SummaryTheorem 5.2.

Blocking and the Core

Last week we defined WEs and WEAs.

We looked at existence, feasibility, Pareto optimality, and thefirst welfare theorem.

We also did some Edgeworth box drawings (feasibility=thebox itself!, contract curve, WEA).

We skipped a crucial theorem which we begin with today.

Martin K. Jensen (U. B’ham) Econ 320B, Set 2

SummaryTheorem 5.2.

Blocking and the Core

Last week we defined WEs and WEAs.

We looked at existence, feasibility, Pareto optimality, and thefirst welfare theorem.

We also did some Edgeworth box drawings (feasibility=thebox itself!, contract curve, WEA).

We skipped a crucial theorem which we begin with today.

Martin K. Jensen (U. B’ham) Econ 320B, Set 2

SummaryTheorem 5.2.

Blocking and the Core

Last week we defined WEs and WEAs.

We looked at existence, feasibility, Pareto optimality, and thefirst welfare theorem.

We also did some Edgeworth box drawings (feasibility=thebox itself!, contract curve, WEA).

We skipped a crucial theorem which we begin with today.

Martin K. Jensen (U. B’ham) Econ 320B, Set 2

SummaryTheorem 5.2.

Blocking and the Core

Last week we defined WEs and WEAs.

We looked at existence, feasibility, Pareto optimality, and thefirst welfare theorem.

We also did some Edgeworth box drawings (feasibility=thebox itself!, contract curve, WEA).

We skipped a crucial theorem which we begin with today.

Martin K. Jensen (U. B’ham) Econ 320B, Set 2

SummaryTheorem 5.2.

Blocking and the Core

Theorem 5.2.∗

(Properties of Excess Demand Functions)

If for each consumer i , ui satisfies assumption 5.1., then for allp� 0.

1 Continuity: z(p) will be continuous in p.

2 Homogeneity: z(λp) = z(p) for all λ > 0

3 Walras’ law: pz(p) = 0.

Martin K. Jensen (U. B’ham) Econ 320B, Set 2

SummaryTheorem 5.2.

Blocking and the Core

z will be continuous in p because it is defined as the sum ofcontinuous functions (the demand functions) minus aconstant term (the aggregate endowment). Note: Continuityof demand functions is part of Theorem 5.1. from last week.

Homogeneity is also quite easy. The conclusion followsdirectly from the fact that x i (p,pei ) = x i (λp, λpei ). To seethat this is so, note that:

Martin K. Jensen (U. B’ham) Econ 320B, Set 2

SummaryTheorem 5.2.

Blocking and the Core

z will be continuous in p because it is defined as the sum ofcontinuous functions (the demand functions) minus aconstant term (the aggregate endowment). Note: Continuityof demand functions is part of Theorem 5.1. from last week.

Homogeneity is also quite easy. The conclusion followsdirectly from the fact that x i (p,pei ) = x i (λp, λpei ). To seethat this is so, note that:

Martin K. Jensen (U. B’ham) Econ 320B, Set 2

SummaryTheorem 5.2.

Blocking and the Core

max ui (x i1, . . . , x

in)

s.t.

{ ∑k pkx i

k ≤∑

k pke ikx ik ≥ 0 for k = 1, . . . , n

(1)

...is entirely equivalent to:

max ui (x i1, . . . , x

in)

s.t.

{λ∑

k pkx ik ≤ λ

∑k pke ik

x ik ≥ 0 for k = 1, . . . , n

(2)

...indeed, we can just cancel out λ !

Martin K. Jensen (U. B’ham) Econ 320B, Set 2

SummaryTheorem 5.2.

Blocking and the Core

max ui (x i1, . . . , x

in)

s.t.

{ ∑k pkx i

k ≤∑

k pke ikx ik ≥ 0 for k = 1, . . . , n

(1)

...is entirely equivalent to:

max ui (x i1, . . . , x

in)

s.t.

{λ∑

k pkx ik ≤ λ

∑k pke ik

x ik ≥ 0 for k = 1, . . . , n

(2)

...indeed, we can just cancel out λ !

Martin K. Jensen (U. B’ham) Econ 320B, Set 2

SummaryTheorem 5.2.

Blocking and the Core

max ui (x i1, . . . , x

in)

s.t.

{ ∑k pkx i

k ≤∑

k pke ikx ik ≥ 0 for k = 1, . . . , n

(1)

...is entirely equivalent to:

max ui (x i1, . . . , x

in)

s.t.

{λ∑

k pkx ik ≤ λ

∑k pke ik

x ik ≥ 0 for k = 1, . . . , n

(2)

...indeed, we can just cancel out λ !

Martin K. Jensen (U. B’ham) Econ 320B, Set 2

SummaryTheorem 5.2.

Blocking and the Core

max ui (x i1, . . . , x

in)

s.t.

{ ∑k pkx i

k ≤∑

k pke ikx ik ≥ 0 for k = 1, . . . , n

(1)

...is entirely equivalent to:

max ui (x i1, . . . , x

in)

s.t.

{λ∑

k pkx ik ≤ λ

∑k pke ik

x ik ≥ 0 for k = 1, . . . , n

(2)

...indeed, we can just cancel out λ !

Martin K. Jensen (U. B’ham) Econ 320B, Set 2

SummaryTheorem 5.2.

Blocking and the Core

I’ll prove part 3 (Walras’ law) on the board.

Martin K. Jensen (U. B’ham) Econ 320B, Set 2

SummaryTheorem 5.2.

Blocking and the Core

Consider an economy with I consumers, so I = {1, . . . , I} isthe set of consumers. A coalition is a subset of I, S ⊆ I.

Example: Let I = 2 so I = {1, 2}. There are then threecoalitions, namely S1 = {1}, S2 = {2}, and S3 = {1, 2}.

Example: Let I = 3 so I = {1, 2, 3}. There are then sevencoalitions, namely S1 = {1}, S2 = {2}, S3 = {3}, S4 = {1, 2},S5 = {1, 3}, S6 = {2, 3}, and S7 = {1, 2, 3}.You should think of the coalitions as “all possible ways ofsplitting up a country”.

Martin K. Jensen (U. B’ham) Econ 320B, Set 2

SummaryTheorem 5.2.

Blocking and the Core

Consider an economy with I consumers, so I = {1, . . . , I} isthe set of consumers. A coalition is a subset of I, S ⊆ I.

Example: Let I = 2 so I = {1, 2}. There are then threecoalitions, namely S1 = {1}, S2 = {2}, and S3 = {1, 2}.

Example: Let I = 3 so I = {1, 2, 3}. There are then sevencoalitions, namely S1 = {1}, S2 = {2}, S3 = {3}, S4 = {1, 2},S5 = {1, 3}, S6 = {2, 3}, and S7 = {1, 2, 3}.You should think of the coalitions as “all possible ways ofsplitting up a country”.

Martin K. Jensen (U. B’ham) Econ 320B, Set 2

SummaryTheorem 5.2.

Blocking and the Core

Consider an economy with I consumers, so I = {1, . . . , I} isthe set of consumers. A coalition is a subset of I, S ⊆ I.

Example: Let I = 2 so I = {1, 2}. There are then threecoalitions, namely S1 = {1}, S2 = {2}, and S3 = {1, 2}.

Example: Let I = 3 so I = {1, 2, 3}. There are then sevencoalitions, namely S1 = {1}, S2 = {2}, S3 = {3}, S4 = {1, 2},S5 = {1, 3}, S6 = {2, 3}, and S7 = {1, 2, 3}.

You should think of the coalitions as “all possible ways ofsplitting up a country”.

Martin K. Jensen (U. B’ham) Econ 320B, Set 2

SummaryTheorem 5.2.

Blocking and the Core

Consider an economy with I consumers, so I = {1, . . . , I} isthe set of consumers. A coalition is a subset of I, S ⊆ I.

Example: Let I = 2 so I = {1, 2}. There are then threecoalitions, namely S1 = {1}, S2 = {2}, and S3 = {1, 2}.

Example: Let I = 3 so I = {1, 2, 3}. There are then sevencoalitions, namely S1 = {1}, S2 = {2}, S3 = {3}, S4 = {1, 2},S5 = {1, 3}, S6 = {2, 3}, and S7 = {1, 2, 3}.You should think of the coalitions as “all possible ways ofsplitting up a country”.

Martin K. Jensen (U. B’ham) Econ 320B, Set 2

SummaryTheorem 5.2.

Blocking and the Core

S = I is always a coalition, called the grand coalition.

Coalitions consisting only of a single consumer, S1 = {1},S2 = {2}, and so forth, are called singleton coalitions.

Martin K. Jensen (U. B’ham) Econ 320B, Set 2

SummaryTheorem 5.2.

Blocking and the Core

S = I is always a coalition, called the grand coalition.

Coalitions consisting only of a single consumer, S1 = {1},S2 = {2}, and so forth, are called singleton coalitions.

Martin K. Jensen (U. B’ham) Econ 320B, Set 2

SummaryTheorem 5.2.

Blocking and the Core

Definition

(JR Definition 5.2) Let S ⊆ I denote a coalition of consumers.We say that S blocks a feasible allocation x ∈ F (e) if there is anallocation y = (y1, . . . , yI ) such that:

1∑

i∈S yi =∑

i∈S ei . [Coalition Feasibility]

2 ui (yi ) ≥ ui (xi ) for all i ∈ S with at least one strict inequality.[Coalition Pareto Dominance]

If a feasible allocation x ∈ F (e) cannot be blocked by anycoalition, we say that it is an unblocked feasible allocation.

Martin K. Jensen (U. B’ham) Econ 320B, Set 2

SummaryTheorem 5.2.

Blocking and the Core

What does it mean that a feasible allocation is blocked by acoalition S?

Answer: The group of people S could “go solo” (leave theeconomy), and divide their resources among themselves insuch a way that everyone will be at least as well of as beforeand someone strictly better off.

This leads to the core: The set of allocations given which nogroup would want to break away.

Martin K. Jensen (U. B’ham) Econ 320B, Set 2

SummaryTheorem 5.2.

Blocking and the Core

What does it mean that a feasible allocation is blocked by acoalition S?

Answer: The group of people S could “go solo” (leave theeconomy), and divide their resources among themselves insuch a way that everyone will be at least as well of as beforeand someone strictly better off.

This leads to the core: The set of allocations given which nogroup would want to break away.

Martin K. Jensen (U. B’ham) Econ 320B, Set 2

SummaryTheorem 5.2.

Blocking and the Core

What does it mean that a feasible allocation is blocked by acoalition S?

Answer: The group of people S could “go solo” (leave theeconomy), and divide their resources among themselves insuch a way that everyone will be at least as well of as beforeand someone strictly better off.

This leads to the core: The set of allocations given which nogroup would want to break away.

Martin K. Jensen (U. B’ham) Econ 320B, Set 2

SummaryTheorem 5.2.

Blocking and the Core

Definition

(JR Definition 5.3.) The core of an exchange economy withendowments e, denoted C (e), is the set of all unblocked feasibleallocations.

Let’s look at what this means. An allocation in the core(x ∈ C (e)), is not blocked by any coalition.

In particular, not blocked by the grand coalition - what doesthis mean?

Answer: It means that x is Pareto optimal!

Nor can any singleton coalition block (no one is strictly betteroff going solo just munching his initial resources). “Gainsfrom trade” (or at least, no losses). We’ll get back to this.

Martin K. Jensen (U. B’ham) Econ 320B, Set 2

SummaryTheorem 5.2.

Blocking and the Core

Definition

(JR Definition 5.3.) The core of an exchange economy withendowments e, denoted C (e), is the set of all unblocked feasibleallocations.

Let’s look at what this means. An allocation in the core(x ∈ C (e)), is not blocked by any coalition.

In particular, not blocked by the grand coalition - what doesthis mean?

Answer: It means that x is Pareto optimal!

Nor can any singleton coalition block (no one is strictly betteroff going solo just munching his initial resources). “Gainsfrom trade” (or at least, no losses). We’ll get back to this.

Martin K. Jensen (U. B’ham) Econ 320B, Set 2

SummaryTheorem 5.2.

Blocking and the Core

Definition

(JR Definition 5.3.) The core of an exchange economy withendowments e, denoted C (e), is the set of all unblocked feasibleallocations.

Let’s look at what this means. An allocation in the core(x ∈ C (e)), is not blocked by any coalition.

In particular, not blocked by the grand coalition - what doesthis mean?

Answer: It means that x is Pareto optimal!

Nor can any singleton coalition block (no one is strictly betteroff going solo just munching his initial resources). “Gainsfrom trade” (or at least, no losses). We’ll get back to this.

Martin K. Jensen (U. B’ham) Econ 320B, Set 2

SummaryTheorem 5.2.

Blocking and the Core

Definition

(JR Definition 5.3.) The core of an exchange economy withendowments e, denoted C (e), is the set of all unblocked feasibleallocations.

Let’s look at what this means. An allocation in the core(x ∈ C (e)), is not blocked by any coalition.

In particular, not blocked by the grand coalition - what doesthis mean?

Answer: It means that x is Pareto optimal!

Nor can any singleton coalition block (no one is strictly betteroff going solo just munching his initial resources). “Gainsfrom trade” (or at least, no losses). We’ll get back to this.

Martin K. Jensen (U. B’ham) Econ 320B, Set 2

SummaryTheorem 5.2.

Blocking and the Core

Definition

(JR Definition 5.3.) The core of an exchange economy withendowments e, denoted C (e), is the set of all unblocked feasibleallocations.

Let’s look at what this means. An allocation in the core(x ∈ C (e)), is not blocked by any coalition.

In particular, not blocked by the grand coalition - what doesthis mean?

Answer: It means that x is Pareto optimal!

Nor can any singleton coalition block (no one is strictly betteroff going solo just munching his initial resources). “Gainsfrom trade” (or at least, no losses). We’ll get back to this.

Martin K. Jensen (U. B’ham) Econ 320B, Set 2

SummaryTheorem 5.2.

Blocking and the Core

Theorem∗ (All WEAs are in the Core) Consider an exchange economy(ui , ei )i∈I . If each consumer’s utility function ui is strictlyincreasing on Rn

+, then every Walrasian equilibrium allocation is inthe core.

If we denote the set of WEAs by W (e), we can write theconclusion of the previous theorem more compactly as:

W (e) ⊆ C (e)

Note that the first welfare theorem follows from this resultsince any allocation in the core is Pareto optimal (it is notblocked by the grand coalition).

Martin K. Jensen (U. B’ham) Econ 320B, Set 2

SummaryTheorem 5.2.

Blocking and the Core

Theorem∗ (All WEAs are in the Core) Consider an exchange economy(ui , ei )i∈I . If each consumer’s utility function ui is strictlyincreasing on Rn

+, then every Walrasian equilibrium allocation is inthe core.

If we denote the set of WEAs by W (e), we can write theconclusion of the previous theorem more compactly as:

W (e) ⊆ C (e)

Note that the first welfare theorem follows from this resultsince any allocation in the core is Pareto optimal (it is notblocked by the grand coalition).

Martin K. Jensen (U. B’ham) Econ 320B, Set 2

SummaryTheorem 5.2.

Blocking and the Core

Theorem∗ (All WEAs are in the Core) Consider an exchange economy(ui , ei )i∈I . If each consumer’s utility function ui is strictlyincreasing on Rn

+, then every Walrasian equilibrium allocation is inthe core.

If we denote the set of WEAs by W (e), we can write theconclusion of the previous theorem more compactly as:

W (e) ⊆ C (e)

Note that the first welfare theorem follows from this resultsince any allocation in the core is Pareto optimal (it is notblocked by the grand coalition).

Martin K. Jensen (U. B’ham) Econ 320B, Set 2