Answer to Question #270380 in Microeconomics for Econ

Question

Suppose a certain city has a monopoly cable-television company. This company has total costs TC = 0.25Q² + 30Q + 70. (Hint: using calculus, this means MC = 0.5Q + 30 since MC is the derivative of TC with respect to output.) 

The demand in the community is approximated by the equation Q_d = 60 - P/2 (alternatively, you can write the demand equation as Q_d = 60 – 0.5P).

·       Graphically depict the demand curve as well as the marginal cost (MC) curve. 

·       If the cable company is free to choose its own price P_m and quantity Q_m, graphically depict the monopoly equilibrium price and quantity. Add any other curve(s) to your diagram that may be required to obtain this outcome.

·       Compute and state the exact monopolist equilibrium price P_m and quantity Q_m that you depicted graphically.

Answer 1

Profit maximizing quantity that the firm would produce would be where marginal revenue equals marginal cost.

Answer

1)  

In the above graph,

D = Demand curve

MC = Marginal cost curve

 

2) 

In the above graph,

D = Demand curve

MC = Marginal cost curve

MR = Marginal revenue curve

 

Profit maximizing quantity that the firm would produce would be where marginal revenue equals marginal cost.

Marginal revenue equals marginal cost at quantity Qm.

Thus quantity that the firm would produce is Qm and price that the firm would charge is Pm.

 

3)

Q_d = 60 – 0.5P

Q = 60 - 0.5 P

0.5 P = 60 - Q

P = 120 - 2Q

Total revenue = P * Q

TR = P * Q

TR = [120 - 2Q] * Q

TR = 120Q - 2Q²

Differentiating the above total revenue function with respect to Q, we get marginal revenue function.

dTR/dQ = MR

MR = 120 - 4Q

 

MC = 0.5Q + 30

 

MR = MC

120 - 4Q = 0.5Q + 30

120 - 30 = 0.5Q +4Q

90 = 4.5Q

Q = 90/4.5

Q = 20

Value of Qm is 20 units.

 

Substitute Q = 20 in P = 120 - 2Q

P = 120 - 2*20

P = 120 - 40

P = 80

 

Value of Pm is 80.