## Examples of production functions

### Fixed proportions

An important family of production functions models technologies involving a single technique of production. The only way to produce a unit of output, for example, may be to use 1 machine and 2 workers; if the firm has available 2 machines and 2 workers then the extra machine simply sits idle, and if it wants to produce two units of output then it has to use 2 machines and 4 workers. Such a production function has fixed proportions.

How can we describe such a technology precisely? If 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 minimum of 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 some constants a and b. The technology this production function models involves a single technique that produces one unit of output from 1/a units of input 1 and 1/b units of input 2, and, more generally, 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 there are 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}.

### 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 US beef, or 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.

### A production function modeling smooth but not perfect substitution between inputs

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 since some is now wasted. But if the speed is increased again, reducing the amount of labor time needed to 30 minutes, the amount of raw material needed may increase by much more than 50 units, since wastage may greatly increase.

A class of production functions 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, contains functions of the form

F (z1, z2) = Az1uz2v
for some constants A, u, and v. Such a production function is known as a Cobb-Douglas production function.

An example of such a function is

F (z1, z2) = z11/2z21/2.

### Exercise

What production function models each of the following technologies?
1. One computer can be made from two 32 megabyte memory chips or a single 64 megabyte chip.
2. Every course that is taught requires 1 instructor, 2 teaching assistants, and 1 lecture room.
[Solution]

### Exercise

A firm's production function is given by
F (z1z2) = 2z11/4z2.
How much output can it produce if it has 16 units of input 1 and 2 units of input 2?

[Solution]