Question

Consider the following two demand segments: 120−4Q = P and 100−4Q = P assuming there’s a constant marginal cost to serving either group, C(Q) = 60Q.

1. Find the optimal price, quantity and profits when the firm is able to practice third degree price discrimination.

2. Find the optimal price, quantity and profits when the firm is unable to practice third degree price discrimination.

3. Compute the price elasticity of demand associated with each optimal monopoly output in each segment. Do these results match what we would expect?

Answer 1

1. MC = dC/dQ = 60
P = 120 - 4Q
TR = P*Q = (120-4Q)*Q = 120Q - 4Q2
MR = d(TR)/dQ = 120 - 2(4Q) = 120 - 8Q
So, profit is maximized where MR = MC.
So, 120 - 8Q = 60
So, 8Q = 120 - 60 = 60
So, Q = 60/8 = 7.5
P = 120 - 4(7.5) = 120 - 30 = 90
Profit = TR - TC = P*Q - 60Q = (90*7.5) - (60*7.5) = 675 - 450 = 225

P = 100 - 4Q
TR = P*Q = (120-4Q)*Q = 100Q - 4Q2
MR = d(TR)/dQ = 100 - 2(4Q) = 100 - 8Q
So, profit is maximized where MR = MC.
So, 100 - 8Q = 60
So, 8Q = 100 - 60 = 40
So, Q = 40/8 = 5
P = 100 - 4(5) = 100 - 20 = 80
Profit = TR - TC = P*Q - 60Q = (80*5) - (60*5) = 400 - 300 = 100

b. P = 120 - 4Q
So, 4Q = 120 - P
So, Q = (120/4) - (P/4) = 30 - 0.25P

P = 100 - 4Q
So, 4Q = 100 - P
So, Q = (100/4) - (P/4) = 25 - 0.25P

Total demand, Q = 30 - 0.25P + 25 - 0.25P = 55 - 0.5P
So, 0.5P = 55 - Q
So, P = (55/.5) - (Q/.5) = 110 - 2Q
TR = P*Q = (110 - 2Q)*Q = 110Q - 2Q2
So, MR = d(TR)/dQ = 110 - 2(2Q) = 110 - 4Q
So, MR = MC gives,
110 - 4Q = 60
So, 4Q = 110 - 60 = 50
So, Q = 50/4 = 12.5
P = 110 - 2Q = 110 - 2(12.5) = 110 - 25 = 85
Profit = TR - TC = P*Q - 60Q = (85*12.5) - (60*12.5) = 1062.5 - 750 = 312.5

c. Price elasticity of demand, e = (dQ/dP)*(P/Q)

When 120−4Q = P
So, dP/dQ = -4 or dQ/dP = -1/4 = -0.25
So, e = (-0.25)*(90/7.5) = -3

When 100−4Q = P
So, dP/dQ = -4 or dQ/dP = -1/4 = -0.25
So, e = (-0.25)*(80/5) = -4

Yes, these results match with our expectation because we can see that lower price(80) is charged in market with high elatsicity of demand(-4).