Page 368 - DMTH201_Basic Mathematics-1

Unit 14: Business Applications of Maxima and Minima

Now, price p = 20 4 1.9 12.4 (when x = 1.9) Notes
20 12.4
Tax revenue T = 1.9 3.93 .
100 1.2

Example: Show that a monopolist with constant total cost and downward sloping demand
curve will maximise his profits at a level of output where elasticity of demand is unity.

Solution:
Let p = f(x) be the inverse demand function facing a monopolist and c (a constant) be his total cost.
x
f
x
profit p(x) = x.f(x) c and x f ( ) x . ( ) 0 0 for max. p
f ( )
x
f ( ) = x . ( ) or xf ( ) 1
x
f
x
x
Thus, h = 1, where h denotes the elasticity of demand.
Second order condition:
f
x
x
For max. p, we should have x 2 ( ) xf ( ) 0.
x
f ( ) < 2 ( ) x
f
x
Since R.H.S of the above inequality is positive, the above result will hold if either the demand
2
curve is concave f x 0 or if convex then f ( ) f ( ) .
x
x
x
Notes Marginal cost of a monopolist, under normal conditions of production, is always
non-negative since an additional unit of a commodity can be produced only at some
additional cost. Thus we shall always have MR 0 at the profit maximising point, implying
there by that a profit maximising monopolist will never have his equilibrium on any
point that lies on the inelastic portion of the demand curve.

k
Example: Suppose that the demand facing a monopolist is x p , where k > 1, and his
total cost function C ax 2 bx c .
(i) Find the profit maximising output of the monopolist as k 1.
(ii) What restrictions on the constants a, a, b and c are required for the answer to be economically
meaningful?
(iii) Find the supply function, if possible? Is this supply function consistent with your answer
to part (i)?
Solution:
1
k 1 k 1
(i) Total revenue TR = px = .x k .x k
x

1 k 1
Profit p = k .x k ax 2 bx c

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