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.

Table 1: Processes of Production
InputIron IndustryCorn Industry
Labor(1/8) Person-Yr.u(1/2)
Iron(1/2) Tonv2
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


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


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
RegionSwitch PointsTechniques
2OneBeta, Alpha
4OneAlpha, 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


  • 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.
  • 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.

Table 1: Technology
InputsCorn-Producing Processes
Type I Landc100
Type II Land0c20
Type III Land00c3
Outputs1 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

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 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.

Table 1: Processes For Producing Corn
InputCorn Industry
Labora0α = 1 Person-Yr.a0β Person-Yr.
Landbα = 10 Acres of Type Ibβ = 20 Acres of Type II
Cornaα = (1/4) Bushelsaβ = (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.

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"

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.

Wednesday, November 29, 2017

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
1AlphaOne technique cost-minimizing.
2Alpha, Beta"Normal" switch point.
3Beta, Alpha, BetaReswitching. Switch pt. at
highest r is "perverse".
4BetaOne technique cost-minimizing.
5Alpha, Gamma"Normal" switch point.
6Alpha, Gamma, Beta"Normal" switch points.
7Beta, Alpha,
Gamma, Beta
Recurrence of techniques.
Switch pt. at highest r is
8Beta, Alpha, Beta,
Gamma, Beta
Two reswitchings, two
"perverse" switch pts.
9GammaOne technique cost-minimizing.
10Gamma, Beta"Normal" switch point.
11Beta, Gamma, BetaReswitching. 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.)

Saturday, November 25, 2017

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
Corn Industry
Labor10.899650.71733 et1.28237 et
Iron0.450.0250.00176 et0.03375 et
Corn20.10.53858 et0.13499 et

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
1AlphaOne technique cost-minimizing.
2Alpha, Beta"Normal" switch point.
3Beta, Alpha, BetaReswitching. Switch pt. at
highest r is "perverse".
4Beta, Alpha, Gamma, BetaRecurrence of techniques. Switch
pt. at highest r is "perverse".
5Beta, Gamma, BetaReswitching. Switch pt. at
highest r is "perverse".
6Gamma, Beta"Normal" switch point.
7GammaOne technique cost-minimizing.
Maximum r approaches an

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).

Wednesday, November 22, 2017

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.

Thursday, November 16, 2017

Two Techniques, One Linear Wage Curve

Coefficients for Iron-Production in the Leontief Input-Output Matrix

I have uploaded a working paper with the post title.

Abstract: This note demonstrates that the special case condition, needed for a simple labor theory of value, of equal organic compositions of capital does not suffice to determine technology. Prices do not vary across techniques for both techniques in a numeric example of a two-commodity linear model of production, and they are proportional to labor values. Both techniques yield the same wage curve, in which the wage is an affine function of the rate of profits. This indeterminancy generalizes to models with more than two produced commodities.

Friday, November 10, 2017

An Example With Two Fluke Switch Points

Figure 1: Fluke Switch Points on Each Axis
1.0 Introduction

I have developed an approach for finding examples in which either two fluke switch points exist on the wage frontier or a switch point is a fluke in more than one way. This post presents a numerical example with two fluke switch points on the frontier. Not all examples generated by this approach are necessarily interesting, although I find the approach of interest. I don't think the example in this approach is all that fascinating. I had thought that examples of real Wicksell effects of zero were somewhat interesting, but I have received disagreement.

Anyways, what I have been doing is drawing bifurcation diagrams for examples in which coefficients of production vary. The bifurcation diagram partitions a parameter space into regions in which the sequence of switch points does not vary, even though their specific locations on the wage frontier may. The loci dividing regions with topologically equivalent wage frontiers specify fluke cases. A point in the parameter space in which more than one such loci intersect specifies an example which is a fluke in more than one way.

2.0 Technology

The example is a numerical instantiation of the Samuelson-Garegnani model. A single consumption good, corn is produced from inputs of corn and one of three capital goods. Table 1 lists the coefficients of production for production processes for producing corn. Each production process in this example requires a year to complete and exhibits Constant Returns to Scale. A column in Table 1 lists the physical inputs for that process required per unit corn produced at the end of the year. Workers labor over the course of the year, and the inputs of the capital good are totally used up in the process. Managers of firms also know of a process for producing each capital good (Table 2). For a given capital good, the process for producing it requires inputs of labor and the services of that capital good.

Table 1: Processes For Producing Corn
InputCorn Industry

Table 2: Processes For Manufacturing Capital Goods

Any one of three techniques can be adopted to sustainably produce corn. The Alpha technique consists of the iron-producing process and the corresponding, labelled process for producing corn. The Beta technique consists of the copper-producing process and corresponding for producing corn. And similarly for the Gamma technique.

3.0 Prices and the Wage Frontier

For each technique, a system of two equations arises. I take corn as the numeraire and assume that labor is paid out of the surplus at the end of the year. The equations show the same rate of profits being earned for both processes comprising a technique. Given an externally specified rate of profits, the equations are a linear system. They can be solved for the wage and the price of the capital good, as functions of the rate of profits. For the wage, this function is known as the wage curve. All three wage curves, one for each technique, are graphed in Figure 1 above.

The wage frontier consists of the outer envelope of all wage curves. The curve(s) on the frontier at a given rate of profits correspond(s) to the cost-minimizing technique(s) at that rate of profits. The Gamma technique is cost-minimizing at low rates of profits, and the Beta technique is cost-minimizing at high rates of profits. These two techniques are tied - that is, both cost-minimizing - at the switch point dividing these two regions of the rate of profits.

The Alpha technique is only cost-minimizing at the switch points on the wage axis and on the axis for the rate of profits. And, it is tied, with the Gamma and Beta techniques, respectively, at these switch points. A switch point appearing on the wage axis or the axis for the rate of profits is a fluke case. So both switch points with the Alpha technique are flukes. Having Alpha participate in two fluke switch points is even more of a fluke case. For what it is worth, the fluke switch point on the axis for the rate of profits exhibits capital-reversing.

Saturday, October 28, 2017

Braess' Paradox

Figure 1: An Example Of Braess' Paradox

Braess' paradox arises in transport economics, a field for applied research in economics. I was inspired by the example in Fujishige et al. (2017) for the example in this post. Under Braess' paradox, an improvement to a transport network, and thus an increase in the number of choices available to users of the network, results in decrease performance. In reliability engineering, one says such a transport network is not a coherent system.

A transport network, for a single mode (for example, air, rail, road, or water) can be specified by:

  • A network, where a network consists of links between nodes. Links can be one-way or two way.
  • A cost for traversing each link. The cost can be a function of the demand (that is, the amount of traffic traversing that link). Cost can have a stochastic component, such as a (perceived) standard deviation for the distribution of the time to traverse a link.
  • Demands on the network, as specified by source nodes for users and the destination of each user.
  • Objective functions for the users, such as the minimization of trip time or the maximization of the probability that total trip time will not exceed a given maximization. The probability for the latter objective function is known as trip reliability.

In my example (Figure 1), two road networks are specified. The network on the right differs from the one on the left in that an additional road, between nodes A and B has been added. All links are two ways. The cost for each link is specified as the number of minutes needed to travel across the link, where two links have a cost that depends on the traffic, thus modeling the effect of congestion. The parameters XSA and XSA denote the number of vehicles traversing the respective links. Thus, the number of minutes to travel across these links is proportional to the amount of traffic, with a proportionality constant of unity. The demand is assumed to be unchanged by the addition of the new link. One hundred users want to drive their vehicles from the source node S to the node destination node D. Each driver wants to minimize their total trip time.

Table 1: Costs for Each Link
SAXSA Minutes
SB110 Minutes
AD110 Minutes
BDXBD Minutes
ABEither infinity or 5 Minutes

Each user has a choice of two routes, ignoring purposeless cycles, in the network on the left. These routes pass through nodes S, A, and D, or through nodes S, B, and D. The addition of the "short-cut" provides two additional routes, through nodes S, A, B, and D, and through nodes S, B, A, and D.

My method of analysis is an equilibrium assignment of users to routes. John G. Waldrop created this notion of equilibrium, as I understand it. It is an application of Nash equilibrium to transport economics. Bell and Iida call this equilibrium a Deterministic User Equilibrium. The equilibrium assignments in the example are shown as green lines in the figure. On the left, 50 drivers choose each of the two routes, and each driver's trip requires 160 minutes. On the right, all 100 drivers choose the route S, A, B, and D. Each driver takes 205 minutes to complete their trip.

To see why these are equilibria, consider what happens if a single driver deviates from the equilibrium assignment. For example, suppose a driver of the left who has previously chosen the route S, A, and D selects the route S, B, D. The cost for the congested link BD will rise from 50 minutes to 51 minutes, and his total trip time will now be 161 minutes, an increase from the previous 160 minutes. In this model, a driven will not choose to be worse off in this way. Symmetrically, a driver assigned to the route S, B, and D will not decide to switch to the route S, A, and D.

Once the shortcut, AB, has been added, the analysis requires tabulating a few more trips. Suppose a driver swithes from the equilibrium route on the right to the route:

  • S, A, and D or S, B, and D: In each case, the new route includes one congested link which all 100 drivers still traverse. The total trip time is 210 minutes, an undesirable increase over the equilibrium trip time of 205 minutes.
  • S, B, A, and D: All links in this route have a fixed cost. Total trip time is 225 minutes, also an increase over the equilibrium trip time.

So here is a (long-established) case in which improvements to a transport network result in optimizing individuals becoming worse off.

  • Satoru Fujishige, Michel X. Goemans, Tobias Harks, Britta Peis, and Rico Zenklusen (2017). Matroids are immune to Braess' Paradox. Mathematics of Operation Research. V. 42, Iss. 3: 745-761.
  • M. G. H. Bell and Y. Iida (1997). Transportation Network Analysis. New York: John Wiley & Sons.

Tuesday, October 24, 2017

Structural Economic Dynamics with a Choice of Technique: A Numerical Example

A Bifurcation Diagram with Two Temporal Paths
I have a working paper with the post title. Here's the 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 change 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 rates is always cost-minimizing. During the transition between these positions, reswitching, the recurrence of techniques, and capital-reversing can arise. This example emphasizes the importance of fluke switch points and illustrates possible variations in the existence of Sraffa effects.

Tuesday, October 17, 2017


  • A July 24 Jonathan Schlefer article, "Market Parables and the Economics of Populism: When Experts are Wrong, People Revolt", in Foreign Affairs. Schlefer cites the Cambridge Capital Controversy as a demonstration that the neoliberal political project of remaking the world around unembedded markets is doomed to failure.
  • A September 11 interview with Daniel Kahneman in which he basically credits Richard Thaler with inventing behavioral economics. (In his memoirs, Misbehaving, Thaler is also explicit about the disciplinary boundaries between economics and psychology.)
  • Richard Thaler's anomaly columns in the Journal of Economic Perspectives
  • I have not read Nancy Maclean's Democracy in Chains. Marshall Steinbaum reviews this book in Boston Review. Henry Farrell & Steven Teles respond.

Another ongoing brouhaha is about Alice and Wu's undergraduate paper documenting the sexism on Economic Job Market Rumors.

Wednesday, October 11, 2017

Others With Points Of View Like Sraffa's

In Production of Commodities by Means of Commodities, Sraffa writes:

"others have from time to time independently taken up points of view which are similar to one or other of those adopted in this paper and have developed them further or in different directions from those proposed here." -- P. Sraffa (1960): pp. vi - vii.

Who is Sraffa talking about? I suggest the following, and their works, at least:

  • Tjalling C. Koopmans (1957). Three Essays on the State of Economic Science. New York: McGraw-Hill
  • Wassily W. Leontief (1941). Structure of the American Economy, 1919-1929.
  • Jacob T. Schwartz (1961). Lectures on the Mathematical Method in Economics. New York: Gordon & Breach.
  • John Von Neumann (1945-1946). A Model of General Economic Equilibrium, A Model of General Economic Equilibrium. V. 13, No. 1: pp. 1-9.

I thought about listing David Hawkins and Herbert Simon, given how frequently the Hawkins-Simon condition is cited in expositions of Leontief input-output analysis. I might also mention Nicholas Georgescu-Roegen, the creator of the non-substitution theorem. The work of Ladislaus Bortkiewicz, Georg von Charasoff, Vladimir K. Dmitriev, and Robert Remak, as I understand it, mostly predates Sraffa's long period of preparation of his masterpiece.

Sunday, October 08, 2017

Economic Impact Of Regional Disasters: A Job For Input-Output Analysis?

This post, unfortunately, is inspired by current events.

Economists can provide guidance on disaster recovery - for example, from earthquakes and hurricanes.

Economists, for a long time, have been developing input-output models of local economies and interactions between them. I think of Walter Isard as a pioneer here. Such models are of practical importance to my post topic.

Regional input-output models can describe disasters with either a supply-side or demand-side approach. In a supply-side approach, the output of an industry is reduced because the inputs into that industry are not available at the pre-disaster level. Some of the outputs of that industry are inputs into other industries. Other outputs satisfy final demands, for example, for household consumption. Input-output modeling can help trace these consequences.

In a demand-side approach, an industry's output is constrained because those who purchase its outputs cannot do so at the pre-disaster level. If those industries who purchase your products are suddenly reduced in size, you must need cut back your output, too.

Some issues arise here. Can supply-side and demand-side approaches be combined without double counting? How should one model the effects of external infusions of aid? Multiplier effects seem to sit comfortably with the demand side approach. The assumption of fixed coefficients in the Leontief input-output approach seems to be an important restriction here. When it comes to modeling resiliency, I think of the work of Adam Rose. Apparently, some use Computational General Equilibrium (CGE) models for this reason.

I do not know enough to have a firm opinion of CGE models. I have the impression that "Computational" is a misnomer; it does not relate to computational theory and Turing machines, as studied in computer science. I am also not sure that the GE in CGE is what I understand as GE. Anyways, practical considerations interact here with the ideological demands impacting economic theory. I like to think that economists are useful in the hard problem of what to do when disaster strikes.

  • Walter Isard. 1951. Interregional and Regional Input-Output Analysis: A Model of a Space-Economy. Review of Economics and Statistics. Vol. 33, No. 4: pp. 318-328.

Saturday, September 30, 2017

Dean Baker's Rigged And Robert Reich's Saving Capitalism

Dean Baker has a new book out: Rigged: How Globalization and the Rules of the Modern Economy Were Structured to Make the Rich Richer. It recounts how laws that define property, markets, and so on have been rewritten, over the last fifty years, to accomplish an upward redistribution of income. This bias for the rich contrasts with the effects of the rules of the game in the half-century golden age following World War II. This is the same theme as Robert Reich's Saving Capitalism: For the Many, Not the Few. And both books are targeted for the common reader. Thus, a review of Baker's book can usefully compare and contrast it with Reich's book. (I have previously reviewed Reich's book.)

Both books focus on a few areas in which the rules have been rewritten to distribute income upward. For example, consider intellectual property rights, especially the extension of patents and copyrights to last longer and to cover more. Baker, I think, discusses the international dimension more than Reich. The United States has been trying to ensure that patent laws are consistent throughout the world. The largest impact of this attempt, perhaps, is on the price of drugs in developing countries, and the subsequent consequences for health and life.

Both discuss how changes in laws have provided companies with more market power and have protected monopolies and oligopolies. Baker has more of a focus on upper-class professionals, such as doctors, dentists, and lawyers. Baker especially emphasizes how they are protected from international competition. So called free trade treaties, like the North America Free Trade Agreement (NAFTA) are selective in who they subject to the rigors of international competition.

Both discuss corporate governance and the impact on the pay of Corporate Executive Officers (CEOs) and top managers. CEO pay went from 20 times averages wages in the 1960s to over 250 times average wage nowadays. In general, CEO pay is set by a committee appointed by the board of directors who, in turn, are appointed by the CEO. Stockholders have little say, even after the Dodd-Frank bill gives stockholders the right to have an up-or-down non-binding vote on pay packages. Baker extends his critique of CEO pay to heads of foundations and to university presidents, for example.

Reich writes more about contracts, bankruptcy, and enforcement. Baker, on the other hand, writes more about the macroeconomic setting. For example, the Federal Reserve is overly focused on the threat of inflation and not so much on unemployment. I know something of the importance of macroeconomic policy from James Galbraith, who has been writing on this theme for a long time. Baker follows his chapter on macroeconomics with a chapter on the financial sector.

I find Baker more analytical and less polemical than Reich. Baker adopts an interesting trope for putting in context large numbers. He frequently converts dollar flows into multiples of the yearly cost of welfare, that is, the yearly outlay on the Supplemental Nutrition Assistance Program (SNAP). He doesn't always carry through this conversion. I suppose a comparison of doctors' incomes among specialities is only one illustration of health care costs and may not require this contextualization.

I think both books contain a similar tension. Part of their point is that a contrast between non-government intervention in markets and more regulation is a false choice. I think Reich is better on ideological critique of, say, marginal productivity theory or the exploded theory of skills-biased technological change. Baker seems less interested in abstract economic theory, although he does ask whether one can really believe CEOs have gotten so much more productive since the 1950s so as to justify the increased inequality in their pay. But Baker keeps on contrasting legislated barriers to competition with what a free market would produce, that is, less rent. He is too accepting, at least for rhetorical purposes, of traditional economic theory for my tastes. Reich is more consistent with emphasizing that no such thing as a free market can ever exist, absent laws defining markets. Baker does start and conclude with this point.

Baker proposes any number of innovative policies throughout the book, and gathers them together in the second-to-last chapter. For example, Baker suggests that corporations be given an option of issuing non-voting stock to the government, instead of paying corporate income tax. Inventors could be given the option of competing for contest prizes, where a requirement of signing up is that they cannot receive patents over a number of years. (As far as I am aware, existing prize contests have no such connection to the patent system.) He also suggests that governments can pay for medical tourism, where those needing operations travel to other countries in search of cheaper prices. Baker has thought about how some of his policy proposals could first be implemented on a small scale. In general, I find both Baker and Reich too voluntaristic in policy proposals. But I do not know how to avoid that in today's general dismal climate.

I guess my conclusion is that Reich's book is broader, but that Baker's book is generally better in areas of Baker's focus.

Monday, September 18, 2017

Another Example Of A Real Wicksell Effect Of Zero

Figure 1: A Reswitching Example with a Fluke Switch Point
1.0 Introduction

A switch point that occurs at a rate of profits of zero is a fluke. This post presents a two-commodity example with a choice between two techniques, in which restitching occurs, and one switch point is such a fluke. Total employment per unit of net output is unaffected by the choice of technique. Furthermore, the numeraire-value of capital per unit net output is also unaffected by the mix of techniques adopted at a switch point with a positive rate of profits. This is not the first example I present in a draft paper.

2.0 Technology

Consider the technology illustrated in Table 1. The managers of firms know of three processes of production. These processes exhibit Constant Returns to Scale. The column for the iron industry specifies the inputs needed to produce a ton of iron. Two processes are known for producing corn, and their coefficients of production are specified in the last two columns in the table. Each process is assumed to require a year to complete and uses up all of its commodity inputs. (The coefficients were "nicer" fractions before I started perturbing it. Octave code was useful.)

Table 1: The Technology

This technology presents a problem of the choice of technique. The Alpha technique consists of the iron-producing process and the corn-producing process labeled Alpha. Similarly, the Beta technique consists of the iron-producing process and the corn-producing process labeled Beta.

3.0 Quantity Flows

Quantity flows can be analyzed independently of prices. Suppose the economy is in a self-replacing state, with a net output consisting only of corn. Table 2 displays quantity flows for the Alpha technique, when the net output consists of a bushel of corn. The last row shows gross outputs, for each industry. The entries in the three previous rows are found by scaling the coefficients of production in Table 1 by these gross outputs. Table 3 displays corresponding quantity flows for the Beta technique.

Consider the quantity flows for the Alpha technique. The row for iron shows that each year, the sum (9/356) + (11/356) = 5/89 tons are used as iron inputs in the iron and corn industries. These inputs are replaced at the end of the year by the output of the iron industry, with no surplus iron left over. In the corn industry, the sum 10/89 + 11/89 = 21/89 bushels are used as corn inputs in the two industries. When these inputs are replaced out of the output of the corn industry, a surplus of one bushel of corn remains. The net output of the economy, when these processes are operated in these proportions, is one bushel corn. The table allows one to calculate, for each technique, the labor aggregated over all industries per net unit output of the corn industry. Likewise, one can find the aggregate physical quantities of capital goods per net unit output of corn.

Table 2: Quantity Flows for Alpha Technique
Labor5/89 ≈ 0.0562 Person-Yrs.57,101/51,353 ≈ 1.11 Person-Yrs.
Iron9/356 ≈ 0.0253 Tons11/356 ≈ 0.0309 Tons
Corn10/89 ≈ 0.112 Bushels11/89 ≈ 0.124 Bushels
Output5/89 ≈ 0.0562 Tons110/89 ≈ 1.24 Bushels

Table 3: Quantity Flows for Beta Technique
Labor3/577 ≈ 0.00520 Person-Yrs.671/577 ≈ 1.16 Person-Yrs.
Iron27/11,540 ≈ 0.00234 Tons33/11,540 ≈ 0.00286 Tons
Corn6/577 ≈ 0.0104 Bushels2,519/2885 ≈ 0.873 Bushels
Output3/577 ≈ 0.00520 Tons5,434/2,885 ≈ 1.88 Bushels

4.0 Prices and the Choice of Technique

The choice of technique is analyzed based on prices of production and cost-minimization. Labor is assumed to be advanced, and wages are paid out of the surplus product at the end of the year. Corn is taken as the numeraire. Figure 1 graphs the wage curve for the two techniques. The cost-minimizing technique, at a given rate of profits, maximizes the wage. That is, the cost-minimizing techniques form the outer envelope, also known as, the wage frontier, from the wage curves. In the example, the Beta technique is cost minimizing for high rates of profits, while the Alpha technique is cost-minimizing between the two switch points. At the switch points, any linear combination of the two techniques is cost-minimizing.

One switch point is a fluke; it occurs for a rate of profits of zero. Any infinitesimal variation in the coefficients of production would result in the switch point no longer being on the wage axis. This intersection between the wage curves would then either occur at a negative or positive rate of profits. In the former case, the example would be one with a single switch point with a non-negative, feasible rate of profits, and the real Wicksell effect would be negative at that switch point. In the latter case, it would be a reswitching example, with the Beta technique uniquely cost-minimizing for low and high rates of profits. The real Wicksell effect would be negative at the first switch point and positive at the second.

5.0 Aggregates

In calculating wage curves, one can also find prices for each rate of profits. Table 5 shows certain aggregates, as obtained from Tables 2 and 3 and the price of iron at the switch point with a positive rate of profits. (Table 4 shows this price.) The numeraire value of capital per person-year, for a given technique and a given rate of profits, is the additive inverse of the slope of a line joining the intercept of the technique's wage curve with the wage axis to a point on the wage curve at the specified rate of profits. The capital-labor ratio, for a given technique, varies with the rate of profits, unless the wage curve is a straight line. Since a switch point occurs on the wage axis, the capital-labor ratio for both techniques at the other switch point is identical. As seen in Table 5, it does not vary among the two cost-minimizing techniques at the switch point with a positive rate of profits. The real Wicksell effect is zero at this switch point.

Table 4: Price Variables at Switch Point with Real Wicksell Effect of Zero
Rate of Profits125,483/209,727 ≈ 59.8 Percent
Wage9,226,807/24,957,513 ≈ 0.370 Bushels per Person-Yr.
Wage7,558/595 ≈ 12.7 Bushels per Ton

Table 5: Aggregates at Switch Point with Real Wicksell Effect of Zero
Net Output1 Bushel Corn
Labor674/577 ≈ 1.17 Person-Years
Physical Capital5/89 Tons Iron3/577 Tons Iron
21/89 Bushels Corn2,549/2,885 Bushels Corn
Financial Capitl113/119 ≈ 0.945 Bushels Corn
Capital-Labor Ratio65,201/80,206 ≈ 0.813 Bushels per Person-Yr.

6.0 Implications

A certain sort of indeterminacy arises in the example. For a given quantity of corn produced net, the ratio of labor employed in corn production to labor employed in iron production varies, at the switch point with a positive rate of profits, from around 1/5 to just over 223 to one. A change in technique leaves total employment unchanged, given net output, even as it alters the allocation of labor among industries. At the switch point, a change in technique, given net output, leaves the total value of capital unchanged, while, once again, altering its allocation among industries.

Suppose the economy is in a stationary state with the wage slightly below the wage at the switch point with a real Wicksell effect of zero. The Beta technique is in use. Consider what happens if a positive shock to wages result in a wage permanently higher than the wage at the switch point. The shock might be, for example, from an unanticipated increase in the minimum wage. Prices and outputs will be out of proportion, and a perhaps long disequilibrium adjustment process begins. Suppose that, eventually, after all this folderol, the economy, once more, attains another stationary state. The Alpha technique will now be in use. Labor hired per unit net output will be unchanged. The only variation in the value of capital goods per unit labor is a result of price changes, independent of the change in technique.