Extensive Rent, Absolute Rent, and Markup Pricing

Figure 1: Variation of Technique with Relative Markups

1.0 Introduction

This is a rewrite of a previous post with somewhat 'nicer' values for coefficients of production. I also expand on it with some observations on absolute rent. As far as I know, these posts are the first explicit presentation in the post-Sraffian tradition of a model of the prices of production with extensive rent and markup pricing.

These posts explore the conflict over distribution among workers, capitalists, and landlords. In a model of extensive rent, perturbations of relative market power among industries can create or destroy reswitching of the orders of fertility or of rentability. Extensive rent, called ‘differential rent of the first kind’ by Marx, arises when different types of land are cultivated. The same commodity, 'corn', is produced on each type of land, with a different mixture of labor and material inputs on each. An industrial commodity, 'iron', is produced in the numerical example explored in this article, thereby ensuring that capital consists of heterogeneous products.

2.0 Technology

Table 1 presents the technology for the example. The second column shows the inputs of labor, iron, and corn needed to produce a ton of iron. The remaining three columns to the right are the coefficients of production for processes to produce corn. A unit level of operation of a process in agriculture produces a bushel corn and requires an input of one of three types of land, as shown. Constant returns to scale prevail, although the level of operation of the processes producing corn is limited by the available acreage. This example has the same structure as a previous example, with different numbers.

Table 1: The Coefficients of Production

Input	Industry
	Iron	Corn
	I	II	III	IV
Labor	1 person-yr.	9/10 person-yrs.	6/10 person-yr.	29/50 person-yr.
Type I Land	0	1 acre	0	0
Type II Land	0	0	49/50 acre	0
Type III Land	0	0	0	2/5 acre
Iron	9/20 ton	1/40 ton	3/2000 ton	29/500 ton
Corn	2 bushels	1/10 bushel	9/20 bushel	13/100 bushel

In the three processes for producing corn, process III requires more labor per acre of land than process II, and process IV requires even more. Output per acre of land also increases across these three processes. Process III requires less seed corn per acre than process II, and process IV requires even less. Given these contrasts, processes II, III, and IV cannot be ranked by physical efficiency alone. Iron inputs per acre do not even vary monotonically among processes II, III, and IV, further illustrating the need for prices to rank lands by efficiency.

The given data includes the land available and the requirements for use. These are such that all three type of land must be at least partially farmed. Specifically, 100 acres of each type of land exist, and net output consists of 300 bushels corn. Three hundred bushels of corn is taken as the numeraire. The given data are in principle observable at a single moment in time. Different types of land are distinguished by how corn is grown on them. No need exists to imagine marginal adjustments (Gehrke 2021).

Three techniques, Alpha, Beta, and Gamma, can feasibly satisfy requirements for use. In all three techniques, all four processes are operated. One of the types of land is not fully cultivated in each technique (Table 2). The choice of technique is based on cost-minimization or profit maximization.

Table 2: Techniques of Production

Technique	Land
	Type 1	Type 2	Type 3
Alpha	Partially farmed	Fully farmed	Fully farmed
Beta	Fully farmed	Partially farmed	Fully farmed
Gamma	Fully farmed	Fully farmed	Partially farmed

3.0 Prices of Production

Prices of production are here defined for a given ratio of markups in the industrial and agriculture sectors. The rate of profits in the process producing iron is s₁ r, while the rate of profits in each of the three processes producing corn is s₂ r. I call r the scale factor for the rate of profits. Prices of production satisfy the following system of equations:

(p₁ a_1,1 + p₂ a_2,1)(1 + s₁ r) + w a_0,1 = p₁

(p₁ a_1,2 + p₂ a_2,2)(1 + s₂ r) + ρ₁ c_1,2 + w a_0,2 = p₂

(p₁ a_1,3 + p₂ a_2,3)(1 + s₂ r) + ρ₂ c_2,3 + w a_0,3 = p₂

(p₁ a_1,4 + p₂ a_2,4)(1 + s₂ r) + ρ₃ c_3,4 + w a_0,4 = p₂

I am assuming that wages and rents are paid out of the surplus at the end of the period of production. The relative market power of industry over industry, or vice versa, is expressed by the ratio s₁/s₂. When this ratio is unity, the equations characterize a competitive capitalist economy.

Each of the processes in the technology contributes an equation to the system of equations defining prices of production. The rate of profits is calculated on the value of the capital goods advanced. Rent and wages are paid out of the surplus. Four equations are defined in terms of seven variables, the prices of iron and corn, the rents per acre on each of the three types of land, the wage, and the scale factor for the rate of profits. The coefficients of production and the markups s₁ and s₂ are taken as given.

The following equation specifies that the rent on at least one type of land is zero:

ρ₁ ρ₂ ρ₃ = 0

Specifying the numeraire removes one degree of freedom:

300 p₂ = 1

All rents must be non-negative and one type of land must pay no rent. This is the type of land not fully cultivated. This constraint removes another degree of freedom. The following equation specifies that the rent on at least one type of land is zero:

ρ₁ ρ₂ ρ₃ = 0

This constraint removes another degree of freedom. The system of equations for prices of production has one degree of freedom for this model of markup pricing with extensive rent. This degree of freedom can be expressed as a function showing how the wage decreases with the scale factor for the rate of profits. With the rate of profits somehow specified, the wage, the price of iron, and rent on the scarce land are determined.

4.0 The Choice of Technique: An Example with Competitive Markets

Figure 2 illustrates wage and rent curves when s₁ and s₂ are unity. Markets are competitive, and the scale factor for the rate of profits is merely the rate of profits. The cost-minimizing technique corresponds to the wage curve on the inner frontier.

Consider a rate of profits of zero or just barely positive. Alpha is cost-minimizing. If the requirements for use were small enough that they could be satisfied by only farming Type 3 land, a technique with same wage curve as Gamma would be cost-minimizing. No land would pay rent and the wage would be as shown on the highest wage curve. With somewhat greater requirements for use, Type 2 land would be taken into cultivation, and the wage would be as shown on the wage curve for Beta. Type 3 land would be fully cultivated and pay a rent. Finally, with the originally postulated requirements for use, Type 1 land is cultivated. In the range for the smallest rate of profits, the order of fertility, from most fertile to least fertile land, is Type 3, Type 2, Type 1.

Figure 2: Wage and Rent Curves with Competitive Markets

Alpha is still cost-minimizing for a rate of profits greater than that at the first intersection on the outer frontier for the wage curves, but smaller than that at the switch point on the inner frontier of the wage curves. For a given rate of profits the order of fertility is from the type of land associated with the technique with the highest wage curve downwards. In this range of the rate of profits, the order of fertility is Type 2, Type 3, Type 1.

This is an example of the reswitching of the order of fertility. When Gamma is cost-minimizing, the order of fertility varies from Type 2, Type 1, Type 3 lands to Type 1, Type 2, Type 3 lands and back. The wage curve for the Beta technique is never on the inner frontier, and Beta is never cost-minimizing.

Rent curves are graphed on the right pane in Figure 2. For each switch point for the wage curves, a pair of points, vertically stacked, are shown on the graph of rent curves. For the switch points on the inner frontier of the wage curves, two rent curves intersect at a rent of zero. Type 1 and Type 3 lands pay no rent at this switch point. The switch points for the intersections on the outer frontier of wage curves are not striking on the graph of the rent curves.

The intersections for the rent curves occur at rates of profits different from those for which wage curves intersect. When Alpha is cost-minimizing, the order of rentability varies from Type 3, Type 2, Type 1 to Type 2, Type 3, Type 1. The order of fertility first matches the order of rentability, then deviates from it at a higher rate of profits, then matches again at a still higher rate of profits. Whether or not the orders of fertility and rentability match also varies with the rate of profits in the range where Gamma is cost-minimizing and Type 3 land pays no rent.

Figure 3: Distribution of National Income with Competitive Markets

Given net output and the rate of profits, the levels of operation of each process, the prices of their inputs, and the value of the components of net output are defined. Figure 3 plots total wages, rent, and profits as functions of the rate of profits. Since net output is taken as numeraire, these components add up to unity, whatever the rate of profits. Wages decrease and profits increase with the rate of profits, but rents do not vary monotonically. When Type 1 land is non-scarce, total rents decrease with the rate of profits. When Type 3 land is non-scarce, on the other hand, total rents increase with the rate of profits.

5.0 Perturbing Relative Markups

The non-competitive case can differ qualitatively from the competitive case presented in Section 4. In a non-competitive case, with agriculture having sufficient market power, Beta is sometimes cost-minimizing. This is never so in the competitive case. Throughout the range for the scale factor for the rate of profits in which the Alpha technique is cost-minimizing in this non-competitive case, the orders of efficiency and rentability do not vary. The same is true for the range of the scale factor in which Gamma is cost-minimizing. The opposite is true in the competitive case, and sometimes the order of rentability does not match the order of fertility when Alpha or Gamma are cost-minimizing.

One can construct an overall picture of how these changes come about by analyzing the full range of possible ratios of the markup in industry to the markup in agriculture. The heavy, solid lines in Figure 1, at the top of this post, are switch points on the inner frontier of the wage curves and the maximum scale factor for the rate of profits. The cost-minimizing technique is labeled. The dashed lines are intersections on the outer frontier of the wage curves. The order of fertility varies across dashed lines. Dotted lines are intersections of rent curves. The order of rentability varies across dotted lines. Dashed lines are hard to perceive to the right, when industry has more market power than agriculture.

Figure 4: Enlargement of Variation with Relative Markups

Thin vertical lines partition the axis for the ratio of markups. Each vertical line corresponds to a fluke case. Figure 4 enlarges the left part of the figure, where agriculture has the most market power. For the first fluke case, the rent curves for Type 1 and Type 3 lands, when Beta is cost-minimizing, become tangent at the indicated ratio of markups. The partition between the second and third region is a fluke case where the second intersection for the rent curves for Type 1 and Type 3 lands occurs at the maximum value for the scale factor for the rate of profits. At the ratio for markups for the third fluke case, the wage curves for Beta and Gamma techniques intersect with a wage of zero. The fourth fluke case is a ratio of markups such that all three wage curves intersect at a single switch point. The rents of all three types of land are zero at that switch point. The fifth fluke case, at a partition between Regions 5 and 6, is such that the wage curves for Alpha and Beta intersect at the maximum scale factor for the Gamma technique. The wage curves for the Alpha and Beta techniques are tangent at a switch point for the ratio of markups at the last fluke case.

Table 3 summarizes this numeric example. Variations in the cost-minimizing technique, in the order of fertility, and in the order of rentability with the scale factor for the rate of profits are indicated. In Region 1, agriculture has the most extreme level of market power over industry. The wage curve for Alpha intersects the wage curve for Beta on the inner frontier and then the wage curve for Gamma on the outer frontier. The rent curves do not intersect in the appropriate ranges of the scale factor for the rate of profits. The order of rentability only varies with the cost-minimizing technique.

Table 3: Variations in the Cost-Minimizing Technique

Region	Range	Technique	Order of Fertility	Order of Rentability
1	0 ≤ r ≤ r1	Alpha	Type 3, 2, 1	Type 3, 2, 1
	r1 ≤ r ≤ r2	Beta	Type 3, 1, 2	Type 3, 1, 2
	r2 ≤ r ≤ rmax, β		Type 1, 3, 2
2	0 ≤ r ≤ r1	Alpha	Type 3, 2, 1	Type 3, 2, 1
	r1 ≤ r ≤ r2	Beta	Type 3, 1, 2	Type 3, 1, 2
	r2 ≤ r ≤ r*		Type 1, 3, 2
	r* ≤ r ≤ r**			Type 1, 3, 2
	r** ≤ r ≤ rmax, β			Type 3, 1, 2
3	0 ≤ r ≤ r1	Alpha	Type 3, 2, 1	Type 3, 2, 1
	r1 ≤ r ≤ r2	Beta	Type 3, 1, 2	Type 3, 1, 2
	r2 ≤ r ≤ r*		Type 1, 3, 2
	r* ≤ r ≤ rmax, β			Type 1, 3, 2
4	0 ≤ r ≤ r1	Alpha	Type 3, 2, 1	Type 3, 2, 1
	r1 ≤ r ≤ r2	Beta	Type 3, 1, 2	Type 3, 1, 2
	r2 ≤ r ≤ r*		Type 1, 3, 2
	r* ≤ r ≤ r3			Type 1, 3, 2
	r3 ≤ r ≤ rmax, γ	Gamma	Type 1, 2, 3	Type 1, 2, 3
5	0 ≤ r ≤ r1	Alpha	Type 3, 2, 1	Type 3, 2, 1
	r1 ≤ r ≤ r*		Type 2, 3, 1
	r* ≤ r ≤ r2			Type 2, 3, 1
	r2 ≤ r ≤ r**	Gamma	Type 2, 1, 3	Type 2, 1, 3
	r** ≤ r ≤ r3			Type 1, 2, 3
	r3 ≤ r ≤ rmax, γ		Type 1, 2, 3
6	0 ≤ r ≤ r1	Alpha	Type 3, 2, 1	Type 3, 2, 1
	r1 ≤ r ≤ r*		Type 2, 3, 1
	r* ≤ r ≤ r2			Type 2, 3, 1
	r2 ≤ r ≤ r**	Gamma	Type 2, 1, 3	Type 2, 1, 3
	r** ≤ r ≤ r3			Type 1, 2, 3
	r3 ≤ r ≤ r4		Type 1, 2, 3
	r4 ≤ r ≤ rmax, γ		Type 2, 1, 3
7	0 ≤ r ≤ r1	Alpha	Type 3, 2, 1	Type 3, 2, 1
	r1 ≤ r ≤ r*		Type 2, 3, 1
	r* ≤ r ≤ r2			Type 2, 3, 1
	r2 ≤ r ≤ r**	Gamma	Type 2, 1, 3	Type 2, 1, 3
	r** ≤ r ≤ rmax, γ			Type 1, 2, 3

Region 2, and the fluke cases bounding it, illustrates that a persistent change in the relative market power among industries can bring about the reswitching of the order of rentability. The rent curves for Type 1 and Type 3 land intersect twice in the range of the scale factor of profits in which Beta is cost-minimizing. The rent curves for both types of land are increasing functions of the scale factor in this range.

The rent curve for Type 3 land becomes downward-sloping in Region 3. It is not always monotone. The order of rentability at the highest range of the scale factor for the rate of profits in Region 2 has disappeared in Region 3. With some thought, one can see how the variations between regions in the ranges of the scale factor for the cost-minimizing technique, the order of fertility, and the order of rentability relate to the fluke cases dividing these regions. The range of the scale factor in which Beta is cost-minimizing has disappeared in Region 5, after the switch point at which all three wage curves intersect. Switch points between Alpha and Beta and between Beta and Gamma are thenchforth on the outer wage frontier and are the occasion of a change in the order of fertility. Region 6 is an example of the reswitching of the order of fertility, and is illustrated in Section 4 with the competitive case. In Region 7, the last in the table, the order of fertility does not vary when Gamma is cost-minimizing.

6.0 Conclusion

The numerical example illustrates complications that can arise in the conflict over the distribution of the surplus among workers, capitalists in various industries, and owners of types of land. I take the coefficients of production as frozen in considering this conflict. That is, I do not consider the conflict over the length of the working day, the intensity with which workers work, the care they must take to prevent waste, the right to urinate on the job, and so on. Nor do I consider some conflicts between capitalists and landlords, outside of distribution. For example, landlords would like leases as short as possible so as to be able to raise rents for more-or-less permanent improvements brought about by the capitalists, while the capitalists would like longer rents to prevent this outcome.

The model illustrated above is one in which relative markups are given. It demonstrates that in the competitive case, the order of fertility of lands can vary with the rate of profits, where the rate of profits is taken as given from outside the model. The order of fertility can vary both with and without the cost-minimizing technique varying. The order of lands from high rent to low rent lands also varies, in general, with the rate of profits. One may find that ownership of one type of asset provides greater returns than another, even though the latter asset is more efficient at the going rate of profits.

Persistent differences in markups among industries does not alter these conclusions, but does provide the possibility of qualitative and quantitative variation in details. In the example, if agriculture has sufficient market power over industry, the example no longer exhibits a reswitching of the order of fertility. A range of the scale factor for the rate of profits emerges in which type 2 land no longer obtains a rent. An even further increase in the market power of agriculture can lead to the appearance and the disappearance of the reswitching of the order of rentability

No simple picture emerges from this analysis. The workers can only obtain a larger share of the surplus product if the capitalists get less. Whether or not the landlords are able to obtain a larger share with either an increased rate of profits or increased market power for some sectors varies

Ludwig Von Mises Being Stupid

Are those who obtain more income better in some way, perhaps more intelligent than others? Are they luckier? According to Ludwig Von Mises, they are superior in intelligence:

"The entrepreneur is the agency that prevents the persistence of a state of production unsuitable to fill the most urgent wants of the consumers in the cheapest way. All people are anxious for the best possible satisfaction of their wants and are in this sense striving after the highest profit they can reap. The mentality of the promoters, speculators, and entrepreneurs is not different from that of their fellow men. They are merely superior to the masses in mental power and energy. They are the leaders on the way toward material progress. They are the first to understand that there is a discrepancy between what is done and what could be done. They guess what the consumers would like to have and are intent upon providing them with these things. In the pursuit of such plans they bid higher prices for some factors of production and lower the prices of other factors of production by restricting their demand for them. In supplying the market with those consumers' goods in the sale of which the highest profits can be earned, they create a tendency toward a fall in their prices. In restricting the output of those consumers' goods the production of which does not offer chances for reaping profit, they bring about a tendency toward a rise in their prices. All these transformations go on ceaselessly and could stop only if the unrealizable conditions of the evenly rotating economy and of static equilibrium were to be attained." -- Von Mises, Human Action, Chapter XVI, Section 3.

You can see why he is popular among plutocrats. His confusion (not demonstrated in this post) about how capitalist economies can work is a feature, not a bug.

Some Notes On Marx On Rent

Marx writes about rent extensively in Part II of Theories of Surplus Value and in volume 3 of Capital. I read Theories of Surplus Value decades ago. I have been trying to read the chapters of volume 3 on rent, that is, chapters 37 to 47.

In general, these chapters do not use Hegelian terminology, but are just a matter of mathematical economics. Marx conflates analyses I would keep separate. Maybe this is a matter of a dynamic analysis set in historical time. I suppose I do not have standing to complain about everything being argued through numerical examples.

In Chapter 37, Marx clarifies that he is discussing ground rent on unimproved land or land with permanent improvements. Capitalist farmers are assumed, along with landed property. He talks about the incentives to tenants not to make improvements. Once a lease runs out, improvements will allow the landlord to raise rents. If the improvements are permanent, these improvements will henceforth enter into ground rents. Marx notes the accounting where the price of land is equal to the discounted value of yearly rents. Even so, the portion of surplus value paid out in rent should not be confused with interest.

Table 1: My Preferred Terminology and Marx's

My Technical Term	Marx's Technical Term
Extensive rent	Differential rent I
Intensive rent	Differential rent II
?	Absolute rent
External intensive rent	?

Chapter 38 starts the analysis of differential rent of the first kind, and Chapter 39 has numeric examples in tables. Marx discusses the case of waterfalls providing power to a factory. Those capitalists without access to the waterfalls, I guess, use steam power. He combines an analysis of physical fertility with nearness to markets. Marx says the most fertile land is not cultivated first. Improvements in transportation can change the ranking of lands. I like the example of the Erie Canal. More than one kind of agricultural product can be produced, but one is dominant in Marx's analysis. Anyways, he considers different arbitrary orders in which lands may be cultivated when discussing extensive rent. Also he considers cases where more of a type of land is discovered; supplies of land are not fixed in the very long run. Prices of production prevail. Capital investments are in monetary terms, despite Marx having shown in earlier chapters that prices of production of, say, seed, fertilizers, ploughs may vary with distribution.

Marx has something like a long-run demand curve for consumer goods:

The ... assumption is that total demand keeps pace with the increase in the total product. First, one need not imagine such an increase coming about abruptly, but rather gradually... Secondly, it is not true that the consumption of necessities of life does not increase as they become cheaper. The abolition of the Corn Laws in England proved the reverse to be the case (F. Newman, Lectures on Political Economy, London, 1851, p.158. — Ed.); the opposite view stems solely from the fact that large and sudden differences in harvests, which are mere results of weather, bring about at one time an extraordinary fall, at another an extraordinary rise, in grain prices. While in such a case the sudden and short-lived reduction in price does not have time to exert its full effect upon the extension of consumption, the opposite is true when a reduction arises from the lowering of the regulating price of production itself, i.e., is of a long-term nature. Thirdly, a part of the grain may be consumed in the form of brandy or beer; and the increasing consumption of both of these items is by no means confined within narrow limits. Fourthly, the matter depends in part upon the increase in population and in part on the fact that the country may be grain-exporting, as England still was long after the middle of the 18th century, so that the demand is not solely regulated within the confines of national consumption. Finally, the increase and price reduction in wheat production may result in making wheat, instead of rye or oats, the principal article of consumption for the masses, so that the demand for it may grow if only for this reason, just as the opposite may take place when production decreases and prices rise. -- Karl Marx, Capital, volume 3, Chapter 39.

Chapter 40 is the start of Marx's discussion of intensive rent, or differential rent of the second kind. He does not discuss it separately from differential rent, but adds it to his analysis. Intensive rent involves the returns to capital, in monetary units, when added to land of a given type. Marx does not treat in agriculture in a complete system. The approach I take to intensive rent is quite different. I am not sure I could always justify Marx's examples with a consistent system of equations, where the number of price and rent variables matches the number of processes, and the wage and the rate of profit frontier provides one degree of freedom. I allow for non-existence and non-uniqueness in these models.

With Marx's approach, he must consider in his numeric examples the possibility of the discovery of any type of land, changes in technology on any type, and the results of additional does of 'capital' on any type of land. He breaks these possibilities into three overarching subcases: A constant price of production of the dominant agriculture product, a falling price of production, and a rising price of production. These cases are discussed in Chapters 41, 42, and 43, respectively. Marx has subcases, in which an additional dose of capital has constant, decreasing, or rising productivity. Which land is on the margin can change. To tell the truth, I did not pay much attention to the details. In Chapter 42, Engels says an error exists in the tables that does not influence the conclusions.

I will update this post when I read further. I am interested if Marx justifies the existence of no cultivated land with no rent by the theory of intensive rent. And where does absolute rent come from?

Extensive Rent and Markup Pricing In A Complicated Example

Figure 1: Variation of Technique with Relative Markups

"That 'diminishing returns' was not an essential element in the surplus-based theory emerged in Marx's criticisms of Ricardo. Sraffa (1960), in his short chapter on land, the implications of which have yet to be developed, shows how the classical view of rents need not necessarily rest on the conception of 'the law of diminishing returns' or need not suggest necessarily any functional relationship between output and cost, or even presume a unique rank ordering of lands according to productivity. In fact, correcting the classical writers, her shows that differential rents as well as the 'no-rent' land depend upon the rate of profit (or wages)." -- Krishna Bharadwaj (1984) Piero Sraffa: A Tribute Economic and Political Weekly 19 (30-1): 1236-1235.

1.0 Introduction

This example illustrates how the struggle over distribution can be complicated. I start with a fluke case, in which three wage curves intersect at a switch point for a model of competitive capitalism with wage labor and extensive rent.

I assume markup pricing, with a stable ratio of the rate in profits in industry to the rate of profits in agriculture. For each such ratio, consider how the choice of the cost-minimizing technique varies with an exogeneous rate of profits. The wage varies, but so do rents per acre, including even which lands pay rent and which do not, because they are not scarce.

This post demonstrates that the introduction of markup pricing into a model of extensive can result in the reswitching of the order of fertility and the reswitching of the order of rentability.

I think I have demonstrated that one can conduct more research in price theory.

2.0 Technology

Table 1 presents the technology for the example. The second column shows the inputs of labor, iron, and corn needed to produce a ton of iron. The remaining three columns to the right are the coefficients of production for processes to produce corn. A unit level of operation of a process in agriculture produces a bushel corn and requires an input of one of three types of land, as shown. Constant returns to scale prevail, although the level of operation of the processes producing corn is limited by the available acreage. This example has the same structure as a previous example, with different numbers.

Table 1: The Coefficients of Production

Input	Industry
	Iron	Corn
	I	II	III	IV
Labor	1 person-yr.	0.89965 person-yrs.	0.60518 person-yr.	0.58085 person-yr.
Type I Land	0	1 acre	0	0
Type II Land	0	0	0.98020 acre	0
Type III Land	0	0	0	0.38724 acre
Iron	9/20 ton	1/40 ton	0.0014882 ton	0.058085 ton
Corn	2 bushels	1/10 bushel	0.45438 bushel	0.12908 bushel

The given data also include the land available and the requirements for use. These are such that all three type of land must be at least partially farmed. Specifically, 100 acres of each type of land exist, and net output consists of 300 bushels corn. Three hundred bushels of corn is taken as the numeriare.

Three techniques, Alpha, Beta, and Gamma, can feasibly satisfy requirements for use. In all three techniques, all four processes are operated. One of the types of land is not fully cultivated in each technique. Table 2 displays which type of land is only partially cultivated in each technique. The choice of technique is based on cost-minimization or profit maximization.

Table 2: Techniques of Production

Technique	Land
	Type 1	Type 2	Type 3
Alpha	Partially farmed	Fully farmed	Fully farmed
Beta	Fully farmed	Partially farmed	Fully farmed
Gamma	Fully farmed	Fully farmed	Partially farmed

2.0 Prices of Production

Prices of production are here defined for a given ratio of markups in the industrial and agriculture sectors. The rate of profits in the process producing iron is s₁ r, while the rate of profits in each of the three processes producing corn is s₂ r. I call r the scale factor for the rate of profits.

Each of the processes in the technology contributes an equation to the system of equations defining prices of production. The rate of profits is calculated on the value of the capital goods advanced. Rent and wages are paid out of the surplus. Four equations are defined in terms of seven variables, the prices of iron and corn, the rents per acre on each of the three types of land, the wage, and the scale factor for the rate of profits. The coefficients of production and the markups s₁ and s₂ are taken as given.

Specifying the numeraire removes one degree of freedom. All rents must be non-negative and one type of land must pay no rent. This is the type of land not fully cultivated. This constraint removes another degree of freedom. The system of equations for prices of production has one degree of freedom for this model of markup pricing with extensive rent. This degree of freedom can be expressed as a function showing how the wage decreases with the scale factor for the rate of profits. With the rate of profits somehow specified, the wage, the price of iron, and rent on the scarce land are determined.

3.0 The Choice of Technique: An Example with Agriculture Having Market Power

Figure 2 illustrates wage curves for the specified markups. Since the ratio s₁/s₂ is less than unity, this is a case in which agriculture has more market power than industry. For a given scale factor for the rate of profits, the order of fertility is from the type of land associated with the technique with the highest wage curve downwards. The cost minimizing technique corresponds to the wage curve on the inner frontier.

Figure 2: Wage Curves with Agriculture Having Market Power

Consider a small scale factor for the rate of profits is such that Alpha is cost-minimizing. If the requirements for use were small enough that they could be satisfied by only farming Type 3 land, Gamma would be cost-minimizing. No land would pay rent and the wage would be as shown on the highest wage curve. With somewhat greater requirements for use, Type 2 land would be taken into cultivation, and the wage would be as shown on the wage curve for Beta. Type 3 land would be fully cultivated and pay a rent. Finally, with the originally postulated requirements for use, Type 1 land is cultivated. In the range for the smallest scale factor for the rate of profits, the order of fertility, from most fertile to least fertile land, is Type 3, Type 2, Type 1.

When Beta is cost-minizing, the order of fertility is Type 3, Type 1, and Type 2 land or Type 1, Type 3, Type 2 land, depending on whether the scale factor for the rate of profits is less than or greater then the scale factor for the rate of profits at the switch point on the outer frontier. For a higher range of the scale factor for the rate of profits, Gamma is cost minimizing, and the order of fertility is Type 1, Type 2, and Type 3 lands.

Figure 3 graphs the rent curves for the specified markups. For each switch point for the wage curves, a pair of points, vertically stacked are shown on the graphs. For the switch points on the inner frontier of the wage curves, two rent curves intersect at a rent of zero. For the first such switch point, Type 2 and Type 1 land pay no rent at the switch point. For the switch point between the Beta and Gamma techniques, Type 3 and Type 2 lands pay no rent at the switch point.

Figure 3: Rent Curves with Agriculture Having Market Power

The switch point for the intersection on the outer frontier between the Alpha and Gamma wage curves is not striking on the graph of the rent curves. One sees one point on one rent curve for the Beta technique above the other.

The intersection for the two rent curves for the Beta technique occurs at a different scale factor for the rate of profits. It is not a scale factor for the rate of profits for which two wage curves intersect. When Beta is cost minimizing, the order of rentability is Type 3, Type 1, Type 2 for scale factors for the rate of profits less than the scale factor at which the rent curves intersect. For greater scale factors, the order of rentability is Type 1, Type 3, Type 2 when Beta is cost-minimizing.

This example demonstrates that the cost-minimizing technique depends on the scale factor for the rate of profits. The order of fertility of types of lands is not uniquely determined by the cost-minimizing technique. Neither is the order of rentability. And these orders do not need to match. One type of land can obtain more rent per acre than another, even when it is less fertile at the given ratio of markups and scale factor for the rate of profits.

4.0 The Choice of Technique: An Example with Industry Having Market Power

Figure 4 presents another graph of wage curves. Here the ratio of markups, s₁/s₂, exceeds unity. Industry has more market power than agriculture. The wage curve for the Beta technique is never on the inner frontier, and Beta is never cost-minimizing. This is an example of the reswitching of the order of fertility. When Gamma is cost-minimizing, the order of fertility varies from Type 2, Type 1, Type 3 lands to Type 1, Type 2, Type 3 lands and back.

Figure 4: Wage Curves with Industry Having Market Power

Figure 5 shows the rent curves for the given markups. Here the order of rentability varies both when the Alpha technique is cost-minimizing amd when the Gamma technique is cost-minimizing.

Figure 5: Rent Curves with Industry Having Market Power

5.0 Perturbing Relative Markups

Figure 1, at the top of this post is an attempt to present a synthetic picture of the example. The heavy, solid lines are switch points on the inner frontier of the wage curves and the maximum scale factor for the rate of profits. The cost minimizing technique is labeled. The dashed lines are intersections on the outer frontier of the wage curves. The order of fertility varies across dashed lines. Dotted lines are intersections of rent curves. The order of rentability varies across dotted lines. Dashed and dotted lines are hard to perceive to the right, when industry has more market power than agriculture.

Thin vertical lines parition the ratio of markups. Each vertical line corresponds to a fluke case. Section 3 is a point in region 4. Section 4 is a point in region 6. Figure 6 enlarges the left of the figure, where agriculture has the most market power. For the first fluke case, the rent curves for Type 1 and Type 3 lands, when Beta is cost-minimizing, become tangent at the indicated ratio of markups. The partition between the second and third case is a fluke care where the second intersection for the rent curves for Type 1 and Type 3 lands occurs at the maximum value for the scale factor for the rate of profits. The second region, and the fluke cases bounding it, illustrates that a persistent change in the relative market power among industries can bring about the reswitching of the order of rentability.

Figure 6: Enlargement of Variation with Relative Markups

In some sense, the ordinate in Figures 1 and 6 should not have units. I drew the graphs by varying s₁, while keeping s₂ constant at unity. But one could double both markups, while cutting the scale factor for the rate of profits in half. The graphs would be unchanged.

Anyways, Table 3 summarizes this numeric example. Variations in the cost-minimizing technique, in the order of fertility, and in the order of rentability with the scale factor for the rate of profits are indicated.

Table 3: Variations in the Cost-Minimizing Technique

Region	Range	Technique	Order of Fertility	Order of Rentability
1	0 ≤ r ≤ r1	Alpha	Type 3, 2, 1	Type 3, 2, 1
	r1 ≤ r ≤ r2	Beta	Type 3, 1, 2	Type 3, 1, 2
	r2 ≤ r ≤ rmax, β		Type 1, 3, 2
2	0 ≤ r ≤ r1	Alpha	Type 3, 2, 1	Type 3, 2, 1
	r1 ≤ r ≤ r2	Beta	Type 3, 1, 2	Type 3, 1, 2
	r2 ≤ r ≤ r*		Type 1, 3, 2
	r* ≤ r ≤ r**			Type 1, 3, 2
	r** ≤ r ≤ rmax, β			Type 3, 1, 2
3	0 ≤ r ≤ r1	Alpha	Type 3, 2, 1	Type 3, 2, 1
	r1 ≤ r ≤ r2	Beta	Type 3, 1, 2	Type 3, 1, 2
	r2 ≤ r ≤ r*		Type 1, 3, 2
	r* ≤ r ≤ rmax, β			Type 1, 3, 2
4	0 ≤ r ≤ r1	Alpha	Type 3, 2, 1	Type 3, 2, 1
	r1 ≤ r ≤ r2	Beta	Type 3, 1, 2	Type 3, 1, 2
	r2 ≤ r ≤ r*		Type 1, 3, 2
	r* ≤ r ≤ r3			Type 1, 3, 2
	r3 ≤ r ≤ rmax, γ	Gamma	Type 1, 2, 3	Type 1, 2, 3
5	0 ≤ r ≤ r1	Alpha	Type 3, 2, 1	Type 3, 2, 1
	r1 ≤ r ≤ r*		Type 2, 3, 1
	r* ≤ r ≤ r2			Type 2, 3, 1
	r2 ≤ r ≤ r**	Gamma	Type 2, 1, 3	Type 2, 1, 3
	r** ≤ r ≤ r3			Type 1, 2, 3
	r3 ≤ r ≤ rmax, γ		Type 1, 2, 3
6	0 ≤ r ≤ r1	Alpha	Type 3, 2, 1	Type 3, 2, 1
	r1 ≤ r ≤ r*		Type 2, 3, 1
	r* ≤ r ≤ r2			Type 2, 3, 1
	r2 ≤ r ≤ r**	Gamma	Type 2, 1, 3	Type 2, 1, 3
	r** ≤ r ≤ r3			Type 1, 2, 3
	r3 ≤ r ≤ r4		Type 1, 2, 3
	r4 ≤ r ≤ rmax, γ		Type 2, 1, 3
7	0 ≤ r ≤ r1	Alpha	Type 3, 2, 1	Type 3, 2, 1
	r1 ≤ r ≤ r*		Type 2, 3, 1
	r* ≤ r ≤ r2			Type 2, 3, 1
	r2 ≤ r ≤ r**	Gamma	Type 2, 1, 3	Type 2, 1, 3
	r** ≤ r ≤ rmax, γ			Type 1, 2, 3

6.0 Conclusion

The numerical example illustrations complications that can arise in the conflict over the distribution of the surplus among workers, capitalists in various industries, and owners of types of land. I take the coefficients of production as frozen in considering this conflict. That is, I do not consider the conflict over the length of the work day, the intensity with which workers work, the care they must take to prevent waste, the right to urinate on the job, and so on. It is hard to summarize the example. For example, for middling values of the scale factor for the rate of profits, owners of Type 2 land receive no rent if agriculture has somewhat more market power than industry. This possibility disappears when industry has more market power than agriculture. Anyways, this is another example in which obtaining a higher rent per acre can be connected with owning a less productive asset.

The coefficients of production in Table 1 are only approximate. My next step is to pick somewhat 'nicer' fractions' where the fluke three-technique switch point need not occur in the competitive special case.

Marginalism As A Distortion Of Classical Rent Theory

I want to note some instances in the literature that argue that one strain in the marginal revolution was an extension of (a misunderstanding of) the theory of intensive rent in classical political economy to all factors of production, particularly capital. This this is not of only historical interest. It suggests that another approach, that of classical political economy, to value and distribution exists. Furthermore, a treatment of intensive rent exists that is opposed to marginalism, in some sense.

The classical theory of rent first became widely known through pamphlets published in February 1815 by Thomas Robert Malthus, David Ricardo, Robert Torrens, and Edward West. James Anderson was an eighteenth century predecessor. Was Karl Marx to first point out that Anderson was overlooked? I have previously provided a taxonomy of some kinds of rent. Pasinetti (2015) has an interesting essay on these pamphlets. He argues Ricardo's was most comprehensive.

Krishna Bharadwaj has an article in which she writes about Marshall's reading of J. S. Mill:

"It is in the third class of commodities, marked by the coexistence of 'several costs of production' and by costs which vary with changes in supply, that provided the basis for the later marginalist conception of a supply function. The law of value in their case is stated by [J. S.] Mill thus: The value is determined by the cost of that portion of the supply which is produced and brought to markt at the greatest expense. (Mill, 1871 [1929], p. 471). To so-called 'law of diminishing returns' was to interest Jevons⁹ and Marshall and to play a significant role in the analytical development of marginalism, through its generalization to situations other than that of explaining rent: 'The law of diminishing marginal utility' was to be its counterpart in the theory of consumption, while it led to a more general notion of diminishing returns to a variable factor in the theory of production (see Bharadwaj, 1978). The Early Writings of Marshall reveal his great interest in the rent doctrine; as he himself remarked, 'improvements in cultivation decided me to adopt curves as an engine' (Marshall, 1975, I, pp. 40-1, p. 41, n. 12). In this third law of value, the functional link between costs and scale of output was clearly evident and provided the ground for a generalized 'supply function'.

⁹ Jevons wrote in the Preface to the first edition of his Theory of Political Economy (included in the fourth edition): 'There are many portions of Economical doctrine which appear to me as scientific in form as they are consonant with facts. I would especially mention the Theories of Population and Rent, the latter a theory of distinctly mathematical character, which seems to give a clue to the correct mode of treating the whole science (Jevons, 1911, p. vi, italics added)." -- Krishna Bharadwaj (1978).

Gehrke (2019) reports on some presentations at the Nationalökonomische Gesellschaft (NOeG), an interwar Vienna circle (not that one). Apparently, Oskar Morgenstern induced them to discuss Sraffa's 1925 article. Karl Menger, the mathematician son of the economist Carl Menger, was inspired by this and a conversation in which Ludwig Von Mises claimed that the law of diminishing returns could be proven. Karl Menger looked at the literature, and found it full of mistakes. The law of diminishing marginal returns could not be validly deduced from the assumptions that marginalist economists made:

"Yesterday ... Menger gave a brilliant paper on the law of dimishing returns. It was an exemplary performance for showing the need for exact reasoning in economics. interestingly, Haberler completely failed in the discussion; I very much noticed this. Of all these exact things he still does not understand the essence. Mises uttered pure nonsense." -- Oskar Morgenstern (31 December 1935, as quoted by Gehrke).

Some decades later Ronald Shepard introduced some arbitrary assumptions 'which are contrived to obtain the result'. Apparently, textbook presentations of increasing supply curves in partial equilibrium analysis of individual markets for consumer goods are still quite confused.

References

• Krishna Bharadwaj. 1978. The subversion of classical analysis: Alfred Marshall's early writings on value. Cambridge Journal of Economics 2(3): 253-271. (Reprined in Krishna Bharadwaj. 1989. Themes in value and distribution. Unwin Hyman)
• Christian Gehrke. 2019. The outstanding NOeG presentations: Morgenstern, Viner, and Menger on the laws of costs and returns. Empirica
• Christian Gehrke. 2021. Differences, switches and 'consistently side by side': Krishna Bharadwaj and the Sraffian critique of economic theory. Indian Economic Journal I-22.
• Luigi L. Pasinetti. 2015. On the origin of the theory of rent in economics. In Resources, Production and Structural Dynamics (ed. by Mauro L. Baranzini, Claudi Rotondi and Roberto Scazzieri). Cambridge University Press.
• Beth Stratford. 2023. Rival definitions of economic rent: historical origins and normative implications. New Political Economy 28(3): 347-362.

Competitive Capitalism Rewards Inefficiency: The Production of Commodities with Extensive Rent and Markup Pricing

Figure 1: Order of Rentability Varying with Relative Markups

1.0 Introduction

Ownership is not productive, as Joan Robinson informs us. But, at least under competitive conditions one might hope, more productive assets earn their owners more than less productive assets. And this applies to scarce skills as well. But none of this is necessarily true, either. This article presents a numerical example in which, among scarce lands, rent per acre is higher on more fertile land only when capitalists in agriculture have more market power than industrial capitalists. Only the unenlightened would assess one’s worth by how much income they are able to obtain from what they own.

A less efficient scarce asset can receive a higher rent. The proof of this statement is provided by a model of the production of commodities by means of commodities with land. No imperfections, such as externalities, asymmetric information, transactions costs, and so on exist in the model. More fertile lands are more efficient. The order of efficiency of types of land is the order in which they are introduced into cultivation as requirements for use expand, given the wage. Lands can also be ordered by rent per acre, with non-scarce lands being fallow or only partially cultivated. Lands that are not scarce receive a rent of zero.

The existence of market power is shown by markup pricing. When markups in agriculture and industry are equal and the economy is competitive, here the order of efficiency is opposite from the order of rentability for scarce lands. The order of efficiency varies when agriculture has sufficient market power. Only when agriculture has sufficient market power does the order of efficiency match the order of rentability.

As far as I know, this is the first explicit presentation of a model of the prices of production with extensive rent. Kurz & Salvadori (1995), Quadrio-Curzio (1980), and Schefold (1989) treat rent, including extensive rent. D'Agata (2018), Steedman (1981), Vienneau (2023), and Zambelli (2018) modify Sraffa's equations for prices of production to include markup pricing.

2.0 Technology

Table 1 presents the technology for the example. The second column shows the inputs of labor, iron, and corn needed to produce a ton of iron. The remaining three columns to the right are the coefficients of production for processes to produce corn. A unit level of operation of a process in agriculture produces a bushel corn and requires an input of one of three types of land, as shown. Constant returns to scale prevail, although the level of operation of the processes producing corn is limited by the available acreage.

Table 1: The Coefficients of Production

Input	Industry
	Iron	Corn
	I	II	III	IV
Labor	a0,1 = 1	a0,2 = 11/25	a0,3 = 91/250	a0,4 = 67/100
Type I Land	0	c1,2 = 49/100	0	0
Type II Land	0	0	c2,3 = 59/100	0
Type III Land	0	0	0	c3,4 = 9/20
Iron	a1,1 = 9/20	a1,2 = 1/100	a1,3 = 9/10000	a1,4 = 67/1000
Corn	a2,1 = 2	a2,2 = 6/125	a2,3 = 27/100	a2,4 = 3/20

The given data also include the land available and the requirements for use. These are such that all three type of land must be at least partially farmed. Specifically, 100 acres of each type of land exist, and net output consists of 400 bushels corn. Four hundred bushels of corn is taken as the numeriare.

Three techniques, Alpha, Beta, and Gamma, can feasibly satisfy requirements for use. In all three techniques, all four processes are operated. One of the types of land is not fully cultivated in each technique. Table 2 displays which type of land is only partially cultivated in each technique. The choice of technique is based on cost-minimization or profit maximization.

Table 2: Techniques of Production

Technique	Land
	Type 1	Type 2	Type 3
Alpha	Partially farmed	Fully farmed	Fully farmed
Beta	Fully farmed	Partially farmed	Fully farmed
Gamma	Fully farmed	Fully farmed	Partially farmed

Table 3: Definition of Variables

Variable	Definition
s1	Relative markup in industry.
s2	Relative markup in agriculture.
r	Scale factor for the rate of profits.
ρ1	The rent on Type 1 land.
ρ2	The rent on Type 2 land.
ρ3	The rent on Type 3 land.
w	The wage.

3.0 Prices of Production

Prices of production are specified by a system of equations with one degree of freedom. Table 3 defines the variables in these equations. Prices of production satisfy the following system of equations:

(p₁ a_1,1 + p₂ a_2,1)(1 + s₁ r) + w a_0,1 = p₁

(p₁ a_1,2 + p₂ a_2,2)(1 + s₂ r) + ρ₁ c_1,2 + w a_0,2 = p₂

(p₁ a_1,3 + p₂ a_2,3)(1 + s₂ r) + ρ₂ c_2,3 + w a_0,3 = p₂

(p₁ a_1,4 + p₂ a_2,4)(1 + s₂ r) + ρ₃ c_3,4 + w a_0,4 = p₂

I am assuming that wages and rents are paid out of the surplus at the end of the period of production. The relative market power of industry over industry, or vice versa, is expressed by the ratio s₁/s₂. When this ratio is unity, the equations characterize a competitive capitalist economy. The following equation specifies that the rent on at least one type of land is zero:

ρ₁ ρ₂ ρ₃ = 0

Finally, let the price of the commodity basket consisting of the numeraire be unity:

400 p₂ = 1

This completes the exposition of the equations specifying prices of production.

The above system of equations can be solved given the technique, the markups for industry and agriculture, and the wage. For a given technique, one of the equations for agriculture processes is picked out as being operated at a positive level with a rent of zero. The equation for the industrial process and for this process is a system of equations in four variables: the scale factor for the rates of profit, the wage, the price of iron, and the price of corn. (I am taking the scale factors as given.) The numeraire in this case fixes the price of corn. It is convenient to take the scale factor as given to obtain the wage and the price of iron as functions of the scale factor of the rate of profits. Each of the other equations for the agricultural processes can then be used to obtain rent per acres on scarce lands as functions of either the scale factor or the wage.

4.0 The Choice of Technique

Figures 2 and 3 illustrate prices of production for specific markups, in which agriculture has some market power, as compared to industry. Figure 2 displays the wage as a function of the scale factor for the rate of profits. It is a general property in a model of extensive rent that these wage curves slope down to the right. Thus, one can also view this graph as showing the scale factor as a function of the wage, given markups and the technique. Figure 3 graphs rent per acre, for the scarce lands when the Gamma technique is adopted, as a function of the wage, with markups given.

Figure 2: Wage Curves with Agriculture Having Market Power

Figure 3: Rent Curves with Agriculture Having Market Power

At a given rate wage, order the wage curves on the left pane in Figure 2 from the right to the left. This is the order of fertility, also known as the order of efficiency. A specific type of land with a rent of zero is associated with each technique. If requirements for use could be satisfied by farming just one type of land, type 1 land would be the only land farmed at a feasible wage above the switch point, given the markups. That is the economically most efficient process for growing corn maximizes the rate of profits, given the wage. Economic efficiency, generally, varies from technical efficiency and varies with distribution.

But suppose two types of land must be farmed to satisfy requirements for use. Then, given high wages, the land associated with the Beta technique would be brought under cultivation. Type 2 land would be farmed and obtain no rent.

Under the assumption that all three types of land must be farmed to satisfy requirements for use, the Gamma technique is adopted for any feasible wage. The maximum feasible wage is lower than when two types of lands can satisfy requirements for use. The order of efficiency for high wages is Type 1, Type 2, and Type 3 lands for high wage. It is Type 2, Type 1, and Type 3 for low wages.

Since only the Gamma technique is cost minimizing in the illustrated case, only the rent curves for Gamma are graphed in Figure 3. The order of rentability, from high rent per acre down, is Type 1, Type 2, and Type 3 lands. The order of efficiency does not match the order of rentability for low wages.

5.0 Economic Efficiency Varying with Relative Markups and the Wage

How does the above analysis vary with relative markups? Consider Figure 1 at the top of this post. For this numerical example, Gamma is always cost-minimizing for all feasible wages, whatever relative markups in industry and agriculture. Thus, the maximum feasible wage is a horizontal line at the top of the graph. The rent curves never intersect with variation in relative markups. The order of rentability does not change. The order of efficiency does change, however. Figure 1 plots the wage at which the Alpha and Beta wage curves intersect.

The thin vertical lines in Figure 1 are fluke cases. The left-most case arises when the Alpha and Beta wage curves intersect at a wage of zero. The rightmost case occurs when these wage curves intersect at the maximum feasible wage for Gamma.

Anyways the order of efficiency does not vary with the wage for sufficiently high market power for agriculture. On the left in Figure 1, the order of efficiency matches the order of rentability for all feasible wages. On the right, including the competitive case, the order of efficiency differs from the order of rentability. The variation of the order of efficiency with the wage is an intermediate case.

This example does not illustrate some of the possibilities of the theory of extensive rent. In general, the cost-minimizing technique varies with the wage. The order of efficiency varies with the technique. As illustrated, the order of efficiency can vary with the wage, even in a range of the wage in which a single technique is cost-minimizing. The order of rentability also varies with the technique, including which lands are scarce. And the order of rentability can vary in a range of the wage in which a single technique is cost-minimizing. That is the rent curves for a single technique may intersect.

6.0 Conclusion

This article builds on Ricardian theory done right. Karl Marx, in the third volume of Capital, does not go deep enough. He does, however, have some insights on the dynamics of rent. He is correct in his mocking of the trinty formula. Whether you want to say that the variation of the dependence of rent on wages on relative markups is an illustration of his concept of absolute rent is a matter of taste, I guess.

Anyways, I have proven my original claim. An asset, for example, a type of land, may obtain a larger rent than another asset even though processes that use the latter asset are more economically efficient. It is only when agriculture has sufficient market power in the example that, given the wage, the ordering of lands by economic efficiency matches the ordering of lands by rent.

References

• Deepankar Basu, 2022. A Reformulated Version of Marx’s Theory of Ground-Rent Shows that there Cannot be any Absolute Rent. Review of Radical Political Economy 54(4)
• D'Agata, A. 2018. Freeing Long-Period Prices from the Uniform Profit Rate Hypothesis: A General Model of Long-Period Positions. Metroeconomica 69 (4): 847–861.
• Kurz, H. D. and N. Salvadori (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.
• Schefold, B. (1989) Mr. Sraffa on Joint Production and other Essays, London: Unwin-Hyman.
• Steedman, I. 1981. Marx After Sraffa. London: Verso.
• Vienneau, R. L. (2023) Characteristics of labor markets varying with perturbations of relative markups, Review of Political Economy To appear.
• Zambelli, S. 2018. Production of Commodities by Means of Commodities and Non-Uniform Rates of Profits. Metroeconomica 69 (4): 791–819.

An Answer To Ludwig Von Mises

The Start Of This Video Is An Anecdote About George Dantzig

1.0 Introduction

Ludwig Von Mises popularized the Socialist Calculation Problem and brought it to wider attention. (This problem is also known as the Economic Calculation Problem.) In Von Mises' 1920 paper, he argues rational economic planning is impossible without market prices for capital goods and unproduced resources. Thus, anybody advocating socialism is advocating a system that cannot obtain a high material standard of living. In this post, I want to offer a simple solution to Von Mises' statement of the problem.

2.0 Informal Statement of the Problem

Von Mises thinks that socialism must imply central planning. He assumes that markets for consumer goods exist. Households are given tokens with which they can use to purchase consumer goods, including from one another. He describes a process that will eventually result in equilibrium on markets for consumer goods. The planning authority can use these prices in their planning.

But markets do not exist for capital goods, for goods of higher order in the jargon of the Austrian school. Von Mises asserts that without prices on such markets, the planning authority cannot engage in rational economic accounting. It is an impossibility argument based on economic theory.

Many have provided models of socialism, including with full awareness of the SCP. Oskar Lange had the most well-known answer. Both Von Mises and Lange based themselves on marginalist equilibrium theory.

Price ... may have the generalized meaning of 'terms on which alternatives are offered.' ...It is only prices in the generalized sense which are indispensable to solving the problem of allocation of resources. The economic problem is a problem of choice among alternatives. To solve the problem three data are needed: (1) a preference scale which guides the act of choice; (2) knowledge of the 'terms on which alternatives are offered'; and (3) knowledge of the amount of resources available. These three data being given, the problem of choice is soluble.

Now it is obvious that a socialist economy may regard the data under 1 and 3 as given, in at least as great a degree as they are given in a capitalist economy. The data under 1 may be given by the demand schedules of the individuals or be judged by the authorities administering the economic system. The question remains whether the data under 2 are accessible to the administrators of a socialist economy. Professor Mises denies this. However, a careful study of price theory and of the theory of production convinces us that, the data under 1 and 3 being given, the 'terms on which alternatives are offered' are determined ultimately by the technical possibilities of transformation of one commodity into another, i.e., by the production functions. The administrators of a socialist economy will have exactly the same knowledge, or lack of knowledge, of the production functions as the capitalist entrepreneurs have. -- Oskar Lange

In this post, I, too, base myself on marginalist theory, but on a Linear Programming formulation. In demonstrating Von Mises mistaken, I put aside that he, like marginalists everywhere, does not have a price theory that applies to actually-existing capitalist economies.

Table 1: Parameters and Variables

Symbol	Variable or Parameter	Definition
k	Parameter	The number of consumer goods, also known as the number of goods for final demand.
m	Parameter	The number of activities or processes.
n	Parameter	The number of resources.
p	Parameter	A k-element row vector of prices of consumer goods.
x	Parameter	A n-element column vector of the available quantities of resources.
A	Parameter	n x m matrix. A column of A represents the resources used up by an activity, that is, a production process.
B	Parameter	A k x m matrix. A column of B represents the final goods produced by that activity.
q	Variable	A m-element column vector of levels of operations of the processes.
w	Variable	A n-element column vector of resource prices.

3.0 A Linear Programming Solution

I now turn to a formal statement and solution of the problem. I assume that the central planner knows the parameters listed in Table 1. The planner's task is to set the values of the elements of the vector q. It is not required to operate all processes. Some elements of q can be set to zero. Likewise, not all resources must be fully used. Some resources may be left in excess supply. But the planner cannot use more of a resource than exists. An important requirement of the problem is that the planner does not know w. Resources that are inputs to production do not have prices.

So the planner must choose q to maximize:

p B q

such that:

A q ≤ x

x ≥ 0

The above is a Linear Program. It has a dual program. This is where shadow prices come in. The dual problem is to choose w to minimize:

x^T w

such that:

A^T w ≥ B^T p^T

w ≥ 0

The value of the objective functions are equal for solutions of the primal and dual problems. Thus the solution of the primal, which does not require knowledge of resource prices, assigns prices to those resources. Von Mises did not understand duality theory.

Some theorems draw more connections between the solutions to the primal and dual problems. Suppose that a constraint in the primal problem is met with an inequality in the solution. Some resource is not fully used. Then its price in the solution to the dual is zero. And, contrawise, suppose a constraint in the dual is met with an inequality in its solution. The cost of running a process with the chosen prices exceeds the value of the goods produced by that process. Then that process will not be operated in the solution to the primal.

How is this marginalism? This is certainly an example of the choice of the allocation of scarce resources among alternatives. Consider the impact on the solution of an incremental increase in the quantity of a specified resource. This increase introduces slack in the corresponding constraint in the primal. The value of the objective function is increased. The shadow price of that resource, if I recall correctly, is the value of the marginal product of the resource.

This math relates to an intervention in economics by Von Neumann and to Leontief's input-output analysis. The simplex algorithm is widely used to solve the above problems. Koopmans and Kantorovich shared the economics pseudo Nobel prize for Linear Programming. Leontief received a stand-alone pseudo Nobel.

4.0 Caveats

The above answers the problem, as stated by Von Mises (1920). Von Mises did not understand duality theory.

I can think of some difficulties to the above formulation of the problem. How would one accomodate economic growth? Can final demand include commodities not destined for consumption, and, if so, where would their prices come from? I suppose the planner could include such commodities, based on last mix of inputs last year in the industries that one wants to increase the output of. And their prices could then be last year's shadow prices. This approach leans toward input-output analysis, a better approach than the marginalist approach of Von Mises.

Another question is where do the consumers get their money-like tokens with which they purchase consumer goods? Are these labor vouchers? Some sort of income paid out of the total surplus regardless of contributions? Von Mises allows for a range of institutions for compensating various types of labor. Total equality need not be enforced.

Oskar Lange felt he had to go further than state the equations that must be solved. He had the planning authority simulating a tatonnement process, a trial and error solution. The managers of factories are instructed to treat the 'prices' issued by the central planner as given parameters. Lange, like others of his time, did not understand the difficulties in modeling capitalist markets by a static general equilibrium model.

Lange proposed a trial and error solution partly because he was responding to Robbins:

"On paper we can conceive the problem to be solved by a series of mathematical calculations... But in practice this solution is quite unworkable. It would necessitate the drawing up of millions of equations on the basis of millions of statistical data based on many more of millions of individual computations. By the time the equations were solved, the information on which they were based would have become obsolete and they would need to be calculated anew. The suggestion that a practical solution of the problem on planning is possible on the basis of the Paretian equations simply indicates that those who put it forward have not grasped what those equations mean." -- Lionel Robbins, as quoted by Lange (1938)

The analysis of the SCP nowdays should draw on the theory of computational complexity. One can count the computations that must be done to solve the problem. Presumably the matrices in the formulation of the problem are sparse. Cottrell and Cockshott are the authors I look to for these questions. I see the 2008 Gödel prize was awarded for work on the simplex method beyond me.

Von Mises does not consider the time or computational complexity for solving the equations. He says no calculations are possible. Likewise, Von Mises' objections are not about the difficulties of the central planner acquiring the data on prices for consumer goods, technological possibilities in physical units (in natura), or the difficulties in articulating dispersed tacit knowledge. These difficulties are later stated by Hayek when he and others changed the objection to an argument about the practicality, not possibility, of rational economic accounting under socialism. Along with this change in the statement of the problem, the Austrian school began to differentiate themselves from other marginalists. Von Mises attempted to pose the problem on a common ground shared by all marginalist economists.

References

• Allin Cottrell and W. Paul Cockshott. 1993. Calculation, Complexity and Planning: The Socialist Calculation Debate Once Again. Review of Political Economy 5: 73-112.
• Allin Cottrell and W. Paul Cockshott. 2007. Against Hayek. MPRA Working Paper No. 6062.
• Karras J. Lambert and Tate Fegley. 2023. Economic calculation in light of advances in big data and artificial intelligeence. Journal of Economic Behavior & Organization 206: 243-250.
• Oskar Lange and Fred M. Taylor. 1938. On the Economic Theory of Socialism. (ed. by Benjamin E. Lippincott). University of Minnesota Press.
• Tiago Camarinha Lopes. 2021. Technical or political? The socialist economic calculation debate. Cambridge Journal of Economics 45(4): 787-810.
• Ludwig Von Mises. 1920. Economic calculation in the socialist commonwealth. (Trans. and reprinted in Collectivist Economic Planning (ed. by F. A. Hayek). Routledge and Kegan Paul, 1935).
• Luigi L. Pasinetti. 1977. Lectures on the Theory of Production. Columbia University Press.
• John E. Roemer. 1994. A Future for Socialism. Harvard University Press.