## Wednesday, May 03, 2006

### An Example of Capital Reversing (Part 1)

I thought I would, for my own amusement, create an example of capital-reversing. I go through the example, emphasizing the implications for the labor "market". Any detailed example is lengthy, even for me. So I break the exposition up into three, maybe four, parts. (I haven't yet finished the last part.)

This part discusses the data on technology, which is presented in Figure 1-1. This table presents coefficients of production for three processes which exhibit Constant Returns to Scale. The inputs must be available starting at the beginning of the year. The inputs provide their services over the course of the year, and the iron and corn inputs are totally used up in production. The outputs become available at the end of the year.

Figure 1-1: Production Processes Known By Firm Managers
INPUTS HIRED
AT START OF
YEAR
FIRST IRON-
PRODUCING
PROCESS
SECOND IRON-
PRODUCING
PROCESS
CORN-
PRODUCING
PROCESS
Labor1 Person-Year305/494 Person-Year1 Person-Year
Iron1/10 Ton229/494 Ton2 Ton
Corn1/40 Bushel3/1,976 Bushel2/5 Bushel
OUTPUTS1 Ton Iron1 Ton Iron1 Bushel Corn

This type of data on Constant-Returns-to-Scale production processes is enough to construct production functions. Consider the production function for iron, for example. Let:
• X be the gross amount of iron produced.
• X1 be the gross amount of iron produced with the first iron-producing process.
• X2 be the gross amount of iron produced with the second iron-producing process.
• Q0 be the Person-Years of labor available for iron production.
• Q1 be the Tons of iron available as input into iron production.
• Q2 be the Bushels of corn available as input into iron production.

For purposes of constructing the production function, the quantities of labor, iron and corn available as input to iron production are taken as given. The amount of each input needed to produce iron by any specified combination of levels of the iron-producing production processes can be found from the coefficients of production. These combinations of levels are constrained by the necessity of not having input requirements that exceed the available inputs. The following Linear Program expresses the problem of maximizing the amount of iron produced while meeting these constraints:
Choose X1 and X2
To Maximize X = X1 + X2
Such that
1 X1 + (305/494) X2 does not exceed Q0
(1/10) X1 + (229/494) X2 does not exceed Q1
(1/40) X1 + (3/1,976) X2 does not exceed Q2
X1 and X2 are both nonnegative.

One can express the value of the objective function in the solution to this Linear Program as a funtion of the parameters specified by the available inputs. Let f( Q0, Q1, Q2 ) be this function. This is the production function for iron. It exhibits Constant Returns to Scale and non-increasing marginal products for each input. Any “smooth” well-behaved microeconomic production expressed in terms of physical inputs can be approximated by enough production processes, as above. “Capital”, as measured in monetary or numeraire units, is not a physical input.

(In the third part, I present dual Linear Programs characterizing the firm’s choice of level of operations of the production processes. Neither of these later Linear Programs repeat the above Linear Program. For example, the firm does not have predefined resources devoted to iron-production, as in the constraints in the above Linear Program. How much labor, for example, the firm managers want to hire for iron production and for corn production are decision variables.)

This discrete model of iron production is not characterized by a Leontief or fixed-coefficient production function. The production function for corn, though, is Leontief. Neither the discrete nature of this model of iron production nor the Leontief production function for corn, however, drive capital-reversing. Capital-reversing can arise in models with "smooth" production functions (Steedman 2002). Since I am interested in behavior around a switch point (defined in Part 3), the assumption of fixed coefficients for corn production involves no loss of generality. Typically, techiques on two sides of a switch point differ in only one process (see, for example, the theorem on page 542 of Bruno et. al 1966). In the case of the above technology, those processes are iron-producing processes.

I think that in a model with more sectors, the costs of the various processes available for producing one or another commodity can vary more dramatically than in this example. Perhaps some day I will blog about the empirical evidence that technology is not unlikely to be such that “Sraffa effects” can arise.

Now to pose questions that are answered in the next part. Define a technique to consist of a positive level of operation of exactly one iron-producing process and the corn-producing process. For each of the two techniques so defined, at what levels can the two processes in the technique be operated so that the net ouput consists solely of corn? (The net output is what remains after replacing the iron and corn used up in production. If the net output is consumed, production can continue on the same scale for the adopted technique.) How much labor is employed per Bushel corn in net output for each of the two techniques?

References
• Bruno, Michael; Edwin Burmeister; and Eytan Sheshinski (1966). "The Nature and Implications of the Reswitching of Techniques", Quarterly Journal of Economics, V. 80, N. 4 (Nov.): 526-553.
• Steedman, Ian (2002). "Non-Monotonic c(r) Relations in the Abscence of Complementarity", Metroeconomica, V. 53, N. 1: 25-36