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Chapter 32 NAME
Exchange
Introduction. The Edgeworth box is a thing of beauty. An amazing
amount of information is displayed with a few lines, points and curves.
In fact one can use an Edgeworth box to show just about everything
there is to say about the case of two traders dealing in two commodities.
Economists know that the real world has more than two people and more
than two commodities. But it turns out that the insights gained from this
model extend nicely to the case of many traders and many commodities.
So for the purpose of introducing the subject of exchange equilibrium, the
Edgeworth box is exactly the right tool. We will start you out with an
example of two gardeners engaged in trade. You will get most out of this
example if you fill in the box as you read along.
Example: Alice and Byron consume two goods, camelias and dahlias.
Alice has 16 camelias and 4 dahlias. Byron has 8 camelias and 8 dahlias.
They consume no other goods, and they trade only with each other. To
Dahlias Byron
12
6
0 6 12 18 24
Alice Camelias
Any feasible allocation of flowers between Alice and Byron is fully
described by a single point in the box. Consider, for example, the alloca-
tion where Alice gets the bundle (15,9) and Byron gets the bundle (9,3).
This allocation is represented by the point A=(15,9) in the Edgeworth
box, which you should draw in. The distance 15 from Ato the left side of
the box is the number of camelias for Alice and the distance 9 from Ato
392 EXCHANGE (Ch. 32)
It is also useful to mark the initial allocation in the Edgeworth box,
which, in this case, is the point E=(16,4). Now suppose that Alice’s
utility function is U(c, d)=c+2dand Byron’s utility funtion is U(c, d)=
cd. Alice’s indifference curves will be straight lines with slope −1/2. The
The Pareto set or contract curve is the set of points where Alice’s
indifference curves are tangent to Byron’s. There will be tangency if the
slopes are the same. The slope of Alice’s indifference curve at any point is
−1/2. The slope of Byron’s indifference curve depends on his consumption
Some problems ask you to find a competitive equilibrium. For an
economy with two goods, the following procedure is often a good way to
calculate equilibrium prices and quantities.
•Since demand for either good depends only on the ratio of prices of
•With the price of good 1 held at 1, calculate each consumer’s demand
•Write an equation that sets the total amount of good 2 demanded by
•Solve this equation for the value of p2that makes the demand for
•Plug this price into the demand functions to determine quantities.
Example: Frank’s utility function is U(x1,x
2)=x1x2and Maggie’s is
U(x1,x
2)=min{x1,x
2}. Frank’s initial endowment is 0 units of good 1
Set p1= 1 and find Frank’s and Maggie’s demand functions for good
2 as a function of p2. Using the techniques learned in Chapter 6, we
find that Frank’s demand function for good 2 is m/2p2,wheremis his
income. Since Frank’s initial endowment is 0 units of good 1 and 10 units
NAME 393
budget constraint implies that Maggie’s demand function for good 2 is
m/(1 + p2). Since her endowment is 20 units of good 1 and 5 units of
good 2, her income is 20 + 5p2. Therefore at price p2, Maggie’s demand
Solving this equation, one finds that the equilibrium price is p2=2. At
the equilibrium price, Frank will demand 5 units of good 2 and Maggie
will demand 10 units of good 2.
32.1 (0) Morris Zapp and Philip Swallow consume wine and books.
Morris has an initial endowment of 60 books and 10 bottles of wine. Philip
has an initial endowment of 20 books and 30 bottles of wine. They have
no other assets and make no trades with anyone other than each other.
For Morris, a book and a bottle of wine are perfect substitutes. His utility
function is U(b, w)=b+w,wherebis the number of books he consumes
and wis the number of bottles of wine he consumes. Philip’s preferences
are more subtle and convex. He has a Cobb-Douglas utility function,
U(b, w)=bw. In the Edgeworth box below, Morris’s consumption is
measured from the lower left, and Philip’s is measured from the upper
right corner of the box.
0204060
80
20
40
Books
PhilipWine
Morris
e
Blue curve
Red curve
Black
line
(a) On this diagram, mark the initial endowment and label it E.Usered
ink to draw Morris Zapp’s indifference curve that passes through his initial
394 EXCHANGE (Ch. 32)
(b) At any Pareto optimum, where both people consume some of each
good, it must be that their marginal rates of substitution are equal. No
matter what he consumes, Morris’s marginal rate of substitution is equal
bP.Use black ink on the diagram above to draw the locus of Pareto
optimal allocations.
(c) At a competitive equilibrium, it will have to be that Morris consumes
(d) At the equilibrium prices you found in the last part of the question,
what is the value of Philip Swallow’s initial endowment? 50. At these
(e) At the competitive equilibrium prices that you found above, Morris’s
income is 70. Therefore at these prices, the cost to Morris of con-
(f) Suppose that an economy consisted of 1,000 people just like Morris
and 1,000 people just like Philip. Each of the Morris types had the same
endowment and the same tastes as Morris. Each of the Philip types had
the same endowment and tastes as Philip. Would the prices that you
found to be equilibrium prices for Morris and Philip still be competitive
NAME 395
32.2 (0) Consider a small exchange economy with two consumers, Astrid
and Birger, and two commodities, herring and cheese. Astrid’s initial
endowment is 4 units of herring and 1 unit of cheese. Birger’s initial en-
dowment has no herring and 7 units of cheese. Astrid’s utility function is
U(HA,C
A)=HACA. Birger is a more inflexible person. His utility func-
tion is U(HB,C
B)=min{HB,C
B}.(HereHAand CAare the amounts
of herring and cheese for Astrid, and HBand CBare amounts of herring
and cheese for Birger.)
(a) Draw an Edgeworth box, showing the initial allocation and sketching
in a few indifference curves. Measure Astrid’s consumption from the lower
0246
8
2
4
Cheese
BirgerHerring
Astrid
eBlue curves
Red curves
Black
line
(b) Use black ink to show the locus of Pareto optimal allocations. (Hint:
Since Birger is kinky, calculus won’t help much here. But notice that
because of the rigidity of the proportions in which he demands the two
goods, it would be inefficient to give Birger a positive amount of either
good if he had less than that amount of the other good. What does that
(c) Let cheese be the numeraire (with price 1) and let pdenote the price
of herring. Write an expression for the amount of herring that Birger
will demand at these prices. 7/(p+1) (Hint: Since Birger initially
owns 7 units of cheese and no herring and since cheese is the numeraire,
the value of his initial endowment is 7. If the price of herring is p,how
many units of herring will he choose to maximize his utility subject to his
budget constraint?)
396 EXCHANGE (Ch. 32)
32.3 (0) Dean Foster Z. Interface and Professor J. Fetid Nightsoil ex-
change platitudes and bromides. When Dean Interface consumes TIplat-
itudes and BIbromides, his utility is given by
UI(BI,T
I)=BI+2
TI.
When Professor Nightsoil consumes TNplatitudes and BNbromide, his
utility is given by
TN.
Dean Interface’s initial endowment is 12 platitudes and 8 bromides. Pro-
fessor Nightsoil’s initial endowment is 4 platitudes and 8 bromides.
04812
16
4
8
12
Bromides
Platitudes
16
Nightsoil
Interface
eRed curve
Pencil curve
Blue line
3.2
(a) If Dean Interface consumes TIplatitudes and BIbromides, his mar-
(b) On the contract curve, Dean Interface’s marginal rate of substitution
equals Professor Nightsoil’s. Write an equation that states this condition.
NAME 397
(c) From this equation we see that TI/TN=1/4at all points on the
(d) But we also know that along the contract curve it must be that TI+
(e) Solving these two equations in two unknowns, we find that everywhere
(f) In the Edgeworth box, label the initial endowment with the letter
E. Dean Interface has thick gray penciled indifference curves. Profes-
sor Nightsoil has red indifference curves. Draw a few of these in the
Edgeworth box you made. Use blue ink to show the locus of Pareto op-
(g) Find the competitive equilibrium prices and quantities. You know
what the prices have to be at competitive equilibrium because you know
what the marginal rates of substitution have to be at every Pareto
32.4 (0) A little exchange economy has just two consumers, named Ken
and Barbie, and two commodities, quiche and wine. Ken’s initial endow-
ment is 3 units of quiche and 2 units of wine. Barbie’s initial endowment
is 1 unit of quiche and 6 units of wine. Ken and Barbie have identical util-
ity functions. We write Ken’s utility function as, U(QK,W
K)=QKWK
and Barbie’s utility function as U(QB,W
B)=QBWB,whereQKand
WKare the amounts of quiche and wine for Ken and QBand WBare
amounts of quiche and wine for Barbie.
(a) Draw an Edgeworth box below, to illustrate this situation. Put quiche
on the horizontal axis and wine on the vertical axis. Measure goods for
Ken from the lower left corner of the box and goods for Barbie from the
upper right corner of the box. (Be sure that you make the length of the
398 EXCHANGE (Ch. 32)
24
2
4
6
8
0
Ken Quiche
Wine Barbie
w
ce
Black line
Red
curve
Blue
curve
Pareto
efficient
points
(b) Use blue ink to draw an indifference curve for Ken that shows alloca-
(c) At any Pareto optimal allocation where both consume some of each
good, Ken’s marginal rate of substitution between quiche and wine must
(d) On your graph, show the locus of points that are Pareto efficient.
(Hint: If two people must each consume two goods in the same proportions
(e) In this example, at any Pareto efficient allocation, where both persons
consume both goods, the slope of Ken’s indifference curve will be −2.
(f) In competitive equilibrium, Ken’s consumption bundle must be 2
392 EXCHANGE (Ch. 32)
It is also useful to mark the initial allocation in the Edgeworth box,
which, in this case, is the point E=(16,4). Now suppose that Alice’s
utility function is U(c, d)=c+2dand Byron’s utility funtion is U(c, d)=
cd. Alice’s indifference curves will be straight lines with slope −1/2. The
The Pareto set or contract curve is the set of points where Alice’s
indifference curves are tangent to Byron’s. There will be tangency if the
slopes are the same. The slope of Alice’s indifference curve at any point is
−1/2. The slope of Byron’s indifference curve depends on his consumption
Some problems ask you to find a competitive equilibrium. For an
economy with two goods, the following procedure is often a good way to
calculate equilibrium prices and quantities.
•Since demand for either good depends only on the ratio of prices of
•With the price of good 1 held at 1, calculate each consumer’s demand
•Write an equation that sets the total amount of good 2 demanded by
•Solve this equation for the value of p2that makes the demand for
•Plug this price into the demand functions to determine quantities.
Example: Frank’s utility function is U(x1,x
2)=x1x2and Maggie’s is
U(x1,x
2)=min{x1,x
2}. Frank’s initial endowment is 0 units of good 1
Set p1= 1 and find Frank’s and Maggie’s demand functions for good
2 as a function of p2. Using the techniques learned in Chapter 6, we
find that Frank’s demand function for good 2 is m/2p2,wheremis his
income. Since Frank’s initial endowment is 0 units of good 1 and 10 units
NAME 393
budget constraint implies that Maggie’s demand function for good 2 is
m/(1 + p2). Since her endowment is 20 units of good 1 and 5 units of
good 2, her income is 20 + 5p2. Therefore at price p2, Maggie’s demand
Solving this equation, one finds that the equilibrium price is p2=2. At
the equilibrium price, Frank will demand 5 units of good 2 and Maggie
will demand 10 units of good 2.
32.1 (0) Morris Zapp and Philip Swallow consume wine and books.
Morris has an initial endowment of 60 books and 10 bottles of wine. Philip
has an initial endowment of 20 books and 30 bottles of wine. They have
no other assets and make no trades with anyone other than each other.
For Morris, a book and a bottle of wine are perfect substitutes. His utility
function is U(b, w)=b+w,wherebis the number of books he consumes
and wis the number of bottles of wine he consumes. Philip’s preferences
are more subtle and convex. He has a Cobb-Douglas utility function,
U(b, w)=bw. In the Edgeworth box below, Morris’s consumption is
measured from the lower left, and Philip’s is measured from the upper
right corner of the box.
0204060
80
20
40
Books
PhilipWine
Morris
e
Blue curve
Red curve
Black
line
(a) On this diagram, mark the initial endowment and label it E.Usered
ink to draw Morris Zapp’s indifference curve that passes through his initial
394 EXCHANGE (Ch. 32)
(b) At any Pareto optimum, where both people consume some of each
good, it must be that their marginal rates of substitution are equal. No
matter what he consumes, Morris’s marginal rate of substitution is equal
bP.Use black ink on the diagram above to draw the locus of Pareto
optimal allocations.
(c) At a competitive equilibrium, it will have to be that Morris consumes
(d) At the equilibrium prices you found in the last part of the question,
what is the value of Philip Swallow’s initial endowment? 50. At these
(e) At the competitive equilibrium prices that you found above, Morris’s
income is 70. Therefore at these prices, the cost to Morris of con-
(f) Suppose that an economy consisted of 1,000 people just like Morris
and 1,000 people just like Philip. Each of the Morris types had the same
endowment and the same tastes as Morris. Each of the Philip types had
the same endowment and tastes as Philip. Would the prices that you
found to be equilibrium prices for Morris and Philip still be competitive
NAME 395
32.2 (0) Consider a small exchange economy with two consumers, Astrid
and Birger, and two commodities, herring and cheese. Astrid’s initial
endowment is 4 units of herring and 1 unit of cheese. Birger’s initial en-
dowment has no herring and 7 units of cheese. Astrid’s utility function is
U(HA,C
A)=HACA. Birger is a more inflexible person. His utility func-
tion is U(HB,C
B)=min{HB,C
B}.(HereHAand CAare the amounts
of herring and cheese for Astrid, and HBand CBare amounts of herring
and cheese for Birger.)
(a) Draw an Edgeworth box, showing the initial allocation and sketching
in a few indifference curves. Measure Astrid’s consumption from the lower
0246
8
2
4
Cheese
BirgerHerring
Astrid
eBlue curves
Red curves
Black
line
(b) Use black ink to show the locus of Pareto optimal allocations. (Hint:
Since Birger is kinky, calculus won’t help much here. But notice that
because of the rigidity of the proportions in which he demands the two
goods, it would be inefficient to give Birger a positive amount of either
good if he had less than that amount of the other good. What does that
(c) Let cheese be the numeraire (with price 1) and let pdenote the price
of herring. Write an expression for the amount of herring that Birger
will demand at these prices. 7/(p+1) (Hint: Since Birger initially
owns 7 units of cheese and no herring and since cheese is the numeraire,
the value of his initial endowment is 7. If the price of herring is p,how
many units of herring will he choose to maximize his utility subject to his
budget constraint?)
396 EXCHANGE (Ch. 32)
32.3 (0) Dean Foster Z. Interface and Professor J. Fetid Nightsoil ex-
change platitudes and bromides. When Dean Interface consumes TIplat-
itudes and BIbromides, his utility is given by
UI(BI,T
I)=BI+2
TI.
When Professor Nightsoil consumes TNplatitudes and BNbromide, his
utility is given by
TN.
Dean Interface’s initial endowment is 12 platitudes and 8 bromides. Pro-
fessor Nightsoil’s initial endowment is 4 platitudes and 8 bromides.
04812
16
4
8
12
Bromides
Platitudes
16
Nightsoil
Interface
eRed curve
Pencil curve
Blue line
3.2
(a) If Dean Interface consumes TIplatitudes and BIbromides, his mar-
(b) On the contract curve, Dean Interface’s marginal rate of substitution
equals Professor Nightsoil’s. Write an equation that states this condition.
NAME 397
(c) From this equation we see that TI/TN=1/4at all points on the
(d) But we also know that along the contract curve it must be that TI+
(e) Solving these two equations in two unknowns, we find that everywhere
(f) In the Edgeworth box, label the initial endowment with the letter
E. Dean Interface has thick gray penciled indifference curves. Profes-
sor Nightsoil has red indifference curves. Draw a few of these in the
Edgeworth box you made. Use blue ink to show the locus of Pareto op-
(g) Find the competitive equilibrium prices and quantities. You know
what the prices have to be at competitive equilibrium because you know
what the marginal rates of substitution have to be at every Pareto
32.4 (0) A little exchange economy has just two consumers, named Ken
and Barbie, and two commodities, quiche and wine. Ken’s initial endow-
ment is 3 units of quiche and 2 units of wine. Barbie’s initial endowment
is 1 unit of quiche and 6 units of wine. Ken and Barbie have identical util-
ity functions. We write Ken’s utility function as, U(QK,W
K)=QKWK
and Barbie’s utility function as U(QB,W
B)=QBWB,whereQKand
WKare the amounts of quiche and wine for Ken and QBand WBare
amounts of quiche and wine for Barbie.
(a) Draw an Edgeworth box below, to illustrate this situation. Put quiche
on the horizontal axis and wine on the vertical axis. Measure goods for
Ken from the lower left corner of the box and goods for Barbie from the
upper right corner of the box. (Be sure that you make the length of the
398 EXCHANGE (Ch. 32)
24
2
4
6
8
0
Ken Quiche
Wine Barbie
w
ce
Black line
Red
curve
Blue
curve
Pareto
efficient
points
(b) Use blue ink to draw an indifference curve for Ken that shows alloca-
(c) At any Pareto optimal allocation where both consume some of each
good, Ken’s marginal rate of substitution between quiche and wine must
(d) On your graph, show the locus of points that are Pareto efficient.
(Hint: If two people must each consume two goods in the same proportions
(e) In this example, at any Pareto efficient allocation, where both persons
consume both goods, the slope of Ken’s indifference curve will be −2.
(f) In competitive equilibrium, Ken’s consumption bundle must be 2
