Lecture Notes in Ma 20 (Calculus): Marginal Cost as a Derivative, 6/30/08

.
Lecture Notes in Ma 20 (Calculus for Economics)
30 June 2008


Topic: Marginal Cost as a Derivative

Suppose that a firm which makes electric light bulbs knows that in order to produce q light bulbs it will have to pay out C(q) dollars in wages, materials, overheads and so on. We say that C is the firm’s cost function.

In this case a change in production from q to q + 1 light bulbs is relatively small, and may be described as ‘marginal’. The corresponding increase in cost is C(q + 1) – C(q), and this can be thought of as the ‘marginal cost’ of making one more lightbulb, when the level of production is q. In the delta notation, we see that the marginal cost is the change delta C corresponding to a change delta q = 1. In general, the relationship between delta q and delta C is given by the approximation

delta C is approximately equal to C’(q) delta q

and so on, when delta q = 1, the resulting delta C is approximately C’(q) . For this reason it makes sense to define the marginal cost function to be the derivative of the cost function C.

(Note: In the traditional language of economics, the derivative of any function F is referred to as the marginal of F, and is often denoted by MF.)

What we have shown is that if the units are small, as for example in the production of light bulbs, C’(q) represents the cost of producing one more unit when the q units are being produced. The concept of marginal cost is equally useful when the units are larger, provided we remember this basic idea.

EXAMPLE:

Suppose that the costs of a firm making bicycles are $50,000 per week in overheads and $25 for every bicycle made. Then its cost function (in dollars) is

C(q) = 50,000 + 25q

So that C’(q) = 25

In this case the marginal cost is 25 dollars, independent of the level of production q.

Slightly more realistically, suppose that in order to produce a substantially larger weekly output of bicycles it would be necessary for the firm to incur extra costs, possibly because its increased consumption of a raw material would drive the price of that raw material upwards.

We can account for this by introducing an additional cost term, say 0.001q^2, which is trivial when q is small but which is more significant as q increases. In this case we have

C(q) = 5000 + 25q + 0.001q^2

So that

C’(q) = 25 + 0.002q

Thus the marginal cost is $25.2 if the output is 100 bicycles per week, but it would rise to $45 if the output were 10,000 bicycles per week.

Reference:

Martin Anthony and Norman Biggs (1996). Mathematics for Economics and Finance. Cambridge University Press, pp. 59 - 60.
.