Introduction
The Model
Equilibrium
Welfare
Long-Run Equilibrium
Advanced Microeconomics
Partial and General Equilibrium
Giorgio Fagiolo
giorgio.fagiolo@sssup.it
http://www.lem.sssup.it/fagiolo/Welcome.html
LEM, Sant’Anna School of Advanced Studies, Pisa (Italy)
Part 2
Conclusions
Introduction
The Model
Equilibrium
Welfare
Long-Run Equilibrium
Industry Partial Equilibrium Analysis
Studying a small part of the overall economy
One good (G) vs other (L − 1) markets
Industry of commodity (G) is small
Basic Assumptions
Consumers spend a small (negligible) part of their total income for G
When consumer wealth increases, demand for G does not increase (no
wealth effect)
Substitution effects are negligible and dispersed: when pG increases, no
effects on demand for other commodities
Prices of other commodities can be considered as given
(L − 1) goods as composite commodity (money or numeraire)
Formally: Consumers have quasi-linear preferences
Conclusions
Introduction
The Model
Equilibrium
Welfare
Long-Run Equilibrium
Conclusions
Quasi-Linear Preferences
Consider an economy with L = 2 commodities. The representative
consumer holds a preference relation with associated utility function:
u(x, m) = φ(x) + m
where x ≥ 0 is the consumption of commodity 1 and m ∈ R is the
consumption of commodity 2. We also assume that φ0 > 0, φ00 < 0 and
that φ(0) = 0.
Commodity 1 is a consumption good, while commodity 2 can be defined
as ’everything else’, i.e. money left apart by the consumer for purchasing
all other goods, after having made the optimal choice for commodity 1.
The latter can be referred to as the ’consumption good’, while commodity
2 is the ’numeraire’.
Thus, we normalize prices so that px = p > 0 and pm = 1. Notice that
we allow ’money’ to be negative.
Introduction
The Model
Equilibrium
Welfare
Long-Run Equilibrium
Conclusions
Quasi-Linear Preferences: No Wealth Effects
The most important feature of the quasi-linear preference is that an increase
in consumer’s income has no effect on the demand of the consumption good.
To see that let’s solve the consumer program:
max φ(x) + m, s.t. px + m = y
x≥0,mR
for an income’s level y .
The Lagrangean is: L(x, m, λ) = φ(x) + m − λ[y − px − m] and FOCs for an
interior solution are:
φ0 (x(p, y )) ≡ p
p · x(p, y ) + m(p, y ) ≡ y
Notice that FOCs are also sufficient because φ00 < 0.
Introduction
The Model
Equilibrium
Welfare
Long-Run Equilibrium
Quasi-Linear Preferences: No Wealth Effects
Differentiating FOCs with respect to y gives:
∂x(p, y )
∂y
∂x(p, y )
∂m(p, y )
p·
+
∂y
∂y
φ00 (x(p, y )) ·
=
0
=
1
which hold if and only if:
∂m(p, y )
∂x(p, y )
= 0 and
= 1.
∂y
∂y
Hence an increase in y has no effect on the demand of x and all wealth
increase has a 1:1 effect on the numeraire.
Conclusions
Introduction
The Model
Equilibrium
Welfare
Long-Run Equilibrium
Conclusions
Quasi-Linear Preferences and UPS
Consider an economy with I consumers, each one holds a quasilinear utility
function ui (xi , mi ) = φi (xi ) + mi , i = 1, ..., I, an initial endowment of the
numeraire ωmi : Σi ωmi = ωm > 0 and an initial endowment of the consumption
good ωxi : Σi ωxi = ωx > 0.
Then the utility possibility set (UPS) is defined as:
X
X
U = {(u1 , ..., uI ) ∈ RI : ui ≤ φi (xi ) + mi ,
mi = ωm ,
xi = ωx }
i
i
that is the set of all attainable utility levels by consumers given resource
constraints.
Introduction
The Model
Equilibrium
Welfare
Long-Run Equilibrium
Conclusions
Quasi-Linear Preferences and UPF
By summing up the inequalities ui ≤ φi (xi ) + mi over all consumers and by
using the resource constraint, one gets:
Σi ui ≤ Σi φi (xi ) + ωm
Hence, given any feasible allocation, the utility possibility frontier (UPF) reads:
UPF = {(u1 , ..., uI ) ∈ RI : Σi ui = max{Σi φi (xi ) + ωm , Σi xi = ωx }}
is an hyperplane in the utility space and all points in this boundary are
associated with consumption allocations that differ only in the distribution of
the amount of the numeraire ωm among the I consumers.
Introduction
The Model
Equilibrium
Welfare
Long-Run Equilibrium
Conclusions
Quasi-Linear Preferences: UPS and UPF
Example: I = 2 and φi (xi ) = log xi . Then:
U
=
{(u1 , u2 ) ∈ R2 : u1 ≤ log x1 + m1 ,
u2
≤
log x2 + m2 , m1 + m2 = ωm , x1 + x2 = ωx }
This implies that x2 = ωx − x1 , m2 = ωm − m1 and
m1 ≤ log(ωx − x1 ) + ωm − u2 . Hence:
U = {(u1 , u2 ) ∈ R2 : u1 + u2 ≤ log[x1 (ωx − x1 )] + ωm , 0 ≤ x1 ≤ ωx }
whose boundary is the line u1 + u2 = log[x1 (ωx − x1 )] + ωm . Thus, by
changing the demand of the consumption good by consumer 1, one simply
shifts the boundary in a parallel manner. It is intuitive that when the allocation
is Pareto Efficient, the set U will be extended as far out as possible. In this
case, as both consumers have the same preferences over the consumption
good, the Pareto efficient allocation is (ωx /2, ωx /2), which is the value of x1
which maximizes log[x1 (ωx − x1 )]. Hence, the boundary of U (i.e. the UPF) in
this particular example is given by:
Bd(U) = {(u1 , u2 ) ∈ R2 : u1 + u2 = log[ωx2 /4] + ωm }.
Introduction
The Model
Equilibrium
Welfare
Long-Run Equilibrium
Conclusions
The Model
Two-good economy
Good 1: Commodity under study `; price: p` = p; agent i’s consumption: xi
Good 2: Numeraire (money) m; price: pm = 1; agent i’s consumption: mi
Consumers: i = 1, . . . , I
Utility: ui (xi , mi ) = φi (xi ) + mi , mi ∈ R, xi ∈ R+
Assumptions: φ0i > 0, φ00
i < 0, φi (0) = 0
Firms: j = 1, . . . , J
Employ good m to produce `. Since pm = 1, if firm j uses zj units of m to
produce `, then its cost is zj
Production sets: Yj = {(−zj , qj ) : qj ≤ fj (zj ), qj ≥ 0} =
{(−zj , qj ) : zj ≥ cj (qj ), qj ≥ 0}, where fj0 > 0, cj0 > 0 and cj00 > 0 (strict
convexity)
NB: Since J is fixed, we are in the SR!
Endowments and Shares
Good `: no endowments ω` = 0 (must be produced)
Good m: ωmi , i = 1, . . . , I, ωm = Σi ωmi
Each consumer i owns a share θij of firm j, Σi θij = 1
Introduction
The Model
Equilibrium
Welfare
Long-Run Equilibrium
Conclusions
Supply Behavior
artial Equilibrium Competitive Analysi
Competitive equilibria:
�
At price p ∗ , firm j’s equilibrium output level qj∗ must solve
max p ∗ qj − cj (qj )
qj ≥0
with necessary and sufficent FOC:
p ∗ ≤ cj� (qj ), with equality if qj∗ > 0
�
consumer i’s equilibrium consumption vector (mi∗ , xi∗ ) must
solve
max m ∈R mi + φi (xi )
Introduction
with necessary
and sufficent
FOC:Welfare
The Model
Equilibrium
Long-Run Equilibrium
Conclusions
Demand Behavior p ∗ ≤ cj� (qj ), with equality if qj∗ > 0
�
consumer i’s equilibrium∗ consumption
vector (mi∗ , xi∗ ) must
Each consumer i given p and πj∗ = p∗ qj∗ − cj (qj∗ ) solves:
solve
max mi ∈R mi + φi (xi )
xi ∈R+
�
J
∗ ∗
∗ ))
mi +
i ≤ ωmi +
Partial s.t.
Equilibrium
Competitive
j=1 θij (p qj − cj (qjAnalysis
p∗x
Competitive equilibria:
� . . . since the budget constraint must hold with equality:


J
�
max φi (xi ) − p ∗ xi + ωmi +
θij (p ∗ qj∗ − cj (qj∗ ))
xi ≥0
j=1
with necessary and sufficent FOC:
φ�i (xi∗ ) ≤ p ∗ , with equality if xi∗ > 0
Introduction
The Model
Equilibrium
Welfare
Long-Run Equilibrium
Conclusions
Equilibrium
It is an allocation (x1∗ , . . . , xI∗ ; q1∗ , . . . , qJ∗ ) and a price (scalar) p∗ such
that:
1
All firms maximize profit given technology and p∗ , i.e. cj0 (qj∗ ) ≥ p∗ (= if
qj∗ > 0)
2
All consumers maximize utility s.t. budget constraint given p∗ and firms’
optimal profits, i.e. φ0i (xi∗ ) ≤ p∗ (= if xi∗ > 0)
Market for good ` clears: Σi xi∗ = Σj qj∗
3
Remarks
Since all utilities satisfy non-satiation, all budget constraints are satisfied
with equality and therefore all markets must clear (with equality)
Check that mi∗ and zj∗ can be recovered using BC and production function
Does equilibrium condition imply that market for numeraire also clears?
Check that using budget constraints. . .
X
X
X
X
X
zj∗ =
ωmi
mi∗ +
qj∗ =
xi∗ ⇒
j
i
i
j
i
Introduction
The Model
Equilibrium
Welfare
Long-Run Equilibrium
Conclusions
Existence
Does an interior equilibrium always exist? No!
Sufficient condition: If maxi φ0i (0) > minj cj0 (0) then an interior equilibrium
Σi xi∗ = Σj qj∗ > 0 does exist
Why? Suppose not, then there exists an equilibrium such that
Σi xi∗ = Σj qj∗ = 0, meaning that xi∗ = qj∗ = 0 for all i, j. From conditions (1)
and (2) we have: φ0i (0) ≤ cj0 (0). This implies: maxi φ0i (0) ≤ minj cj0 (0),
which contradicts the assumption.
Consequences
In any interior solution: φ0i (xi∗ ) = p∗ = cj0 (qj∗ ). Thus equilibrium price:
is equal to firm marginal benefit in selling one additional unit of good (i.e. firm
marginal cost in producing one additional unit)
is equal to consumer marginal cost in buying one additional unit of good (i.e.
consumer marginal benefit in consuming one additional unit of good)
Conditions 1-3 not affected by the distribution of endowments and ownership
shares (because of quasi-linearity of utility functions)
Introduction
The Model
Equilibrium
Welfare
Long-Run Equilibrium
Aggregate Supply and Aggregate Demand
What does all that have to do with the traditional demand=supply
scheme?
Consumer behavior: xi∗ = xi (p) for any given level p
Firm behavior: qj∗ = qj (p) for any given level p
Market clearing condition ⇒ aggregate demand = aggregate supply
Aggregate Supply and Aggregate Demand
S(p) = Σj qj (p), increasing in p if p > minj cj0 (0)
D(p) = Σi xi (p), decreasing in p if p < maxi φ0i (0)
Conclusions
Introduction
Aggregate
Supply
Function
for Long-Run
Good
�
The Model
Equilibrium
Welfare
Equilibrium
AS vs. AD: Graphical Construction
Aggregate Demand Function for Good �
Conclusions
Introduction
The Model
AS vs. AD: Equilibrium
Equilibrium
Welfare
Long-Run Equilibrium
Conclusions
Introduction
The Model
Equilibrium
Welfare
Partial Equilibrium: When does it fail to exist?
See exercise!
Long-Run Equilibrium
Conclusions
Introduction
The Model
Equilibrium
Welfare
Long-Run Equilibrium
Conclusions
Comparative Statics
How does a change in underlying conditions affect the equilibrium
outcome?
exogenous parameters α ∈ R M affecting consumer’s preferences: φi (xi , α)
exogenous parameters β ∈ R S affecting firm’s technology: cj (qj , β)
exogenous tax and subsidy parameters t ∈ R K affecting price: p̂i (p, t) (paid
by consumers) and p̂j (p, t) (received by firms)
Typical exercise:
the equilibrium allocation and price are functions of (α, β, t)
if functions are differentiable, then the implicit function theorem can be used
to derive the marginal change in equilibrium allocation and price in response
to a differential change in parameters
Introduction
The Model
Equilibrium
Welfare
Long-Run Equilibrium
Conclusions
Comparative statics effects of a sales tax
Suppose to introduce a sales tax such that consumers must pay t ≥ 0
for each unit of good `
Price received by producers is p(t)
Price paid by consumers is p(t) + t
Equilibrium condition: D(p(t) + t) ≡ S(p(t))
Differentiating and solving for p0 (t):
d(p(t))
D 0 (p(t) + t)
= p0 (t) = − 0
∈ [−1, 0)
dt
D (p(t) + t) − S 0 (p(t))
d(p(t) + t)
= p0 (t) + 1 ∈ [0, 1)
dt
Total quantities produced and consumed fall
But how much this change is felt by consumers and producers mainly
depends on the steepness of supply function: the steeper S, the more
rigid production to prices, the more producers feel the burden of the sale
tax
Introduction
The Model
Equilibrium
Welfare
Long-Run Equilibrium
Comparative statics effects of a sales tax: Three cases
The larger |S 0 (p(t))|, the flatter the S schedule
If |S 0 (p(t))| very large (i.e. supply very elastic, i.e. horizontal) then S is
flat and all the burden is felt by consumers
If |S 0 (p(t))| is very small (i.e. supply very rigid, i.e. vertical) then S is
vertical and all the burden is felt by producers
Conclusions
Introduction
The Model
Equilibrium
Welfare
Long-Run Equilibrium
Conclusions
Main questions
1
Suppose that for a given endowment distribution and owner shares the market
mechanisms reaches an equilibrium allocation (x1∗ , . . . , xI∗ ; q1∗ , . . . , qJ∗ ) and a
price (scalar) p∗ . Is that allocation Pareto optimal?
2
Suppose we start from a Pareto optimal allocation (x1o , . . . , xIo ; q1o , . . . , qJo ). Is
there a redistribution of initial endowments that allows to sustain that allocation as
a market equilibrium? I.e. is there a way to redistribute initial endowments in such
a way that a given PO allocation will be reached by market forces without any
additional intervention?
∗
∗
The allocation (x )i , (y )j maximizes
social welfare for some
choice of weights λ∗
Convexity
The price-allocation
{p∗ , (x∗ )i , (y ∗ )j }
is a competitive equilibrium
?
∗
∗
The allocation (x )i , (y )j is a
Pareto optimum
Introduction
The Model
Equilibrium
Welfare
Long-Run Equilibrium
Conclusions
The Utility Possibility Set (UPS)
The UPS for the partial-equilibrium economy is defined as
U = {(u1 , . . . , uI ) :
I
X
i=1
ui ≤
I
X
i=1
φi (xi ) + ωm −
J
X
cj (qj )}
j=1
for
P any given
P feasible allocation (x1 , . . . , xI ; q1 , . . . , qJ ), i.e. such that
i xi =
j qj .
Recall: The boundary of the UPS (i.e. the UPF) is in a 1:1 relation with
all
suchPthat u ∈
PIPO allocations.
PI Therefore all allocations
PJ
P{(u1 , . . . , uI ) :
i=1 ui = max{
i=1 φi (xi ) + ωm −
j=1 cj (qj ),
i xi =
j qj }}, are PO.
Recall: When consumer preferences are quasilinear, the boundary of
the economy’s utility possibility set is linear, and all points in this
boundary are associated with consumption allocations that differ only in
the distribution of the numeraire (ωi1 , . . . , ωIm ) s.t. ωm = Σi ωim
Introduction
The Model
Equilibrium
The Utility Possibility Set (UPS)
Welfare
Long-Run Equilibrium
Conclusions
Introduction
The Model
Equilibrium
Welfare
Long-Run Equilibrium
Partial Equilibrium Analysis: First Welfare Theorem
Theorem
If the price p∗ and the allocation (x ∗ , q ∗ ) = (x1∗ , . . . , xI∗ , q1∗ , . . . , qJ∗ )
constitutes a competitive (Walrasian) equilibrium, then (x ∗ , q ∗ ) is Pareto
optimal
Proof.
We prove it for the case of an interior allocation. An allocation
(x1+ , . . . , xI+ , q1+ , . . . , qJ+ ) is Pareto optimal if given the set U, it lies in the
boundary of the UPS, i.e. if it maximizes
I
X
φi (xi ) + ωm −
i=1
subject to feasibility constraints, i.e.
J
X
cj (qj )
j=1
P
i
xi =
P
j
qj .
Conclusions
Introduction
The Model
Equilibrium
Welfare
Long-Run Equilibrium
Partial Equilibrium Analysis: First Welfare Theorem
Theorem
If the price p∗ and the allocation (x ∗ , q ∗ ) = (x1∗ , . . . , xI∗ , q1∗ , . . . , qJ∗ )
constitutes a competitive (Walrasian) equilibrium, then (x ∗ , q ∗ ) is Pareto
optimal
Proof (Cont’d).
FOCs for this problem read (in an interior solution):
φ0i (xi+ ) = µ = cj0 (qj+ )
X
i
xi+ =
X
qj+
j
Note that FOCs are sufficient because φ00i < 0 and cj00 > 0. Thus an
allocation that solves FOCs is PO. But FOCs are equivalent to those for an
interior competitive equilibrium when µ = p∗ . Therefore, if p∗ and
(x1∗ , . . . , xI∗ , q1∗ , . . . , qJ∗ ) satisfy FOCs for µ = p∗ , they also satisfy sufficient
conditions for PO.
Conclusions
Introduction
The Model
Equilibrium
Welfare
Long-Run Equilibrium
Conclusions
Partial Equilibrium Analysis: Second Welfare Theorem
Theorem
For any Pareto optimal levels of utility (u1∗ , . . . , uI∗ ) ∈
PUPF , there are transfers
of the numeraire commodity (T1 , . . . , TI ) satisfying i Ti = 0, such that a
competitive equilibrium reached from the endowments
(ωm1 + T1 , . . . , ωmI + TI ) yields precisely the utilities (u1∗ , . . . , uI∗ )
Argument: consumption levels, production levels and firms’s profits are
unaffected by changes in consumers’ wealth levels; thus, ex-ante transfers of
the numeraire causes each equilibrium consumption of the numeraire to
change by exactly the amount of the transfer; hence, they allow to reach any
utility vector in the boundary of the utility possibility set
Note: convexity assumptions about technologies and preferences are central
to show this result!
Introduction
The Model
Equilibrium
Welfare
Long-Run Equilibrium
Equilibrium and Welfare
∗
∗
The allocation (x )i , (y )j maximizes
social welfare for some
choice of weights λ∗
Convexity
The price-allocation
{p∗ , (x∗ )i , (y ∗ )j }
∗
∗
The allocation (x )i , (y )j is a
Pareto optimum
Co
nv
ex
ity
is a competitive equilibrium
Conclusions
Introduction
The Model
Equilibrium
Welfare
Long-Run Equilibrium
Conclusions
Free-Entry and Long-Run Competitive Equilibria
So far: Short run competitive equilibria
Number of firms J is fixed
Firms hold heterogeneous technologies and cannot change them
Long run: free entry/exit model
Let J (number of firms) in the market be endogenous: firms can freely
enter/exit in response to π opportunities: a firm will enter if it can earn a
positive profit upon entry; it exits if it gets negative profits at the current price
p for any q > 0
There is a potentially infinite set of firms all having access to technology
c(q) s.t. c(0) = 0 (no sunk costs in the LR)
Firms are price takers: in the equilibrium each firm must earn zero profits
(why?)
Let the aggregate demand be X (p), with X 0 < 0
Long run equilibrium. It is a triple (p∗ , q ∗ , J ∗ ) such that:
1
2
3
π-max: q ∗ solves maxq≥0 {p∗ q − c(q)}
AD=AS: X (p∗ ) = J ∗ · q ∗
Free entry: π(p∗ ) = p∗ q ∗ − c(q ∗ ) = 0
Introduction
The Model
Equilibrium
Welfare
Long-Run Equilibrium
Long-Run Aggregate Supply (LR-AS) Correspondence
Define LR-AS as:

 ∞ if π(p) > 0
{Q ≥ 0 : Q = Jq for some J ≥ 0 and q ∈ q(p)} if π(p) = 0
Q(p) =

0 if π(p) < 0
Intuition:
If π(p) > 0 then every firm wants to supply a strictly positive amount of
product, therefore AS is ∞
If π(p) = 0 and for some integer J we have Q = Jq(p), then there are J
firms in the market each producing q(p) at price p. All other firms stay out
and produce 0 (which is a profit maximizing choice as c(0) = 0)
If π(p) < 0 all firms produce 0
LR equilibrium: alternative characterization
The price level p∗ is a LR competitive equilibrium if and only if
X (p∗ ) = Q(p∗ )
Proof: See MWG, p. 336, footnote 27
Conclusions
Free-Entry and Long-Run Competitive
Equilibria
Example 1: LR Equilibrium with CRTS technology
Introduction
The Model
Equilibrium
Welfare
Constant returns to scale
Q(p) =


∞
[0, ∞)

0
if p > c
if p = c
if p < c
Long-Run Equilibrium
Conclusions
Equilibria
Introduction
The Model
Equilibrium
Welfare
Long-Run Equilibrium
Example 2: LR Equilibrium with Strictly DRTS technology
Non existence of long run competitive equilibrium with
strictly convex costs
Q(p) =
�
∞
0
if p > c � (0)
if p ≤ c � (0)
Conclusions
Free-Entry and Long-Run Competitive
Example
3: LR Equilibrium with U-shaped Costs
Equilibria
Introduction
The Model
Equilibrium
Welfare
Long-Run Equilibrium
Average costs exhibit a strictly positive efficient scale

 ∞ if p > c̄
{Q ≥ 0 : Q = J q̄ for some integer J ≥ 0 if p = c̄
Q(p) =

0 if p < c̄
Conclusions
Introduction
The Model
Equilibrium
Welfare
Long-Run Equilibrium
Conclusions
Conclusions
Partial Equilibrium Approach
From the positive side, it allows to determine the equilibrium outcome in one
market in isolation from all other markets
From the normative side, it allows to prove the 1st and 2nd welfare theorems
(a sort of formal expression of Adam Smith’s “invisible hand”)
Problems
Prices of all other goods remain fixed
There are no wealth effects in the market under study
What about issues that are inherently general-equilibrium ones?
It is crucial to consider a setup where all prices are simultaneously
determined so as to take into account economy-wide feedbacks!
Questions: Do welfare theorems still hold?