Question

Problem 2:
Let’s look again at the two small-town driveway paving companies, Asphalt, Inc. and Blacktop Bros. The inverse demand curve for
paving services is: P = 1,600 – 2 Q, where quantity is measured in pave jobs per month and price is measured in dollars per job.
Assume Asphalt, Inc. has a marginal cost of $400 per driveway and Blacktop Bros. has a marginal cost of $200.
Answer the following questions:
a. Determine each firm’s reaction curve and graph it.
b. How many oil changes will each firm produce in Cournot equilibrium?
c. What will the market price of an oil change be?
d. How much profit does each firm earn?

Answer 1

P = 1600 - 2Q = 1600 - 2Q1 - 2Q2 [Where Q1: Quantity by Asphalt and Q2: Quantity by Blacktop and Q = Q1 + Q2]

(a)

For Asphalt,

Total revenue (TR1) = P x Q1 = 1600Q1 - 2Q1² - 2Q1Q2

Marginal revenue (MR1) = TR1 / Q1 = 1600 - 4Q1 - 2Q2

Equating MR1 and MC1,

1600 - 4Q1 - 2Q2 = 400

4Q1 + 2Q2 = 1200

2Q1 + Q2 = 600.........(1) [Reaction curve, Asphalt]

For Blaktop,

Total revenue (TR2) = P x Q2 = 1600Q2 - 2Q1Q2 - 2Q2²

Marginal revenue (MR2) = TR2 / Q2 = 1600 - 2Q1 - 4Q2

Equating MR2 and MC2,

1600 - 2Q1 - 4Q2 = 200

2Q1 + 4Q2 = 1400.........(2) [Reaction curve, Blacktop]

From Asphalt's reaction curve,

When Q1 = 0, Q2 = 600 (Vertical intercept) & when Q2 = 0, Q1 = 600/2 = 300 (Horizontal intercept)

From Blacktop's reaction curve,

When Q1 = 0, Q2 = 1400/4 = 350 (Vertical intercept) & when Q2 = 0, Q1 = 1400/2 = 700 (Horizontal intercept)

The reaction curves are graphed as follows.

(b) Cournot equilibrium is obtained by solving (1) and (2).

2Q1 + Q2 = 600......... (1)

2Q1 + 4Q2 = 1400........(2)

(2) - (1) results: 3Q2 = 800

Q2 = 267 (Assuming Q2 is integer)

Q1 = (600 - Q2) / 2 [From (1)] = (600 - 267) / 2 = 333 / 2 = 167 (Assuming Q1 is integer)

(c) P = 1600 - 2 x (167 + 267) = 1600 - (2 x 434) = 1600 - 868 = 732

(d)

Asphalt's profit = Q1 x (P - MC1) = 167 x (732 - 400) = 167 x 332 = 55,444

Blackrock's profit = Q2 x (P - MC2) = 267 x (732 - 200) = 267 x 532 = 142,044