Question

Cournot Competition. Consider the problem of the Angels and Dodgers, two teams in the MLB. Before the season begins, the Angels and Dodgers simultaneously choose the number of wins WA and WD they are each going to get this season by constructing a roster of players. Assume that the average revenue (inverse demand) for wins is given by: AR(W ) = P (W ) = 240 − 2W where W = W_A + W_D is the sum of the two teams’ wins. Baseball fans in Los Angeles will choose one team to support (i.e. buy) which determines the ‘price’ of a win, and fans don’t have preferences for either team before the season begins. It is costly for each team to increase the number of wins because teams must acquire better talent. The cost of an additional win is constant at MC = 180 for each team.

a) Suppose that each team chooses their own W_A and W_D to maximize profits. Find the best response function for each team and then find the Nash equilibrium number of wins for each team. Find the equilibrium price and profits of each team.

b) Suppose that each team chooses their own W_A and W_D to maximize the number of wins. Find the best response function for each team and characterize the Nash equilibrium number of wins. Find the equilibrium price and profits of each team.

c) Suppose that the Angels are maximizing profits while the Dodgers are maximizing the number of wins. At the Nash equilibrium, which team will win more games? Which team gets a higher price? Which team has higher profits? (You can either solve for the new equilibrium values or use your intuition and provide a concise explanation.)

Answer 1

Given,

P(W)=240-2W , W=W_A+W_D

MC=180

Thus, TC=180Wi

a) Profits for both teams individually can be calculated as

Profit for A

  = P(W)*W_A-180W_A

=(240-2W)W_A-180W_A

  =(240-2W-180)W_A

=(240-2W_A-2W_D-180)W_A

  =240W_A-2W_A²-2W_DW_A-180W_A

Solving this you'll get profit for A as

=60W_A-2W_A²-2W_AW_D

Similarly, we can find profit for D

= P(W)*W_D-180W_D

=60W_D-2W_D²-2W_AW_D

on differentiating the two profits and using the first order conditions we can find the best responses functions for the game.

i.e.   = 60-4W_A-2W_D =0

   = 60-4W_D-2W_A =0

SOLVING THESE TWO EQUATIONS WE'LL GET  W_D AND W_A  WHICH ARE THE BEST RESPONSES.

(where our nash equilibrium stratagies are always the best responses functions )

W_D= (30-W_A)/2

W_A =  (30-W_D)/2

NOW, on simplyfying the two equations we'll get values for W_A AND W_D as

W_D^*= (30- ((30-W_D)/2) /2 =10

W_A^* =  (30-10)/2 =10

Thus, W_A^*= W_D^*=10 , which is the required profits for each team.

Total number of wins can be calculated as

W_A^* + W_D^* =10+10 = 20 = W

The euilibrium price can be calculated by substituting the values in equation

P(W)=240-2W

  =240-2 ( W_{A +} W_D)

=240 - 2(20)

=200 = P^*

Since, we know our price, the number of individual wins and the total number of wins , we can find the individual profits as

   ==(240-2W-180)W_A (as W_A= W_D )

=(240 - 40 - 180) 10

= 200