Question

Given a demand function of the monopolist as Q=50-0.5P and cost function of the nature C=50-4Q.

1. Calculate profit maximising output

2. Calculate the price at which profit is maximised

3. Compute the total profit

Answer 1

(1)

$Q=50-0.5P$

$C=50-4Q$

Add 0.5P to both sides of the demand equation:

$Q+0.5P=50-0.5P+0.5P$

$Q+0.5P=50$

Subtract Q from both sides of the above equation:

$Q+0.5P-Q=50-Q$

$0.5P=50-Q$

Dividing both sides of the equation by 0.5:

$\frac{0.5P}{0.5}=\frac{50-Q}{0.5}$

$P=100-2Q$

To determine Total Revenue (TC), multiply both sides of the equation by Q:

$TR=PQ=(100-2Q)\times Q$

$TR=100Q-2Q^2$

$\pi=TR-TC$

$=(100Q-2Q^2)-(50-4Q)$

$=-50+104Q-2Q^2$

Differentiating the above equation w.r.t. Q:

$\frac{d\pi}{dQ}=104-4Q$

$104-4Q=0$

$\implies Q=26.$

(2)

$Q=50-0.5P$

$26=50-0.5P$

$P=48$

(3)

Total Profit:

$\pi=-50+104Q-2Q^2$

$\pi=-50+104(26)-2(26)^2$

$\pi=1,302$