CHAPTER 34: Welfare
TRUE/FALSE
1. According to Arrow’s impossibility theorem, it is impossible to find a social ordering that is complete,
reflexive, and transitive.
2. An allocation is fair if whenever one person envies another, the envied person does not envy the
envier.
3. In a pure exchange economy if the initial allocation is Pareto optimal, then competitive equilibrium is
fair.
4. In a competitive equilibrium, no matter how different their preferences may be, no two people with the
same income will envy each other’s consumption bundles.
5. An allocation that is worse for somebody than the initial allocation cannot be Pareto optimal.
6. If allocation x is Pareto optimal and allocation y is not, then everyone is at least as well off with x as
with y, and someone is better off with x than with y.
7. The utility possibilities frontier is the boundary of the production possibilities set.
8. In a pure exchange economy, if an allocation is Pareto efficient, it is impossible to have two people
who prefer each other’s consumption bundles to their own.
9. If a social welfare function is an increasing function of each person’s utility, then every allocation that
maximizes this social welfare function must be a Pareto optimum.
MULTIPLE CHOICE
1. Mr. Angst has two children, Dick and Jane. Dick is a slow learner and Jane is very bright. If Mr. Angst
spends $X per month on Dicks education, Dick will score a total of X/2 points on his SAT tests. If Mr.
Angst spends $Y per month on Janes education, she will score a total of 2Y on her SAT tests. Mr.
Angst has a utility function U(D, J) = minD, J, where D is Dicks SAT score and J is Janes SAT
score. To maximize his utility, he will spend
a.
equal amounts of money on the two children.
b.
4 times as much money on Dick’s education as on Jane’s.
c.
4 times as much money on Jane’s education as on Dick’s.
d.
between 1 and 2 times as much money on Dick’s education as on Jane’s.
e.
between 1 and 2 times as much money on Jane’s education as on Dick’s.
2. A Borda count is used to decide an election between 3 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 29 voters. 10
voters rank the candidates x first, y second, and z third; 3 voters rank the candidates x first, z second,
and y third; 8 voters rank the candidates z first, y second, and x third; 8 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.
3. A Borda count is used to decide an election between 3 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 28 voters. 5 voters
rank the candidates x first, y second, and z third; 10 voters rank the candidates x first, z second, and y
third; 4 voters rank the candidates z first, y second, and x third; 9 voters rank the candidates y first, z
second, and x third. Which candidate wins?
a.
Candidate y.
b.
Candidate z.
c.
There is a tie between x and y, with z coming in third.
d.
Candidate x.
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 3 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 14 voters. 6 voters
rank the candidates x first, y second, and z third; 3 voters rank the candidates x first, z second, and y
third; 3 voters rank the candidates z first, y second, and x third; 2 voters rank the candidates y first, z
second, and x third. Which candidate wins?
a.
There is a tie between x and y, with z coming in third.
b.
Candidate y.
c.
Candidate x.
d.
Candidate z.
e.
There is a tie between y and z, with x coming in third.
5. A parent has two children living in cities with different costs of living. The cost of living in city B is 4
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 $12,000. The parent wants to give a total of $1,000 to her two children. Her utility
function is U(C A, C B) = C A C B, where C A and C B are the consumptions of the children living in
cities A and B respectively. She will choose to give
a.
each child $500, even though this will buy less goods for the child in city B.
b.
the child in city B 4 times as much money as the child in city A.
c.
the child in city A 4 times as much money as the child in city B.
d.
the child in city B 2 times as much money as the child in city A.
e.
the child in city A 2 times as much money as the child in city B.
6. 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 $2,000 to her two children. Her utility
function is U(C A, C B) = C A C B, where C A and C B are the consumptions of the children living in
cities A and B respectively. She will choose to give
a.
each child $1,000, even though this will buy less goods for the child in city B.
b.
the child in city B 3 times as much money as the child in city A.
c.
the child in city B 1.50 times as much money as the child in city A.
d.
the child in city A 3 times as much money as the child in city B.
e.
the child in city A 1.50 times as much 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 $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(C A, C B) = C A C B, where C A and C B 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.
each child $500, even though this will buy less goods for the child in city B.
d.
the child in city A 3 times as much money as the child in city B.
e.
the child in city A 1.50 times as much money as the child in city B.
8. Suppose that Paul and David have utility functions U = 4A P + O P and U = A D + 4O D, respectively,
where A P and O P are Paul’s consumptions of apples and oranges and A D and O D are David’s
consumptions of apples and oranges. The total supply of apples and oranges to be divided between
them is 20 apples and 12 oranges. The fair allocations consist of all allocations satisfying the following
conditions.
a.
AD = AP and OD = OP.
b.
8AP + 2OP is at least 92 and 2AD + 8OD is at least 68.
c.
4AP + OP is at least 92 and 2AD + 4OD is at least 68.
d.
AD + OD is at least 16 and AS + OS is at least 16.
e.
4AP + OP is at least AD + 4OD and AD + 4OD is at least 4AP + OP.
9. Suppose that Paul and David have utility functions U = 4A P + O P and U = A D + 3O D, respectively,
where A P and O P are Paul’s consumptions of apples and oranges and A D and O D are David’s
consumptions of apples and oranges. The total supply of apples and oranges to be divided between
them is 18 apples and 18 oranges. The fair allocations consist of all allocations satisfying the following
conditions.
a.
AD = AP and OD = OP.
b.
8AP + 2OP is at least 90 and 2AD + 6OD is at least 72.
c.
4AP + OP is at least 90 and 2AD + 3OD is at least 72.
d.
AD + OD is at least 18 and AS + OS is at least 18.
e.
4AP + OP is at least AD + 3OD and AD + 3OD is at least 4AP + OP.
10. Suppose that Paul and David have utility functions U = 4A P + O P and U = A D + 5O D, respectively,
where A P and O P are Paul’s consumptions of apples and oranges and A D and O D are David’s
consumptions of apples and orange. The total supply of apples and oranges to be divided between
them is 20 apples and 14 oranges. The “fair” allocations consist of all allocations satisfying the
following conditions.
a.
4AP + OP is at least 94 and 2AD + 5OD is at least 90.
b.
AD + OD is at least 17 and AS + OS is at least 17.
c.
AD = AP and OD = OP.
d.
8AP + 2OP is at least 94 and 2AD + 10OD is at least 90.
e.
4AP + OP is at least AD + 5OD and AD + 5OD is at least 4AP + OP.
11. Suppose that Romeo has the utility function U = S5RS2J and Juliet has the utility function U = S1RS5J,
where SR is Romeo’s spaghetti consumption and SJ is Juliet’s. They have 36 units of spaghetti to divide
between them.
a.
Romeo would want to give Juliet some spaghetti if he had more than 18 units of spaghetti.
b.
Juliet would want to give Romeo some spaghetti if she had more than 28 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 had more than 26 units of spaghetti.
e.
Juliet would want to give Romeo some spaghetti if she had more than 30 units of
spaghetti.
12. Suppose that Romeo has the utility function U = S 7RS3J and Juliet has the utility function U = S 3RS7J,
where SR is Romeo’s spaghetti consumption and SJ is Juliet’s. They have 60 units of spaghetti to divide
between them.
a.
Romeo would want to give Juliet some spaghetti if he had more than 30 units of spaghetti.
b.
Romeo would want to give Juliet some spaghetti if he had more than 38 units of spaghetti.
c.
Romeo and Juliet would never disagree about how to divide the spaghetti.
d.
Juliet would want to give Romeo some spaghetti if she had more than 40 units.
e.
Juliet would want to give Romeo some spaghetti if she had more than 42 units of
spaghetti.
13. Suppose that Romeo has the utility function U = S 5RS1J and Juliet has the utility function U = S1RS5J,
where SR is Romeo’s spaghetti consumption and SJ is Juliet’s. They have 48 units of spaghetti to divide
between them.
a.
Juliet would want to give Romeo some spaghetti if she had more than 38 units.
b.
Romeo would want to give Juliet some spaghetti if he had more than 24 units of spaghetti.
c.
Romeo and Juliet would never disagree about how to divide the spaghetti.
d.
Romeo would want to give Juliet some spaghetti if he had more than 36 units of spaghetti.
e.
Juliet would want to give Romeo some spaghetti if she had more than 40 units of
spaghetti.
14. 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 28 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.
14 gallons.
b.
4 gallons and spill 20 gallons in the creek.
c.
2 gallons and spill 24 gallons in the creek.
d.
8 gallons and spill the rest in the creek.
e.
1 gallon and spill the rest in the creek.
15. Hatfield and McCoy burn with hatred for each other. They both consume corn whisky. Hatfield’s
utility function is U = WH − W 2M / 40 and McCoy’s utility is U = WM − W 2H / 40, where WH is
Hatfield’s whisky consumption and WM is McCoy’s whisky consumption, measured in gallons. The
sheriff has a total of 60 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.
30 gallons.
b.
20 gallons and spill 20 gallons in the creek.
c.
24 gallons and spill the rest in the creek.
d.
10 gallons and spill 40 gallons in the creek.
e.
5 gallons and spill the rest in the creek.
16. 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 58 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.
2 gallons and spill 54 gallons in the creek.
b.
4 gallons and spill 50 gallons in the creek.
c.
29 gallons.
d.
8 gallons and spill the rest in the creek.
e.
1 gallon and spill the rest in the creek.
17. A tiny fishing village has 3 residents. Ann has a utility of 10, Bruce has a utility of 6, and Charlie has a
utility of 7. If the mayor uses a Rawlsian social welfare function, the social welfare of this tiny village
would be
a.
6.
b.
10.
c.
23.
d.
420.
e.
20.49.
PROBLEM
1. No one is meaner and uglier than Gladys. Someone is meaner and uglier than Harold. Therefore
Gladys is meaner and uglier than Harold. Is this reasoning correct? If so, explain why. If not, explain
why not.
(Assume that people can be ranked from ugliest to least ugly by a complete transitive ordering and that
there are no ties. Likewise assume that people can be ranked from meanest to least mean by a complete
transitive ordering and that there are no ties.)