ABSTRACT
One economics example of a function within a function occurs in the marginal revenue productivity theory of the demand for labour, where a firm’s total revenue depends on output which, in turn, depends on the amount of labour employed. An applied example is explained later. However, we shall first look at what is perhaps the most frequent use of the chain rule, which is to break down an awkward function artificially into two components in order to allow differentiation via the chain rule. Assume, for example, that you wish to find an expression for the slope of the non-linear demand function
p = (150 − 0.2q)0.5 (1)
The basic rules for differentiation explained in Chapter 8 cannot cope with this sort of function. However, if we define a new function
z = 150 − 0.2q (2)
then (1) above can be rewritten as
p = z0.5 (3)
(Note that in both (1) and (3) the functions are assumed to hold for p ≥ 0 only, i.e. negative roots are ignored.)
Differentiating (2) and (3) we get
dz dq
= −0.2 dp dz
= 0.5z−0.5
Thus, using the chain rule and then substituting equation (2) back in for z, we get
dp dq
= dp dz
dz dq
= 0.5z−0.5(−0.2) = −0.1 z0.5
= −0.1 (150 − 0.2q)0.5
Some more examples of the use of the chain rule are set out below.
