Advanced Microeconomics Partial and General … The Model Equilibrium Welfare Long-Run Equilibrium Conclusions Advanced Microeconomics Partial and General Equilibrium Giorgio Fagiolo

Introduction The Model Equilibrium Welfare Long-Run Equilibrium Conclusions

Advanced MicroeconomicsPartial and General Equilibrium

Giorgio [email protected]

http://www.lem.sssup.it/fagiolo/Welcome.html

LEM, Sant’Anna School of Advanced Studies, Pisa (Italy)

Part 2


Industry Partial Equilibrium Analysis

Studying a small part of the overall economyOne good (G) vs other (L− 1) marketsIndustry of commodity (G) is small

Basic AssumptionsConsumers spend a small (negligible) part of their total income for GWhen consumer wealth increases, demand for G does not increase (nowealth effect)Substitution effects are negligible and dispersed: when pG increases, noeffects on demand for other commoditiesPrices of other commodities can be considered as given(L− 1) goods as composite commodity (money or numeraire)Formally: Consumers have quasi-linear preferences


Quasi-Linear Preferences

Consider an economy with L = 2 commodities. The representativeconsumer 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 theconsumption of commodity 2. We also assume that φ′ > 0, φ′′ < 0 andthat φ(0) = 0.

Commodity 1 is a consumption good, while commodity 2 can be definedas ’everything else’, i.e. money left apart by the consumer for purchasingall other goods, after having made the optimal choice for commodity 1.The latter can be referred to as the ’consumption good’, while commodity2 is the ’numeraire’.

Thus, we normalize prices so that px = p > 0 and pm = 1. Notice thatwe allow ’money’ to be negative.


Quasi-Linear Preferences: No Wealth Effects

The most important feature of the quasi-linear preference is that an increasein consumer’s income has no effect on the demand of the consumption good.

To see that let’s solve the consumer program:

maxx≥0,mεR

φ(x) + m, s.t. px + m = y

for an income’s level y .

The Lagrangean is: L(x ,m, λ) = φ(x) + m − λ[y − px −m] and FOCs for aninterior solution are:

φ′(x(p, y)) ≡ p

p · x(p, y) + m(p, y) ≡ y

Notice that FOCs are also sufficient because φ′′ < 0.


Quasi-Linear Preferences: No Wealth Effects

Differentiating FOCs with respect to y gives:

φ′′(x(p, y)) · ∂x(p, y)

∂y= 0

p · ∂x(p, y)

∂y+∂m(p, y)

∂y= 1

which hold if and only if:

∂x(p, y)

∂y= 0 and

∂m(p, y)

∂y= 1.

Hence an increase in y has no effect on the demand of x and all wealthincrease has a 1:1 effect on the numeraire.


Quasi-Linear Preferences and UPS

Consider an economy with I consumers, each one holds a quasilinear utilityfunction ui (xi ,mi ) = φi (xi ) + mi , i = 1, ..., I, an initial endowment of thenumeraire ωmi : Σiωmi = ωm > 0 and an initial endowment of the consumptiongood ωxi : Σiωxi = ωx > 0.

Then the utility possibility set (UPS) is defined as:

U = {(u1, ..., uI) ∈ RI : ui ≤ φi (xi ) + mi ,∑

i

mi = ωm,∑

i

xi = ωx}

that is the set of all attainable utility levels by consumers given resourceconstraints.


Quasi-Linear Preferences and UPF

By summing up the inequalities ui ≤ φi (xi ) + mi over all consumers and byusing the resource constraint, one gets:

Σiui ≤ Σiφi (xi ) + ωm

Hence, given any feasible allocation, the utility possibility frontier (UPF) reads:

UPF = {(u1, ..., uI) ∈ RI : Σiui = max{Σiφi (xi ) + ωm, Σixi = ωx}}

is an hyperplane in the utility space and all points in this boundary areassociated with consumption allocations that differ only in the distribution ofthe amount of the numeraire ωm among the I consumers.


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 andm1 ≤ 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, bychanging the demand of the consumption good by consumer 1, one simplyshifts the boundary in a parallel manner. It is intuitive that when the allocationis Pareto Efficient, the set U will be extended as far out as possible. In thiscase, as both consumers have the same preferences over the consumptiongood, 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) inthis particular example is given by:

Bd(U) = {(u1, u2) ∈ R2 : u1 + u2 = log[ω2x/4] + ωm}.


The Model

Two-good economyGood 1: Commodity under study `; price: p` = p; agent i ’s consumption: xiGood 2: Numeraire (money) m; price: pm = 1; agent i ’s consumption: mi

Consumers: i = 1, . . . , IUtility: ui (xi ,mi ) = φi (xi ) + mi , mi ∈ R, xi ∈ R+

Assumptions: φ′i > 0, φ′′i < 0, φi (0) = 0

Firms: j = 1, . . . , JEmploy good m to produce `. Since pm = 1, if firm j uses zj units of m toproduce `, then its cost is zjProduction sets: Yj = {(−zj , qj ) : qj ≤ fj (zj ), qj ≥ 0} ={(−zj , qj ) : zj ≥ cj (qj ), qj ≥ 0}, where f ′j > 0, c′j > 0 and c′′j > 0 (strictconvexity)NB: Since J is fixed, we are in the SR!

Endowments and SharesGood `: no endowments ω` = 0 (must be produced)Good m: ωmi , i = 1, . . . , I, ωm = ΣiωmiEach consumer i owns a share θij of firm j , Σiθij = 1


Supply Behavior

Partial Equilibrium Competitive Analysis

Competitive equilibria:� At price p∗, firm j ’s equilibrium output level q∗

j must solve

maxqj≥0

p∗qj − cj(qj)

with necessary and sufficent FOC:

p∗ ≤ c �j (qj), with equality if q∗j > 0

� consumer i ’s equilibrium consumption vector (m∗i , x

∗i ) must

solve

maxmi∈Rxi∈R+

mi + φi (xi )

s.t. mi + p∗xi ≤ ωmi +�J

j=1 θij(p∗q∗ − cj(q∗j ))


Demand Behavior

Each consumer i given p∗ and π∗j = p∗q∗j − cj (q∗j ) solves:

Partial Equilibrium Competitive Analysis

Competitive equilibria:� At price p∗, firm j ’s equilibrium output level q∗

j must solve

maxqj≥0

p∗qj − cj(qj)


p∗ ≤ c �j (qj), with equality if q∗j > 0

� consumer i ’s equilibrium consumption vector (m∗i , x

∗i ) must

solve

maxmi∈Rxi∈R+

mi + φi (xi )

s.t. mi + p∗xi ≤ ωmi +�J

j=1 θij(p∗q∗j − cj(q∗

j ))Partial Equilibrium Competitive AnalysisCompetitive equilibria:

� . . . since the budget constraint must hold with equality:

maxxi≥0

φi (xi ) − p∗xi +

ωmi +

J�

j=1

θij(p∗q∗j − cj(q∗

j ))


φ�i (x

∗i ) ≤ p∗, with equality if x∗

i > 0

� market clearing (only for good �, by lemma 10.B.1)

I�

i=1

x∗i =

J�

j=1

q∗j

Note: equilibrium conditions do not depend on the distribution ofendowments and ownership shares


Equilibrium

It is an allocation (x∗1 , . . . , x∗I ; q∗1 , . . . , q

∗J ) and a price (scalar) p∗ such

that:1 All firms maximize profit given technology and p∗, i.e. c′j (q∗j ) ≥ p∗ (= if

q∗j > 0)2 All consumers maximize utility s.t. budget constraint given p∗ and firms’

optimal profits, i.e. φ′i (x∗i ) ≤ p∗ (= if x∗i > 0)3 Market for good ` clears: Σi x∗i = Σj q∗j

RemarksSince all utilities satisfy non-satiation, all budget constraints are satisfiedwith equality and therefore all markets must clear (with equality)Check that m∗i and z∗j can be recovered using BC and production functionDoes equilibrium condition imply that market for numeraire also clears?Check that using budget constraints. . .∑

j

q∗j =∑

i

x∗i ⇒∑

i

m∗i +∑

j

z∗j =∑

i

ωmi


Existence

Does an interior equilibrium always exist? No!Sufficient condition: If maxi φ

′i (0) > minj c′j (0) then an interior equilibrium

Σi x∗i = Σj q∗j > 0 does existWhy? Suppose not, then there exists an equilibrium such thatΣi x∗i = Σj q∗j = 0, meaning that x∗i = q∗j = 0 for all i, j . From conditions (1)and (2) we have: φ′i (0) ≤ c′j (0). This implies: maxi φ

′i (0) ≤ minj c′j (0),

which contradicts the assumption.

ConsequencesIn any interior solution: φ′i (x∗i ) = p∗ = c′j (q∗j ). Thus equilibrium price:

is equal to firm marginal benefit in selling one additional unit of good (i.e. firmmarginal 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 ownershipshares (because of quasi-linearity of utility functions)


Aggregate Supply and Aggregate Demand

What does all that have to do with the traditional demand=supplyscheme?

Consumer behavior: x∗i = xi (p) for any given level pFirm behavior: q∗j = qj (p) for any given level pMarket clearing condition⇒ aggregate demand = aggregate supply

Aggregate Supply and Aggregate DemandS(p) = Σj qj (p), increasing in p if p > minj c′j (0)

D(p) = Σi xi (p), decreasing in p if p < maxi φ′i (0)


AS vs. AD: Graphical Construction

Aggregate Supply Function for Good �

Aggregate Demand Function for Good �


AS vs. AD: Equilibrium

Market Equilibrium

� the inverse industry supply function gives the industry marginalcost function

� the inverse industry demand function gives the marginal socialbenefit of good �


Partial Equilibrium: When does it fail to exist?

See exercise!


Comparative Statics

How does a change in underlying conditions affect the equilibriumoutcome?

exogenous parameters α ∈ RM affecting consumer’s preferences: φi (xi , α)

exogenous parameters β ∈ RS affecting firm’s technology: cj (qj , β)

exogenous tax and subsidy parameters t ∈ RK affecting price: p̂i (p, t) (paidby 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 usedto derive the marginal change in equilibrium allocation and price in responseto a differential change in parameters


Comparative statics effects of a sales tax

Suppose to introduce a sales tax such that consumers must pay t ≥ 0for 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 p′(t):

d(p(t))

dt= p′(t) = − D′(p(t) + t)

D′(p(t) + t)− S′(p(t))∈ [−1, 0)

d(p(t) + t)dt

= p′(t) + 1 ∈ [0, 1)

Total quantities produced and consumed fall

But how much this change is felt by consumers and producers mainlydepends on the steepness of supply function: the steeper S, the morerigid production to prices, the more producers feel the burden of the saletax


Comparative statics effects of a sales tax: Three cases

The larger |S′(p(t))|, the flatter the S schedule

If |S′(p(t))| very large (i.e. supply very elastic, i.e. horizontal) then S isflat and all the burden is felt by consumers

If |S′(p(t))| is very small (i.e. supply very rigid, i.e. vertical) then S isvertical and all the burden is felt by producers


Main questions

1 Suppose that for a given endowment distribution and owner shares the marketmechanisms reaches an equilibrium allocation (x∗1 , . . . , x

∗I ; q∗1 , . . . , q

∗J ) and a

price (scalar) p∗. Is that allocation Pareto optimal?2 Suppose we start from a Pareto optimal allocation (xo

1 , . . . , xoI ; qo

1 , . . . , qoJ ). Is

there a redistribution of initial endowments that allows to sustain that allocation asa market equilibrium? I.e. is there a way to redistribute initial endowments in sucha way that a given PO allocation will be reached by market forces without anyadditional intervention?

The allocation maximizessocial welfare for some

choice of weights

(x∗)i, (y∗)j

λ∗

The allocation is a Pareto optimum

(x∗)i, (y∗)j

Conv

exity

{p∗, (x∗)i, (y∗)j}

The price-allocation

is a competitive equilibrium

?


The Utility Possibility Set (UPS)

The UPS for the partial-equilibrium economy is defined as

U = {(u1, . . . , uI) :I∑

i=1

ui ≤I∑

i=1

φi (xi ) + ωm −J∑

j=1

cj (qj )}

for any given 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 withall PO allocations. Therefore all allocations such that u ∈ {(u1, . . . , uI) :∑I

i=1 ui = max{∑I

i=1 φi (xi ) + ωm −∑J

j=1 cj (qj ),∑

i xi =∑

j qj}}, are PO.

Recall: When consumer preferences are quasilinear, the boundary ofthe economy’s utility possibility set is linear, and all points in thisboundary are associated with consumption allocations that differ only inthe distribution of the numeraire (ωi1, . . . , ωIm) s.t. ωm = Σiωim


The Utility Possibility Set (UPS)


Partial Equilibrium Analysis: First Welfare Theorem

Theorem

If the price p∗ and the allocation (x∗, q∗) = (x∗1 , . . . , x∗I , q

∗1 , . . . , q

∗J )

constitutes a competitive (Walrasian) equilibrium, then (x∗, q∗) is Paretooptimal

Proof.

We prove it for the case of an interior allocation. An allocation(x+

1 , . . . , x+I , q

+1 , . . . , q

+J ) is Pareto optimal if given the set U, it lies in the

boundary of the UPS, i.e. if it maximizes

I∑i=1

φi (xi ) + ωm −J∑

j=1

cj (qj )

subject to feasibility constraints, i.e.∑

i xi =∑

j qj .


Partial Equilibrium Analysis: First Welfare Theorem

Theorem

If the price p∗ and the allocation (x∗, q∗) = (x∗1 , . . . , x∗I , q

∗1 , . . . , q

∗J )

constitutes a competitive (Walrasian) equilibrium, then (x∗, q∗) is Paretooptimal

Proof (Cont’d).

FOCs for this problem read (in an interior solution):

φ′i (x+i ) = µ = c′j (q

+j )

∑i

x+i =

∑j

q+j

Note that FOCs are sufficient because φ′′i < 0 and c′′j > 0. Thus anallocation that solves FOCs is PO. But FOCs are equivalent to those for aninterior competitive equilibrium when µ = p∗. Therefore, if p∗ and(x∗1 , . . . , x

∗I , q

∗1 , . . . , q

∗J ) satisfy FOCs for µ = p∗, they also satisfy sufficient

conditions for PO.


Partial Equilibrium Analysis: Second Welfare Theorem

Theorem

For any Pareto optimal levels of utility (u∗1 , . . . , u∗I ) ∈ UPF, there are transfers

of the numeraire commodity (T1, . . . ,TI) satisfying∑

i Ti = 0, such that acompetitive equilibrium reached from the endowments(ωm1 + T1, . . . , ωmI + TI) yields precisely the utilities (u∗1 , . . . , u

∗I )

Argument: consumption levels, production levels and firms’s profits areunaffected by changes in consumers’ wealth levels; thus, ex-ante transfers ofthe numeraire causes each equilibrium consumption of the numeraire tochange by exactly the amount of the transfer; hence, they allow to reach anyutility vector in the boundary of the utility possibility set

Note: convexity assumptions about technologies and preferences are centralto show this result!


Equilibrium and Welfare

The allocation maximizessocial welfare for some

choice of weights

(x∗)i, (y∗)j

λ∗

The allocation is a Pareto optimum

(x∗)i, (y∗)j

Conv

exity

{p∗, (x∗)i, (y∗)j}

The price-allocation

is a competitive equilibrium

Conv

exity


Free-Entry and Long-Run Competitive Equilibria

So far: Short run competitive equilibriaNumber of firms J is fixedFirms hold heterogeneous technologies and cannot change them

Long run: free entry/exit modelLet J (number of firms) in the market be endogenous: firms can freelyenter/exit in response to π opportunities: a firm will enter if it can earn apositive profit upon entry; it exits if it gets negative profits at the current pricep for any q > 0There is a potentially infinite set of firms all having access to technologyc(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

Long run equilibrium. It is a triple (p∗, q∗, J∗) such that:1 π-max: q∗ solves maxq≥0{p∗q − c(q)}2 AD=AS: X(p∗) = J∗ · q∗3 Free entry: π(p∗) = p∗q∗ − c(q∗) = 0


Long-Run Aggregate Supply (LR-AS) Correspondence

Define LR-AS as:

Q(p) =

∞ if π(p) > 0{Q ≥ 0 : Q = Jq for some J ≥ 0 and q ∈ q(p)} if π(p) = 00 if π(p) < 0

Intuition:If π(p) > 0 then every firm wants to supply a strictly positive amount ofproduct, therefore AS is∞If π(p) = 0 and for some integer J we have Q = Jq(p), then there are Jfirms in the market each producing q(p) at price p. All other firms stay outand produce 0 (which is a profit maximizing choice as c(0) = 0)If π(p) < 0 all firms produce 0

LR equilibrium: alternative characterizationThe price level p∗ is a LR competitive equilibrium if and only ifX(p∗) = Q(p∗)Proof: See MWG, p. 336, footnote 27


Example 1: LR Equilibrium with CRTS technology

Free-Entry and Long-Run CompetitiveEquilibria

Constant returns to scale

Q(p) =

∞ if p > c[0,∞) if p = c

0 if p < c


Example 2: LR Equilibrium with Strictly DRTS technology

Free-Entry and Long-Run CompetitiveEquilibria

Non existence of long run competitive equilibrium withstrictly convex costs

Q(p) =

�∞ if p > c �(0)0 if p ≤ c �(0)


Example 3: LR Equilibrium with U-shaped CostsFree-Entry and Long-Run CompetitiveEquilibria

Average costs exhibit a strictly positive efficient scale

Q(p) =

∞ if p > c̄{Q ≥ 0 : Q = Jq̄ for some integer J ≥ 0 if p = c̄0 if p < c̄


Conclusions

Partial Equilibrium ApproachFrom the positive side, it allows to determine the equilibrium outcome in onemarket in isolation from all other marketsFrom the normative side, it allows to prove the 1st and 2nd welfare theorems(a sort of formal expression of Adam Smith’s “invisible hand”)

ProblemsPrices of all other goods remain fixedThere are no wealth effects in the market under studyWhat about issues that are inherently general-equilibrium ones?It is crucial to consider a setup where all prices are simultaneouslydetermined so as to take into account economy-wide feedbacks!Questions: Do welfare theorems still hold?

Documents

Advanced Microeconomics Partial and General … The Model Equilibrium Welfare Long-Run Equilibrium Conclusions Advanced Microeconomics Partial and General Equilibrium Giorgio Fagiolo