For each of the following monopolists, calculate the profit maximizing price and

Question

For each of the following monopolists, calculate the profit maximizing price and quantity Q^D = 50 - 2P, TC(Q) = 10Q QD = 100 - 1/2 P, TC(Q) = 3Q^2 QD = 6P^-4,TC(Q) = 30Q For each of the following find the optimal pricing strategy and the quantity that will be sold, given the stated form of price discrimination Q^D - 30 - 5P, TC(Q) - Q: First-degree price discrimination Q_i^D = 40 - 2P, TC(Q) = 2Q; Two-Part Tariff^1 Q_m^D = 50 - P, Q_f^D = 40 - P, TC(Q) - Q^2; Third-Degree price discrimination^2

Explanation / Answer

a) Qd = 50 - 2P ; TC(Q) = 10Q

2P = 50 - Qd

P = 25 - 0.2Qd

Revenue = Price X quantity

R = (25 - 0.2Qd ) Q

Marginal revenue = 25 - 0.4Qd

TC(Q) = 10Q

Marginal cost = 10

Marginal revenue = Marginal cost

25 - 0.4Qd = 10

Qd = 37.50 units

Price = 25 - 0.2Qd

Price= $ 17.50

b) Qd = 100 - 0.5P ; TC = 3Q2

P = 200 - 2 Qd

Revenue = Price X quantity

R = (200 - 2 Qd ) Qd

Marginal revenue = 200 - 4Qd

TC = 3Q2

Marginal cost = 6Q

MR = MC

200 - 4Qd   = 6Q

Q = 20 units

Price = 200 - 2Qd

Price = 200 - 2 X 20

Price = $ 160

c) Qd = 6 P-4

Qd = 6 / P4

P4 = 6/Qd

P = 1.56 / Q1/4

Revenue = Price X quantity

R = (1.56 / Q1/4 ) Q

R = 1.56 Q3/4

marginal revenue = 1.56 3/4 / Q1/4

MR = 4.68 / 3Q1/4

MR = 1.56 / Q1/4

Marginal cost = 30

MR = MC

1.56 / Q1/4 = 30

Q1/4 = 0.052

Q = 0.4775 units

P = 1.56 / Q1/4

P = 1.56 / 0.47751/4

P = $ 1.876

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a) Qd = 30 - 5P ; TC = Q

P = 6 - 0.2Q

Revenue = price X quantity

R =( 6 - 0.2Q) Q

Marginal revenue = 6 - 0.4Q

TC = Q

Marginal cost = 1

6 - 0.4Q = 1

Q = 12.50 units

Price = 6 - 0.2Q

P = $ 3.50

b) Qd = 40 - 2P ; TC = 2Q

P = 20 - 5 Q

R = (20 - 5 Q) Q

marginal revenue = 20 - 10Q

Tc = 2Q

Marginal cost = 2

20 - 10Q = 2

18 = 10Q

Q = 1.8 units

Price= 20 - 5Q

Price = $ 11

c) Qm = 50 - P ; Qf = 40 - P ; TC = Q2

Pm = 50 - Qm ; Pt = 40 - Qt

Rm = ( 50 - Qm ) Qm ; Rt = ( 40 - Qt ) Qt

Marginal revenue = 50 - 2Qm ; MRt = 40 - 2Qt

TC = Q2

Marginal cost = 2Q

50 - 2Qm = 2Q

Qm = 12.50 units ;

40 - 2Qt = 2Q

Qt = 10 units

Pm = 50 - Qm

Pm = 50 - 12.50

Pm = $ 37.50

Pt = 40 - Qt

Pt = 40 - 10

Pt = $ 30

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