Page 372 - DMTH201_Basic Mathematics-1

Unit 14: Business Applications of Maxima and Minima

b Notes
Show that the amount X produced and sold is x and that 2 x and p . x
2 a 2
Find the output if the producer monopolist sold directly to the market and show that “bilateral
monopoly” here restricts output and raises price.
Solution:

Total revenue of the merchant = x x x x 2
Total cost of the merchant = x + d
(where d is distributive cost given to be constant.)
Now, profit of the merchant p me x x 2 x d

d P me
We have 2 x 0 , for max. P
dx me
= – 2 x … (1)
Second order condition
2
d P me 2
dx 2 0 (since a > 0)
The producer monopolist sells the output to the merchant at a price equal – 2 x, given by the
equilibrium condition (1). Thus, = – 2ax serves as a demand function facing the producer
monopolist.

The revenue of the producer monopolist = 2 x x , and his total cost = x(ax+b).
Profit P = x 2 x 2 ax 2 bx
mo
d P mo
Thus, = 4 x 2ax b 0, for max. profits.
dx
b
x = 2 a 2 ... (2)
Second order condition:
2
d P mo
dx 2 = 4 2a 0, (since a, a > 0).
If the producer monopolist sold direct to the market, we can write his profit function as P
= x x ax 2 bx .
dP
= 2 x 2ax b 0, for max. profits.
dx
b
or x = ... (3)
2 a
Comparing (2) and (3), we conclude that bilateral monopoly restricts output. Since output price
are inversely related by the demand function, this also implies that price is higher in bilateral
monopoly.

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