## Output as a function of a single input: the total product, marginal product, and average product functions

### The total product function

One way of looking at a production function is to consider how output changes as we vary one input, holding the other inputs fixed. This view of a production function is especially useful if we are studying the decision-making of a firm that can vary only one input.

Consider the production function F (z1, z2). Assume that the amount of input 2 is fixed at k. Then the amount of output the firm can produce as it varies the amount of input 1 is given by the function F (z1,k). For no very good reason, this function is known as a total product function and is denoted TP(z1) (or TPk(z1) when we want to make it clear that z2 is fixed at k).

#### A "typical" total product function

Total product functions may take a wide variety of forms. However, we think of the "typical" total product function as increasing slowly for small values of the input, then increasing more rapidly, then increasing slowly again.

### The marginal product function

A useful concept when thinking about how the output of a firm varies as it changes one input, holding all other inputs fixed, is the rate of increase of the total product, known as the marginal product of the variable input. The marginal product for any value of the variable input is the slope of the total product function at that point. In particular, if the total product function is differentiable, the marginal product is the derivative of the total product function.

### The average product function

The average product of input 1 is defined to be the average output per unit of input 1:
AP(z1) = TP(z1)/z1.

### Total product, marginal product, and average product for a typical production function

Geometrically, the marginal product for any value of z1 is the slope of the total product function at z1; the average product is the slope of a line from the origin to the point (z1,TP(z1)) on the total product function.

Given the relation between marginals and averages, the marginal product and average product curves for a "typical" total product function thus look like this: