## Sunday, December 31, 2017

### 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.

 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:

pα = (1/4)

The wage curve for the Alpha technique is:

wα = (1/2)(1 - 3 r)
3.2 Beta

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

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

The wage curve for the Beta technique is:

wβ = [(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:

wα = wβ

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

+ (120 u - 20 v - 1) (rswitch)2
+ (80 u - 40 v +18) rswitch
+ (-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.

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.

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.

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.

 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.

## Wednesday, December 27, 2017

### 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:
• 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.

## Friday, December 22, 2017

### 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.

## Monday, December 18, 2017

### 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.

## Thursday, December 14, 2017

### 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 tj 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.

 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 (tj, aj, lj, cj) are positive.
• Each input of corn per bushel corn produced, aj, 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:

aj(1 + r) + lj w + cj qj ≥ 1

where w is the wage, r is the rate of profits, and qj 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 - aj)/lj bushels per person-year, with the maximum rate of profits, (1 - aj)/aj, 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/li) bushels of corn. This increased gross output requires an increased input of (∂L ai/li) 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/li) - (1 + r)(∂L ai/li)] 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/li) - ai(1 + r)/liw ≤ (1/li - 1) - ai - 1(1 + r)/li - 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.

## Tuesday, December 12, 2017

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

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 a0β, 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.

 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:

• LI = 100 Acres of Type I land exist.
• LII = 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 - aα) LI/bα = (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 a0β 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 ρI + w = 1
(1/5)(1 + r) + 20 ρII + a0β w = 1

where r, w, ρI, and ρII 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:

ρI ρII = 0

4.0 The Choice of Technique

I consider three solutions of the price equation, each for a different parameter value of a0β.

4.1 First Example

First, suppose a0β 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 a0β 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 a0β 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.

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"

rswitch = (15 a0β - 16)/(5 a0β - 4)

This function asymptotically approaches the maximum rate of profits for the Alpha technique as a0β 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:

ρI = [(15 a0β - 16) + (4 - 5 a0β)r]/(200 5 a0β)

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

ρII = [(16 - 15 a0β) + (5 a0β - 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?

## Thursday, December 07, 2017

### 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.
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