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.