Question

Use the following inverse demand curve P=250-5Q and TC=$6Q+$2Q2 for the product basketballs.

-In a competitive market, what would be the equilibrium price, quantity, and profit?
-With two firms in the market, what would be equilibrium price, quantity, and profit?
-With a monopoloy present in the market, what would be equilibrium price, quantity, and profit?

Answer 1

Competitive Market:

TC = 6Q+2Q²

MC = dTC/dQ = 6 + 4Q

P = 250 - 5Q

A competitive firm optimises at point where P = MC

250-5Q = 6+4Q

244 = 9Q

Q = 27.11

P = 250-5*27.11 = 114.45

Total Revenue = P*Q = 27.11*114.45 = 3102.74

Total Cost = 6*27.11 + 2*27.11² = 162.66 + 1470 = 1632.66

Profit = 3102.74 - 1632.66 = 1470.08

Two Firms

Firm 1:

P = 250 - 5(Q1+Q2)

Total Revenue = P*Q1 = 250Q1 - 5Q1² - 5Q1Q2

MR = dTR/dQ1 = 250 - 10Q1 - 5Q2

MC = 6 + 4Q1

MR = MC

250-10Q1-5Q2 = 6+4Q1

244-5Q2 = 14Q1

Q1 = (244 - 5Q2)/14

Firm 2:

P = 250 - 5(Q1+Q2)

Total Revenue = P*Q2 = 250Q2 - 5Q2² - 5Q1Q2

MR = dTR/dQ1 = 250 - 10Q2 - 5Q1

MC = 6 + 4Q2

MR = MC

250-10Q2-5Q1 = 6+4Q2

244-5Q1 = 14Q2

Q2 = (244 - 5Q1)/14

Put value of Q1 from above and solve

Q2 = (244-5((244 - 5Q2)/14))/14

Q2 = 12.84

Similary for Q1 = (244 - 5Q2)/14

Q1 = (244-5*12.84)/14 = 12.84

Total Output = Q1 + Q2 = 25.68

Price: P = 250-5Q = 250 - 5*25.68 = 121.6

Profit of Firm 1 = TR -TC = P*Q1 - TC = 121.6*12.84 - (6*12.84 + 2*12.84²) = 1561.34 - (77.04 + 329.73) = 1561.34 - 406.77 = 1154.57

Same for Firm 2.

Monopoly

P = 250-5Q

TR = P*Q = 250Q - 5Q²

MR = dTR/dQ = 250-10Q

MC = 6 + 4Q

A monopolist optimises at MR = MC

250-10Q = 6+4Q

244 = 14Q

Q = 17.43

Price: P = 250-5*17.43 = 162.85

Total Revenue =P*Q = 17.43*162.85 = 2838.48

Total Cost = 6*17.43 + 2*17.43² = 104.58 + 607.61 = 712.19

Profit = 2838.48 - 712.19 = 2126.29