Page 369 - DMTH201_Basic Mathematics-1

Basic Mathematics – I

Notes Differentiating w.r.t. x, we get
d 1 k k 1 1 k
dx = . k x 2ax b 0, for maximum profits
Taking limit of the above as k 1, we get

1 k 1 1
lim k .x k 2ax b 0 or x b
k 1 k = 2ax b 2a

The profit at x b 2a will be maximum if

d 2 = lim 1 k . k 1 1 x 1 k 1 2a 2a 0
dx 2 k 1 k k a > 0

(ii) Since x is positive in x p k , > 0.

b
Further, b must be negative in order that x 0 and no restriction is needed for c.
2a
(iii) Since the elasticity of demand is k (constant), we can find the supply function of the
monopolist. The supply function is given by the condition MR = MC. We have
1
1 k 1 1 k k 1 k –1
MR = k x k . p and MC = 2ax + b
k x k k

k 1 k 1
p = 2ax b or 2ax p b
k k
k 1 1 b
or x = p is the required supply function.
k 2a 2a

lim k 1 1 b b
Since x k 1 p k 2a 2a 2a , this supply function is consistent with the answer to part (i).

1
2
Example: A monopolist with the cost function C(x) = x faces a demand curve x 12 p.
2
(i) What will be his equilibrium price and quantity?
(ii) If for some reason the firm behaves as though it were in a perfectly competitive industry,
what will equilibrium price and quantity be? How much money will the firm require to
forgo monopoly profits and behave competitively instead?
Solution:

(i) Total revenue TR = px 12 x x 12x x 2

1 2 3 2
2
Profit = 12x x x 12x x
2 2
d
= 12 3x 0 or x = 4 for maximum p.
dx
d 2
dx 2 = 3 0 , the second order condition is satisfied.

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