Page 371 - DMTH201_Basic Mathematics-1

Basic Mathematics – I

Notes Since x > 0, we consider only x = 1
d A 2
2
dx 2 x 3 0 when x = 1
Also A = 2 when x = 1
A has a minima at (1, 2).
(iii) Average variable cost is AV x . This is the equation of a straight line passing
through origin with slope equal to unity.
(iv) Marginal cost is M = 2x This is the equation of a straight line passing through origin
with slope equal to 2. The diagrams of the above function are shown in Figure 14.1.
(b) The supply function of the individual firm is given by the condition p MC or p 2x i
(Note that whole of MC lies above AVC)
p
x i is the supply function of an individual firm.
2
When there are n firms, the industry supply is
n
X s x i 2 p

dX s p p
dn 2 , which shows that supply increases by a constant 2 with entry of an additional
firm.
(c) The condition for the short term equilibrium is X X
d s
n n 2 104
or 52 – p = p or p 52 or p
2 2 n 2

n 52n
Further, equilibrium output is X p
2 n 2
Profit of the ith firm TR TC
i i i
104 52 52 2 1 2 52 2 52 2 1 52 2 1
n 2 n 2 n 2 n 2 n 2 n 2

(d) In the long run 0
i
52 2
2
or 2 1 or n 2 2 52 or n 2 52
n 2
or n = 50 (no. of firms)

104
Also price p 2
50 2

Example: The market demand for a good X is given by the relation p x . A
monopolist produces X at an average cost ax b for output x and sells to a merchant at a price p
which maximises his profits. The merchant is a monopolist with constant distributive costs and
maximises his profits by selling on the market at price p.

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