Page 361 - DMTH201_Basic Mathematics-1

Basic Mathematics – I

Notes If p is the price at which the firm can sell its output, then total revenue of the firm is R(x) = p.x,
where x is the level of output. We note that total revenue of the firm is a straight line passing
through origin with slope p. Assuming the cost function as C = C(x), we can write the profit of the
firm as p(x) = R(x) – C(x) = px – C(x).
x = p C ( ) 0, for maximum (note that MR = p).
x
x
Thus, p = C ( ) or p = MC(x) is the necessary condition for maximum profits.
Second order condition
x = 0 C ( ) 0, for maximum p.
x
d MC x
x
This condition will hold only if C ( ) or 0 at the stationary value i.e. MC must be
dx
rising at the stationary point.
Break-Even Point

It can be shown that the break-even point of a profit maximising firm under perfect competition
will occur at a level of output where average cost is minimum.
We can write
TR = TC (for break even)
TC
or px = TC or p
x
or MC = AC ( p = MC in equilibrium)

Starting Point

The starting point of a firm is the minimum level of output at which total variable costs (TVC)
of the firm are covered. Therefore we have
TR = TVC, (at the starting point)

TVC
or px = TVC or p AVC
x
or MC = AVC (in equilibrium)
Thus the starting point occurs at the minima of AVC.

Example: A plant produces x tons of steel per week at a total cost of
1 x 3 3x 2 50x 1
10 300. If the market price is fixed at 33 , find the profit maximising output of
3
the plant and the maximum profit. Will the firm continue production?
Solution:

100 1 3 2
We can write R x = x and C x x 3x 50x 300
3 10
100 1 3 2
x = R x C x x x 3 x 50 x 300
3 10

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