Page 363 - DMTH201_Basic Mathematics-1

Basic Mathematics – I

Notes Supply Curve of a Firm under Perfect Competition

The supply curve of a firm, under perfect competition, is that portion of the marginal cost curve
that lies above the average variable cost curve. Let p = MC = min. AVC (average variable cost).
m
We can say that
(i) When p < p , quantity supplied x = 0, and
m
(ii) When p ³ p , quantity supplied is given by the condition p = MC. Solving this equation for
m
x gives the supply function of the firm.

Example: The total cost of a firm under perfect competition is given by
C x 3 6x 2 15x 10. Find the supply function of the firm.
Solution:
First we find the lowest price p below which the supply will be zero.
m
Total variable cost TVC = x 3 6x 2 15x AVC = x 2 6x 15

d AVC
We have = 2x 6 0, for minima Þ x = 3
dx
Now, p = min. AVC = 3 2 6 3 15 6 , below which quantity
m
supplied will be zero.
Further, we write p = MC = 3x 2 12x 15 or 3x 2 12x (15 p ) 0

Solving this quadratic equation for x, we have
12 144 12 15 p 12 12p 36 6 3p 9
x =
6 6 3
Thus, the supply function of the firm is

6 3p 9
x = when p 6,
3
and x = 0 when p < 6.
Note that we have ignored the negative sign because this will give values of x lying on that
portion of MC which lies below AVC.

1
Example: The total cost of a firm is C x 3 6x 2 30x 20. Find the equilibrium output
3
if price is fixed at 10 per unit. What will be the effect of a specific tax of 3 per unit on the
equilibrium output?
Solution:
1 3 2 1 3 2
Profit (x) = 10x x 6x 30x 20 x 6x 20x 20
3 3
x = x 2 12x 20 0, for maximum profit
x 2 12x 20 = 0 or x 10 x 2 0 x = 10 or 2

Further, x = – 2x + 12 = 2 10 12 8 0, when x = 10
and = 2 2 12 8 0, when x = 2

356 LOVELY PROFESSIONAL UNIVERSITY