Otto Neurath And Happiness

I think of Otto Neurath primarily as a member of the Vienna circle. Rudolf Carnap and Moritz Schlick were two ther prominent members. They developed the philosophy of logical empiricalism or logical positivism. No woolly dialectics or cultural criticism for them.

It was a custom, when they were meeting at their preferred coffee-house in 1920s Vienna, to interrupt any speaker who had strayed into much-despised metaphysics. Neurath did this so often that they told him to hold up his hand whenever the speaker said something that was not metaphysics.

When Neurath fled the Nazis, he took his wife and his mistress with him. They got along well.

A number of revolutions, inspired by the Bolsheviks, convulsed central Europe after World War I. I think of Hungary and the Spartacists, in particular. Neurath began implementing his ideas for central planning under the Bavarian Soviet Republic. He advocated planning in kind, without monetary prices.

This was long before the definition of the Gross Domestic Product (GDP). But Neurath would not be interested in optimizing GDP in his planning. He did not want to duplicate capitalism. Rather he looked at a broader array of measures:

"Neurath ... pioneered a measure of living standard. He took variables that are now familiar to economists, such as nutrition, health, life expectancy, housing, clothing, incidence of crime. He was also concerned to build these up in a single measure, and its level, as well as its distribution, were to be the concern of the socialist planner." – Meghnad Desai. 2002. Marx's Revenge: The Resurgence of Capitalism and the Death of Statist Socialism.

This measure is like the United Nations Human Development Index (HDI), Bhutan's Gross National Happiness (GNP), or, say, metrics promoted by Joseph Stiglitz.

I suppose I ought to mention ISOTYPE, Neurath's pictorial language or symbols.

A century has shown that an unregulated, unrestrained capitalism does not deliver a broad-based prosperity for all, in which most can fully develop their capabilities. Suppose you care about the majority of your fellow citizens, not just a few at the top. Empirically, you should vote for socialists.

Sraffa On The Use Of The Notion Of Surplus Value

Sraffa, in his archives in the 1940s and 1950s, is quite appreciative of Karl Marx's analysis of capitalism. This appreciation contrasts with the opinion embodied in the label 'neo-Ricardian', which Bob Rowthorn invented.

I know about the passages below in the Sraffa archives from Riccardo Bellofiore. The archivist, Jonathan Smith, has dated this entry from 1955-1959, late in the writing of Production of Commodities.

I do not want to focus on whether Marx or Sraffa are correct or not. I would want to work out a simple example. Besides, Sraffa seems not convinced of how to analyze the reduction in the working day, when starting at prices.

But I want to note that Sraffa is very much using Marxist concepts: vulgar economics, labor values, prices of production, surplus value, exploitation, and rates of exploitation. And the analysis is based on Marx. Surplus value comes from extending the working day past the point at which workers reproduce their labor power.

"Use of the Notion of Surplus Value

"The prolongation of the working day beyond the point at which the labourer would have produced just an equivalent for the value of his labour-power ..." (Cap., Engels transl. p. 518) cp p. 539 [Chapter Sixteen: Absolute and Relative Surplus-Value]

Put it the other way round. If starting from capitalist society the working day is shortened till there is no surplus value left, this shortening must be equal for all: if it is, the prices of the commodities will change [owing to change in the rate of profits, which vanishes], but the wages will remain unchanged : if it is not, and the working day is reduced to the extent of the profits made in each industry, then prices would remain unchanged* after the shortening [for the number of (shorter) labor days, in industries having a high organic composition of capital, would increase in the same proportion as the fall of profits] but wages would be different.

[Footnote:] *(28.12.41) But profits would be different (after the reduction) in different industries!

[Marginal note:] c/p Letters of M and E 129-32 (letter of M. 2.8.62)

In other words, if we start from profits (as vulgar economy does) we reach the conclusion that the rate of exploitation is different in different industries, being higher in the more highly capitalised ones – which is not [and indeed contrary to] the fact. If we start from surplus value, which is equal in all industries, we get the correct measure of exploitation. The former conclusion is patent nonsense, and no view of exploitation could be based on it.

Note that the former (profits) goes with a theory of prices, the latter, of value (as defined below).

12.11.40 [Price is an exchange ratio which equalises rates of profit on capitals. Value is an exchange ratio which equalises rates of surplus-value on labour. If commodities exchanged at their values, profits would be different for different capitals, and capitals would move: therefore, this competition of capitals causes them to exchange at their prices.

The question is: are the rates of exploitation different? and if so why doesn’t labor move, and restore values and equality of rates of surplus value?]

The starting point is "the prolongation of the working day beyond the point at which the labourer would have produced just an equivalent for the value of his labour power" (Cap., Engels Tr. 518)

This point cannot be determined without reference to the value of the product (unless the labourer produces himself all the commodities he consumes).

But the point varies if we take value and if we take price.

Now, we are comparing the actual state with a hypothetical one in which only the necessary labour is performed.

In the actual state commodities are exchanged at their prices, whilst in the hypothetical state (where there would be nothing to be paid out in profits) at their values.

Which scale should we adopt for both states, in comparing them? It may be said: neither – each state has its own scale and that only is appropriate to it. No comparison can be made directly between the two extreme states. [marginal note says, "wrong, see p. 56.] We cannot imagine to move gradually from the actual state, shortening the working day; as we start from the actual state, we use its own scale, i.e. prices, in determining the ultimate goal towards which we move [and that will give different reductions for different branches of industry; but as we pass to successive other states, with shorter and shorter working days, the scale to be used changes, and prices move nearer (as the rate of profits is reduced) to values – so does the "point" aimed at change; until, on the threshold of the state in which only the necessary labour is performed, the prices will practically coincide with values, and the point aimed at with that determined by the scale of values, i.e. all labourers will have had their hours reduced in the same proportion. [The converse is true: if starting from the hypothetical state we prolong the working day by this method, we reach the actual state, having prolonged it for all labourers proportionally/equally, but through the change in prices having raise the profits in each branch proportionally with its capital]

Note that if we had adopted straightway values, and made the comparison between the two extreme cases, we should have obtained the same, correct, result. But if we had adopted prices, and made that comparison, it would have led us astray: the 'point' indicated by prices [i.e. different reductions in different industries] would have been false when the hyp. state was reached – for on the basis of values some labourers would be working more, and some less, than the necessary hours.

The imaginary process (described above, p. 3 bottom) of gradually shortening the working day, on the basis of the prices appropriate to each intermediate point, and therefore in different proportions for different industries, requires further consideration. As it stands, it is only correct at the wo extreme points [or rather only at the final point], but false at all intermediate ones: for, e.g., on the first step, when the rate of profit is reduced from 6 to 5%, the day of every worker must be reduced in one and the same proportion, and not in different proportions: it is clear that the latter method would give immediately absurd results.

In fact, this shows that the way in which I have argued the point on p. 1 is wrong (too weak). The objection of the vulgar economist is that the surplus produced in each industry (or firm) is measured by its profits. If he agreed to call it exploitation he would say that this is higher (absorbs a larger proportion of the working day) in the industries having more capital per worker. Therefore, he would have to conclude that, if exploitation has to be reduced in the different industries in such proportions as would maintain the rate of profit equal between them, at the lower level, this would require a larger reduction of the working day in the more capitalised industries. It can be shown, by a numerical example, that this is not the case. That on the contrary if the was reduced equally in all industries, the rate of profits would also be reduced equally. This result is made possible by a simultaneous change in prices – those of highly capitalised industries falling (when the rate of profits falls) relatively to the other prices. So that the larger fall in the surplus of such industries has two sources: a) the reduction in working way (common to other industries), (b) the fall in the relative price of their product (peculiar to the highly cap. industries)

29.12.41 [A third source, working in the same direction, would at first sight appear to be the increased depreciation allowances for capital, as the rate of interest falls. However this is a delusion: the 10 loom case shows that it is constant. That is to say, it is constant if real capital has to be maintained intact, though allowing value of semi-used capital to fall as rate of interest falls - and this is the relevant case. Its rise only if money value has to maintained as originally, in case

29.12.41 The previous paragraph is misleading. There is a third source, even if depreciation allowances are regarded as constant. For the value (or rather, in M's sense, the "price") of capital goods falls with the fall in the rate of interest. Therefore, when the rate of s.v. falls, the profits of industries with a large amount of capital per man fall still more owing to this third source - even if the capitals are no more (but no less!) "durable" than in other industries (if they are more durable in the highly capitalised industries than in others, this is a fourth source - for with fall in interest the more durable capitals falls more than that of the less durable ones).

[The whole subject of the "measure of capital" requires investigation in this connection. It has a striking similarity to the contradiction of "prices" and "value" of commodities, and it also depends on the equalisation of the rate of profits. One should start from the "value" of capitals (i.e. quantity of labour necessary to construct them) and see how the requirement of equal rates of profits leads to "prices" of capitals different from their "values".]

30.12.41 This business of 3 or 4 sources is wrong. There are only two sources: a) the reduction of working time in the industry, which reduces the quantity of goods produced; b) The reduction in the price at which the product is sold. The fall in the value of capital of certain industries along with the fall in the general rate of profits and other possible causes, contribute to source b, but don't add anything besides b.

[N.B. The fact that the value of capital (and therefore its "quantity" or magnitude) varies with the rate of profits (and generally cannot be known without knowing prices and rate of profits) makes nonsense of many cornerstones: 1) "Sacrifice of waiting", but how if they don’t know what they are abstaining from? 2) rate of interest, or marg. prod. of cap., as criterion for distribution of resources; but how, if the same resource (in "value") becomes larger or smaller (in "price") according as it used in one way or another?

5.1.42 Those who regard Marx's transition from values to prices, by the necessity of equalising the rate of profit, as a trick, should say the same of Ricardo's (and the whole marginal school) method of determining cost of production by considering only that on the marginal land, by the necessity of equalising the price of all bushels of corn, on whichever land they may be procured. Cannan does so (Rev. of Ec. Theory, p. 178): Ricardo 'did the trick by little more than an arbitrary exercise of the right to define terms ..." -- Piero Sraffa D3.12.46/57r – 63r

Nothing like the above is in Sraffa's book. Connections to Marx are less apparent, although some reviewers perceived them. Counterfactual reasoning is mostly eschewed. The length of the working day is not discussed, but taken as given.

Sraffa does not seem very confident about whether he should start with value or prices and how he should proceed if he adopts the latter. He does see the importance of what was later called price Wicksell effects. I want to note that the next pages in the archive are a draft of the chapter on land in Sraffa's book.

By the way, Ian Steedman has a chapter towards the end of Marx after Sraffa illustrating the analysis of the length of the working day. Consistent with his general approach, he uses data on physical quantity flows and does not take the point at which prices are values and labor is not exploited as a reference point.

Another 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

I have been exploring examples with both intensive and extensive rent. I try to make this a stand-alone post, with possibilities for later elaborations.

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.

This example works like a lot of mine. It is of perhaps the minimum structure needed to make my points in a model with more than one commodity produced, with a circular structure of production and labor inputs in all processes. Yet its elaboration seems complicated. And surprising results arise here or there. They would be more impressive if they occurred in a larger range of the rate of profits.

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 limitation 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	9/10	3/5	29/50	9/20
Type 1 Land	0	1	0	0	0
Type 2 Land	0	0	49/50	0	0
Type 3 Land	0	0	0	2/5	2
Iron	9/20	1/40	3/2000	29/500	2/30
Corn	2	1/10	9/20	13/100	13/100

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 55 tons iron and 55 bushels corn. This completes the specification of the example. The parameters for the example are fairly arbitrary. They are chosen to ensure reswitching of techniques between the Alpha and Beta techniques when net output is small and no land is scarce. The given net output, however, results in all types of land being scarce.

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 Quantity Flows

What techniques are feasible varies as output expands. I claim that, unlike in models with only extensive rent, the order of efficiency depends on the sequence in which techniques are cost-minimizing as net output expands. So I find the following remark dubious:

"No changes in output and (at any rate in Parts I and Il) no changes in the proportions in which different means of production are used by an industry are considered..." (Sraffa 1960)

Sraffa considers land, along with other examples of joint production, in part II.

If net output is low enough, gross outputs can be such that one type of land is partially farmed. The Alpha, Beta, Gamma, and Delta techniques are all feasible. For the example, the Delta technique is the first technique to become infeasible as output expands. It is replaced by the Eta, Kappa, and Phi techniques, in which type 3 land is scarce and obtains a rent. Table 3 shows which techniques become infeasible or feasible as output expands. Without an improvement in technology, the maximum output for the Omicron, Rho, Tau, and Omega techniques provides a hard limit for this economy.

Table 3: Output Regions

Output Region	Feasible Techniques
1	Alpha, Beta, Gamma, Delta
2	Alpha, Beta, Gamma, Eta, Kappa, Phi
3	Alpha, Gamma, Epsilon, Eta, Kappa, Mu, Xi, Phi
4	Gamma, Epsilon, Eta, Theta, Kappa, Lambda, Mu, Nu, Xi, Phi
5	Gamma, Epsilon, Eta, Theta, Lambda, Mu, Nu, Pi, Phi, Chi
6	Gamma, Epsilon, Theta, Lambda, Mu, Pi, Sigma, Phi, Chi, Psi
7	Gamma, Lambda, Mu, Pi, Sigma, Tau, Upsilon, Phi, Chi, Psi
8	Zeta, Iota, Lambda, Mu, Pi, Sigma, Tau, Upsilon, Chi, Psi
9	Zeta, Iota, Lambda, Mu, Tau, Chi, Psi, Omega
10	Zeta, Lambda, Omicron, Tau, Psi, Omega
11	Omicron, Rho, Tau, Omega

4.0 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

4.1 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.

4.2 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 of zero, 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 wage curve for the Omega technique is only shown for the range in which the rents on all three types of land are zero. The enlargement in Figure 3 shows a range of the rate of profits towards the start of the range at which Omega is cost-minimizing.

Figure 2: Wage Curves for Feasible Techniques

Figure 3: Wage Curves for Feasible Techniques (Detail)

Figure 3 plots rent per acre, as a function of the rate of profits, for the feasible techniques in this post. Figure 1, at the top of this post, is an enlargement of part of the range of the rate of profits. Under Rho, both type 1 and type 3 lands are fully farmed. No range of the rate of profits exist in which scarce land under Rho both receive non-negative rates of profits. Thus, Rho, although feasible, can never be cost-minimizing. Omicron, Omega, and Tau are each uniquely cost-minimizing, other than at switch points, for some range of the rate of profits.

Figure 4: Rent Curves for Feasible Techniques

5.0 Cost-Minimizing Techniques and Orders of Efficiency and Rentability

The cost-minimizing technique, at a given level of net output, varies with the rate of profits. Table 4 specifies the cost-minimizing technique in each output region. The rates of profits shown are only approximate.

Table 4: Cost-Minimizing Technique

Output Region	Range	Technique	Order of Efficiency	Order of Rentability
1	0≤r≤15.8%	Delta	3	N/A
	15.8≤r≤18.1%	Beta	2	N/A
	18.1≤r≤48.2%	Alpha	1	N/A
	48.2≤r≤78.5%	Beta	2	N/A
2	0≤r≤6.2%	Phi	3	3
	6.2≤r≤15.8%	Kappa	3,2	3,2
	15.8≤r≤18.1%	Beta	2	N/A
	18.1≤r≤48.2%	Alpha	1	N/A
	48.2≤r≤78.5%	Beta	2	N/A
3	0≤r≤6.2%	Phi	3	3
	6.2≤r≤15.8%	Kappa	3,2	3,2
	15.8≤r≤16.1%	Xi	2,3	2,3
	16.1≤r≤18.1%	Epsilon	2,1	2,1
	18.1≤r≤48.2%	Alpha	1	N/A
	48.2≤r≤78.5%	Epsilon	2,1	2,1
4	0≤r≤6.2%	Phi	3	3
	6.2≤r≤15.8%	Kappa	3,2	3,2
	15.8≤r≤16.1%	Xi	2,3	2,3
	16.1≤r≤18.1%	Epsilon	2,1	2,1
	18.1≤r≤48.2%	Theta	1,2	1,2
	48.2≤r≤78.5%	Epsilon	2,1	2,1
5, 6	0≤r≤6.2%	Phi	3	3
	6.2≤r≤9.5%	Chi	3,2	3,2
	9.5≤r≤15.4%	Pi	3,2,1	3,2,1
	15.4≤r≤15.8%			2,3,1
	15.8≤r≤16.1%		2,3,1
	16.1≤r≤18.1%	Epsilon	2,1	2,1
	18.1≤r≤48.2%	Theta	1,2	1,2
	48.2≤r≤78.5%	Epsilon	2,1	2,1
7	0≤r≤6.2%	Phi	3	3
	6.2≤r≤9.5%	Chi	3,2	3,2
	9.5≤r≤15.4%	Pi	3,2,1	3,2,1
	15.4≤r≤15.8%			2,3,1
	15.8≤r≤16.1%		2,3,1
	16.1≤r≤16.4%	Upsilon	2,1,3	2,1,3
	16.4≤r≤18.1%			1,2,3
	18.1≤r≤24.4%		1,2,3
	24.4≤r≤48.2%	Tau
	48.2≤r≤50.1%		2,1,3
8	0≤r≤6.2%	Iota	3,2	3,2
	6.2≤r≤9.5%	Chi
	9.5≤r≤15.4%	Pi	3,2,1	3,2,1
	15.4≤r≤15.8%			2,3,1
	15.8≤r≤16.1%		2,3,1
	16.1≤r≤16.4%	Upsilon	2,1,3	2,1,3
	16.4≤r≤18.1%			1,2,3
	18.1≤r≤24.4%		1,2,3
	24.4≤r≤48.2%	Tau
	48.2≤r≤50.1%		2,1,3
9	0≤r≤6.2%	Iota	3,2	3,2
	6.2≤r≤9.5%	Chi
	9.5≤r≤12.2%	Omega	3,2,1	3,2,1
	12.2≤r≤12.5%			3,1,2
	12.5≤r≤12.8%			1,3,2
	12.8≤r≤15.8%			1,2,3
	15.8≤r≤16.1%		2,3,1
	16.1≤r≤18.1%		2,1,3
	18.1≤r≤24.4%		1,2,3
	24.4≤r≤48.2%	Tau
	48.2≤r≤50.1%		2,1,3
10, 11	0≤r≤9.5%	Omicron	3,2,1	3,2,1
	9.5≤r≤12.2%	Omega
	12.2≤r≤12.5%			3,1,2
	12.5≤r≤12.8%			1,3,2
	12.8≤r≤15.8%			1,2,3
	15.8≤r≤16.1%		2,3,1
	16.1≤r≤18.1%		2,1,3
	18.1≤r≤24.4%		1,2,3
	24.4≤r≤48.2%	Tau
	48.2≤r≤50.1%		2,1,3

The range of the rate of profits towards its minimum value illustrates that intensive rent can be transient. When the Delta technique, which produces corn only with process V, becomes infeasible, The technique, Phi in which type 3 land is fully farmed with processes IV and V operating side-by-side, becomes cost-minimizing at a rate of profits of zero and nearby.

For a large enough output, Phi is no longer feasible. The cost-minimizing technique at a rate of profits becomes Iota, a technique with only extensive rent. Type 3 land is fully farmed with process IV, and Type 2 land is partially farmed.

Much else can be said about the sequence in which techniques become cost-minimizing, in various ranges of the rate of profits, as net output expands.

5.1 The Order of Efficiency

The order of efficiency is the order in which techniques are adopted with increasing net output at a given wage or rate of profits. The order of efficiency is also known as the order of fertility. Since the order of efficiency varies with the rate of profits, fertility is not defined solely by physical inputs and outputs.

In models with extensive rent, the order of efficiency can be read off the order of wage curves. The range of the rate of profits at which Tau is cost-minimizing in the example illustrates. The order of efficiency here is the sequence of wage curves from the highest to the wage curve for Gamma. Under Gamma, Type 3 land is partially farmed with process IV, as in the Tau technique.

The order of efficiency for Tau varies at the switch point between the Alpha and Beta wage curves. For the lowest levels of net output, this switch point between Alpha and Beta is an example of capital-reversing. But, for Tau, a change in quantity flows is not even associated with this change in the order of efficiency. Capital-reversing does not occur.

Consider the range of the rate of profits between approximately 6.18 and 9.54 percent. As net output expands, the Delta, Kappa, Chi, and Omicron techniques succeed one another. Delta and Kappa operate process V on type 3 land. Chi operates a linear combination of processes IV and V. The sequence of wage curves in this range varies from Delta, Gamma, Beta, Alpha through Delta, Beta, Gamma, Alpha. Yet the order of efficiency does not vary. Even though process IV is operated on type 3 land in both Gamma and Omicron, the switch point between Beta and Gamma has no effect on the order of fertility. Gamma does not matter to the initial historical order in which techniques are introduced here.

For Omega, the switch point between the Alpha and Gamma techniques does not matter for the order of efficiency. More could be set about the variation in the order of efficiency with the rate of profits.

5.2 The Order of Rentability

The order of rentability varies with intersections of rent curves. These variations in the order of rentability are independent from variations in the order of efficiency. The owner of a less fertile land can obtain more rent per acre than the owner of a more fertile land. As usual, a simplistic marginal productivity theory of distribution cannot be sustained.

5.3 The Cost-Minimizing Technique for the Example

Observations have been made above about the cost-miniziming technique. But the interaction between quantity flows, prices, and rents suggest that the cost-minimizing technique cannot always be found by constructing a frontier from the wage curves.

Figure 5 justifies that Omicron is cost-minimizing at a small rate of profits. Capitalists can gain extra profits by adopting process V in the range of the rate of profits at which both Omicron and Omega pay positive rents. Type 5 land becomes fully farmed by combining the two processes on Type 3 land, and the Omega technique results.

Figure 5: Extra Profits at Omicron Prices

Figure 6 justifies that Tau is cost-minizing at a high rate of profits. Here, too, the Omega technique will be adopted in the range of the rate of profits at which both the technique under consideration and Omega pay positive rents.

Figure 6: Extra Profits at Tau Prices

6.0 Conclusion

This post has illustrated that the introduction of natural resources into a model of the production of commodities provides a complicated picture. The concepts of the order of efficiency and the order of rentability apply to a model in which both intensive and extensive rent occur, depending on net output and the range of the rate of profits. The order of efficiency depends on the sequence in which techniques are cost-minimizing as net output expands, in a more complicated way than when just extensive rents are available. Intensive rent can be transient. An expansion of net output may replace a technique in which intensive rent obtains, at a given rate of profits, with a technique with only extensive rent. And the opposite can happen, as well.

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.