In: Economics
Question 1: Monopoly
(a) Given demand function is P = 160-2Q and AC=MC=40
So, the total revenue function is TR = P*Q = 160Q-2Q2 , and marginal revenue function is MR = dTR/dQ = 160-4Q
Total cost function is TC = AC*Q = 40Q
The profit of the monopoly is given by:
π = TR - TC = 160Q-2Q2-40Q = 120Q-2Q2
Profits are maximised at a point where dπ/dQ = 0 or MR = MC
Thus, 120-4Q = 0 or 160-4Q = 40
So, Qm = 30 and Pm = 160-2(30) = 100
and π = (120*30)-[2(30)2] = 3600 - 1800 = 1800
Qm = 30 , Pm = 100 and π = 1800.
(b) The monopoly in part (a) is shown below:
The monopoly equilibrium is depicted by em and perfect competition equilibrium is denoted by ec. The shaded triangle Aemec shows the deadweight loss (DWL) due to monopoly.
In perfect competition, P=MC
160-2Q=40
Qc = 60 and Pc = 40
Thus, DWL = area of triangle Aemec
= (1/2) * (Qc-Qm) * (Pm-Pc)
= (1/2) * (60-30) * (100-40) = (1/2)*30*60 = 900
DWL = 900
(c) Given that profits are π = 600Q-2Q3-1000
We know that π =TR-TC
The only constant term in the profit is '1000' that is it is not dependent on the quantity sold. Thus, these are the total fixed costs which the monopolist has to bear even if it is not producing any output.
TFC = 1000
(d) When Q=17
π = 600Q-2Q3-1000
= (600*17) - [2(17)3] - 1000
= 10200 - 9826 - 1000
= -626
Thus, at Q=17 units the monopolist is incurring a loss of 626.
π = -626
(e) To maximise the profits, the firt should produce at a point where
dπ/dQ = 0
d(600Q-2Q3-1000)/dQ = 0
600 - 6Q2 = 0
Q2 = 100
Q = 10
Thus, the monopolist should produce 10 units to maximise its profits.
Q = 10 units.