Economics

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 ≤ p̂1ω

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) =

! 2mb

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

=