Nonsense In Mankiw's Introductory Textbook

Marginalist economics was shown to be incoherent about two thirds of a century ago. It collapsed just around the issues Marx investigated more than a century and a half ago. How does the ownership of capital goods result in the owner obtaining a return? Mainstream economists address their inadequacy by refusing to talk about their demonstrated inconsistencies.

Those who understand the theory have available a certain form of amusement. They can quickly locate confusion in mainstream textbooks. I happen to have available the eighth edition of N. Gregory Mankiw's Principles of Economics (2018). I may have missed something. Over the course of hundreds of pages, he confuses capital, as a factor of production supplied by households, physical capital goods, deferred consumption, and finance.

Mankiw is careful, I guess, in what he does not say. He has "capital" meaning physical goods, for a while. There seems to be no explanation of the level of interest or dividend payments to households. Households trade consumption between now and later. These savings are not related to changes in the capital stock, although a later section on savings and investment confusingly suggests that some unspecified relationship exists. An aggregate production function has an argument for physical capital, with no discussion of units of measurement. And this all falls by the wayside when he gets to macroeconomics. He presents the obsolete theory of loanable funds, even with silliness about the crowding-out effect of government deficits.

Section 2-1d is "Our first model: the circular flow diagram." With the usual confusion, in one half of the diagram, households supply firms with the factors of production. Capital is "building and machines". At this point, you have a blast furnace in your back yard, which you rent to a steel manufacturer.

Chapter 18 is "The Markets for Factors of Production", and Mankiw emphasizes labor markets. The non-wage part of the national income "went to landowners and to the owners of capital - the economy's stock of equipment and structures - in the form of rent, profit, and interest" (pp. 361-362). Mankiw does not seem to know of any difficulties raised for labor markets or the supposed marginal productivity theory of distribution by the Cambridge capital controversy. "Put simply, highly productive workers are highly paid, and less productive workers are less highly paid" (p. 37).

Capital is like land. "The purchase price of land or capital is the price a person pays to own that factor of production indefinitely. The rental price is the price a person pays to use that factor for a limited period of time" (p. 375). A box on p. 376 is titled "What is capital income?" He brings up interest, dividends, and retained earnings but has no explanation for their levels:

"In our analysis, we have been implicitly assuming that households own the economy’s stock of capital - ladders, drill presses, warehouses, and so on ... In fact, firms usually own the capital they use, and therefore, they receive the earnings from this capital... [I]nstitutional details are interesting and important, but they do not alter our conclusion about the income earned by the owners of capital. Capital is paid according to the value of its marginal product, regardless of whether this income is transmitted to households in the form of interest or dividends or whether it is kept within firms as retained earnings."

Chapter 21 is the theory of consumer choice. Mankiw has the analysis of the trade-off between leisure and work. Section 21-4c treats "How Do Interest Rates Affect Household Saving?" Figure 15 shows the budget constraint and indifference curves for an example of intertemporal choice (p. 444).

Chapter 25 is "Production and Capital" and is part of the treatment of macroeconomics. A box on the production function is on p. 523. Section 25-3a is "Savings and Investment":

"Because capital is a produced factor of production, a society can change the amount of capital it has. If today the economy produces a large quantity of new capital goods, then tomorrow it will have a larger stock of capital and be able to produce more goods and services. Thus, one way to raise future productivity is to invest more current resources in the production of capital. Because resources are scarce, devoting more resources to producing capital requires devoting fewer resources to producing goods and services for current consumption. That is, for society to invest more in capital, it must consume less and save more of its current income. The growth that arises from capital accumulation is not a free lunch: It requires that society sacrifice consumption of goods and services in the present to enjoy higher consumption in the future."

I do not know what skipping my dinner has to do with manufacturing more ladders to outfit employees of firms with orchards and apples to be picked. Neither does Mankiw, of course.

Chapter 26 treats Saving, Investment, and the Financial System. "Now the interest rate is the price that adjusts to balance supply and demand ... for funds in financial markets" (p. 542). Banks and mutual funds are "financial intermediaries" "directing the resources of savers into the hands of borrowers." Mankiw presents the usual national income accounting, with savings and investment in monetary (financial units). "In the language of macroeconomics, investment refers to the purchase of new capital, such as equipment or buildings." He has the crudest loanable funds model. He presents the argument that government deficits crowd out private investment (p. 554) as if it were scientific fact. (On page 590, a box from David Neumark has the usual archaic nonsense about minimum wages causing structural unemployment.)

Mankiw's textbook lacks an explanation of the returns to ownership and an acknowledgement of the existence of this gap. He could argue that this reflects mainstream economics, which is apologetics.

An Example With A Cost-Minimizing Technique With Intensive And Extensive Rent

Figure 1: Detail on Variation of Rent per Acre with Rate of Profits

1.0 Introduction

Consider a model of the production of commodities with non-produced means of production that are unchanged by their use in production. In other words, they are types of land. In a simple model of extensive rent, a single agricultural commodity, 'corn', can be produced, on each type of land, with a single production production. This post expands a simple multi-commodity model to postulate the existence of two production processes on one type of land. The model then combines intensive and extensive rent, depending on the choice of technique.

In the example, all three types of land are at least partially cultivated to satisfy requirements for use. Whether or not all three types of land obtain a rent depends on the level of profits. A mixture of intensive and extensive rent is obtained only for a range of the rate of profits.

I repeat a lot from a previous post so that this post somewhat makes sense by itself.

2.0 Technology, Resources, Final Demand, and Feasibility

A model of the production of commodities is specified by the technology, the endowments of unproduced natural resources, and the requirements for use. Technology is specified, in a discrete technology, by coefficients of production for each production process. Each process is assumed to require the same time to complete and to exhibit constant returns to scale, up to the limited imposed by scarce land. The endowments of each type of land are specifed. Requirements for use are specified by final demand.

Table 1 presents coefficients of production for the example. Two commodities are produced, iron and corn. Aside from the use of land, joint production is not possible. Multiple types of land (that is, three types) exist. Only one agricultural commodity, corn, can be produced on the processes in which land is used. For one type of land, more than one process can be operated on land. Only one process is known for producing iron, the industrial commodity. Each column in Table specifies the person-years of labor, acres of a type of land, tons of iron, and bushels of corn needed to produce a unit output of the specified commodity.

Table 1: The Coefficients of Production

Input	Industry
	Iron	Corn
	I	II	III	IV	V
Labor	1	0.517	91/250	0.67	3/10
Type 1 Land	0	0.49	0	0	0
Type 2 Land	0	0	0.59	0	0
Type 3 Land	0	0	0	9/20	3
Iron	9/20	0.03744	0.0009	0.067	0.08
Corn	2	0.048	0.27	0.15	0.15

Various techniques (Table 2) can be defined with this technology. All twenty-four letters in the Greek alphabet are needed to specify the techniques. Not all techniques are feasible, given technology, endowments, and requirements for use. Land is not scarce for the Alpha, Beta, Gamma, and Delta techniques, and ownership of land obtains no rent. The Epsilon through Upsilon techniques are examples of extensive rent. One type of land obtains a rent in the Epsilon through Xi techniques. All three types are farmed in Omnicro through Upsilon, and two types obtain a rent. Phi is an example of intensive rent. Chi, Psi, and Omega are examples of the combination of intensive and extensive rent.

Table 2: Techniques of Production

Technique	Processes	Land
		Type 1	Type 2	Type 3
Alpha	I, II	Partially farmed	Fallow	Fallow
Beta	I, III	Fallow	Partially farmed	Fallow
Gamma	I, IV	Fallow	Fallow	Partially farmed
Delta	I, V	Fallow	Fallow	Partially farmed
Epsilon	I, II, III	Partially farmed	Fully Farmed	Fallow
Zeta	I, II, IV	Partially farmed	Fallow	Fully Farmed
Eta	I, II, V	Partially farmed	Fallow	Fully Farmed
Theta	I, II, III	Fully Farmed	Partially farmed	Fallow
Iota	I, III, IV	Fallow	Partially farmed	Fully Farmed
Kappa	I, III, V	Fallow	Partially farmed	Fully Farmed
Lambda	I, II, IV	Fully Farmed	Fallow	Partially farmed
Mu	I, III, IV	Fallow	Fully Farmed	Partially farmed
Nu	I, II, V	Fully Farmed	Fallow	Partially farmed
Xi	I, III, V	Fallow	Fully Farmed	Partially farmed
Omnicron	I, II, III, IV	Partially farmed	Fully Farmed	Fully Farmed
Pi	I, II, III, V	Partially farmed	Fully Farmed	Fully Farmed
Rho	I, II, III, IV	Fully Farmed	Partially farmed	Fully Farmed
Sigma	I, II, III, V	Fully Farmed	Partially farmed	Fully Farmed
Tau	I, II, III, IV	Fully Farmed	Fully Farmed	Partially farmed
Upsilon	I, II, III, V	Fully Farmed	Fully Farmed	Partially farmed
Phi	I, IV, V	Fallow	Fallow	Fully Farmed
Chi	I, III, IV, V	Fallow	Fully Farmed	Fully Farmed
Psi	I, II, IV, V	Fully Farmed	Fallow	Fully Farmed
Omega	I, II, III, IV, V	Fully Farmed	Fully Farmed	Fully Farmed

I assume that 100 acres of each of the three types of land are available. Net output consists of 66 tons iron and 88 bushels corn. This completes the specification of the example. The parameters for the example are fairly arbitrary. They are chosen to ensure a reswitching of the order of rentability for the Tau technique and to ensure that the Omega technique is feasible.

Under these assumptions, Omnicron, Rho, Tau, and Omega are feasible. All three types of land are farmed under these three techniques. Type 1 land is only partially farmed under Omnicron, and it is non-scarce and does not obtain a rent. Type 2 land does not obtain a rent under Rho. Type 3 land does not obtain a rent under Tau. All three types are fully farmed under Omega. A linear combination of processesare IV and V are operated side-by-side under Omega. Type 3 land is therefore scarce under Omega. All three types are farmed under Omnicron, with non-scarce Type 3 land only partially farmed.

3.0 Prices of Production

A system of equations specify prices of production for each technique. All operated processes pay the same rate of profits. Rents and wages are paid out of the surplus at the end of the year. A type of land that is only partially farmed is not scarce and pays no rent. I take the net output as the numeraire.

As an example, the system of equations in following five displays specify the prices of production for the Omega technique.

(p₁ a_1,1 + p₂ a_2,1)(1 + r)+ w a_0,1 = p₁

(p₁ a_1,2 + p₂ a_2,2)(1 + r) + rho₁ c_1,2 + w a_0,2 = p₂

(p₁ a_1,3 + p₂ a_2,3)(1 + r) + rho₂ c_2,3 + w a_0,3 = p₂

(p₁ a_1,4 + p₂ a_2,4)(1 + r) + rho₃ c_3,4 + w a_0,4 = p₂

(p₁ a_1,5 + p₂ a_2,5)(1 + r) + rho₃ c_3,5 + w a_0,5 = p₂

Prices of production for the other techniques are specified by a subset of the system of equations for the Omega technique. Each operated process corresponds to an equation in the corresponding system of prices of production. The rent on land that is partially farmed is zero in the corresponding equation, since land in excess supply is not scarce.

The numeraire is specified by a further equation, where the column vector d represents net output.

p₁ d₁ + p₂ d₂ = 1

3.1 On the Solution

A linear combination of the last two equations in the system of prices of production, for the Phi, Chi, Psi, and Omega techniques, eliminates the rent of type 3 land. In the techniques with extensive rent, one of the equations for a corn-producing process does not contain a term for rent either.

This equation for a corn-producing process or the linear combination of the last two equations can be combined with the first equation, for the iron-producing process. This results in a system of two equations in four unknowns, the price of iron, the price of corn, the wage, and the rate of profits. The equation for the numeriare removes one degree of freedom. If the rate of profits is taken as given, this is a linear system which can be solved for prices of produced commodities and the wage.

The rent per acre can be found for each equation remaining in the original system of equations for a technique. The Alpha, Epsilon, Zeta, Eta, Omnicron, and Pi techniques, for example, have the same solution for prices of produced commodities and the wage. Epsilon, Omnicron, Pi have the same rent per acre on type 2 land. Zeta and Omnicron have the same rent per acre on Type 3 land, while Eta and Pi have the same rent per acre on Type 3 land.

3.2 Wage and Rent Curves

Given the technique, the wage is therefore a function of the rate of profits. Likewise the rent on lands that are always fully-farmed with that technique is also a function of the rate of profits.

The wage is a declining function of the rate of profits in the first four techniques and in the 16 techniques with extensive rent alone. A maximum wage corresponds to a rate of profits, and a maximum rate of profits corresponds to a wage of zero. The wage curve can be upward-sloping in models of extensive rent. The wage curves, in the example, happen to be downward-sloping in the example. Figure 2 shows the wage curves for the feasible techniques in the example. The order of efficiency is the order in which techniques are adopted with increasing net output at a given wage or rate of profits. In models with extensive rent, the order of efficiency can be read off the order of wage curves.

Figure 2: Wage Curves for Feasible Techniques

Figure 3 shows the rent curves for the techniques with non-negative rents in the example. Figure 1, at the top of the post, is an enlargement. Rent curves do not need to have any particular slope. They can slope down or up and vary along their extent. The rent curves for Tau are an example of the reswitching of the order of rentability.

Figure 3: Rent Curves for Feasible Techniques

4.0 The Choice of Technique

Only two techniques, Tau and Omega, are feasible in the example and have non-negative rents for scarce lands. Table 3 lists approximate ranges of the rate of profits and which techniques are cost-minimizing in which ranges. The orders of efficiency and the order of rentability are also shown.

Table 3: Cost-Minimizing Technique

Range	Technique	Order of Efficiency	Order of Rentability
0 ≤ r ≤ 29.05 %	Omega	Type 2, 1, 3	Type 1, 2, 3
29.05 ≤ r ≤ 35.50 %	Tau
35.05 ≤ r ≤ 43.76 %			Type 2, 1, 3

Figure 4 justifies which technique is cost-minimizing in which range of the rate of profits. Capitalists can gain extra profits by adopting process V in the range in which Omega pays positive rents. Type 5 land becomes fully farmed by combining the two processes on Type 3 land, and the Omega technique results. For higher rates of profits, Tau is cost-minimizing, up to the maximum for Tau.

Figure 4: Extra Profits at Tau Prices

Intensive and extensive rents are both obtained by landlords when the Omega technique is cost-minimizing. Whenever the Omega technique is cost-minimizing, and in some range of the rate of profits in which Tau is cost-minimizing, the order of efficiency varies from the order of rentability. Type 2 land is more efficienct or more fertile than Type 1 land. Yet ownership of Type 1 land obtains more rent per acre than Type 2 land. Why would one ever expect competitive capitalist markets to reward efficiency?

5.0 Conclusion

This post presents the first concrete example of a case where a cost-minimizing technique combines intensive and extensive rent. It demonstrates that the concepts of the order of effiency and the order of rentability apply to models with intensive rent. As with models with only extensive rent, the order of effiency cannot be generally defined in terms of physical properties alone. And these orders can differ from one another at some given wage or rate of profits.

The example does not illustrate issues that can arise with intensive rent. Wage curves can slope up. The cost-minimizing technique can be non-unique away from switch points. No cost-minimizing technique may exist, even though feasible techniques exist at a given wage or rate of profits (D'Agata 1983).

The analysis can be extended to more kinds of rent and more complicated production models, while still not treating general joint production. Absolute rent, which may not make sense (Basu 2022) and external intensive rent (Kurz and Salvadori 1995) are examples. Rent might be analyzed in models with systematic, persistent variations in the rate of profits among industries (Vienneau 2024). Likewise, a more general model could have some types of lands that are inputs into processes that each produce a different agricultural commodity. Does it make sense to compare and contrast the order of efficiency and the order of rentability in these models?

References

• D’Agata, A. 1983. The existence and unicity of cost-minimizing systems in intensive rent theory, Metroeconomica 35: 147-158.
• Basu, Deepankar. 2022. A reformulated version of Marx's theory of ground rent shows that there cannot be any absolute rent. Review of Radical Political Economics 54(4): .
• Kurz, H. D. and Salvadori N. 1995. Theory of Production: A Long-Period Analysis, Cambridge: Cambridge University Press.
• Quadrio-Curzio, A. 1980. Rent, income distribution, and orders of efficiency and rentability, in Pasinetti, L. L. (ed.) Essays on the Theory of Joint Production, New York: Columbia University Press.
• Quadrio-Curzio, A. and F. Pellizzari. 2010. Rent, Resources, Technologies. Berlin: Springer. [I NEED TO READ THIS TO ENSURE THAT I AM ORIGINAL]
• Vienneau, R. L. 2022. Reswitching in a model of extensive rent. Bulletin of Political Economy 16(2): 133-146.
• Vienneau, R. L. 2024. Characteristics of labor markets varying with perturbations of relative markups. Review of Political Economy (36)2: 827-843.

A Non-Reswitching Theorem Inapplicable To Non-Competitive Markets?

Consider a circulating capital model of the production of commodities. A non-reswitching theorem exists:

Theorem: Suppose a commodity exists which is a basic commodity in all techniques. And a smooth, continuously differentiable production function exists for producing that commodity. Then the reswitching of techniques cannot arise.

Marglin (1984: 285-286) states a theorem like this in which continuously differentiable production functions exist for all commodities. He also states:

"Once again, a result I thought to be original turned out not to have been. Reviewing the literature for these notes, I found a proof of the impossibility of reswitching in a continuous-substitution framework in Burmeister and Dobell (1970)." – Marglin (1984: 542).

Marglin's proof is one by contradiction. Capital-reversing is still possible under the assumption of these theorems.

Pasinetti and Scazzieri (2008) find the theorem in Bruno, Burmeister, and Sheshinski (1966), which I do not recall. They, in turn, attribute the theorem to Martin Weitzman and Robert Solow. Pasinetti and Scazzieri doubt the validity of the theorem.

"It is worth noting that Weitzman–Solow's theorem is simply a consequence of the idea that, in the case of a commodity produced by a neoclassical production function, each set of input–output coefficients ought to be associated in equilibrium with a one-to-one correspondence between marginal productivity ratios and input price ratios. No ratio between marginal productivities would be associated with more than one set of input prices, and this is taken to exclude the possibility that the same technique be chosen at alternative rates of interest, and thus at different price systems. The Weitzman–Solow theorem is at the origin of a line of arguments that has been followed up by a number of other authors, such as David Starrett (1969) and Joseph Stiglitz (1973). These authors have pursued the idea that 'enough' substitutability, by ensuring the smoothness of the production function, is sufficient to exclude reswitching of technique. However, non-reswitching theorems of this type involve that, for each technique of production, the capital stock may be measured either in physical terms or at given prices. For in a model with heterogeneous capital goods, if we allow prices to vary when the rate of interest or the unit wage are changed, there is no reason why the same physical set of input–output coefficients might not be associated with different price systems: even in the case of a continuously differentiable production function, the marginal product of 'social' capital cannot be a purely real magnitude independent of prices. Once it is admitted that 'in general marginal products are in terms of net value at constant prices, and hence may well depend upon what those prices happen to be' (Bliss, 1975, p. 195), it is natural to allow for different marginal productivities of the same capital stock at different price systems. It would thus appear that reswitching of technique does not carry with it any logical contradiction even in the case of a smoothly differentiable production function." Pasinetti and Scazzieri (2008)

I do not know about that. But I have never been clear on how substitutability is supposed to justify marginalist theory.

Suppose rates of profits differ among industries. And that the ratios of rates of profits among industries are stable in the long run. I have shown that the arguments of the Cambridge capital controversy extend to such non-competitive markets. In my paper, I had a reswitching example, in a discrete technology, that did not arise in competitive markets.

I conjecture that the non-reswitching theorem for continuous-substitution does not apply to non-competitive markets.

References

• Bruno, M., Burmeister, E. and Sheshinski, E. 1966. The nature and implications of the reswitching of techniques. Quarterly Journal of Economics 80, 526–53.
• Burmeister, Edwin and A. Rodney Dobell. 1970. Mathematical Theories of Economic Growth. New York: Macmillan.
• Marglin, S. A. 1984. Growth, Distribution, and Prices. Harvard University Press.
• Pasinetti, L. L. and Roberto Scazzieri, R. 2008. Capital theory (paradoxes). The New Palgrave.
• Starrett, D. 1969. Switching and reswitching in a general production model. Quarterly Journal of Economics 83, 673–87.
• Stiglitz, J. 1973. The badly behaved economy with the well-behaved production function. In Models of Economic Growth, ed. J.A. Mirrlees and N.H. Stern. London: Macmillan.

An Example With Intensive And Extensive Rent

Figure 1: Detail on Variation of Rent per Acre with Rate of Profits

1.0 Introduction

This post is the start of an attempt to develop an interesting example with both intensive and extensive rent. A feasible technique exists in the example with both intensive and extensive rent. Yet, it is never cost-minimizing. So this example does not do what I want. I have previously thought about other examples.

The example is an extension of my example of the reswitching of the order of rentability. Such reswitching occurs in this example. But the first switch point of the order of rentability is off the frontier.

I think some perturbation of this example will get me an example where a technique with both intensive and extensive rent is cost-minimizing for some range of the rate of profits. That example will illustrate that the orders of efficiency and of rentability can be analyzed in the context of intensive rent. And these orders need not co-incide in the case of intensive rent too.

2.0 Technology, Resources, Final Demand, and Feasibility

Table 1 presents coefficients of production for the example. Two commodities are produced, iron and corn. Aside from the use of land, joint production is not possible. Multiple types of land (that is, three types) exist. Only one agricultural commodity, corn, can be produced on the processes in which land is used. For one type of land, more than one process can be operated on land. Only one process is known for producing iron, the industrial commodity. Each column in Table specifies the person-years of labor, acres of a type of land, tons of iron, and bushels of corn needed to produce a unit output of the specified commodity.

Table 1: The Coefficients of Production

Input	Industry
	Iron	Corn
	I	II	III	IV	V
Labor	1	0.517	91/250	0.67	3/10
Type I Land	0	0.49	0	0	0
Type II Land	0	0	0.59	0	0
Type II Land	0	0	0	9/20	3
Iron	9/20	0.03744	0.0009	0.067	0.08
Corn	2	0.048	0.27	0.15	0.15

I can define various techniques (Table 2) with this technology. I need all twenty-four letters in the Greek alphabet to specify the techniques. Not all techniques are feasible, given technology, endowments, and requirements for use. Land is not scarce for the Alpha, Beta, Gamma, and Delta techniques, and ownership of land obtains no rent. The Epsilon through Upsilon techniques are examples of extensive rent. One type of land obtains a rent in the Epsilon through Xi techniques. All three types are farmed in Omnicro through Upsilon, and two types obtain a rent. Phi is an example of intensive rent. Chi, Psi, and Omega are examples of the combination of intensive and extensive rent.

Table 2: Techniques of Production

Technique	Processes	Land
		Type 1	Type 2	Type 3
Alpha	I, II	Partially farmed	Fallow	Fallow
Beta	I, III	Fallow	Partially farmed	Fallow
Gamma	I, IV	Fallow	Fallow	Partially farmed
Delta	I, V	Fallow	Fallow	Partially farmed
Epsilon	I, II, III	Partially farmed	Fully Farmed	Fallow
Zeta	I, II, IV	Partially farmed	Fallow	Fully Farmed
Eta	I, II, V	Partially farmed	Fallow	Fully Farmed
Theta	I, II, III	Fully Farmed	Partially farmed	Fallow
Iota	I, III, IV	Fallow	Partially farmed	Fully Farmed
Kappa	I, III, V	Fallow	Partially farmed	Fully Farmed
Lambda	I, II, IV	Fully Farmed	Fallow	Partially farmed
Mu	I, III, IV	Fallow	Fully Farmed	Partially farmed
Nu	I, II, V	Fully Farmed	Fallow	Partially farmed
Xi	I, III, V	Fallow	Fully Farmed	Partially farmed
Omnicron	I, II, III, IV	Partially farmed	Fully Farmed	Fully Farmed
Pi	I, II, III, V	Partially farmed	Fully Farmed	Fully Farmed
Rho	I, II, III, IV	Fully Farmed	Partially farmed	Fully Farmed
Sigma	I, II, III, V	Fully Farmed	Partially farmed	Fully Farmed
Tau	I, II, III, IV	Fully Farmed	Fully Farmed	Partially farmed
Upsilon	I, II, III, V	Fully Farmed	Fully Farmed	Partially farmed
Phi	I, IV, V	Fallow	Fallow	Fully Farmed
Chi	I, III, IV, V	Fallow	Fully Farmed	Fully Farmed
Psi	I, II, IV, V	Fully Farmed	Fallow	Fully Farmed
Omega	I, II, III, IV, V	Fully Farmed	Fully Farmed	Fully Farmed

I assume that 100 acres of each of the three types of land are available. Net output consists of 60 tons iron and 80 bushels corn. This completes the specification of the example.

Under these assumptions, Zeta, Lambda, Omnicron, Pi, Sigma, Tau, Upsilon, and Psi are feasible. Types 1 and 3 land are farmed under Zeta, with process IV being operated on Type 3 land. Which of Types 1 and 3 obtain a rent depends on which land is fully farmed. I should say more here.

3.0 Prices of Production

A system of equations specify prices of production for each technique. All operated processes pay the same rate of profits. Rents and wages are paid out of the surplus at the end of the year. A type of land that is only partially farmed is not scarce and pays no rent. I take the net output as the numeraire.

One degree of freedom exists for the system of equations for each technique. Figure 2, below, shows how the wage varies with the rate of profits for each technique. Figure 3 shows the variation in rent per acre with the rate of profits. Figure 1, at the top of this post, is a detail for an interesting part of Figure 3.

Figure 2: Wage Curves for Feasible Techniques

Figure 3: Rent Curves for Feasible Techniques

4.0 Choice of Technique

A technique is not cost-minimizing if it requires a negative rent to be paid. Rent is negative, under Sigma, for both Type 1 and Type 3 lands. Under Zeta, Omnicron, and Pi, rent on Type 3 land is negtive.

This leaves Lambda, Tau, Upsilon, and Psi as feasible techniques that pay positive rents on scarce lands in some range of the rate of profits. Table 1 lists the cost-minimizing techniques, Upsilon and Tau, in order of an increasing rate of profits.

Table 3: Cost-Minimizing Technique

Range	Technique	Order of Efficiency	Order of Rentability
0 ≤ r ≤ 28.49 %	Upsilon	Type 2, 1, 3	Type 2, 1, 3
28.49 ≤ r ≤ 29.05 %			Type 1, 2, 3
29.05 ≤ r ≤ 35.50 %	Tau
35.05 ≤ r ≤ 43.76 %			Type 2, 1, 3

The order of efficiency, at a given rate of profits, is the order in which different types of land would be brought under cultivation as final demand was increased. This order can be read off of Figure 2 by working downward over the wage curves. Since the wage curves for Sigma and for Zeta, Omnicron, and Pi do not intersect, the order of efficiency does not vary, with the rate of profits, in this example. Type 2 land is partially farmed under Sigma. So Type 2 land is first in the order of efficiency. Type 1 land is partially farmed in Zeta, Omnicron, and Pi. Hence, Type 1 land is next in the order of efficiency for techniques in which all three lands are farmed.

The order of rentability is read off of Figures 1 and 3. The order in which rent per acre decreases varies with the rate of profits. For order of rentability differs from the order of efficiency for rates of profits around the switch point between Upsilon and Tau. A change in the order of rentability occurs around the second intersection between the two rent curves for Tau. This effect is a manifestation of the reswitching of the order of rentability. But the order of rentability varies around any intersection of these curves.

It remains to demonstrate that the above claims about which is the cost-minimizing technique at each rate of profits. Figure 4 shows that Lambda is never cost minimizing. Extra profits can always be obtained at Lambda prices by farming Type 2 land with process III. At low rates of profits, extra profits can also be obtained by farming Type 3 land with the other corn-producing process available for that type of land.

Figure 4: Extra Profits at Lambda Prices

Figures 5 and 6 show that Upsilon is cost-minimizing below the switch point, and that Tau is cost-minimizing at higher rates of profits. In these ranges, extra profits are not available by operating the process not in the technique.

Figure 5: Extra Profits at Upsilon Prices

Figure 6: Extra Profits at Tau Prices

Both intensive and extensive rent are paid when Psi is adopted. Figure 7 demonstrates that Psi is never cost-minimizing. Extra profits are always available, whatever the rate of profits.

Figure 7: Extra Profits at Psi Prices

5.0 Conclusion

This post has illustrated the analysis of the choice of technique in an example with both intensive and extensive rent. Constructing the wage curve is not necessarily the correct method of analysis in models with general joint production. Looking at whether or not extra profits are available for the prices associated with a technique is always applicable.

Elsewhere

• I find Matt McManus has argued in Damage magazine for liberal socialism.
• The liberal socialism developed by Nello and Carlo Rosselli was sort of endorsed by Mussolini. He had them murdered.
• When it comes to Italian exiles from fascism, I could learn more about Pietro Nenni.
• This Ha-Joon Chan article in the Financial Times is behind a paywall and is generating a buzz. "Economics teaching has become the Aeroflot of ideas". But it is available here.

Revolutionary And Reformist Socialists Splitting Around 1900

The second international was the major organization for socialist parties around 1900. The first international had collapsed with struggles for leadership between Marx and anarchists. The German Social Democratic Party, headed by Karl Kautsky, seemed to be the most successful socialist party in the second international. After Engels, Kautsky became the literary executor for Marx. He edited and put out volume 4 of Capital, that is, Theories of Surplus Value.

Eduard Bernstein was Engels' literary executor and therefore a prominent member of the German Social Democratic Party. He had been the editor of Der Sozialdemokrat, the party's newspaper. Perhaps Bernstein was influenced by his acquaintance with members of the Fabian society when he was in exile in London. He looked at the growing wealth of the German workers; the apparent strength of working-class organizations, such as unions; and the SDP representation in the Reichstag. Economic development was not concentrating wealth in a smaller and smaller capitalist class. These trends did not seem to him consistent with the revolution that Marx foresaw. And he said so.

The Preconditions of Socialism and the Tasks of Social Democracy (Die Voraussetzungen des Sozialismus) is the major statement of Bernstein's views, and was published in 1899. It started as articles in Die Neue Ziet, the SDP paper for more theoretical work. Bernstein was called a revisionist, which has ever after been a pejorative among more radical socialists. His theses were argued about at the SDPss Stuttgart Conference, held in October 1898.

Bernstein argued for legislation and peaceful reform in favor of the workers. Socialists should be a parliamentary party. They should agitate for universal suffrage. They should leave businesses, for the most part, in private hands. Socialists should support the development of civil society.

Rosa Luxemburg saw an opportunity to raise her stature in the German SDP. She had already participated in the Stuttgart conference. Others, including Kautsky, also argued against Bernstein. Luxemburg's Reform or Revolution is a classic statement of the radical, anti-revisionist view. She argued against idealism, against petty bourgeois moralism, and against opportunism. Idealism, in this sense, means basing political views purely on intellectual arguments. Emphasizing universal citizenship loses a working-class standpoint. According to Luxemburg, capitalism will inevitably break down. Socialism is a scientific standpoint, given its historical necessity. I do not know that this is in this pamphlet, but Luxemburg famously said that our choice is socialism or barbarianism.

This controversy was echoed in other countries. In France, Jean Jaurès led the reformists. I think of Georges Sorel as an intellectual leader of the revolutionaries. His 1908 book, Reflections On Violence does not strike me as particularly Marxist. You maybe should read 'violence' in the title as what is today called direct action. Sorel, at the time, was an advocate of syndicalism. He was kind of mystical in his emphasis on non-rational motives for mass movements. Hence, his myth of the general strike.

In Italy, Filippo Turati, a founder of the Italian socialist party (PSI), was a reformist. The radicals were called maximalists. They seem to me more positivist than Marx would ever be. They saw the revolution as inevitable, not something to be brought about by political action in the here-and-now. Socialists should organize and educate, holding themselves back until the revolution comes.

In Russia, the split was between the Mensheviks and the Bolsheviks. Apparently, these names mean 'the minority' and 'the majority'. Bolsheviks were only the majority because some supporters of the Mensheviks had walked out of the Second Congress of the RSDLP (Russian Social Democratic Labor Party), at which the split occurred. Julius Martov was an important leader of the moderates, while Lenin headed the Bolsheviks. Lenin's pamphlets, What is to be done? (1902) and Two steps forward, one step back (1904) are essential primary sources here.

Lenin argued for agitation on all fronts, not just for economic improvements for the workers. He wanted an organization of professional revolutionaries, and developed the idea of democratic centralism. The Bolsheviks should have the freest discussion in deciding on policy and tactics. But once a vote has decided the issue, the comrades follow the party line. The establishment of an all-Russian newspaper, Iskra, is the immediate implementation called for in What is to be done?

I ought to say something about Austria.

I have said nothing about Sweden, Denmark, Norway, or Finland. Bernstein provided the intellectual structure for what became democratic socialism and social democracy. I do not know historical details about Scandinavia. He said, "The final goal, no matter what it is, is nothing; the movement is everything."

Some Results From My Research

I have written about my research program before. Here I write about results.

1. A numerical example of a new capital-theoretic paradox, the reswitching of the order of rentabily.
2. A numeric example of, perhaps, another new capital-theoretic paradox, the recurrence of truncation without reswitching or capital-reversing. (I have not yet had this published.)
3. A demonstration that the critique developed in the Cambridge capital controversy extends to non-competitive markets.
4. A diagram illustrating how the analysis of the choice of technique varies with the perturbation of a parameter in models of the production of commodities.
5. A claim that certain capital-theoretic paradoxes are (usually?) only transient in the very long-run, secular time.
6. The discovery and presentation of certain generic structures in parameter spaces, associated with qualitative change in the analysis of the choice of technique. This is the broadest claim in my series of papers.

I do not imagine that Christian Bidard, Heinz Kurz, Neri Salvadori, Bertram Schefold, or Ian Steedman, for example, would be surprised by the first two or three bullet points. I concentrate on producing concrete numerical illustrations. I do not expect ever to state the fifth and sixth point with full mathematical rigor. I know the fifth point cannot be completely general.

Richest Man In The World Kills Hundreds Of Thousands, Will Kill Millions

This post is current events.

I refer to the illegal and unconstitutional elimination of United States Agency for International Development (USAID). This agency was eliminated by the unelected, nazi-saluting Elon Musk, working for the traitor Donald Trump.

Some documentation:

• Cavalcanti et al., 30 June 2025. Evaluating the impact of two decades of USAID interventions and projecting the effects of defunding on mortality up to 2030. The Lancelet.
• Malone. 31 May 2025. Bono criticizes USAid cuts in lengthy interview with podcaster Joe Rogan. Irish Times.
• Ensor. 30 May 2025. DOGE cuts to USAID blamed for 300,000 deaths ' most of them children. The Times.
• Donati et al. 16 May 2025. US aid cuts leave food for millions mouldering in storage. Reuters.
• Belz. 8 May 2025. Aid cuts could disrupt historic drops in child and maternal mortality. Christianity Today.
• Colarossi. 30 March 2025. "It's unacceptable": BU mathematician tracks how many deaths may result from USAID, Medicaid cuts. The Brink.

Still More On Recurrence Of Truncation Without Reswitching

Figure 1: An Example of Structural Economic Dynamics

This post presents a diagram illustrating the effects of a particular kind of technical progress in a model with fixed capital. The example is one with two industries. Machines are produced in the machine industry and used in both the machine and corn industry. All production processes take a year to complete. Machines have a physical lifetime of two years. Managers of firms have a choice in each industry. The economic life of a machine can be one or two years in either industry.

This previous post specifies the technology at time zero. Coefficients of production for inputs and outputs are shown in Tables 1 and 2 in that post. Table 3 defines the four techniques, Alpha, Beta, Gamma, and Delta. Table 1 repeats that specification in a somewhat different format.

Table 1: Specification of Techniques

Technique	Economic Lifetime of Machines
	Machine Industry	Corn Industry
Alpha	One Year	One Year
Beta	Two Years	One Year
Gamma	One Year	Two Years
Delta	Two Years	Two Years

I consider perturbations of coefficients of production that define the amount of circulating capital needed to operate new and old machines, both in producing more new machines and in producing corn. This previous post defines the regions illustrated in the diagram at the top of this post. I repeat the definitions of the regions in Table 2 here. The cost-minimizing techniques, in each region, are listed in the order of an increasing rate of profits or decreasing wage.

Table 2: Overview of Regions

Region	Techniques	Notes
1	Alpha	No switch point.
2	Alpha, Gamma	Lower rate of profits associated with truncation in corn industry, greater output per worker.
3	Alpha, Gamma, Delta	Lower rate of profits associated with truncation, greater output per worker.
4	Alpha, Gamma, Delta, Beta	Recurrence of truncation in corn industry.
5	Alpha, Beta	Lower rate of profits associated with truncation in machine industry, greater output per worker.

I also consider the perturbations of relative markups, yielding a figure much like that at the top of this post. I conclude that both technical progress and changes in market power can have similar effects, in the large. In this example, both can bring about or eliminate the recurrence of the truncation of the economic life of the machine in one industry. This post further illustrates this claim about technical progress.

I consider further perturbations of coefficients of production in this post and this post. Although four coefficients of production decline with technical progress in Figure 1, I have not managed to specify a path through the additional regions in parameter space found in these later posts.

Sewer Socialism Worked In Milwaukee, Wisconsin, In The First Half Of The 20th Century

America's Socialist Experiment

Socialists have been elected mayor, to city councils, and to county commissions in the USA. Emil Seidel was the first socialist mayor in the USA. He was elected mayor of Milwaukee in 1910. Other socialists were elected to the city council. Another socialist, Victor Berger, was elected to the House of Representatives, to represent Milwaukee in Washington, DC.

By concentrating on Milwaukee, I am downplaying the extent to which socialists were elected mayor in cities in the USA. Wikipedia has a list, and so does the University of Washington. More than 130 mayors were socialists. Maybe I want to read David R. Berman's 2022 book, Socialist Mayors in the United States: Governing in an Era of Municipal Reform, 1900-1920.

Wisconsin is a democracy. Up until 1960, the mayor was mostly a socialist. Daniel Hoan, the second socialist mayor, was elected in 1916. Frank Zeider, the third socialist mayor, was elected in 1948.

The socialists focused on cleaning up Milwaukee, both figuratively and literally. For the former, they fought corruption and graft. They obtained their name, Sewer Socialists, from the latter. Introducing sanitation, electric, and power systems was an importance advance in their day.

The name, I guess, was coined by Morris Hillquit, a prominent member of the Socialist Party of America (1901 – 1972). He was tired of the boasting from Milwaukee socialists. But they had good cause to boast.

As I understand it, German immigrants, who fled to the US after participating in the failed 1848 revolutions, were an important current for the development of the sewer socialists. Nicholas Howland, in the link, makes an explicit link to Eduard Bernstein's reformism. Bernie Sanders, the former mayor of Burlington, Vermont, can be said to be in this tradition. Ruth Messinger, a former city council member in New York City, borough president of Manhattan, and co-chair of the Democratic Socialists of America is maybe another example.

Extensive Rent And Labor Values

1.0 Introduction

Do scarce natural resources provide additional difficultes for modern reconstructions of classical and Marxian theories of value? After all land can be sold or rented, and labor cannot produce more land. (I put aside Holland.)

This post presents an exposition of the theory of extensive rent, a start on examining possible difficulties. This type of rent provides the least dificulties, as I understand it, for such modern reconstructions. As usual, I present an example, close to the minimal complexity, needed to make my points. The model can obviously be generalized to include many more produced industrial commodities; many more types of agricultural commodities; and many more types of land, each specialized to support the production of one kind of agricultural commodity.

2.0 Technology

Table 1 specifies the technology for this example. Each column defines the coefficients of production for a process. For example, the only iron-producing process requires a_0,1 person-years of labor, a_1,1 tons of iron, and a_2,1 bushels of corn as inputs for every ton iron produced. I assume that each process requires a year to complete and exhibits constant returns to scale. The corn-producing processes each have an upper limit on how much corn they can produce.

Table 1: A Technology

	Iron Industry	Corn Industry
	Process a	Process b	Process c
Labor	a0,1	a0,2	a0,3
Land, Type 1	c1,1 = 0	c1,2 > 0	c1,3 = 0
Land, Type 2	c2,1 = 0	c2,2 = 0	c2,3 > 0
Iron	a1,1	a1,2	a1,3
Corn	a2,1	a2,2	a2,3
OUTPUTS	1 ton iron	1 bushel corn	1 bushel corn

I assume two types of land exist, distinguished by the processes that can be operated on them. A single corn-producing process can be operated on each type of land. Only a certain number of acres of each type of land exists. Each corn-producing process leaves the land unchanged at the end of operating the process. The given quantities of land limit how much corn can be produced. This model cannot accomodate a positive steady-state rate of growth without technical progress.

A full specification for this model should include requirements for use. I assume that the net output must be such that both types of land are farmed, but only one type is fully farmed. Two techniques for production exist, as shown in Table 2. All three processes are operated in each technique, but only one type of land is fully used.

Table 2: Specification of Techniques

Technique	Type 1 Land	Type 2 Land
Alpha	Partially farmed	Fully farmed
Beta	Fully farmed	Partially farmed

3.0 Parameters and Variables

I have already implicitly defined certain parameters above. Table 3 lists certain parameters I use in this model. Table 4 lists variables that I need. Some assumptions are imposed on the matrices A_α and A_β:

• All produced commodities are basic. Iron and corn enter directly or indirectly into the production of both commodities.
• The technology expressed by these matrices is productive. Each matrix satisfies the Hawkins-Simon condition.

Table 3: Selected Parameters

Symbol	Definition
a0, α	Two-element row vector consisting of first two labor coefficients.
a0, β	Two-element row vector consisting of first and third labor coefficients.
Aα	2x2 matrix, with columns consisting of iron and corn coefficients of production for first and second processes.
Aβ	2x2 matrix, with columns consisting of iron and corn coefficients of production for first and third processes.
d	Two-element column vector consisting of iron and corn quantities in the numeraire.

Table 4: Variables

Symbol	Definition
vα	2-element row vector of labor values when type 1 land is free.
vβ	2-element row vector of labor values when type 2 land is free.
p	2-element row vector of prices of unit quantities of iron and corn.
p1	The price of iron, in numeraire units per ton. The first element of p.
p2	The price of corn, in numeraire units per bushel. The second element of p.
rho1	The rent of type 1 land, in numeraire units per acre.
rho2	The rent of type 2 land, in numeraire units per acre.
w	The wage, in numeraire units per person-year.
r	The rate of profits.

4.0 Labor Values

Given the technique in use, how much additional labor would be employed throughout the economy if the net output was such that one additional unit of iron were produced? This is the labor value of iron, and it easily calculated in the theory. The answer to the same question for corn is its labor value.

Suppose type 1 land is free. Then labor values are:

v_α = a_{0, α} (I - A_α)^-1

Labor values, when type 2 land is free, are the corresponding Leontief employment multipliers for the Beta technique. Variations in net output require varying the amount of the land farmed on the type of land that is not fully farmed.

5.0 Prices of Production

With market prices, some operated processes will be obtaining a higher rate of profits than average, and some will be obtaining a lower rate. These variations in the profit rates are perhaps a signal to capitalists that they should disinvest in some industries or processes and increase investment in others. Models of cross-dual dynamics and other models explore these disequilibria.

Prices of production are such that these signals are absent. All operated processes obtain the same rate of profits. I assume profits, rents, and wages are paid out of the surplus product at the end of the year. The following three equations express the condition that all processes obtain the same rate of profits:

(p₁ a_1,1 + p₂ a_2,1)(1 + r) + w a_0,1 = p₁

(p₁ a_1,2 + p₂ a_2,2)(1 + r) + rho₁ c_1,2 + w a_0,2 = p₂

(p₁ a_1,3 + p₂ a_2,3)(1 + r) + rho₂ c_2,3 + w a_0,3 = p₂

The next equation expresses the condition that the price of the numeraire is unity:

p d = 1

Finally, one of the rents must be zero:

rho₁ rho₂ = 0

The last equation is a defining feature of the theory of extensive rent.

Suppose one of the types of land is rent-free. For deiniteness, let type 1 land be only partially farmed. Then the first four equations are in terms of five variables (p₁, p₂, rho₂, w, r). Just as in the case with only circulating capital, prices of production are specified up to one degree of freedom. In classical political economy, the wage is take as given.

6.0 Choice of Technique

Suppose the wage is non-negative and does not exceed a maximum defined by the technology. The system of equations for prices of production has two solutions. Each solution has the rent on one type of land set to zero. The cost-minimizing technique is the one in which the rent on the other land is positive. If, for a technique, the rent on a type of land is negative, that technique will not be adopted by capitalists. At a switch point, the rents on both types of land are zero.

But the analysis of the choice of technique can be expressed in terms of wage curves. Suppose rents were zero. Consider the first two equations in the system of equations for the prices of production and the equation setting the price of the numeraire to unity. These equations yield a function in which the wage decreases with an increase in the rate of profits. Similarly, the first and third equations yield another decreasing wage curve.

In the case of circulating capital alone, the cost-minimizing technique is found by the wage frontier formed out of the outer envelope of these wage curves. At a given wage, the cost-minimizing technique maximizes the wage.

In this example of extensive rent, the cost-minimizing technique is found by the wage frontier formed out of the inner envelope of the wage curves.

In either case, the appropriate wage frontier shows that a lower rate of profits is associated with a higher wage and vice versa. The maximum wage occurs when the rate of profits is zero. The maximum rate of profits arises when the wage is zero.

7.0 Special Cases

Which land is free and which land pays a rent depends on either the wage or the rate of profits, whichever is taken as exogenous in the system of prices of production. At any rate, a wage frontier exists in which the wage is higher the smaller the rate of profits. This frontier is not the outer frontier of the wage curves for the technique.

Without loss of generality, suppose the Alpha technique is cost-minimizing. Type 1 land is not fully farmed and pays no rent. Then labor values are defined, based on the iron-producing process and the process on type 1 land.

Consider the special case in which a_{0, α} is an eigenvector corresponding to the maximum eigenvector for A_α. Then relative prices of production are equal to relative labor values.

On the other hand, suppose that the numeraire is the standard commodity, as found from a_{0, α} and A_α. Suppose only the standard commodity is produced. In this case, only the process on the rent-free land would be used, in contradiction to the analysis of the choice of technique. And suppose the wage is paid out in the form of the standard commodity. Then the following hold:

• The labor value of gross output is equal to total gross output, evaluated at prices of production.
• The labor value of net output is equal to net output, evaluated at prices of production.
• The labor value of the proportion of the standard commodity paid out in wages is equal to wage goods, evaluated at prices of production.

This special case seems especially forced in the case of extensive rent. Is some reformulation available in which surplus value can be treated as the sum of profits and rent?

I do not address the use of labor values in Marx's account of exploitation, Marx-biased technical change, and so on. The special cases in which the labor theory of value hold make obvious that, for a given technology, a higher rate of profits require a lower wage. And this wage frontier continues to hold in models of extensive rent.

8.0 Conclusion

The inclusion of natural resources, insofar as they can be modeled by extensive rent, does not seem to pose any additional issues for modern formulations of classical and Marxian political economy. It does highlight some issues that arise in models with circulating capital.

Labor values can be calculated for all produced commodities, given the technique in use. They are calculated from the marginal land that receives no rent. But suppose that a choice of technique exists. Then, an analysis at the level of prices of production must be prior to the calculation of labor values. The theory of extensive rent highlights this issue.

As Ricardo and Marx noted, prices of production are generally not proportional to labor values. They are equal in the special case, in which all industries have equal organic compositions of capital, in both models of circulating capital and of extensive rent. In the latter case, the organic composition of capital is found for agriculture from no-rent lands partially farmed.

A commodity of average organic composition is picked out in both models. Total labor values and the labor value of wages are equal to the corresponding aggregates in the system of prices of production when this average commodity is used as numeraire and is produced. These invariants, though, have to restricted to the production of the numeraire with the iron-producing process and the process on no-rent land. It is not clear to me that Marx thought his invariants held in his chapters on rent, given their location towards the end of volume 3 of Capital.

Obviously, these observations on natural resources and rent are just a start. They do seem to match what Ricardo was about in the second chapter of his Principles. The analysis of the choice of technique can be thought of, somewhat, as a critique of Ricardo.

At any rate, prices of production are well-defined in models of extensive rent. And they can be used in an analysis of the choice of technique. As usual, I present the analysis with no mention of utility maximization, preferences, or tastes.

Elsewhere

• Branko Milanovic reviews Nat Dyer's Ricardo's Dream.
• Ben Burgis explains Why Marx is back in fashion.
• Symposium on Michael Heinrich's work.
• Doug Henwood interviews John Cassidy on his new book, Capitalism and Its Critics: A Battle of Ideas in the Modern World. (Apparently there are overviews of critics I have never heard of.)
• James Surowiecki Interviews John Cassidy, also on his new book.
• Nathan Robinson interviews Richard Wolff.

Even More On Recurrence Of Truncation Without Reswitching

Figure 1: Partitions in a Parameter Space with Small a1, 1, Small a1, 3

1.0 Introduction

I have been exploring patitions of parameter spaces by fluke switch points. I always have trouble visualizing higher dimensional spaces. Sometimes, my examples are two-dimensional. If I were doing more formal mathematics, instead of numerical exploration, I would not need to care.

This post is a continuation of the example developed here, here, and here. I present two-dimensional slices of a four-dimensional space.

2.0 Perturbations of Coefficients of Production

As the result of technical change, coefficients of production vary. No variation in labor coefficients or of the output matrix are considered here so as to retain reverse labor substitution at the switch point between Beta and Delta and forward substitution of labor at the switch points between Alpha and Gamma and between Gamma and Delta. Accordingly, consider perturbations of the coefficients of production in the first row of the input matrix. These parameters define a four-dimensional space. How does the analysis of the choice of technique vary with a decrease in these coefficients? The recurrence of truncation turns out be only a transient possibility in secular time with this specific model of technical change. The reverse substitution of labor can occur without the recurrence of truncation, but is also transient. Capital reversing, also transient, occurs in one region of parameter space.

2.1 Small Amounts of Circulating Capital Needed for New Machines

The parameter space is partitioned by parameters corresponding to fluke switch points. Figure 1 shows partitions with the values of a_1,1 and a_1,3 as in this previous post. A switch point is a fluke if it is a knife edge case in which almost all perturbations of model parameters destroy its defining properties. Four of the five partitions shown correspond to a switch point on the axis for the rate of profits. With four techniques, switch points between six pairs are possible. A switch point between Alpha and Delta can only occur as an intersection of all four curves. The same goes for a switch point between Beta and Gamma. These two pairs of techniques correspond to the partition for the switch point at which all four wage curves intersect, called here a four-technique pattern.

The sequence of techniques along the wage frontier is invariant in each numbered region. Table 1 lists the cost-minimizing techniques in each region, in order of an increasing rate of profits. Consider perturbations of the coefficients of production, a_1,2 and a_1,4, that specify the inputs of corn needed for each process in which the machine is run for the second year of its life. In region 1, to the northeast, old machines require so much circulating capital to operate that it is never cost-minimizing to run them for a second year. The Alpha technique is cost-minimizing, whatever the distribution of income. An improvement in the efficiency of old machines in using circulating capital in the machine industry leads to machines being operated for two years in that industry at large rates of profits, as in region 5. A similar improvement in the efficiency of old machines in the corn industry leads to machines being operated for two years in the corn industry, as in region 2. The recurrence of truncation only occurs in region 4.

Table 1: Overview of Regions

Region	Techniques	Notes
1	Alpha	No switch point.
2	Alpha, Gamma	Lower rate of profits associated with truncation in corn industry, greater output per worker.
3	Alpha, Gamma, Delta	Lower rate of profits associated with truncation, greater output per worker.
4	Alpha, Gamma, Delta, Beta	Recurrence of truncation in corn industry.
5	Alpha, Beta	Lower rate of profits associated with truncation in machine industry, greater output per worker.
6	Gamma, Delta	Lower rate of profits associated with truncation in machine industry, greater output per worker.
7	Gamma	No switch point.
8	Beta	No switch point.
9	Delta	No switch point.
10	Delta, Beta	Lower rate of profits associated with extension of economic life in corn industry, reverse substitution of labor.
11	Gamma, Delta, Beta	Lower rate of profits associated with truncation in machine industry, extension of economic life in corn industry, reverse substitution of labor.
12	Alpha, Beta, Delta	Around the Beta vs. Delta switch point, lower rate of profits associated with truncation, smaller output per worker.
13	Alpha, Beta, Delta	Lower rate of profits associated with truncation, greater output per worker.

2.2 Large Amounts of Circulating Capital Needed for New Machines in the Corn Industry

Figure 2 shows the partitions in the parameter space at a higher level of a_1,3. The structure in Figure 1 has moved upwards and to the left. The wedge in which the recurrence of truncation appears, region 4, has widened a bit. A new partition has appeared, for a fluke switch point between Gamma and Delta on the wage axis. For parameters where the partitions between regions 2 and 3 and between regions 2 and 7 intersect, the wage frontier has fluke switch points on both the wage axis and the axis for the rate of profits. For a low enough value of a_1,4, it is no longer cost-minimizing to truncate the machine in both industries at a low rate of profits. Gamma is cost-minimizing, whatever the distribution of income, for a high enough value of a_1,2 and a low enough value of a_1,4.

Figure 2: Parameter Space with Small a1, 1, Large a1, 3

2.3 Large Amounts of Circulating Capital Needed for New Machines in the Machine Industry

Figure 3 shows the partitions in the parameter space at a higher level of a_1,1. The amount of corn needed to operate a new machine in the corn industry, a_1,3, is the same as in Figure 1. Only the partitions and regions to the right are labeled in the figure. This part of the parameter space resembles those slices examined above, with a couple of variations. To the right, the lower boundary of region 5 corresponds to a switch point on the axis for the rate of profits, not a fluke switch point in which all four wage curves intersect. The upper boundary of region 3 now is, to the right, such a four-technique pattern of switch points.

Figure 3: Parameter Space with Large a1, 1, Small a1, 3

Figure 4 is an enlargement of the lower left of this slice of the parameter space. Three new points that are a fluke in multiple ways have appeared. One point corresponds to the intersection of the four wage curves on the wage axis. This point is the intersection of five partitions in the parameter space, just as the point in parameter space corresponding to the intersection of the four wage curves on the axis for the rate of profits. The region in which the recurrence of truncation occurs, region 4, has appeared. The reverse substitution of labor also occurs in regions 10 and 11. Two points in the part of the parameter space depicted correspond to fluke switch points simultaneously lying on both axes.

Figure 4: Parameter Space with Large a1, 1, Small a1, 3 (Enlarged)

Figure 5 enlarges this slice of the parameter space in the middle of Figure 3. The sequence of cost-minimizing techniques, in order of an increasing rate of profits, is Alpha, Beta, and Delta in both region 12 and region 13. The boundary between regions 12 and 13 corresponds to an intersection of the Beta and Delta wage curves on the waxis but not on the frontier. In region 12, the wage curves for Beta and Delta intersect twice in the first quadrant. The second intersection is on the frontier. It is a switch point exhibiting capital-reversing. Around the switch point between Beta and Delta a lower rate of profits is associated with a decreased capital-intensity and decreased net output per worker in the economy as a whole.

Figure 5: Parameter Space with Large a1, 1, Small a1, 3 (Another Enlargement)

3.0 Conclusion

The exploration of partitions in parameter space supports an analysis of technical change in which the circulating capital needed to operate a machine decreases. A trajectory from the upper right to the lower left in the figures and from Figures 2 or 3 to Figure 1 represent this specific form of technical change.

This analysis illustrates how perturbing coefficients of production affects the analysis of the choice of technique. Parameter spaces are partitioned by fluke switch points into regions in which the variation of the cost-minimizing technique with distribution is itself invariant. Some of the structures formed by intersections of partitions are not specific to the example, or even to models of fixed capital. For example, fluke switch points can arise in many models simultaneously on both the wage axis and the axis for the rate of profits. Perturbations of parameters around such a point eliminates or creates the possibility for certain techniques to be cost-minimizing at extreme ranges for the rate of profits.

Likewise, perturbation analysis supports the analysis of specific forms of technical progress. By contrast with fluke switch points, the recurrence of truncation, the reverse substitution of labor, and capital-reversing are not flukes. The reverse substitution of labor occurs in regions in which the recurrence of truncation does not occur. In this sense, the reverse substitution of labor is more general in this example. Capital-reversing arises in a region in which neither of these other 'perverse' phenomena occur. Configurations of parameters in which these phenomena arise, however, define only a few regions of the parameter space. This observation seems capable of generalization to other 'perverse' phenomena in other examples. Other forms of technical progress can be explored.