1. Chapter 6 Exchange Economies
Intermediate Microeconomics:
A Tool-Building Approach
Routledge, UK
© 2016 Samiran Banerjee

2. Partial vs. general equilibrium
Partial equilibrium analysis
Looking at one market in isolation, is there a price so that this market is in equilibrium?
Answer: See Chapter 1
General equilibrium analysis
Looking at all markets, are there prices so that each market is in equilibrium simultaneously?
Since markets are interrelated, it is not obvious if this is even possible!

3. Simplest general equilibrium model
• Two consumers, a and b
• Two goods, x1 and x2
• No production
• Consumers exchange what they bring to the market
Two-person pure exchange economy
• Consumer a’s characteristic: ea = (ua, ωa)
• Consumer b’s characteristic: eb = (ub, ωb)
• The economy: e = (ea, eb)
Utility function
and
endowment
List of consumers’ characteristics

4. Pure exchange economy Consumer a • ωa = (5, 2) • ua = x1ax2a
Consumer b
• ωb = (7, 8)
• ub = 2x1b + x2b
a’s origin
b’s origin

5. Pure exchange economy • Rotate b’s graph counterclockwise
• Slide to the left until ωa and ωb coincide

6. Length and height of the Edgeworth box
Pure exchange economy
• Every point in the Edgeworth box is a feasible allocation
• The point ω is the initial endowment of this economy
• ω = (ωa, ωb) = ((5, 2), (7, 8))
• Aggregate endowment, Ω = (12, 10)
Length and height of the Edgeworth box
b’s origin
Edgeworth box
a’s origin

7. Individually rational allocations
• An allocation (xa, xb) is individually rational means everyone is at least as well off as at their endowment:
ua(xa) ≥ ua(ωa) and ub(xb) ≥ ub(ωb)
• Draw a’s indifference curve through ω
• Draw b’s indifference curve through ω
• The lens-shaped area, IR, is the set of individually rational allocations

8. Pareto superior allocations
• An allocation S is Pareto superior to (or a Pareto improvement over) R means everyone at S is at least as well off as at R and at least one person is better off.
Both a and b are better off at S
a is better off at S’,
b is better off at S”

9. Pareto efficient allocations
• An allocation E is Pareto efficient if no other allocation is Pareto superior.
MRSa = MRSb at an interior Pareto efficient allocation
• Fix b’s indifference curve arbitrarily
• Maximize a’s utility given b’s indifference curve
• Point E is Pareto efficient
• Check zones I–IV for any
Pareto superior allocations

10. Derive the Pareto set graphically
• Fix one individual’s indifference curve, say, for a
• Maximize b’s utility keeping a on this indifference curve
• Pick a higher indifference curve for a
• Maximize b’s utility again
• Pick an even higher indifference curve for a
• Maximize b’s utility again (a corner solution at T)
• Join all these Pareto efficient points, PE
• The set of Pareto efficient allocations is called the contract curve

11. Derive the Pareto set algebraically
Interior
Derive the Pareto set algebraically
• Consumer a: ωa = (10, 20), ua = x1a (x2a)2
• Consumer b: ωb = (50, 10), ub = x1b (x2b)2
• Set MRSa = MRSb to obtain x2a/x1a = x2b/x1b
• Use the supply constraints
x1a + x1b = 60 and
x2a + x2b = 30
to get rid of all b terms:
x2a/x1a = (30 – x2a)/(60 – x1a)
• Solve to obtain the contract curve: x2a = 0.5x1a

12. Walras equilibrium concept
• Market mechanism
• Walras prices: (p1, p2)
• Walras allocation: (xa, xb)
Given the Walras prices, ˆ
a and b maximize their utilities ˆ
1. xa maximizes a’s utility given the budget constraint
p1xa1 + p2xa2 ≤ p1ωa1 + p2ωa2
2. xb maximizes b’s utility given the budget constraint
p1xb1 + p2xb2 ≤ p1ωb1 + p2ωb2
3. Demand equals supply for each good:
xa1 + xb1 = ωa1 + ωb1
xa2 + xb2 = ωa2 + ωb2 ˆ ˆ ˆ
The markets for goods 1 and 2 clear ˆ

13. Walras (dis)equilibrium–
• Fix prices arbitrarily at (p1, p2)
• Draw a’s budget line through ωa
• Maximize a’s utility at A
• S1a = supply of 1 by a
• D2a = demand for 2 by a
• Maximize b’s utility at B
• S2b = supply of 2 by b
• D1b = demand for 1 by b
• S1a < D1b implies p1 increases
• D2a < S2b implies p2 decreases
p1/p2 increases!

14. Both markets clear at new prices!
Walras equilibrium
• Change old prices
• Since new prices are (p1, p2), where p1/p2 > p1/p2, therefore
new budget pivots around ω
• Maximize a’s utility at E
• S1a = supply of 1 by a
• D2a = demand for 2 by a
• Maximize b’s utility at E
• S2b = supply of 2 by b
• D1b = demand for 1 by b
• S1a = D1b
• D2a = S2b ˆ ˆ –
Both markets clear at new prices!

15. Features of Walras equilibrium
First Welfare Theorem:
The Walras allocation is IR and PE (in the absence of externalities, asymmetric information, and public goods)
• Walras prices: (p1, p2)
• Walras allocation: E
• E is individually rational: it is preferred to ω by both
• E is Pareto efficient:
MRSa = MRSb
• (Walras’ Law corollary) If the market for good 1 clears, the market for good 2 clears automatically (and vice versa)
• The Walras equilibrium depends on relative prices,
p1/p2 ˆ ˆ

16. Deriving a Walras equilibrium
Consumer a
• ωa = (6, 4)
• ua = x1ax2a
• h1a = ma/(2p1)
• h2a = ma/(2p2)
Consumer b
• ωb = (2, 8)
• ub = (x1b)2 x2b
• h1b = 2mb/(3p1)
• h2b = mb/(3p2)
Normalize one price, say, p2 = 1. Then
ma = 6p1 + 4 and mb = 2p1 + 8
2. Select any one good for market clearing, say, good 1
3. Set total demand equal to total supply for good 1:
h1a + h1b = 6 + 2
Solve to obtain p1 = 2
4. Substitute prices in demands to find the Walras allocation: (xa, xb) = ((4, 8), (4, 4)) ˆ ˆ ˆ

17. Two persons, three goods
1. Normalize one price, say, p3 = 1.
2. Select any two goods for market clearing, say, 1 and 2
3. Set total demand equal to total supply for goods 1 and 2,
obtaining two equations in two unknowns, p1 and p2
4. Solve simultaneously to obtain p1 and p2
5. Substitute prices in demands to find the Walras allocation
At any interior Pareto efficient allocation, we require
MRS12a = MRS12b
MRS13a = MRS13b
MRS23a = MRS23b