Advanced Microeconomics II

by Jinwoo Kim

October 6, 2010

Contents

I General Equilibrium and Social Welfare 3

1 General Equilibrium Theory 3

1.1 Basic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Pareto Efficient Allocations . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Walrasian Equilibrium and Core . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Example: 2 × 2 Pure Exchange Economy . . . . . . . . . . . . . . . . . . . 9

2 Equilibrium Analysis 13

2.1 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Welfare Properties of WE . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Uniqueness and Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4 Equivalence between Core and WE . . . . . . . . . . . . . . . . . . . . . . 22

3 Equilibrium in Production Economy 26

3.1 Profit Maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 Utility maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.3 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1

3.4 First and Second Welfare Theorems . . . . . . . . . . . . . . . . . . . . . 29

4 General Equilibrium under Uncertainty 32

4.1 Arrow-Debreu Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.2 Asset Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5 Public Goods and Externality 36

5.1 Public Goods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.2 Externality and Lindahl Equilibrium . . . . . . . . . . . . . . . . . . . . . 39

6 Social Choice and Welfare 42

6.1 Arrow’s Impossibility Theorem . . . . . . . . . . . . . . . . . . . . . . . . 42

6.2 Some Possibility Results: Pairwise Majority Voting in Restricted Domain 44

2

Part I

General Equilibrium and Social Welfare

1 General Equilibrium Theory

1.1 Basic model

• Consider an economy in which there are n goods traded.

• The initial resources, or endowment, of the economy is given as a vector e = (e1, · · · , en) ∈Rn

+.

Consumers

• Assume that there are I consumers:

– I = {1, · · · , I} : Set of consumers.

– xik : Consumer i’s consumption of good k, xi = (xi

1, · · · , xin) ∈ X i, where X i is

i’s consumption set and assumed to be Rn+.

– eik : i’s endowment for good. Let ei = (ei1, · · · , ein) and e = (e1, · · · , eI). Notethat ek =

∑i∈I e

ik.

– ≽i: i’s preference defined on X i, which we assume can be represented by afunction ui : X i → R.

• Assumption C: For any i ∈ I,(i) ui is continuous.(ii) ui is strictly quasiconcave: For all x, y ∈ Rn

+,

ui(λx+ (1− λ)y) > min{ui(x), ui(y)},∀λ ∈ (0, 1).

(iii) ui is strictly monotone: If x > y (that is xk ≥ yk, ∀k and xk > yk for some k),then ui(x) > ui(y).

3

Producers

• Assume that there are J producers:

– J ={1, · · · , J}: Set of producers.

– yjk : Firm j’s output (or input) for good k, yj = (yj1, · · · , yjn) ∈ Rn : j’s productionplan.

∗ If yjk > (<)0, then good k is produced (used) as output (input).

– Y j ⊂ Rn : Set of feasible production plan for j, called production possibility set.

∗ Call any production plan yj ∈ Y j feasible.

∗ Assume that there exists a function F j : Rn → R such that Y j = {y ∈Rn |F j(y) ≤ 0}.

∗ {y ∈ Rn |F j(y) = 0} : Production possibility frontier for firm j.

– Y ≡ {y∣∣ y =

∑j∈J yj, where yj ∈ Y j,∀j ∈ J }: Aggregate production possibil-

ity set.

• Assumption P: For any j ∈ J ,

(i) 0 ∈ Y j.

(ii) Y j is closed and bounded.(iii) Y j is strictly convex.

Example 1.1. Suppose that there are one input and one output.

.

.y2

.ProductionPossibility Set

.y1.O

4

Feasible Allocations

• An allocation in this economy is denoted by (x, y) = (x1, · · · , xI , y1, · · · , yJ) ∈ RnI+ ×

RnJ .

• Given e = (e1, · · · , eI), an allocation (x, y) is feasible if∑i∈I

xik ≤ ek +

∑j∈J

yjk, ∀k = 1, · · · , n. (1)

• Let F (e, Y ) denote the set of all feasible allocations i.e. set of all (x, y)’s satisfying(1).

1.2 Pareto Efficient Allocations

Definition 1.2. An allocation (x, y) ∈ F (e, Y ) is Pareto efficient (PE) if there does notexist any other allocation (x, y) ∈ F (e, Y ) such that ui(xi) ≥ ui(xi),∀i ∈ I with at leastone strict inequality.

• With fixed utility levels, u2, · · · , uI , let us solve the following maximization problem:

max(x,y)∈RnI

+ ×RnJu1(x1

1, · · · , x1n) (2)

s.t. ui(xi1, · · · , xi

n) ≥ ui, i = 2, · · · , I (3)∑i∈I

xik ≤ ek +

∑j∈J

yjk, k = 1, · · · , n (4)

F j(yj1, · · · , yjn) ≤ 0, j = 1, · · · , J (5)

– One can show the following: An allocation (x, y) is Pareto efficient if and onlyif it solve (2) for some utility levels, u2, · · · , uI .

5

First Order Conditions for Pareto Efficiency

• Letting δi,µk, and γj denote a nonnegative multiplier for (3), (4), (5), respectively,the Lagrangian function for problem (2) can be set up as

L =u1(x11, · · · , x1

n) +I∑

i=2

δi[ui(xi1, · · · , xi

n)− ui]

+n∑

k=1

µk[ek +∑j∈J

yjk −∑i∈I

xik]

+J∑

j=1

γj[−F j(yj1, · · · , yjn)].

– Define δ1 ≡ 1. Assuming the interior solution (i.e. xi > 0, ∀i), the first orderconditions for maximizing the Lagrangian are

δi∂ui

∂xik

= µk, ∀i, k or δi∇ui = µ, ∀i (6)

γj ∂Fj

∂yjk= µk, ∀j, k or γj∇F j = µ, ∀j, (7)

where ∇ui = ( ∂ui

∂xi1, · · · , ∂ui

∂xin), ∇F j = (∂F

j

∂yj1, · · · , ∂F j

∂yjn), and µ = (µ1, · · · , µn).

– The conditions (6) and (7) imply the followings:

µk

µk′=

∂ui/∂xik

∂ui/∂xik′

=∂ui′/∂xi′

k

∂ui′/∂xi′k′

for all i, i′, k, k′

that is, MRS for any pair of goods must be equalized across consumers.

µk

µk′=

∂F j/∂yjk∂F j/∂yjk′

=∂F j′/∂yj

′

k

∂F j′/∂yj′

k′

for all j, j′, k, k′

that is, MRT for any pair of goods must be equalized across producers.

∂ui/∂xik

∂ui/∂xik′

=∂F j/∂yjk∂F j/∂yjk′

for all i, j, k, k′

that is, MRS and MRT for any pair of goods must be equalized across consumersand producers.

6

1.3 Walrasian Equilibrium and Core

From here on, we focus on the exchange economy in which there is no production i.e.Y j = {0},∀j ∈ J . We denote an exchange economy by E = (ui, ei)i∈I.

Walrasian (or Competitive) Equilibrium

• Let us introduce a perfectly competitive market system:

– Consumers see themselves as not being able to affect prevailing prices in themarkets.

– Consumers only need to look at the market prices and not worry about whatother consumers might demand or how demands are satisfied.

• Suppose that price of good k is given as pk > 0 with p = (p1, · · · , pn) ∈ Rn++ being a

price vector.

– p · ei =∑n

k=1 pkeik: Market value of i’s endowment i.e. i’s wealth.

– Consumer i’s budget set is

Bi(p) = {xi ∈ Rn+ | p · xi ≤ p · ei}.

• Given market price vector p ∈ Rn++, each consumer i has to solve

maxxi∈Rn

+

ui(xi) s.t. xi ∈ Bi(p). (8)

– Let xi(p, p·ei) = (xi1(p, p·ei), · · · , xi

n(p, p·ei)) denote the optimal bundle/bundles(or demand function/correspondence) that solves (8).

– Note that xi(p, p · ei) is not continuous in p in all of Rn+ since demand will be

infinite if some price is zero.

• The aggregate excess demand function for good k is defined as

zk(p) ≡∑i∈I

xik(p, p · ei)− ek.

7

– If zk(p) > (<)0, we say that good k is in excess demand (supply).

– Define z(p) ≡ (z1(p), , · · · , zn(p)).

Definition 1.3. Walrasian equilibrium (WE) is a price vector p∗ ∈ Rn++ such that zk(p∗) =

0, ∀k.

Core

The core is another equilibrium concept that has its foundation in the cooperative gametheory and assumes more centralized market than the Walrasian equilibrium does.

• Let S ⊂ I denote a coalition of consumers. We say S blocks x if there is an allocationx such that(i)∑

i∈S xi =

∑i∈S e

i

(ii) xi ≽ xi for all i ∈ S with at least one preference strict.

– Note that a feasible allocation x is Pareto efficient if and only if it is not blockedby S= I.

Definition 1.4. A feasible allocation x is in the core if and only if x is not blocked by anycoalition of consumers, i.e. the core is the set of allocations that are not blocked by anycoalitions.

The following result shows the relationship between allocations in WE and core.

Theorem 1.5. If each ui is strictly monotone, then any WE allocation belongs to the core.

Proof. Letting p∗ and x denote WE price and allocation vectors, suppose to the contrarythat x does not belong to the core. Then, we can find a coalition S ⊂ I and anotherallocation x such that∑

i∈S

xi =∑i∈S

ei and ui(xi) ≥ ui(xi),∀i ∈ S with at least one inequality strict.

By the strict monotonicity, this implies

p∗ · xi ≥ p∗ · ei, ∀i ∈ S with at least one inequality strict,

8

which can be added up across consumers to yield

p∗ ·∑i∈S

xi > p∗ ·∑i∈S

ei.

This, however, contradicts with∑

i∈S xi =

∑i∈S e

i.

1.4 Example: 2 × 2 Pure Exchange Economy

Suppose that n = 2 and I = 2 and call a feasible allocation nonwasteful if it satisfies (1)with equality.

• The set of nonwasteful allocations can be depicted by an Edgeworth box.

.

.p1/p2

.p1/p2

.endowmentpoint

.e11

.e21

.e22

.e12

.O2

.O1

• How to find the set of PE allocations

– Fix an indifference curve for consumer 2 and identify allocation(s) on that curvethat maximizes consumer 1’s utility.

– Do this for all possible indifference curves of consumer 2, which will give us theset of all PE allocations, often called contract curve.

9

.

.Contract curve

.O1

.O2

• How to find the WE allocations

– Trace out optimal bundles for each consumer i by varying p = (p1, p2) and obtainan offer curve denoted OCi.

..O1

.O2

.e

.OC1

– Any intersection of OC1 and OC2, which is different from the endowment point,corresponds to an equilibrium allocation.

10

.

.O1

.O2

.e

.OC1

.OC2

.E

– Note that WE allocation is always PE (First Welfare Theorem).

• The core is the segment of contract curve that lies within the lens formed by twoindifference curves.

.

.Core

.Contract

.curve

.O1

.O2

.e

Example 1.6 (Calculating WE). Consider an exchange economy with two consumers andtwo goods:

e1 = (1, 0) and e2 = (0, 1)

11

u1(x1) = (x11)

a(x12)

1−a and u2(x2) = (x21)

b(x22)

1−b, a, b ∈ (0.1).

Setting p1 = 1 for normalization, consumer 1 solves

maxx1

u1(x1) subject to x11 + p2x

12 = 1,

which yieldsx11(1, p2) = a and x1

2(1, p2) =1− a

p2.

Likewise, for consumer 2, we obtain

x21(1, p2) = bp2 and x2

2(1, p2) = 1− b

The market clearing condition is

z1(1, p2) = x11(1, p2) + x2

1(1, p2)− 1 = a+ bp2 − 1 = 0,

which yields p2 = 1−ab

. Note that given p2 = 1−ab

, the market for good 2 is cleared, i.e.z2(1, p2) = 0, as can be expected by the Walras’ Law.

Example 1.7 (Calculating PE allocation). Consider the same setup as in Example 1.6.We can set up the Lagrangian for calculating the PE allocations as

L =(x11)

a(x12)

1−a + δ2[(x21)

b(x22)

1−b − u2]

+ µ1[1− x11 − x2

1] + µ2[1− x12 − x2

2],

whose first-order conditions, assuming the interior solution, are

a(x11)

a−1(x12)

1−a = µ1 = δ2b(x21)

b−1(x22)

1−b (9)

(1− a)(x11)

a(x12)

−a = µ2 = δ2(1− b)(x11)

a(x22)

−b (10)

x11 + x2

1 = 1 = x12 + x2

2. (11)

Dividing the RHS and LHS of (9) and (10) side by side, we obtain

a

1− a

x12

x11

=b

1− b

x22

x21

=b

1− b

1− x12

1− x11

,

where the second equality follows from applying (11). Rearranging this yields

x11 =

a(1− b)x12

b(1− a) + (a− b)x12

. (12)

So the set of PE allocations or contract curve contains all the points in the Edgeworthbox that satisfy (12). Try to see for yourself what the contract curves look like for varyingvalues of a and b.

12

2 Equilibrium Analysis

2.1 Existence

The proof of WE existence relies much on the properties possessed by the excess demandfunctions, which thus we first explore.

Properties of Excess Demand Functions

• Under Assumption C, the following basic properties hold: For any i ∈ I,

(P0) Uniqueness: The problem (8) has a unique solution, i.e. xi(p, p ·ei) is a functionof p.

– Follows from the strict quasiconcavity of ui.

(P1) Continuity: The demand function xi(p, p · ei), and thus excess demand functionzi(p), is continuous in p on Rn

++.

– Follows from the continuity of ui and Berge’s maximum theorem.

(P2) Homogeneity: zi(p) is homogeneous of degree 0 i.e. zi(λp) = zi(p),∀p, ∀λ > 0.

– Follows from the fact that Bi(λp) = Bi(p).

• Under Assumption C, we can prove some further properties:

(P3) Walras’ Law: p · z(p) = 0,∀p ∈ Rn++.

Proof. Write down the budget constraint for each consumer i ∈ In∑

k=1

pk[xik(p, p · ei)− eik] = 0.

Summing across consumers yields

0 =∑i∈I

n∑k=1

pk[xik(p, p · ei)− eik]

13

=n∑

k=1

pk

[∑i∈I

[xik(p, p · ei)− eik]

]

=n∑

k=1

pkzk(p) = p · z(p).

– This result implies that if n−1 markets clear, then the remaining 1 market mustalso clear.

(P4) Bounded below: zk(p) ≥ ek,∀k (Obvious).

(P5) Unbounded above: Suppose that e ≫ 0. Then, if {pm}∞m=1 is a sequence of pricevectors in Rn

++ converging to p = 0 and pk = 0 for some k, then we must havezk′(p

m) → ∞ for some k′ with pk′ = 0.

Proof. Since p · [∑

i∈I ei] = p · e > 0, there is at least one i ∈ I such that

p · ei > 0. Now let xm ≡ xi(pm, pm · ei). Suppose for a contradiction that{xm}∞m=1 is bounded. Then, there must exist some x∗ ≪ ∞ such that xm → x∗.

Note that as m → ∞,

pm · xm = pm · ei → p · x∗ = p · ei > 0.

Let x = (x∗1, · · · , x∗

k + 1, · · · , x∗n) and then ui(x) > ui(x∗) (by strict monotonic-

ity) and p · x = p · ei > 0. Since ui is continuous, there exists t ∈ (0, 1) suchthat

ui(tx) > ui(x∗) and pm · (tx) < pm · ei.

Thus, for large enough m,

ui(tx) > ui(xm) and pm · (tx) < pm · ei,

which is a contradiction since xm maximizes i’s utility given price vector pm.

So, there must be some k′ such that {xmk′} is unbounded. Also,

pm · xm = pm · ei → p · ei < ∞ implies pk′ = limm→∞

pmk′ = 0.

14

Proof of Existence

• The existence in two goods case can be proved in the following steps:

1. By (P2), we can normalize one of two prices to 1 say p2 = 1.

2. By (P3), we can focus on one market, say good 1 market.

3. For some p′1 w 0, we must have z1(p′1, 1) > 0 by (P5).

4. For some p′′1 w ∞, we must have z1(p′′1, 1) < 0 by (P3) and (P5).

5. By (P2) and mean value theorem, there is some p∗1 ∈ (p′1, p′′1) such that z1(p∗1, 1) =

0.

.

.p1.p∗∗1

.p∗1 .p∗∗∗1

.z1(·)

.O .p′1

.p′′1

The above argument, however, cannot be used when there are more than two goods. Sowe rely on the fixed point theorem, the version of Brower.

Theorem 2.1 (Borwer’s Fixed Point Theorem). Let A be a nonempty, compact, andconvex subset of Rn and f : A → A be continuous. Then, f has a fixed point, that is thereexists x∗ ∈ A such that f(x∗) = x∗.

Using the fixed point theorem and the properties above, we are able to prove that WEalways exists.

Theorem 2.2 (Existence of WE). If Assumption C holds and e ≫ 0, then there exists atleast one price vector, p∗ ≫ 0, such that z(p∗) = 0.

15

Proof. For each k, let zk(p) ≡ min(zk(p), 1), ∀p ≫ 0 and let z(p) ≡ (z1(p), · · · , zn(p)). Sozk(p) ≤ 1,∀p, ∀k.

Fix ϵ ∈ (0, 1) and let

Sϵ ≡ {p |n∑

k=1

pk = 1 and pk ≥ϵ

1 + 2n, ∀k}.

Note that Sϵ is compact, convex, and nonempty.

For each k and every p ∈ Sϵ, define fk(p) as

fk(p) ≡ϵ+ pk +max{0, zk(p)}

nϵ+ 1 +∑n

m=1max{0, zm(p)}

and let f(p) = (f1(p), · · · , fn(p)). Hence, for all p ∈ Sϵ,

n∑k=1

fk(p) = 1 and fk(p) ≥ϵ

nϵ+ 1 + n≥ ϵ

1 + 2n,

which implies that f is a mapping from Sϵ to itself. Since each zk is continuous in p on Sϵ,fk and f are also continuous in p on Sϵ. Appeal to the fixed point theorem to find pϵ ∈ Sϵ

such that f(pϵ) = pϵ or fk(pϵ) = pϵk, ∀k, which means

pϵk

[nϵ+

n∑m=1

max{0, zm(pϵ)}

]= ϵ+max{0, zk(pϵ)}. (13)

Now, let ϵ → 0 and find a sequence of ϵ such that pϵ → p∗ for some p∗(since the sequencebelongs to a compact set). We argue that p∗ ≫ 0. Suppose not for a contradiction. Then,p∗k = 0 for some k and, by (P5), zk′(pϵ) → ∞ for some k′ with p∗k′ = 0, which contradicts(13).

Thus, pϵ → p∗ ≫ 0 as ϵ → 0, which implies by the continuity of z that for all k

p∗k

n∑m=1

max{0, zm(p∗)} = max{0, zk(p∗)}.

Multiplying zk(p∗) to both sides and summing up for all k’s, we obtain

0 = p∗ · z(p∗)

(n∑

m=1

max{0, zm(pϵ)}

)=

n∑k=1

zk(p∗)max{0, zk(p∗)}. (14)

Since zk and zk have the same sign and p∗ · z(p∗) = 0, we must have zk(p∗) = 0,∀k due to

(14).

16

• WE may not exist in cases

– Preferences are not strictly monotone: Suppose that

u1(x1,x2) = x1 and u2(x1,x2) = x1x2

and e1 = (1, 0) and e2 = (0, 1). It is straightforward to verify that given p2 = 1,there exists no price p1that clears the markets.

– Preferences are nonconvex

.

.slope = p′1

.slope = p′′′1

.slope = p′′1

.p2 = 1

.e

.O1

.O2

.

.z21(·)

.z11(·)

.p1.p′1

.p′′1 .p′′′1.O

.

.

Corollary 2.3. If Assumption C holds and e ≫ 0, then the core is nonempty.

Proof. This is immediate from combining Theorem 1.5 and Theorem 2.2.

2.2 Welfare Properties of WE

First Welfare Theorem

The first welfare theorem is a positive answer to the question of whether the competitivemarket mechanism always yields an efficient allocation.

Theorem 2.4 (First Welfare Theorem). If each ui is strictly increasing, then every WEallocation is PE.

Proof. Immediate from Theorem 1.5 and the fact that all core allocations are PE.

17

Second Welfare Theorem

The second welfare theorem is the converse of the first welfare theorem, i.e. any PEallocation can be achieved as an equilibrium allocation in a competitive market (withappropriate redistribution).

Theorem 2.5 (Second Welfare Theorem). Consider an exchange economy E = (ui, ei)i∈I

satisfying Assumption C and e ≫ 0. Then, for every PE allocation x, there are a pricevector p and a redistribution of wealth (w1, · · · , wI) ∈ RI such that(i)∑

i∈I wi = 0 (budget balance) and

(ii) For every i, xi ∈ argmax ui(xi) subject to p · xi ≤ p · ei + wi (utility maximization).

Proof. Since x is feasible, we have∑

i∈I xi = e ≫ 0. Thus, by Theorem 2.2, we can

find a WE price vector p and WE allocation x for the economy E = (ui, xi)i∈I . Now setwi = p · (xi − ei), which satisfies (i) since∑

i∈I

wi = p ·∑i∈I

(xi − ei) = 0.

To see that (ii) is satisfied, note first that ui(xi) ≥ ui(xi), ∀i ∈ I since in the economyE , consumers must be maximizing their utilities with endowments (xi)i∈I . Since x is PE,however, we must have ui(xi) = ui(xi), which implies (by the strict quasiconcavity of ui)that xi = xi. Since p and x are WE prices and allocation for the economy E , we have forall i

xi = xi ∈ argmax ui(xi) subject to p · xi ≤ p · xi = p · ei + wi,

as desired.

Example 2.6. The following graph illustrates how the second welfare theorem can failwhen the preference is nonconvex:

18

.

.O1

.O2

2.3 Uniqueness and Stability

Uniqueness

When there are more than one equilibria, it is harder for us as economists to provide anunambiguous prediction about what will happen in the economy. Also, the second welfaretheorem loses its power since, if the economy resulting from the redistribution admitsmultiple equilibria, then some unwanted allocation may arise in equilibrium. Here we lookfor conditions under which WE is unique.

Definition 2.7. The excess demand function z(·) satisfies the weak axiom of revealedpreference (WARP) if for any pair of price vectors p and p′, we have

z(p) = z(p′) and p · z(p′) ≤ 0 implies p′ · z(p) > 0.

• Letting x ≡∑

i∈I xi(p, p · ei) and x′ ≡

∑i∈I x

i(p′, p′ · ei), WA can be rewritten as

p · x ≥ p · x′ and x = x′ implies p′ · x > p′ · x′.

– So, WARP requires that the aggregate demand behaves as if it is generated bya single consumer.

19

– Can show that WARP is satisfied if the indirect utility function for each con-sumer i with income mi, denoted vi(p,mi), takes a Gorman form

vi(p,mi) = ai(p) + b(p)mi.

Proposition 2.8. If z(·) satisfies the WARP, then the set of WE is convex (and so, if theset of WE is finite, there can be at most one WE).

Note: In a generic economy, the number of WE is finite and thus WE, if exists, is unique.

Proof. Suppose for a contradiction that for some p, p′, and α ∈ (0, 1), we have

z(p) = z(p′) = 0 = z(αp+ (1− α)p′).

Let p′′ = αp+ (1− α)p′. Since z(p) = 0 = z(p′′) and p′′ · z(p) = 0, WA implies

p · z(p′′) > 0. (15)

Also, since z(p′) = 0 = z(p′′) and p′′ ˙·z(p′) = 0, WA implies

p′ · z(p′′) > 0. (16)

Combining (15) and (16) yields

p′′ · z(p′′) = αp · z(p′′) + (1− α)p′ · z(p′′) > 0,

which contradicts the Walras’ Law.

Here I introduce another condition that guarantees a uniqueness of WE.

Definition 2.9. The function z(·) has the gross substitute (GS) property if whenever p′

and p are such that, for some ℓ, p′ℓ > pℓ and p′k = pk for k = ℓ, we have zk(p′) > zk(p) for

k = ℓ.

Proposition 2.10. An aggregate excess demand function z(·) that satisfies the GS propertyhas at most one WE.

Proof. Suppose for a contradiction that for some p∗ and p = αp∗ for any α > 0, we havez(p∗) = z(p) = 0. Since z(·) is homogeneous of degree 0, we can find some α > 0 andp′ = αp∗ such that p′ ≥ p and p′ℓ = pℓ for some ℓ. Now consider changing p′ to obtain p inn− 1 steps, lowering (or keeping unchanged) the price of every good k = ℓ one at a time.By GS, the excess demand of good ℓ cannot decrease in any step, and, because p = p′, itwill actually increase in at least one step. Hence, zℓ(p) > zℓ(p

′), a contradiction.

20

Tatonnement Stability

Let us suppose that an economy is in disequilibrium state. Can we be sure that the economywill eventually evolve into some equilibrium state?

• Let us assume that the dynamic path of price takes the following form of differentialequation: For every k,

dpkdt

= ckzk(p), (17)

where dpk/dt is the rate of change of the price for good k and ck > 0 is a constantaffecting the speed of adjustment.

– This adjustment process can be better considered as a tentative trial-and-errorprocess taken by an auctioneer who tries to find the equilibrium prices.

– According to (17), the excess demand (supply) for good k causes its price tomove upward (downward).

– Suppose that there is a unique WE p∗. We say that p∗ is globally stable if everytrajectory satisfying (17) converges to p∗ (or αp∗ for some α > 0 ).

Proposition 2.11. Suppose that z(p∗) = 0 and p∗ · z(p) > 0 for every p not proportionalto p∗. Then, p∗ is globally stable.

Proof. We borrow a result from Lyapunov’s stability theory. To do so, consider a dynamicalsystem

x = f(x)

where f : Rn → Rn, and a unique equilibrium point x∗ satisfying f(x∗) = 0.

Lyapunov Function: A differentiable function V : Rn → R is called Lyapunov function ifit satisfies (i) V (x∗) = 0; (ii) V (x) > 0 for all x = x∗; and (iii) dV (x(t))

dt< 0 for all x(t) = x∗,

i.e. ∇V (x(t)) · f(x(t)) < 0 for x(t) = x∗.

We can prove

Lyapunov’s Theorem: If one can find a Lyapunov function for a dynamical system, thenthe unique equilibrium x∗ is globally stable.

Remark. The condition in Proposition 2.11 that p∗ · z(p) > 0 for every p not proportionalto p∗ is satisfied if GS property holds or if WARP holds for a generic the economy.

21

Proof. To apply this result, let us set a candidate Lyavnov function as

V (p) =n∑

k=1

1

ck(pk − p∗k)

2.

Obviously, this function satifies (i) and (ii) above. As for (iii), we have

dV (p(t))

dt= 2

n∑k=1

1

ck(pk(t)− p∗k)

dpk(t)

dt

=n∑

k=1

1

ck(pk(t)− p∗k)ckzk(p(t))

= −p∗ · z(p(t)) < 0

if p(t) = p∗, as desired.

2.4 Equivalence between Core and WE

In general, the set of core is larger than that of WE. Edgeworth conjectured that thedifference between the two would disappear if the economy gets ‘large’, which is where theassumption of price-taking behavior makes most sense. Here, we follow Debreu and Scarfto formalize the idea of ‘large’ economy and establish the equivalence between core andWE.

Definition 2.12 (r-Replica Economy). Given an exchange economy E = (ui, ei)i∈I , ther-fold replica economy, denoted Er, is the economy with r consumers of each type for atotal of rI consumers.

We assume that the preference of each type satisfies Assumption C.

• An allocation in Er is denoted as

x = (x11, · · · , x1r, · · · , xI1, · · · , xIr),

where xiq denotes the bundle for the qth consumer of type i.

– The allocation is feasible ifI∑

i=1

r∑q=1

xiq = re.

22

Theorem 2.13 (Equal Treatment in the Core). If x belongs to the core of Er, then forevery i, xiq = xiq′ ,∀q, q′.

Proof. Suppose for a contradiction (and without loss of generality) that x1q = x1q′ for someq and q′. We show that x can be blocked by a coalition of I distinct types of consumers,each of whom is treated worst within its type. Suppose wlog that S = {11, · · · , I1} is thatcoalition i.e. for each i = 1, · · · , I,

ui(xi1) ≤ ui(xiq),∀q = 1, · · · , r.

Define xi = 1r

∑rq=1 x

iq and give xi to each consumer i1 in S. Note that by the strictquasi-concavity of ui, we have

ui(xi) = ui(1

r

r∑q=1

xiq) ≥ minq

ui(xiq) = ui(xi1)

and also u1(x1) > u1(x11) since x1q = x1q′ for some q and q′. So every consumer in S isweakly better off while type 1 consumer in S is strictly better off.

We will be done if (xi)i∈I turns out to be feasible within S, which is true since

I∑i=1

xi =1

r

I∑i=1

r∑q=1

xiq =1

rre = e,

where the second equality is due to the feasibility of x in Er.

Thanks to Theorem 2.13, we may (and will do so) let (xi)i∈I denote an allocation in thecore of any replica economy. We want to show that as r gets large, the core of Er shrinksto the set of WE in the original economy E . Before that, we first look at the example ofEdgeworth box economy.

Example 2.14. Suppose that there are two types of consumers. Some allocation in thecore of E is not in the core of E2. Consider, for instance, allocation x as below, where x isa middle point between x and e:

23

.

.Contract curve

.O1

.O2

.e

.x

.x

Consider a candidate blocking coalition S = {11, 12, 21}, and give a bundle x1 = 12(e1+

x1) to each of 11 and 12, a bundle x2 = x2 to 21. So, type 1 consumers in S are better offwhile type 2 consumer is indifferent. Also, this allocation is feasible within S since

2x1 + x2 = (e1 + x1) + x2 = e1 + (x1 + x2) = e1 + (e1 + e2) = 2e1 + e2.

Theorem 2.15. Suppose that for each i, ui is differentiable and ∇ui(x) ≫ 0,∀x ∈ Rn++. If

(xi)i∈I ≫ 0 belongs to the core of Er for every r, then it is a WE of E .

Proof. We first prove the following claim.

Claim. For each i ∈ I,

ui((1− t)xi + tei) ≤ ui(xi),∀t ∈ [0, 1].

Proof. Suppose not for a contradiction. Then, we must have some type i and t ∈ (0, 1)

such thatui((1− t)xi + tei) > ui(xi).

The strict quasi-concavity of ui then implies

ui((1− t)xi + tei) > ui(xi),∀t ∈ (0, t].

24

Thus, we can find a positive integer r large enough that 1r∈ (0, t] and thus

ui((1− 1

r)xi +

1

rei) > ui(xi). (18)

Consider Er and a blocking coalition S that consists of all type i consumers and (r − 1) oftype j consumer for all j = i. Give a bundle xi = (1 − 1

r)xi + 1

rei to each of type i and a

bundle xj = xj to each of type j = i, which makes type 1 better (due to (18)) while makingother types indifferent. Also, this allocation is feasible within S since

rxi + (r − 1)∑j =i

xj = (r − 1)xi + ei + (r − 1)∑j =i

xj = (r − 1)e+ ei = rei + (r − 1)∑j =i

ej,

which establishes a contradiction.

From the above claim, for each i ∈ I,

0 ≥ limt→0+

ui((1− t)xi + tei)− ui(xi)

t= ∇ui(xi) · (ei − xi). (19)

Since xi is in the core of E and thus Pareto efficient, the first order condition (6) impliesthat the gradient vectors are proportional at (xi)i∈I across consumers, i.e. for all i = 1,

∇ui(xi) = λi∇u1(x1) for some λi > 0.

Letting p∗ ≡ ∇u1(x1) and λ1 ≡ 1, we have

∇ui(xi) = λip∗,∀i ∈ I. (20)

Then, (19) and (20) implyp∗ · xi ≥ p∗ · ei, ∀i ∈ I. (21)

The inequality here must be satisfied as equality, for otherwise adding up the inequalitiesacross consumers yields

p∗ ·∑i∈I

xi > p∗ · e,

which is impossible since∑

i∈I xi = e. Now, (20) and (21) as equality correspond to the

first-order (necessary) condition for the utility maximization, which is also sufficient giventhe assumption that each consumer’s preference is strictly quasi-concave. Thus, p∗ and(xi)i∈I constitute WE prices and allocation.

25

3 Equilibrium in Production Economy

3.1 Profit Maximization

Given a price vector p ≫ 0, each firm j ∈ J solves

maxyj∈Y j

p · yj.

Let yj(p) ≡ argmaxyj∈Y j p · yj and πj(p) = p · yj(p).

Example 3.1. Continuing on the Example 1.1, the profit maximization problem can beillustrated by the following graph:

.

.y2

.y(p)

.ProductionPossibility Set

.p1y1 + p2y2 = π(p)

.y1

.π(p)p2

.π(p)p1

.O

Theorem 3.2. If Y jsatisfies Assumption P, then for any p ≫ 0, yj(p) is unique andcontinuous on Rn

+. Also, πj(p) is continuous on Rn+.

Theorem 3.3 (Aggregate Profit Maximization ). Fix prices at p. Then, p · y ≥ p ·y, ∀y ∈ Y

if and only if there is some yj ∈ Y j for each j ∈ J such that y =∑

j∈J yj and p · yj ≥p · yj,∀yj ∈ Y j.

3.2 Utility maximization

• The firms are owned by consumers:

26

– θij ∈ [0, 1]: Consumer i’s share in firm j →∑

i∈I θij = 1, ∀j ∈ J .

– A production economy is described by a vector (ui, ei, θij, Y j)i∈I,j∈J .

• Given prices p, each consumer i has two sources of income:

– Endowment income =p · ei

– Share income=∑

j∈J θijπj(p).

– Let mi(p) ≡ p · ei +∑

j∈J θijπj(p) denote the total income of consumer i.

• Consumer i solvesmaxxi∈Rn

+

ui(xi) subject to p · xi ≤ mi(p),

whose solution is given as xi(p,mi(p)).

3.3 Equilibrium

• The aggregate excess demands are defined as

zk(p) =∑i∈I

xik(p,m

i(p))−∑i∈I

eik −∑j∈J

yjk(p) and

z(p) = (z1(p), · · · , zn(p)).

Theorem 3.4 (Existence of competitive equilibrium ). Consider the economy (ui, ei, θij, Y j)i∈I,j∈J .

Under Assumption C, Assumption P, and e + y ≫ 0 for some y =∑

j∈J yj ∈ Y, thereexists at lest one price vector p∗ such that z(p∗) = 0.

Proof. It suffices to establish that z(p) satisfies the properties (P0) to (P5). The onlynontrivial part is to prove that z(p) is “unbounded above” that is, for any sequence of pricevectors pm → p = 0 with pk = 0 for some k, we must have zk′(p

m) → ∞ for some k′. Theproof can be done following the same steps as in the exchange economy once it is shownthat ∃i ∈ I such that mi(p) > 0 (or some consumer has a positive income given p). To thisend, note that

∑i∈I

mi(p) =∑i∈I

(p · ei +

∑j∈J

θijπj(p)

)

27

=∑i∈I

p · ei +∑j∈J

πj(p)

≥∑i∈I

p · ei +∑j∈J

p · yj

= p · (e+ y) > 0,

which implies that ∃i such that mi(p) > 0.

Example 3.5 (Robinson Crusoe Economy ). Robinson, who lives in a desert island, woksin the daytime (Robinson producer or RP) and consumes “consumption” and “leisure” inthe remaining time (Robinson consumer or RC).

• RC’s utility: U(h, y) = h1−βyβ, β ∈ (0, 1), where h denotes the amount of leisure timeand y the amount of consumption.

– RC’s initial endowment of time and consumption is T and 0, respectively.

– RC owns the firm for which he is the only provider of labor.

• RP’s production function: y = ℓα, α ∈ (0, 1), where ℓ ≥ 0 denotes the supply of labor.

Letting p and w denote the price of consumption and wage, resp., RP solves

maxℓ≥0

pℓα − wℓ (22)

while RC solvesmaxh,y

h1−βyβ subject to py + wh = wT + π(w, p). (23)

Solving (22) yields

ℓ(w, p) =(αpw

) 11−α and π(w, p) = w

(1− α

α

)(αpw

) 11−α

.

Solving (23) yields h(w, p) = (1−β)(wT+π(w,p))w

. Thus, setting p = 1 for normalization, themarket clearing requires

ℓ(w, 1) + h(w, 1) = T

or (αw

) 11−α

+(1− β)(wT + π(w, 1))

w= T,

28

from which

w = α

(αβ + (1− β)

αβ

)1−α

T 1−α.

Graphically,

.

.OC .OP

.yC .yP

.E

.ProductionPossibility Set

.Iso-Profit Line= Budget Line

.h.−ℓ.w

.π(w, 1)

.T

.h(w, 1) .ℓ(w, 1)

3.4 First and Second Welfare Theorems

Theorem 3.6 (First Welfare Theorem with Production ). If each ui is strictly increasingon Rn

+, then any competitive equilibrium allocation must be Pareto efficient.

Proof. Suppose that (x, y) is a competitive equilibrium but not Pareto efficient. Then,∃(x, y) ∈ F (e, Y ) such that ui(xi) ≥ ui(xi), ∀i ∈ I with at least one strict inequality, whichimplies that letting p∗ denote the competitive equilibrium price vector,

p∗ · xi ≥ p∗ · xi,∀i ∈ I with at least one strict inequality

⇒ p∗ ·∑i∈I

xi > p∗ ·∑i∈I

xi

⇒ p∗ ·

(∑i∈I

ei +∑j∈J

yj

)> p∗ ·

(∑i∈I

ei +∑j∈J

yj

)⇒ p∗ ·

∑j∈J

yj > p∗ ·∑j∈J

yj,

29

which implies p∗ · yj > p∗ · yj for some j ∈ J , contradicting the profit maximization.

Theorem 3.7 (Second Welfare Theorem with Production ). Assume (i) Assumption Cand Assumption P (ii) y+

∑i∈I e

i ≫ 0 for some y ∈ Y , and (iii) (x, y) is Pareto efficient.Then, ∃T1, · · · , TI with

∑i∈I Ti = 0 and p such that

1. xi ∈ argmaxxi∈Rn+ui(xi) such that p · xi ≤ mi(p) + Ti,∀i ∈ I

2. yj ∈ argmaxyj∈Y j p · yj,∀j ∈ J .

Proof. Let Y j ≡ Y j − yj and then each Y j satisfies A4, as can be easily verified. Considera (hypothetical) economy (ui, xi, θij, Y j)i∈I,j∈J . Then, ∃ a competitive equilibrium p ≫ 0

and (x, y) for this economy. We first prove the following claim.

Claim. (xi, yj + yj)i∈I,j∈J ∈ F (e, Y ).

Proof. Note first that yj + yj ∈ Y j, as can be easily seen. From the fact that (x, y) ∈F (x, Y ), we also have ∑

i∈I

xi =∑i∈I

xi +∑j∈J

yj

=∑i∈I

xi −∑j∈J

yj +∑j∈J

(yj + yj)

=∑i∈I

ei +∑j∈J

(yj + yj),

as desired.

The above claim, together with the Pareto efficiency of (x, y) in the original economy,implies that ui(xi) = ui(xi), ∀i ∈ I so xi = xi,∀i ∈ I. Thus,

xi = xi ∈ arg maxxi∈Rn

+

ui(xi) subject to p · xi ≤ p · xi +∑j∈J

θij p · yj,∀i ∈ I, (24)

which implies that∑

j∈J θij p · yj = 0. Since this is true for all consumer i ∈ I, we musthave

∑j∈J p · yj = 0 so p · yj = 0,∀j ∈ J . Then,

0 = p · yj ≥ p · (yj − yj),∀yj ∈ Y j,∀j ∈ J

⇒ p · yj ≥ p · yj, ∀yj ∈ Y j,∀j ∈ J . (25)

30

Now, letting Ti = p · xi −mi(p), (24) and (25) imply

xi = arg maxxi∈Rn

+

ui(xi) subject to p · xi ≤ mi(p) + Ti = p · xi, ∀i ∈ I

yj = arg maxyj∈Y j

p · yj, ∀j ∈ J .

It remains to verify that∑

i∈I Ti = 0, which holds since

∑i∈I

Ti =∑i∈I

p · xi −

(∑i∈I

p · ei +∑i∈I

∑j∈J

θijπj(p)

)

=∑i∈I

p · xi −

(∑i∈I

p · ei +∑i∈I

∑j∈J

θij p · yj)

=∑i∈I

p · xi −

(∑i∈I

p · ei +∑j∈J

p · yj)

= p ·

(∑i∈I

xi −∑i∈I

ei −∑j∈J

yj

)= 0.

31

4 General Equilibrium under Uncertainty

We now apply the general equilibrium framework to economic situations involving uncer-tainty, which may be about technologies, endowments, or preferences.

• Let s ∈ S = {1, · · · , S} denote the state of the world, which will realize at time t = 1.

– At t = 0, each consumer i is uncertain about s and believes it will occur withprobability πi

s..

– x = (x1, · · · , xS) ∈ RnS+ : Contingent consumption vector (or contingent com-

modity), where xs = (x1s, · · · , xns) ∈ Rn+ is the commodity vector to be con-

sumed if state s occurs at t = 1.

• Each consumer i’s endowment and preference may depend on the state of the world.

– ei = (ei1, · · · , eiS): Consumer i’s endowment vector.

– ≽i: Consumer i’s (rational) preference relation, which is represented by a utilityfunction U i defined on RnS

+ .

∗ For instance, if the consumer i has Bernoulli (state-dependent) utility func-tion ui

s(xs) in state , then his preference will be given as follow:

x ≽i x′ if and only if U i(x) =∑s∈S

πisu

is(xs) ≥

∑s∈S

πisu

is(x

′s) = U i(x′).

4.1 Arrow-Debreu Equilibrium

Consider an exchange economy with no production for simplicity.

Definition 4.1. An allocation x∗ = (x1∗, · · · , xI∗) ∈ RnSI and prices p = (p11, · · · , pnS) ∈RnS constitute an Arrow-Debreu equilibrium (ADE) if:(i) ∀i ∈ I, xi∗ ≽i xi for any xi ∈ {xi | p · xi ≤ p · ei}.(ii)

∑i∈I x

i∗s =

∑i∈I e

is, ∀s ∈ S.

Example 4.2. Suppose that I = 2, n = 1, and S = 2. Also, each consumer has a state-independent Bernoulli utility function ui while e1 = (1, 0) and e2 = (0, 1) so that there

32

is no aggregate risk. At the (interior) optimum xi for consumer i = 1, 2, the first-orderconditions yield

π11

du1(x11)

dx

/π12

du1(x12)

dx= MRS1

12 =p1p2

= MRS212 = π2

1

du2(x21)

dx

/π22

du2(x22)

dx. (26)

Assuming that π1s = π2

s = πs, ∀s = 1, 2, we have xi1 = xi

2, ∀i = 1, 2 and p1/p2 = π1/π2. Asshown here, the contingent commodity serves the purpose of transferring wealth across thestates of the world.

Example 4.3. Consider the same economy as in Example 4.2 except that e1 + e2 = (2, 1)

so there is an aggregate risk. Assuming π1s = π2

s = πs,∀s = 1, 2, (26) implies that xi1 >

xi2, ∀i = 1, 2, which in turn implies p1/p2 < π1/π2.

4.2 Asset Market

In reality, not all goods perform the wealth-transferring role. There are some commoditiescalled assets, or securities, that are designed specifically for transferring wealth acrossstates.

Definition 4.4. A unit of asset or security is the title to receive an amount rsunits ofgood 1 at t = 1 if state s ∈ S occurs. Thus, an asset is denoted by r = (r1, · · · , rS) ∈ RS.

Example 4.5. Here are the examples of securities:

1. r = (1, · · · , 1) : Noncontingent delivery of one unit of good 1 →Commodity futures.

2. r = (0, · · · , 0, 1, 0, · · · , 0) : Contingent delivery of one unit of good 1 at a certain state(Arrow security).

3. r(c) = (max{0, r1 − c}, · · · ,max{0, rS − c}) for a primary asset r ∈ RS: Call optionwith the primary asset r and strike price c→ Option to exercise r at price c ∈ R(assuming that the spot-market price of good 1 is normalized to 1 in every state).

• At t = 0, only the asset market opens in which K assets are traded.

– q = (q1, · · · , qK) : Asset price vector.

33

– z = (z1, · · · , zK) ∈ RK : Asset trade vector or portfolio.

• At t = 1, the spot market opens at each state s ∈ S in which all goods are traded.

– ps = (p1s, · · · , pns) : Spot prices at state s.

– xs = (x1s, · · · , xns) : Spot trades at state s.

Definition 4.6. A collection {q, (ps)s∈S , (zi∗)i∈I , (xi∗)i∈I} constitutes a Radner equilibrium(RE) if:

(i) ∀i ∈ I, (zi∗, xi∗) solves

maxx∈RnS

+

z∈RK

U i(x) subject to (a)K∑k=1

qkzk ≤ 0

(b) ps · xs ≤ ps · eis +k∑

k=1

p1szkrsk, ∀s ∈ S. (27)

(ii)∑

i∈I zi∗k ≤ 0, ∀k and

∑i∈I x

i∗s ≤

∑i∈I e

is,∀s ∈ S.

Normalize p1s = 1 for each s ∈ S and then (27) can be writtenp1 · (x1 − ei1)

...pS · (xS − eiS)

≤

r11 · · · r1K... . . . ...

rS1 · · · rSK

z1...zK

= R · z,

where R is a return matrix.

Theorem 4.7. Suppose that rank R = S or there are at least S assets with linearly inde-pendent returns. Then, the set of allocations in ADE is the same as that of allocations inRE.

Proof. Refer to the MWG, p. 705.

Example 4.8. Suppose that n = 2, I = 2, S = 2, and π1 = π2. Suppose also thatu1 = u2 = u and e12 = e21 = (0, 0) while e = e11 = e22 ≫ 0. In this economy, two assets areavailable: r1 = (1, 0) and r2 = (0, 1).

34

.

.E = 12e

.O1

.O2

.α

.α

In ADE, we must have xi∗1 = xi∗

2 = 12e, i = 1, 2. The corresponding RE is as follows:

q1 = q2 = 1 and z1∗2 = z2∗1 = α for some α > 0 while z1∗1 = z2∗2 = −α.

35

5 Public Goods and Externality

5.1 Public Goods

• There are one private good and one pubic good (PG): Letting yi and xi denote theamount of private and public goods for consumer i ∈ I,

– Consumer i’s utility: ui(yi, X), where X =∑

i∈I xi

– Consumer i’s endowment: ei = (wi, 0).

– G(y,X) ≥ 0 : Production technology; ∂G∂y

> 0 and ∂G∂X

< 0.

– The set of feasible allocations is given as

F (e) =

{(yi, X)i∈I

∣∣∣∣∣G(∑i∈I

wi −∑i∈I

yi, X) ≥ 0

}.

Pareto Efficiency

• A Pareto efficient allocation solves

max(y1,··· ,yI ,X)

u1(y1, X) subject to ui(yi, X) ≥ ui,∀i = 2, · · · , I (µi)

G(∑i∈I

wi −∑i∈I

yi, X) ≥ 0. (λ)

– Assuming an interior solution and letting µ1 = 1, the first-order condition isgiven as

yi : µi∂ui

∂y− λ

∂G

∂y= 0

X :∑i∈I

µi ∂ui

∂X+ λ

∂G

∂X= 0

⇒ λ

(∑i∈I

∂ui

∂X

/∂ui

∂y+

∂G

∂X

/∂G

∂y

)= 0

⇒∑i∈I

∂ui

∂X

/∂ui

∂y= − ∂G

∂X

/∂G

∂y, (28)

which is so called “Samuelson Condition”.

36

Competitive Equilibrium and Inefficiency

Normalize the price of private good to 1 and let p denote the price of PG.

• An allocation (yi∗, xi∗)i∈I is a competitive equilibrium allocation if(i) xi∗ ∈ argmaxxi∈[0,wi/p] u

i(wi − pxi, xi +∑

j =i xj∗)

(ii) (∑

i∈I wi −∑

i∈I yi∗,∑

i∈I xi∗) ∈ argmax(y,X) pX − y subject to G(y,X) ≥ 0.

– From (i), the first-order condition is given as

− ∂G

∂X

/∂G

∂y= p (29)

– From (ii), the first-order condition (or Kuhn-Tucker condition) is given as

∂ui

∂y(−p) +

∂ui

∂X≤ 0

= 0 if xi∗ > 0.

(30)

– Combining (30) and (29) yields

∂ui

∂X

/∂ui

∂y≤ p = − ∂G

∂X

/∂G

∂y

= p = − ∂G

∂X

/∂G

∂yif xi∗ > 0,

which is different from (28). So. a competitive equilibrium is not PE in general.

Lindahl Equilibrium and Efficiency

Lindahl suggested an idea that the inefficiency caused by the PG can be fixed by creatinga market for each individual consumer.

• To explain, take an example with two consumers and the production technology givenas G(y,X) = y − cX :

– Let each consumer i pay an individualized price qi ∈ [0, 1] for the PG.

– So, the producer of PG collects q1 + q2 for selling 1 unit of PG, which impliesq1 + q2 = c at the equilibrium.(why?)

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– Stipulate that the transaction of PG only occurs if two consumers agree on theamount of PG.

– Then, each consumer i solves

max(yi,X)

ui(yi, X) subject to yi + qiX = wi,

which yields the first-order condition

∂ui

∂X

/∂ui

∂y≤ qi

= qi if X > 0.

– In the equilibrium, it must be that∑i=1,2

∂ui

∂X

/∂ui

∂y= q1 + q2 = c

so the efficiency is restored.

In general case where q = (q1, · · · , qI) be a vector of individualized prices for PG,

Definition 5.1. A collection {(qiL)i∈I , (yiL)i∈I , XL}is a Lindahl equilibrium if letting qL =∑i∈I q

iL

(i) (∑

i∈I wi −∑

i∈I yiL, XL) ∈ argmax(y,X) q

LX − y subject to G(y,X) ≥ 0

(ii) For all i ∈ I, (yiL, XL) ∈ argmax(yi,X) ui(yi, X) subject to yi+qiLX ≤ wi+θiπ(qL),

where π is the (maximized) profit function.

Theorem 5.2. A Lindahl equilibrium is Pareto efficient.

Proof. Suppose not and then we must have some allocation (y1, · · · , yI , X) such that

ui(yi, X) ≥ ui(yiL, xL),∀i ∈ I with at least one strict inequality

⇒ yi + qiLX ≥ wi + θiπ(qL),∀i ∈ I with at least one strict inequality

⇒∑i∈I

yi + qLX >∑i∈I

wi + π(qL)

or (∑i∈I

qiL)X − (∑i∈I

wi −∑i∈I

yi) > π(qL),

a contradiction.

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5.2 Externality and Lindahl Equilibrium

• Suppose that there are two consumers who both consume “nuts” and one of whom,say consumer 1, consumes “smoking”:

– Bi : Amount of nuts consumed by consumer i; S : Amount of smoking byconsumer 1

– wi : Endowment of nuts for consumer i

– ui(Bi, S) : Consumer i’s utility; ∂ui

∂B> 0,∀i = 1, 2 and ∂u2

∂S< 0.

– Graphically,

.

.A

.B

.ω1 .ω2

.O1 .O2

.S .S

• In an autarky,

– if consumer 1 (2 resp.) has the property right, then B (A resp.) results;

– neither A nor B is PE.

• The competitive equilibrium allocation is also inefficient if two consumers face thecommon price for smoking.(why?)

• Introduce the Lindahl market where 2 has the property right:

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– qi : Price of smoking for consumer i

– Since the production of smoking takes no cost, we must have q1 + q2 = 0 in theequilibrium.

– In the Lindahl equilibrium,

(BiL, SL) ∈ argmax ui(Bi, S) subject to Bi + qiLS ≤ wi.

Graphically,

.

.O1 .O2.A

.E

.S .S

.−1/q1L = 1/q2L

– When 1 has the property right,

40

.

.O1 .O2

.E′

.S.S

.B

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6 Social Choice and Welfare

• Let X denote the set of social alternatives.

– R : Set of all rational preference relations on X

– P : Set of all strict preference relations. Then, P ⊂ R

– A : Admissible domain of preference profiles. Assume that either A = RI orA = PI that is the largest domains.

Definition 6.1. A social welfare function F : A → R is a rule to assign a social prefer-ence relation F (%1, · · · ,%I) ∈ R to any profile of individual rational preference relations(%1, · · · ,%I) ∈ A.

– If xF (%1, · · · ,%I)y, then we say x is socially at least as good as y

– If xF (%1, · · · ,%I)y but not yF (%1, · · · ,%I)x, then we say x is socially preferred y, inwhich case we write xFp(%1, · · · ,%I)y.

6.1 Arrow’s Impossibility Theorem

• Consider the following properties for a social welfare function F :

– (Weak) Paretian: xFp(%1, · · · ,%I)y whenever x ≻i y,∀i ∈ I

– Independence of irrelevant alternatives (IIA): For any pair of alternatives x, y ∈X and for any pair of preference profiles (%1, · · · ,%I) , (%′

1, · · · ,%′I) ∈ A, if for

all i ∈ I, x %i y ⇐⇒ x %′i y, then xF (%1, · · · ,%I)y ⇐⇒ xF (%′

1, · · · ,%′I)y.

– Dictatorial: There is an agent h such that for any x, y ∈ X and any profile(%1, · · · ,%I) ∈ A, xFp(%1, · · · ,%I)y whenever x ≻h y.

Theorem 6.2 (Arrow’s Impossibility Theorem). Suppose that |X| ≥ 3 and that eitherA = RI or A = PI . Then, every social welfare function that is Paretian and satisfies IIAis necessarily dictatorial.

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Proof. Refer to pages 796-799 of MGW.

Example 6.3 (Pairwise Majority Voting and Condorcet Paradox). The pairwise majorityvoting (PMV) is defined as follows: xF (%1, · · · ,%I)y if

#{i ∈ I : x ≻i y} ≥ #{i ∈ I : y ≻i x}. (31)

Though widely used, this social preference is not transitive. Consider the following example:I = {1, 2, 3} and X = {x, y, z}; Preferences are given as

x ≻1 y ≻1 z

y ≻2 z ≻2 x

and z ≻3 x ≻3 y

It is easy to check that x is socially preferred to y,y to z, and z to x, violating the transitivity.

Example 6.4 (Borda Rule). Given a preference relation %i∈ R, assign a number ci(x) toeach alternative x ∈ X: If x ∈ X is kth ranked in the ordering of %i, then assign ci(x) = k.

(If there are ties among alternatives, then assign the average rank of those alternatives.)The Borda rule is defined as follows: xF (%1, · · · ,%I)y if∑

i∈I

ci(x) ≤∑i∈I

ci(y).

It is easy to check that for any profile (%1, · · · ,%I), the social preference F (%1, · · · ,%I)

is complete and transitive. Also, F is Paretian. However, F violates IIA, as shown in thefollowing example: I = {1, 2} and X = {x, y, z}; One preference profile is given as

x ≻1 z ≻1 y and y ≻2 x ≻2 z

while the other as

x ≻′1 y ≻′

1 z and y ≻′2 z ≻′

2 x.

Note that the ranking between x and y is preserved across two profiles. However, under theformer profile, c(x) = 3 < 4 = c(y) while under the latter, c(x) = 4 > 3 = c(y), violatingIIA.

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6.2 Some Possibility Results: Pairwise Majority Voting in Re-

stricted Domain

• Suppose that X is linearly ordered, that is elements of X are ordered as real numbersare ordered.

• Given a profile (%1, · · · ,%I), let F (%1, · · · ,%I) denote the social preference from thepairwise majority voting, which is defined in Example 6.3.

Definition 6.5. The preference relation % is single-peaked if there exists a peak x ∈ X

such that

if x ≥ z > y, then z ≻ y

and if y > z ≥ x, then z ≻ y.

Let Rs = {%∈ R :% is single-peaked} and xi denote i’s peak.

Definition 6.6. Agent h ∈ I is a median agent (or voter) for the profile (%1, · · · ,%I) if

#{i ∈ I : xi ≥ xh} ≥ I

2and #{i ∈ I : xh ≥ xi} ≥ I

2.

Note that it is always possible to find a median agent.

Proposition 6.7. Suppose that each agent has a single-peaked preference. Let h ∈ I be amedian agent. Then, xhF (%1, · · · ,%I)y for every y ∈ X.

Note: An alternative that is not defeated by any other alternative in the PMV is calleda Condorcet winner. So the result says that Condorcet winner always exists and it is themedian agent’s peak.

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Proof. Take any y < xh and we show that y does not defeat xh. (The same argumentapplies to y > xh.) Consider any agent i ∈ I whose peak xi is larger than or equal to xh.

That is, xi ≥ xh > y so the single-peakedness implies xh ≻i y. Since h is a median agent,we have #{i ∈ I : xi ≥ xh} ≥ I

2so #{i ∈ I : xh ≻i y} ≥ I

2, which means that (31) holds

between xh and y.

Certainly, the PMV is Paretian and satisfies IIA. (Verify this for yourself.) Also, thesocial preference relation generated by the PMV is complete.

Proposition 6.8. Suppose that I is odd. Then, given any profile (%1, · · · ,%I) of single-peaked and strict preferences, F (%1, · · · ,%I) is transitive.

Proof. Suppose that xF (%1, · · · ,%I)y and yF (%1, · · · ,%I)z. Consider a set X ′ = {x, y, z}.Restrict each agent’s preference to X ′ and it is still single-peaked. So there exists a Con-dorcet winner, which can neither y nor z, and thus has to be x. This let us concludexF (%1, · · · ,%I)z.

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