Page 367 - DMTH201_Basic Mathematics-1

Basic Mathematics – I

Notes Also, the profit maximising price p 50 6 44.
(ii) Post-tax situation:
Profit x = 50x x 2 20 2x 3x 2 5x 4x 2 43x 20

43
x = 8x 43 0 x 5.375
8
Second order condition:

x = 8 0, x 5.375 is the profit maximising output in
post-tax situation. Also, price = 50 - 5.375 = 44.625
(iii) When rate of excise tax is t per unit
Profit t x = 50x x 2 20 2x 3x 2 tx 4x 2 48 t x 20

48 t
x = 8x 48 t 0, for max. x
t 8
48 t t 2
Now tax revenue T = .t 6t
8 8

dT t
We have = 6 0, for max. T t 24
dt 4
Second order condition:
2
d T 1
Since 2 0, hence, T is maximum when rate of excise-tax is 24 per unit.
dt 4

Example: Suppose that the demand and total cost functions of a monopolist are p 20 4x
and C 4x 2 respectively, where p is price x is quantity. If the government imposes tax @ 20%
of sales, determine the total tax revenue.
Solution:

p
We have p = p s 0.2p s 1 0.2 p or p s 1.2
s
p 20 4x x
TR = p x x
s
1.2 1.2
20 4x x 20x 4x 2
Thus, profit p(x) = 4x 2 4x 2
1.2 1.2
20 8x
Now x = 4 0, for max.
1.2
20 – 8x = 4.8 or x 15.2 8 1.9

Second order condition
8
x = 0, is max. at x = 1.9
1.2

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