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?