This is another example of reswitching and capital reversing. In this example, a capital good can only be used with a fixed quantity of labor in producing the single consumption good. If entrepreneurs would like to employ a different quantity of labor per unit output of the consumption good, they must use the services of some other capital good. In other words, this is an example of fixed coefficients, in some sense. But the existence of reswitching and capital-reversing are compatible with variable coefficients and the possibility of marginal adjustments. I think many economists may be confused on this point.
This example can also be seen as an illustration of a generalization of Paul Samuelson's "Surrogate Production Function" model. Samuelson assumed that alternative processes for producing the consumption good each require the use of a different capital good. But Samuelson required the special-case assumption that, for a given capital good, the desired ratio of physical units of the capital good to labor was invariant among processes producing the consumption good and producing the capital good. As Garegnani pointed out, this special case effectively collapses Samuelson's model to a one-good model. And so Samuelson's attempted defense of the aggregate production function fails.
2.0 The Technology and Quantity Flows
Consider a firm whose managers are aware of the technology shown in Table 1. Each column defines a Constant-Returns to Scale process for producing the output indicated by the column heading. The managers know of two processes for producing corn, a good used only in consumption. Steel and labor are each used in producing either more steel or in producing corn. Similarly for tin. Each process requires a year to complete, and the capital good is totally used up in producing the output. In the jargon, this is an example with no fixed capital; all capital is circulating capital.
Two techniques are available for the vertically-integrated firm to produce corn. Each technique consists of two processes operated in parallel. The first technique, called the alpha technique, consists of the steel-producing process and the first of the two corn-producing processes. Suppose these processes are operated at a scale to produce 6/5 tons of steel gross and one bushel corn. Of the output of the steel-producing process, 1/5 tons replaces the steel inputs used up in producing the steel. The remaing ton replaces the steel used up in producing the corn. Thus, at this scale, the net output of the vertically-integrated firm consists of one bushel corn. Since 7/5 person-years are used across both processes, the ratio of the capital-inputs in physical terms to worker is 6/7 tons steel per person-year. Output per worker consists of 5/7 bushels per person-year.
The second technique, called the beta technique, consists of the tin-producing process and the second corn-producing process. When these processes are used to produce 1/3 tons tin gross and one bushel corn, the net output consists of one bushel corn. At this scale of operations, 5/3 person-years are employed. The ratio of the capital-inputs to worker is 1/5 tons tin per person-year. Output per worker is 3/5 bushels per person-year.
The alpha technique produces more output per worker. In the traditional and incorrect neoclassical analysis of the production function, greater output per worker for a known technology is achieved by the use of more capital per worker. Firms choose to use a less labor-intensive process, in this incorrect analysis, when consumers choose to save more and the interest rate (called the rate of profits below) consequently falls. Notice that one cannot tell from the above arithmetic with the example whether the capital-to-labor ratio is higher in the alpha or in the beta technique. The units of comparison so far are incommensurable.
The adoption of each technique in a steady-state competitive capitalist economy specifies certain relationships among price variables.
3.1 The Alpha Technique
If the alpha technique is cost-minimizing, the following equations must hold:
(1/6) ps (1 + r) + (1/3) w = ps
ps (1 + r) + w = 1where ps is the price of steel in units of bushels corn per ton steel, w is the wage in units of bushels corn per person year, and r is the rate of profits. These equations incorporate the assumption that wages are paid out of the surplus at the end of the year. This system of equations shows that the same rate of profits is earned in producing both steel and corn.
This is a system of two equations in three variables. As Sraffa notes, it has one degree of freedom. Two of the variables can be found in terms of the third. I take the rate of profits as the independent variable. The wage is then:
wα(r) = (5 - r)/(7 + r)And the price of steel is:
ps(r) = 2/(7 + r)The wage-rate of profits curve is a declining function in the first quadrant of the wage-rate of profits space (Figure 1). Since this is a model of the production of commodities by means of commodities, the wage-rate of profits curve cuts both axes. The maximum rate of profits in the alpha system, found when the workers live on air, is r = 500%. The maximum wage, where the capitalists receive none of the surplus, is 5/7 bushels per person-year.
The price of steel can be used to evaluate the capital goods used per worker in the alpha technique:
Iα(r) = 12/[7 (7 + r)] bushels per person-yearThe price Wicksell effect is the variation, given the technique, in the value of capital per worker with the rate of profits.
|Figure 1: Wage-Rate Of Profits Curves|
3.2 The Beta Technique
The price equations for the beta technique consist of:
(1/4) pt (1 + r) + (1/2) w = pt
(1/4) pt (1 + r) + (3/2) w = 1where pt is the price of tin in units of bushels corn per ton tin. The solution of this system of equations is:
wβ(r) = (3 - r)/(5 - r)
pt(r) = 2/(5 - r)The maximum rate of profits in the beta system is r = 300%, which is lower than the maximum in the alpha system. The maximum wage in the beta system is 3/5 bushels per person-year, which is also lower than in the alpha system. The value of capital per worker is:
Iβ(r) = 2/[5 (5 - r)] bushels per person-year
3.3 Switch Points
The technique with the highest wage, at a given rate of profits, is the cost-minimizing technique at that rate of profits. The wage-rate of profits frontier in models like the example is the outer envelope in Figue 1 of all wage-rate of profits curves. The alpha technique is cost-minimizing both for rates of profits between 0% and 100% and for rates of profits between 200% and 500%. The beta technique is cost minimizing for rates of profits between 100% and 200%.
At a switch point, more than one technique is cost minimizing. In the example, switch points are at (r, w) = (100%, 1/2) and at (200%, 1/3).
3.4 Selected Consequences for Capital and Labor "Markets"
The analysis of the choice of technique allows one to graph the value of capital per worker in a steady state versus the rate of profits (Figure 2). The graph displays a combination of price and real Wicksell effects. Consider the horizontal lines at the rate of profits of 100% and 200% for the switch points. The real Wicksell effect is the variation in the in the value of capital per worker with the technique at a rate of profits for a switch point. The price Wicksell effect is shown by the curves not being vertical between switch points. In Figure 2, the switch point with a positive real Wicksell effect is indicated. The example demonstrates the logical invalidity of traditional neoclassical theory, in which real Wicksell effects are always negative.
|Figure 2: Investment Function|
The analysis of the choice of technique also allows one to graph the level of employment firms offer at each wage, given the level of net output. Figure 2 indicates the switch point around which firms will attempt to hire more labor if the wage is increased.
|Figure 3: Labor Employed per Unit Net Output as a Function of Wages|
As an aside, I wonder if the following quotation includes an allusion to the Cambridge Capital Controversy:
"For output in goods-producing industries to increase, the quantity of intermediate goods used to produce the output must also be increased (e.g. cloth to produce a shirt, ham to produce a ham sandwich, etc.). However, the concept of the marginal product of labor (or capital) requires that as the input of labor (or capital) is increased, all other inputs must be held constant. But this is not possible for intermediate inputs in goods-producing industries. Therefore, the concept of the marginal product of labor (or capital) is not possible when there are intermediate goods in the production function. Again, we prefer not to teach such logical errors to our students." -- Bernard Guerrien and Emmanuelle Benicourt (2008)
So much for the logical validity of typical arguments for improved labor market flexibility and for the belief that minimum wages cause unemployment.
- Bernard Guerrien and Emmanuelle Benicourt (2008) "Is Anything Worth Keeping in Microeconomics?" Review of Radical Political Economics, V. 40, N. 3 (Summer): 317-323
- P. Garegnani (1970) "Heterogeneous Capital, the Production Function and the Theory of Distribution", Review of Economic Studies, V. 37, N. 3 (July): 407-436.
- Paul A. Samuelson (1962) "Parable and Realism in Capital Theory: The Surrogate Production Function", Review of Economic Studies, V. 29, N. 3 (June): 193-206