A Four-Technique Pattern

Figure 1: Partition of the Parameter Space

1.0 Introduction

I here provide some notes on a perturbation of an example from Salvadori and Steedman (1988).

Consider an economy in which n commodities are produced in n industries. In each industry, a single commodity is produced from inputs of labor and the services of previously produced capital goods. Suppose the technology can be represented in each industry by a continuously-differentiable production function. The wage-rate of profits frontier for such a model does not contain any switch points. In other words, for each feasible rate of profits, a single technique is cost minimizing. Nevertheless, the cost-minimizing technique varies continuously with the rate of profits. Furthermore, the process associated with the cost-minimizing technique in each industry also varies continuously with the rate of profits.

Suppose, instead, that the processes in each industry were represented by a set of fixed-coefficient processes, instead of a smooth production function. What would hold in a discrete model that is in the spirit of the neoclassical model? I suggest that at each switch point on the frontier, 2ⁿ wage curves would intersect. In a model with two produced commodities and two processes available in each industry, four wage curves would intersect at the single switch point. With three produced commodities, eight wage curves would intersect. The natural properties for a neoclassical model - if that is what this is - are flukes to several degrees.

I do not necessarily claim anything revelatory from the details of this post. I am testing the applicability of my pattern analysis by trying it out for various examples. Although you cannot tell from my presentation, the graphs I draw rely less on numerical approximations than in many of my earlier examples. This example is the first I have seen where a pattern with a co-dimension of two or higher happens to form a one-dimensional locus (curved line) in the two-dimensional slice of the parameter space I graph. Salvadori and Steedman could have varied their example in an infinite number of ways and still had an example where all processes varied at a switch point.

2.0 Technology

I make my usual assumptions about technology. At a given point in time, managers of firms know of a number of production processes (Table 1). A single commodity - a ton iron or a bushel corn in the example - is the output of each process. Each process lasts a year and exhibits constant returns to scale. Inputs are defined in physical units. For example, labor inputs are specified in terms of person-years per ton iron output or per bushel corn output. All inputs are used up in production; there is no fixed capital or joint production.

Table 1: The Technology for a Two-Industry Model

Input	Iron Industry		Corn Industry
	(a)	(b)	(c)	(d)
Labor	1 e1 - σ t	2 e1 - φ t	1	2
Iron	0	0	2/3	1/2
Corn	(2/3) e1 - σ t	(1/2) e1 - φ t	0	0

To produce a self-sustaining net output with this technology, both iron and corn must be produced. Four techniques can be defined with this technology (Table 2).

Table 2: Techniques in a Two-Commodity Model

Technique	Processes
Alpha	a, c
Beta	b, d
Gamma	a, d
Delta	b, c

I have defined the technology such that coefficients of production decrease with time in both processes for producing iron. The rate at which they decrease differs between the two processes. A more general case would allow for technical process in each of the processes for producing corn.

3.0 A Temporal Path

I first consider the variation with time of prices of production for a special case. Consider:

σ = φ = 1

I make the usual assumptions for prices. Relative spot prices are stationary, such that the same rate of profits is earned in both industries if the technology at a given point of time had prevailed over the year. I assume labor is advanced, and wages are paid out of the surplus at the end of the year. A bushel corn is taken as the numeraire. Supernormal profits cannot be made for either process comprising the chosen technique(s). No process in use incurs extra costs.

Figure 2 shows how cost-minimizing techniques, the maximum rate of profits, and switch points vary with time. In the region label 1, the Beta technique is cost-minimizing for all feasible rates of profits. The Gamma technique is cost-minimizing for high wages and low rates of profits in Reqion 2. A single switch arises, where wage curves for the Beta and Gamma techniques intersect on the frontier. In the language of the technical terminology I have been introducing, the boundary between Regions 1 and 2 is a pattern across the wage axis. Other patterns are labeled in the diagram.

Figure 2: Variation of Switch Points with Time

When t = 1, this model reduces to Salvadori and Steedman's example. A single switch exists, with a rate of profits, r₀, of 20 percent and a wage of (1/5) bushel per person-year. The wage curves for all four techniques intersect at the switch point. I call the boundary between Regions 5 and 7 a four technique pattern.

I argue that a four technique pattern is of co-dimension two, in my jargon. Each pattern is defined for a switch point. So, in a pattern, at least two wage curves intersect at a switch point:

w_α(r₀) = w_γ(r₀)

The co-dimension is the number of additional conditions that must be satisfied for the pattern. Here are two more conditions:

w_β(r₀) = w_δ(r₀)

w_α(r₀) = w_β(r₀)

In this example, for any switch point between the Alpha and Beta techniques, all processes are cost-minimizing. Thus, all techniques are cost-minimizing at such a switch point. For any set of parameters (σ, φ, t) at which there exists a switch point on the frontier between Alpha and Gamma and between Beta and Delta, all techniques are cost-minimizing. In the example, the first two conditions imply the third because of the processes of which the techniques are composed. I think this implication does not hold in general, for all technologies. So I think the definition of a four technique pattern must include three equalities.

4.0 Partition of the Parameter Space

The above analysis can be generalized, to consider any combination of (σ t) and (φ t). Figure 1, at the top of the post, partitions the parameter space into seven regions. In any given region, the switch points and the wage curves along the frontier do not vary qualitatively. (Maximum wage, maximum rate of profits, and rate of profits for switch points may vary.) Table 3 lists the switch points and wage curves along the wage frontier, for each region.

Table 3: Cost-Minimizing Techniques

Region	Switch Points	Techniques
1	None	Beta
2	Between Beta & Gamma	Gamma, Beta
3	None	Gamma
4	Alpha & Gamma	Alpha, Gamma
5	Alpha & Gamma, Beta & Gamma	Alpha, Gamma, Beta
6	Beta & Delta	Delta, Beta
7	Alpha & Delta, Beta& Delta	Alpha, Delta, Beta

As an aid to visualization, I present some specific configuration of wage curves. Consider the point in the parameter space that is simultaneously on the boundary of Regions 1, 2, 5, 6, and 7. At this point, all techniques are cost-minimizing for a rate of profits of zero. It is simultaneously a four-technique pattern and patterns across the wage axis. Figure 3 shows the wage curves in this case. For feasible positive rates of profits, the Beta technique is uniquely cost-minimizing.

Figure 3: Patterns over the Wage Axis

Figure 1 shows loci for four wage patterns intersecting at the point in the parameter space with wage curves illustrated above. Since six pairs of (unordered) techniques can be chosen from four techniques, one might think that six wage patterns should intersect at this point. But I am only defining patterns for switch points on the frontier. To illustrate, consider figure 4, which shows wage curves for a point in Region 5. The wage curves for the Gamma and Delta techniques intersect on the wage axis. Neither, however, are cost-minimizing here; the Alpha technique is cost-minimizing for a rate of profits of zero.

Figure 4: Wage Frontier in Region 5

Region 7 is the other region in three techniques are cost-minizing along the wage frontier. Figure 5 illustrates Region 7. For this particular set of parameters, the wage curves for the Gamma and Delta techniques are tangent at a point within the wage frontier. As far as I can tell, no reswitching patterns arise in this example, for switch points on the frontier.

Figure 5: Wage Frontier in Region 7

It is also the case that if one extends Figure 1 to the right, the locus for the four-technique pattern never ends. There is not some set of parameter values where the wage curves for all techniques intersect at the maximum rate of profits.

In a perturbation of the example, one can find a set of parameters at which the wage curves for all four techniques intersect at a switch point for a rate of profits of zero. And the parameters can be varied such that the rate of profits for a switch point for all four techniques can be any positive rate of profits.

Reference

• Salvadori, Neri and Ian Steedman. 1988. No Reswitching? No Switching! Cambridge Journal of Economics, 12: 481-486.

From Odo's Prison Letters

For we each of us deserve everything, every luxury that was ever piled in the tombs of the dead kings, and we each of us deserve nothing, not a mouthful of bread in hunger. Have we not eaten while another starved? Will you punish us for that? Will you reward us for the virtue of starving while others ate? No man earns punishment, no man earns reward. Free your mind of the idea of deserving, the idea of earning, and you will begin to be able to think. -- Ursula K. Le Guin (21 October 1929 - 22 January 2018)

Labor Values Taken As Given

I have been considering a case in which a simple Labor Theory of Value (LTV) is a valid theory of prices of production. When, for each technique, all processes have the same organic composition of capital, prices of production are proportional to labor values. Given labor values and direct labor coefficients in each industry, an uncountably infinite number of techniques - as specified by a Leontief input-output specified in terms of physical inputs per physical outputs - satisfies these conditions.

In outlining this mathematics, I start with labor values and derive technical conditions of production as a detour on the way to prices of production. (I have also considered a perturbation of this possibility, as an application of my pattern analysis.)

Has anybody commenting on Marx actually started with labor values, taken as given, in this way? If this is a straw person, I am good company. Ian Steedman (1977) makes something like the same accusation. See the section, "A spurious impression", in Chapter 4, "Value, Price, and Profit Further Considered", of his book.

But I have found examples of other approaching Marx in something like this way. I refer to von Bortkiewicz (1907) and Seton (1957), two authors taken as a precursor to the Sraffian reading of Marx. The fact that Steedman can be read as criticizing such authors complicates the claim that this literature exhibits continuity. I think others have also argued that some novelty arises in Steedman's critique insofar as he argues that labor values are redundant, since prices of production are properly calculated from technical data on production and the physical composition of wage goods.

Perhaps my examples of Bortkiewicz and Seton should not be read as propounding any large claim that Marx takes labor values as more fundamental, in some sense, than physical conditions in production processes. Rather, Bortkiewicz started from the schemes of simple and expanded reproduction at the end of Volume 2 of Capital. Since Seton, and other authors, were generalizing and commenting on Bortkiewicz, they, as a matter of path dependence, happened to keep the assumption of given labor values. One wanting to argue for a reading of Marx that I seem to be stumbling into, without any firm commitment, needs to deal with Volume 1.

I have two additional notes on rereading these references. First, I like to talk about Marx' invariants in the transformation problem. I thought I had taken this term from formal modeling in computer science. Edsger Dijkstra and C. A. R. Hoare talk about loop invariants, and I sometimes even comment my code with explicit statements of invariants. But Seton has a section titled "Postulates of Invariance".

Is Steedman disappointed in the reception of his book? Obviously, his points about the transformation problem, including the possibility of negative surplus value being consist with positive profits, under a case of joint production, have been widely discussed. But consider his exposition of simple examples intended to demonstrate that Sraffa's analysis can take into account all sorts of issues that some had argued were ignored. Consider letting how much work capitalists can get out of labor being a variable, heterogeneous types of abstract labor not reducible to one and the possibility of workers of each type exploiting others, wages being paid, say, weekly, during processes that take a year to complete, how wages relate to the rate of exploitation when a choice of technique exists, the treatment depreciation of capital, and the existence of a retail sector for circulating produced commodities. How many of these analyses have been taken up and continued by those building on Sraffa? (I think some have.)

References

• Eugen von Bohm-Bawerk (1949). Karl Marx and the Close of his System: Bohm-Bawerk's Criticism of Marx. Edited by P. M. Sweezy.
• Ladislaus von Bortkiewicz (1907). On the Correction of Marx's Fundamental Theoretical Construction in Third Volume of Capital, Trans. by P. M. Sweezy. In Bohm-Bawer (1949).
• F. Seton (1957). The "Transformation" Problem. Review of Economic Studies, 24 (3): 149-160.
• Ian Steedman (1981, first edition 1977). Marx After Sraffa, Verso.

Start of a Catalogue of Flukes of Fluke Switch Points

I claim that the pattern analysis I have defined can be used to generate additional fluke switch points. I am particularly interested in switch points that are flukes in more than one way (local patterns of co-dimension higher than one) and fluke switch points that are combined with other fluke switch points or some aspect of other switch points (global patterns). I have already generated some examples, not always with pattern analysis.

• Fluke switch points of higher co-dimension
• A switch point that is simultaneously a pattern across the wage axis and a reswitching pattern (a case of a real Wicksell effect of zero).
• A switch point combining four three-technique patterns (due to Salvadori and Steedman).
• Fluke switch points combined with other switch points
• A reswitching example with one switch point being a pattern across the wage axis (another case of a real Wicksell effect of zero).
• An example with a pattern across the wage axis and a pattern over the axis for the rate of profits.
• Two switch points with both being reswitching patterns can be found from a partition of a parameter space where two loci for reswitching patterns intersect.
• A pattern across the point where the rate of profits is negative one hundred percent, combined with a switch point, for the same techniques, with a positive rate of profits (of interest for the reverse substitution of labor).
• An example where every point on the frontier is a switch point.

The above list is not complete. More types of fluke switch points exist. Some, like the examples of a real Wicksell effect of zero, I thought, should be of interest for themselves to economists. Others show examples of parameters where the appearance of the wage frontier, at least, changes with perturbations of the parameters. I would like to see that in at least some cases, short run dynamics changes qualitatively with such perturbations. But this seems to be beyond my capabilities.

Maybe I'll update this post some day, if I create more examples.

A Pattern For The Reverse Substitution Of Labor

Figure 1: Variation of Switch Points with Time

1.0 Introduction

This post presents another local pattern of co-dimension one. I have conjectured that only four types of local patterns of co-dimension one exist (a reswitching pattern, a three-technique pattern, a pattern across the wage axis, and a pattern over the axis for the rate of profits). In this conjecture, I meant to implicitly limit the rate of profits at which switch points occur to be non-negative and not exceeding the maximum rate of profits. The pattern illustrated in this post is a pattern around the rate of profits of -100 percent. (I was prompted to develop this example by an anonymous comment, as I was also prompted for this post.)

Although this is a local pattern, its interest comes from global effects. Suppose another switch point exists, other than the one at a rate of profits of -100 percent. This other switch point involves the same two techniques and occurs at a positive rate of profits. A perturbation of coefficients of production around the pattern changes the other switch point from one exhibiting a conventional substitution of labor to the reverse substitution of labor. Han and Schefold (2005) describe empirical examples of the reverse substition of labor.

2.0 Technology

Table 1 specifies the technology for this example. I make the usual assumptions. Each column lists inputs per unit output for each process. Each process exhibits constant returns to scale. Each process requires a year to complete, and there are no joint products. Inputs of capital goods are totally used up in production.

Table 1: Processes of Production

Input	Iron Industry	Corn Industry
		Alpha	Beta
Labor	1 Person-Yr.	9/10	0.994653826 e-σ t
Iron	(7/10) Ton	1/40	0.002444903 e-σ t
Corn	2 Bushels	1/10	0.746512055 e-σ t

In this example, technical change occurs in the Beta process for producing corn. I assume that σ is 1/10. So coefficients of production fall at a rate of ten percent.

At any moment of time, two techniques can be created out of these processes. The Alpha technique consists of the iron-producing process and the corn-producing process labeled Alpha. Likewise, the Beta technique consists of the iron-producing process and the corn-producing process labeled Beta.

3.0 Prices and Structural Economic Dynamics

I consider a common system for defining prices of production. Relative spot prices are assumed to be constant, and the same rate of profits is earned in both industries for the cost-minimizing technique. Labor power is advanced, and wages are paid out of the surplus product at the end of the year. For a technique that is not cost-minimizing, the costs for operating the corn-producing process in this technique, as evaluated at prices of production, exceed the revenues. I take a bushel corn as the numeraire.

These assumptions allow one to construct wage curves for each technique. The cost-minimizing technique at, say, a given rate of profits is found from the outer envelopes of the wage curves, that is, the wage frontier. Switch points arise at rate of profits for which both techniques are cost-minimizing. For this example, I do not present wage curves at selected moments of time. Figure 1 graphs the rate of profits at switch points and the maximum rate of profits against time. Patterns, including a pattern for the reverse substitution of labor, are indicated on the graph.

3.1 A Superficial Neoclassical Story

Suppose one limits one's analysis to non-negative rates of profits. Figure 1 shows that technical progress leads to a switch point at the maximum rate of profits. As the wage curve for the Beta technique continues to move outward, this switch point falls below the maximum rate of profits. For rate of profits lower than at the switch point, the Alpha technique is cost-minimizing. The Beta technique is cost-minimizing at higher rates of profits. Eventually, the switch point disappears across the wage axis, and only the Beta technique is cost-minimizing.

This story initially seems to correspond to exploded neoclassical intuition about technical change. Reswitching and capital-reversing - two phenomena much emphasized in the Cambridge Capital Controversy - never occur. Around the switch point with a positive rate of profits, the Beta technique is cost-minimizing at a notionally smaller wage, and the Alpha technique is cost-minimizing at a notionally higher wage. A lower wage is associated with a technique in which greater labor inputs, aggregated across both industries, are employed per bushel of corn produced net.

3.2 A Region in which the Reverse Substitution of Labor Occurs

But consider what happens when the analysis is extended to a rate of profits of -100 percent. A switch point with a positive rate of profits exists only for time between the patterns over the axis for the rate of profits and across the wage axis. Figure 2 graphs the difference in the labor coefficients with time. After the pattern for the reverse substitution of labor, the labor coefficient for the Alpha process in producing corn exceeds the labor coefficient for the Beta process in producing corn. That is, around the switch point, the adoption of the cost-minimizing technique at a lower wage results in less labor being employed in corn production per unit corn produced gross. How is this consistent with the textbook account of labor demand functions?

Figure 2: Change in Labor Input per Unit Gross Output in Corn

4.0 Conclusion

The more I investigate price theory, the less I understand how economists can teach neoclassical microeconomics.

Labor Values As A Foundation

Figure 1: Physical Production Data as a Side Route

1.0 Introduction

One way of reading the first volume of Marx's Capital is that labor values provide a foundation, upon which the structure of prices of production and, eventually market prices are based. I find that, for example, Joseph Schumpeter presents Marx's work in this way.

Another reading takes both labor values and prices as founded on physical data specifying the technique in use. Ian Steedman, as illustrated in Figure 2, argues for such a reading. Furthermore, Steedman argues that one cannot get from the system of labor values to prices. Labor values are not needed for analyzing prices of production; they are redundant.

Figure 2: Labor Values as a Side Route

These are not the only possible ways of reading Marx. Another reading might emphasize the bits on commodity fetishism. Nothing is hidden. In selling produced commodities on the market, the concrete work activities that go into making commodities are abstracted from and treated as commensurable. This is crazy, but according to Marx, this is how capitalism works.

I seem to have stumbled on some mathematics supporting the first reading. I consider the question of what physical data is consistent with given labor values and direct labor inputs, under the condition that the organic composition of capital does not vary among industries. The issue is not that there is no way to go from labor values, through data on physical production, to prices. Rather, there are too many routes - in fact, an infinite number of them.

Figure 1 is not quite how I present my results in my draft paper. I end up with the wage curve for the price system; unlike in the above diagram, I do not close the system. I am not sure I am correct on how I specify distributive variables in the figure. I end up with the wage as a vector, where the same money wage is earned in each industry. I found it natural to close the system with the rate of profits when going from labor values to prices. On the other hand, I found it more convenient to specify the wage in going from physical data to prices. Perhaps these closures need more thought.

A substantial issue is whether it makes any sense to talk about labor values prior to and independently of physical data on processes of production. Steedman asserts it is not possible. Marx, in the first volume of Capital goes back and forth between labor values and prices. I might need to think a little more about how money, or the choice of a numeraire, fits into this, but I seem to be arguing for this possibility, at least under the conditions in which a simple labor theory of value holds as a theory of price.

Perturbation Of An Example With A Continuum Of Switch Points

Figure 1: A Partitioning Of The Parameter Space

1.0 Introduction

I consider here a case where two different techniques have the same wage curve. A simple labor of theory of value describes prices in the case under consideration. I treat the labor coefficient and another coefficient of production for a process in one technique as parameters. And I look at what happens when they vary.

A note on terminology: on the basis of expert advice and peer review, I am no longer using the term "bifurcation" for a pattern of switch points where a perturbation of model parameters, such as coefficients of production, removes or adds a switch point to the wage frontier. Instead, I am calling such a configuration a "pattern."

2.0 Technology

Table 1 specifies the technology for this example. I make the usual assumptions. Each column lists inputs per unit output for each process. Each process exhibits constant returns to scale. Each process requires a year to complete, and there are no joint products. Inputs of capital goods are totally used up in production.

Table 1: Processes of Production

Input	Iron Industry		Corn Industry
	Alpha	Beta
Labor	(1/8) Person-Yr.	u	(1/2)
Iron	(1/2) Ton	v	2
Corn	(1/16) Bushel	(1/80)	(1/4)

Two techniques can be created out of these processes. The Alpha technique consists of the iron-producing process labeled Alpha and the corn-producing process. Likewise, the Beta technique consists of the iron-producing process labeled Beta and the corn-producing process.

I think I'll say something about how I created this example. A simple labor theory of value applies to the Alpha technique. The wage curves associated with the Alpha and Beta techniques are identical when u = (1/8) person-year and v = (7/10) ton. This special case is an application of some math I have set out in a working paper.

3.0 Prices

I take corn as the numeraire and assume labor is advanced. Wages are paid out of the surplus at the end of year.

3.1 Alpha

The price of production for iron, when the Alpha technique is in use, is:

p_α = (1/4)

The wage curve for the Alpha technique is:

w_α = (1/2)(1 - 3 r)

3.2 Beta

The price of production for iron, when the Beta technique is in use, is:

p_β = [(1 + 120u) + (1 - 40u)r)]/{80[(4u - v)r + (1 + 4u - v)]}

The wage curve for the Beta technique is:

w_β = [(10v - 1)r² -4(5v + 3)r + (29 - 30v)]

/{20[(4u - v)r + (1 + 4u - v)]}

3.3 Switch Points

One finds switch points by equating the two wage curves:

w_α = w_β

One obtains a quadratic equation in the rate of profits, r:

+ (120 u - 20 v - 1) (r_switch)²

+ (80 u - 40 v +18) r_switch

+ (-40 u - 20 v + 19) = 0

This equation can be factored:

(r + 1)[(120 u - 20 v - 1)r + (-40 u - 20 v + 19)] = 0

4.0 Special Cases

4.1 A Continuum of Switch Points

I first want to check the special case u = (1/8) and v = (7/10). Recall, this example was created so that the wage curves for the two techniques would be identical in this case. In this special case, the two coefficients for the second factor of the Left Hand Side of the above quadratic equation reduce to zero. So that equation is identically true for all feasible rates of profits. Every point on the wage frontier is a switch point.

4.2 A Pattern Over the Wage Axis

In this pattern, a switch point exists on the wage frontier for a rate of profits of zero. That is, one must be able to factor out r from the left-hand side of the above quadratic equation. In other words, the constant term for the second factor must be zero. One thereby obtains:

-40 u - 20 v + 19 = 0

Or

v = - 2 u + (19/20)

4.3 A Pattern Over the Axis for the Rate of Profits

In this pattern, the wage curves have a switch point at the maximum rate of profits. I did not start with the quadratic equation for this special case. The maximum rate of profits for the Alpha and Beta techniques are equal when the two wage curves are identical. The maximum rate of profits for the Beta technique does not depend on the value of labor coefficients. Thus, the condition for this pattern is:

v = (7/10)

You can check that above condition yields a switch point at a rate of profits of (1/3).

4.4 A Reswitching Pattern

In a reswitching pattern, the wage curves for two techniques are tangent at a switch point. For the example in which only two commodities are produced, the quadratic equation obtain by equating the wage curves for two techniques has two repeated roots. In other words, the discriminant for this quadratic equation must be equal to zero. Some algebra gives (some Octave code was useful here):

400 (8 u - 1)² = 0

Or:

u = (1/8)

If v is not equal to (7/10), the above value of u results at repeated roots for a rate of profits of -1. But I am only considering non-negative rates of profits. Thus, a reswitching pattern does not exist for this example at feasible rates of profits.

5.0 Visualization

I can bring the above observations together with various pictures here.

5.1 Variation of Switch Points with Coefficients of Production

Consider how the wage frontier varies with v, given a particular parameter value of u. (After reading this section, one might consider how the wage frontier varies with u, given a particular value of v.)

Suppose u is smaller than the special case in which the wage curves for the two techniques are identical for a specific value of v. Figure 2 illustrates this case. In a certain region of variation in v, the wage curves for the Alpha and Beta techniques appear on the frontier, with a single switch point between them.

Figure 2: Variation of v, Case 1

As u increases, towards 1/8, the interval for v in which both wage curves appear on the frontier gets smaller and smaller. Figure 3 shows that, in the limit, this interval narrows to a width of zero. Both wage curves are identical. The wage frontier consists of a continuum of switch points.

Figure 3: Variation of v, Case 2

As u increases beyond 1/8, an interval for v once again appears in which the wage frontier contains a single switch point. As shown in Figure 4, the the endpoints of the interval have become interchanged, in some sense.

Figure 4: Variation of v, Case 3

5.2 A Partitioning of the Parameter Space

Figure 1, at the top of this post, graphs the parameter space for u and v. The patterns across the wage axis and the over the axis for the rate of profits divide the parameter space into the four numbered regions. Table 2 lists the switch points and the techniques along the wage frontier, in each region, in order of an increasing rate of profits.

Table 2: The Wage Frontier By Region

Region	Switch Points	Techniques
1	None	Alpha
2	One	Beta, Alpha
3	None	Beta
4	One	Alpha, Beta

My methods for pattern analysis and visualization apply in this case generalizing an instance in which two techniques have identical wage curves.

6.0 Conclusions

I have conjectured that four types of patterns of co-dimension one exist (the three-technique pattern, the pattern over the wage axis, the pattern over the axis for the rate of profits, and the reswitching pattern). This example of two techniques having identical wage curves is not a counter-example to this conjecture. It is simultaneously a a pattern over the wage curve and a pattern over the axis for the rate of profits. Thus, it is at least of co-dimension two.

The conditions for those two patterns, however, are not sufficient for this pattern. They are merely necessary. One could have two wage curves with switch points on the wage axis and on the axis for the rate of profits, but differing for all positive rate of profits less than the maximum rate of profits. The example does make me wonder about my distinction between local and global patterns; this is not the type of global pattern I had in mind when I came up with the idea. And what is the co-dimension for this pattern? Is it of an uncountably infinite co-dimension?

I can see why some might think my write-up is not all that exciting. Likewise, there is a certain amount of tedium in performing the analysis documented above. Nevertheless, I was intrigued to find the above picture emerging. I think I have stumbled upon a vast unexplored landscape in which complicated fluke cases can fit.

Elsewhere

• Steve Keen and others, in a showy bit of performance art in London, have called for a reformation of economics. Imitating Luther, they have nailed some theses to a door. Here's some links:
• Guardian article
• Letters to the editor in response.
• New Weather Institute blog post.
• 33 Theses.
• An article by Ben Chu, in the Independent saying, more or less, let's not get carried away.
• I do not know who Charles Mudede is or what his platform is. His style is more popular and very different from mine. Examples:
• On Seattle's minimum wage, in which he brings up an imperfectionist thesis related to the Cambridge Capital Controversy.
• On Cornel West vs. Ta-Nehisi Coates. I think the idea that identity politics associated with post modernism accommodates neoliberalism is not new (see references below). I don't want to box Coates in, but the way he writes about the Black body in Between the World and Me is definitely a post modern trope. But he writes about it, I guess, because it make sense of his lived experience.
• I stumble upon a tweet by Duncan Weldon, in which he says he resolves every year to try and understand the Cambridge Capital Controversy.

References

• Samir Amin (1998). Spectres of Capitalism: A Critique of Current Intellectual Fashions, Monthly Review Press.
• Terry Eagleton (1996). The Illusions of Postmodernism, Blackwell.

Richard Thaler Confused On Microeconomics

Richard Thaler espouses an incorrect imperfectionist viewpoint. If only all markets were competitive, agents did not suffer from limitations in calculating and lack of information, etc., all markets would clear. Or so he says, at least when it comes to the labor market:

Perceptions of fairness ... help explain a long-standing puzzle in economics: in recessions, why don't wages fall enough to keep everyone employed? In a land of Econs, when the economy goes into a recession and firms face a drop in the demand for their goods and services, their first reaction would not be to simply lay off employees. The theory of equilibrium says that when demand for something falls, in this case labor, prices should also fall enough for supply to equal demand. So we should expect to see that firms would reduce wages when the economy tanks, allowing them to also cut the price of their products and still make a profit. But this is not what we see: wages and salaries appear to be sticky. When a recession hits, either wages do not fall at all or they fall too little to keep everyone employed. -- Richard H. Thaler, Misbehaving: The Making of Behavioral Economics

Of course, equilibrium theory says no such thing. It is weird that I should know more about some bits of price theory than a Nobel laureate. (By the way, I take the term imperfectionist from John Eatwell and Murray Milgate.)

The book from which this quote is from is very much an intellectual memoir. We do not see Thaler getting married, raising children, or having cultural interests, except as it impacts on the development of his research. So I do not know whether he thinks the Distillery is a fine place to hang out in Rochester, whether he enjoyed listening to the Rochester Philharmonic in Highland Park Bowl, what his favorite wine from the Finger Lakes region is (presumably a white, maybe riesling), or whether he's ever played chess outside in that mall in downtown Ithaca. I did learn that Buffalo once had a professional basketball team - I knew that Syracuse had. And there's quite a bit about Greek Peak, a small ski resort that Thayer tried to help promote season tickets before he had his ideas fully worked out.

Thaler, in this book, is very aware of the challenges in getting mainstream economists to accept new ideas. How people say they will react to a choice is not counted as evidence. I think of the Allais paradox, for example. Thaler has an example of a friend and him deciding not to drive during a blizzard to Buffalo to see a basketball game. The friend says, "If we had not got these tickets for free, we would go." Likewise, surveys are also not counted as evidence. Nor are anecdotes. So he spent a lot of time in devising experiments, with real money at stake.

Elsewhere And Some Time Ago

• Unlearning Economics adds a comment to a long-ago thread on the ignorance of Henry Hazlitt.
• Tim Worstall lies (I informed Worstall at least a decade ago of the existence of economists who do not agree):

"These old things about supply and demand in Econ 101 really are true, when the price of something rises then people do tend to buy less of it. Force up the minimum wage and people will buy less minimum wage labour.

All economists would agree to the basic idea, the discussion becomes at what wage does that effect start to predominate?"

• In a comment on Reddit, glenra ignorantly asserts that "nobody except [me] is likely to refer to an argument [by Pierangelo Garegnani or Arrigo Opocher & Ian Steedman] as 'building on the work of' Piero Sraffa."
• On some discussion site for video gamers, a thread devolves to, apparently, some academic economist asserting that I am "quite keen on a topic only kept barely alive by fifth-rate Marxists". But this ignorant economist admittedly has nothing substantial to say.
• Eric Lonergan argues that there is no equilibrium real rate of interest. Although he does not note this, this is an implication of the Cambridge Capital Controversy.

A Neoclassical Labor Demand Function?

Figure 1: A Labor Demand Function

1.0 Introduction

I am not sure the above graph works. I could draw three-dimensional graphs in PowerPoint, for models specified with algebra, where relative sizes are indefinite. But, I would need to be able to draw parallel lines, and so on.

This post presents a model of extensive rent, with one produced commodity. A labor demand function, for a given rate of profits, graphs real wages versus employment. The resulting function is a non-increasing step function. Net output, in the model, varies with employment.

This post was inspired by Exercise 7.5 of Chapter 10 (p. 312) and Section 1 of Chapter 14 (pp. 428-432) of Kurz and Salvadori (1995). I gather one can advance the same sort of argument in a model with intensive rent or with a mixture of intensive and extensive rent. I conclude with some observations about generalizing this approach to models with multiple produced commodities.

2.0 Technology

Land is in fixed supply in this model. Three types of land exist. I assume t_j acres of Type j land are available. Capitalists know of a single process for producing corn on each type of land. Table 1 displays the coefficients of production for each process.

Table 1: Technology

Inputs	Corn-Producing Processes
	Alpha	Beta	Gamma
Labor	l1	l2	l3
Type I Land	c1	0	0
Type II Land	0	c2	0
Type III Land	0	0	c3
Corn	a1	a2	a3
Outputs	1 Bushel Corn

I make a number of assumptions:

• Each process exhibits constant returns to scale.
• All processes require a year to complete and totally consume their capital (seed corn).
• Wages and rent are paid out of the surplus product at the end of the year.
• All parameters (t_j, a_j, l_j, c_j) are positive.
• Each input of corn per bushel corn produced, a_j, is less than one.
• Without loss of generality, I assume:

(1 - a₁)/l₁ > (1 - a₂)/l₂ > (1 - a₃)/l₃

• In this specific case:

a₁ < a₃ < a₂

3.0 Price Equations

Prices must be such that, for j = 1, 2, and 3, the following inequality holds:

a_j(1 + r) + l_j w + c_j q_j ≥ 1

where w is the wage, r is the rate of profits, and q_j is the rent on land of Type j. When the above is a strict inequality, the corn-producing process with the given index incurs extra costs and will not be operated.

In a self-sustaining state, the above equation will be met with equality for at least some processes. For almost all feasible levels of employment, the equality will be met with one type of land, known as the marginal land, paying no rent. The marginal land will be partially in use, but some of it will be in excess supply. Other types of land, if any, that pay a rent will be fully used.

4.0 The Choice of Technique

The problem becomes to determine the order in which land is cultivated, as employment increases; the marginal land; and the corresponding wage and rents. I take the rate of profits as given in this analysis.

Consider a vertical line (not necessarily just the ones shown) on the wage-rate of profits plane, with employment set to zero. This line should be drawn at a given rate of profits. Three wage curves are drawn on this plane, each for an equality in the above equation, with rent set to zero. Each line connects the maximum wage, (1 - a_j)/l_j bushels per person-year, with the maximum rate of profits, (1 - a_j)/a_j, for the corresponding process.

The intersections of the wage curves, on this plane, with the vertical line you have drawn, working downward, establishes an order of types of land. I have given the assumptions such that this order is Type 1, Type 2, and Type 3 land when the rate of profits is zero. At the switch point, Type 2 and Type 3 land are tied in this order. For a somewhat larger rate of profits, the order is Type 1, Type 3, and Type 2.

This is the order in which lands are cultivated as output expands. Accordingly, I have drawn labor demand curves as step functions in planes parallel to the wage-employment plane. The height of these steps are determined by the wage that is paid on the marginal land. The height decreases as the rate of profits increases. The width of each step corresponds to how much employment is needed to fully use that land.

The lands for the steps higher and to the left of any point on the step function for the demand function for labor pay a rent when employment and wages are as at that point, in a self-reproducing equilibrium. The lands for the steps lower and to the right pay no rent and are not farmed. If the point is somewhere on the horizontal portion of a step, that land is marginal. Some of it lies fallow, and it pays no rent.

5.0 The Marginal Productivity of Labor

I might as well explain in what sense the wage is equal to the marginal productivity of labor at any point along the demand curve for labor. For the sake of argument, take the rate of profits, r, as fixed. I assume types of land have been re-indexed in order of cultivation, as described above. An ordered pair (L, w) on the labor demand function is either on a horizontal step or a vertical line segment between steps.

First, consider a horizontal step. An increment of labor, ∂L, results in an increased gross output of (∂L/l_i) bushels of corn. This increased gross output requires an increased input of (∂L a_i/l_i) bushels of seed corn. The increased net output would be the difference between the increment of gross output and the increment of seed corn if these changes occurred at the same moment in time. Either the increased output (and the wage) must be discounted back to the start of the year or the increment in seed corn must be costed up for the end of the year. Adopting the latter alternative, an increment of labor results in a marginal increase in net output of [(∂L/l_i) - (1 + r)(∂L a_i/l_i)] bushels of corn.

Second, consider a vertical drop. Then, the marginal net product of labor is specified by an interval. In linear models of production, the "equality" of the wage with the marginal product of labor is expressed by an interval bounding the wage:

(1/l_i) - a_i(1 + r)/l_i ≤ w ≤ (1/l_{i - 1}) - a_{i - 1}(1 + r)/l_{i - 1}

The marginal product of labor is not a physical quantity, independent of prices. It depends on the rate of profits, an important variable in any model of distribution.

6.0 Conclusion

The above is an exposition of a modern analysis of a special case of Ricardo's theory of extensive rent. Mainstream microeconomics can be viewed, after 1870, as (mostly) an unwarranted extension of Ricardo's theory of rent, especially his theory of intensive rent.

Explaining equilibrium prices and quantities by intersections of well-behaved supply and demand functions makes no sense, in general. In particular, wages and employment cannot be explained by supply and demand functions. The above example fails to illustrate this result.

Two limitations of this example, which do not generalize to a model with multiple produced commodities, perhaps account for this failure. First, no distinction can be drawn in the model between demand for labor in the corn-producing sector and demand for labor in the economy as a whole. Increased employment results in both increased gross and increased net output of corn. It is impossible, in this model, for another process to be adopted in an industrial sector (which does not require land as input) such that less corn is required for gross output in the corn sector, for a greater net output in the economy as a whole.

Second, corn capital and output are homogeneous with one another. Different wage levels may result in the adoption of a different process on (newly) marginal land. But no possibility arises in the model for components of capital to vary in relative price with one another. (Prices must vary for the long-period method to be applied in the analysis of labor demand. But prices, other than wages, cannot vary for (some) conceptions of the neoclassical long-period labor demand function. See Vienneau (2005).)

Oppocher and Steedman's 2015 book expands on these points. I was interested to find out that various mainstream economists had developed a new long-period theory of the firm, in the late 1960s and early 1970s, in which a variation in one price must be compensated for by a variation in other prices.

An Example of Bifurcation Analysis with Land and the Choice of Technique

Figure 1: A Bifurcation Diagram

1.0 Introduction

I have been looking at how bifurcation analysis can be applied to the choice of technique in models in which all capital is circulating capital. In my sense, a bifurcation occurs when a switch point appears or disappears off the wage frontier. A question arises for me about how to apply or visualize bifurcations in models with land, fixed capital, and so on.

This post starts to investigate this question by looking at a numerical example of a overly simple model with land and extensive rent.

2.0 Parameters and Assumptions for the Model

Table 1 specifies the technology for this example. One parameter, the labor coefficient a₀^β, is left free. Managers of firms know of two processes for producing corn from inputs of labor, (a type of) land, and seed corn. Each process is defined in terms of coefficients of production. All processes exhibit constant returns to scale; require a year to complete; and totally use up, in producing their output, the capital good required as input. Land, of the specified type, exits the production process as good as it was at the start of the year.

Table 1: Processes For Producing Corn

Input	Corn Industry
	Alpha	Beta
Labor	a0α = 1 Person-Yr.	a0β Person-Yr.
Land	bα = 10 Acres of Type I	bβ = 20 Acres of Type II
Corn	aα = (1/4) Bushels	aβ = (1/5) Bushels

Each type of land is in fixed supply:

• L_I = 100 Acres of Type I land exist.
• L_II = 100 Acres of Type II land exist.

The assumptions so far impose some limits on the quantity of net output that can be produced. If only Type I land is seeded, and that land is fully used, net output consists of:

(1 - a_α) L_I/b_α = (15/2) bushels

Likewise, if only Type II land is seeded, net output consists of 4 bushels. If net output exceeds (15/2) bushels (that is, the maximum of 15/2 and 4 bushels), both types of land will need to be seeded. If net output is less than (23/2) bushels (that is, the sum of 15/2 and 4 bushels), at least one type of land will not be fully used. Accordingly, assume:

(15/2) bushels < y < (23/2) bushels

where y is net output. Under these assumptions, one type of land is in excess supply and pays no rent.

I consider prices of production to determine rent and to find out which land is free. Since net output is taken here as a constant, no matter how much a₀^β may fall, I am assuming increased productivity (per worker) is taken in the form of decreased employment.

3.0 Price Equations

I take corn to be numeraire, and I assume rent and wages are paid out of the surplus at the end of the period. Prices of production must satisfy the following system of equations:

(1/4)(1 + r) + 10 ρ_I + w = 1

(1/5)(1 + r) + 20 ρ_II + a₀^β w = 1

where r, w, ρ_I, and ρ_II are the rate of profits, the wage, the rent on Type I land, and the rent of Type II land. All four of these distribution variables are assumed to be non-negative. The condition that at least one type of land pays a rent of zero is expressed by a third equation:

ρ_I ρ_II = 0

4.0 The Choice of Technique

I consider three solutions of the price equation, each for a different parameter value of a₀^β.

4.1 First Example

First, suppose a₀^β is (6/5) person-years per bushel. Each process yields a wage curve, under the assumption that the corresponding type of land pays no rent. Figure 1 graphs both wage curves. A simple generalization of this model would be to multiple produced commodities, with land only used in one industry. Each process in that industry would be associated with a technique, and the associated wage curve could be of any convexity, with the convexity possibly varying throughout its extent.

Figure 2: Each Type of Land Sometimes Pays Rent

In this example, in which both types of land must be used to produce the given net output, the relevant frontier is the inner frontier, shown as a solid black line in the figure. This, too, does not generalize to a multi-commodity model with more types of land. In that case, one would work from the outer frontier inward until the successive types of land could produce, at least, the given net output. This order might depend on whether the wage or the rate of profits was taken as given. Or perhaps some other theory of distribution could be analyzed.

Anyways, the type of land associated with the technique on the inner frontier, in this example, pays no rent. For low rates of profits or high wages, Type II land pays no rent. For high rates of profits or low wages, Type I land pays no rent. At the switch point, both types of land pay no rent. If the wage were given, rent on the type of land associated with the process further from the origin would come out of the super profits that would otherwise be earned on that process. If the rate of profits were given, one might see a conflict between workers and landlords. This analysis is a matter of competitive markets, inasmuch as capitalists can move their investments among industries and processes.

4.2 Bifurcation Over Wage Axis

I next consider a parameter value for a₀^β of (16/15) person-years per bushel. As shown in Figure 2, this is a case of a bifurcation over the wage axis. You cannot see the wage curve for the Alpha technique in the figure because it is always on the inner frontier. For any distribution of the surplus, Type I land pays no rent. If the rate of profits is zero, Type II land also pays no rent. For any positive rate of profits, landlords obtain a rent on Type II rent.

Figure 3: A Bifurcation Over the Wage Axis

4.3 Type II Land Always Pays Rent

For a final case, let a₀^β be one person-years per bushel. The wage curve for the Alpha technique has now rotated downwards counter clockwise so far that it never intersects the wage curve for the Beta technique. Whatever the distribution, Type I land pays no rent, and owners of Type II land receive a rent.

Figure 4: Wage Curves Never Intersect

4.4 Bifurcation Diagram

So this simple example can be illustrated with a bifurcation diagram, as seen at the top of this post. The rate of profits for the switch point is"

r_switch = (15 a₀^β - 16)/(5 a₀^β - 4)

This function asymptotically approaches the maximum rate of profits for the Alpha technique as a₀^β increases without bound. The wage curve for Alpha continues to become steeper and steeper. I suppose wage for the switch point approaches the wage on the wage curve for the Beta technique when the rate of profits is 300 percent.

One can also solve for the rents. When the rent on Type I land is non-negative, it is:

ρ_I = [(15 a₀^β - 16) + (4 - 5 a₀^β)r]/(200 5 a₀^β)

When the rent on Type II land is non-negative, it is:

ρ_II = [(16 - 15 a₀^β) + (5 a₀^β - 4)]/400

5.0 Conclusions

I am partly interested in bifurcation analysis because one can draw neat graphs to visualize the economics. For the numerical example, I would like to be able to draw three-dimensional diagrams. Imagine an axis coming out of the page for the bifurcation digram at the top of this post. I then could have a surface where the rent on one of the types of land is graphed against the rate of profits and the coefficient of production being varied parametrically.

It seems like all four of the normal forms for bifurcations of co-dimension one that I have defined may arise in examples of extensive rent. These are a bifurcation over the wage axis, a bifurcation over the axis for the rate of profits, a three-technique bifurcation, and a restitching bifurcation. They will not necessarily be on the outer frontier, however.

I think another type of bifurcation may be possible. Suppose productivity increases because coefficients of production decreases for land inputs or inputs of capital goods. Given net output, could such an increase in productivity result in some type of land that formerly paid rent (for some range of the rate of profits) becoming rent-free? Could all types of land become non-scarce? How would this sort of bifurcation look on an appropriate bifurcation diagram? Would the distinction between the order of rentability and efficiency be reflected in bifurcation analysis? Can I draw a bifurcation diagram with a discontinuity?

Infinite Number of Techniques, One Linear Wage Curves

Coefficients for First Column in Leontief Input-Output Matrix

I have uploaded a draft paper with the post title to my SSRN site.

Abstract:This note demonstrates that the special case condition, needed for a simple labor theory of value (LTV), of equal organic compositions of capital does not suffice to determine technology. A model of the production of commodities, with circulating capital and all commodities basic, is analyzed. Given direct labor coefficients and labor values, an uncountably infinite number of Leontief input-output matrices yield the same wage curve under the conditions in which prices of production are proportional to labor values.

This paper is an update of a previous draft paper. I have posed the problem better that I am addressing, have deleted an error in my previously most general formulation, replaced the numerical example by algebra, and shortened my paper. I hope I am not restating something that I did not absorb decades ago in reading John Roemer or Michio Morishima. As of today, I think I am subjectively original.

Bifurcation Analysis of a Two-Commodity, Three-Technique Technology

Figure 1: A Bifurcation Diagram

This post expands on this previous post. The technology is the same, but the rates of decrease of the coefficients of production in the Beta and Gamma corn-producing processes are not fixed. Instead, I consider the full range of parameter values. (I find the graphs produced by bifurcation analysis interesting for this case, but I think a two-commodity example can be found with more pleasing diagrams.)

Anyways, Figure 1 shows a bifurcation diagram for the parameter space in this example. The region numbered 8 is not visible on the graph. Accordingly, Figure 2 below shows a much expanded picture of the parameter space around that region. The specific parameter values in the previous post lead to a temporal path along the dashed ray extending from the origin in Figure 1. (The numbering of regions in this post and the previous post do not correspond.) Although it is not obvious, the locus of points bifucating regions 9 and 10 eventually, somewhere to the right of the region shown in Figure 1 eventually decreases in slope and intercepts the dashed ray.

Figure 2: Blowup of a Part of the Bifurcation Diagram

As usual, each numbered region corresponds to a definite sequence of cost-minimizing techniques contributing wage curves along the wage frontier. Table 1 lists this sequence for each region. Some notes on switch points are provided. A switch point is called "normal" merely if it conforms to outdated neoclassical intuition. In other words, such a switch point exhibits negative real Wicksell effects. In the example, regions also exist where switch points exhibit positive real Wicksell effects.

Table 1: Cost-Minimizing Techniques by Region

Region	Cost-Minimizing Techniques	Notes
1	Alpha	One technique cost-minimizing.
2	Alpha, Beta	"Normal" switch point.
3	Beta, Alpha, Beta	Reswitching. Switch pt. at highest r is "perverse".
4	Beta	One technique cost-minimizing.
5	Alpha, Gamma	"Normal" switch point.
6	Alpha, Gamma, Beta	"Normal" switch points.
7	Beta, Alpha, Gamma, Beta	Recurrence of techniques. Switch pt. at highest r is "perverse".
8	Beta, Alpha, Beta, Gamma, Beta	Two reswitchings, two "perverse" switch pts.
9	Gamma	One technique cost-minimizing.
10	Gamma, Beta	"Normal" switch point.
11	Beta, Gamma, Beta	Reswitching. Switch pt. at highest r is "perverse".

One can compare and contrast the above bifurcation diagram with the one in this post. The latter bifurcation diagram is for a specific instance of the Samuelson-Garegnani model, in which the basic commodity varies among techniques. (I have a more recent write-up of that bifurcation analysis linked to here.)

Reswitching Without a Reswitching Bifurcation

Figure 1: A Bifurcation Diagram

This post presents another example of bifurcation analysis applied to structural economic dynamics with a choice of technique. This example illustrates:

• Two reswitching examples appear and disappear without a reswitching bifurcation ever occurring, at least on the wage frontier.
• Two bifurcations over the wage axis arise. At the time each bifurcation of this type occurs, another switch point for the same techniques exhibits a real Wicksell effect of zero. Thus, for each, a switch point transitions from being a "normal" switch point to a "perverse" one exhibiting capital-reversing.
• Each of the four types of bifurcations of co-dimension one that I have identified have no preferred temporal order. For example, a bifurcation over the wage axis can add a switch point to the wage frontier. And another such bifurcation can remove a switch point, as time advances.
• The maximum rate of profits approaches an asymptote from below as time increase without bound.

Table 1 specifies the technology for this example, in terms of two parameters, σ and φ. Managers of firms know of one process for producing iron and of three processes for producing corn. Each process is defined in terms of coefficients of production, which specify the quantities of labor, iron, and corn needed to produce a unit output for that process. All processes exhibit constant returns to scale; require a year to complete; and totally use up, in producing their output, the capital goods required as input. I consider the special case in which the rate of decrease of the coefficients of production in the Beta corn-producing process, σ, is 5 percent, and the rate of decrease of coefficients in the Gamma corn-producing process, φ, is 10 percent.

Table 1: Processes For Producing Iron and Corn

Input	Iron Industry	Corn Industry
		Alpha	Beta	Gamma
Labor	1	0.89965	0.71733 e-σ t	1.28237 e-φ t
Iron	0.45	0.025	0.00176 e-σ t	0.03375 e-φ t
Corn	2	0.1	0.53858 e-σ t	0.13499 e-φ t

Three techniques are available for producing a net output of, say, corn, while reproducing the capital goods used as input. The Alpha process consists of the iron-producing process and the corn-producing process labeled Alpha. And so on for the Beta and Gamma techniques.

The choice of technique is analyzed in the usual way. I assume that labor is advanced, and wages are paid out of the surplus product at the end of the year. Corn is taken as numeraire. A wage curve can be drawn for each technique, given the coefficients of production prevailing at a given moment in time. Figure 1 illustrates a case of the recurrence of techniques in the example. The cost-minimizing technique is found by constructing the outer frontier of the wage curves. In Figure 2, the cost-minimizing techniques are Beta, Alpha, Gamma, and Beta, in that order. The switch point at approximately 57 percent exhibits capital-reversing. Around the switch point, a higher wage is associated with the adoption of a more labor-intensive technique. If prices of production prevail, firms will find it cost-minimizing to hire more workers at a higher wage, given net output.

Figure 2: Wage Curves in Region 4

Figure 3 illustrates the analysis of the choice of technique for all time. Switch points along the frontier and the maximum rate of profits are plotted versus time. Figure 1, at the top of this post, is a blowup of Figure 3 from time zero to a time of five years. These pictures show which technique is cost-minimizing at each rate of profits, at each moment in time. Bifurcations are also shown. Table 2 lists the cost-minimizing techniques in each region between the bifurcations.

Figure 3: An Extended Bifurcation Diagram

Table 2: Cost-Minimizing Techniques by Region

Region	Cost-Minimizing Techniques	Notes
1	Alpha	One technique cost-minimizing.
2	Alpha, Beta	"Normal" switch point.
3	Beta, Alpha, Beta	Reswitching. Switch pt. at highest r is "perverse".
4	Beta, Alpha, Gamma, Beta	Recurrence of techniques. Switch pt. at highest r is "perverse".
5	Beta, Gamma, Beta	Reswitching. Switch pt. at highest r is "perverse".
6	Gamma, Beta	"Normal" switch point.
7	Gamma	One technique cost-minimizing. Maximum r approaches an asymptote.

I suppose I can extend this example to partition the complete parameter space, as in this example, with an updated write-up here. That analysis will demonstrate, by example, that this sort of bifurcation analysis applies to cases in which multiple commodities are basic in multiple techniques. It is not confined to the special case of the Samuelson-Garegnani model. I am also thinking that I could perform a bifurcation analysis where parameters that vary include the ratio of the rates of profits in various industries, as in these examples of a model of oligopoly. Maybe such an analysis will yield an empirically relevant tale of the evolution of economic duality (also known as segmented markets).

Bifurcation Analysis Applied to Structural Economic Dynamics with a Choice of Technique

Variation of Switch Points with Technical Progress in Two Industries

I have a new working paper - basically an update of one I have previously described.

Abstract: This article illustrates the application of bifurcation analysis to structural economic dynamics with a choice of technique. A numerical example of the Samuelson-Garegnani model is presented in which technical progress is introduced. Examples of temporal paths through the parameter space illustrate variations of the wage frontier. A single technique is initially uniquely cost-minimizing for all feasible rates of profits. Eventually, the technique for which coefficients of production decrease at the fastest rate is always cost-minimizing. This example illustrates possible variations in the existence of Sraffa effects, which arise during the transition between these positions.