Question

TC = 100Q - 10Q^2 +5Q^3

To find optimal level of output

Answer 1

The optimal level of output is output that maximizes a firm's profit. This level of output exists where marginal cost equals marginal revenue, that is, MC = MR.

Given the TC function, differential calculus is applied to find MC.

$MC = \dfrac {d}{dQ}(TC)$

$=\dfrac {d}{dQ}(100Q-10Q^2+5Q^3)$

$= 100-20Q+15Q^2$

Since, MR is not given, Let us Assume MR = $95.

$Note:$ if MR is low, for example, $MR \leq \$90$ , output cannot be found. Hence, MR = $95 is a better estimate.

The optimum output occurs where:

$MC = MR$

$100 - 20Q+15Q^2 =95$ $=>15Q^2 -20Q+100 -95 =0$

$=>15Q^2-20Q+5 = 0$

Dividing by 5 each term gives:

$=> 3Q^2 - 4Q + 1 = 0$

$=> 3Q^2-3Q-Q+1=0$

$=> 3Q(Q-1)-1(Q-1)=0$

$=>(3Q-1)(Q-1)=0$

$\therefore \space Q = \dfrac {1}{3} \space or \space Q=1$

Thus, the optimal quantity, given that MR = $95, is 1 unit.