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.

Balderdash In Teaching Of Labor Economics

Borjas (2016) seems to be a standard textbook in labor economics. It has the usual obsolete idea that labor markets would clear, as shown by supply and demand, under ideal conditions. Unemployment is to be explained by deviations from these ideal conditions. Why would a professor in such a backwoods place like Harvard be expected to know about results in capital theory from two-thirds of a century ago?

The last chapter, chapter 12, is on unemployment. It concludes with this summary:

• Although the unemployment rate in the United States drifted upward between 1960 and 1990, the economic expansion of the 1990s reduced the unemployment rates substantially.
• Even a well-functioning competitive economy experiences frictional unemployment because some workers will unavoidably be 'in between' jobs. Structural unemployment arises when there is an imbalance between the supply of workers and the demand for workers.
• The steady-state rate of unemployment depends on the transition probabilities among employment, unemployment, and the nonmarket sector.
• Although most spells of unemployment do not last very long, most weeks of unemployment can be attributed to workers who are in very long spells.
• The asking wage makes the worker indifferent between continuing his search activities and accepting the job offer at hand. An increase in the benefits from search raises the asking wage and lengthens the duration of the unemployment spell; an increase in search costs reduces the asking wage and shortens the duration of the unemployment spell.
• Unemployment insurance lengthens the duration of unemployment spells and increases the probability that workers are laid off temporarily.
• The intertemporal substitution hypothesis argues that the huge shifts in labor supply observed over the business cycle may be the result of workers reallocating their time so as to purchase leisure when it is cheap (that is, during recessions).
• The sectoral shifts hypothesis argues that structural unemployment arises because the skills of workers cannot be easily transferred across sectors. The skills of workers laid off from declining industries have to be retooled before they can find jobs in growing industries.
• Efficiency wages arise when it is difficult to monitor workers' output. The above-market efficiency wage generates involuntary unemployment.
• Implicit contract theory argues that workers prefer employment contracts under which incomes are relatively stable over the business cycle, even if such contracts imply reductions in hours of work during recessions.
• A downward-sloping Phillips curve can exist only in the short run. In the long run, there is no trade-off between inflation and unemployment.
• The combination of high unemployment insurance benefits, employment protection restrictions, and wage rigidity probably accounts for the high levels of unemployment observed in Europe in the 1980s and 1990s.

Borjas does note that the claim about European unemployment seems to be empirically untrue in that European institutions did not change much, and yet unemployment became lower. I suppose Borjas feels obligated to present theories of imperfections and rigidities that keep wages above the supposed market-clearing wage. Note some of these blame workers, who prefer leisure, would otherwise take advantage of information asymmetries by shirking, or obtain unemployment benefits. The science regerettably shows why we need to beat up on workers.

But, of course, this is pure ideology. Borjas teaches the obsolete theory that wages and employment would be successfully modeled in the long run by the intersection of well-behaved supply and demand curves, absent imperfections. He probably does not know better.

Reference

George J. Borjas. 2016. Labor Economics. Seventh edition. McGraw Hill.

Innovation Hurting Workers Or Capitalists

Figure 1: The Wage Frontier Is The Inner Envelope Of The Wage Curves For Feasible Techniques

1.0 Introduction

This post presents a solution to the homework problem 7.13 in Kurz & Salvadori (1995), Chapter 10. They assign credit for this problem to Antonio D'Agata. I extend it to include a negligible industrial commodity, as in my outline of a model with rent, multiple lands, and multiple agricultural commodities.

Two agricultural commodities, wheat and rye, are produced by the example economy. If only processes II and V existed, each commodity could only be produced on one type of land. With the given final demand and the given endowments of land, no land would be scarce and no landlords could obtain rent. Suppose innovations introduce new processes, III and IV, so that each commodity could be produced on each type of land. As a result, no long period exists in a range of the rate of profits towards its maximum, and landlords obtain rent at a lower rate of profits.

Nobody else, as far as I know, has considered the orders of efficiency and of rentability in a model with multiple agricultural commodities. Maybe I need to read further in Quadrio Curzio & Pellizzari's book.

2.0 Technology, Endowments, Final Demands, and Techniques

Table 1 shows the inputs and outputs for each process known to the managers of firms. 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 one person-year, the services of one acre of type 1 land, a tiny fraction of a ton iron, 3/10 bushels wheat, and 1/10 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 = 0.0001	a0,2 = 1	a0,3 = 3/2	a0,4 = 1/10	a0,5 = 1/2
Type 1 Land	0	c1,2 = 1	0	c1,4 = 1	0
Type 2 Land	0	0	c2,3 = 5	0	c2,5 = 2
Iron	a1,1 = 0.00001	a1,2 = 0.00001	a1,3 = 0.00001	a1,4 = 0.00001	a1,5 = 0.00001
Wheat	a2,1 = 0.00001	a2,2 = 3/10	a2,3 = 1/10	a2,4 = 1/10	a2,5 = 1/5
Rye	a3,1 = 0.00001	a3,2 = 1/10	a3,3 = 3/10	a3,4 = 1/5	a3,5 = 1/10
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 15 bushels wheat and 35 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 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

Suppose only processes I, II, and V are known by managers of firms. Then only the Beta technique is available. Wheat is grown on type 1 land, and rye on type 2 land. The endowments of land provide an upper limit on the level of final demand that can be feasibly satisfied. The innovations that make processes III and IV available provide a choice of technique, including which grains should be grown on which lands. The limits to feasible final demands are increased.

3.0 Quantity Flows

Beta, Kappa, Lambda, and Nu are feasible for the given final demand. Figure 2 shows which techniques are feasible for final demand consisting of any combination of specified quantities of wheat and rye. One type of land is farmed under both Alpha and Delta, with wheat and rye each produced on that type. The maximum final demand for each of these techniques is a downward-sloping straight line. The outer limits for Beta and Gamma have segments where constraints for each type of land kick in.

Figure 2: Feasible Final Demands

Techniques in which extensive rent is paid extend the two techniques with non-scarce land in which both lands are farmed. Epsilon and Theta become feasible when Gamma is no longer feasible. They differ in which type of land, still non-scarce, becomes used to grow both wheat and rye. The other type of land is cultivated to the extent of its endowment. Eventually the non-scarce land, on which both wheat and rye are produced, becomes scarce Zeta and Eta have the same relationship to Beta.

The limits of the final demand for techniques which pay intensive rent also relate to the boundaries on final demand for the other techniques. The maximum final demand for Alpha is the minimum for Iota and Kappa. Iota and Kappa both grow wheat and rye on type 1 land, as in Alpha, but to the full extent of its endowment. They vary with whether non-scarce type 2 land is used to produce wheat or rye. In the same way, the maximum final demand for Delta is the minimum for Lambda and Mu. When the maximum final demand for Iota is the maximum for Gamma (the minimum for Theta), type 2 land is not a constraint. When the maximum for Iota is the maximum for Epsilon (the minimum for Nu), both types of land are constraints. The maximum final demand for Kappa relates to the maximum for Beta and for Zeta in the same way. Likewise, the maximum for Lambda relates to the maximum for Beta and Eta. Finally, the maximum Mu is either the maximum for Gamma or Theta.

4.0 Price Systems

A system of equations for prices is associated with each technique. The going rate of profits is made in each process operated in the technique. I assume wages are paid out of the surplus product at the end of the period required to operate a technique. Rent is also paid on scarce land at the end of the period. A final equation sets the price of the numeraire to unity.

The variables defined by the price system consist of the rate of profits; the wage; the prices of iron, wheat, and rye; and the rents per acre of the two types of land. They are defined up to a degree of freedom. As usual, I take the dependence of the wage on the rate of profits as expressing that degree of freedom. Figure 1, at the top of this post, plots the wage curves for the four feasible technques. Figures 3 plots the rent curves.

Figure 3: Rent Curves

5.0 The Choice of Technique

For a technique to be cost-minimizing at a given rate of profits, the following must be true:

• It must be feasible.
• No price of a commodity, wage, or rent of a type of land can be negative.
• Extra profits cannot be obtained, at the prices associated with the technique, by operating a process not in the technique.

Extra profits are obtained if the difference between revenues and costs for a process, with the going rate of profits charged on advances for capital goods, is positive. Accordingly, I check whether a technique is cost-minimizing by plotting the difference between revenues and costs, for each process, with the prices of that technique.

Process IV pays extra profits throughout the range of the rate of profits in which the price system for Beta has positive orices and a positive wage (Figure 4). Gamma results from replacing the rye-producing process in Beta with process IV. Zeta and Kappa result from producing rye of both types of land. Gamma would be cost-minimizing at a rate of profits greater than approximately 65 percent if it were feasible. But only Kappa, out of Gamma, Zeta, and Kappa, is feasible. Above a rate of profits of approximately 100 percents, processes III and IV both obtain extra profits under Beta. So Beta is not cost-minimizing at any rate of profits.

Figure 4: Extra Profits at Beta Prices

At Kappa prices, process III obtains extra profits at a rate of profits above approximately 18.17 percent. Kappa is cost-minimizing only for rates a profits below this switch point.

Process IV obtains extra profits at Lambda prices for the entire range at which the rent on type 2 land is non-negative for the Lambda price system. Lambda is never cost-minimizing.

All processes are operated under the Nu technique. So extra profits cannot be obtained. But the rent on type 2 land is positive under Nu only when the rate of profits is greater than 18.7 percent. Thus, Nu is not cost-minimizing for a low rate of profits.

The maximum rate of profits for Nu is approximately 153.8 percent. The maximum for Beta is approximately 167.9 percent. Between these limits, Beta is feasible. The wage and the prices of the three produced commodities are positive in the price system for Beta. Both lands are free. Nevertheless, Beta is not cost-minimizing, and a long period position does not exist.

6.0 The Orders of Efficiency and Rentability

The wage curve for the Alpha technique lies on the outer envelope curve, for small rates of profits. The second wage curve, up to a rate of profits of approximately 65 percent, is Gamma's. After that rate of profits, up to the maximum, the wage curve for Gamma is the outermost. These wage curves are not shown in Figure 1.

Only type 1 land is farmed under Alpha. Both types are farmed in Gamma. As final demand expands for a low rate of profits, first type 1 land is cultivated and then both types are farmed. For a larger rate of profits, both types are initially cultivated together. Thus, the order of efficiency varies from type 1, 2 lands to an order in which they are tied (Table 3).

The order from high rent to low rent lands is type 1, 2, whether Kappa or Nu is cost-minimizing. Under Kappa, type 2 land is not scarce and is free. The orders of efficiency and rentability match for low rates of profits, up to a rate of profits in the range in which Nu is cost-minimizing. They differ for higher rates of profits insofar as the order of rentability is not tied.

Table 3: The Choice of Technique

Rate of Profits (Percent)		Technique	Order of Efficiency	Order of Rentability
Minimum	Maximum
0	18.2	Kappa	Type 1, 2	Type 1, 2
18.2	65.1	Nu
65.1	153.8		Type 1 and Type 2 tied

Suppose you take the order of efficiency as showing which lands, at a given rate of profits, contribute most to production. Since the order of rentability can differ from the order of efficiency, prices in competitive markets do not necessarily reward you for the contributions of the factors of production that you own. This conclusion is aside from the doctrine of Henry George.

This example is extremely restricted, when it comes to examining the orders of efficiency. The posibility of ties in the order of efficiency is a new possibility raised by the existence of multiple agricultural commodities. (I suppose you could look for different techniques having identical wage curves in a model of extensive rent and one agricultural comodity.)

7.0 Conclusion

The example shows how innovations create complications in the analysis of long run positions. In the example, innovations lead to the possibility of a class of landlords to come into existence. I doubt this happened like this anywhere. For a certain range of the rate of profits, a long period position no longer exists.

Kurz and Salvadori (1995) have additional numeric examples with multiple agricultural commodities. I should be able to create more. I am interested in examples with the reswitching of the order of fertility, as well as examples in which techniques with extensive and intensive rent are simultaneously feasible.

An Algorithm Trace For The Truncation Of Fixed Capital

1.0 Introduction

This post revisits my example of the recurrence of truncation without reswitching. In this example, the choice of technique consists of deciding on the economic life of a machine in each industry. I present an application of an algorithm to find the cost-minimizing technique, given the rate of profits. The algorithm needs more elaboration. A trace of the algorithm is a dynamic path through the space of techniques.

2.0 Technology and Techniques

I repeat the parameters that define the example in this section.

Tables 1 and 2 show the inputs and outputs for each process known to the managers of firms. For example, the inputs for the first process, at a unit level of operation, consist of 1/10 person-years, 1/16 bushels corn, and one new machine. The outputs, available after a year, are two new machines and one machine a year older.

Table 1: Inputs for The Technology

Input	Industry
	Machine		Corn
	I	II	III	IV
Labor	1/10	8	43/40	1
Corn	1/16	3/20	1/8	53/200
New Machines	1	0	1	0
One-Year Old Machines (1st type)	0	1	0	0
One-Year Old Machines (2nd type)	0	0	0	1

Table 2: Outputs for The Technology

Output	Industry
	Machine		Corn
	I	II	III	IV
Corn	0	0	1	14/25
New Machines	2	5/2	0	0
One-Year Old Machines (1st type)	1	0	0	0
One-Year Old Machines (2nd type)	0	0	1	0

With this specification of the technology, the economic life of the machine must be chosen in each industry. Table 3 lists the available techniques. The machine is truncated in both industries in the Alpha technique. The machine is operated for its full physical life in both industries in the Delta technique. In Beta and Gamma, the machine is truncated in one industry and operated for its full physical life in the other.

Table 3: Specification of Techniques

Technique	Processes	Notes
Alpha	I, III	Machines truncated in both industries.
Beta	I, II, III	Machines truncated in machine-production.
Gamma	I, III, IV	Machines operated at full physical life in both industries.
Delta	I, II, III, IV	Machines truncated in corn-production.

3.0 An Algorithm for Fixed Capital

I now present a hand-waving, incomplete specification of an algorithm for the choice of technique. This algorithm is supposed to apply when the choice of technique consists exclusively of the choice of the economic life of a machine in various industries.

1. Solve price system, given the rate of profits, for each technique.
2. Identify technique in which machines are operated for two years (longest in example).
• Beta and DELTA in the machine industry
• Gamma and DELTA in the corn industry
3. Find price of old machine in each industry. If it is negative, truncate to longest time in which it is first negative.
4. If a machine is truncated in any industry, repeat previous step.
5. For COST-MINIMIZING technique, prices of old machines are non-negative in all industries.

4.0 Traces

Which order should industries be considered? This is one way the above specification is incomplete. Maybe I should say this is a non-deterministic algorithm. Anyways, Table 4 shows the application of this algorithm starting with the first industry in the example.

Table 4: The Algorithm, Starting with the Machine Industry

Calculate the price of an old machine in the machine industry with Delta prices.
Price negative for 0 ≤ r < 71.2 percent		Price positive for 71.2 percent < r ≤ Rδ
Truncate to Gamma		Keep Delta
Calculate the price of an old machine in the corn industry with Gamma prices.		Calculate the price of an old machine in the corn industry with Delta prices.
Price negative for 0 ≤ r < 70.2 percent	Price positive for 70.2 < r < 71.2 percent	Price positive for 71.2 < r < 87.5 percent	Price negative for 87.5 percent < r < Rδ
Truncate to Alpha	Keep Gamma	Keep Delta	Truncate to Beta
Calculate the price of an old machine in the machine industry with Beta prices.			Calculate the price of an old machine in the machine industry with Beta prices.
Price negative for 0 ≤ r < 70.2 percent			Price positive for 87.5 < r ≤ Rδ
Keep Alpha			Keep Beta

Perhaps the termination criterion for the algorithm should include that a longer economic life of machines has been considered in each industry. In the range of profits in which Alpha is cost-minimizing, I consider extending the economic life of the machine in the corn industry, for the start of the last three rows. These steps extend the algorithm in section 3. Are these steps necessary? When I find a negative price for an old machine in such an extension, can I stop? Or should I, in other examples, continue consider extensions up to the physical life of the machine? I know that truncation can jump from three years, for example, to one year.

Table 5 shows the application of the algorithm starting with the second industry in the example. These two tables illustrate that it does not matter which industry is considered first. I suppose this algorithm, like Christian Bidard's market algorithm, could be distributed across industries, with steps being executed in parallel. I think that if somebody was going to elaborate on this claim, they should consider a specification of market algorithms in a language designed for parallel processing, such as Tony Hoare's Communicating Sequential Processes.

Table 5: The Algorithm, Starting with the Corn Industry

Calculate the price of an old machine in the corn industry with Delta prices.
Price positive for 0 ≤ r < 87.5 percent			Price negative for 87.5 percent < r ≤ Rδ
Keep Delta			Truncate to Beta
Calculate the price of an old machine in the machine industry with Delta prices.			Calculate the price of an old machine in the machine industry with Beta prices.
Price negative for 0 ≤ r < 71.2 percent		Price positive for 71.2 < r < 87.5 percent	Price positive for 87.5 < r ≤ Rδ
Truncate to Gamma		Keep Delta	Keep Beta
Calculate the price of an old machine in the machine industry with Gamma prices.
Price negative for 0 ≤ r < 70.2 percent	Price positive for 70.2 < r < 71.2 percent
Truncate to Alpha	Keep Gamma
Calculate the price of an old machine in the machine industry with Beta prices.
Price negative for 0 ≤ r < 70.2 percent
Keep Alpha

5.0 Conclusion

How should the algorithm be modified for a rate of profits towards the maximum? Can a proof be found that the convergence of the algorithm does not depend on the order in which industries are considered? Once a machine is truncated, is it true, the extension of the economic life a machine need never be considered in any industry?

Socialism Works in Kerala, India

Socialists and communists have been elected in many places, for significant periods of time. And those places did not necessarily turn into dysopian tyrannies. Often, they did not get further than social democratic policies, improving the lives of most citizens. 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 one of those places that I do not know much about.

Kerala is a state in India. Currently, the governing party in Kerala is a coalition, the Left Democratic Front (LDF). The Communist Party of India (Marxist) (CPI(M)) is the dominant party. Communists have traded off governing with another coalition, the United Democratic Front, since Kerala's founding, in 1957. They have democratic elections.

The UDF is dominated by the Congress party, which governed India for a long time. I recollect Indira Gandhi, for example. I have no idea how the current national government, headed by Narendra Modi and the Bharatiya Janata Party (BJP), impacts Kerala. My impression is that the BJP raises the question of what is fascism? (By the way, I do not necessarily understand the political positions of some of these newspapers I link to.)

Compared to the rest of India, Kerala has a high Human Development Index (HDI), high literacy, high life expectancy, low poverty rates. For some reason, I have the impression that it has a well-established Information Technology industry.