When Did The Marginalist Theory Of Labor Markets Become Obsolete?

Chemists once believed, before Lavoisier and Priestly discovered oxygen, in the theory of phlogiston. Physicists, before Galileo, believed in the impetus theory of motion. Academic economists once believed that, in competitive markets, wages and employment tend to the point of intersection of supply and demand curves. The supply curve is supposed to slope up, showing that with a higher real wage, the hours offered for employment increase. The demand curve slopes down, modeling a smaller quantity demanded of labor services at higher wages. A short-run and long-run version of the theory existed.

When did this theory become obsolete? Some candidates:

• With the 2015 publication of Arrigo Opocher and Ian Steedman's Full Industry Equilibrium.
• With Pierangelo Garegnani's 1970 paper, Heterogeneous capital, the production function and the theory of distribution.
• With Piero Sraffa's 1960 book, The Production of Commodities by Means of Commodities.
• With the 1936 publication of John Maynard Keynes' The General Theory of Employment, Interest qand Money.
• With G. F. Shove's 1933 review of J. R. Hicks' The Theory of Wages.

Empirical work went along with this timeline, whether that includes the discovery that firms use markup pricing or the use of natural experiments showing that minimum wages do not decrease employment.

Oh, what’s that you say? You have not heard that the theory of supply and demand is obsolete? Well, not everybody can be expected to understand the periodic table or laws of motion.

Employment And Wages Not Determined By The Supply And Demand Of Labor

Figure 1: The Demand for Labor

1.0 Introduction

Wages and employment are not determined in competitive markets by the interaction of well-behaved supply and demand curves, as portrayed in much introductory economics. At least, no reason exists to thinks so. Every once in a while I like to recall that this is an implication of the Cambridge Capital Controversy.

2.0 Technology

Consider a very simple competitive capitalist economy in which corn and iron are produced from inputs of labor, iron, and corn. All production processes in this example require a year to complete. The managers of firms know of two processes for producing corn and two processes for producing iron (Table 1). The processes a and b, for producing corn, require the tabulated inputs to be available at the beginning of the year for each bushel corn produced and available at the end of the year. Similarly, process c, for example, requires one person-year, 1/40 bushels corn, and 1/10 tons iron to be available at the beginning of the year for each ton of iron produced by this process. This is an example of circulating capital; all inputs of corn and iron are used up during the year in producing the gross output.

Table 1: Coefficients of Production

Input	Industry
	Corn		Iron
	a	b	c	d
Labor	a0, 1(a) = 1	a0, 1(b) = 1	a0, 2(c) = 1	a0, 2(d) = 275/464
Corn	a1, 1(a) = 2/5	a1, 1(b) = 3/5	a1, 2(c) = 1/40	a1, 2(d) = 0
Iron	a2, 1(a) = 2	a2, 1(b) = 1/2	a2, 2(c) = 1/10	a2, 2(d) = 113/232

A technique consists of a process for producing corn and a process for producing iron. Thus, there are four techniques in this example. They are defined in Table 3.

Table 2: Techniques of Production

Technique	Corn Process	Iron Process
Alpha	a	c
Beta	a	d
Gamma	b	c
Delta	b	d

3.0 Quantity Flows

Suppose the net output of the economy is some multiple c of the numeraire. I let d₁ be the bushels corn in the numeraire, and d₂ be the tons iron. How can one find, for a given technique, how much labor must be employed throughout the economy to produce, say, a bushel corn?

Let the coefficients of production be as above for a given technique. Let d₁ be unity, and d₂ be zero. The bushels corn y₁ and tons iron y₂ in net output are now specified. The question becomes what are the gross quantities q₁ of corn and q₂ of iron that need to be produced for the given net output.

With this specification, the following equation must be satisfied for the production of the given net output of corn:

y₁ = c d₁ = q₁ - (a_{1, 1} q₁ + a_{1, 2} q₂)

The following equation is for the given net output of iron:

y₂ = c d₂ = q₂ - (a_{2, 1} q₁ + a_{2, 2} q₂)

The labor L employed throughout the economy with this net output is:

L = a_{0, 1} q₁ + a_{0, 2} q₂

One can set L to unity and solve the above system of equations for c, q₁, and q₂. The person-years of labor needed to produce a net output of one unit of the numeraire is then the reciprocal of c.

Or one can set c to unity and solve for L, q₁, and q₂. This, too, will find the person-years employed throughout the economy, with the given technique, to produce one unit of the numeraire net.

4.0 Prices of Production

Which technique will the firm adopt, if any? The answer depends, in this analysis, on which is more profitable. So one has to consider prices. I here assume that inputs of iron and corn are charged at the start of the year. The wages for labor are paid out of the surplus at the end of the year.

Select a technique.

• p₁: The price of a bushel corn.
• p₂: The price of a ton iron.
• w: The wage for hiring a person-year of labor.
• r: The rate of profits

The corn-producing process gives one equation for specifying prices of production:

(p₁ a_1,1 + p₂ a_2,1)(1 + r) + w a_{0, 1} = p₁

The iron-producing process specifies another equation:

(p₁ a_1,2 + p₂ a_2,2)(1 + r) + w a_{0, 2} = p₂

The price of the numeraire is unity:

p₁ d₁ + p₂ d₂ = 1

You can solve the above system, to find the price of corn, the price of iron, and the wage as (rational) functions of the rate of profits. You actually want to solve for functions of (1 + r). Each technique yields a set of functions.

5.0 The Choice of Technique

Figure 2 graphs the wage curves for each of the four techniques. The outer frontier shows the cost-minimizing technique. The sequence of cost-minimizing techniques, as the wage increases, is Beta, Alpha, Gamma, and Delta. Apparently, I constructed this example, decades ago, to have all four techniques on the frontier.

Figure 2: Wage Curves and the Their Outer Frontier

The construction of the outer frontier in the analysis of the cost-minimizing technique is a derived result. Figure 3 shows the extra profits, for any wage, to be obtained when Alpha or Beta prices prevail. You can see Beta is cost-minimizing at the smallest range of wages and Alpha at the next largest range. Suppose Alpha was in operation at the lowest wages. Prices would signal to managers of firms that they should make iron with process d, not process c. So they would adopt the Beta technique. Market prices would no longer match prices of production. A disequilibrium process would presumably end up with Beta prices prevailing at the given wage.

Figure 3: Extra Profits at Alpha and Beta Prices

In a model of circulating capital, it does not matter at which system of prices you start at. Away from switch points, only one technique is cost-minimizing. This uniqueness does not necessarily hold for general models of joint production.

By the way, this is also an instance of process recurrence, as well as of capital-reversing. Beta is cost-minimizing at the lowest wage, and Delta at the highest wage. Both operate the second process in iron production. This process recurs. Process recurrence can arise with neither capital reversing nor the reswitching of techniques. I suppose this independence is more apparent if more that two goods are being produced.

6.0 The 'Demand' for Labor

The above has briefly outlined how to find the cost-minimizing technique for any given wage, up to a maximum. And I have also outlined how to find employment for any given technique and net output. Figure 1, at the top of this post, puts these results together to present a graph for the long-period demand for labor. It is not downward-sloping throughout. The existence of capital-reversing implies that labor-demand curves can slope up.

7.0 Conclusion

I really do not know how to explain what I understand most economists teach their students. I suppose some students might greet a upward-sloping labor demand curve by talking about how some make mistakes and it takes time for managers of firms to learn. They might bring up an evolutionary process. Or principal agent problems, transactions costs, and information asymetries. And on and on.

But these imperfections and frictions are off-point. The basic logic of the textbooks is wrong. And this has been known for over a half-century.

Wages, Employment Not Determined By Supply And Demand

1.0 Introduction

I do not think I have presented an introductory example in a while in which an increased wage is associated with firms wanting to employ more labor, given the level of net output. This example is presented as a matter of accounting for a vertically integrated firm.

Exact calculations with rational numbers are tedious in this example. I expect that if anybody bothers to check this, they would use a spreadsheet. As far as I can tell, Microsoft Excel uses double precision floats.

2.0 Technology

The managers of a competitive, vertically-integrated firm for producing corn know of the four production processes listed in Table 1. Corn is a consumption good and also a capital good, that is, a produced commodity used in the production of other commodities. In fact, iron, steel, and corn are capital goods in this example. The first process produces iron, the second process produces steel, and the last two processes produce corn. Each process exhibits Constant Returns to Scale (CRS) and is characterized by coefficients of production. Coefficients of production (Table 1) specify the physical quantities of inputs required to produce the specified unit output in the specified industry. All processes require a year to complete, and the inputs of iron, steel, and corn are all consumed over the year in providing their services so as to yield output at the end of the year. The data on technology are taken from a larger example.

Table 1: Technology

Input	Process
	a	d	e	f
Labor	1/3 person-year	7/20 person-year	1 person-year	3/2 person-year
Iron	1/6 ton	1/100 ton	1 ton	0 tons
Steel	1/200 ton	3/10 ton	0 tons	1/4 ton
Corn	1/300 bushel	0 bushel	0 bushels	0 bushels
Output	1 ton iron	1 ton steel	1 bushel corn	1 bushel corn

The managers of the firm have available two techniques for producing corn from inputs of labor, with intermediate inputs being constantly replaced. The iron-producing, steel-producing, and first corn-producing processes are operated in the Gamma technique. The second corn-producing process, as well as the iron and steel-producing processes, are operated in the Delta technique. Iron, steel, and corn all enter, either directly or indirectly, into the production of corn in both techniques. Vertically-integrated firms can also operate a linear combination of the Gamma and Delta technique.

3.0 Quantity Flows

One can consider various levels of operations in each of the processes for each of the technique. I consider two examples of snychronized production, in which inputs of labor simultaneously produce a net output of corn for consumption. A structure of production, consisting of specific capital goods, intervenes between the inputs and output. The labor input reproduces that structure, as well as producing the output.

3.1 Gamma Quantity Flows

Suppose 14,000/11,619 ≈ 1.205 tons iron are produced with the first process, 100/11,619 ≈ 0.0086 tons steel are produced with the second process, and 34,997/34,857 ≈ 1.004 bushels corn are produced with the third process. Then the quantity flows illustrated in Table 2 result. 14,000/11,619 tons iron are used as inputs among the three industries. These inputs are replaced by the output of the iron-producing process. 100/11,619 tons of steel are used as inputs among the three industries, and these inputs are replaced by the output of the steel-producing process. 140/34,857 bushels of corn are used as inputs among the three industries, leaving a net output of one bushel corn. In short, these quantity flows are such that 49,102/34,857 ≈ 1.409 person-years produce one bushel corn net. Obviously, I did not pick a very good set of coefficients for this example to support exact calculations in rational numbers.

Table 2: Vertically-Integrated Production with the Gamma Technique

Input	Process
	a	d	e
Labor	14,000/34,857 person-year	35/11,619 person-year	34,997/34,857 person-year
Iron	7000/34,857 ton	1/11,619 ton	34,997/34,857 ton
Steel	70/11,619 ton	30/11,619 ton	0 tons
Corn	140/34,857 bushel	0 bushel	0 bushels
Output	14,000/11,619 ton iron	100/11,619 ton steel	34,997/34,857 bushel corn

3.2 Delta Quantity Flows

Suppose 100/23,331 ≈ 0.00429 tons iron are produced with the first process, 25,000/69,993 ≈ 0.3572 tons steel are produced with the second process, and 69,994/69,993 ≈ 1.00001 bushels corn are produced with the fourth process. By the same logic as above, these quantity flows are such that 1807/1111 ≈ 1.626 person-years produce one bushel corn net.

Table 3: Vertically-Integrated Production with the Delta Technique

Input	Process
	a	d	f
Labor	100/69,993 person-year	1,250/9,999 person-year	34,997/23,331 person-year
Iron	50/69,993 ton	250/69,993 ton	0 ton
Steel	1/46,662 ton	7,500/69,993 ton	34,997/139,986 tons
Corn	1/69,993 bushel	0 bushel	0 bushels
Output	100/23,331 ton iron	25,000/69,993 ton steel	69,994/69,993 bushel corn

4.0 Prices

Which technique will the managers of the firm choose to adopt? By assumption, they take the price of corn and the wage as given on the consumer and labor markets. For simplicity, assume that price of a bushel corn is unity. That is firms treat the price of the consumer good as numeraire. At the end of the year, firms own a stock of iron, steel, and corn. They sell some of the corn to consumers. They retain the iron, steel, and enough corn to continue production the next year.

In a consistent accounting scheme, the price of iron and steel are such that:

• The same (accounting) rate of profits is obtained in all operated processes.
• The cost of the inputs, per bushel corn produced gross, for the corn-producing process not operated for a technique does not fall below that for the operated process.

The first condition specifies prices of intermediate goods and the rate of profits the accountants register. The second condition states that no pure economic profits can be obtained. Under these conditions, the managers of the firm can price their capital stock at the end of any year.

4.1 Prices at a Low Wage

Suppose the wage is w = 19,296/352,547 ≈ 0.05473 bushels per person-year. The accountants set the price of iron at p₁ = 6,860/27,119 ≈ 0.2530 bushels per ton iron and the price of steel at p₂ = 76,454/27,119 ≈ 2.819 bushels per ton steel. Table 4 shows the cost per unit output for each process and the resulting rate of profits obtained by operating each process. In constructing the tables for price systems, wages are assumed to be advanced. Under these assumptions, the rate of profits is 9/4, that is 225 percent, in each process comprising the Gamma technique. A lower rate of profits is obtained in the remaining corn-producing process, and it will not be operated. This is a consistent accounting system for the vertically-integrated firm, given the wage.

Table 4: Costs and the Rate of Profits at a Low Wage

Process	Cost	Rate of Profits
a	(1/6)p1 + (1/200)p2 + (1/300) + (1/3)w = 27,440/352,547	225 percent
d	(1/100)p1 + (3/10)p2 + (7/20)w = 305,816/352,547	225 percent
e	p1 + w = 2,308/7,501	225 percent
f	(1/4)p2 + (3/2)w = 554,839/705,094	≈ 27.1 percent

4.2 Prices at a Higher Wage with the Original Technique

Now suppose the wage is higher, namely w = 1,332/5,197 ≈ 0.2563 bushels per person-year. Consider prices of p₁ ≈ 0.2622 bushels per ton iron and p₂ ≈ 0.4167 bushels per ton steel. Table 5 shows cost accounting for these prices.

Table 5: Costs and the Rate of Profits at a High Wage (Incomplete)

Process	Cost	Rate of Profits
a	0.141 Bushels per ton iron	85.9 percent
d	0.2241 Bushels per ton steel	85.9 percent
e	0.5379 Bushels per bushel	85.9 percent
f	0.5178 Bushels per bushel	93.1 percent

Notice the same rate of profits is obtained in operating the first three processes. But the cost of producing a bushel corn with the last process is lower than in producing corn with process e. A larger rate of profits is obtained in operating that process. The managers of the firm will realize that their accounting implies that the Delta technique should be operated. If this firm were not vertically integrated and iron and steel were purchased on the market, a market algorithm would also lead to the Delta technique being adopted at this wage.

4.3 Prices at the Higher Wage with the Cost-Minimizing Technique

Continue to consider a wage of w = 1,332/5,197 ≈ 0.2563 bushels per person-year. The accountants report prices of p₁ = 1,420/5,197 ≈ 0.2732 bushels per ton iron and p₂ = 2,402/5,197 ≈ 0.4622 bushels per ton steel. Table 6 shows costs per unit output for the five processes under these prices.

Table 6: Costs and the Rate of Profits at a High Wage

Process	Cost	Rate of Profits
a	710/5,197	100 percent
d	1,201/5,197	100 percent
e	2,752/5,197	≈ 88.8 percent
f	1/2	100 percent

With this set of prices, the Delta technique is operated, and a rate of profits of 100 percent is obtained. The cost of operating the first corn-producing process exceeds the cost of operating the corn-producing process in the Delta technique. With a higher wage, the managers of a cost-minimizing firm will choose to operate a corn-producing process that requires more labor per bushel corn produced gross. (3/2 person-years is greater than 1 person-year.) More labor will also be hired per bushel corn produced net.

5.0 Conclusion

Table 7 summarizes these calculations. The ultimate result of a higher wage in the range considered is the adoption of a more labor-intensive technique. If this firm continues to produce the same level of net output and maximizes profits, its managers will want to employ more workers at the higher of the two wages considered. So much for the theory that, given competitive markets, wages and employment are determined by the interaction of well-behaved supply and demand curves on the labor market.

Table 7: A More Labor-Intensive Technique at a Higher Wage

Wage	Technique	Labor Intensity
0.05473 bushels per person-year	Gamma	1.409 person-years per bushel
0.2563 bushels per person-year	Delta	1.626 person-years per bushel

This example can be generalized in many ways. Different types of labor can be introduced. More intermediate produced capital goods can be included. Any number of processes can be available for producing each good, including an uncountable infinity. The use of fixed capital introduces more complications. The introductory marginalist textbook story about wages and employment in competitive markets is without foundation.

Why do so many economists teach nonsense?

The Emergence Of The Reverse Substitution Of Labor

Figure 1: A Wage Frontier With Two Fluke Switch Points

This post is a rewrite of this, without the attempt to draw a connection to structural economic dynamics.

This post presents an example with circulating capital alone. Table 1 presents the technology for an economy in which two commodities, iron and corn, are produced. Managers of firms know of one process for producing iron and two for producing corn. Each process is specified by coefficients of production, that is, the required physical inputs per unit output. The Alpha technique consists of the iron-producing process and the first corn-producing process. Similarly, the Beta technique consists of the iron-producing process and the second corn-producing process. At any time, managers of firms face a problem of the choice of technique.

Table 1: The Coefficients of Production

Input	Iron Industry	Corn Industry
		Alpha	Beta
Labor	a0,1 = 1	a0,2α = 16/25	a0,2β
Iron	a1,1 = 9/20	a1,2α = 1/625	a1,2β
Corn	a2,1 = 2	a2,2α = 12/25	a2,2β = 27/400

Two parameters are not given numerical values in this specification of technology. The approach taken here is to examine a local perturbation of parameters in a two-dimensional slice of the higher dimensional parameter space defined by the coefficients of production in particular numeric examples. With wages paid out of the surplus product at the end of the period of production, the wage curves for the two techniques are depicted in Figure 1 for a particular parametrization of the coefficients of production. The Beta technique is cost-minimizing for any feasible distribution of income. If the wage is zero and the workers live on air, the Alpha technique is also cost-minimizing.

A switch point is defined in this model of circulating capital to be an intersection of the wage curves. These switch points, for the particular parameter values illustrated in Figure 1, are fluke cases. Almost any variation in the model parameters destroys their interesting properties. A switch point exists at a rate of profits of -100 percent only along a knife edge in the parameter space (Figure 2). Likewise, a switch point exists on the axis for the rate of profits only along another knife edge. The illustrated example, with two fluke switch points, arises at a single point in the parameter space, where these two partitions intersect.

Figure 2: The Parameter Space for the Reverse Substitution of Labor

Figure 2 depicts a partition of the parameter space around the point with these two fluke switch points. Below the horizontal line, the switch point on the axis for the rate of profits has disappeared below the axis. The Beta technique is cost-minimizing for all feasible non-negative rates of profits. Above this locus, the Alpha technique is cost-minimizing for a low enough wage or a high enough feasible rate of profits.

In the northwest, the switch point at a negative rate of profits occurs at a rate of profits lower than 100 percent. Around the switch point at a positive rate of profits, a lower wage is associated with the adoption of the corn-producing process with a larger coefficient for labor. That is, at a higher wage, employment is lower per unit of gross output in the corn industry.

In the northeast of Figure 2, the switch point for a positive rate of profits exhibits the reverse substitution of labor. Around this switch point, a higher wage is associated with the adoption of a process producing the consumer good in which more labor is employed per unit of gross output. The other switch point exists for a rate of profits between -100 percent and zero. Steedman (2006) presents examples with this phenomenon in models with other structures.

Qualitative changes in the wage frontier exist in the parameter space away from the part graphed in Figure 2. The analysis presented here is of local perturbations of the depicted fluke case.

Reminder: Wages, Employment Not Determined By The Supply And Demand Of Labor

1.0 Introduction

Over a half-century ago, economists reached a consensus. The model in which employment and real wages are explained by the intersection of a downwards-sloping labor demand function and a supply function is incoherent, not even wrong. This incoherence was demonstrated under the assumptions of perfect competition and of firms that have adjusted their plant and other capital inputs. I do not know what Greg Mankiw and Jonathan Gruber are doing, but it certainly is not education.

Anyways, I have not recently gone through a simple example, without some of my innovations. Maybe I will repost this some time with graphs and more references.

2.0 Technology

Consider a very simple vertically-integrated (representative) firm that produces a single consumption good, corn, from inputs of labor, iron, and (seed) corn. All production processes in this example require a year to complete. The managers of the firm know of two processes for producing corn and two processes for producing iron. The processes A and B, for producing corn, require the tabulated inputs to be available at the beginning of the year for each bushel corn produced and available at the end of the year. Similarly, process C, for example, requires one person-year, 1/40 bushels corn, and 1/10 tons iron to be available at the beginning of the year for each ton of iron produced by this process. This is an example of circulating capital; all inputs of corn and iron are used up during the year in producing the gross output.

Table 1: Coefficients of Production

Input	Industry
	Corn		Iron
	A	B	C	D
Labor	a0, 1(A) = 1	a0, 1(B) = 1	a0, 2(C) = 1	a0, 2(D) = 275/464
Corn	a1, 1(A) = 2/5	a1, 1(B) = 3/5	a1, 2(C) = 1/40	a1, 2(D) = 0
Iron	a2, 1(A) = 2	a2, 1(B) = 1/2	a2, 2(C) = 1/10	a2, 2(D) = 113/232

Apparently, inputs of iron and corn can be traded off in producing corn outputs. Likewise, inputs of corn and iron can be traded off in producing iron. The iron-producing process that uses less iron and more corn, however, also requires a greater quantity of labor input.

2.0 Techniques

A technique consists of a process for producing corn and a process for producing iron. Thus, there are four techniques in this example. They are defined in Table 3.

Table 2: Techniques of Production

Technique	Corn Process	Iron Process
Alpha	A	C
Beta	A	D
Gamma	B	C
Delta	B	D

3.0 Quantity Flows

I want to consider a couple of different levels at which this firm can operate the processes comprising the techniques. First, consider the quantity flows in Table 3, in which Process A is used to produce 1 41/49 Bushels corn, and Process C is used to produce 4 4/49 Tons iron. When the firm operates these processes in parallel, it requires a total of 41/49 bushels corn as input. The output of the corn-producing process can replace this input, leaving a net output of one bushel corn. Notice that the total input of iron are 3 33/49 + 20/49 = 4 4/49 tons iron, which is exactly replaced by the output of Process C. So Table 4 shows a technique in which 5 45/49 person-years labor are used to produce a net output of one bushel corn. The firm, when operating this technique can produce any desired output of corn by scaling both processes equally.

Table 3: The Alpha Technique

Inputs	Process A	Process C
Labor	a0, 1(A) qA = 1 41/49 person-yrs.	a0, 2(C) qC = 4 4/49 person-yrs.
Corn	a1, 1(A) qA = 36/49 bushels	a1, 2(C) qC = 5/49 bushels
Iron	a2, 1(A) qA = 3 33/49 tons	a2, 2(C) qC = 20/49 tons
Outputs	qA = 1 41/49 bushels	qC = 4 4/49 tons

Table 5 shows the application of the same sort of arithmetic to the Beta technique. The labor-intensity of the Beta technique is 5 185/357 person-years per bushel. Neither the Gamma nor the Delta technique are profit-maximizing for the prices considered below.

Table 4: The Beta Technique

Inputs	Process A	Process D
Labor	a0, 1(A) qA = 1 2/3 person-yrs.	a0, 2(D) qD = 3 304/357 person-yrs.
Corn	a1, 1(A) qA = 2/3 bushels	a1, 2(D) qD = 0 bushels
Iron	a2, 1(A) qA = 3 1/3 tons	a2, 2(D) qD = 3 59/357 tons
Outputs	qA = 1 2/3 bushels	qD = 6 178/357 tons

4.0 Prices

Which technique will the firm adopt, if any? The answer depends, in this analysis, on which is more profitable. So one has to consider prices. I assume throughout that inputs of iron, corn, and labor are charged at the start of the year. Corn is the numeraire; its price is unity throughout. Two different levels of wages are considered.

4.1 Prices with a Low Wage

Accordingly, assume wages are initially 3/2780 bushels per person-year. Under the assumptions of perfect competition, this price of labor is a given for the firm.

If all corn-producing firms are vertically integrated, a market price for iron is not available. At the end of the year, the firm will have a stock of produced corn and iron. Even though the managers of the firm intend all of the iron to be used as an input to further production, the question arises for accountants of how to evaluate the stock of gross output. I suggest the accountants set a price of iron such that the firm is making the same rate of profits in all of the processes that it is operating. According let the price of iron, p, be 55/1112 bushels per ton.

Table 5 shows accounting with these prices. The column labeled "cost" shows the cost of the inputs needed to produce one unit output, a bushel corn or a ton iron, depending on the process. Accounting profits for a unit output are the difference between the price of a unit output and this cost. The rate of (accounting) profits, shown in the last column, is the ratio of accounting profits to the cost. The rate of profits is independent of the scale at which each process is operated.

Table 5: Costs and the Rate of Profits at a Low Wage

Process	Costs	Rate of Profits
A	a1, 1(A) + a2, 1(A) p + a0, 1(A) w = 1/2	100 percent
B	a1, 1(B) + a2, 1(B) p + a0, 1(B) w = 6959/11120	59.8 percent
C	a1, 2(C) + a2, 2(C) p + a0, 2(C) w = 69/2224	59.4 percent
D	a1, 2(D) + a2, 2(D) p + a0, 2(D) w = 55/2224	100 percent

These prices are compatible with the use of the Beta technique to produce a net output of corn. The Beta technique specifies that process A be used to produce corn and process D be used to produce iron. Notice that process B is more expensive than process A, and that process C is more expensive than process D. These prices do not provide signals to the firm that processes outside the Beta technique should be adopted. The vertically-integrated firm is making a rate of profit of 100 percent in producing corn with the Beta technique. The same rate of profits are earned in producing corn and in reproducing the used-up iron by an iron-producing process.

4.2 One Set of Prices with a Higher Wage

Suppose this firm faces a wage more than 25 times higher, namely 109/4040 bushels per person-year. Consider what happens if the firm doesn't revalue the price of iron on its books. Table 6 shows this case. Since labor enters into each process, the rate of profits has declined for all processes. The ratio of labor to the costs of the other inputs is not invariant across processes. Thus, the rate of profits has declined more in some processes than in others. Notice especially, than the rate of profits is no longer the same in the processes, A and D, that comprise the Beta technique.

Table 6: Costs and the Rate of Profits at a Higher Wage

Process	Costs	Rate of Profits
A	a1, 1(A) + a2, 1(A) p + a0, 1(A) w ≈ 0.5259	90.1 percent
B	a1, 1(B) + a2, 1(B) p + a0, 1(B) w ≈ 0.6517	53.4 percent
C	a1, 2(C) + a2, 2(C) p + a0, 2(C) w ≈ 0.05693	-13.1 percent
D	a1, 2(D) + a2, 2(D) p + a0, 2(D) w ≈ 0.04008	23.4 percent

This accounting data does not reveal the firm's rate of return in operating the Beta technique. The firm cannot be simultaneously making both 23 percent and 90 percent in operating that technique. Furthermore, this data provides a signal to the firm to withdraw from iron production and make only corn. So this data says that something must change.

4.3 Another Set of Prices with a Higher Wage

Perhaps all that is needed is to re-evaluate iron on the firm's books. Higher wages have made iron more valuable. Table 7 shows costs and the rate of profits when iron is evaluated at an accounting price of approximately 0.10569124 bushels per ton.

Table 7: Costs and the Rate of Profits with Iron Repriced

Process	Costs	Rate of Profits
A	a1, 1(A) + a2, 1(A) p + a0, 1(A) w ≈ 0.6384	56.7 percent
B	a1, 1(B) + a2, 1(B) p + a0, 1(B) w ≈ 0.6798	47.1 percent
C	a1, 2(C) + a2, 2(C) p + a0, 2(C) w ≈ 0.06255	69.0 percent
D	a1, 2(D) + a2, 2(D) p + a0, 2(D) w ≈ 0.06747	56.7 percent

This revaluation of iron reveals that the firm makes a rate of profits of 57 percent in operating the Beta technique. The firm makes the same rate of profits in producing corn and in producing its input of iron. But the manager of the iron-producing process would soon notice that the cost of operating process C is cheaper.

4.4 A Final Set of Prices with a Higher Wage

So the firm would ultimately switch to using process C to produce iron. The price of iron the firm would enter on its books would fall somewhat, but still be higher than the original price at the low wage. Table 8 shows the accounting with a price of iron of 10/101 Bushels per Ton. The firm has adopted the cheapest process for producing iron, and the rate of profits is the same in both corn-production and iron-production. The accounting for this vertically-integrated firm is internally consistent.

Table 8: Costs and the Rate of Profits at a High Wage

Process	Costs	Rate of Profits
A	a1, 1(A) + a2, 1(A) p + a0, 1(A) w = 5/8	60 percent
B	a1, 1(B) + a2, 1(B) p + a0, 1(B) w ≈ 0.6765	47.8 percent
C	a1, 2(C) + a2, 2(C) p + a0, 2(C) w = 25/404	60 percent
D	a1, 2(D) + a2, 2(D) p + a0, 2(D) w ≈ 0.06422	54.2 percent

5.0 Conclusion

Table 9 summarizes these calculations. The ultimate result of a higher wage is the adoption of a more labor-intensive technique. If this firm continues to produce the same level of net output and maximizes profits, its managers will want to employ more workers at the higher of the two wages considered.

Table 9: A More Labor-Intensive Technique at a Higher Wage

Wage	Technique	Labor-Intensity
3/2780 ≈ 0.00108 bushels per person-year	Beta	5 185/357 ≈ 5.52 person-years per bushel
109/4040 ≈ 0.0270 bushels per person-year	Alpha	5 45/49 ≈ 5.92 person-years per bushel

Economists, such as Edwin Burmeister, have investigated what conditions on technology might be necessary to rule out the illustrated effects. They know that no such conditions are known, and would be extremely restrictive anyways. A marginalist special case has not been specified for the case in which more than one commodity is produced.

So much for the theory that wages and employment are determined by the interaction of well-behaved supply and demand curves on the labor market.

Appendix: Production Functions

The data above allow for the specification of two well-behaved production functions, one for corn and the other for iron. For illustration, I outline how to construct the production function for corn.

Let L be the person-years of labor, Q₁ be bushels corn, and Q₂ be tons labor allocated as inputs for corn-production during the production period (a year). Let X₁ be the bushels corn produced with Process A, and X₂ be the bushels corn produced with Process B. The production function for corn is the solution of an optimization problem in which as much corn as possible is produced from the given inputs.

Choose X₁, X₂

To maximize X = X₁ + X₂

subject to

a_{0, 1}(A) X₁ + a_{0, 1}(B) X₂ ≤ L

a_{1, 1}(A) X₁ + a_{1, 1}(B) X₂ ≤ Q₁

a_{2, 1}(A) X₁ + a_{2, 1}(B) X₂ ≤ Q₂

X₁ ≥ 0, X₂ ≥ 0,

Let f(L, Q₁, Q₂) be the solution of this Linear Program, that is, the production function for corn. (This production function is not Leontief.) The production functions constructed in this manner exhibit properties typically assumed in marginalist economics. In particular, they exhibit Constant Returns to Scale, and the marginal product, for each input, is a non-increasing step function. The production functions are differentiable almost everywhere.

The point of this example, that sometimes a vertically integrated firm will want to hire more labor per unit output at higher wages, is compatible with the existence of many more processes for producing each commodity. As more processes are used to construct the production functions, the closer they come to smooth, continuously-differentiable production functions. The point of this example seems to be compatible with smooth production functions. It also does not depend on the circular nature of production in the example, in which corn is used to produce more corn.

References

• Pierangelo Garegnani. 1970. Heterogeneous capital, the production function and the theory of distribution. The Review of Economic Studies 37(3): 407-436.
• Arrigo Opocher and Ian Steedman. 2015. Full Industry Equilibrium: A Theory of the Industrial Long Run. Cambridge University Press.
• K. Sharpe. 1999. Notes and comment. On Sraffa's price system. Cambridge Journal of Economics 23(1): 93-1010.
• Paul A. Samuelson. 1966. A summing up. Quarterly Journal of Economics 80(4): 568-583.
• Ian Steedman. 1985. On input "demand curves". Cambridge Journal of Economics 9(2): 165-172.
• Robert L. Vienneau. 2005. On labour demand and equilibria of the firm, Manchester School 73(5): 612-619.

The Emergence Of The Reverse Substitution Of Labor

Figure 1: A Wage Frontier With Two Fluke Switch Points

This post presents an example with circulating capital alone. Table 1 presents the technology for an economy in which two commodities, iron and corn, are produced. One process is known for producing iron, and two are known for producing corn. Each process is specified by coefficients of production, that is, the required inputs per unit output. The Alpha technique consists of the iron-producing process and the first corn-producing process. Similarly, the Beta technique consists of the iron-producing process and the second corn-producing process. At any time, managers of firms face a problem of the choice of technique.

Table 1: The Coefficients of Production

Input	Iron Industry	Corn Industry
		Alpha	Beta
Labor	a0,1 = 1	a0,2α = 0.7174 e-σ t	a0,2β = 1.282 e-φ t
Iron	a1,1 = 9/20	a1,2α = 0.001764 e-σ t	a1,2β = 0.3375 e-φ t
Corn	a2,1 = 2	a2,2α = 0.5386 e-σ t	a2,2β = 0.135 e-φ t

Technical progress, as in structural economic dynamics (Pasinetti 1981), results in decreasing coefficients of production. The coefficients in each corn-producing process decrease at the same rate, but vary between the processes. With wages paid out of the surplus product at the end of the period of production, the wage curves for the two techniques are depicted in Figure 1 for a particular parametrization of the coefficients of production. At this moment in time, the Beta technique is cost-minimizing for any feasible distribution of income. If the wage is zero and the workers live on air, the Alpha technique is also cost-minimizing.

A switch point is defined in this model to be an intersection of the wage curves. These switch points are fluke cases in that almost any variation in the model parameters destroys their interesting properties. A switch point exists at a rate of profits of -100 percent only along a knife edge in the parameter space (Figure 2). Likewise, a switch point exists on the axis for the rate of profits only along another knife edge. The illustrated example, with two fluke switch arises at a single point in the parameter space, where these two partitions intersect.

Figure 2: The Parameters Space for the Reverse Substitution of Labor

Figure 2 depicts a partition of the parameter space around the point with these two fluke switch points. Above the more steeply-sloping locus, the switch point on the axis for the rate of profits has disappeared below the axis. The Beta technique is cost-minimizing for all feasible non-negative rates of profits. Below this locus, the Alpha technique is cost-minimizing for a low enough wage or a high enough feasible rate of profits.

In the north east, the switch point at a negative rate of profits occurs at a rate of profits lower than 100 percent. Around the switch point at a positive rate of profits, a lower wage is associated with a corn-producing process with the larger coefficient for labor. That is, a_0,2^α(σ t) > a_0,2^β(φ t).

In the south east of Figure 2, the switch point for a positive rate of profits exhibits the reverse substitution of labor. Around this switch point, a lower wage is associated with the adoption of a process producing the consumer good in which less labor is employed per unit of gross output. The other switch point exists for a rate of profits between -100 percent and zero. Steedman (2006) presents examples with this phenomenon in models with other structures.

Qualitative changes in the wage frontier exist in the parameter space away from the part graphed in Figure 2. The analysis presented here is only local to the depicted fluke case.

Reminder: Wages, Employment Not Determined By Supply And Demand For Labor

Figure 1: The Wage as Functions of Employment by Industry

1.0 Introduction

This post repeats a common theme of mine. It builds on an example I have previously gone on about. I use this example to graph, given the wage, the amount of labor firms would like to employ in each industry, per unit of gross output in each industry. These graphs are derived for an economy in which three commodities are produced: iron, steel, and corn. I also graph the amount of labor firms would like to employ across all industries, given that the net output of the economy consists of a unit quantity of corn. The value of this function is called an employment multiplier.

No doubt, in actual capitalist economies, some firms in some places have market power in hiring workers. Workers incur search costs in trying to find jobs whose requirements match well with their skills. Owners and managers of firms face principal agent problems. Owners, managers, workers, etc. have their own information sets at any given instant, and doubtless they are not all identitical. But, before exploring these complications, if would be nice if so many leading mainstream economists were not clueless about price theory. One might be more interested in institutions and the history of the labor movement.

2.0 Technology

Consider an economy in which three commodities, iron, steel, and corn, are produced. Two processes, as seen in Table 1 are available to produce each commodity from inputs of labor, iron, steel, and corn. Each process exhibits constant returns to scale and takes a year to produce. Each column in Table 1 specifies the inputs needed to produce a unit quantity of the commodity produced by that process. This is a model of circulating capital. All physical inputs in each process are used up in the course of the year in producing the commodity output by that process.

Table 1: The Technology

Input	Iron Industry		Steel Industry		Corn Industry
	a	b	c	d	e	f
Labor	1/3	1/10	5/2	7/20	1	3/2
Iron	1/6	2/5	1/200	1/100	1	0
Steel	1/200	1/400	1/4	3/10	0	1/4
Corn	1/300	1/300	1/300	0	0	0

A technique consists of a process in each industry. Table 2 specifies the eight techniques that can be formed from the processes specified by the technology. If you work through this example, you will find that to produce a net output of one bushel corn, inputs of iron, steel, and corn all need to be produced to reproduce the capital goods used up in producing that bushel.

Table 2: Techniques

Technique	Processes
Alpha	a, c, e
Beta	a, c, f
Gamma	a, d, e
Delta	a, d, f
Epsilon	b, c, e
Zeta	b, c, f
Eta	b, d, e
Theta	b, d, f

Each technique is represented by coefficients of production. For the Alpha technique, let a_{0, α} be a three-element row vector representing the labor coefficients, and let A_α be the 3 x 3 Leontief matrix for this technique. The first element of a_{0, α}, (1/3) person-years per ton, represents the labor input needed to produce a ton of iron. The first column of A_α represents the inputs of iron, steel, and corn needed to produce a ton of iron. A parallel notation is used for the other seven techniques.

Suppose the net output of the economy is a bushel corn. A bushel corn is also the numeraire.

3.0 The Price System

Prices of production are defined to be constant spot prices that allow the smooth reproduction of the economy. Suppose Alpha is the cost-minimizing technique. Let p be the three-element row matrix designating the prices of iron, steel, and corn. I make the assumption that markets are such that the rate of profits in the iron, steel, and corn industries are (r s₁), (r s₂), and (r s₃), respectively. Suppose S is a diagonal matrix with the obvious elements along the diagonal, and I designates the identity matrix. Then prices of production satisfy the following system of equations:

p_α A_α (I + r S) + w_α a_{0, α} = p_α

I choose a bushel of corn to be the numeraire. If e₃ is the last column of the identity matrix, the following equation specifies the numeraire:

p_α e₃ = 1

As is not surprising, the above system of equations has one degree of freedom. One can solve for the wage, w_α(r), as a function of the scale factor for the rate of profits, r. The wage curve is a downward-sloping curve that intercepts both the axis for the wage and the scale factor at positive values. A similar function can be derived the other techniques, and they can be graphed in the same diagram.

4.0 The Choice of Technique

Figure 2 graphs the wage curves for the techniques that are cost-minimizing for some feasible wage, given markups by industry. The outer envelope is the wage frontier. The cost-minimizing technique at a given wage is the technique with the right-most wage curve at that wage. The cost-minimizing techniques at each wage and the switch points between techniques are noted on the figure.

Figure 2: The Wage Frontier

5.0 Wages and Employment

For each technique, one can calculate the employment required across all three industries to produce a net product of a bushel corn. In these calculations, the processes in a technique are operated at a level so as to replace the iron, steel, and corn used up in producing that bushel of corn. Since which technique is cost-minimizing at a given wage is shown above, one can plot the wage against employment, as in Figure 3. In some sense, this is a macroeconomic labor demand function. On the other hand, if one does not get well-behaved supply and demand functions for labor, one might want to say that supply and demand does not apply here. Notice the switch point between the Gamma and Delta techniques. Around this switch point, a higher wage is associated with firms wanting to employ more workers.

Figure 3: The Wage as a Function of Employment Across Industries

The labor coefficient in each industry is specified along with each technique. Figure 1, at the top of this post, graphs employment in each industry per unit gross product. Here, a higher wage around the switch point between the Gamma and Delta techniques is associated with firms wanting to employ more labor per bushel corn produced as gross output in the corn industry. This reverse substitution of labor can occur around a switch point in which capital-reversing does not occur and vice versa.

6.0 The Effects of Markups

In the above story, the markup in the steel industry is less than the markups in the iron and corn industries. One might think of this as a deviation from competitive markets. In this conception, markets are competitive when markups are unity in all industries.

Figure 4 illustrates how the sequence of techniques along the wage frontier varies with the markup in the steel industry. The result of the specific markups used above is that the Beta technique is cost-minimizing at a low enough wage. That is the second process in the corn-producing industry recurs. The first corn-producing process also recurs.

Figure 4: The Variation of the Wage Frontier with the Markup in the Steel Industry

If those investing in the iron and corn industries are able to persistently impose even greater barriers to entry, the markup in the steel industry would be even lower. Evenually, the Alpha and the Gamma techniques would not be cost-minimizing at any wage. Neither process in the corn industry would recur. The instance of capital-reversing would also be destroyed. The same follows if the markup in the steel industry exceeds the markups in the iron and corn industry sufficiently.

7.0 Conclusion

As far as I know, mainstream economists have been teaching what has been known to be, at best, incorrect for half a century. Are they fools or knaves? What accounts for this extraordinary intellectual bankruptcy?

On David Card's Nobel

The Sveriges Riksbank prize in economic sciences in memory of Alfred Nobel this year goes to David Card, Joshua Angrist, and Guido Imbens. I cannot say much about instrumental variables, Angrist, or Imbens. Since I have been pointing to Card's work with Alan Krueger on minimum wages for decades, I thought I might say somthing about his half of the prize.

I do not have much new to say. I find both natural experiments and meta-analysis intriguing.

Both Card and Krueger's natural experiments with minimum wages and their meta-analysis have been superceded. Maybe 'transcended' or 'replicated' would be better terminology. That is why, in my 2019 paper in Strucutral Change and Economic Dynamics, I reference Andrajit Dube and his colleagues, not Card and Krueger. Also, David Neumark's quibbles with Card are currently uninteresting. (Any reporter talking to Neumark should note he started out with funding from a consortium of fast food joints.)

I object to attempts to explain the lack of impact of minimum wages on employment by the theory of monopsony. Economists have known, for over half a century, that wages and employment cannot, even under ideal conditions, be explained by the interaction of well-behaved supply and demand curves in the labor market. In marginalist theory, the supply of labor is derived from utility-maximizing households trading off leisure and commodities to consume. The demand for labor is supposed to be derived from profit-maximizing firms. But no such valid derivation goes through if firms produce some commodities with the use of previously produced commodities, that is, capital goods. This well-established result is widely ignored, with no pretence at justification.

Bushwa From Jeffrey Clemens In The Journal of Economic Perspectives

"The labor supply curve slopes upward, reflecting differences in workers’ reservation wages (as driven by outside opportunities related, perhaps, to leisure, home production, and economic assistance that can be received while out of work). The labor demand curve slopes downward, tracing out the relationship between the quantity of labor employed and the marginal revenue product of that labor. This, in turn, reflects the assumption of a constant price (due, perhaps, to a perfectly competitive market for the firm’s output) and a production function in which, holding capital and technology fixed, labor has diminishing marginal productivity.

In a perfectly competitive labor market, a freely set wage will adjust to equilibrate supply and demand..."

-- Jeffrey Clemens. 2021. How do firms respond to minimum wage increases? Understanding the relevance of non-employment margins. Journal of Economic Perspectives (Winter): 51 - 72.

Clemens then considers shifts in demand and supply curves for labor with changes in prices due to market power in final goods, changes in benefits, and other aspects of jobs. He never notes his framework is balderdash. Empirical evidence, which Clemens cites, cannot make up for his basic incoherence. Mayhaps, Clemens could read Fabio Petri's textbook when it is published.

I do not expect to read articles about the speed of the ether in absolute space in physics journals. Nor do I expect to read about the weight of phlogiston in chemistry journals. Why does the American Economic Association publish articles that make astrologists look good?

Vienneau (2005) Is A Necessary Resource For Arguments About A Minimum Wage

Maybe, perhaps, that is a bit hyperbolic. But it has been known for at least half a century that, even in competitive markets, wages and employment cannot be explained by the interaction of well-behaved supply and demand curves for labor. If you do not want to read me, check out, for example, Garegnani (1970) or Opocher and Steedman (2015). Shove (1933) illustrates how far awareness of the difficulties go. White (2001) is a demonstration that I am not the only one to draw practical conclusions from the theory.

Cohort after cohort, generation after generation, in the supposedly best schools promulgate falsehoods, ignorance, and incoherent nonsense.

References

• Garegnani, Pierangelo. 1970. Heterogeneous capital, the production function and the theory of distribution. Review of Economic Studies 37(3): 407-436.
• Opocher, Arrigo and Ian Steedman. 2015. Full Industry Equilibrium: A Theory of the Industrial Long Run Cambridge: Cambridge University Press.
• Shove, G. F. 1933. Review of The Theory of Wages. Economic Journal (Sep.)
• Vienneau, Robert L. 2005. On labour demand and equilibria of the firm. Manchester School 73(5): 612-619.
• White, Graham. 2001. The poverty of conventional economic wisdom and the search for alternative economic and social policies Austrlian Review of Public Affairs

"When Economists Are Wrong"

In a blog associated with the Frankfurter Allegmeine, Gerald Braunberger criticizes the effects of Sraffian political economy on Italian policy in the 1970s. I rely on google translate and subject matter expertise to make some sense out of this. By the way, Bertram Schefold shows up in the comments. I would like to know more about the motivations behind this. Does Braunberger think the public is increasingly aware that mainstream economics is broken?

Before I disagree, I note Braunberger seems well-informed on some points. I know of grumbles about Garegnani's treatment of Sraffa's archives. And I have heard that the Trieste summer schools were torn between those who emphasized long period logic and Post Keynesians who emphasized uncertainty, money, and historical time. On another point, I do not see why Sraffians giving policy advice should care about whether their advice is consistent with the advice Ricardo had for Britain during and after the Napoleonic war. I would think Sraffians in Italy during the 1970s would be more interested in Marx's views, anyways.

I do not understand what non-Sraffian theory Braunberger thinks exists. All economists should (but do not) recognize no reason exists to think that a lower real wage, in a time of depression, will encourage firms to adopt more labor-intensive techniques and thereby increase employment. In parallel, higher real wages need not decrease employment through the adoption of less labor-intensive techniques. Supply and demand just is not a logically-consistent model, in which conclusions follow from assumptions, of wages and employment.

This does not mean that real wages can be increased willy-nilly, without any consequences. Income effects could be important. And one might want to worry about the possibility of a capital strike. I agree that the political slogan, "The wage as the independent variable" draws directly on Sraffa.

More than economics was involved in the going-ons in Italy in the 1970s. When activists are kidnapping and executing the prime minister and the government is imprisioning leftists without discrimination, firms, government, and unions are unlikely to come to a peaceful agreement about distribution.

Furthermore, Italy was not isolated from the wider world. Were the lira and the mark pegged to the dollar before Nixon ended Bretton Woods? The world-wide rise in oil prices was not the result of Sraffian policy advice. Wage-push inflation arose in many countries; it was not just an Italian problem. Sraffa's colleagues had worked out an explanation of stagflation long before the event. I would think this is an example of when (heterodox) economists are right. (A Tax-based Income Policy (TIP) is a policy idea I associate with American Post Keynesians, like Sidney Weintraub, that might have been worth trying in some countries in the 1970s.)

A Fluke Case With Two Fluke Switch Points

Figure 1: Switch Points On The Axis For The Rate Of Profits And At r = -100 Percent

This is an example of a fluke case in the analysis of the choice of technique. The interest in flukes, for me, is that they show how the characteristics of markets can change. They provide insight into structural economic dynamics, as Luigi Pasinetti calls it.

I have previously shown a fluke case, with a switch point on the axis for the rate of profits with a real Wicksell effect of zero. A perturbation of the example can lead to a reswitching example. The switch point at a wage of zero (when the workers live on air) then becomes one at a positive wage. And around that switch point, a higher wage is associated with cost minimizing firms hiring more workers to produce a given net output.

In the example in this post, the switch point on the axis for the rate of profits exhibits neither a forward nor a reverse substitution of labor. The labor coefficient in the corn industry does not vary with the processes in the technique. The Alpha technique has a ghostly presence. It can only be chosen, and not even uniquely so, when the wage is zero. A perturbation of this example can lead to one of the reverse substitution of labor. The switch point on the axis for the rate of zero would also become one at a positive wage. And that switch point might be the only switch point on the frontier at a non-negative rate of profits. Around that switch point, a higher wage is associated with cost-minimizing firms hiring more workers to produce a given gross output of corn. The labor coefficient in the corn-producing process for the technique preferred at a higher wage is larger.

Table 1: Coefficients of Production for The Technology

Input	Iron	Corn Industry
		Alpha	Beta
Labor	1	0.64097922	0.64097922
Iron	9/20	0.00157618	0.01686787
Corn	2	0.48125981	0.0674715

Table 1 specifies the technology in my usual way. I assume labor is advanced, and wages are paid out of the product at the end of a production cycle. I take a unit of corn as numeraire. Prices of production are here defined with a uniform rate of profits between the industry. I found this example with numerical exploration, so there is some round-off error in the figures.

This post is another demonstration that explaining wages and employment by supply and demand, even under ideal competitive conditions, is incoherent nonsense.

Martin Weitzman's The Share Economy

I happen to have one book by Marty Weitzman (1942 - 2019) on my bookshelf. So I thought I would write a bit about The Share Economy: Conquering Stagflation.

This is an ill-timed book. It proposes that firms negotiate with workers to pay them a percentage of revenues, instead of, say, an hourly money wage. It argues that such a change will address the widespread macroeconomic problem, throughout the 1970s, of simultaneously high unemployment and high inflation. But, by the time the book came out, stagflation had been "solved", in an extremely reactionary way. The countervailing power of organized labor was being abolished. Labor unions were being crushed, and workers would, by and large, no longer see their wages increase with productivity. Instead of unemployment being addressed, workers would just have to get used to long-lasting higher unemployment.

Maybe some day, we will get back to a setting where Weitzman's book is socially relevant. Even so, it is worth exploring how macroeconomic performance is affected by microeconomic structures.

Although I think of Weitzman as a mainstream economist, his view of the microeconomic setting at the time of his writing was not that far away from Post Keynesianism. He thinks of the "tone" of "modern industrial capitalism" as set by "a relatively small number of large-scale firms", such as those in the Fortune 500. These firms are described by the theory of monopolistic competition. (quotes on p. 11). These firms are characterized by constant costs over a wide range of levels of production below limits set by capacity. They set their prices at a markup over cost. The theory of profit maximization, under these assumptions, yields a markup based on elasticity of consumer demand.

Weitzman explicitly rejects a theory of monopsony for labor markets:

"...If your aim is to focus in on fine close-up details and you wish to do justice to the facts, you must rely on a heavily institutional approach. But I think the unique long-run substitutability of labor among different uses actually makes the competitive theory a rather good description of long-run tendencies in the labor market...

In this book I am primarily interested in the general theory of wage determination... ...at least the labor market behaves 'as if' it is competitive, in the sense that countervailing power between buyers and sellers of labor is sufficiently balanced that neither party has a clear upper hand and both possess approximately equal bargaining strength. The economy-wide real wage is not very different from what would be determined by competitive forces in the labor market." (pp. 29-30.)

I am not sure that Weitzman's account of firms is consistent with firms operating multiple plants and producing multiple products. I think of Alfred Eichner's theory of the megacorp here. I also doubt that theories of full cost, markup, or administered prices should be developed based on markups determined by elasticities. Rather, the markup might be theorized as based on firm's plans for growth.

Weitzman sees that firms will respond to fluctuations of demand by adjusting quantities, not prices. He cites Janos Kornai's contrast of planned, socialist economies with capitalist economies. In the United States, firms must attend to making the consumer's shopping experience as pleasant as possible, while in the Soviet Union, establishments do not care and consumers wait in queue. On the other hand, establishments in the Soviet Union cater to the worker. Weitzman argues his share economy would change the dynamics of the labor market such that firms in the United States would also worry more about the worker's experience.

Wietzman sees the contemporary practice of firms awarding year-end bonuses as a start towards his share economy. He includes Eastman Kodak as an example. Kodak is now bankrupt, and Kodak Park in Rochester, NY, is mostly empty and decaying. In my anecdotal experience, bonuses are often experienced as a present that cannot be planned or depended on. Maybe it would be different with more transparency from your employer, as resulting from a union contract, representatives from the union sitting on the board of directors, an Employee Stock Ownership Plan (ESOP), or some such.

Overall, I find The Share Economy intriguing. It illustrates how good economists will not develop an universal theory, but will address problems of the economist's own time and place.

(A propos of nothing in particular, Branko Milanovic has a post coming close to an endorsement of Neo-Ricardianism.)

• Martin L. Weitzman (1983). The Share Economy: Conquering Stagflation. Harvard University Press.

All Combinations of Real Wicksell Effects, Substitution of Labor

Figure 1: A Pattern Diagram

Consider an example of the production of commodities, in which many commodities are produced within capitalist firms. Suppose two techniques are available to produce a given net output. These techniques use the same set of capital goods, albeit in different proportions. They differ in process in use for only one industry. Given the qualification about the same capital goods, generic (non-fluke) switch points are the intersection of the intersection of the wage curves for two techniques that differ in exactly one process.

Suppose that, due to technological progress, some coefficients of production decrease in the process unique to the Alpha technique. Figure 1 shows a possible pattern diagram for this generalization of a previous example. Here, switch points and the maximum rate of profits are plotted against the rate of profits. As time goes by, a reswitching pattern leads to a reswitching example. The switch point created at the larger rate of profits exhibits, after t = 1/2, a negative real Wicksell effect and a reverse substitution of labor. A pattern over the axis for the rate of profits then results in the existence of another switch point at an even higher rate of profits. Technological progress can bring about, in a single example, the combination of both non-zero directions of real Wicksell effects with both non-zero directions of the substitution of labor.

The regions in Figure 1 in which reswitching occurs also illustrate process recurrence. Process recurrence is more general, inasmuch as it can arise even without reswitching.

Since all four possible combinations, of nonzero-real Wicksell effects and the substitution of labor, are possible, the direction of real Wicksell effects and the direction of the substitution of labor are independent of one another. The choice of technique results in variation in gross outputs in multiple industries, for given net outputs. (The question of returns to scale is of interest in this context.) These variations in gross outputs also result in variation in the amount of labor firms want to employ. Around a switch point with a positive real Wicksell effect, firms want to employ more labor, per unit of net output, in the aggregate across all industries. A necessary consequence is that they want to employ more labor in at least one industry. This variation in aggregate employment is consistent with any direction in the variation in the labor coefficient of production in the industry with the varying process.