Due in 48 hours, all questions are in the first file. Please show calculations steps and graphs. All reading that related this assignment are the rest three files.
48 hours urgent
1)
a) Show an Edgeworth box for a general case of two agents who both have Cobb-Douglas preferences. (Make sure to label axis and curves.)
b) Briefly explain what the contract curve is.
2) For two agents, a and b, with the following utility functions over goods x and y
a) Determine the slope of the contract curve in the interior of an Edgeworth box that would show this two-person two-goods situation.
b) For initial endowments ωa=(12,7) and ωb=(9,10), what is the Walras allocation between the two agents a and b? (Remember that it is the relative price of the goods that matters in this consideration. Also remember that all the units of goods that exist here are will end up with one or the other agent; so, the overall 21 units of x and the 17 units of y will be fully allocated between the two. )
c) Calculate the utility level before and after trading.
3) For two agents, a and b, with the following utility functions over goods x and y
a) Determine the slope of the contract curve in the interior of an Edgeworth box that would show this two-person two-goods situation.
b) For initial endowments ωa=(9,8) and ωb=(6,7), what is the Walras allocation between the two agents a and b?
c) Calculate the utility level before and after trading.
4) For two agents, a and b, with the following utility functions over goods x and y
a) Determine the slope of the contract curve in the interior of an Edgeworth box that would show this two-person two-goods situation.
b) For initial endowments ωa=(4,5) and ωb=(3,2), what is the Walras allocation between the two agents a and b? (Remember that it is the relative price of the goods that matters in this consideration.)
c) Calculate the utility level before and after trading.
Exchange Economies 91
The length of an Edgeworth box shows the total supply of good 1, while
the height shows the total supply of good 2. Given the Edgeworth box and
the initial endowment, any exchange of goods between the consumers en-
tails a movement to another allocation inside the box. Starting from any
allocation inside the Edgeworth box — say, the center, C
=
((6, 5), (6, 5))
— to an allocation to its northeast makes consumer a better off and b worse
off because both consumers’ preferences are strictly monotonic. Conversely,
any allocation to the southwest of the box makes b better off and a worse off.
6.2 Properties of Allocations
Given the preferences of the individuals and the initial endowment, we can
now discuss properties of allocations. Some allocations may be more desir-
able than others. We explore two different notions of desirability.
6.2.1 Individually rational allocations
Individual rationality embodies the idea that if two people trade voluntar-
ily, that trade must leave each person at least as well off as before they trade;
if trade hurts either consumer, they will have no incentive to engage in such
an exchange of goods.
We define an allocation (xa, xb) to be individually rational if
ua(xa) ≥ ua(ωa) and ub(xb) ≥ ub(ωb), (6.3)
i.e., each person’s utility at her consumption bundle xi is at least as great as
her utility from her endowment ωi, where i = a, b. Thus, the movement from
the endowment bundle ωa to the bundle xa leaves consumer a no worse off
than initially, and similarly for consumer b.
In Figure 6.3, the individually rational allocations lie in the blue lens-
shaped area (labeled IR) between the indifference curves of each consumer
that pass through the initial endowment. For example, in moving from
ω
to A, both consumers are better off than initially because A lies on a higher
indifference curve for each consumer. At an allocation such as B, consumer
a remains on her initial indifference curve and so remains as well off, but
consumer b is on a higher indifference curve. You can verify this by drawing ¶b
b’s indifference curve through point B. At C, consumer b is as well off as
initially but a is better off.
/: C 8: B : 3 M =B:M 4B B . 9 E / BE=B .II : A 7 ME = M 2 D 0 M :
E
AMMI D M :E I M EB = = M:BE : MB -= 31,
0 :M = ? =
0
IP
B
AM
Q
7
ME
=
.
EE
B
AM
=
92 Chapter 6
x1
a
x2
b
x2
a
x1
b
Oa
Ob
I
R
A
B
C
E
F
D
ω
Figure 6.3 Individually rational allocations
Note that any allocation inside the Edgeworth box but outside of the IR
area places at least one consumer behind her indifference curve, signifying
that she is worse off than at ω. For example, at D, consumer a is worse off;
at E, b is worse off, and at F, both consumers are worse off. If we expect
the consumers to barter and trade with each other starting at ω, the only
allocations that they would agree to move to voluntarily must lie within the
IR area since neither is made worse off by such a move; indeed, it is quite
possible for one or even both of them to be better off.
Individually rational allocations inside the Edgeworth box can be found
by following the three steps summarized below.
1. Identify the initial endowment, ω, in the Edgeworth box.
2. Draw an indifference curve for consumer a that passes through ω, us-
ing arrows to show the direction in which her utility is increasing. Do
the same for consumer b.
3. The area between the indifference curve for consumer a and that for
consumer b (including the indifference curves themselves) is the set of
individually rational allocations.
/: C 8: B : 3 M =B:M 4B B . 9 E / BE=B .II : A 7 ME = M 2 D 0 M :E
AMMI D M :E I M EB = = M:BE : MB -= 31,
0 :M = ? =
0
IP
B
AM
Q
7
ME
=
.
EE
B
AM
=
Exchange Economies 93
6.2.2 Pareto efficient allocations
Pareto efficiency (or more traditionally, Pareto optimality) embodies the
idea of non-wastefulness in allocating the total supply of goods at our dis-
posal among consumers.3 Given an allocation, if it is possible to reallocate
the goods so as to make at least one person happier and no one worse off,
then the original allocation is wasteful in the sense that there is scope for im-
proving on it. At a Pareto efficient allocation, it is not possible to reallocate
the goods so as to make one consumer better off without hurting someone
else, so it is non-wasteful.
To illustrate this idea simply, suppose we have an apple and a banana to
allocate between two persons. Consumer a is indifferent between an apple
and a banana, but consumer b has an aversion to bananas and strictly prefers
apples over bananas. Then the allocation that gives a the apple and b the
banana is wasteful because it is possible to make at least one person better
off without hurting the other. Simply give the banana to consumer a and the
apple to b; then a is as well off, but b is better off. Giving the banana to a
and the apple to b is a Pareto efficient allocation because it is not possible to
reallocate the goods and make at least one person happier without hurting
the other.
Before we can define what a Pareto efficient allocation is formally, we
need another definition. Starting from an allocation (xa, xb), the allocation
(x̄a, x̄b) is said to be Pareto superior to (or a Pareto improvement over)
(xa, xb) if nobody is worse off at (x̄a, x̄b) and at least one person is better off.
In other words, if we started with the initial allocation (xa, xb) and moved
to (x̄a, x̄b), then that would constitute an improvement because nobody is
hurt and someone is happier. An allocation (x̂a, x̂b) is Pareto efficient if there
is no other allocation that is Pareto superior to (x̂a, x̂b). In other words, at a
Pareto efficient allocation, it is not possible to make at least one person hap-
pier without hurting anyone else — any reallocation of goods either hurts
somebody, or leaves everyone as well off.
Graphical representation
Typically an Edgeworth box will have many Pareto efficient allocations. These
Pareto efficient allocations can be found by following this algorithm.
3Pareto efficiency is named after Vilfredo Pareto, an influential economist and sociologist.
The phrase “non-wastefulness” was coined by Leonid Hurwicz.
/: C 8: B : 3 M =B:M 4B B . 9 E / BE=B .II : A 7 ME = M 2 D 0 M :E
AMMI D M :E I M EB = = M:BE : MB -= 31,
0 :M = ? =
0
IP
B
AM
Q
7
ME
=
.
EE
B
AM
=
94 Chapter 6
1. Fix the utility of one consumer, say individual b, at some arbitrary level
ūb inside the Edgeworth box.
2. Maximize the utility of consumer a while keeping b on the indifference
curve ūb. Then the allocation reached is a Pareto efficient allocation.
3. To find other Pareto efficient allocations, repeat the process by picking
a different utility level for b in step 1.
To find one Pareto efficient allocation and understand how this algorithm
works, arbitrarily fix b’s utility at ūb shown by the green ūb indifference curve
in Figure 6.4. Maximize a’s preferences while keeping b on her green indiffer-
ence curve, yielding the allocation A. Then A is a Pareto efficient allocation.
To check this, consider the different regions of the Edgeworth box where an
x1
a
x2
b
x2
a
x1
b
Oa
Ob
A
I
II
III
IV
ub
ua
Figure 6.4 A Pareto efficient allocation
alternative allocation could be picked. Any allocation in region I (which lies
to the southwest of the green indifference curve) makes consumer a worse
off. In regions II and III, both a and b are worse off as they are behind their
indifference curves ūa and ūb. In region IV (which lies to the northeast of the
orange indifference curve ūa), b is worse off. Therefore, beginning with A,
there is no Pareto superior allocation in the Edgeworth box, and hence A is
Pareto efficient.
/: C 8: B : 3 M =B:M 4B B . 9 E / BE=B .II : A 7 ME = M 2 D 0 M :E
AMMI D M :E I M EB = = M:BE : MB -= 31,
0 :M = ? =
0
IP
B
AM
Q
7
ME
=
.
EE
B
AM
=
Exchange Economies 95
Two remarks are in order. First, the fact that we fix the utility of b in step 1
is totally arbitrary. In other words, the same set of Pareto efficient allocations
can be found by reversing the roles of a and b, namely, fixing the utility of a
instead in step 1, and maximizing b’s utility while keeping a at this utility in
step 2.
Second, unlike individually rational allocations, Pareto efficient alloca-
tions do not depend on the initial endowment as a reference point. They
only depend on the consumers’ preference and the aggregate supplies of the
goods, Ω. In other words, given the consumers’ preferences and the dimen-
sions of the Edgeworth box, the set of Pareto efficient allocations would re-
main unchanged if the initial endowment were to be some other point inside
the Edgeworth box.
Algebraic derivation
The algorithm to find the Pareto efficient allocations graphically is tedious
since there are infinitely many utility levels that could be picked in the first
step. The alternative algebraic method presented here holds the promise of
finding many, if not all, the Pareto efficient allocations in the interior of the
Edgeworth box at once.
The algebraic derivation is motivated by Figure 6.4 which suggests that
at an interior Pareto efficient allocation, the tangency of the consumers’ in-
difference curves is a necessary condition, i.e., if (x̄a, x̄b) is Pareto efficient,
then MRSa(x̄a) = MRSb(x̄b). When preferences are strictly monotonic and
convex, the tangency of the indifference curves is also sufficient to guaran-
tee Pareto efficiency, i.e., if MRSa(x̄a) = MRSb(x̄b), then (x̄a, x̄b) is Pareto
efficient. Therefore, the tangency of the indifference curves is often a way to
find (interior) Pareto efficient allocations algebraically, or to verify whether
a given allocation in the interior of the Edgeworth box is Pareto efficient.
To find the interior Pareto efficient allocations algebraically for the econ-
omy in section 6.1, set the marginal rate of substitution for a equal to that for
b to obtain
MRSa = xa2/x
a
1 = MR
S
b = 2.
Then xa2 = 2x
a
1, which means that when the two consumers’ indifference
curves are tangent, person a consumes twice as much of good 2 as good 1.
Plot the equation xa2 = 2x
a
1 in Figure 6.5 beginning from O
a, joining interior
Pareto efficient allocations such as R and S where the consumers’ indiffer-
ence curves are tangent as shown.
/: C 8: B : 3 M =B:M 4B B . 9 E / BE=B .II : A 7 ME = M 2 D 0 M :E
AMMI D M :E I M EB = = M:BE : MB -= 31,
0 :M = ? =
0
IP
B
AM
Q
7
ME
=
.
EE
B
AM
=
96 Chapter 6
x1
a
x2
b
x2
a
x1
b
Oa
Ob
R
T
S
PE
Figure 6.5 The Pareto set or contract curve
However, there are other Pareto efficient allocations in addition to the
allocations that lie along the line xa2 = 2x
a
1. For instance, verify by inspec-b·
tion that a point like T = ((9, 10), (3, 0)) which is on the edge (and not
the interior) of the Edgeworth box is also Pareto efficient. Generally, the
tangency condition will not hold at Pareto efficient allocations along the
edges of the Edgeworth box. For instance, at T, MRSa(9, 10) = 0.9 while
MRSb(3, 0) = 2.4 The set of all Pareto efficient allocations (often called the
contract curve) for this economy is labeled PE.
When the contract curve consists of allocations in the interior of the Edge-
worth box, it is possible to find an equation for it by following these three
steps.
1. Set MRSa = MRSb.
2. From the supply constraints for the two goods, xa1 + x
b
1 = Ω1 and x
a
2 +
xb2 = Ω2, derive x
b
1 = Ω1 − x
a
1 and x
b
2 = Ω2 − x
a
2. Use these to eliminate
xb1 and x
b
2 in the equation from step 1.
3. Solve the equation from step 2 to write xa2 as a function of x
a
1. Then this
is the equation for the contract curve with Oa as the origin.
4In general, at a Pareto efficient allocation that lies on the left hand or top edge of the
Edgeworth box, it will be the case that MRSa ≤ MRSb; the inequality will be reversed for a
Pareto efficient allocation that lies on the right hand or bottom edge of the Edgeworth box.
/: C 8: B : 3 M =B:M 4B B . 9 E / BE=B .II : A 7 ME = M 2 D 0 M :E
AMMI D M :E I M EB = = M:BE : MB -= 31,
0 :M = ? =
0
IP
B
AM
Q
7
ME
=
.
EE
B
AM
=
Exchange Economies 97
To illustrate, suppose both consumers have Cobb-Douglas preferences,
where a’s utility function is ua(xa1, x
a
2)
=
x
a
1 x
a
2 while b’s utility is u
b(xb1, x
b
2) =
(xb1)
2 xb2. Suppose that there are 10 units of each good in this economy, i.e.,
Ω = (10, 10). Then from step 1, we get
xa2
xa1
=
2xb2
xb1
.
From step 2, xb1 = 10 − x
a
1 and x
b
2 = 10 − x
a
2. Substituting these into the
equation above and solving, we get the contract curve ¶b
xa2 =
20xa1
10 + xa1
,
where 0 ≤ xa1 ≤ 10.
Finally, to end this section on Pareto efficiency, note that in moving from
one Pareto efficient allocation to another, there will typically be a change in
the distribution of the goods that makes one person better off at the expense
of another. In other words, no Pareto efficient allocation can be Pareto supe-
rior to another Pareto efficient allocation. For example, the extreme situation
where consumer a gets the aggregate endowment (at the point Ob) or its
polar opposite where consumer b gets everything (at the point Oa) are both
Pareto efficient. Thus, the notion of Pareto efficiency is insensitive to distri-
butional concerns.
6.3 Walras Equilibrium
We will now consider the possibility of the two consumers trading goods 1
and 2 in markets at a per-unit price of p1 and p2. Even though there are only
two consumers for now, we will assume that each takes the market prices
as given and outside of their control.5 Given these prices, each consumer
decides how much she wishes to buy or sell of each good. The markets are
said to clear if the quantity demanded of good 1 by both consumers equals its
supply, and likewise for good 2. Then the question that Léon Walras asked
in the 1870s in the context of our Edgeworth box economy is: does there exist
a price pair (p̂1, p̂2) for which both markets clear? We explore this question
graphically to uncover the basic insights and then fill in the more technical
details.
5This assumption would of course be more plausible if there were a very large number of
consumers.
/: C 8: B : 3 M =B:M 4B B . 9 E / BE=B .II : A 7 ME = M 2 D 0 M :
E
AMMI D M :E I M EB = = M:BE : MB -= 31,
0 :M = ? =
0
IP
B
AM
Q
7
ME
=
.
EE
B
AM
=
98 Chapter 6
6.3.1 Graphical representation
We begin with a definition. A Walras equilibrium (or a competitive equilib-
rium) consists of prices (p̂1, p̂2) and an allocation (x̂a, x̂b) = ((x̂a1, x̂
a
2), (x̂
b
1, x̂
b
2))
such that:
(a) the consumption bundle x̂a maximizes ua subject to the budget con-
straint p̂1 xa1 + p̂2 x
a
2 ≤ p̂1
ω
a
1 + p̂2ω
a
2;
(b) the consumption bundle x̂b maximizes ub subject to the budget con-
straint p̂1 xb1 + p̂2 x
b
2 ≤ p̂1ω
b
1 + p̂2ω
b
2; and
(c) the markets for goods 1 and 2 clear:
x̂a1 + x̂
b
1 = ω
a
1 + ω
b
1 and x̂
a
2 + x̂
b
2 = ω
a
2 + ω
b
2.
Therefore a Walras equilibrium is a pair of prices and a pair of consumption
bundles at which each consumer maximizes her utility given her budget con-
straint, and the total demand for each good equals its supply.
Note that the right hand side of consumer i’s budget constraint in (a) and
(b) above represent her income which is merely the value of i’s endowment
at the equilibrium prices, i.e.,
m̂i = p̂1ωi1 + p̂2ω
i
2.
Therefore (a) and (b) are an alternative way of saying that x̂i is the bundle
demanded by consumer i when the prices are the equilibrium ones and her
income is m̂i:
x̂i = hi(p̂1, p̂2, m̂i).
Before we see what happens in equilibrium, consider an arbitrary pair of
prices (p̄1, p̄2) set by a mythical Walrasian auctioneer whose job is to find
the equilibrium prices. In Figure 6.6, the blue budget line with slope −p̄1/ p̄2
is shown passing through the initial endowment, ω. Viewed from origin Oa,
this is the endowment budget6 for consumer a, while the same line is the
endowment budget for consumer b when viewed from origin Ob. Note that
the slope of this budget line is −p̄1/ p̄2 irrespective of whether you view it
using Oa as your origin, or whether you turn the page upside down and
view it with Ob as your origin.
6See section 2.3.1 and Figure 2.5.
/: C 8: B : 3 M =B:M 4B B . 9 E / BE=B .II : A 7 ME = M 2 D 0 M :E
AMMI D M :E I M EB = = M:BE : MB -= 31,
0 :M = ? =
0
IP
B
AM
Q
7
ME
=
.
EE
B
AM
=
Exchange Economies 99
x1
a
x2
b
x2
a
x1
b
Oa
Ob
S1
a
D2
a
D1
b
A
B
p2
–
p1
ω
S2
b
Figure 6.6 Demand and supply at (p̄1, p̄2)
Given this budget, consumer a demands the bundle at point A. In other
words, starting from ω, she is willing to supply Sa1 units of good 1 (shown
by the solid magenta arrow) in exchange for Da2 units of good 2 (shown by
the dashed magenta arrow) to move to the bundle at A. Likewise, consumer
b would like to move from ω to point B, supplying Sb2 units of good 2 in
exchange for Db1 units of good 1. But the market for good 1 does not clear
at these prices: consumer a would like to supply Sa1 units but consumer b
demands more, Db1. Similarly, the market for good 2 does not clear either as
the demand for good 2, Da2, is less than its supply, S
b
2.
Assume now that the Walrasian auctioneer raises p1 which makes con-
sumer a wish to supply more and consumer b to demand less of good 1,
and/or lowers p2 which makes consumer a demand more of good 2 and
consumer b supply less of it. In other words, beginning with the initial dot-
ted blue budget line in Figure 6.7, the auctioneer can raise the relative price
ratio, p1/ p2, to find a set of prices (p̂1, p̂2) shown by the steeper, solid blue
budget line. Note that this new budget pivots around the endowment ω
as the relative price ratio increases, and equates Sa1 = D
b
1 for good 1, and
Sb2 = D
a
2 for good 2. Then, (p̂1, p̂2) are the Walras prices, the prices at which
the consumers attain the Walras allocation, E = (x̂a, x̂b), where each per-
son is maximizing her utility given her budget (at the Walras prices) and the
demand for each good equals its supply.
/: C 8: B : 3 M =B:M 4B B . 9 E / BE=B .II : A 7 ME = M 2 D 0 M :E
AMMI D M :E I M EB = = M:BE : MB -= 31,
0 :M = ? =
0
IP
B
AM
Q
7
ME
=
.
EE
B
AM
=
100 Chapter 6
x1
a
x2
b
x2
a
x1
b
Oa
Ob
S1
a
D2
a
D1
b
S2
b
E
—
p1
p2
ˆ
ˆ
ω
Figure 6.7 Walras equilibrium
There are three insights regarding Walras equilibria that can be gleaned
from Figure 6.7:
(1) whenever the market for one good is in equilibrium, the other must
also be in equilibrium;
(2) what matters for bringing about equilibrium is the relative price ratio,
not the absolute price levels; and
(3) the Walras allocation is both individually rational and Pareto efficient.
Insight (1) follows from the fact that in moving from the initial endow-
ment ω to the Walras allocation E in Figure 6.7, the quantities that each
consumer wants to buy and sell are opposite sides of a rectangle (shown
with the solid and dashed magenta arrows). It is not possible, for example,
for the market for good 1 to clear but not that of good 2. Mathematically,
this follows from Walras’ Law7 which states that the value of everyone’s
consumption expenditures must always add up to the value of the aggre-
gate endowment. A consequence of Walras’ Law is that if there are ℓ goods
with prices p̂1, p̂2, . . . , p̂ℓ so that every market but one is in equilibrium, then
that remaining market must also be in equilibrium. Since here there are two
goods (ℓ = 2), this corollary to Walras’ Law guarantees that finding prices
7Section 6.5.1 below presents a formal statement and proof.
/: C 8: B : 3 M =B:M 4B B . 9 E / BE=B .II : A 7 ME = M 2 D 0 M :E
AMMI D M :E I M EB = = M:BE : MB -= 31,
0 :M = ? =
0
IP
B
AM
Q
7
ME
=
.
EE
B
AM
=
Exchange Economies 101
to bring about equilibrium in one market ensures that the other market is
automatically in equilibrium.
Insight (2) follows from the fact that in going from the initial prices of
(p̄1, p̄2) to the Walras equilibrium prices of (p̂1, p̂2), what equilibrates the
two markets is the steeper slope of the latter budget. If the slope of the bud-
get at the Walras prices is −2 for example, there are infinitely many price
combinations that give rise to this slope. Therefore, the absolute levels of the
prices is indeterminate at a Walras equilibrium. To peg the level of the Wal-
ras prices, we normalize the price of one good to $1; this good is then called
the numéraire good and the prices of all other goods are measured in terms
of this numéraire. For instance, if a pack of chewing gum is the numéraire,
then the price of a shirt worth $30 would be priced at 30 packs of gum —
packs of gum are the unit of account.
Finally, regarding insight (3), individual rationality holds since each con-
sumer is on a higher indifference curve at E as compared to ω. Indeed, since
trade is voluntary, neither consumer would wish to move to the Walras allo-
cation from ω unless they are at least as well off as initially. Pareto efficiency
of the Walras allocation follows from the tangency of the consumers’ indif-
ference curves at E. This result, known as the First Welfare Theorem, is one
of the key insights of microeconomic theory and is a precise modern restate-
ment of the idea attributed to Adam Smith that the greatest social good arises
when individuals follow their self-interest in free markets.
6.3.2 Algebraic derivation
Consider a two-person economy where the utilities are Cobb-Douglas and
given by
ua = xa1 x
a
2 and u
b = (xb1)
2 xb2
and endowments are
ωa = (6, 4) and ωb = (2, 8).
Then the demand functions for each consumer (using the formulas in equa-
tion (4.11)) are
ha(p1, p2, ma) =
!
ma
2p1
,
ma
2p2
”
and hb(p1, p2, mb) =
!
2
mb
3p1
,
mb
3p2
”
,
where ma = 6p1 + 4p2 and mb = 2p1 + 8p2 are the values of each consumer’s
endowment.
/: C 8: B : 3 M =B:M 4B B . 9 E / BE=B .II : A 7 ME = M 2 D 0 M :E
AMMI D M :E I M EB = = M:BE : MB -= 31,
0 :M = ? =
0
IP
B
AM
Q
7
ME
=
.
EE
B
AM
=
Chapter 6
Exchange Economies
One of the significant advances in economic theory in the 20th century has
been the development of general equilibrium analysis which explores the
possibility of simultaneous equilibrium in multiple markets, as opposed to
the older partial equilibrium analysis of Alfred Marshall which studies the
possibility of equilibrium in a single market in isolation. Today, much of
modern macroeconomic theory is developed in a general equilibrium frame-
work. In this chapter, we take up the simplest possible general equilibrium
model with two consumers and two goods. Because there is no production,
the consumers may only choose to trade the available supplies of the goods;
ergo, such an economic environment is called a pure exchange economy.
6.1 The Edgeworth Box
Suppose there are only two consumers, a and b, and two goods, 1 and 2. We
will use the superscript i to refer to either individual, and the subscript j to
refer to either good. Each consumer i has a characteristic ei which consists
of two pieces of information specific to her, namely, her preferences and her
individual endowment. Her preferences are represented by a utility func-
tion, ui, over the two goods; her individual endowment, ωi, is a commodity
bundle which shows the total amounts of the two goods that she possesses
initially, i.e., ωi
=
(ωi1,
ω
i
2). Then i’s characteristic is written as
ei = (ui, ωi) (6.1)
8
8
/: C 8: B : 3 M =B:M 4B B . 9 E / BE=B .II : A 7 ME = M 2 D 0 M :E
AMMI D M :E I M EB = = M:BE : MB -= 31,
0 :M = ? =
0
IP
B
AM
Q
7
ME
=
.
EE
B
AM
=
Exchange Economies 89
which summarizes all the relevant information about this consumer. Finally,
an economy, e, is a list of the characteristics of all consumers:
e = (ea, eb) = ((ua, ωa), (ub, ωb)). (6.2)
This economy e is our prototype of a two-person private goods pure ex-
change economy.1
5
2
8
7
Oa
x2
a
x1
a
Ob
x2
b
x1
b
ωa
ωb
Figure 6.1 Characteristics of consumers a and b
To make things more concrete, suppose consumer a’s characteristic ea is
given by a Cobb-Douglas utility ua = xa1 x
a
2 and an endowment ω
a = (5, 2),
while eb is given by a linear utility ub = 2xb1 + x
b
2 and ω
b = (7, 8). The
left panel of Figure 6.1 shows consumer a’s origin, Oa, a couple of her or-
ange indifference curves and her endowment bundle, ωa. The right panel
of Figure 6.1 shows b’s origin, Ob, a couple of her linear green indifference
curves and her endowment bundle, ωb. By adding the endowment of each
consumer, we obtain the aggregate endowment, Ω (read as ‘capital omega’),
which shows the total supply of all goods in the economy:
Ω = ωa + ωb = (5, 2) + (7, 8) = (12, 10).
Any list of consumption bundles (xa, xb) for the two consumers is called
an allocation. Suppose the total supplies of both goods are divided between
1A good is said to be private if one person’s consumption of a good precludes it being
consumed by someone else, and if others can be excluded from consuming it. See Chapter 16
for more details.
/: C 8: B : 3 M =B:M 4B B . 9 E / BE=B .II : A 7 ME = M 2 D 0 M :E
AMMI D M :E I M EB = = M:BE : MB -= 31,
0 :M = ? =
0
IP
B
AM
Q
7
ME
=
.
EE
B
AM
=
90 Chapter 6
x1
a
x2
b
x2
a
x1
b
Oa
Ob
5
7
8
2
ω
C
Figure 6.2 The Edgeworth box
the two consumers so that a receives the bundle x̄a = (4, 7) while b receives
the remainder, x̄b = (8, 3). Then we say that the pair of consumption bun-
dles (x̄a, x̄b) = ((4, 7), (8, 3)) is a feasible allocation, meaning that this al-
location is actually possible given the total supply of the goods. In fact any
pair (xa, xb) is a feasible allocation so long as xa + xb ≤ Ω.
In order to better understand allocations, take the right panel of Figure
6.1, rotate it counterclockwise by 180◦, and place it over the left panel so that
the bundles ωa and ωb coincide as shown by the point ω in Figure 6.2. The
rectangle contained between the origins Oa and Ob is known as an Edge-
worth box named after Francis Edgeworth.2
Any point inside this box represents a feasible allocation, where the con-
sumption bundle for individual a is read from her origin, Oa, while that of
b is read (upside down!) from the perspective of b’s origin, Ob. For exam-
ple, the point ω = (ωa, ωb) is the allocation ((5, 2), (7, 8)). This is called the
initial endowment for this Edgeworth box economy; it shows the consump-
tion bundle each person starts out with before any trade takes place. The
allocation corresponding to Ob is ((12, 10), (0, 0)) where individual a gets
everything while b gets nothing. Conversely, the allocation corresponding to
Oa is ((0, 0), (12, 10)).
2It is also known as an Edgeworth-Bowley box, after the English statistician and economist
Arthur Bowley who popularized it.
/: C 8: B : 3 M =B:M 4B B . 9 E / BE=B .II : A 7 ME = M 2 D 0 M :E
AMMI D M :E I M EB = = M:BE : MB -= 31,
0 :M = ? =
0
IP
B
AM
Q
7
ME
=
.
EE
B
AM
=