Page 356 - DMTH201_Basic Mathematics-1

Unit 14: Business Applications of Maxima and Minima

dx dx dx t 1 Notes
f x t t g x t 1 or
dt dt dt f x t g x t
Since demand function is assumed to be downward sloping, the denominator of the above
dx
expression is negative. Thus t 0, which implies that equilibrium output decreases as tax rate
dt
increases.

14.2 Maximisation of Output

Assuming that labour is the only variable factor, we can write the production function of a firm
as x = f(L), where x denotes total product of labour which will be denoted as TP .
L
TP f L
The average product of labour is AP = L , the marginal product of labour is MP =
L L L L
dx f L d TP L dx MP 0
dL ( ) and necessary condition for maximum output is dL dL L

Often we are interested in finding that level of employment of labour at which its average
product is maximum.
For maxima of AP , we have
L
d AP . L f L f L
L 0
dL L 2
L
f ( )
L
L
Lf ( ) f ( ) or ( )f L or MP = AP . Thus, the marginal and average products of a factor are
L L L
equal at the maxima of the later.
Maximisation of Total Revenue Product
If p is price of a unit of output, the total revenue of the firm is, TR = p.x. This total revenue, when
expressed as a function of L, using production function x = f(L), is called the total revenue product
d TR
of labour (TRP ). Units of L, to be employed, for maximum TR is given by the equation = 0.
L dL
d TR
The derivative is known as the marginal revenue product of labour.
dL
Since TR is a function of x and x is a function of L, using chain rule, we can write an expression for
marginal revenue product in terms of marginal revenue and marginal product.
d TR d TR dx
MRP = . = MR.MP
L dL dx dL L

Example: The short-run production function of a manufacturer is given as
3
x 11L 16L 2 L .
(i) Find the average product function, AP , the marginal product function, MP , and show
L L
that MP = AP where AP is maximum.
L L L

LOVELY PROFESSIONAL UNIVERSITY 349