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.
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 describe
the possible allocations of ﬂowers, we ﬁrst draw a box whose width is
the total number of camelias and whose height is the total number of
dahlias that Alice and Byron have between them. The width of the box
is therefore 16 + 8 = 24 and the height of the box is 4 + 8 = 12.
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 allocation
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 A to the left side of
the box is the number of camelias for Alice and the distance 9 from A to
the bottom of the box is the number of dahlias for Alice. This point also
determines Byron’s consumption of camelias and dahlias. The distance
9 from A to the right side of the box is the total number of camelias
consumed by Byron, and the distance from A to the top of the box is the
number of dahlias consumed by Byron. Since the width of the box is the
total supply of camelias and the height of the box is the total supply of
dahlias, these conventions ensure that any point in the box represents a
feasible allocation of the total supply of camelias and dahlias.
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+2d and Byron’s utility funtion is U(c, d) =
cd. Alice’s indiﬀerence curves will be straight lines with slope −1/2. The
indiﬀerence curve that passes through her initial endowment, for example,
will be a line that runs from the point (24, 0) to the point (0, 12). Since
Byron has Cobb-Douglas utility, his indiﬀerence curves will be rectangular
hyperbolas, but since quantities for Byron are measured from the upper
right corner of the box, these indiﬀerence curves will be ﬂipped over as in
the Edgeworth box diagrams in your textbook.
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
of the two goods. When Byron is consuming the bundle (cB, dB), the slope
of his indiﬀerence curve is equal to his marginal rate of substitution, which
is −dB/cB. Therefore Alice’s and Byron’s indiﬀerence curves will nuzzle
up in a nice tangency whenever −dB/cB = −1/2. So the Pareto set in
this example is just the diagonal of the Edgeworth box.
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
good 1 to good 2, it is convenient to set the price of good 1 equal to
1 and let p2 be the price of good 2.
• With the price of good 1 held at 1, calculate each consumer’s demand
for good 2 as a function of p2.
• Write an equation that sets the total amount of good 2 demanded by
all consumers equal to the total of all participants’ initial endowments
of good 2.
• Solve this equation for the value of p2 that makes the demand for
good 2 equal to the supply of good 2. (When the supply of good 2
equals the demand of good 2, it must also be true that the supply of
good 1 equals the demand for good 1.)
• Plug this price into the demand functions to determine quantities.
Frank’s utility function is U(x1, x2) = x1x2 and Maggie’s is U(x1, x2) =
min{x1, x2}. Frank’s initial endowment is 0 units of good 1 and 10 units
of good 2. Maggie’s initial endowment is 20 units of good 1 and 5 units
of good 2. Let us ﬁnd a competitive equilibrium for Maggie and Frank.
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, where m is his
income. Since Frank’s initial endowment is 0 units of good 1 and 10 units
of good 2, his income is 10p2. Therefore Frank’s demand for good 2 is
10p2/2p2 = 5. Since goods 1 and 2 are perfect complements for Maggie,
she will choose to consume where x1 = x2. This fact, together with her
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
is (20 + 5p2)/(1 + p2). Frank’s demand plus Maggie’s demand for good 2
adds up to 5 + (20 + 5p2)/(1 + p2). The total supply of good 2 is Frank’s
10 unit endowment plus Maggie’s 5 unit endowment, which adds to 15
units. Therefore demand equals supply when
5 +
(20 + 5p2)
(1 + p2)
= 15.
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.
31.1 (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, where b is the number of books he consumes
and w is 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.
Wine Philip
40
20
0 20 40 60 80
Morris Books
(a) On this diagram, mark the initial endowment and label it E. Use red
ink to draw Morris Zapp’s indiﬀerence curve that passes through his initial
endowment. Use blue ink to draw in Philip Swallow’s indiﬀerence curve
that passes through his initial endowment. (Remember that quantities
for Philip are measured from the upper right corner, so his indiﬀerence
curves are “Phlipped over.”)
(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
to When Philip consumes the bundle, (bP , wP ), his MRS is
Therefore every Pareto optimal allocation where both
consume positive amounts of both goods satisﬁes the equation
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
some books and some wine. But in order for him to do so, it must be that
the ratio of the price of wine to the price of books is Therefore
we know that if we make books the numeraire, then the price of wine in
competitive equilibrium must be .
(d) At the equilibrium prices you found in the last part of the question,
what is the value of Philip Swallow’s initial endowment? At
these prices, Philip will choose to consume books and
bottles of wine. If Morris Zapp consumes all of the books and all of the
wine that Philip doesn’t consume, he will consume books and
bottles of wine.
(e) At the competitive equilibrium prices that you found above, Morris’s
income is Therefore at these prices, the cost to Morris of consuming
all of the books and all of the wine that Philip doesn’t consume is
(the same as, more than, less than) his income. At these
prices, can Morris aﬀord a bundle that he likes better than the bundle
(55, 15)?
(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
equilibrium prices? If each of the Morris types and each of
the Philip types behaved in the same way as Morris and Philip did above,
would supply equal demand for both wine and books? .
31.2 (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 endowment
has no herring and 7 units of cheese. Astrid’s utility function is
U(HA, CA) = HACA. Birger is a more inﬂexible person. His utility function
is U(HB, CB) = min{HB, CB}. (Here HA and CA are the amounts
of herring and cheese for Astrid, and HB and CB are 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
left and Birger’s from the upper right. In your Edgeworth box, draw two
diﬀerent indiﬀerence curves for each person, using blue ink for Astrid’s
and red ink for Birger’s.
(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 tell you about where the Pareto eﬃcient locus has
to be?)
.
(c) Let cheese be the numeraire (with price 1) and let p denote the price
of herring. Write an expression for the amount of herring that Birger
will demand at these prices. (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?)
(d) Where the price of cheese is 1 and p is the price of herring, what is
the value of Astrid’s initial endoment? . How much herring
will Astrid demand at price p?
31.3 (0) Dean Foster Z. Interface and Professor J. Fetid Nightsoil exchange
platitudes and bromides. When Dean Interface consumes TI platitudes
and BI bromides, his utility is given by
UI(BI, TI) = BI + 2 TI.
When Professor Nightsoil consumes TN platitudes and BN bromide, his
utility is given by
UN (BN , TN ) = BN + 4 TN .
Dean Interface’s initial endowment is 12 platitudes and 8 bromides. Professor
Nightsoil’s initial endowment is 4 platitudes and 8 bromides.
Platitudes Nightsoil
16
12
8
4
0 4 8 12 16
Interface Bromides
(a) If Dean Interface consumes TI platitudes and BI bromides, his marginal
rate of substitution will be If Professor Nightsoil consumes
TN platitudes and BN bromides, his marginal rate of substitution
will be .
(b) On the contract curve, Dean Interface’s marginal rate of substitution
equals Professor Nightsoil’s. Write an equation that states this condition.
This equation is especially simple because each person’s
marginal rate of substitution depends only on his consumption of
platitudes and not on his consumption of bromides.
(c) From this equation we see that TI/TN = at all points on the
contract curve. This gives us one equation in the two unknowns TI and
TN .
(d) But we also know that along the contract curve it must be that TI +
TN = , since the total consumption of platitudes must equal the
total endowment of platitudes.
(e) Solving these two equations in two unknowns, we ﬁnd that everywhere
on the contract curve, TI and TN are constant and equal to
and .
(f) In the Edgeworth box, label the initial endowment with the letter
E. Dean Interface has thick gray penciled indiﬀerence curves. Professor
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 optimal
points. The contract curve is a (vertical, horizontal, diagonal)
line in the Edgeworth box.
(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 optimum.
.
31.4 (0) A little exchange economy has just two consumers, named Ken
and Barbie, and two commodities, quiche and wine. Ken’s initial endowment
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 utility
functions. We write Ken’s utility function as, U(QK, WK) = QKWK
and Barbie’s utility function as U(QB, WB) = QBWB, where QK and
WK are the amounts of quiche and wine for Ken and QB and WB are
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
box equal to the total supply of quiche and the height equal to the total
supply of wine.) Locate the initial allocation in your box, and label it W.
On the sides of the box, label the quantities of quiche and wine for each
of the two consumers in the initial endowment.
(b) Use blue ink to draw an indiﬀerence curve for Ken that shows allocations
in which his utility is 6. Use red ink to draw an indiﬀerence curve
for Barbie that shows allocations in which her utility is 6.
(c) At any Pareto optimal allocation where both consume some of each
good, Ken’s marginal rate of substitution between quiche and wine must
equal Barbie’s. Write an equation that states this condition in terms of
the consumptions of each good by each person. .
(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
as each other, and if together they must consume twice as much wine as
quiche, what must those proportions be?)
(e) In this example, at any Pareto eﬃcient allocation, where both persons
consume both goods, the slope of Ken’s indiﬀerence curve will be
Therefore, since we know that competitive equilibrium must be
Pareto eﬃcient, we know that at a competitive equilibrium, pQ/pW =
.
(f) In competitive equilibrium, Ken’s consumption bundle must be
How about Barbie’s consumption bundle?
(Hint: You found competitive equilibrium prices
above. You know Ken’s initial endowment and you know the equilibrium
prices. In equilibrium Ken’s income will be the value of his endowment at
competitive prices. Knowing his income and the prices, you can compute
his demand in competitive equilibrium. Having solved for Ken’s consumption
and knowing that total consumption by Ken and Barbie equals the
sum of their endowments, it should be easy to ﬁnd Barbie’s consumption.)
(g) On the Edgeworth box for Ken and Barbie, draw in the competitive
equilibrium allocation and draw Ken’s competitive budget line (with black
ink).
31.5 (0) Linus Straight’s utility function is U(a, b) = a + 2b, where a
is his consumption of apples and b is his consumption of bananas. Lucy
Kink’s utility function is U(a, b) = min{a, 2b}. Lucy initially has 12 apples
and no bananas. Linus initially has 12 bananas and no apples. In the
Edgeworth box below, goods for Lucy are measured from the upper right
corner of the box and goods for Linus are measured from the lower left
corner. Label the initial endowment point on the graph with the letter
E. Draw two of Lucy’s indiﬀerence curves in red ink and two of Linus’s
indiﬀerence curves in blue ink. Use black ink to draw a line through all
of the Pareto optimal allocations.
Bananas Lucy
12
6
0 6 12
Linus Apples
(a) In this economy, in competitive equilibrium, the ratio of the price of
apples to the price of bananas must be .
(b) Let aS be Linus’s consumption of apples and let bS be his consumption
of bananas. At competititive equilibrium, Linus’s consumption will have
to satisfy the budget constraint, as+ bS = This gives us
one equation in two unknowns. To ﬁnd a second equation, consider Lucy’s
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 − as apples
and −bs bananas. At a competitive equilibrium, Lucy will be
consuming at one of her kink points. The kinks occur at bundles where
Lucy consumes apples for every banana that she consumes.
Therefore we know that 12−as
12−bs
= .
(c) You can solve the two equations that you found above to ﬁnd the
quantities of apples and bananas consumed in competitive equilibrium
by Linus and Lucy. Linus will consume units of apples and
units of bananas. Lucy will consume units of apples
and 3 units of bananas.
31.6 (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 A has a marginal
rate of substitution between the two goods of −2. What is consumer B’s
marginal rate of substitution between these two goods? .
31.7 (0) Charlotte loves apples and hates bananas. Her utility function
is U(a, b) = a − 1
4 b2
, where a is the number of apples she consumes and
b is 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
this point. Use blue ink to draw the indiﬀerence curve for Wilbur that
passes through this point.
Apples Wilbur
16
12
8
4
0 4 8 12 16
Charlotte Bananas
(b) If Charlotte hates bananas and Wilbur likes them, how many bananas
can Charlotte be consuming at a Pareto optimal allocation? On
the graph above, use black ink to mark the locus of Pareto optimal allocations
of apples and bananas between Charlotte and Wilbur.
(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 bananas. If Wilbur is consuming this number of bananas,
his marginal utility for bananas will be and his marginal
utility of apples will be If apples are the numeraire, then the
only price of bananas at which he will want to consume exactly 16 bananas
is In competitive equilibrium, for the Charlotte-Wilbur
economy, Wilbur will consume bananas and apples
and Charlotte will consume bananas and apples.
31.8 (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}, where m is the amount of milk and j is 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.
(a) Sketch an Edgeworth box for Mutt and Jeﬀ. Use blue ink to show a
couple of indiﬀerence curves for each. Use red ink to show the locus of
Pareto optimal allocations. (Hint: Look for boundary solutions.)
Juice Jeﬀ
8
6
4
2
0 2 4 6 8
Mutt Milk
31.9 (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.
Milk
16
14
12
10
8
6
4
2
0 2 4 6 8 10 12 14 16
Cookies
(a) On the graph, shade in the area consisting of combinations of cookies
and milk that both Tommy and his mother agree are better than 7 cookies
and 8 glasses of milk, where “better” for Mrs. Twit means she thinks it
is healthier, and where “better” for Tommy means he likes it better.
(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.
Twit thinks is worse for him. The locus of Pareto optimal points that you
just drew should consist of two line segments. These run from the point
(8,4) to the point and from that point to the point .
31.10 (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.
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 I units 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 N1 old creatures and N2 young creatures.
The consumption bundles of interest to creatures are pairs (c1, c2),
where c1 is cake in period 1 and c2 is 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, c2) = ca
1c1−a
2 , where
a is 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
bundle (c1, c2). Write down the present value of
income for old creatures and for young creatures
The budget line for any creature is determined by the condition that the
present value of its consumption bundle equals the present value of its
income. Write down this budget equation for old creatures:
and for young creatures: .
(b) If the interest rate is r, write down an expression for an old creature’s
demand for cake in period 1 and in period 2
Write an expression for a young creature’s demand for cake
in period 1 and in period 2 (Hint:
If its budget line is p1c1 + p2c2 = W and 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?
For what value of a would it choose the same amount in each
period if the interest rate is zero? If a = .55, what would
r have to be in order that young creatures would want to consume the
same amount in each period? .
(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 N1 old creatures and each earns I units of cake, so this total is N1I.
Similarly, the total supply of cake in period 2 equals the total amount
earned by young creatures. This amount is .
(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 and the
demand for period-1 cake by each young creature is Since
there are N1 old creatures and N2 young creatures, the total demand for
period-1 cake at interest rate r is .
(e) Using the results of the last section, write an equation that sets the
demand for period-1 cake equal to the supply.
Write a general expression for the equilibrium value of r, given
N1, N2, I, and I∗
. Solve this equation for the special
case when N1 = N2 and I = I∗
and a = 11/21. .
(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.)
.