambridge University Press
• Fratini, Saverio M. 2010. Reswitching and decreasing demand for capital. Metroeconomica 61 (4): 676-682.
• Han, Zonghie and Bertram Schefold. 2006. An empirical investigation of paradoxes: reswitching and reverse capital deepening in capital theory. Cambridge Journal of Economics 30: 737-765.
• Kersting, Götz and Bertram Schefold. 2020. Best techniques leave little room for substitution: a new critique of the production function. Centro Sraffa Working Paper n. 47.
• Ochoa, E. M. 1987. Is reswitching empirically relevant? US wage-profit-rate frontiers, 1947-1972. Economic Forum 16: 45-67.
• Pasinetti, Luigi L. 1977. Lectures on the Theory of Production New York: Columbia University Press.
• Schefold Bertram. 2013. Approximate surrogate production functions. Cambridge Journal of Economics 37 (5): 1161-1184.
• Schefold Bertram. 2016. Profits equal surplus value on average and the significance of this result for the Marxian theory of accumulation.. Cambridge Journal of Economics 40 (1): 165-199.
• Zambelli, Stefano. 2018. The aggregate production function is NOT neoclassical. Cambridge Journal of Economics 42: 383-426.

Books To Make You More Muddled

I have not read all of these, and you might think I am being unfair with this post title. If you want critiques of post modernism, try the Amin and Eagleton referenced at the end of this post. Sokal, after the cited book, participated in interesting colloquia with those who were scholars of what he was attacking and mocking. If you want to see how little I know about this area, you can look at my posts on Gramsci, Foucault, Wittgenstein, or Zizek.

• Gellner, Ernest. 1959. Words and Things: A Critical Account of Linguistic Philosophy and a Study in Ideology. London: Gollantz.
• Gross, Paul R. and Norman Levitt. 1998. Higher Superstition: The Academic Left and its Quarrels with Science. Baltimore: John Hopkins Press.
• Hicks, Stephen R. C. 2004. Explaining Postmodernism: Skepticism and Socialism from Rousseau to Foucault. New Berlin: Scholarly Publishing.
• Pluckrose, Helen and James A. Lindsay. 2020. Cynical Theories: How Activist Scholarship Made Everything about Race, Gender, and Identity - and Why This Harms Everybody. Pitchstone Publishing.
• Sokal, Alan and Jean Bricmont. 1998. Fashionable Nonsense: Postmodern Intellectuals Abuse of Science. New York: Picador USA.

John Roemer's Reproducible Solution

Can I adapt Roemer's work, suitably taking into account later work by D'Agata and Zambelli, to found this approach to markup pricing? As a start, I here quote Roemer on a reproducible solution (RS), before he takes into account unequal rates of profits and a choice of technique. Given the role of endowments, is this a neoclassical approach, like Hahn's 1984 CJE paper? Even so, is it a valid justification for Sraffa's price equations? Notice there are no subscripts for time below.

"There are N capitalists; the νth one is endowed with a vector of produced commodity endowments ω^ν ... Capitalist ν starts with capital ω^ν, which he seeks to turn in more wealth at the highest rate of return. Thus the program of capitalist ν is

Facing prices p, to

choose x^ν ≥ 0 to

max (p - (p A + L)) x^ν

s.t. (p A + L) x^ν ≤ p ω^ν

(The constraint says that the inputs costs can be covered by current capital.) Let us call A^ν(p) the set of solution vectors to this program." -- Roemer (1981: 18-19, I made changes for typesetting mathematics).

Roemer defines a RS:

"Definition 1.1: A price vector p is a reproducible solution for the economy {A, L; b; ω¹, ..., ω^N} if:

• For all ν, there exists x^ν in A^ν(p), such that (profit maximization)
• x = Σ x^ν and x ≥ A x + (L x) b (reproducibility)
• p b = 1 (subsistence wage)
• A x + (L x) ≤ ω = Σ ω^ν (feasibility)

We shall also refer to the entire set {p, x¹, ..., x^N} as a reproducible solution." -- Roemer (1981: 19-20, with for math).

A RS can only exist if the elements of the endowment vector are in certain proportions:

"Theorem 1.2: Let the model {A, L, b} be given with A productive and indecomposable, and the rate of exploitation e > 0. Let {p, x¹, ..., x^N} be a nontrivial RS. (i.e., Σ x^ν = x ≠ 0). Then the vector of prices p is the E[qual] P[rofit] R[ate] vector p^*. Furthermore, a RS exists if and only if omega is an element of C^*, where C^* is a particular convex cone in [the space of n-dimensional real vectors] containing the balanced growth path of {A, L, b}. (C^* is specified precisely below.)" -- Roemer (1981: 20, with changes for math).

Even though endoments are taken as given in defining the firm's LP, endowments are endogenous in the sense that they must lie close to those on a balanced growth path. I like to have labor advanced and wages paid out of the surplus, instead of vice versa as above. The above does not allow for a choice of technique. Roemer has at least some of this in later chapters.

References

• John E. Roemer. 1981. Analytical Foundations of Marxian Economic Theory. Cambridge University Press.

23 February 1981: King Juan Carlos Becomes A Spanish National Hero

I only know about this at the level of a Wikipedia article. Or maybe a short newspaper article. Some of you doubtlessly know more.

Some Spanish military officers, pining for the certainty of a fascist authortarian state, assaulted the Congress of Deputies in 1981. They held the deputies hostage. Some showed real physical courage. The prime minister and deputy prime minister refused to sit down when ordered so, despite having guns pointed at them.

I'd like to conclude that, despite this failed coup attempt, Spain is a thriving democracy today. But I think political parties today are addressing problems more connected with austerity after 2008 than with nostalgia for Franco. I conclude with a couple references about violence in politics.

• Hannah Arendt. 1969. On violence. In Crises of the Republic New York: Harcourt Brace Jovanovich.
• Georges Sorel. 1950. Reflections on Violence (Trans. by T. E. Hulme) London: Collier-Macmillan.

The Tractor-Corn Model: A Start

1.0 Introduction

In my ROBE article, I consider fluke switch points arising from perturbations of coefficients of production in the Samuelson-Gargenani model, but in the case with only circulating capital. An obvious generalization is to consider fixed capital. This generalization is simplified by restricting oneself to the case in which machines operate with constant efficiency. Steedman (2020) analyzes this case, and this post is a start on working through elements of the corn-tractor model he leaves as homework. I do not know how far I will go in rewriting my paper for this case.

2.0 Technology for a Technique

In the model, corn is produced by labor working working with a specified type of tractor. And that type of tractor is itself produced by labor working with that type of tractor.

Each type of tractor defines a technique, where a technique is specified by six parameters:

• a: The number of tractors (of a given age) whose services are used for a year in producing a new tractor.
• b: The person-years of labor needed to work with tractors (of a given age) to produce a new tractor.
• n: The number of years a tractor lasts when used in producing new tractors.
• α: The number of tractors (of a given age) whose services are used for a year in producing a bushel of corn.
• β: The person-years of labor needed to work with tractors (of a given age) to produce corn.
• ν: The number of years a tractor lasts when used in producing corn.

The notation is Steedman's, borrowed from J. R. Hicks. I see that if I keep this notation, I will have to drop my usual practice, in honor of Joan Robinson, of using lowercase Greek letters to refer to a technique.

Consider, for a technique, the (n + ν)-element row vector of labor coefficients a₀, the (n + ν) x (n + ν) matrix A of input coefficients, and the (n + ν) x (n + ν) matrix B of output coefficients. This vector and these matrices have a block structure:

a0 =

b

b (uT)1,n - 2

b

β

β (uT)1,ν - 2

β

A =	0	01,n - 2	0	0	01,ν - 2	0
	a	01,n - 2	0	α	01,ν - 2	0
	0n - 2, 1	a In - 2,n - 2	0n - 2, 1	0n - 2,1	0n - 2,ν - 2	0n - 2,1
	0	01,n - 2	a	0	01,ν - 2	0
	0ν - 2,1	0ν - 2,n - 2	0ν - 2,1	0ν - 2,1	α Iν - 2,ν - 2	0ν - 2,1
	0	01,n - 2	0	0	01,ν - 2	α

B =	0	01,n - 2	0	1	(uT)1,ν - 2	1
	1	(uT)1,n - 2	1	0	01,ν - 2	0
	a	01,n - 2	0	0	01,ν - 2	0
	0n - 2,1	a In - 2,n - 2	0n - 2, 1	0n - 2, 1	0n - 2,ν - 2	0n - 2, 1
	0	01,n - 2	0	α	01,ν - 2	0
	0ν - 2,1	0ν - 2,n - 2	0ν - 2,1	0ν - 2,1	α Iν - 2,ν - 2	0ν - 2,1

Obviously, HTML defeated me here. I is the identity matrix, and u is a column unit vector.

Each element of a₀ and each column of A and B correspond to a process of production. The first n columns constitute the tractor sector, and the remaining ν columns are the corn sector. I assume constant returns to scale and that each process requires a year to complete. a_{0, j} is the person-years of labor that enters the jth process per unit-level of operations. The jth column of A is the inputs consumed by the process, and the jth column of B is the outputs. The first row index is for corn. The first row of A is zero, since corn is not used as an input in any process. The second row index is for new tractors. The remaining row indices are for old tractors. Once a tractor is used in tbe production of tractors, it can no longer be used in producing corn. Likewise, a tractor used in the corn sector cannot be transferred to the tractor sector.

3.0 An Annuity

Consider an annuity c_n(r) bought for a dollar at the start of a year. This annuity pays out the sum c_n(r) at the end of the first year, at the end of the second year, and so on through the end of the nth year. This arrangement implicitly specifies an interest rate r which equates the cost and the present value of the payments:

1 = c_n(r)/(1 + r) + c_n(r)/[(1 + r)²] + ... + c_n(r)/[(1 + r)ⁿ]

A bit of algebra reveals that the payments for the annuity are given by the following formula:

c_n(r) = r (1 + r)ⁿ/[(1 + r)ⁿ - 1]

The limit as the interest rate approaches zero can be found by L'Hôpital's rule. It is:

c_n(0) = 1/n

I need these formulas below.

4.0 The Quantity System

Now I want to consider a steady state in which the economy grows at a uniform rate of 100 g percent. Let the column vector q specify the level of operation of each process. I postulate that q has the following form:

q^T = [q₁, q₁/(1 + g), ..., q₁/(1 + g)^{n - 1}, q₂, q₂/(1 + g), ..., q₂/(1 + g)^{ν - 1}]

where q₁ and q₂ are variables to be determined. Let e₁ be the first column of the identity matrix. Consumption in a steady-state is c(g) e₁, where:

c(g) e₁ = [B - (1 + g) A] q

Expanding the first element of the column vectors on both sides, one gets:

c(g) = (1 + g) q₂/c_ν(g)

The second element yields:

0 = {[(1 + g)/c_n(g)] - (1 + g) a} q₁ - (1 + g) α q₂

Or:

0 = [1 - a c_n(g)] q₁ - α c_n(g) q₂

Given g, the above is a linear equation in q₁ and q₂. New tractors do not enter into consumption. Quantity flows are specified such that one person-year of labor is employed:

a₀ q = 1

Or:

(1 + g) b q₁/c_n(g) + (1 + g) β q₂/c_ν(g) = 1

Or:

b c_ν(g) q₁ + β c_n(g) q₂ = c_n(g) c_ν(g)/(1 + g)

A linear system of two equations in two unknowns, given the rate of growth, has now been derived.

The system is easily solved:

q₁ = α c_n(g) c_ν(g) /{[β + αbc_ν(g) - aβc_n(g)](1 + g)}

q₂ = [1 - ac_n(g)] c_ν(g)/{[β + αbc_ν(g) - aβc_n(g)](1 + g)}

Consumption per worker (in units of bushels corn per person-year) is:

c(g) = [1 - ac_n(g)]/[β + αbc_ν(g) - aβc_n(g)]

In a comparison of steady states, consumption per worker is higher if the rate of growth is lower. The dependence of the denominator on the rate of growth vanishes under the special case in which:

a c_n(g)/b = α c_ν(g)/β

Somehow, the above says that the organic composition of capital does not vary between the tractor and the corn sectors. The tradeoff, however, between consumption per worker and the rate of growth is still not linear. The maximum rate of growth, G, is the smallest non-negative real solution to:

0 = 1 - a c_n(G)

Consumption per worker in a stationary state is:

c(0) = [n - a]ν/[nνβ + αbn - aβν]

One might use the above to discuss the capital-intensity of a technique. If the technique with one type of tractor is more capital-intensive than the technique with another type, one would expect c(0) to be higher with the first type.

5.0 The Price System

I now consider prices. Let p be a row vector of prices, w the wage, and r the rate of profits. In matrix form, the price equations are:

p A (1 + r) + w a₀ = p B

A bushel corn is the numeraire:

p e₁ = 1

The above consists of a system of (n + ν + 1) equations for (n + ν + 2) variables. The system has one degree of freedom. Labor is advanced, and wages are paid out of the surplus at the end of the year. A tractor of each age and history has a seperate price.

I now rewrite the price equations for the first time. The price of a bushel cotn is unity, and p represents thd price of a new machine. p_m,j is the price of a j-year old tractor in the tractor sector. p_c,j is the price of a j-year old tractor in the corn sector. The n equations for the tractor sector are:

p a (1 + r) + w b = p + p_m,1 a

p_m,1 a (1 + r) + w b = p + p_m,2 a

...

p_{m,n - 1} a (1 + r) + w b = p

The ν equations for the corn sector are:

p α (1 + r) + w β = 1 + p_c,1 α

p_c,1 α (1 + r) + w β = 1 + p_c,2 a

...

p_{c,ν - 1} α (1 + r) + w β = 1

Consider the equations for the machine sector. Multiply the first equation by (1 + r)^{n - 1}, the second equation by (1 + r)^{n - 2}, and so on, until the last equation is multiplied by (1 + r)⁰.Sum these equations:

p a (1 + r)ⁿ + w b [1 + (1 + r) + ... + (1 + r)^{n - 1}] = p [1 + (1 + r) + ... + (1 + r)^{n - 1}]

The prices for old tractors appear on both sides of successive equations with the same coefficient and drop out. A similiar procedure for the corn sector yields:

p α (1 + r)^ν + w β [1 + (1 + r) + ... + (1 + r)^{ν - 1}] = [1 + (1 + r) + ... + (1 + r)^{ν - 1}]

So far, this procedure works if tractors do not have constant efficiency. The next step requires that, though. The price equations become:

p a c_n(r) + w b = p

p α c_ν(r) + w β = 1

Fpr both the quantity and the price system, a set of (n + ν) equations is reduced to two equations in which (quantities or prices) of old tractors do not enter. The charge for a tractor is that of an annuity that pays out for each year of the tractor's life.

The price system is easily solved. The price of a new tractor is:

p = b/[β + αbc_ν(r) - aβc_n(r)]

Under the special case of equal organic compositions of capital, the ratio of a price of a new tractor to a bushel corn is the ratio of direct labor inputs. Presumably, prices are also proportional to labor values in this special case. The wage curve is:

w = [1 - ac_n(r)]/[β + αbc_ν(r) - aβc_n(r)]

I have already discussed the wage curve under the guise of the tradeoff between consumption per worker and the rate of growth. The maximum rate of profits R is identical to the maximum rate of growth G.

6.0 Conclusion

Steedman (2020) avoids writing about almost all of the above or leaves it as an exercise for the reader. Basically, I have derived Steedman's first five numbered equations. Some of this is in Chapter 10 of Sraffa (1960).

References

• Gargenani, Pierangelo. 1970. Heterogeneous capital, the production function and the theory of distribution. Review of Economic Studies 37 (3): 407-436..
• Samuelson, Paul A. 1962. Parable and realism in capital theory: the surrogate production function. Review of Economic Studies 29 (3): 193-206.
• Steedman, Ian. 2020. Fixed capital in the corn-tractor model. Metroeconomica 71: 49-56.
• Vienneau, Robert L. 2018. Normal forms for switch point patterns. Review of Behavioral Economics 5 (2): 169-195.