Page 366 - DMTH201_Basic Mathematics-1

Unit 14: Business Applications of Maxima and Minima

Notes
2
2
2
b t b t b t b t
= b t c c
4 a 2 a 4 a 2 a
b t 2
= c
4 a
b t t bt t 2
The tax revenue T = . t x . t 2 a 2 a

dT b 2t 1
= 0, for max. T t b
dt 2 a 2
Second order condition

2
d T 2 0
dt 2 = 2 a (since a, a > 0)
Figure 14.1

Fig. 5.13
Thus, maximum tax revenue is given by

1 b 1 2 b 1 b 2
T = b
2 2 a 8 a

Example: A firm under non-perfect competition has the following total cost and demand
functions:
2
x
C 20 2x 3 , p 50 x
(i) Find the values of p and x that maximise profit.
(ii) An excise tax is imposed @ 5 per unit. Compute the profit maximising values of p and x
in the post-tax situation.

(iii) Find the rate of excise tax t that would fetch maximum tax revenue to the government.
Solution:
(i) Profit x = 50x x 2 20 2x 3x 2 4x 2 48x 20
Now, x = 8x 48 0,for max. x = 6

Second order condition:
x = 8 0, x = 6 is the profit maximising output.

LOVELY PROFESSIONAL UNIVERSITY 359