Question

Suppose major league baseball consists if two types of teams: profit maximizing teams (e.g., Rays) and win maximizing teams (e.g., Red Sox). Before the season, each team decides the number of expected wins they are going to get that season by constructing a roster of players. Fans enjoy wins, but the additional revenue of each win is decreasing and given by the following average revenue (inverse demand) curve: AR(W) 320-2w It is costly to increase the expected number of wins because the team has to acquire better talent. The cost of an additional win is constant at: MC(W) 100 You can think of this as being in millions of dollars in order to make it (slightly) more realistic. The total number of games in a season is 162.

Answer 1

a) Average revenue for both profit maximizing teams and win maximizing teams are given as

AR(W) = 320-2W

Therefore, total revenue is

TR(W) = 320W-2W²

and marginal revenue function is

MR(W) = 320-4W......(i)

Also marginal cost of additional win is

MC(W) =100 .......(ii)

For profit maximizing wins,

               MR(W) = MC(W)

     or, 320-4W=100 or, 4W= 220 or W=55

Therefore, expected number of wins of profit maximizing teams = 55

b) The win maximizing condition of team is MR(W)=0

      or, 320-4W=0 or, W=80

Therefore, expected number of wins of win maximizing teams = 80

c) the salary cap does not achieve parity between expected number of wins of both teams. Total number of games in a season is 162, expected number of wins of profit maximizing teams is 55 and expected number of wins of win maximizing teams = 80. Therefore, 162-55-80= 27 games would end up in a draw.

d) The average revenue function of expected number of wins is

AR(W) = 420- 2W

Therefore, Marginal revenue function of expected number of wins is

MR(W) = 420-4W

Also Marginal cost of additional win (W) is

MC(W) = 100

Therefore for profit maximizing condition,

                     MR(W) =MC(W)

          or, 420-4W=100 or , W=80

Therefore, expected number of wins of profit maximizing teams is 80

Again for win maximizing condition,

                              MR(W) = 420-4W= 0

      or, W= 105 but the maximum number of expected wins =80.

Therefore, expected number of wins of win maximizing teams is 80.

In this case, salary cap achieve parity of expected number of wins of both teams. Because win maximizing teams will not try to win more than 80 games as it will increase their costs of additional wins.