CHAPTER 34: Welfare
MULTIPLE CHOICE
1. A Borda count is used to decide an election between three candidates, x, y, and z, where a score of 1 is
awarded to a first choice, 2 to a second choice, and 3 to a third choice. There are 25 voters. 10
voters rank the candidates x first, y second, and z third; 7 voters rank the candidates x first, z second,
and y third; 5 rank the candidates z first, y second, and x third; and 3 voters rank the candidates y first,
z second, and x third. Which candidate wins?
a.
Candidate x.
b.
Candidate y.
c.
Candidate z.
d.
There is a tie between x and y, with z coming in third.
e.
There is a tie between y and z, with x coming in third.
2. A Borda count is used to decide an election between three candidates, x, y, and z, where a score of 1 is
awarded to a first choice, 2 to a second choice, and 3 to a third choice. There are 26 voters. 9
voters rank the candidates x first, y second, and z third; 9 voters rank the candidates x first, z second,
and y third; 6 rank the candidates z first, y second, and x third; and 2 voters rank the candidates y first,
z second, and x third. Which candidate wins?
a.
Candidate x.
b.
There is a tie between x and y, with z coming in third.
c.
Candidate y.
d.
Candidate z.
e.
There is a tie between y and z, with x coming in third.
3. A Borda count is used to decide an election between three candidates, x, y, and z, where a score of 1 is
awarded to a first choice, 2 to a second choice, and 3 to a third choice. There are 25 voters. 3
voters rank the candidates x first, y second, and z third; 10 voters rank the candidates x first, z second,
and y third; 6 rank the candidates z first, y second, and x third; and 6 voters rank the candidates y first,
z second, and x third. Which candidate wins?
a.
Candidate x.
b.
Candidate z.
c.
There is a tie between x and y, with z coming in third.
d.
Candidate y.
e.
There is a tie between y and z, with x coming in third.
4. A Borda count is used to decide an election between three candidates, x, y, and z, where a score of 1 is
awarded to a first choice, 2 to a second choice, and 3 to a third choice. There are 20 voters. 4
voters rank the candidates x first, y second, and z third; 5 voters rank the candidates x first, z second,
and y third; 8 rank the candidates z first, y second, and x third; and 3 voters rank the candidates y first,
z second, and x third. Which candidate wins?
a.
Candidate y.
b.
There is a tie between x and y, with z coming in third.
c.
Candidate z.
d.
Candidate x.
e.
There is a tie between y and z, with x coming in third.
5. A Borda count is used to decide an election between three candidates, x, y, and z, where a score of 1 is
awarded to a first choice, 2 to a second choice, and 3 to a third choice. There are 23 voters. 3
voters rank the candidates x first, y second, and z third; 5 voters rank the candidates x first, z second,
and y third; 10 rank the candidates z first, y second, and x third; and 5 voters rank the candidates y first,
z second, and x third. Which candidate wins?
a.
Candidate x.
b.
There is a tie between x and y, with z coming in third.
c.
Candidate z.
d.
Candidate y.
e.
There is a tie between y and z, with x coming in third.
6. A parent has two children living in cities with different costs of living. The cost of living in city B is 2
times the cost of living in city A. The child in city A has an income of $5,000 and the child in city B
has an income of $10,000. The parent wants to give a total of $4,000 to her two children. Her utility
function is U(CA, CB) = CACB, where CA and CB are the consumptions of the children living in cities A
and B respectively. She will choose to give
a.
each child $2,000, even though this will buy less goods for the child in city B.
b.
the child in city B twice as much money as the child in city A.
c.
the child in city A twice as much money as the child in city B.
d.
the child in city B the same amount of money as the child in city A.
e.
the child in city A the same amount of money as the child in city B.
7. A parent has two children living in cities with different costs of living. The cost of living in city B is 3
times the cost of living in city A. The child in city A has an income of $3,000 and the child in city B
has an income of $9,000. The parent wants to give a total of $2,000 to her two children. Her utility
function is U(CA, CB) = CACB, where CA and CB are the consumptions of the children living in cities A
and B respectively. She will choose to give
a.
the child in city B 3 times as much money as the child in city A.
b.
each child $1,000, even though this will buy less goods for the child in city B.
c.
the child in city A 3 times as much money as the child in city B.
d.
the child in city B 1.50 times as much money as the child in city A.
e.
the child in city A 1.50 times as much money as the child in city B.
8. A parent has two children living in cities with different costs of living. The cost of living in city B is 3
times the cost of living in city A. The child in city A has an income of $2,000 and the child in city B
has an income of $6,000. The parent wants to give a total of $1,000 to her two children. Her utility
function is U(CA, CB) = CACB, where CA and CB are the consumptions of the children living in cities A
and B respectively. She will choose to give
a.
the child in city B 1.50 times as much money as the child in city A.
b.
the child in city A 3 times as much money as the child in city B.
c.
the child in city B 3 times as much money as the child in city A.
d.
each child $500, even though this will buy less goods for the child in city B.
e.
the child in city A 1.50 times as much money as the child in city B.
9. A parent has two children living in cities with different costs of living. The cost of living in city B is 3
times the cost of living in city A. The child in city A has an income of $3,000 and the child in city B
has an income of $9,000. The parent wants to give a total of $2,000 to her two children. Her utility
function is U(CA, CB) = CACB, where CA and CB are the consumptions of the children living in cities A
and B respectively. She will choose to give
a.
the child in city B 3 times as much money as the child in city A.
b.
the child in city B 1.50 times as much money as the child in city A.
c.
the child in city A 3 times as much money as the child in city B.
d.
each child $1,000, even though this will buy less goods for the child in city B.
e.
the child in city A 1.50 times as much money as the child in city B.
10. A parent has two children living in cities with different costs of living. The cost of living in city B is 3
times the cost of living in city A. The child in city A has an income of $3,000 and the child in city B
has an income of $9,000. The parent wants to give a total of $4,000 to her two children. Her utility
function is U(CA, CB) = CACB, where CA and CB are the consumptions of the children living in cities A
and B respectively. She will choose to give
a.
the child in city B 1.50 times as much money as the child in city A.
b.
the child in city A 3 times as much money as the child in city B.
c.
the child in city B 3 times as much money as the child in city A.
d.
each child $2,000, even though this will buy less goods for the child in city B.
e.
the child in city A 1.50 times as much money as the child in city B.
11. Suppose that Paul and David from Problem 7 have utility functions U = 4AP + OP and U = AD + 5OD,
respectively, where AP and OP are Paul’s consumptions of apples and oranges and AD and OD are
David’s consumptions of apples and oranges. The total supply of apples and oranges to be divided
between them is 16 apples and 16 oranges. The “fair” allocations consist of all allocations satisfying
a.
AD = AP and OD = OP.
b.
8AP + 2OP is at least 80 and 2AD + 10OD is at least 96.
c.
4AP + OP is at least 80 and 2AD + 5OD is at least 96.
d.
AD + OD is at least 16 and AS + OS is at least 16.
e.
4AP + OP is at least AD + 5OD and AD + 5OD is at least 4AP + OP.
12. Suppose that Paul and David from Problem 7 have utility functions U = 5AP + OP and U = AD + 3OD,
respectively, where AP and OP are Paul’s consumptions of apples and oranges and AD and OD are
David’s consumptions of apples and oranges. The total supply of apples and oranges to be divided
between them is 14 apples and 14 oranges. The “fair” allocations consist of all allocations satisfying
a.
10AP + 2OP is at least 84 and 2AD + 6OD is at least 56.
b.
5AP + OP is at least 84 and 2AD + 3OD is at least 56.
c.
AD + OD is at least 14 and AS + OS is at least 14.
d.
AD = AP and OD = OP.
e.
5AP + OP is at least AD + 3OD and AD + 3OD is at least 5AP + OP.
13. Suppose that Paul and David from Problem 7 have utility functions U = 5AP + OP and U = AD + 3OD,
respectively, where AP and OP are Paul’s consumptions of apples and oranges and AD and OD are
David’s consumptions of apples and oranges. The total supply of apples and oranges to be divided
between them is 16 apples and 14 oranges. The “fair” allocations consist of all allocations satisfying
a.
5AP + OP is at least 94 and 2AD + 3OD is at least 58.
b.
AD = AP and OD = OP.
c.
AD + OD is at least 15 and AS + OS is at least 15.
d.
10AP + 2OP is at least 94 and 2AD + 6OD is at least 58.
e.
5AP + OP is at least AD + 3OD and AD + 3OD is at least 5AP + OP.
14. Suppose that Paul and David from Problem 7 have utility functions U = 3AP + OP and U = AD + 2OD,
respectively, where AP and OP are Paul’s consumptions of apples and oranges and AD and OD are
David’s consumptions of apples and oranges. The total supply of apples and oranges to be divided
between them is 10 apples and 20 oranges. The “fair” allocations consist of all allocations satisfying
a.
AD = AP and OD = OP.
b.
6AP + 2OP is at least 50 and 2AD + 4OD is at least 50.
c.
3AP + OP is at least 50 and 2AD + 2OD is at least 50.
d.
AD + OD is at least 15 and AS + OS is at least 15.
e.
3AP + OP is at least AD + 2OD and AD + 2OD is at least 3AP + OP.
15. Suppose that Paul and David from Problem 7 have utility functions U = 5AP + OP and U = AD + 3OD,
respectively, where AP and OP are Paul’s consumptions of apples and oranges and AD and OD are
David’s consumptions of apples and oranges. The total supply of apples and oranges to be divided
between them is 12 apples and 20 oranges. The “fair” allocations consist of all allocations satisfying
a.
AD = AP and OD = OP.
b.
AD + OD is at least 16 and AS + OS is at least 16.
c.
10AP + 2OP is at least 80 and 2AD + 6OD is at least 72.
d.
5AP + OP is at least 80 and 2AD + 3OD is at least 72.
e.
5AP + OP is at least AD + 3OD and AD + 3OD is at least 5AP + OP.
16. Suppose that Romeo in Problem 8 has the utility function U = S 8RS4J and Juliet has the utility function
U = S4RS 8J, where SR is Romeo’s spaghetti consumption and SJ is Juliet’s. They have 120 units of
spaghetti to divide between them.
a.
Romeo would want to give Juliet some spaghetti if he had more than 60 units of spaghetti.
b.
Juliet would want to give Romeo some spaghetti if she has more than 78 units.
c.
Romeo and Juliet would never disagree about how to divide the spaghetti.
d.
Romeo would want to give Juliet some spaghetti if he has more than 76 units of spaghetti.
e.
Juliet would want to give Romeo some spaghetti if she has more than 80 units of
spaghetti.
17. Suppose that Romeo in Problem 8 has the utility function U = S 6RS4J and Juliet has the utility function
U = S4RS 6J, where SR is Romeo’s spaghetti consumption and SJ is Juliet’s. They have 70 units of
spaghetti to divide between them.
a.
Romeo would want to give Juliet some spaghetti if he had more than 35 units of spaghetti.
b.
Juliet would want to give Romeo some spaghetti if she has more than 40 units.
c.
Romeo and Juliet would never disagree about how to divide the spaghetti.
d.
Romeo would want to give Juliet some spaghetti if he has more than 38 units of spaghetti.
e.
Juliet would want to give Romeo some spaghetti if she has more than 42 units of
spaghetti.
18. Suppose that Romeo in Problem 8 has the utility function U = S 3RS1J and Juliet has the utility function
U = S1RS 3J, where SR is Romeo’s spaghetti consumption and SJ is Juliet’s. They have 36 units of
spaghetti to divide between them.
a.
Romeo and Juliet would never disagree about how to divide the spaghetti.
b.
Romeo would want to give Juliet some spaghetti if he had more than 18 units of spaghetti.
c.
Juliet would want to give Romeo some spaghetti if she has more than 25 units.
d.
Romeo would want to give Juliet some spaghetti if he has more than 23 units of spaghetti.
e.
Juliet would want to give Romeo some spaghetti if she has more than 27 units of
spaghetti.
19. Suppose that Romeo in Problem 8 has the utility function U = S 5RS 2J and Juliet has the utility function
U = S 2RS 5J, where SR is Romeo’s spaghetti consumption and SJ is Juliet’s. They have 70 units of
spaghetti to divide between them.
a.
Romeo and Juliet would never disagree about how to divide the spaghetti.
b.
Romeo would want to give Juliet some spaghetti if he has more than 46 units of spaghetti.
c.
Romeo would want to give Juliet some spaghetti if he had more than 35 units of spaghetti.
d.
Juliet would want to give Romeo some spaghetti if she has more than 48 units.
e.
Juliet would want to give Romeo some spaghetti if she has more than 50 units of
spaghetti.
20. Suppose that Romeo in Problem 8 has the utility function U = S 5RS 2J and Juliet has the utility function
U = S 2RS 5J, where SR is Romeo’s spaghetti consumption and SJ is Juliet’s. They have 42 units of
spaghetti to divide between them.
a.
Romeo would want to give Juliet some spaghetti if he had more than 21 units of spaghetti.
b.
Romeo would want to give Juliet some spaghetti if he has more than 26 units of spaghetti.
c.
Juliet would want to give Romeo some spaghetti if she has more than 28 units.
d.
Romeo and Juliet would never disagree about how to divide the spaghetti.
e.
Juliet would want to give Romeo some spaghetti if she has more than 30 units of
spaghetti.
21. Hatfield and McCoy burn with hatred for each other. They both consume corn whisky. Hatfield’s
utility function is U = WH − W 2M /8 and McCoy’s utility is U = WM − W 2H /8, where WH is Hatfield’s
whisky consumption and WM is McCoy’s whisky consumption, measured in gallons. The sheriff has a
total of 48 gallons of confiscated whisky which he could give back to them. For some reason, the
sheriff wants them both to be as happy as possible and he wants to treat them equally. The sheriff
should give them each
a.
24 gallons.
b.
4 gallons and spill 40 gallons in the creek.
c.
2 gallons and spill 44 gallons in the creek.
d.
8 gallons and spill the rest in the creek.
e.
1 gallon and spill the rest in the creek.
22. Hatfield and McCoy burn with hatred for each other. They both consume corn whisky. Hatfield’s
utility function is U = WH − W 2M /32 and McCoy’s utility is U = WM − W2H /32, where WH is Hatfield’s
whisky consumption and WM is McCoy’s whisky consumption, measured in gallons. The sheriff has a
total of 52 gallons of confiscated whisky which he could give back to them. For some reason, the
sheriff wants them both to be as happy as possible and he wants to treat them equally. The sheriff
should give them each
a.
16 gallons and spill 20 gallons in the creek.
b.
20 gallons and spill the rest in the creek.
c.
8 gallons and spill 36 gallons in the creek.
d.
26 gallons.
e.
4 gallons and spill the rest in the creek.
23. Hatfield and McCoy burn with hatred for each other. They both consume corn whisky. Hatfield’s
utility function is U = WH − W 2M /16 and McCoy’s utility is U = WM − W2H /16, where WH is Hatfield’s
whisky consumption and WM is McCoy’s whisky consumption, measured in gallons. The sheriff has a
total of 66 gallons of confiscated whisky which he could give back to them. For some reason, the
sheriff wants them both to be as happy as possible and he wants to treat them equally. The sheriff
should give them each
a.
8 gallons and spill 50 gallons in the creek.
b.
33 gallons.
c.
12 gallons and spill the rest in the creek.
d.
4 gallons and spill 58 gallons in the creek.
e.
2 gallons and spill the rest in the creek.
24. Hatfield and McCoy burn with hatred for each other. They both consume corn whisky. Hatfield’s
utility function is U = WH − W 2M /32 and McCoy’s utility is U = WM − W2H /32, where WH is Hatfield’s
whisky consumption and WM is McCoy’s whisky consumption, measured in gallons. The sheriff has a
total of 82 gallons of confiscated whisky which he could give back to them. For some reason, the
sheriff wants them both to be as happy as possible and he wants to treat them equally. The sheriff
should give them each
a.
8 gallons and spill 66 gallons in the creek.
b.
16 gallons and spill 50 gallons in the creek.
c.
41 gallons.
d.
20 gallons and spill the rest in the creek.
e.
4 gallons and spill the rest in the creek.
25. Hatfield and McCoy burn with hatred for each other. They both consume corn whisky. Hatfield’s
utility function is U = WH − W 2M /8 and McCoy’s utility is U = WM − W 2H /8, where WH is Hatfield’s
whisky consumption and WM is McCoy’s whisky consumption, measured in gallons. The sheriff has a
total of 48 gallons of confiscated whisky which he could give back to them. For some reason, the
sheriff wants them both to be as happy as possible and he wants to treat them equally. The sheriff
should give them each
a.
24 gallons.
b.
2 gallons and spill 44 gallons in the creek.
c.
4 gallons and spill 40 gallons in the creek.
d.
8 gallons and spill the rest in the creek.
e.
1 gallon and spill the rest in the creek.