The theory of the firm and industry equilibrium

Martin J. Osborne
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1.1 The production function

One way of characterizing the technology available to a firm is to specify a “production function”, which associates an output with every combination of inputs. Suppose that the firm uses n inputs to produce output, and denote the amounts of the inputs by the vector (list) (z1, ..., zn), where zi is the amount of input i. The production function F associates with every such vector an output F(z1, ..., zn).
Example: Fixed proportions
Suppose that a single “technique” of production is available. The only way to produce a unit of output, for example, may be to use 1 machine and 2 workers. If such a firm has available 2 machines and 2 workers then the extra machine simply sits idle; if it wants to produce two units of output then it has to use 2 machines and 4 workers. We say that such a production function has “fixed proportions”.

How can we describe this technology precisely? Suppose that the only way to produce y units of output is to use y machines and 2y workers. Then the output from z1 machines and z2 workers is

min{z1, z2/2},
the smaller of the two numbers z1 and z2/2. Check out the logic of this formula by considering the output it assigns to various combinations of machines and workers.
  • 1 machine and 2 workers yield min{1, 2/2} = min{1, 1} = 1 unit of output
  • 2 machines and 2 workers yield min{2, 2/2} = min{2, 1} = 1 unit of output
  • 2 machines and 4 workers yield min{2, 4/2} = min{2, 2} = 2 units of output.
A general fixed proportions production function for two inputs has the form
min{az1, bz2}
for positive constants a and b. The technology this production function models involves a single technique that produces y units of output from y/a units of input 1 and y/b units of input 2. Extra units of either input cannot be put to use. For example, if the firm has y/a units of input 1 and more than y/b units of input 2—say z2 units—then its output is min{a(y/a), bz2} = min{y, bz2} = y, since z2 > y/b.

If the firm uses more than two inputs, a single-technique technology can be modeled by a production function with a similar form. For example, if four wheels, one engine, and one body are needed to make a car, and no substitution between the inputs is possible, the number of cars that may be produced from the vector (z1, z2, z3) of inputs, where input 1 is wheels, input 2 is engines, and input 3 is bodies, is

min{z1/4, z2, z3}.
Example: Perfect substitutes
A technology whose character is exactly the opposite to that of a fixed proportions technology allows one input to be substituted freely for another at a constant rate. For example, one hamburger may be made with 100g of Canadian beef, or with 50g of Canadian beef and 50g of Argentinian beef, or with any combination of the two inputs that sums to 100g. In this case we can describe the technology precisely by the production function
F(z1, z2) = z1 + z2.

More generally, any production function of the form

F(z1, z2) = az1 + bz2
for some nonnegative numbers a and b is one in which the inputs are perfect substitutes. Such a production function models a technology in which one unit of output can be produced from 1/a units of input 1, or from 1/b units of input 2, or from any combination of z1 and z2 for which az1 + bz2 = 1. That is, one input can be substituted for the other at a constant rate.
Example: Cobb–Douglas production function
Many technologies allow inputs to be substituted for each other, but not at a constant rate. Suppose that one person operating a machine for an hour can produce 100 units of output using 100 units of raw material. Perhaps if the speed of the machine is increased, the same 100 units of output can be produced in 45 minutes using 150 units of raw material, more raw material being needed because some is now wasted. But if the speed is increased again, reducing the amount of labor time needed to 30 minutes, perhaps the amount of raw material needed increases by more than 50 units.

One class of production functions with two inputs that models situations in which inputs can be substituted for each other to produce the same output, but cannot be substituted at a constant rate, consists of functions of the form

F(z1, z2) = Azu
1zv
2
for some positive constants A, u, and v with 0 < u < 1 and 0 < v < 1. Such a production function is known as a Cobb–Douglas production function.

An example of such a function is

F(z1, z2) = z1/2
1z1/2
2.