Let\'s say the cost function and the demand function of a monopolistic firm are

Question

Let's say the cost function and the demand function of a monopolistic firm are as follows C=10+Q^2 P=20-Q What is the price and output this monopolistic firm will produce? What will be the price and quantity if the government mandated that this monopolist behaves as a perfect competitor (charges price of a perfect competitor and sells quantity of a perfect competitor)? What is the PS, CS, and total welfare when in perfect competition? Compute the numerical value of PS, CS and TW. What is the PS, CS, and Total welfare is monopoly? Compute numerical values. Compute the deadweight loss when you move from perfect competition to monopoly? If the government sets a price of p=10, what is the new price and quantity?

Explanation / Answer

C = 10+ Q2

Marginal Cost = dC/dQ => 2Q

P = 20 -Q

TR = P*Q => 20Q - Q2

Marginal Revenue = dTR/dQ => 20 - 2Q

Equilibrium output is achieved when MR = MC

20 - 2Q = 2Q => 4Q = 20

Q =5

Substituting value of Q in demand function => 20 - 5 => 15

a) A monoplistic firm produces 5 units at $15.

Under Perfect Competition Price will be same as Marginal Revenue because perfectly competitive firms earn normal profits.

Quantity will remain same as 5 here because profit maximization condition is same for both type of firms.

Marginal Revenue = 20 - 2Q => 20 - 2 (5) => 10

b) A perfectly competitive firm will produce 5 units at $10.

c) Producer and consumer surplus is equal to the area of triangle demarcated by Supply and Demand curve respectuvely at equilibrium price and quantity.

Area of Triangle = 1/2 Base x Height

Here Base is Quantity and Height is equal to price

Height and base is same for both so PS and CS will be same.

PS = CS => 1/2 x 5 x 10 = 25

TW or Total welfare = CS + PS = 25+ 25 => 50

(D) At 0 units of Q, P = 20 which the highest willingness to pay by consumer

Price under monopoly = 15

Height = 20 - 15 => 5 And Base i.e., quantity = 5

Consumer Surplus = 1/2 x 5 x 5 => 12.5

Produce Surplus

Base is same as 5

It is composed of a rectangle and triangle

AREA of Rectangle = Lenght x Breadth

Length = Monopolist Price - Perfectly Competitve Price => 15 - 10 = 5

AREA of Rectangle = 5x5 = 25

Area of Triangle = 25 as calculated according to Perfect Competition earlier.

PS = 25+ 25 => 50

Total Welfare = 50 + 12.5 => 62.5

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