NAME 399
400 EXCHANGE (Ch. 32)
consumption. In competitive equilibrium, total consumption of apples
equals the total supply of apples and total consumption of bananas equals
the total supply of bananas. Therefore Lucy will consume 12 −asapples
(c) You can solve the two equations that you found above to find the
quantities of apples and bananas consumed in competitive equilibrium
32.6 (0) Consider a pure exchange economy with two consumers and
two goods. At some given Pareto efficient allocation it is known that both
consumers are consuming both goods and that consumer Ahas a marginal
rate of substitution between the two goods of −2. What is consumer B’s
32.7 (0) Charlotte loves apples and hates bananas. Her utility function
is U(a, b)=a−1
4b2,whereais the number of apples she consumes and
bis the number of bananas she consumes. Wilbur likes both apples and
bananas. His utility function is U(a, b)=a+2√b. Charlotte has an initial
endowment of no apples and 8 bananas. Wilbur has an initial endowment
of 16 apples and 8 bananas.
(a) On the graph below, mark the initial endowment and label it E.Use
red ink to draw the indifference curve for Charlotte that passes through
NAME 401
04812
16
4
8
12
Bananas
Apples
16
Wilbur
Charlotte
e
Red
line
Blue line
Black line
(b) If Charlotte hates bananas and Wilbur likes them, how many bananas
can Charlotte be consuming at a Pareto optimal allocation? 0.
(c) We know that a competitive equilibrium allocation must be Pareto
optimal and the total consumption of each good must equal the total
supply, so we know that at a competitive equilibrium, Wilbur must be
consuming 16 bananas. If Wilbur is consuming this number of
32.8 (0) Mutt and Jeff have 8 cups of milk and 8 cups of juice to
divide between themselves. Each has the same utility function given by
u(m, j)=max{m, j},wheremis the amount of milk and jis the amount
of juice that each has. That is, each of them cares only about the larger
of the two amounts of liquid that he has and is indifferent to the liquid
of which he has the smaller amount.
402 EXCHANGE (Ch. 32)
(a) Sketch an Edgeworth box for Mutt and Jeff. Use blue ink to show a
0246
8
2
4
6
Milk
Juice
8
Jeff
Mutt
Red
point
Red
point
Blue
curves
(Jeff)
Blue
curves
(Mutt)
32.9 (1) Remember Tommy Twit from Chapter 3. Tommy is happiest
when he has 8 cookies and 4 glasses of milk per day and his indifference
curves are concentric circles centered around (8,4). Tommy’s mother,
Mrs. Twit, has strong views on nutrition. She believes that too much
of anything is as bad as too little. She believes that the perfect diet for
Tommy would be 7 glasses of milk and 2 cookies per day. In her view,
a diet is healthier the smaller is the sum of the absolute values of the
differences between the amounts of each food consumed and the ideal
amounts. For example, if Tommy eats 6 cookies and drinks 6 glasses of
milk, Mrs. Twit believes that he has 4 too many cookies and 1 too few
glasses of milk, so the sum of the absolute values of the differences from
her ideal amounts is 5. On the axes below, use blue ink to draw the locus
of combinations that Mrs. Twit thinks are exactly as good for Tommy
as (6,6). Also, use red ink to draw the locus of combinations that she
thinks is just as good as (8,4). On the same graph, use red ink to draw an
indifference “curve” representing the locus of combinations that Tommy
likes just as well as 7 cookies and 8 glasses of milk.
NAME 403
2 4 6 8 10 12 14 16
2
4
6
8
10
12
14
16
Cookies
Milk
0
Black
line
Red curve
Blue curve
Tommy’s red
curve
(a) On the graph, shade in the area consisting of combinations of cookies
(b) Use black ink to sketch the locus of “Pareto optimal” bundles of
cookies and milk for Tommy. In this situation, a bundle is Pareto optimal
if any bundle that Tommy prefers to this bundle is a bundle that Mrs.
32.10 (2) This problem combines equilibrium analysis with some of the
things you learned in the chapter on intertemporal choice. It concerns the
economics of saving and the life cycle on an imaginary planet where life
is short and simple. In advanced courses in macroeconomics, you would
study more-complicated versions of this model that build in more earthly
realism. For the present, this simple model gives you a good idea of how
the analysis must go.
404 EXCHANGE (Ch. 32)
On the planet Drongo there is just one commodity, cake, and two
time periods. There are two kinds of creatures, “old” and “young.” Old
creatures have an income of Iunits of cake in period 1 and no income in
period 2. Young creatures have no income in period 1 and an income of I∗
units of cake in period 2. There are N1old creatures and N2young crea-
tures. The consumption bundles of interest to creatures are pairs (c1,c
2),
where c1is cake in period 1 and c2is cake in period 2. All creatures, old
and young, have identical utility functions, representing preferences over
cake in the two periods. This utility function is U(c1,c
2)=ca
1c1−a
2,where
ais a number such that 0 ≤a≤1.
(a) If current cake is taken to be the numeraire, (that is, its price is
set at 1), write an expression for the present value of a consumption
(b) If the interest rate is r, write down an expression for an old creature’s
demand for cake in period 1 c1=aI andinperiod2 c2=
(1−a)I(1+r).Write an expression for a young creature’s demand
(1−a)I∗.(Hint: If its budget line is p1c1+p2c2=Wand its utility
function is of the form proposed above, then a creature’s demand function
for good 1 is c1=aW/p and demand for good 2 is c2=(1−a)W/p.) If
the interest rate is zero, how much cake would a young creature choose in
period 1? aI∗.For what value of awould it choose the same amount
(c) The total supply of cake in period 1 equals the total cake earnings of
all old creatures, since young creatures earn no cake in this period. There
are N1old creatures and each earns Iunits of cake, so this total is N1I.
Similarly, the total supply of cake in period 2 equals the total amount
NAME 405
(d) At the equilibrium interest rate, the total demand of creatures for
period-1 cake must equal total supply of period-1 cake, and similarly the
demand for period-2 cake must equal supply. If the interest rate is r, then
the demand for period-1 cake by each old creature is aI and the
demand for period-1 cake by each young creature is aI∗/(1+r).
Since there are N1old creatures and N2young creatures, the total demand
(f) In the special case at the end of the last section, show that the interest
rate that equalizes supply and demand for period-1 cake will also equalize
supply and demand for period-2 cake. (This illustrates Walras’s law.)
400 EXCHANGE (Ch. 32)
consumption. In competitive equilibrium, total consumption of apples
equals the total supply of apples and total consumption of bananas equals
the total supply of bananas. Therefore Lucy will consume 12 −asapples
(c) You can solve the two equations that you found above to find the
quantities of apples and bananas consumed in competitive equilibrium
32.6 (0) Consider a pure exchange economy with two consumers and
two goods. At some given Pareto efficient allocation it is known that both
consumers are consuming both goods and that consumer Ahas a marginal
rate of substitution between the two goods of −2. What is consumer B’s
32.7 (0) Charlotte loves apples and hates bananas. Her utility function
is U(a, b)=a−1
4b2,whereais the number of apples she consumes and
bis the number of bananas she consumes. Wilbur likes both apples and
bananas. His utility function is U(a, b)=a+2√b. Charlotte has an initial
endowment of no apples and 8 bananas. Wilbur has an initial endowment
of 16 apples and 8 bananas.
(a) On the graph below, mark the initial endowment and label it E.Use
red ink to draw the indifference curve for Charlotte that passes through
NAME 401
04812
16
4
8
12
Bananas
Apples
16
Wilbur
Charlotte
e
Red
line
Blue line
Black line
(b) If Charlotte hates bananas and Wilbur likes them, how many bananas
can Charlotte be consuming at a Pareto optimal allocation? 0.
(c) We know that a competitive equilibrium allocation must be Pareto
optimal and the total consumption of each good must equal the total
supply, so we know that at a competitive equilibrium, Wilbur must be
consuming 16 bananas. If Wilbur is consuming this number of
32.8 (0) Mutt and Jeff have 8 cups of milk and 8 cups of juice to
divide between themselves. Each has the same utility function given by
u(m, j)=max{m, j},wheremis the amount of milk and jis the amount
of juice that each has. That is, each of them cares only about the larger
of the two amounts of liquid that he has and is indifferent to the liquid
of which he has the smaller amount.
402 EXCHANGE (Ch. 32)
(a) Sketch an Edgeworth box for Mutt and Jeff. Use blue ink to show a
0246
8
2
4
6
Milk
Juice
8
Jeff
Mutt
Red
point
Red
point
Blue
curves
(Jeff)
Blue
curves
(Mutt)
32.9 (1) Remember Tommy Twit from Chapter 3. Tommy is happiest
when he has 8 cookies and 4 glasses of milk per day and his indifference
curves are concentric circles centered around (8,4). Tommy’s mother,
Mrs. Twit, has strong views on nutrition. She believes that too much
of anything is as bad as too little. She believes that the perfect diet for
Tommy would be 7 glasses of milk and 2 cookies per day. In her view,
a diet is healthier the smaller is the sum of the absolute values of the
differences between the amounts of each food consumed and the ideal
amounts. For example, if Tommy eats 6 cookies and drinks 6 glasses of
milk, Mrs. Twit believes that he has 4 too many cookies and 1 too few
glasses of milk, so the sum of the absolute values of the differences from
her ideal amounts is 5. On the axes below, use blue ink to draw the locus
of combinations that Mrs. Twit thinks are exactly as good for Tommy
as (6,6). Also, use red ink to draw the locus of combinations that she
thinks is just as good as (8,4). On the same graph, use red ink to draw an
indifference “curve” representing the locus of combinations that Tommy
likes just as well as 7 cookies and 8 glasses of milk.
NAME 403
2 4 6 8 10 12 14 16
2
4
6
8
10
12
14
16
Cookies
Milk
0
Black
line
Red curve
Blue curve
Tommy’s red
curve
(a) On the graph, shade in the area consisting of combinations of cookies
(b) Use black ink to sketch the locus of “Pareto optimal” bundles of
cookies and milk for Tommy. In this situation, a bundle is Pareto optimal
if any bundle that Tommy prefers to this bundle is a bundle that Mrs.
32.10 (2) This problem combines equilibrium analysis with some of the
things you learned in the chapter on intertemporal choice. It concerns the
economics of saving and the life cycle on an imaginary planet where life
is short and simple. In advanced courses in macroeconomics, you would
study more-complicated versions of this model that build in more earthly
realism. For the present, this simple model gives you a good idea of how
the analysis must go.
404 EXCHANGE (Ch. 32)
On the planet Drongo there is just one commodity, cake, and two
time periods. There are two kinds of creatures, “old” and “young.” Old
creatures have an income of Iunits of cake in period 1 and no income in
period 2. Young creatures have no income in period 1 and an income of I∗
units of cake in period 2. There are N1old creatures and N2young crea-
tures. The consumption bundles of interest to creatures are pairs (c1,c
2),
where c1is cake in period 1 and c2is cake in period 2. All creatures, old
and young, have identical utility functions, representing preferences over
cake in the two periods. This utility function is U(c1,c
2)=ca
1c1−a
2,where
ais a number such that 0 ≤a≤1.
(a) If current cake is taken to be the numeraire, (that is, its price is
set at 1), write an expression for the present value of a consumption
(b) If the interest rate is r, write down an expression for an old creature’s
demand for cake in period 1 c1=aI andinperiod2 c2=
(1−a)I(1+r).Write an expression for a young creature’s demand
(1−a)I∗.(Hint: If its budget line is p1c1+p2c2=Wand its utility
function is of the form proposed above, then a creature’s demand function
for good 1 is c1=aW/p and demand for good 2 is c2=(1−a)W/p.) If
the interest rate is zero, how much cake would a young creature choose in
period 1? aI∗.For what value of awould it choose the same amount
(c) The total supply of cake in period 1 equals the total cake earnings of
all old creatures, since young creatures earn no cake in this period. There
are N1old creatures and each earns Iunits of cake, so this total is N1I.
Similarly, the total supply of cake in period 2 equals the total amount
NAME 405
(d) At the equilibrium interest rate, the total demand of creatures for
period-1 cake must equal total supply of period-1 cake, and similarly the
demand for period-2 cake must equal supply. If the interest rate is r, then
the demand for period-1 cake by each old creature is aI and the
demand for period-1 cake by each young creature is aI∗/(1+r).
Since there are N1old creatures and N2young creatures, the total demand
(f) In the special case at the end of the last section, show that the interest
rate that equalizes supply and demand for period-1 cake will also equalize
supply and demand for period-2 cake. (This illustrates Walras’s law.)