Supply And Demand Breaking Down Half A Century Ago: The Sonnenschein-Mantel-Debreu Theorem

"[M]ainstream economists [divide] into effective 'castes', with only a tiny but exalted subset of the profession undertaking the detailed mathematical work needed to discover the weaknesses in the theory. The vast majority of economists believe that this high caste, the mathematical economists, did their work properly, and proved that the theory is internally consistent. The caste has indeed done its work properly, but it has proved precisely the opposite: that the theory is consistent only under the most restrictive and specious of assumptions." - Steve Keen, Debunking Economics

1.0 Introduction

Economists like to tell stories about supply and demand, in which a higher price of a good signals that it is more scarce and encourages agents to substitute other goods for the more scarce good. Mainstream economists have known for more than half a century that these stories have no justification in the most rigorous versions of their theory. Their stories are ad hoc and arbitrary.

I have summarized the Cambridge Capital Controversy before. Here I concentrate on the Sonnenschein-Mantel-Debreu (SMD) theorem.

If General Equilibrium Theory (GET) were to have empirical implications, it would restrict what was possible for market behavior. It turns out that, however, supply and demand functions can have almost any shape. No reason exists, in the theory, for equilibria to be unique or stable. As Andreu Mas Colell and his co-authors put it, anything goes.

I rely more on Alan Kirman's presentations than the original papers for the SMD theorem.

2.0 General Equilibrium Theory (GET)

Leon Walras invented GET and set out its canonical problems: the existence of an equilibrium, its uniqueness, and its stability. For the latter, he invented the tatonnement process, an auction in which no transactions are allowed until prices are found in which demand and supplies are equal. The Arrow-Debreu-McKenzie model is the current canonical statement of GET. For purposes of this post, you can consider a pure exchange economy.

Supply and demand are functions. For example, the quantity demanded and supplied of butter are depicted as functions of its price. The difference between demand and supply is an excess demand function.

Expressing the supply and demand of butter as only a function of its price seems inadequate. Should the demand not also depend on the price of margarine? If the price of bread fell and consumers consumed more bread, would not their demand for butter also rise? Would not the supply of bread, and thus the demand for butter, be impacted by decisions of farmers between growing wheat and producing crops for ethanol?

GET attempts to model all these interactions. Households, in a competitive pure exchange economy, are assumed to start with given endowments, with a certain basket of goods. They also are assumed to have preferences among these goods and to face given prices. The households decide how much of each good in their endowment to sell on the market and how much more to buy. In the jargon, they maximize their utility subject to a wealth constraint.

So for any set of prices, the model describes the difference, for each household, between the quantity demanded on the market of each good and their endowment of each good. This is the household's excess demand function. Under certain general and non-restrictive assumptions, individual excess demand functions have certain supposedly intuitive properties. I think the demonstration that demand functions slope down, if substitution effects dominate income effects, applies to the analysis of a household's maximization problem.

Aggregate or market excess demand functions are found by summing over all households. (Aggregate demand, in this sense, is not the aggregate demands in macroeconomics. They are specified for each of thousands of goods, not somehow summed over all goods.) Suppose the market excess demand for some good was positive at some price vector. Then the households would be trying to buy more of that good than exists. This is a disequilibrium.

An equilibrium exists when the prices are such that utility-maximization decisions of the households are mutually consistent. No good exists in which the households want to buy more than the aggregate endowment of that good.

3,0 Characterization of Market Excess Demand Functions

Arrow & Debreu and McKenzie proved that, under fairly general conditions, an equilibrium exists. I am unsure if the first welfare theorem, from GET, is the theoretical justification for claims that an unregulated capitalism can be efficient. Debreu always denied this interpretation, as I understand it. Debreu (1959) provides no attempt to describe how an equilibrium can be achieved. This remains an unsolved problem (see Fisher 1983).

Almost any functions can be excess demand functions. The restrictions are that the functions be continuous, homogeneous of degree zero, and satisfy Walras' law. Also, we only consider the functions bounded an arbitrarily small distance away from zero. That is, the behavior of the function when all prices are zero is not considered.

Homogenity here means only relative, not absolute prices matter. It does not matter if prices are denominated in dimes or dollars, euros or yuan.

Walras law states that if the excess demand for some good is positive, at disequilibrium prices, then some other markets have excess supplies. The disequilibria cancel out, in some sense.

The conclusion is that GET has no empirical implications at the level of markets.

4.0 Failed Attempts at Workarounds

Market excess demand functions can inherit nice properties on individual excess demand functions if all individuals are identical and have homothetic preferences. The latter implies that Engel curves are linear functions. Your relative demands for different goods, for say, chicken or lobster, does not depend on your income.

These assumptions were typical of macroeconomists for a long time after the so-called rational expectations revolution. They talk a lot about micro foundations, but their models lack them. They could not accommodate individuals with different tastes or with tastes that varied in some way with income.

Kirman may have been sensitized to the importance of the SMD theorem by his attempts, with co-authors to relax these assumptions. What happens if individuals have homothetic preferences, but individuals vary among themselves in their preferences? The same class of functions can still be excess demand functions, with the above extremely limited constraints. How about if individuals have identical preferences, but they are not necessarily homothetic? This does not help. Nor does it help to include production.

5.0 Conclusion

Kirman suggests, as I understand it, that part of the problem is that individuals interact in the model only through markets. Maybe some sort of norms or fashions shape preferences to provide some sort of coordination. Or maybe economists should consider broad classes of households as having common preferences. This type of approach is like that of the classical political economists who assumed, for example, that workers consume all their income (they do not have much), capitalists save, and landlords indulge in spending on luxuries.

References

• Gerard Debreu (1959) Theory of Value: An Axiomatic Analysis of Economic Equilibrium.
• Gerard Debreu (1974) Excess demand functions. Journal of Mathematical Economics 1(1): 15-21.
• Franklin M. Fisher (1983) Disequilibrium Foundations of Equilibrium Economics.
• Alan Kirman (1989) The intrinsic limits of modern economic theory: the emperor has no clothes. Economic Journal 99(395): 126-139.
• Alan Kirman (1992). Whom or what does the representative individual represent? Journal of Economic Perspectives 6(2): 117-136.
• Alan Kirman and K. J. Koch (1986) Market excess demand in exchange economies with identical preferences and collinear endowments. Review of Economic Studies 53(3): 457-463.
• Rolf R. Mantel (1974) On the characterization of excess demand functions. Journal of Economic Theory 7(3): 348-353.
• Hugo Sonnenschein (1972) Market excess demand functions. Econometrica 40(3): 549-563.
• Leon Walras(1874/1877) Elements of Pure Economics.

The Centre Of The Solving Subsystem In A Model With Fixed Capital And Scarce Land

1.0 Introduction

This post revisits my example with fixed capital and two types of land. It presents, by means of an example, the concept of the centre of a solving subsystem. Quadrio Curzio & Pellizzari (2010) introduce the solving subsystem in models of rent so as to first solve the price equations without rent. Schefold (1989) introduces the centre of the price system for a pure fixed capital model to, following Sraffa, initially eliminate the prices of old machines from price equations. As far as I know, nobody has combined these concepts before.

The concept of a solving subsystem clarifies how a switch point can lie along a single wage curve. A system of equations for prices is associated with each technique. Each operated process contributes an equation equating revenues and costs. The revenues can include the prices of joint products, and costs include a charge for the rate of profits on advanced capital goods. A last equation specifies the value of the numeraire as unity. In models of extensive rent, a subsystem can be formed from the processes that characterize industrial processes, with no inputs from land, and processes run on land that are not scarce. The resulting subsystem, with the equation for the numeraire concatenated, can be solved, given the rate of profits, for the wage and the prices of produced commodities. In models of intensive rent, the solving subsystem includes the equations for industrial processes and a linear combination of the equations for the processes that operate on one type of land to the limits of its endowment. As Sraffa (1960) explains, a variable for rent is eliminated by this linear combination. In the case of extensive rent, with no joint production otherwise, the solving subsystem also applies to a model of single production. In any case, the solution to the solving subsystem can then be used to find rents. The example in this post, extends the concept of a solving subsystem to a case with extensive rent and fixed capital. I do not know if the concept of a solving subsystem can usefully apply to joint production more generally

The centre of a pure fixed capital system (Schefold 1989) helps solve the price system of a pure fixed capital system. Joint utilization of machines does not exist in any process in a model of pure fixed capital. Old machines are not consumer goods. In the example, a single commodity is a consumption good and acts as numeraire. Old machines may be freely disposed of; no cost arises in junking a machine, including before its technical life is complete. Nice properties of single production systems generalize to such cases of fixed capital. In particular, the "determination of the cost-minimising technique is independent of the structure of requirements for use" (Huang, 2019). The cost-minimizing technique can be determined by the construction of the wage frontier. These properties are not retained in the combination of pure fixed capital with scarce land. The centre still helps solve the price system.

2.0 Technology, Endowments, Final Demand

Tables 1 and 2 specify the technology. This technology extends an example of fixed capital from Baldone (1974). Labor uses circulating capital to manufacture a machine in process I. The machine has a physical life of three years. Labor uses circulating capital and the machine to produce corn on type 1 land in processes II, III, and IV. The machine is operated on type 2 land in processes V, VI, and VII. A process that produces corn jointly produces a machine one year older than the machine used as input, up to its physical life. One hundred acres of each type of land are assumed to exist. Final demand is for 87 bushels corn, a level that ensures one or the other type of land is scarce. The numeraire is a bushel of corn.

Table 1: Inputs for Processes Comprising the Technology

Input	Processes
	I	II	III	IV	V	VI	VII
Labor	a0,1 = 0.4	a0,2 = 0.2	a0,3 = 0.6	a0,4 = 0.4	a0,5 = 0.23	a0,6 = 0.59	a0,7 = 0.39
Type 1 Land	0	c1,2 = 1	c1,3 = 1	c1,4 = 1	0	0	0
Type 2 Land	0	0	0	0	c2,5 = 1	c2,6 = 1	c2,7 = 1
Corn	a1,1 = 0.1	a1,2 = 0.4	a1,3 = 0.578	a1,4 = 0.6	a1,5 = 0.39	a1,6 = 0.59	a1,7 = 0.61
New Machines	0	1	0	0	1	0	0
Type 1 1-Yr. Old Machines	0	0	1	0	0	0	0
Type 1 2-Yr. Old Machines	0	0	0	1	0	0	0
Type 2 1-Yr. Old Machines	0	0	0	0	0	1	0
Type 1 2-Yr. Old Machines	0	0	0	0	0	0	1

Table 2: Outputs for Processes Comprising the Technology

Input	Processes
	I	II	III	IV	V	VI	VII
Corn	0	b1,2 = 1	b1,3 = 1	b1,4 = 1	b1,5 = 1	b1,6 = 1	b1,7 = 1
New Machines	1	0	0	0	0	0	0
Type 1 1-Yr. Old Machines	0	1	0	0	0	0	0
Type 1 2-Yr. Old Machines	0	0	1	0	0	0	0
Type 2 1-Yr. Old Machines	0	0	0	0	1	0	0
Type 1 2-Yr. Old Machines	0	0	0	0	0	1	0

3.0 Techniques

Tables 3, 4, and 5 specify the techniques that may be chosen with this technology. Alpha, Beta, and Gamma differ in the economic life of the machine on non-scarce, type 1 land. No processes are operated on type 2 land. Under Delta, Epsilon, and Zeta, on the other hand, type 1 land is not farmed at all, and the economic life of the machine varies among the techniques in the processes operated on type 2 land. The remaining techniques fully cultivate one or the other type of land and require rent to be paid to landlords

Table 3: Techniques of Production with Non-Scarce Land

Technique	Processes	Type 1 Land	Type 2 Land
Alpha	I, II	Partially farmed	Fallow
Beta	I, II, III	Partially farmed	Fallow
Gamma	I, II, III, IV	Partially farmed	Fallow
Delta	I, V	Fallow	Partially farmed
Epsilon	I, V, VI	Fallow	Partially farmed
Zeta	I, V, VI, VII	Fallow	Partially farmed

Table 4: Techniques of Production with Type 1 Land Scarce

Technique	Processes	Type 1 Land	Type 2 Land
Eta	I, II, V	Fully farmed	Partially farmed
Theta	I, II, III, V	Fully farmed	Partially farmed
Iota	I, II, III, IV, V	Fully farmed	Partially farmed
Kappa	I, II, V, VI	Fully farmed	Partially farmed
Lambda	I, II, III, V, VI	Fully farmed	Partially farmed
Mu	I, II, III, IV, V, VI	Fully farmed	Partially farmed
Nu	I, II, V, VI, VII	Fully farmed	Partially farmed
Xi	I, II, III, V, VI, VII	Fully farmed	Partially farmed
Omicron	I, II, III, IV, V, VI, VII	Fully farmed	Partially farmed

Table 5: Techniques of Production with Type 2 Land Scarce

Technique	Processes	Type 1 Land	Type 2 Land
Pi	I, II, V	Partially farmed	Fully farmed
Rho	I, II, III, V	Partially farmed	Fully farmed
Sigma	I, II, III, IV, V	Partially farmed	Fully farmed
Tau	I, II, V, VI	Partially farmed	Fully farmed
Upsilon	I, II, III, V, VI	Partially farmed	Fully farmed
Phi	I, II, III, IV, V, VI	Partially farmed	Fully farmed
Chi	I, II, V, VI, VII	Partially farmed	Fully farmed
Psi	I, II, III, V, VI, VII	Partially farmed	Fully farmed
Omega	I, II, III, IV, V, VI, VII	Partially farmed	Fully farmed

Under techniques Eta through Omicron, type 1 land is fully farmed and pays rent. Under Eta, Theta, and Iota, the machine is operated for only one year on type 2 land and then discarded. The techniques differ on the economic life of the machine on type 1 land. Under Kappa, Lambda, and Mu, the machine is operated for two years on type 2 land, while it is operated for its full physical life of three years under Nu, Xi, and Omicron. Under Pi through Omega, type 2 land is scarce and pays rent. Each technique between Eta and Omicron corresponds to a technique between Pi and Omega in which the same processes are operated. The economic life of the two types of machines are the same in these corresponding techniques. The scale at which the processes are run varies so as to vary which type of land is fully farmed.

4.0 The Price System for Omicron

I consider the price equations for Omicron to illustrate the concepts of the solving subsystem and of the centre. All seven processes are operated under Omicron, and type 1 land is scarce. The following seven displays, in obvious notation, specify the price system for Omicron:

a_1,1(1 + r) + w a_0,1 = p₀

(a_1,2 + p₀)(1 + r) + rho₁ c_1,2 + w a_0,2 = b_1,2 + p_1,1

(a_1,3 + p_1,1)(1 + r) + rho₁ c_1,3 + w a_0,3 = b_1,3 + p_1,2

(a_1,4 + p_1,2)(1 + r) + rho₁ c_1,4 + w a_0,4 = b_1,4

(a_1,5 + p₀)(1 + r) + w a_0,5 = b_1,5 + p_2,1

(a_1,6 + p_2,1)(1 + r) + w a_0,6 = b_1,6 + p_2,2

(a_1,7 + p_2,2)(1 + r) + w a_0,7 = b_1,7

Revenues for operating each process at a unit level are shown on the right-hand side of these equations. Revenues for the first process are obtained by selling new machines. Revenues for the second process result from products of both corn and a type 1 one-year old machine. That type 1 machine, in turn, enters into the advanced costs of the third process, and so on. Type 1 land obtains a rent, and type 2 land is free.

The first equation and the last three of the seven constitute the solving subsystem for Omicron. Given the rate of profits, the solving subsystem specifies the wage, the price of a new machine, and the prices of one-year old and two-year old machines when operated on free type 2 land. The remaining three equations can then be used to find the rent on type 1 land and the prices of one-year old and two-year old machines when operated on type 1 land. The solving subsystem for Omicron is also the solving subsystem for Zeta, Nu, and Xi. In all these techniques, the machine is run for its full physical life of three years on free type 2 land.

The prices of old type 2 machines can be eliminated from the solving subsystem for Omicron. Multiply both sides of the second equation of the solving subsystem by (1 + r)²:

(a_1,5 + p₀)(1 + r)³ + w a_0,5(1 + r)² = b(1 + r)²_1,5 + p_2,1(1 + r)²

Multiply both sides of the third equation of the solving subsystem by (1 + r):

(a_1,6 + p_2,1)(1 + r)² + w a_0,6(1 + r) = b_1,6(1 + r) + p_2,2(1 + r)

Add these two equations and the last equation of the solving subsystem:

where the row vector and matrix in this system of equations is as follows:

The ordered pair consisting of this row vector and matrix is the centre (Schefold 1989) for the solving subsystem for Omicron. Given the rate of profits, this system of matrix equations can be solved for the wage and the price of a new machine. This price system has the form of a price system for a circulating capital model, with the exception of the dependence of the Leontief input matrix and the vector of direct labor coefficients on the rate of profits. Unlike in the model of circulating capital, the wage curve derived from the centre of a pure fixed capital system can slope up for part of its range. The wage frontier of a pure fixed capital system, however, decreases throughout its length (Baldone 1974, Varri 1974).

The prices of old type 1 machines can be similarly eliminated from the full price system for Omicron.

5.0 Conclusion and Questions

The above illustrates the centre of a solving subsystem. In the example, the solving subsystem shows that a system of seven equations for a price system can be decomposed such that a system of four equations is solved first. And the centre of the solving subsystem shows that that system of four equations can be further decomposed so that a system of two equations is solved first.

Perhaps the centre of a solving subsystem can be used to address a theoretical question. Is the wage frontier always decreasing in a model combining fixed capital and rent? Can the wage frontier sometimes slope up?

In a model of extensive rent, the wage frontier is not the outer envelope of the wage curves for the technique. But it is always decreasing. Each wage curve is found from a solving subsystem. And the solving subsystem is from a related circulating capital model. So the wage curves inherit the properties of circulating capital models. The wage frontier is formed from the wage curves of the cost-minimizing techniques and always is decreasing.

In a pure fixed capital model, the wage frontier is the outer envelope of the wage curves for the techniques and is always decreasing. Individual wage curves can be increasing, but the ranges of the rate of profits at which they are increasing is never on the frontier.

I suspect the wage frontier for a model combining extensive rent and fixed capital can be increasing over some range of the rate of profits. This suspicion should be validated by constructing a numerical example. On the other had, if the wage frontier is alwys decreasing in such a model, that should be capable of a proof. And such a proof, if it exists, will probably use the concept of the centre of a solving subsystem.

References

• Baldone, S. (1974), Il capitale fisso nello schema teorico di Piero Sraffa, Studi Economici, XXIV(1): 45-106. Trans. in Pasinetti (1980).
• Huang, B. 2019. Revisiting fixed capital models in the Sraffa framework. Economia Politica 36: 351-371.
• Pasinetti, L.L. 1980. (ed.), Essays on the Theory of Joint Production, New York, Columbia University Press.
• Quadrio Curzio, Alberto. 1980. Rent, income distribution, and orders of efficiency and rentability (in Pasinetti 1980).
• Quadrio Curzio, Alberto and Fausta Pellizzari. 2010. Rent, Resources, Technologies. Berlin: Springer.
• Schefold, Bertram. 1989. Mr. Sraffa on Joint Production and other Essays, London, Unwin-Hyman.
• Sraffa, Piero. 1960. The Production of Commodities by Means of Commodities: A Prelude to a Critique of Economic Theory. Cambridge: Cambridge University Press.
• Varri, P. 1974. Prezzi, saggio del profitto e durata del capitale fisso nello schema teorico di Piero Sraffa, Studi Economici, XXIX(1): 5-44. Trans. in Pasinetti (1980).

Old Papers On Rent And One New One

This post annotates some papers that I want to remind myself of.

Montani (1975) references Quadrio Curzio (in Italian), defines the order of fertility and rentability, notes that they are different, and has something like the reswitching of the order of fertility. He does not have the reswitching of the order of rentability. He treats both extensive and intensive rent, but does not combine them. He notes the wage frontier can slope up under intensive rent. I have to read more closely to see if he already has multiple cost-minimizing techniques. I am under the impression that D'Agata first notice this possibility.

Montet (1979) criticizes Metcalfe and Steedman in that their perversities are more general than they know. Land provides another degree of freedom. They have a wage, rent, rate of profits frontier. I generally do not set equations for natural resources out this way. I once set out an example with heterogeneous labor, relabeling 'land' as 'skilled labor'.

Gibson & McLeod (1983) look at extensive, intensive, and external intensive rent. They go into difficulties of defining basics in joint production. One definition is about the decomposability of matrices and the other is about the rank of some sort of block matrix. They define quasi-basics for the latter. D’Agata has some sort of objection to this. They have interchanges in both the CJE and the RRPE.

Erreygers (1995) considers joint production. Toward the end of his paper, he shows how extensive rent fits into this framework. He wants to avoid setting out another equation in the quantity system to constrain levels of operations of processes from requiring more land to be farmed than exist. And rents should be part of the price vector in the price system, not seperate variables. Kurz & Salvadori (1995) show how to define certain block structured matrices to achieve this end. I think Erreygers may have created this approach.

Ianni (2026) is about international trade, not rent. The theory of intensive and extensive rent can show why most lands are specialized, so the theory may have implications for the theory of international trade. Also, my way of analyzing the choice of technique with long-lasting and given ratios of the rate of profits among industries may have implications for trade. Different countries may be modeled as having different rates of profits.

References

• Erreygers, Guido. 1995. On the uniqueness of cost-minimizing techniques. The Manchester School 63: 145-166
• Gibson, Bill and Darryl McLeod. 1983. Non-produced means of production in Sraffa's system: basics, non-basics and quasi-basics. Cambridge Journal of Economics 7: 141-150.
• Ianni, Guido. 2026. Class struggle and the international division of surplus: The distributive surface in the open economy. Metroeconomica
• Montani, Guido. 1975. Scarce natural resources and income distribution. Metroeconomica 27(1): 68=101.
• Montet, C. 1979. Reswitching and primary input use: a comment. Economic Journal 89 (355): 642-647.

Factor Demand Curves For An Example With Fixed Capital And Rent

Figure 1: Demand Curve for Labor

I have created and worked through an example in which a machine with a physical life of three years can be used in producing an agricultural commodity on one of two types of land.

My example is one of capital-reversing. It occurs to me that I have not plotted the demand for so-called factors of production in this example. Accordingly, Figure 1 plots the wage against the employment firms want to offer, given final demand. Switch points are horizontal line segments in this graph. Around the 'perverse' switch point, a higher wage is associated with firms wanting to employ more workers.

Given final demand and the rate of profits, a price system is defined for each technique. I can add up the value of the capital goods that must exist at the start of the year to produce the given final demand. Prices of production are used to aggregate heterogeneous goods. Figure 2 shows the demand for capital, in some sense. Here, too, the 'perverse' switch point is indicated for a step function approximation for an increasing demand curve. The value of capital varies between switch points because of price Wicksell effects.

Figure 2: Demand Curve for Capital

A model with both fixed capital and the rent of natural resources is a step towards realism if you want. It is also a step beyond what can be found from empirical Leontief matrices, as I understand it. Still, wages and employment, for example, cannot be explained in the long run by the interactions of well-behaved supply and demand functions in the labor market.

Some Phenomena In Price Theory

I occasionally list theoretical possibilities that I think interesting. Outside of a working paper at Centro Sraffa, I have not managed to publish papers detailing the possibilities listed in this post. Some I have not even written up outside of blog posts. I now know that:

• The recurrence of truncation can occur without the reswitching of techniques. This possibility arises in an example of pure fixed capital, with long-lived machines used in both industries that exist in the example.
• A switch point can lie along a single wage curve, with no other wage curve intersecting at the switch point. This possibility occurs in an example with both fixed capital and rent.
• The order of rentability can be completely opposite the order of efficiency. This possibility can arise in a model that combines extensive and intensive rent.
• The partitioning of parameter spaces by fluke switch points is useful in the analysis of structural economic dynamics with a choice of technique.
• Capital-theoretic paradoxes are transient, in many instances, in secular time (also known as the very long run).

I have some difficulties in writing these up. First, my status as an independent researcher creating examples as a hobby should make reviewers be a bit skeptical. Second, many may not be interested in these refinements. Does not Kurz and Salvadori (1995) provide a definitive statement of post Sraffian price theory? You need to have mastered quite a bit of that to understand the point of any of these. Third, I try to put each in a somewhat more general framework I cast the first, the recurrence of truncation, as an example of the last. I suggest that the second, a switch point along a single wage curve, is an anomalous switch point, a concept I am introducing. I want to say that the third is an example of a special case of a model of intensive and extensive rent in which 'nice' properties of models of extensive rent obtain; wage curves slope down and no issues of the non-existence or multiplicity of cost-minimiing techniques away from switch points arise. Last, when I make such generalizations, I have trouble casting my results into the abstract theorem-proof form needed to be precise.

Is the analysis of structural economic dynamics with a choice of technique an interesting problem? Maybe a book of bookprints never exists at a point of time. Capitalists do not have option of costlessly choosing another page. When a new technique is introduced, it typically dominates the existing technique. On the other hand, I have trouble with part II of Sraffa's book preceding part III. Part II treats joint production, including rent and fixed capital. Part III treats the choice of technique. Which lands to cultivate and what economic lives of machines to adopt are part of the choice of technique. So maybe I should limit my program to aspects of joint production. But I also have some consideration of Harrod-neutral technical progress.

It seems I still have years of work.

Murray Rothbard Muddled And Confused

1.0 Introduction

I try to read Rothbard's 'Toward a reconstruction of utility and welfare economics' (in On Freedom and Free Enterprise: The Economics of Free Enterprise (ed. Mary Sennholz), 1956). It does not go far toward its declared goal.

Some Austrian fanboys point to this paper to show Rothbard with a good understanding of the technical details of economics. And it fails.

2.0 Demonstrated Preference and Indifference

Rothbard proposes a concept, 'demonstrated preference', but never explains it clearly. He cites Ludwig Von Mises, among others, as a forerunner. He says that "Actual choice reveals, or demonstrates, a man’s preferences."

Rothbard asserts, like Von Mises, that marginal utilities can be ranked. You will find it difficult to identify anybody outside the Austrian school who agree today. I do not see that confining oneself to discrete increments of a single good helps Rothbard make his case.

I find strange Rothbard's rejection of an indifference relation. He writes:

"Indifference can never be demonstrated by action. Quite the contrary. Every action necessarily signifies a choice, and every choice signifies a definite preference. Action specifically implies the contrary of indifference."

And his arguments are quite curious. I do not find this persuasive:

"It is immaterial to economics whether a man chooses alternative A to alternative B because he strongly prefers A or because he tossed a coin. The fact of ranking is what matters for economics, not the reasons for the individuals arriving at that rank."

I would think that if I use a coin flip to decide, I am demonstrating that I do not care which way the decision comes out.

"The other attempt to demonstrate indifference classes rests on the consistency - constancy fallacy, which we have analyzed above. Thus, Kennedy and Walsh claim that a man can reveal indifference if when asked to repeat his choices between A and B over time, he chooses each alternative 50 percent of the time.

The above is silly. Would you say that the agent is indifferent if his preferences were constant over the observed time? Refusing to accept the hypothesis does not answer the question.

Does getting rid of the indifference relation hinder the use of 'demonstrated preference' to derive individual demand functions, whether defined on a discrete space or not? Maybe a primitive relation of 'not preferred to' is all that is needed. But Rothbard does not say.

3.0 Praxeology and Logic

I understand logic to be about the form of an argument or deduction. The content or meaning of propositions that appear in an argument are not supposed to matter for its validity.

Rothbard attempts to clarify a different conception:

"...a fundamental epistemological error ... pervades modern thought: the inability of modern methodologists to understand how economic science can yield substantive truths by means of logical deduction (that is, the method of 'praxeology')."

Rothbard asserts that his starting axioms must be true:

"...economics, or praxeology, has full and complete knowledge of its original and basic axioms. These are the axioms implicit in the very existence of human action, and they are absolutely valid so long as human beings exist. But if the axioms of praxeology are absolutely valid for human exisence, then so are the consequents which can logically be deduced from them. Hence, economics, in contrast to physics, can derive absolutely valid substantive truths about the real world by deductive logic."

We now know that Rothbard is incorrect on the his axioms. But never mind that.

"...mathematical logic is uniquely appropriate to physics, where the various logical steps along the way are not in themselves meaningful; for the axioms and therefore the deductions of physics are in themselves meaningless, and only take on meaning 'operationally,' insofar as they can explain and predict given facts. In praxeology, on the contrary, the axioms themselves are known as true and are therefore meaningful. As a result, each step-by-step deduction is meaningful and true. Meanings are far better expressed verbally than in meaningless formal symbols."

I have no idea what formal logic has to do with physics. As far as I know, the conception that logic is about the form of an argument goes back to Plato and Aristotle. Hegel may have had a different idea. Frege was writing about the foundations of arithmetic, not about physics.

4.0 Von Neumann-Morgenstern Cardinal Utility

Rothbard has a few remarks on the Von Neumann-Morgenstern definition of utility. Their exposition goes along with the development of a theory of measurement. A measurement scale is such that statements about things measured along that scale are only meaningful up to a set of transformations.

But according to Rothbard, "Measurement, on any sensible definition, implies the possibility of a unique assignment of numbers which can be meaningfully subjected to all the operations of arithmetic." "No arithmetical operations whatever can be performed on ordinal numbers." But non-parametric statistics was already being developed then. I think of the Mann-Whitney-Wilcoxon statistic, for example. In fact, the first edition of Sidney Siegel's textbook, Non-Parametric Statistics for the Behavioral Sciences, dates from 1956.

Rothbard tells us that those who follow Von Neumann and Morgenstern only apply probability to repeatable events: "... unique events are not repeatable. Therefore, there is no sense in applying numerical probability theory to such events. It is no coincidence that, invariably, the application of the neo-cardinalists has always been to lotteries and gambling. It is precisely and only in lotteries that probability theory can be applied." And Rothbard also asserts that "The leading adherents of the Neumann-Morgenstern approach are Marschak, Friedman, Savage, and Samuelson". But Leonard Savage, in his 1954 book, starts the development of his personalistic approach to probability with unique events. His application of personalistic probability to small worlds is supposed to apply numeric probabilities to unique events there. So, again, Rothbard is mistaken. (I take no position on whether unique events can meaningfully be assigned probabilities, either in a small world or not.)

5.0 Conclusion

Rothbard makes a lot of other dubious or incorrect statements. I concede that his references are wide ranging.

Rothbard's undergraduate degree was in mathematics. I pity the fool.

Socialism Worked In Bologna, Italy, For Decades

Socialists and communists have been elected in many places, for significant periods of time. Often they introduced policies that improved the lives of most citizens and increased their freedoms. If I were a member of some of those polities, I would almost certainly have disagreements with details of some policy or other. This post is about a place that I do not know much about.

After the end of World War II, Europeans who had resisted fascism in the underground had a certain prestige. That included the Italian Communist Party (PCI). The PCI became the governing party in Bologna and Florence in much of the time after WW II. You can also look to the government of Emilia-Romagna, a region of Italy that includes Bologna.

The PCI did not enter the national government partly as a consequence of Italian foreign policy. They needed to be in alliance with the USA. Perhaps the CIA was involved in interventions to Italian domestic politics.

The PCI introduced a host of reforms including free busing, better health care, better education, housing cooperatives, and generally good government. I have never been to Bologna. Did the PCI have something to do with the maintenance of the Renaissance character of downtown Florence? The Reggio Emilia Approach approach to childcare is still used elsewhere.

I gather that the PCI never was officially part of a national governing coalition, even after Enrico Berlinguer's historic compromise and championing of Eurocommunism. During the years of lead, the PCI found themselves to the right of those, many young, inspired by Operaismo (workerism). This part of my fragments of a story is uninspiring for those who want to pursue an electoral path.

Misinformation From Economists

I have found another source of economists confidently spouting mistakes, Economics Stack Exchange. This has been around for more than a decade.

If I went back in time, I think I would have trouble convincing my 20 year old self that standard introductory textbooks are incoherent nonsense, never corrected.

I quickly found questions on the Cambridge Capital Controversy. What technology do we need to have reswitching to occur? Why is reswitching and reverse capital-deepening a problem exactly? Why did the Cambridge Capital Controversy have no impact on economic modelling? The participants do not seem to have much to say on the topic.

Ten years, ten moths ago, a question was posed: Can capital still be paid its marginal product in the absence of a homogeneous capital stock? This question was inspired by a Krugman answer to critics of Piketty. One answer was offered:

Different sorts of capital used as separate production technologies prevent clean aggregation to a representative form of capital but does not prevent capital from being paid its marginal product...

On the margin the two sorts of capital don't have the same product and so aggregation doesn't make sense here. But in this setting, it is likely that the rental rate on capital would be equated (r₁ = r₁) because why would you buy one sort of capital when the other sort paid more?

The answer is foolish. The variables are supposed to be "rental prices". They might be in units of numeraire units per year per services of ton iron and numeraire units per year per services of square meters of the services of steel sheets, where the latter are of a specified thickness. You could change their values by a change in units. For example, the latter could be in square yards, not square meters. So it makes no sense to equate these values.

I suspect I can find more confusion.

Eurocommunism and Communist Parties In Coalition Governments In Europe

Eurocommunism was a tendency in communist parties in Europe during the 1970s. The Soviet suppression of the Prague Spring cast communist parties in Western Europe in a bad light. How could they follow Moscow's ead after that? So they started articulating their own path and asserted their independence from the Soviet Union.

This tendency was a moderating tendency. Ernest Mandel, a follower of Trotsky and therefore a critic of Stalin, decried this tendency. He called Eurocommunism "the bitter fruits of socialism in one country."

Anyways, two instances of these "bitter fruits" stand out to me. One is the historic compromise, led by Enrico Berlinguer, the leader of the Italian Communist Party (PCI). This involved support for the Christian Democrats (DC). I guess that the PCI did not enter the government in the elections of 1976, but refused to vote against the DC on no-confidence votes in parliament. In some sense, the communists were to the right of the socialists, let alone the workerists outside of the parties.

Another case is Francois Mitterand, a socialist, who was elected president of France in 1981. He took the French Communist Party (PCF) into his governing coalition. The communists did not do well, being sort of domesticated.

That was a while ago. But take a look at Portugal. Antonio Costa, a socialist, was elected Prime Minister in 2015 and served to 2024. This was a coalition government, called the Left Bloc. The Portuguese Communist Party and the Greens were also coalition members. Costa is now President of the European Council. Since 2024, Portugal's Prime Minister is Luís Montenegro, head of a more right-leaning coalition. To confuse me, his party is the Social Democratic Party. I think the names of the parties suggests, to an American, that they are more left-wing than they are now. As of February, the Portuguese president is Jose Seguro, a socialist. I gather that his election was a matter of staving off the far right, in some sense.

So the history of socialism and communism includes typical parliamentary machinations, compromises, coalitions, and so on. To understand Portugal, I should know more about Salazar and the Carnation Revolution.

A Switch Point Along The Same Wage Curve With Multiple Agricultural Commodities

Figure 1: Wage Curves Around An Anomalous Switch Points

This post presents another anomalous switch point. A switch point is anomalous in that it has properties that cannot hold for a switch point in a model of single production, with inputs of labor and circulating capital alone.

This example is one with multiple agricultural commodities and intensive and extensive rent. The technology and the endowments of land are the same as in this example.

Required net output, that is, final demand, varies. I start by postulating that final demand consists of 28 bushels wheat and 28 bushels rye. Under this assumption, Alpha, Beta, Epsilon, and Lambda are feasible techniques.

The cost-minimizing technique at a given rate of profits must be:

• Feasible.
• Have non-negative prices for all commodities produced under the technique, have a non-negative wage, and have non-negative rents on all scarce lands.
• Such that no process not operated under the technique obtains extra profits.

Epsilon is cost-minimizing up to a rate of profits of approximately 223.6 percent. A reswitching of the order of efficiency occurs over the range at which Epsilon is cost-minimizing. After the switch point, as illustrated in Figure 1, Alpha is cost-minimizing.

Figure 1 also illustrates a fake switch point at a rate of profits of approximately 219.0 percent. The wage curves for Alpha and Delta intersect at the fake switch point. The wage curve for Alpha is also the wage curves for Epsilon and Zeta. Likewise, the wage curve for Delta is also the wage curves for Eta and Theta. Epsilon is the unique cost-minimizing technique at and around this fake switch point. The prices of produced commodities (iron, wheat, and rye) differ, at the switch point, between the techniques for the two intersecting wage curves. In this sense, the fake switch point resembles the one in the example from Bidard and Klimovsky. The rent on type 2 land is positive under Epsilon, which would not be the case is this switch point was non-fake. Nor are the rents on type 1 land zero under Eta and Theta at this fake.

But consider again the switch point between Epsilon and Alpha. Under Alpha, only type 1 land is farmed, but only partially. Epsilon extends Alpha to produce wheat on type 2 land, to the extent of its endoment. The switch point lies along a single wage curve, which is anomalous.

Suppose that final demand was small enough that both Alpha and Gamma were feasible. For example, Alpha, Beta, Gamma, and Delta are the only feasible techniques when required net output consists of 10 bushels wheat and 10 bushels rye. Then Gamma is cost-minimizing from before the switch point, from approximately 176.8 percent. Alpha is cost-minimizing after the switch point. As with Epslion, under Gamma wheat is produced on type 2 land. But, unlike Epsilon, type 2 land is not farmed under Gamma to the extent of its endowment and the process in which wheat is produced on type 1 land is no longer operated. With this final demand, the example is one of reswitching between Gamma and Delta, at a lower rate of profits than shown.

Or suppose final demand consisted of 30 bushels wheat and 30 bushels rye. Then Beta, Epsilon, Iota, Kappa, and Lambda are feasible. Then this is a switch point between Epsilon and Iota. The same processes are operated under Epsilon and Iota, but which land is scarce varies. Iota is a technique in which landlords obtain intensive rent on type 1 land. The rent on type 1 land is negative under Iota before the switch point.

I have now found switch points:

• Along a single wage curve, with no intersecting wage curve; with variation in which processes are operated between the cost-minimizing techniques.
• In which wage curves intersect, but no variation occurs in which processes are operated between the cost-minimizing techniques.

The above switch point between Epsilon and Alpha combines these two phenomena, in some sense. I have also found fake switch points:

• In which wage curves intersect; the cost-minimizing technique does not vary.

These results suggest that concept of a switch point is not tightly tied to intersections of wage curves in models of joint production.

Michael Overton Interviewing Margaret Wright On Operations Research

I do not know what order these should be in or even if this is the entire interview. I did not know of either Margaret Wright or Michael Overton before stumbling on this. Apparently she was once president of the Society of Industrial and Applied Mathematicians, worked with George Dantzig at Stanford, and so on.

• Is linear programming polynomial time?
• Karmaker's algorithm
• Creating algorithms and writing software
• First job at GTE Sylvania
• Early years in California, Arizona and choosing Stanford
• Operations research at Stanford
• Faculty in math and computer science at Stanford
• Committees and SIAM
• Advisory Committees and the NAS
• Unexpected connections
• Accomplishements and progress for women

I think I have not appreciated how much progress has been made in the last few decades:

"... but look back and see the progress we've made. So in optimization at various points people have said well, for nonlinear problems, you can solve problems with hundreds of variables. Of course, that's a very imprecise statement. They never say what kind of problems or whatever. Today we had a talk where the person was talking about hundreds of thousands or millions of variables. That's an amazing change. It's an amazing change in capability. And it's not all due to computer hardware. I think sometimes people think, oh, machines are faster; machines are bigger. But it's smarter and better algorithms, which is the area you and I work in, right? Not making faster machines. We try to take advantage of them, but we don't make them run faster. And to me, that's astonishing. And the great part of it, I think there's still interesting problems that don't have many variables in this non-derivative optimization area." -- Margaret Wright

I do not understand interior point methods or why the Simplex algorithm works so well on average.

Anomalous Switch Points

This post is to remind me that I have discovered some anomalous switch points. I am introducing the concept of an anomalous switch point here.

Consider a switch point in a model of single production, with inputs of labor and circulating capital alone. At a (generic) switch point, one process replaces another in an industry that produces a commodity that exists in both techniques. Other processes can be introduced or removed if some capital goods are used only in one of the two processes for the common industry. The switch point is on the wage frontier formed by the outer envelope of the wage curves for all techniques. Two wage curves intersect at the switch point.

An anomalous switch point differs in some property from a generic switch point in models of circulating capital alone.

The anomalous switch points under consideration here are not flukes. A fluke switch point is one in which any permutation of some parameters destroys the qualitative property under consideration for the switch point. A switch point in which two wage curves are tangent on the frontier is an example of a fluke. A switch point in which two processes are replaced, one in each of two industries that exist in the techniques with intersecting wage curves. In this example with both types of flukes, and more, four wage curves intersect at switch point for the second kind of fluke.

Anomalous switch points include:

• A switch point in which the same processes are operated for both techniques: This example is one of extensive rent. The levels of operation of the processes and the land that is scarce vary around the switch point.
• Another switch point in which the same processes are operated for both techniques: This example is one of intensie and extensive rent with multiple agricultural commodities.
• A switch point along a single wage curve: This example combines fixed capital and extensive rent. Two techniques, Nu and Omicron, have the same 'solving system' and, hence, the same wage curve. The techniques differ in the economic life of machine when operated on scarce land that pays extensive rent.

Other switch points can be considered anomalous. D'Agata (1983) has examples with non-unique and non-exisitng cost-minimizing techniques in a model of intensive rent. Two techniques are cost-minimizing before a switch point. No cost-minimizing technique exists after the switch point. I think D'Agata's examples are more challenging to Sraffa's price theory than mine. Woods (1990) also has an example in joint production, without rent.

A non-fluke switch point in a model of joint production in whcih three wage curves intersect is another anomalous switch point. Bidard and Klimovsky have a genuine switch point like this in the paper in which they introduce the concept of a fake switch point. I think I also have examples in my models that combine intensive and extensive rent.

I do not consider anomalous a switch point on the wage frontier, in which the frontier is not the outer envelope of wage curves. The wage frontier must be the outer envelope in models of single production, but researchers in joint production established last century that this property need not hold in models of joint production, including in models of extensive rent.

Fake switch points can also be anomalous. With joint production, a question arises about which process should be replaced when a new process is introduced. A fake switch point, in a model of joint production, is an intersection of (non-tangent) wage curves in which the cost-minimizing technique does not change. The intersection's location and existence depend on the numeraire. The price of a commodity produced under both techniques whose wage curves intersect varies among techniques.

A fake switch point can be anomalous in that it deviates from properties of Biard and Klimovsky's example:

• Example fixed capital and extensive rent: Prices of commodities produced under both techniques do not vary among techniques at switch points. Price and rent vary between techniques only for non-commodities (that are free) under the non-cost-minimizing technique.

Technical terms: Switch point, Anomalous switch point, Fake switch point, Anomalous fake switch point, Fluke switch point.

References

• D’Agata, A. 1983. The existence and unicity of cost-minimizing systems in intensive rent theory, Metroeconomica 35: 147-158.
• Bidard, Christian and Edith Klimovsky. 2004. Switches and fake switches in methods of production. Cambridge Journal of Economics 28 (1): 88-97.
• Vienneau, Robert L. 2024. Characteristics of labor markets varying with perturbations of relative markups. Review of Political Economy 36(2): 827-843.
• J. E. Woods (1990) The Production of Commodities: An Introduction to Sraffa, Humanities Press International

A Reswitching Example With Extensive And Intensive Rent And Multiple Agricultural Commodities

Figure 1: Wage Curves For Feasible Techniques

1.0 Introduction

This post demonstrates a novel aspect of the reswitching of techniques. The cost-minimizing techniques in the example do not differ in which processes are operated. They differ in which lands are fully cultivated and thus obtain rent. In one technique, two commodities are produced, by distinct processes on the type of land that is fully farmed. In the other, one process, producing one commodity, is operated on the land that pays rent. In other words, the techniques that reswitch pay extensive and intensive rent, respectively.

The reswitching example depends on more than one agricultural commodity being produced. When the technique with extensive rent is cost-minimizing, two processes are operated on type 2 land. Type 2 land is not fully farmed. Two processes producing the same commodity cannot be operated on non-scarce land, away from a switch point, when prices of production prevail.

2.0 Technology, Endowments, Final Demands, and Techniques

Table 1 shows the inputs and outputs for each process known to the managers of firms. Iron is an industrial commodity, produced with no inputs from land. Two types of land are available for producing the agricultural commodities, wheat and rye. Wheat is produced by two processes, each operating on a different type of land. The same is true for rye. Inputs and outputs are specified in physical terms. For example, the inputs for process II, per bushel wheat produced, are 5/2 person-year, the services of one acre of type 1 land, 1/200 ton iron, 1/4 bushels wheat, and 1/300 bushels rye. Each process exhibits constant returns to scale (CRS), up to the limits imposed by the endowments of the lands.

Table 1: Processes Comprising the Technology

Inputs	Industries
	Iron	Wheat		Rye
	I	II	III	IV	V
Labor	a0,1 = 1/3	a0,2 = 5/2	a0,3 = 7/20	a0,4 = 1	a0,5 = 3/2
Type 1 Land	0	c1,2 = 1	0	c1,4 = 2	0
Type 2 Land	0	0	c2,3 = 5	0	c2,5 = 1
Iron	a1,1 = 1/6	a1,2 = 1/200	a1,3 = 1/100	a1,4 = 1	a1,5 = 0
Wheat	a2,1 = 1/200	a2,2 = 1/4	a2,3 = 3/10	a2,4 = 0	a2,5 = 1/4
Rye	a3,1 = 1/300	a3,2 = 1/300	a3,3 = 0	a3,4 = 0	a3,5 = 0
OUPUTS	1 ton iron	1 bushel wheat	1 bushel wheat	1 bushel rye	1 bushel rye

The specification of the problem is completed by defining the available endowments of land and the level and composition of final demand. Accordingly, assume 100 acres of each type of land are available. Suppose the required net output, also known as final demand, consists of 8 bushels wheat and 60 bushels rye.

Table 2 shows the available techniques for these parameters. Land is free for techniques Alpha, Beta, Gamma, and Delta. Techniques Epsilon, Zeta, Eta, and Theta pay extensive rent. Intensive rent is obtained by landlords for Iota, Kappa, Lambda, Mu, and Nu. Under Nu, intensive rent is obtained on both types of land.

Table 2: Technique

Name	Processes	Type 1 Land	Type 2 Land
Alpha	I, II, IV	Partially Farmed	Fallow
Beta	I, II, V	Partially Farmed	Partially Farmed
Gamma	I, III, IV	Partially Farmed	Partially Farmed
Delta	I, III, V	Fallow	Partially Farmed
Epsilon	I, II, III, IV	Partially Farmed	Fully Farmed
Zeta	I, II, IV, V	Partially Farmed	Fully Farmed
Eta	I, II, III, V	Fully Farmed	Partially Farmed
Theta	I, III, IV, V	Fully Farmed	Partially Farmed
Iota	I, II, III, IV	Fully Farmed	Partially Farmed
Kappa	I, II, IV, V	Fully Farmed	Partially Farmed
Lambda	I, II, III, V	Partially Farmed	Fully Farmed
Mu	I, III, IV, V	Partially Farmed	Fully Farmed
Nu	I, II, III, IV, V	Fully Farmed	Fully Farmed

Consider the Delta technique, for an example of relationships between these techniques. Under Delta, both wheat and rye are grown on type 2 land, but generally not to the limit imposed by the endowments of land. These same wheat and rye-producing processes are operated in Eta and Theta. In both of these techniques with extensive rent, type 2 land is still not farmed to the extent of its endowment. In Eta, wheat is produced on type 1 land to the extent of the endowment of type 1 land. In Theta, rye is produced on type 1 land to the extent of its endowment. But compare Theta to Mu. In Mu, the processes in Delta are also supplemented by growing rye on type 1 land, as in Theta. Type 1 land is not totally farmed, however, under Mu. Type 2 land is totally farmed to the extent of its endowment, with both wheat and rye contining to be produced in parallel on that type of land.

3.0 Quantity Flows and Feasible Techniques

The space of the final demand vector can be partitioned into regions by where each technique is feasible. Assume the final demand for iron is zero. Figure 2 shows a partition of the two-dimensional space for net output of wheat and rye. The point of final demand is indicated for the reswitching example in this post, with a final demand of 8 bushels wheat and 60 bushels rye.

Figure 2: Final Demands for Feasible Techniques

At the lowest level of final demand, Alpha, Beta, Gamma, and Delta are feasible. Land is in excess supply, and the model reduces to a model of circulating capital. As output expands, towards the specified final demand for the example, Delta becomes infeasible. At the limit of Delta's fesibility, net outputs of wheat and rye can be traded off. Lambda and Mu become feasible, with type 2 land obtaining intensive rent.

As final demand continues to expand, Alpha becomes infeasible. Here, too, at the limit for Alpha, the net outputs of wheat and rye can be traded off. Iota and Kappa are the other techniques that pay intensive rent, and they relate to Alpha in the same way that Lambda and Mu relate to Delta. The feasible techniques are now Beta, Gamma, Iota, Kappa, Lambda, and Mu.

As output continues to expand towards the specified point of final demand, Gamma becomes infeasible, with a constraint imposed by the endowment of type 1 land. Iota becomes infeasible, as well. Epsilon and Theta, which pay extensive rent, become feasible. For the reswitching example, Beta, Theta, Kappa, Lambda, and Mu are feasible.

4.0 Price Equations

A system of equations is defined for prices of production for each technique. I take Mu as an example.. Processes I, III, IV, and V are operated under Mu. Processes III and V combine to bring the entire endowment of type 2 land under cultivation. The prices of production for Mu, in obvious notation, satisfy the following equations

(p₁ a_1,1 + p₂ a_2,1 + p₃ a_3,1)(1 + r) + w a_0,1 = p₁

(p₁ a_1,3 + p₂ a_2,3 + p₃ a_3,3)(1 + r) + rho₂ c_2,3 + w a_0,3 = p₂

(p₁ a_1,4 + p₂ a_2,4 + p₃ a_3,4)(1 + r) + w a_0,4 = p₃

(p₁ a_1,5 + p₂ a_2,5 + p₃ a_3,5)(1 + r) + rho₂ c_2,5 + w a_0,5 = p₃

Each equation applies to a process operated under Mu. It is assumed that wages and rents are paid out of the surplus product, at the end of the period of production. The prices of advanced capital goods incur a charge for the rate of prices. A final equation equates the price of the numeraire to unity.

p₁ d₁ + p₂ d₂ + p₃ d₃ = 1

The above equations specify prices of production for Mu. A similar price system characterizes prices of production for each of the other techniques in the example.

The price system for Mu consists of five equations in six variables, r, w, p₁, p₁, p₁, and rho₂. The solution has one degree of freedom. If the rate of profits is taken as given, each of the other five variables can be expressed as a function of the rate of profits. The wage curve is a function of the rate of profits for a given technique. The rent curve is the corresponding function for rent.

The wage curves for the feasible techniques are plotted in Figure 3, at the top of this post. Wage curves are only shown for a technique for the ranges of the rate of profits in which rents are positive. Figure 3 shows graphs of the rent curves for the feasible technique. No rent is paid under Beta, and Beta is never cost-minimizing. Theta is cost-minimizing at high and low rates of profits. Mu is cost-minimizing for intermediate rates of profits.

Figure 3: Rent Curves for Feasible Techniques

5.0 Cost-Minimizing Techniques

Assertions above about the ranges of the rates of profits in which Theta and Mu are cost-minimizing have yet to be demonstrated. The cost-minimizing technique at a given rate of profits is:

• Feasible
• Such that no price of a produced commodity, wage, rate of profits, or rent is negative
• Such that extra profits cannot be obtained, at prices associated with the technique, by operating processes outside the technique

Extra profits exist when the difference between revenues and costs for a process is positive. Costs include a charge on the prices of the advanced capital goods for the given rate of profits, and the difference is taken for a unit level of operation of the process. Prices of production for a technique are such that extra profits are zero for the processes operated by that technique.

Figure 4 illustrates extra profits for Beta. Processes III and IV are not operated by Beta. Extra profits can always be obtained, under Beta, by operating process IV, whatever the rate of profits for which prices are found. Beta is never cost-minimizing.

Figure 4: Beta is Never Cost-Minimizing

Figure 5 shows that Theta is cost-minimizing whenever rent is positive for prices of production for the technique. The graph also makes obvious that the conditions, that rent be non-negative and that extra profits are not available, are independent of one another.

Figure 5: Theta is Cost-Minimizing at High and Low Rates of Profits

Figures 6 and 7 demonstrate that Kappa and Lambda are never cost-minimizing. Figure 8 demonstrates that Mu is cost-minimizing for the range of the rate of profits in which rent is positive for prices of production associated with the technique.

Figure 6: Kappa is Never Cost-Minimizing

Figure 7: Lambda is Never Cost-Minimizing

Figure 8: Mu is Cost-Minimizing at Intermediate Rates of Profits

6.0 The Demand for Labor and for Capital

The analysis of the choice of technique need make no mention of supply and demand functions. But the results allow us to plot the real wage against employment, for the given final demand (Figure 9). The relation can be interpreted as an economy-wide demand curve for labor. The switch points appear as horizontal segments in this graph. The 'perverse' switch point is a step function approximation to an upward-sloping demand curve for labor. The widespread tendency to draw downward-sloping labor demand functions, given ideal assumptions such as competitive markets, no search costs, and so on, lacks a coherent justification.

Figure 9: The Demand for Labor

The rate of profits can also be plotted against the value of advanced capital goods (Figure 10). Here, too, switch points are horizontal segments. The value of capital is the iron, wheat, and rye advanced at the start of the production period and evaluate at prices of production. The variation of the value of capital between switch points is known as a price Wicksell effect. The variation at a switch point across techniques is the real Wicksell effect. What happens at the 'perverse' switch point has also been called reverse capital deepening. At any rate, justification for drawing downward-sloping demand curves for capital, by default, is also lacking.

Figure 10: The Demand for Capital

7.0 Conclusion

This post extends the well-established critique of economic theory to which Sraffa (1960) is a prelude. In this example, the techniques that reswitch do not differ in which processes are operated. They differ in the scale at which these processes are operated, thereby resulting in a variation in which lands are scarce. The usual consequences of 'perverse' switch points appear in which, for example, a higher wage is associated with firms wanting to employ more workers to produce a given final demand.