Question

28. A monopolist faces two separate demand curves in two separate markets: P1 = 78 - 3Ql and P2 = 86 - 2Q2. The total cost curve is TC = 6 + 6Q. Find Q1, Q2, P1, P2, the price elasticities at the two profit maximizing points, and the Lerner Index at the two profit maximizing points.

Answer #1

Marginal cost (MC) = dTC/dQ = 6

In market 1, profit is maximized when MR1 = MC.

Total revenue (TR1) = P1 x Q1 = 78Q1 - 3Q1^{2}

MR1 = dTR1/dQ1 = 78 - 6Q1

Equating with MC,

78 - 6Q1 = 6

6Q1 = 72

Q1 = 12

P1 = 78 - (3 x 12) = 78 - 36 = 48

From demand function, 3Q1 = 78 - P1, therefore Q1 = 26 - (P1/3)

Price elasticity = (dQ1/dP1) x (P1/Q1) = - (1/3) x (48/12) = - 1.33

Lerner index = - 1 / Price elasticity = -1 / -1.33 = 0.75

In market 2, profit is maximized when MR2 = MC.

Total revenue (TR2) = P2 x Q2 = 86Q2 - 2Q2^{2}

MR2 = dTR2/dQ2 = 86 - 4Q2

Equating with MC,

86 - 4Q2 = 6

4Q2 = 80

Q2 = 20

P2 = 86 - (2 x 20) = 86 - 40 = 46

From demand function, 2Q2 = 86 - P2, therefore Q2 = 43 - 0.5P2

Price elasticity = (dQ2/dP2) x (P2/Q2) = - 0.5 x (46/20) = - 1.15

Lerner index = - 1 / Price elasticity = -1 / -1.15 = 0.87

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