The idea of assigning numerical values to determine a preference ordering over a set of objects is not limited in application to commodity bundles. The Bill James Baseball Abstract argues that a baseball player’s batting average is not an adequate measure of his offensive productivity. Batting averages treat singles just the same as extra base hits. Furthermore they do not give credit for “walks,” although a walk is almost as good as a single. James argues that a double in two at-bats is better than a single, but not as good as two singles. To reflect these considerations, James proposes the following index, which he calls “runs created.” Let A be the number of hits plus the number of walks that a batter gets in a season. Let B be the number of total bases that the batter gets in the season.
(Thus, if a batter has S singles, W walks, D doubles, T triples, and H home runs, then A = S+D+T +H+W and B = S+W+2D+3T +4H.) Let N be the number of times the batter bats. Then his index of runs created in the season is defined to be AB/N and will be called his RC.
(a) In 1987, George Bell batted 649 times. He had 39 walks, 105 singles, 32 doubles, 4 triples, and 47 home runs. In 1987, Wade Boggs batted 656 times. He had 105 walks, 130 singles, 40 doubles, 6 triples, and 24 home runs. In 1987, Alan Trammell batted 657 times. He had 60 walks, 140 singles, 34 doubles, 3 triples, and 28 home runs. In 1987, Tony Gwynn batted 671 times. He had 82 walks, 162 singles, 36 doubles, 13 triples, and 7 home runs. We can calculate A, the number of hits plus walks, B the number of total bases, and RC, the runs created index for each of these players. For Bell, A = 227, B = 408, RC = 143. For Boggs, A = 305, B = 429, RC = 199. For Trammell, A = 265, B = 389, RC = 157. For
Gwynn, A = ____, B = _____ , RC = _____.
(b) If somebody has a preference ordering among these players, based only on the runs-created index, which player(s) would she prefer to Trammell? ________
(c) The differences in the number of times at bat for these players are small, and we will ignore them for simplicity of calculation. On the graph below, plot the combinations of A and B achieved by each of the players.
Draw four “indifference curves,” one through each of the four points you have plotted. These indifference curves should represent combinations of
A and B that lead to the same number of runs-created.