Page 365 - DMTH201_Basic Mathematics-1

Basic Mathematics – I

Notes Second order condition:
We have x = 6x 114
= 6 35 114 96 0 , at x = 35
= 6 3 114 96 0, at x = 3

Therefore, profits are maximised when 35 units of the commodity are produced.
Further, the equilibrium price p 1000 2 35 = 930.

Example: The total cost of a monopolist is C ax 2 bx c (a, b, c > 0) and the inverse
demand function is p x , 0 . Find his equilibrium output, price and net revenue
(profit). How will these values change if a tax of t per unit is levied? Also determine the tax rate
that maximises the tax revenue. Find the maximum tax revenue.
Solution:
Profit x = x x 2 ax 2 bx c a x 2 b x c

b
x = 2 a x b 0, for max. , x
2 a

b
We note that x = 2 a 0. Therefore, x 2 a is the profit maximising output.

The equilibrium price

b 2a 2 b 2a b
p =
2 a 2 a 2 a

1 b 2 b b
Maximum net revenue a 2 c
4 a 2 a

b 2 b 2 b 2 b 2
= c 1 2 c c
4 a 2 a 4 a 4 a
After a specific tax of t per unit is imposed, the profit function can be written as
x a x 2 b x c tx
t
x = 2 a x b t 0 or 2 a x b t for max.
t
b t
x =
2 a
The second order condition is same as before.
b t 2a b t
The post-tax price p =
2 a 2 a

The max. net revenue is given by
b t 2 b t b t
= a 2 b c t
4 a 2 a 2 a

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