Chapter 10

General
Equilibrium and
Economic
Welfare
 Topics

1. General Equilibrium.

2. Trading Between Two People.

3. Competitive Exchange.

4. Production and Trading.

5. Efficiency and Equity.

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-2

 Pareto Principle

• A value criterion used in evaluating welfare

in general equilibrium.
• Used to rank different allocations of goods
and services for which no interpersonal
comparisons need to be made.
• Pareto efficient - describing an allocation
of goods or services such that any
reallocation harms at least one person.

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-3

 Market Equilibrium

• Partial-equilibrium analysis - an
examination of equilibrium and changes in
equilibrium in one market in isolation.
• General-equilibrium analysis - the study
of how equilibrium is determined in all
markets simultaneously.

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-4

 Solved Problem 10.1

• The coffee and tea demand functions depend on

both prices. Suppose the demand curves for coffee
and tea are Qc = 120 – 2pc + pt and Qt = 90 – 2pt
+ pc, where Qc is the quantity of coffee, Qt is the
quantity of tea, pc is the price of coffee, and pt is
the price of tea. These crops are grown in separate
parts of the world, so their supply curves are not
interrelated. We assume that the short-run,
inelastic supply curves for coffee and tea are Qc =
45 and Qt = 30. Solve for the equilibrium prices and
quantities. Now suppose that a freeze shifts the
short-run supply curve of coffee to Qc = 30. How
does the freeze affect the prices and quantities?
Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-5
 Solved Problem 10.1: Answer

• Equate the quantity demanded and supplied

for both markets.
• Substitute the expression for pt from the
coffee equation into the tea equation and
solve for the price of coffee, then use that
result to obtain pt
• Repeat the analysis for Qc = 30.

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-6

 Trading Between Two People:
Scenario
• Jane and Denise live near each other in the
wilds of Massachusetts when a snowstorm
strikes, isolating them from the rest of the
world.
– They must either trade with each other or
consume only what they have at hand.

• Collectively, they have 50 cords of firewood

and 80 bars of candy and no way of
producing more of either good.

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-7

 Trading Between Two People:
Endowments

• Endowment - an initial allocation of goods.

• Jane’s endowment is 30 cords of firewood
and 20 candy bars.
• Denise’s endowment is 20 (= 50 − 30)
cords of firewood and 60 (= 80 − 20) candy
bars.
– So Jane has relatively more wood, and Denise
has relatively more candy.

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-8

 Figure 10.3(a) Endowments in
an Edgeworth Box

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-9

 Figure 10.3(b) Endowments in
an Edgeworth Box

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-10

 Figure 10.3(c) Endowments in
an Edgeworth Box

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-11

 Mutually Beneficial Trades

• Four assumptions about their tastes and

behavior:
– Utility maximization
– Usual-shaped (imperfect substitutes)
indifference curves
– Nonsatiation (unsatisfied)
– No interdependence

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-12

 Figure 10.4 Contract Curve

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-13

 Mutually Beneficial Trades

• We can make four equivalent statements

about allocation f:
1. The indifference curves of the two parties
are tangent at f.
2. The parties’ marginal rates of substitution
are equal at f.
3. No further mutually beneficial trades are
possible at f.
4. The allocation at f is Pareto efficient: One
party cannot be made better off without
harming the other.
Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-14
 Contract Curve

• Contract curve - the set of all Pareto-

efficient bundles.
• Only at these bundles are the parties
unwilling to engage in further trades or
contracts – these allocations are the final
contracts.
• A move from any bundle on the contract
curve would harm at least one person.

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-15

 Bargaining Ability

• Where on the contract curve between points

b and c will Jane and Denise end up?
• It depends on their respective bargaining
abilities.

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-16

 Solved Problem 10.3

• Are allocations a and g in Figure 10.4 part of

the contract curve?

• Answer:
– By showing that no mutually beneficial trades
are possible at those points, demonstrate that
those bundles are Pareto efficient.

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-17

 Competitive Exchange

• Two desirable properties:

– The competitive equilibrium is efficient.
•First Theorem of Welfare Economics
– Any efficient allocations can be achieved by
competition.
•Second Theorem of Welfare Economics

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-18

 Competitive Equilibrium

• If there were a large number of people with

tastes and endowments like Jane’s and a
large number of people with tastes and
endowments like Denise’s, each person
would be a price taker in the two goods.
• In a competitive market, prices adjust until
the quantity supplied equals the quantity
demanded.

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-19

 Figure 10.5(a)
Competitive Equilibrium

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-20

 Figure 10.5(b)
Competitive Equilibrium
(b) Prices That Do Not Lead to a Competitive Equilibrium
Denise’s candy
80 60 43 0d
50

Denise’s wood
45

I d1
30 20
I d2 e
j
22
d
32
I 2j
Jane’s wood

I 1j

a Price line
50
0j 20 30 60 80
Jane’s candy

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-21

 The Efficiency of Competition

• In a competitive equilibrium:
pc
MRS j    MRS d
pw
– Thus, we have demonstrated the First
Theorem of Welfare Economics:

Any competitive equilibrium is Pareto efficient.

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-22

 Obtaining Any Efficient
Allocation Using Competition

• Any Pareto-efficient bundle x can be

obtained as a competitive equilibrium if the
initial endowment is x.
• That allocation can also be obtained as a
competitive equilibrium if the endowment
lies on a price line through x, where the
slope of the price line equals the marginal
rate of substitution of the indifference
curves that are tangent at x.

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-23

 Obtaining Any Efficient Allocation
Using Competition (cont.)

• We’ve demonstrated the Second Theorem of

Welfare Economics:

Any Pareto-efficient equilibrium can be

obtained by competition, given an
appropriate endowment.

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-24

 Solve Problem:

• 𝑈𝐴 = 𝑋𝐴 𝑌𝐴 , 𝑈𝐵 = 𝑋𝐵 𝑌𝐵
• The initial allocation of Endowments is:
𝑋𝐴 = 90 , 𝑌𝐴 = 35
𝑋𝐵 = 30 , 𝑌𝐵 = 25
Find the efficient allocation between A & B.

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-25

 Answer
𝑃𝑥
• 𝑌𝐴 = 𝑋𝐴 , and
• For person A: 𝑃𝑦
𝑃𝑦
• 𝑀𝑎𝑥 𝑈𝐴 = 𝑋𝐴 𝑌𝐴 • 𝑋𝐴 = 𝑌𝐴
𝑃𝑥
• 𝑆𝑡: 𝑀𝐴 = 𝑃𝑋 𝑋𝐴 + 𝑃𝑌 𝑌𝐴 1) Sub. 𝑌𝐴 in 𝑀𝐴 to get
𝑀𝑈𝑥 𝑌𝐴
• 𝑀𝑅𝑆𝐴 = − =− 𝑋𝐴∗
𝑀𝑈𝑦 𝑋𝐴
𝑃𝑥
2) Then Sub. 𝑋𝐴 in 𝑀𝐴
• 𝑀𝑅𝑇𝐴 = −
𝑃𝑦 to get 𝑌𝐴∗
• to max. utility, ∗ 𝑀𝐴
3)From (1), 𝑋𝐴 =
2𝑃 𝑋
• 𝑀𝑅𝑆𝐴 = 𝑀𝑅𝑇𝐴
∗ 𝑀𝐴
𝑌𝐴 𝑃𝑥 4)From(2), 𝑌𝐴 =
• = 2𝑃
𝑌
𝑋𝐴 𝑃𝑦

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-26

 Answer …

• By given an • ∗ 𝑃𝑋 90+𝑃𝑌 35
𝑋𝐴 =
2𝑃
appropriate 𝑋
𝑃𝑦
Endowments of • 𝑋𝐴∗ = 45 + 17.5 …(5)
𝑃𝑥
person A, his own
∗ 𝑃𝑋 90+𝑃𝑌 35
income is: • 𝑌𝐴 =
2𝑃 𝑌
• 𝑀𝐴 = 𝑃𝑋 90 + 𝑃𝑌 35 • 𝑌𝐴∗ = 45
𝑃𝑥
+ 17.5…(6)
𝑃𝑦
• Sub. 𝑀𝐴 in (3) and
(4) to get the • Equations 5 and 6
demand equations of called the demand
person A. equations of person
A.
Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-27
 Answer …

𝑃𝑥
• For person B: • 𝑌𝐵 = 𝑋𝐵 , and
𝑃𝑦
• 𝑀𝑎𝑥 𝑈𝐵 = 𝑋𝐵 𝑌𝐵 𝑃𝑦
• 𝑆𝑡: 𝑀𝐵 = 𝑃𝑋 𝑋𝐵 + 𝑃𝑌 𝑌𝐵 • 𝑋𝐵 = 𝑌𝐵
𝑃𝑥
𝑀𝑈𝑥 𝑌𝐵 1) Sub. 𝑌𝐵 in 𝑀𝐵 to get
• 𝑀𝑅𝑆𝐵 = − =−
𝑀𝑈𝑦 𝑋𝐵 𝑋𝐵∗
𝑃𝑥
• 𝑀𝑅𝑇𝐵 = − 2) Then Sub. 𝑋𝐵 in 𝑀𝐵
𝑃𝑦
to get 𝑌𝐵∗
• to max. utility,
∗ 𝑀𝐵
• 𝑀𝑅𝑆𝐵 = 𝑀𝑅𝑇𝐵 3)From (1), 𝑋𝐵 =
2𝑃 𝑋
𝑌𝐵 𝑃𝑥 ∗ 𝑀𝐵
• = 4)From(2), 𝑌𝐵 =
𝑋𝐵 𝑃𝑦 2𝑃
𝑌
Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-28
 Answer …

• By given an • ∗ 𝑃𝑋 30+𝑃𝑌 25
𝑋𝐵 =
2𝑃
appropriate 𝑋
𝑃𝑦
Endowments of • 𝑋𝐵∗ = 15 + 12.5 …(5)
𝑃𝑥
person B, his own
∗ 𝑃𝑋 30+𝑃𝑌 25
income is: • 𝑌𝐵 =
2𝑃 𝑌
• 𝑀𝐵 = 𝑃𝑋 30 + 𝑃𝑌 25 • 𝑌𝐵∗ = 15
𝑃𝑥
+ 12.5…(6)
𝑃𝑦
• Sub. 𝑀𝐵 in (3) and
(4) to get the • Equations 5 and 6
demand equations of called the demand
person B. equations of person
B.
Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-29
 Answer …
• To find the optimal
allocation of goods
• Now, the demand between persons,
equations for both we have to calculate
A&B are: the relative prices
𝑃𝑋
𝑃𝑦 𝑃𝑌
• 𝑋𝐴∗ = 45 + 17.5 from the given
𝑃𝑥
𝑃𝑥 endowments and
• 𝑌𝐴∗ = 45 + 17.5
𝑃𝑦 demand equations:
𝑃𝑦
• 𝑋𝐵∗ = 15 + 12.5 • 𝑋𝐴∗ + 𝑋𝐵∗ = 90 + 30 =
𝑃𝑥
𝑃𝑥
120 or
• 𝑌𝐵∗ = 15 + 12.5 • 𝑌𝐴∗ + 𝑌𝐵∗ = 35 + 25 = 60
𝑃𝑦

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-30

• 𝑋𝐴∗ + 𝑋𝐵∗ = 120
𝑃𝑦 𝑃𝑦
• 45 + 17.5 + 15 + 12.5 = 120
𝑃𝑥 𝑃𝑥
𝑃𝑦 𝑃𝑦 𝑃𝑥 1
• 30 = 60 → = 2 𝑎𝑛𝑑 = 𝑃𝑦 = 0.5
𝑃𝑥 𝑃𝑥 𝑃𝑦
𝑃𝑥

• Or
• 𝑌𝐴∗ + 𝑌𝐵∗ = 60
𝑃𝑥 𝑃𝑥
• 45 + 17.5+15 + 12.5= 60
𝑃𝑦 𝑃𝑦
𝑃𝑋 𝑃𝑥 30
• 60 = 30 → = = 0.5
𝑃𝑌 𝑃𝑦 60
•
Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-31
 Production and Trading

• Scenario: Jane and Denise can produce

candy or chop firewood using their own
labor. They differ, however, in how much of
each good they produce from a day’s work.

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-32

 Production Possibility Frontier

• Jane can produce either 3 candy bars or 6

cords of firewood in a day.
• Denise can produce up to 3 cords of wood or
6 candy bars in a day.

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-33

 Production Possibility Frontier
(cont.)

• Production Possibility Frontier - shows the

maximum combinations of two goods that
can be produced from a given amount of
input.
–The slope of the production
possibility frontier is the marginal
rate of transformation (MRT).

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-34

 Figure 10.6 Comparative Advantage
and Production Possibility Frontiers

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-35

 Production Possibility Frontier
(cont.)

• Comparative advantage - the ability to

produce a good at a lower opportunity cost
than someone else.
• Because of the difference in their marginal
rates of transformation, Jane and Denise
can benefit from a trade.

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-36

 Solved Problem 10.4

• How does the joint production possibility

frontier in panel c of Figure 10.6 change if
Jane and Denise can also trade with Harvey,
who can produce 5 cords of wood, 5 candy
bars, or any linear combination of wood and
candy in a day?

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-37

 Solved Problem 10.4: Answer

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-38

 The Number of Producers

• With many producers the PPF is a smooth

concave curve.
• Because the PPF is concave, the marginal
rate of transformation decreases (in
absolute value) as we move up the PPF.
• Also,
MCc
MRT  
MC w
– where MCc and MCw are the marginal costs of
producing candy and wood respectively.

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-39

 Figure 10.7 Optimal Product Mix

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-40

 Efficient Product Mix

• If a single person were to decide on the

product mix, that person would pick the
allocation of wood and candy along the PPF
that maximized his or her utility.
– For each consumer:
MRS = MRT,
– if the economy is to produce the optimal mix of
goods for each consumer.

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-41

 Competition

• Each consumer picks a bundle of goods so,

Pc
MRS  
Pw
• Consumption efficiency - we can’t redistribute
goods among consumers to make one
consumer better off without harming another
one.
• The competitive equilibrium lies on the
contract curve.
Copyright © 2012 Pearson Addison-Wesley. All rights reserved.

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-42

 Competition (cont.)

• If candy and wood are sold by competitive

firms,
pc = MCc
pw = MCw
– Therefore,
Pc MCc

Pw MC w
Pc
MRT  
Pw
Pc
MRT    MRS
10 - 43 Pw
Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-43
 Competition (cont.)

• Since,
Pc
MRT    MRS
Pw

• A competitive equilibrium achieves an:

– efficient product mix - the rate at which firms
can transform one good into another equals the
rate at which consumers are willing to substitute
between the goods, as reflected by their
willingness to pay for the two goods.

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-44

 Figure 10.8 Competitive
Equilibrium

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-45

 Efficiency and Equity

• How well various members of society live

depends on how society deals with efficiency
(the size of the pie) and equity (how the pie
is divided).
• The actual outcome depends on choices by
individuals and on government actions.

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-46

 Role of the Government

• By altering the efficiency with which goods

are produced and distributed and the
endowment of resources, governments help
determine how much is produced and how
goods are allocated.

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-47

 Efficiency

• The Pareto criterion ranks allocation x over

allocation y if some people are better off at x
and no one else is harmed.
– If that condition is met, we say that x is Pareto
superior to y.
• Any policy change that leads to a Pareto-
superior allocation must increase W
(welfare).
– However, some policy changes that increase W
are not Pareto superior: There are both winners
and losers.

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-48

 Equity

• Social welfare function - combines various

consumers’ utilities to provide a collective
ranking of allocations.
– Sort of like a utility function for society.
• Utility possibility frontier (UPF): the set of
utility levels corresponding to the Pareto
efficient allocations along the contract
curve.

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-49

 Figure 10.9 Welfare
Maximization

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-50

 Voting

• Sometimes voting does not work well, and

the resulting social ordering of allocations is
not transitive.

Table 10.2 Preferences over Allocations of Three People

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-51

 Arrow’s Impossibility Theorem

• A social welfare function should satisfy the

following criteria:
– Social preferences should be complete and
transitive, like individual preferences.
– If everyone prefers Allocation a to Allocation b, a
should be socially preferred to b.
– Society’s ranking of a and b should depend only
on individuals’ ordering of these two allocations,
not on how they rank other alternatives.
– Dictatorship is not allowed; social preferences
must not reflect the preferences of only a single
individual.

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-52

 Arrow’s Impossibility Theorem
(cont.)

• It is impossible to find a social decision-

making rule that always satisfies all of these
criteria.
– Result indicates that democratic decision making
may fail—not that democracy must fail.

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-53

 Social Welfare Functions

• Utilitarian philosophers: suggested that

society should maximize the sum of the
utilities of all members of society.
– Their social welfare function is the sum of the
utilities of every member of society.
• If Ui is the utility of Individual i and there
are n people, the utilitarian welfare function
is:
W = U1 + U2 + . . . + Un.

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-54

 Social Welfare Functions (cont.)

• The Rawlsian welfare function is:

W = min {U1, U2, . . . , Un}.

– Rawls’ rule leads to a relatively egalitarian

distribution of goods.

Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-55

 Efficiency versus Equity

• Given a particular social welfare function,

society might prefer an inefficient
allocation to an efficient one.
• By most of the well-known social welfare
functions, but not all, there is an efficient
allocation that is socially preferred to an
inefficient allocation.
• Competitive equilibrium may not be very
equitable even though it is Pareto
efficient.
Copyright ©2016 Pearson Education, Ltd. All rights reserved. 10-56