Theory of Markets
Complete Course Summary
Microeconomic Theory: Competitive Markets, Welfare Theorems,
Externalities & Public Goods, Market Power, General Equilibrium,
Uncertainty & Asset Markets, and Strategic Market Interaction
Lecture Notes, Step-by-Step Solution Guides & Worked Exam Exercises
,Contents
I Lecture Notes & Worked Exercises 3
1 Competitive Markets 3
1.1 Speci
cation of the Economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Competitive Market Economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Partial Equilibrium: Two-Good General Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Welfare Theorems in Partial Equilibrium 4
2.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Pareto E
cient Allocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Competitive Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.4 Welfare Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.5 Welfare Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.6 Long-run Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Externalities 6
3.1 Market Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2 Externalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.3 Competitive Outcome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.4 Pareto E
cient Outcome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.5 Internalizing the Externality: Coasean Bargaining . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4 Public Goods 7
4.1 Public Goods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4.2 Finding Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4.3 Solutions to the Free Rider Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
5 Market Power 8
5.1 Monopoly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
5.2 Monopoly with Price Discrimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
6 Oligopoly 9
6.1 Bertrand Competition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
6.2 Cournot Competition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
6.3 Capacity Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
6.4 Product Dierentiation (Hotelling) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
7 Repeated Interaction 10
7.1 Repeated Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
8 General Equilibrium Theory 10
8.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
8.2 Robinson Crusoe Economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
9 Equilibrium and Welfare Properties 11
9.1 Pure Exchange Economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
9.2 Walrasian Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
9.3 First Fundamental Theorem of Welfare Economics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
9.4 Second Welfare Theorem for an Exchange Economy . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
10 Walrasian Equilibrium (Existence) 13
10.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
10.2 Existence Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
11 Local Uniqueness 14
11.1 The Big Question: Global Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
12 Uncertainty & Asset Markets 16
12.1 Two Models of Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
12.2 Quick Reference: Common Utility Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
12.3 Arrow-Debreu Equilibrium with Expected Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1
,Theory of Markets Complete Course Summary 2
12.4 Key Formulas: Incomplete Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
II Quick-Reference Step-by-Step Guides 18
13 Guide: Perfect Competition & Monopoly 18
14 Guide: Externalities & Public Goods 20
15 Guide: Exchange Economies (Cobb-Douglas, Leontief, Min+Max, Linear) 22
16 Guide: Regularity, Uniqueness & Aggregate Excess Demand 24
17 Guide: Uncertainty & Asset Markets 25
III Additional Worked Exam Exercises 26
18 Production Economies & Robinson Crusoe Extensions 27
19 Oligopoly Extensions: Capacity, Product Dierentiation & Government Policy 27
20 Uncertainty, Portfolio Choice & Incomplete Markets 28
,Theory of Markets Complete Course Summary 3
Part I
Lecture Notes & Worked Exercises
1 Competitive Markets
1.1 Speci
cation of the Economy
We have L commodities (products) i.e. apples
We have consumers i = 1, . . . , I for which the consumption set (what they consume) is speci
ed as Xi ⊆ RL
Preferences ≿i on Xi of consumer i are represented by utility function ui (·)
The total amount of each commodity initially available is the endowment ,→ endowment of good ℓ is given by
ωℓ ≥ 0, ℓ = 1, . . . , L
Firms j = 1, . . . , J , with
rm j characterised by production set Yj ⊆ RL
,→ e.g. yj = (hours of work, #apples, #apple pies, #apple juice) = (−10, −60, 3, 2) means: with 10 hours of work
and 60 apples (input), I can produce 3 apple pies and 2 apple juices (output)
De
nition 10.B.1: An economic allocation (x1 , . . . , xI , y1 , . . . , yJ ) is a speci
cation of a consumption vector
xi ∈ Xi for each consumer i = 1, . . . , I and a production vector yj ∈ Yj for each
rm j = 1, . . . , J . The allocation
is feasible if
I
X J
X
xiℓ ≤ ωℓ + yjℓ for ℓ = 1, . . . , L
i=1 j=1
what is consumed ≤ what was already available (initial endowment) + what is produced
De
nition 10.B.2: A feasible allocation (x1 , . . . , xI , y1 , . . . , yJ ) is Pareto optimal if there is no other feasible
allocation (x′1 , . . . , x′I , y1′ , . . . , yJ′ ) s.t. ui (x′i ) ≥ ui (xi ) ∀i and ui (x′i ) > ui (xi ) for some i.
1.2 Competitive Market Economy
For a competitive market economy, we have that each good has a price and each consumer i owns a share of
rms
θij ∈ [0, 1].
In a competitive market, we have a Competitive Equilibrium if:
1. Pro
t maximisation: maxyj ∈Yj p · yj
Utility maximisation: maxxi ∈Xi ui (xi ) s.t. p · xi ≤ p · ωi + θij p · yj∗
P
2. j
Market clearing: x∗i = ω + yj∗
P P
3. i j
2 characteristics of a competitive equilibrium:
1. Prices are relative: p∗ = (p∗1 , . . . , p∗L ) is an equilibrium vector, then so is αp∗ ∈ (0, ∞), α ∈ R++ .
,→ normalize: prices can be normalized by
xing one good as numeraire (pL = 1)
2. Lemma 10.B.1:
condition ( © If the allocation
) for all goods ℓ ̸= k ,
(x1 , . . . , xI , y1 , . . . , yJ ) and the price vector
and if every consumer's budget is satis
ed with equality, s.t.
p≫0 satisfy the market clearing
X
p · xi = p · ωi + θij p · yj , ∀i
j
then the market for good k also clears.
,→ corollary: when searching for competitive equilibria it is su
cient to
nd prices s.t. L−1 markets clear.
In competitive equilibrium typically all markets are related to each other. However, it is often convenient to look at the market
in isolation (partial equilibrium analysis). This is reasonable because the market is small, because the size of substitution and
wealth eects are limited.
1.3 Partial Equilibrium: Two-Good General Model
Two-good quasilinear model: in this model, we look at good ℓ in the small market in isolation compared to all goods
in other markets.
, Theory of Markets Complete Course Summary 4
Other markets: we call the goods the numeraire combined good under which price all other goods are aggregated
(all other prices, wealth, incomes, etc.)
Small isolated market: good ℓ, we say price is p
Model: Consumer: i ωi ; Firm:
P P P
i xi = j yj
Consumer i has endowment ωiℓ = 0, so total endowment of commodity ℓ is 0. Firm j produces good ℓ from good m
(numeraire).
p: price good ℓ
qj : units produced good ℓ
tj : units used to produce good ℓ
cj (qj ): costs in producing good ℓ hence amount of numeraire good to produce good ℓ
The production sets for
rm j (producing ℓ from numeraire good) is given by: Yj = {(−tj , qj ) : qj ≥ 0 and tj ≥
cj (qj )}
How to
nd equilibrium (x∗1 , . . . , x∗I , y1∗ , . . . , yJ∗ )
1. Solve pro
t maximization,
nd qj∗ :
max p · qj − cj (qj ) FOC: p − c′j (qj ) = 0 SOC: − c′′j (qj ) ≤ 0
qj
2. Solve utility maximization,
nd (x∗i , m∗i ):
max ϕi (xi ) + mi s.t. pxi + mi ≤ pωi + mi + πi (p) (equality by nonsatiation)
xi ,mi
FOC: ϕ′i (xi ) − p = 0
Calculate market equilibrium: ∗ ∗
P P
3. i xi = j qj the market equilibrium condition is independent of the distri-
bution of endowments and ownership shares; therefore the equilibrium price and allocation is also independent, thus
the markets can be studied in isolation.
Aggregate demand: X(p) = ϕ′ (x) = ϕ′1 (x1 ) + · · · + ϕ′I (xI )
P
i xi (p) ⇔ inverse aggregate demand
X(p) = ϕ′ (x)−1
Aggregate supply: C ′ (q) = c′1 (q1 ) = · · · = c′J (qJ )
Unique equilibrium exists if a unique price p∗ solves maxq≥0 Φ(q) − C(q), with Φ′ (0) > 0 and C ′ (0) ≥ 0:
from the goods and demand functions we get a unique equilibrium quantity and price.
Exercises Week 1 Two-Good Quasilinear Model
Consider an economy with I consumers and J
rms. Consumer i has quasilinear utility ui (xi , mi ) = ϕi (xi )+mi ,
where ϕi (xi ) = Axi − 12 Bx2i . Firms have cost function cj (qj ) = 21 qj2 .
1. Individual demand: maximize ui (xi , mi ) = ϕi (xi ) + mi s.t. budget: pxi + mi ≤ pωi + mi
′ A−p
FOC: ϕi (xi ) − p = 0 ⇒ xi (p) = (individual demand)
B
2. Individual supply for each
rm: solve M C = AC for q ; minimum AC : p =P
s
AC(q ) s
3. Equilibrium under perfect competition:
nd aggregate supply Q(p) = j qj (p); market clearing
∗ ∗
condition demand = supply; solve for p ;
ll in p in xi , qj , πj
4. How many active
rms with free entry? free entry condition: πj∗ = 0 ⇒ p = min AC ⇒ then solve for
number of active
rms
2 Welfare Theorems in Partial Equilibrium
2.1 Model
We consider two types of goods:
1. A numeraire good: some money, or a good that is always traded at price 1
2. A non-numeraire good x that generates utility
Complete Course Summary
Microeconomic Theory: Competitive Markets, Welfare Theorems,
Externalities & Public Goods, Market Power, General Equilibrium,
Uncertainty & Asset Markets, and Strategic Market Interaction
Lecture Notes, Step-by-Step Solution Guides & Worked Exam Exercises
,Contents
I Lecture Notes & Worked Exercises 3
1 Competitive Markets 3
1.1 Speci
cation of the Economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Competitive Market Economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Partial Equilibrium: Two-Good General Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Welfare Theorems in Partial Equilibrium 4
2.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Pareto E
cient Allocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Competitive Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.4 Welfare Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.5 Welfare Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.6 Long-run Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Externalities 6
3.1 Market Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2 Externalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.3 Competitive Outcome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.4 Pareto E
cient Outcome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.5 Internalizing the Externality: Coasean Bargaining . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4 Public Goods 7
4.1 Public Goods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4.2 Finding Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4.3 Solutions to the Free Rider Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
5 Market Power 8
5.1 Monopoly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
5.2 Monopoly with Price Discrimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
6 Oligopoly 9
6.1 Bertrand Competition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
6.2 Cournot Competition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
6.3 Capacity Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
6.4 Product Dierentiation (Hotelling) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
7 Repeated Interaction 10
7.1 Repeated Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
8 General Equilibrium Theory 10
8.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
8.2 Robinson Crusoe Economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
9 Equilibrium and Welfare Properties 11
9.1 Pure Exchange Economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
9.2 Walrasian Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
9.3 First Fundamental Theorem of Welfare Economics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
9.4 Second Welfare Theorem for an Exchange Economy . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
10 Walrasian Equilibrium (Existence) 13
10.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
10.2 Existence Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
11 Local Uniqueness 14
11.1 The Big Question: Global Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
12 Uncertainty & Asset Markets 16
12.1 Two Models of Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
12.2 Quick Reference: Common Utility Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
12.3 Arrow-Debreu Equilibrium with Expected Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1
,Theory of Markets Complete Course Summary 2
12.4 Key Formulas: Incomplete Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
II Quick-Reference Step-by-Step Guides 18
13 Guide: Perfect Competition & Monopoly 18
14 Guide: Externalities & Public Goods 20
15 Guide: Exchange Economies (Cobb-Douglas, Leontief, Min+Max, Linear) 22
16 Guide: Regularity, Uniqueness & Aggregate Excess Demand 24
17 Guide: Uncertainty & Asset Markets 25
III Additional Worked Exam Exercises 26
18 Production Economies & Robinson Crusoe Extensions 27
19 Oligopoly Extensions: Capacity, Product Dierentiation & Government Policy 27
20 Uncertainty, Portfolio Choice & Incomplete Markets 28
,Theory of Markets Complete Course Summary 3
Part I
Lecture Notes & Worked Exercises
1 Competitive Markets
1.1 Speci
cation of the Economy
We have L commodities (products) i.e. apples
We have consumers i = 1, . . . , I for which the consumption set (what they consume) is speci
ed as Xi ⊆ RL
Preferences ≿i on Xi of consumer i are represented by utility function ui (·)
The total amount of each commodity initially available is the endowment ,→ endowment of good ℓ is given by
ωℓ ≥ 0, ℓ = 1, . . . , L
Firms j = 1, . . . , J , with
rm j characterised by production set Yj ⊆ RL
,→ e.g. yj = (hours of work, #apples, #apple pies, #apple juice) = (−10, −60, 3, 2) means: with 10 hours of work
and 60 apples (input), I can produce 3 apple pies and 2 apple juices (output)
De
nition 10.B.1: An economic allocation (x1 , . . . , xI , y1 , . . . , yJ ) is a speci
cation of a consumption vector
xi ∈ Xi for each consumer i = 1, . . . , I and a production vector yj ∈ Yj for each
rm j = 1, . . . , J . The allocation
is feasible if
I
X J
X
xiℓ ≤ ωℓ + yjℓ for ℓ = 1, . . . , L
i=1 j=1
what is consumed ≤ what was already available (initial endowment) + what is produced
De
nition 10.B.2: A feasible allocation (x1 , . . . , xI , y1 , . . . , yJ ) is Pareto optimal if there is no other feasible
allocation (x′1 , . . . , x′I , y1′ , . . . , yJ′ ) s.t. ui (x′i ) ≥ ui (xi ) ∀i and ui (x′i ) > ui (xi ) for some i.
1.2 Competitive Market Economy
For a competitive market economy, we have that each good has a price and each consumer i owns a share of
rms
θij ∈ [0, 1].
In a competitive market, we have a Competitive Equilibrium if:
1. Pro
t maximisation: maxyj ∈Yj p · yj
Utility maximisation: maxxi ∈Xi ui (xi ) s.t. p · xi ≤ p · ωi + θij p · yj∗
P
2. j
Market clearing: x∗i = ω + yj∗
P P
3. i j
2 characteristics of a competitive equilibrium:
1. Prices are relative: p∗ = (p∗1 , . . . , p∗L ) is an equilibrium vector, then so is αp∗ ∈ (0, ∞), α ∈ R++ .
,→ normalize: prices can be normalized by
xing one good as numeraire (pL = 1)
2. Lemma 10.B.1:
condition ( © If the allocation
) for all goods ℓ ̸= k ,
(x1 , . . . , xI , y1 , . . . , yJ ) and the price vector
and if every consumer's budget is satis
ed with equality, s.t.
p≫0 satisfy the market clearing
X
p · xi = p · ωi + θij p · yj , ∀i
j
then the market for good k also clears.
,→ corollary: when searching for competitive equilibria it is su
cient to
nd prices s.t. L−1 markets clear.
In competitive equilibrium typically all markets are related to each other. However, it is often convenient to look at the market
in isolation (partial equilibrium analysis). This is reasonable because the market is small, because the size of substitution and
wealth eects are limited.
1.3 Partial Equilibrium: Two-Good General Model
Two-good quasilinear model: in this model, we look at good ℓ in the small market in isolation compared to all goods
in other markets.
, Theory of Markets Complete Course Summary 4
Other markets: we call the goods the numeraire combined good under which price all other goods are aggregated
(all other prices, wealth, incomes, etc.)
Small isolated market: good ℓ, we say price is p
Model: Consumer: i ωi ; Firm:
P P P
i xi = j yj
Consumer i has endowment ωiℓ = 0, so total endowment of commodity ℓ is 0. Firm j produces good ℓ from good m
(numeraire).
p: price good ℓ
qj : units produced good ℓ
tj : units used to produce good ℓ
cj (qj ): costs in producing good ℓ hence amount of numeraire good to produce good ℓ
The production sets for
rm j (producing ℓ from numeraire good) is given by: Yj = {(−tj , qj ) : qj ≥ 0 and tj ≥
cj (qj )}
How to
nd equilibrium (x∗1 , . . . , x∗I , y1∗ , . . . , yJ∗ )
1. Solve pro
t maximization,
nd qj∗ :
max p · qj − cj (qj ) FOC: p − c′j (qj ) = 0 SOC: − c′′j (qj ) ≤ 0
qj
2. Solve utility maximization,
nd (x∗i , m∗i ):
max ϕi (xi ) + mi s.t. pxi + mi ≤ pωi + mi + πi (p) (equality by nonsatiation)
xi ,mi
FOC: ϕ′i (xi ) − p = 0
Calculate market equilibrium: ∗ ∗
P P
3. i xi = j qj the market equilibrium condition is independent of the distri-
bution of endowments and ownership shares; therefore the equilibrium price and allocation is also independent, thus
the markets can be studied in isolation.
Aggregate demand: X(p) = ϕ′ (x) = ϕ′1 (x1 ) + · · · + ϕ′I (xI )
P
i xi (p) ⇔ inverse aggregate demand
X(p) = ϕ′ (x)−1
Aggregate supply: C ′ (q) = c′1 (q1 ) = · · · = c′J (qJ )
Unique equilibrium exists if a unique price p∗ solves maxq≥0 Φ(q) − C(q), with Φ′ (0) > 0 and C ′ (0) ≥ 0:
from the goods and demand functions we get a unique equilibrium quantity and price.
Exercises Week 1 Two-Good Quasilinear Model
Consider an economy with I consumers and J
rms. Consumer i has quasilinear utility ui (xi , mi ) = ϕi (xi )+mi ,
where ϕi (xi ) = Axi − 12 Bx2i . Firms have cost function cj (qj ) = 21 qj2 .
1. Individual demand: maximize ui (xi , mi ) = ϕi (xi ) + mi s.t. budget: pxi + mi ≤ pωi + mi
′ A−p
FOC: ϕi (xi ) − p = 0 ⇒ xi (p) = (individual demand)
B
2. Individual supply for each
rm: solve M C = AC for q ; minimum AC : p =P
s
AC(q ) s
3. Equilibrium under perfect competition:
nd aggregate supply Q(p) = j qj (p); market clearing
∗ ∗
condition demand = supply; solve for p ;
ll in p in xi , qj , πj
4. How many active
rms with free entry? free entry condition: πj∗ = 0 ⇒ p = min AC ⇒ then solve for
number of active
rms
2 Welfare Theorems in Partial Equilibrium
2.1 Model
We consider two types of goods:
1. A numeraire good: some money, or a good that is always traded at price 1
2. A non-numeraire good x that generates utility
