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Introduction. Here you will look at various ways of determining social
preferences. You will check to see which of the Arrow axioms for ag-
gregating individual preferences are satisfied by these welfare relations.
You will also try to find optimal allocations for some given social welfare
functions. The method for solving these last problems is analogous to
solving for a consumer’s optimal bundle given preferences and a budget
constraint. Two hints. Remember that for a Pareto optimal allocation
inside the Edgeworth box, the consumers’ marginal rates of substitution
will be equal. Also, in a “fair allocation,” neither consumer prefers the
other consumer’s bundle to his own.
Example: A social planner has decided that she wants to allocate income
between 2 people so as to maximize √Y1+√Y2where Yiis the amount of
income that person igets. Suppose that the planner has a fixed amount
of money to allocate and that she can enforce any income distribution
such that Y1+Y2=W,whereWis some fixed amount. This planner
would have ordinary convex indifference curves between Y1and Y2and
a “budget constraint” where the “price” of income for each person is 1.
Therefore the planner would set her marginal rate of substitution between
income for the two people equal to the relative price which is 1. When you
solve this, you will find that she sets Y1=Y2=W/2. Suppose instead
that it is “more expensive” for the planner to give money to person 1 than
to person 2. (Perhaps person 1 is forgetful and loses money, or perhaps
person 1 is frequently robbed.) For example, suppose that the planner’s
budget is 2Y1+Y2=W. Then the planner maximizes √Y1+√Y2subject
to 2Y1+Y2=W. Setting her MRS equal to the price ratio, we find that
√Y2
√Y1=2. SoY2=4Y1. Therefore the planner makes Y1=W/5and
Y2=4W/5.
34.1 (2) One possible method of determining a social preference relation
is the Borda count, also known as rank-order voting. Each voter is asked
to rank all of the alternatives. If there are 10 alternatives, you give your
first choice a 1, your second choice a 2, and so on. The voters’ scores for
each alternative are then added over all voters. The total score for an
alternative is called its Borda count. For any two alternatives, xand y,
if the Borda count of xis smaller than or the same as the Borda count
for y,thenxis “socially at least as good as” y. Suppose that there are
a finite number of alternatives to choose from and that every individual
has complete, reflexive, and transitive preferences. For the time being,
let us also suppose that individuals are never indifferent between any two
different alternatives but always prefer one to the other.
418 WELFARE (Ch. 34)
(a) Is the social preference ordering defined in this way complete? Yes.
Reflexive? Yes. Transitive? Yes.
(b) If everyone prefers xto y, will the Borda count rank xas socially
(c) Suppose that there are two voters and three candidates, x,y,and
z. Suppose that Voter 1 ranks the candidates, xfirst, zsecond, and y
third. Suppose that Voter 2 ranks the candidates, yfirst, xsecond, and z
z?5. Now suppose that it is discovered that candidate zonce
lifted a beagle by the ears. Voter 1, who has rather large ears himself,
is appalled and changes his ranking to xfirst, ysecond, zthird. Voter
2, who picks up his own children by the ears, is favorably impressed and
changes his ranking to yfirst, zsecond, xthird. Now what is the Borda
(d) Does the social preference relation defined by the Borda count have
the property that social preferences between xand ydepend only on how
people rank xversus yand not on how they rank other alternatives? Ex-
plain. No. In the above example, the ranking
34.2 (2) Suppose the utility possibility frontier for two individuals is
given by UA+2UB= 200. On the graph below, plot the utility frontier.
NAME 419
0 50 100 150 200
50
100
150
UA
UB
200
Blue line
Black line
Red line
Utility frontier
(a) In order to maximize a “Nietzschean social welfare function,”
W(UA,U
B)=max{UA,U
B}, on the utility possibility frontier shown
(b) If instead we use a Rawlsian criterion, W(UA,U
B)=min{UA,U
B},
then the social welfare function is maximized on the above utility possi-
(c) Suppose that social welfare is given by W(UA,U
B)=U1/2
AU1/2
B.In
this case, with the above utility possibility frontier, social welfare is max-
want to think about the similarities between this maximization problem
and the consumer’s maximization problem with a Cobb-Douglas utility
function.)
(d) Show the three social maxima on the above graph. Use black ink
to draw a Nietzschean isowelfare line through the Nietzschean maximum.
Use red ink to draw a Rawlsian isowelfare line through the Rawlsian
maximum. Use blue ink to draw a Cobb-Douglas isowelfare line through
the Cobb-Douglas maximum.
34.3 (2) A parent has two children named A and B and she loves both
of them equally. She has a total of $1,000 to give to them.
420 WELFARE (Ch. 34)
(a) The parent’s utility function is U(a, b)=√a+√b,whereais the
amount of money she gives to A and bis the amount of money she gives
(b) Suppose that her utility function is U(a, b)=−1
a−1
b. How will she
(c) Suppose that her utility function is U(a, b)=loga+logb. How will
(d) Suppose that her utility function is U(a, b)=min{a, b}. How will she
(e) Suppose that her utility function is U(a, b)=max{a, b}. How will she
versa.
(Hint: In each of the above cases, we notice that the parent’s problem is
to maximize U(a, b) subject to the constraint that a+b=1,000. This is
just like the consumer problems we studied earlier. It must be that the
parent sets her marginal rate of substitution between aand bequal to 1
since it costs the same to give money to each child.)
(f) Suppose that her utility function is U(a, b)=a2+b2. How will
she choose to divide the money between her children? Explain why she
doesn’t set her marginal rate of substitution equal to 1 in this case.
34.4 (2) In the previous problem, suppose that A is a much more efficient
shopper than B so that A is able to get twice as much consumption
goods as B can for every dollar that he spends. Let abe the amount of
consumption goods that A gets and bthe amount that B gets. We will
measure consumption goods so that one unit of consumption goods costs
$1 for A and $2 for B. Thus the parent’s budget constraint is a+2b=
1,000.
(a) If the mother’s utility function is U(a, b)=a+b, which child will get
more money? A. Which child will consume more goods? A.
NAME 421
(b) If the mother’s utility function is U(a, b)=a×b, which child will get
(c) If the mother’s utility function is U(a, b)=−1
a−1
b, which child will
(d) If the mother’s utility function is U(a, b)=max{a, b}, which child
will get more money? A. Which child will get to consume more?
A.
(e) If the mother’s utility function is U(a, b)=min{a, b}, which child will
get more money? B. Which child will get to consume more?
Calculus 34.5 (1) Norton and Ralph have a utility possibility frontier that is given
by the following equation, UR+U2
N= 100 (where Rand Nsignify Ralph
and Norton respectively).
(a) If we set Norton’s utility to zero, what is the highest possible utility
Ralph can achieve? 100. If we set Ralph’s utility to zero, what is
the best Norton can do? 10.
(b) Plot the utility possibility frontier on the graph below.
0 5 10 15 20
25
50
75
Norton's utility
Ralph's utility
100
422 WELFARE (Ch. 34)
(c) Derive an equation for the slope of the above utility possibility curve.
dUR
(d) Both Ralph and Norton believe that the ideal allocation is given by
maximizing an appropriate social welfare function. Ralph thinks that
UR= 75, UN= 5 is the best distribution of welfare, and presents the
maximization solution to a weighted-sum-of-the-utilities social welfare
function that confirms this observation. What was Ralph’s social welfare
function? (Hint: What is the slope of Ralph’s social welfare function?)
(e) Norton, on the other hand, believes that UR= 19, UN=9isthe
best distribution. What is the social welfare function Norton presents?
34.6 (2) Roger and Gordon have identical utility functions, U(x, y)=
x2+y2. There are 10 units of xand 10 units of yto be divided between
them. Roger has blue indifference curves. Gordon has red ones.
(a) Draw an Edgeworth box showing some of their indifference curves and
mark the Pareto optimal allocations with black ink. (Hint: Notice that
the indifference curves are nonconvex.)
010
10
Roger
Gordon
Black lines
Black lines
Red curves
Blue
curves
Fair
Fair
y
x
(b) What are the fair allocations in this case? See diagram.
34.7 (2) Paul and David consume apples and oranges. Paul’s util-
ity function is UP(AP,O
P)=2AP+OPand David’s utility function is
NAME 423
UD(AD,O
D)=AD+2OD,whereAPand ADare apple consumptions for
Paul and David, and OPand ODare orange consumptions for Paul and
David. There are a total of 12 apples and 12 oranges to divide between
Paul and David. Paul has blue indifference curves. David has red ones.
Draw an Edgeworth box showing some of their indifference curves. Mark
the Pareto optimal allocations on your graph.
12
Apples
Oranges
0
Red curves
Blue
curves
Blue shading
Red shading
Pareto
optimal
Pareto optimal
Fair
Paul
David
12
(a) Write one inequality that says that Paul likes his own bundle as well
as he likes David’s and write another inequality that says that David likes
his own bundle as well as he likes Paul’s. 2AP+OP≥2AD+OD
(d) On your Edgeworth box, mark the fair allocations.
34.8 (3) Romeo loves Juliet and Juliet loves Romeo. Besides love,
they consume only one good, spaghetti. Romeo likes spaghetti, but he
424 WELFARE (Ch. 34)
also likes Juliet to be happy and he knows that spaghetti makes her
happy. Juliet likes spaghetti, but she also likes Romeo to be happy and
she knows that spaghetti makes Romeo happy. Romeo’s utility function
is UR(SR,S
J)=Sa
RS1−a
Jand Juliet’s utility function is UJ(SJ,S
R)=
Sa
JS1−a
R,whereSJand SRare the amount of spaghetti for Romeo and
the amount of spaghetti for Juliet respectively. There is a total of 24 units
of spaghetti to be divided between Romeo and Juliet.
(a) Suppose that a=2/3. If Romeo got to allocate the 24 units of
spaghetti exactly as he wanted to, how much would he give himself?
16. How much would he give Juliet? 8. (Hint: Notice that this
problem is formally just like the choice problem for a consumer with a
Cobb-Douglas utility function choosing between two goods with a budget
constraint. What is the budget constraint?)
(b) If Juliet got to allocate the spaghetti exactly as she wanted to, how
(c) What are the Pareto optimal allocations? (Hint: An allocation
will not be Pareto optimal if both persons’ utility will be increased by
(d) When we had to allocate two goods between two people, we drew an
Edgeworth box with indifference curves in it. When we have just one
good to allocate between two people, all we need is an “Edgeworth line”
and instead of indifference curves, we will just have indifference dots.
Consider the Edgeworth line below. Let the distance from left to right
denote spaghetti for Romeo and the distance from right to left denote
spaghetti for Juliet.
(e) On the Edgeworth line you drew above, show Romeo’s favorite point
and Juliet’s favorite point.
NAME 425
(f) Suppose that a=1/3. If Romeo got to allocate the spaghetti, how
much would he choose for himself? 8. If Juliet got to allocate
the spaghetti, how much would she choose for herself? 8. Label
the Edgeworth line below, showing the two people’s favorite points and
the locus of Pareto optimal points.
(g) When a=1/3, at the Pareto optimal allocations what do Romeo and
Juliet disagree about? Romeo wants to give spaghetti
to Juliet, but she doesn’t want to take it.
34.9 (2) Hatfield and McCoy hate each other but love corn whiskey.
Because they hate for each other to be happy, each wants the other to
have less whiskey. Hatfield’s utility function is UH(WH,W
M)=WH−W2
M
and McCoy’s utility function is UM(WM,W
H)=WM−W2
H,whereWM
is McCoy’s daily whiskey consumption and WHis Hatfield’s daily whiskey
consumption (both measured in quarts). There are 4 quarts of whiskey
to be allocated.
(a) If McCoy got to allocate all of the whiskey, how would he allocate it?
(b) If each of them gets 2 quarts of whiskey, what will the utility of each
of them be? −2.If a bear spilled 2 quarts of their whiskey and they
divided the remaining 2 quarts equally between them, what would the
utility of each of them be? 0. If it is possible to throw away some
of the whiskey, is it Pareto optimal for them each to consume 2 quarts of
whiskey? No.
426 WELFARE (Ch. 34)
(c) If it is possible to throw away some whiskey and they must consume
equal amounts of whiskey, how much should they throw away? 3
418 WELFARE (Ch. 34)
(a) Is the social preference ordering defined in this way complete? Yes.
Reflexive? Yes. Transitive? Yes.
(b) If everyone prefers xto y, will the Borda count rank xas socially
(c) Suppose that there are two voters and three candidates, x,y,and
z. Suppose that Voter 1 ranks the candidates, xfirst, zsecond, and y
third. Suppose that Voter 2 ranks the candidates, yfirst, xsecond, and z
z?5. Now suppose that it is discovered that candidate zonce
lifted a beagle by the ears. Voter 1, who has rather large ears himself,
is appalled and changes his ranking to xfirst, ysecond, zthird. Voter
2, who picks up his own children by the ears, is favorably impressed and
changes his ranking to yfirst, zsecond, xthird. Now what is the Borda
(d) Does the social preference relation defined by the Borda count have
the property that social preferences between xand ydepend only on how
people rank xversus yand not on how they rank other alternatives? Ex-
plain. No. In the above example, the ranking
34.2 (2) Suppose the utility possibility frontier for two individuals is
given by UA+2UB= 200. On the graph below, plot the utility frontier.
NAME 419
0 50 100 150 200
50
100
150
UA
UB
200
Blue line
Black line
Red line
Utility frontier
(a) In order to maximize a “Nietzschean social welfare function,”
W(UA,U
B)=max{UA,U
B}, on the utility possibility frontier shown
(b) If instead we use a Rawlsian criterion, W(UA,U
B)=min{UA,U
B},
then the social welfare function is maximized on the above utility possi-
(c) Suppose that social welfare is given by W(UA,U
B)=U1/2
AU1/2
B.In
this case, with the above utility possibility frontier, social welfare is max-
want to think about the similarities between this maximization problem
and the consumer’s maximization problem with a Cobb-Douglas utility
function.)
(d) Show the three social maxima on the above graph. Use black ink
to draw a Nietzschean isowelfare line through the Nietzschean maximum.
Use red ink to draw a Rawlsian isowelfare line through the Rawlsian
maximum. Use blue ink to draw a Cobb-Douglas isowelfare line through
the Cobb-Douglas maximum.
34.3 (2) A parent has two children named A and B and she loves both
of them equally. She has a total of $1,000 to give to them.
420 WELFARE (Ch. 34)
(a) The parent’s utility function is U(a, b)=√a+√b,whereais the
amount of money she gives to A and bis the amount of money she gives
(b) Suppose that her utility function is U(a, b)=−1
a−1
b. How will she
(c) Suppose that her utility function is U(a, b)=loga+logb. How will
(d) Suppose that her utility function is U(a, b)=min{a, b}. How will she
(e) Suppose that her utility function is U(a, b)=max{a, b}. How will she
versa.
(Hint: In each of the above cases, we notice that the parent’s problem is
to maximize U(a, b) subject to the constraint that a+b=1,000. This is
just like the consumer problems we studied earlier. It must be that the
parent sets her marginal rate of substitution between aand bequal to 1
since it costs the same to give money to each child.)
(f) Suppose that her utility function is U(a, b)=a2+b2. How will
she choose to divide the money between her children? Explain why she
doesn’t set her marginal rate of substitution equal to 1 in this case.
34.4 (2) In the previous problem, suppose that A is a much more efficient
shopper than B so that A is able to get twice as much consumption
goods as B can for every dollar that he spends. Let abe the amount of
consumption goods that A gets and bthe amount that B gets. We will
measure consumption goods so that one unit of consumption goods costs
$1 for A and $2 for B. Thus the parent’s budget constraint is a+2b=
1,000.
(a) If the mother’s utility function is U(a, b)=a+b, which child will get
more money? A. Which child will consume more goods? A.
NAME 421
(b) If the mother’s utility function is U(a, b)=a×b, which child will get
(c) If the mother’s utility function is U(a, b)=−1
a−1
b, which child will
(d) If the mother’s utility function is U(a, b)=max{a, b}, which child
will get more money? A. Which child will get to consume more?
A.
(e) If the mother’s utility function is U(a, b)=min{a, b}, which child will
get more money? B. Which child will get to consume more?
Calculus 34.5 (1) Norton and Ralph have a utility possibility frontier that is given
by the following equation, UR+U2
N= 100 (where Rand Nsignify Ralph
and Norton respectively).
(a) If we set Norton’s utility to zero, what is the highest possible utility
Ralph can achieve? 100. If we set Ralph’s utility to zero, what is
the best Norton can do? 10.
(b) Plot the utility possibility frontier on the graph below.
0 5 10 15 20
25
50
75
Norton's utility
Ralph's utility
100
422 WELFARE (Ch. 34)
(c) Derive an equation for the slope of the above utility possibility curve.
dUR
(d) Both Ralph and Norton believe that the ideal allocation is given by
maximizing an appropriate social welfare function. Ralph thinks that
UR= 75, UN= 5 is the best distribution of welfare, and presents the
maximization solution to a weighted-sum-of-the-utilities social welfare
function that confirms this observation. What was Ralph’s social welfare
function? (Hint: What is the slope of Ralph’s social welfare function?)
(e) Norton, on the other hand, believes that UR= 19, UN=9isthe
best distribution. What is the social welfare function Norton presents?
34.6 (2) Roger and Gordon have identical utility functions, U(x, y)=
x2+y2. There are 10 units of xand 10 units of yto be divided between
them. Roger has blue indifference curves. Gordon has red ones.
(a) Draw an Edgeworth box showing some of their indifference curves and
mark the Pareto optimal allocations with black ink. (Hint: Notice that
the indifference curves are nonconvex.)
010
10
Roger
Gordon
Black lines
Black lines
Red curves
Blue
curves
Fair
Fair
y
x
(b) What are the fair allocations in this case? See diagram.
34.7 (2) Paul and David consume apples and oranges. Paul’s util-
ity function is UP(AP,O
P)=2AP+OPand David’s utility function is
NAME 423
UD(AD,O
D)=AD+2OD,whereAPand ADare apple consumptions for
Paul and David, and OPand ODare orange consumptions for Paul and
David. There are a total of 12 apples and 12 oranges to divide between
Paul and David. Paul has blue indifference curves. David has red ones.
Draw an Edgeworth box showing some of their indifference curves. Mark
the Pareto optimal allocations on your graph.
12
Apples
Oranges
0
Red curves
Blue
curves
Blue shading
Red shading
Pareto
optimal
Pareto optimal
Fair
Paul
David
12
(a) Write one inequality that says that Paul likes his own bundle as well
as he likes David’s and write another inequality that says that David likes
his own bundle as well as he likes Paul’s. 2AP+OP≥2AD+OD
(d) On your Edgeworth box, mark the fair allocations.
34.8 (3) Romeo loves Juliet and Juliet loves Romeo. Besides love,
they consume only one good, spaghetti. Romeo likes spaghetti, but he
424 WELFARE (Ch. 34)
also likes Juliet to be happy and he knows that spaghetti makes her
happy. Juliet likes spaghetti, but she also likes Romeo to be happy and
she knows that spaghetti makes Romeo happy. Romeo’s utility function
is UR(SR,S
J)=Sa
RS1−a
Jand Juliet’s utility function is UJ(SJ,S
R)=
Sa
JS1−a
R,whereSJand SRare the amount of spaghetti for Romeo and
the amount of spaghetti for Juliet respectively. There is a total of 24 units
of spaghetti to be divided between Romeo and Juliet.
(a) Suppose that a=2/3. If Romeo got to allocate the 24 units of
spaghetti exactly as he wanted to, how much would he give himself?
16. How much would he give Juliet? 8. (Hint: Notice that this
problem is formally just like the choice problem for a consumer with a
Cobb-Douglas utility function choosing between two goods with a budget
constraint. What is the budget constraint?)
(b) If Juliet got to allocate the spaghetti exactly as she wanted to, how
(c) What are the Pareto optimal allocations? (Hint: An allocation
will not be Pareto optimal if both persons’ utility will be increased by
(d) When we had to allocate two goods between two people, we drew an
Edgeworth box with indifference curves in it. When we have just one
good to allocate between two people, all we need is an “Edgeworth line”
and instead of indifference curves, we will just have indifference dots.
Consider the Edgeworth line below. Let the distance from left to right
denote spaghetti for Romeo and the distance from right to left denote
spaghetti for Juliet.
(e) On the Edgeworth line you drew above, show Romeo’s favorite point
and Juliet’s favorite point.
NAME 425
(f) Suppose that a=1/3. If Romeo got to allocate the spaghetti, how
much would he choose for himself? 8. If Juliet got to allocate
the spaghetti, how much would she choose for herself? 8. Label
the Edgeworth line below, showing the two people’s favorite points and
the locus of Pareto optimal points.
(g) When a=1/3, at the Pareto optimal allocations what do Romeo and
Juliet disagree about? Romeo wants to give spaghetti
to Juliet, but she doesn’t want to take it.
34.9 (2) Hatfield and McCoy hate each other but love corn whiskey.
Because they hate for each other to be happy, each wants the other to
have less whiskey. Hatfield’s utility function is UH(WH,W
M)=WH−W2
M
and McCoy’s utility function is UM(WM,W
H)=WM−W2
H,whereWM
is McCoy’s daily whiskey consumption and WHis Hatfield’s daily whiskey
consumption (both measured in quarts). There are 4 quarts of whiskey
to be allocated.
(a) If McCoy got to allocate all of the whiskey, how would he allocate it?
(b) If each of them gets 2 quarts of whiskey, what will the utility of each
of them be? −2.If a bear spilled 2 quarts of their whiskey and they
divided the remaining 2 quarts equally between them, what would the
utility of each of them be? 0. If it is possible to throw away some
of the whiskey, is it Pareto optimal for them each to consume 2 quarts of
whiskey? No.
426 WELFARE (Ch. 34)
(c) If it is possible to throw away some whiskey and they must consume
equal amounts of whiskey, how much should they throw away? 3
