Table 6.6
$$\begin{array} { | c | c | c | }
\hline{ \text { Quantity } } & { \text { Total Utility } } \
& \text { Games of Billiards } & \text { Games of Bowling } \
\hline 0 & 0 & 0 \
\hline 1 & 100 & 70 \
\hline 2 & 180 & 130 \
\hline 3 & 240 & 180 \
\hline 4 & 272 & 210 \
\hline 5 & 288 & 218 \
\hline 6 & 292 & 222 \
\hline
\end{array}$$
-Refer to Table 6.6,which shows John's total utility derived from billiards and bowling games.Assume John has $30 to spend on a game of billiards and/or a game of bowling.A game of billiards costs him $4,and a game of bowling costs him $2.Which of the following is John's utility-maximizing combination of the games of billiards and the games of bowling?
A)four games of billiards and four games of bowling
B)five games of billiards and two games of bowling
C)three games of billiards and eight games of bowling
D)five games of billiards and five games of bowling
E)twelve games of bowling

Question

Table 6.6
 $$\begin{array} { | c | c | c | } 
\hline{ \text { Quantity } } & { \text { Total Utility } } \
& \text { Games of Billiards } & \text { Games of Bowling } \
\hline 0 & 0 & 0 \
\hline 1 & 100 & 70 \
\hline 2 & 180 & 130 \
\hline 3 & 240 & 180 \
\hline 4 & 272 & 210 \
\hline 5 & 288 & 218 \
\hline 6 & 292 & 222 \
\hline
\end{array}$$
-Refer to Table 6.6,which shows John's total utility derived from billiards and bowling games.Assume John has $30 to spend on a game of billiards and/or a game of bowling.A game of billiards costs him $4,and a game of bowling costs him $2.Which of the following is John's utility-maximizing combination of the games of billiards and the games of bowling?
A)four games of billiards and four games of bowling
B)five games of billiards and two games of bowling
C)three games of billiards and eight games of bowling
D)five games of billiards and five games of bowling
E)twelve games of bowling

Quizplus · Accepted Answer

To find John's utility-maximizing combination of the games of billiards and the games of bowling, we need to calculate the marginal utility per dollar for each game.

For billiards, the marginal utility per dollar is found by dividing the increase in total utility from one game to the next by the cost of the game:
- From 0 to 1 game: (100 - 0)/4 = 25 
- From 1 to 2 games: (80 - 100)/4 = -5 
- From 2 to 3 games: (60 - 80)/4 = -5
- From 3 to 4 games: (32 - 60)/4 = -7
- From 4 to 5 games: (16 - 32)/4 = -4
- From 5 to 6 games: (4 - 16)/4 = -3

For bowling, the marginal utility per dollar is found by dividing the increase in total utility from one game to the next by the cost of the game:
- From 0 to 1 game: (70 - 0)/2 = 35
- From 1 to 2 games: (60 - 70)/2 = -5
- From 2 to 3 games: (50 - 60)/2 = -5
- From 3 to 4 games: (30 - 50)/2 = -10
- From 4 to 5 games: (8 - 30)/2 = -11
- From 5 to 6 games: (6 - 8)/2 = -1
- From 6 to 7 games: (4 - 6)/2 = -1
- From 7 to 8 games: (2 - 4)/2 = -1

As we can see, the marginal utility per dollar for billiards is negative after 2 games, while the marginal utility per dollar for bowling is negative after 4 games. Therefore, John should continue to play billiards until the marginal utility per dollar is equal to that of bowling, and then switch to bowling.

The total cost of 2 games of billiards is $8, leaving $22 for bowling. Dividing $22 by the cost of a game of bowling ($2) gives us 11 games of bowling. Therefore, John's utility-maximizing combination is 2 games of billiards and 11 games of bowling, or D.

Table 66 -Refer to Table 6