Question

For a monopolist firm the demand and the total cost functions are given as Q = 20-

0.5P and TC= 4Q2-8Q+15, respectively. 

Answer 1

Optimum quantity

$Q = 20 - 0.5p$

$0.5p = 20 - Q$

$P = 20-Q /0.5$

$P = 40 - 2Q$

$TR=P×Q$

$TR=(40−2Q)Q$

$MR = DTR/DQ$

$MR=40−4Q$

$TC=4Q^2 - 8Q$

$MC = DTC/DQ = 8Q - 8$

Based on equilibrium,

$MR = MC$

$40 - 4Q = 8Q-8$

$40+8= 8Q + 4Q$

$48 = 12Q$

$Q = 48/12 = 4$

Optimum quantity

$Q = 4$

$P = 40 - 2(Q)$

$P = 40-2(4)$

$P = 40 - 8 = 32$

$TR = P × Q = 32 × 4 = 128$

$TC=4Q^2 −8Q+15$

$TC = 4(4^2) -8(4) + 15$

$TC = 64 - 32 + 15$

$TC = 47$

$Profit = TR-TC$

$Profit = 128-47$

$Profit = 41$