Economics

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)

88

/: 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

7Oa

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

=