Question

Suppose that Grinch and Grubb go into the wine business in a small country where imports are prohibited and wine is difficult to grow. The industry consists of just the two duopolists, Grinch and Grubb. The demand for wine is given by p = 420 − y, where p is the price and y is the total quantity sold (in hectoliters). The cost function for Grinch is ?1 (?1 ) = ?1² and for Grubb it is ?2 (?2 ) = ?2² + 21?2.

a) How much is the output of each duopolist in Cournot-Nash equilibrium? Find the price and profits of each duopolist in the equilibrium.

b) Suppose that Grinch and Grubb make a tacit collusion to maximize their joint profits. Calculate the output and profits of each of them. What would be the price in case of a collusion?

c) Grinch entered the market before Grubb. If he took advantage of the opportunity to choose his output first, how much he should sell to maximize his profits? What would be the output of Grubb in the Stackelberg equilibrium? Calculate the price and profits of each duopolist in the equilibrium.

Answer 1

a) The output, price and profit of each firm can be found out by maximising the profit function of each firm with respect to its own profit and finding out the best response function of each firm.

FIRM GRINCH

Profit = Total Revenue - Total Cost

= (Price x y₁) - y₁²

= [ 420 - (y₁ + y₂)] x y₁ - y₁²

= [ 420 - y₁ - y₂] x y₁ - y₁²

= 420y₁ - y₁² - y₂ y₁ - y₁²

= 420y₁ - 2y₁² - y₂ y₁

Differentiating this profit function with respect to y₁ keeping y₂ constant to maximise the profits of Grinch and obtaining the best response function by setting it equal to 0.

420 - 4y₁ - y₂ = 0

y₁ = (420 - y₂ ) / 4

y₁ = 105 - y₂ / 4 .........(1)

FIRM GRUBB

Profit = Total Revenue - Total Cost

= ( Price x y₂) - y₂² - 21 y₂

₌ [ 420 - (y₁ + y₂)] x y₂ - y₂² - 21 y₂

= [ 420 - y₁ - y₂] x y₂ - y₂² - 21 y₂

= 420y₂ - y₁y₂ - y₂² - y₂² - 21 y₂

= 399y₂ - y₁y₂ - 2y₂²

Differentiating this function with respect to y₂ keeping y₁ as constant to maximise the profits of Grubb and obtaining the best response function by setting it equal to 0.

399 - y₁ - 4y₂ = 0

y₁ + 4y₂ = 399

y₂ = (399 - y₁) / 4 .......(2)

Putting (1) in (2)

y₂ = 399/4 - (105 - y₂/4) / 4

y₂ = 399/4 - 105/4 + y₂/16

y₂ - y₂/16 = 294/4

15y₂/16 = 294/4

y₂ = 78 approx is the answer.

y₁ = 105 - y₂ / 4

= 105 - 78/4

= 105 - 19.5

y₁  = 85.5 or 86 approx is the answer.

Price = 420 - (y₁ + y₂)

=  420 - y₁ - y₂

= 420 - 86 - 78

= 256 is the answer.

PROFIT OF GRINCH = (Price x y₁) - y₁²

= 256 X 86 - 86²

= 22016 - 7296

= 14620 IS THE ANSWER.

PROFIT OF GRUBB = ( Price x y₂) - y₂² - 21 y₂

= 256 X 78 - 78² - 21 X 78

= 19968 - 6084 - 1638

= 12246 IS THE ANSWER.