Wages, Employment Not Determined By Supply And Demand

by
Robert Vienneau
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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.

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.

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.

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.

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.

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.

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.

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 ≈ 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.

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.

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.

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.

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.

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.

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?