Page 370 - DMTH201_Basic Mathematics-1

Unit 14: Business Applications of Maxima and Minima

Equilibrium price p = 12 4 8 Notes
(ii) When the firm behaves as a perfectly competitive firm, we can write TR px where p is
constant

1
p = px x 2
2
d
dx = p x 0 or p = x for maximum
,
Substituting p = 12 x we get 12 x x or x = 6, also p = 6
d 2
dx 2 = 1 0, the second order condition is satisfied.

3
Monopoly profit, = 12 4 16 24 and
m 2

1
Profit under competition, c 6 6 2 36 18
Profit forgone = 24 – 18 = 6

Example: An industry consists of a number of profit maximising firms, each with a long
run total cost function C(x) = x 2 1.
(a) On the same diagram sketch the total cost, average cost, average variable cost and marginal
cost curves.

(b) Derive the supply function of the individual firm as a function of price (p). The industry
supply X is the sum of the firm supplies. Obtain X as a function of p if there are n firms in
s s
the industry. How does the X curve change if the number of firms changes?
s
.
(c) The market demand is given by X d 52 p For given n, what will be the (short period)
equilibrium price and output for the industry, and what profits will each firm earn?
(d) With free entry, the long run equilibrium price will be the lowest short-run equilibrium
price compatible with non-negative profits. Determine this price and the number of firms
in the industry in long run equilibrium.
Solution:
2
(a) (i) The total cost function is C x 2 1 or C 1 x . This is a parabola with axis pointing
vertically upward and vertex at (0, 1).
dC
Alternatively, we can determine its turning point by equating 0 x 0. Also C
dx
= 1 when x = 0.

2
d C
Further, 2 2 0 C has a minima at x = 0.
dx
1
(ii) Average cost is A x
x

dA 1
To determine its turning point 1 2 0 or x 2 1 or x 1
dx x

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