## Wednesday, May 03, 2006

### An Example of Capital Reversing (Part 2)

This post continues an examination of the technology described in Part 1 Let
• 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.
• X3 be the gross amount of corn produced (with the corn-producing process).
In a technique, as defined in this series of posts, either X1 or X2 is zero. The other variable is positive, and corn is also produced at a positive level. Consider, for example, the technique in which X2 is zero. Suppose that the iron output of this technique just replaces the iron used up in production, and the net output of corn is one Bushel. These conditions yield the following system of equations:
0 = X1 - (1/10) X1 - 2 X3

1 = X3 - (1/40) X1 - (2/5) X3

The conditions on output provide the left-hand side of this system of equations. The coefficients of production for the processes comprising the technique provide the constants on the right hand side. This is a system of two linear equations in two unknowns, and it is easily solved. Economists talk about the Leontief inverse in describing the solution in a more general problem. The remainder of this part considers the solution to the above system and to the system comprising the other technique.

Consider a firm that produces X1 = 200/49 Tons of iron with the first iron-producing process and X3 = 90/49 Bushels of corn. This firm requires inputs of (1/10) X1 + 2 X3 = 200/49 Tons of iron and of (1/40) X1 + (2/5) X3 = 41/49 Bushels of corn. Notice that the iron produced by this firm just replaces the iron inputs used-up in making the firm's output. On the other hand, the firm has a net output of one Bushel corn, after replacing the corn used up in production. Thus, this firm is a vertically-integrated corn-producing firm using a technique consisting of the first iron-producing process and the corn producing process. The firm employs (1) X1 + (1) X3 = 290/49 Person-Years of labor per bushel corn produced net.

Consider a second firm. This second firm produces X2 = 1,976/315 Tons of iron with the second iron-producing process and X3 = 106/63 Bushels of corn. Similar calculations show that this firm has a net output of no iron and one bushel corn. In other words, this is a vertically-integrated corn-producing firm using a technique containing the second iron-producing process. This firm employs 50/9 Person-Years of labor per bushel corn produced net. Notice that this firm employs 160/441 Person-Years per bushel corn less than the first firm.

This part has described the amount of labor used per net bushel of corn produced by each of the techniques. But for what combinations of prices would the firm want to adopt any particular combination of production processes, if any? The next part answers this question.