Economics

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

IR

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 = MRS

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

=