# STRUCTURED HOMEWORK ASSIGNMENT 3

PART 1

Question 1

(A) First, invert the demand function QD= 8,300 - 2.1P into the price function, so that price is on the left hand side on its own. QD= 8,300 - 2.1P→1QD/2.1 = 8,300/2.1 – 2.1P/2.1

0.5QD = 3,952.4 – P →P = 3,952.40 – 0.5QD

TR = P*Q→TR = (3,952.40 – 0.5Q) *Q →TR = 3,952.40Q – 0.5Q^2 MR = 3,952.40 – Q

(B) Profit = TR – TC

Profit = 3,952.40Q – 0.5Q^2 – (2,200 + 480Q + 20Q^2)

Profit = -2,200 + 3,472.40Q – 20.5Q^2

Marginal Profit = 3,472.40 – 41Q→MP = 0

3,472.40 – 41Q = 0→41Q = 3,472.40

→ Optimal Quantity = 85 (rounded off)

Therefore, Coolidge Corporation should produce and sell 85 lasers each month to maximise profit.

(C) Substitute Q = 85 into Profit = -2,200 + 3,472.40Q – 20.5Q^2 to determine Optimal Profit Profit = -2,200 + (3,472.40 * 85) – (20.5 * (85)^2)

→ Profit = $144,841.50

Question 2

C = 100 + 2Q^2P = 90 – 2Q

(A) TR = P*Q→TR = (90 – 2Q) * Q→TR = 90Q – 2Q^2 Profit = TR – TC→Profit = 90Q – 2Q^2 – (100 + 2Q^2) Profit = -100 + 90Q – 4Q^2

Marginal Profit = 90 – 8Q→MP = 0

90 – 8Q = 0→8Q = 90

→ Monopoly Quantity = 11.25

Substitute Q = 11.25 into P = 90 – 2Q to determine Monopoly Price P = 90 – (2 * 11.25)

→ P = $67.50

Substitute Q = 11.25 into Profit = -100 + 90Q – 4Q^2 to determine Monopoly Profit Profit = -100 + (90 * 11.25) – (4 * (11.25)^2)

→ Profit = $406.25

(B) Price = Marginal Cost in a competitive industry, therefore, set P = MC to determine the Optimal Quantity at which profit is maximised. Cost = 100 + 2Q^2→Marginal Cost = 4Q

P = MC→90 – 2Q = 4Q→6Q = 90

→ Optimal Quantity = 15

Substitute Q = 15 into P = 90 – 2Q to determine the Optimal Price P = 90 – (2 * 15)

→ P = $60

Substitute Q = 15 into Profit = -100 + 90Q – 4Q^2 to determine Optimal Profit Profit = -100 + (90 * 15) – (4 * (15)^2)

→ Profit = $350

(C) The profit that is lost by having the firm produce at the competitive industry as compared to the monopoly is the difference of the two profit levels as calculated in parts (a) and (b): $406.25 – 350 = $56.25. On the graph below, this difference is represented by the lost profit area, which is the triangle below the marginal cost curve and above the marginal revenue curve, between the quantities of 11.25 and 15. This is lost profit because for each of these 3.75 units, extra revenue earned was less than extra cost incurred. This area is (0.5)*(3.75)*(60 – 30) = $56.25.

Question 3

C = 100 – 5Q + Q^2, and demand is P = 55 – 2Q.

(A) TR = P*Q→TR = (55 – 2Q) * Q→TR = 55Q – 2Q^2 Profit = TR – TC→Profit = 55Q – 2Q^2 – (100 – 5Q + Q^2) Profit = -100 + 60Q – 3Q^2

Marginal Profit = 60 – 6Q→MP = 0

60 – 6Q = 0→6Q = 60

→ Monopoly Quantity = 10

Substitute Q = 10 into P = 55 – 2Q to determine Monopoly Price P = 55 – (2 * 10)

→ P = $35

Substitute Q = 10 into Profit = -100 + 60Q – 3Q^2 to determine Monopoly Profit Profit = -100 + (60 * 10) – (3 * (10)^2)

→ Profit = $200

Consumer surplus is equal to one-half times the profit-maximizing quantity, 10, times the difference between the demand intercept (55) and the monopoly price (35): Consumer Surplus (CS) = 0.5 * ($55 – $35) * 10 units

CS = $100

(B) C = 100 – 5Q + Q^2→Marginal Cost = -5 + 2Q

P = MC→55 – 2Q = -5 + 2Q→4Q = 60

→ Optimal Quantity = 15

Substitute Q = 15 into P = 55 – 2Q to determine Optimal Price P = 55 – (2 * 15)

→ P = $25

Substitute Q = 15 into Profit = -100 + 60Q – 3Q^2 to determine Optimal Profit Profit = -100 + (60 * 15) – (3 * (15)^2)

→ Profit = $125

Consumer Surplus (CS) = 0.5 * ($55 – $25) * 15 units

CS = $225

(C) The deadweight loss is equal to the area below the demand curve, above the marginal cost curve, and between the quantities of 10 and 15

And numerically;

Deadweight Loss (DWL) = 0.5 * ($35 – $15) * (15 – 10)

(DWL) = $50

Question 4

(A) Break-even (units) = Fixed Costs/(Selling price – Variable Cost) $30,000/($25 – $10)→...

Bibliography: (8th edition), Competitive Firms vs Monopoly, D. L. Robert S. Pindyck, Microeconomics, Pearson

Please join StudyMode to read the full document