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 TechnologyConsider 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.
| 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 TechniquesA 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.
| 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.
| 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.
| 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 WageAccordingly, 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.
| 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 WageSuppose 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.
| 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 WagePerhaps 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.
| 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 WageSo 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.
| 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.
| 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 FunctionsThe 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
a0, 1(A) X1 + a0, 1(B) X2 ≤ L
a1, 1(A) X1 + a1, 1(B) X2 ≤ Q1
a2, 1(A) X1 + a2, 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.
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