Microeconomics: Barter Economy and Core

Microeconomics: Barter Economy and Core

Ram Singh

Lecture 1

September 14, 2015

Barter: Goods for Goods I

Ref: Advanced Microeconomic Theory by Geoffrey Jehle and Philip Reny (2nd Ed.)

We will consider an exchange economy:

There is no production - it has already taken place

There areNindividuals - an individual is indexed byi,i =1,2, ...,N There areM commodities - a commodity is indexed byj,j =1,2, ...,M Each individual is endowed with a bundle of commodities

First consider a 2×2 economy. Let, initial endowments be(e¹,e²), where

e¹ = (e¹₁,e¹₂) = (8,2) e² = (e²₁,e²₂) = (0,7)

Barter: Goods for Goods II

Definition

Allocation: An allocation is a distribution of available goods among the individuals consistent with the initial endowments

E.g.,

whene¹= (8,2)ande²(0,7),

a possible allocation is: x¹= (3,5), andx²= (5,3).

another possible allocation is:x¹= (3,5), andx²= (5,4).

In general, let

x¹= (x₁¹,x₂¹)denote the bundle allocated to person 1 x²= (x₁²,x₂²)denote the bundle allocated to person 2

Barter: Goods for Goods III

Allocationx= (x¹,x²)is feasible

x₁¹+x₁² ≤ e₁¹+e²₁ x₂¹+x₂² ≤ e₂¹+e²₂ Allocationx= (x¹,x²)is non-wasteful if

x₁¹+x₁² = e₁¹+e²₁ x₂¹+x₂² = e₂¹+e²₂ Question

Under what conditions the individuals will end up non-wasteful allocations?

Barter: Goods for Goods IV

In a N-person and M-good economy, an endowment is a vector of vectors:

e= (e¹,e², ...,e^N),

wheree¹= (e¹₁,e¹₂, ...,e¹_M),...,eⁱ = (e₁ⁱ,eⁱ₂, ...,e_Mⁱ ),...,e^N= (e^N₁,e^N₂, ...,e^N_M).

So, the total endowment of goods can be written as:

good 1: e¹₁+e²₁+...+e^N₁ =

N

X

i=1

eⁱ₁

good j: e_j¹+e²_j +...+e^N_j =

N

X

i=1

eⁱ_j

good M: e¹_M+e²_M+...+e_M^N =

N

X

i=1

eⁱ_M

Barter: Goods for Goods V

For the entire economy, an allocation is x= (x¹, ...,x^N), where

x¹= (x₁¹,x₂¹, ...,x_M¹) xⁱ = (x₁ⁱ,x₂ⁱ, ...,x_Mⁱ ) x^N= (x₁^N,x₂^N, ...,x_M^N) Allocationx= (x¹, ...,x^N)is non-wasteful w.r.t. good 1 if

N

X

i=1

x₁ⁱ =

N

X

i=1

eⁱ₁

Non-wasteful Allocations

Definition

Allocationx= (x¹, ...,x^N)is non-wasteful if

N

X

i=1

x_jⁱ

total consumption

=

N

X

i=1

eⁱ_j

total availablility

for allj=1, ...,M Definition

Set of (non-wasteful) allocations is the set {x= (x¹, ...,x^N)} such that x_jⁱ ≥0 for alliandj, and

PN

i=1x_jⁱ =PN

i=1eⁱ_j for allj=1, ...,M.

Assumptions

We will make the following assumption about the choice sets and the individual preferences:

The set of alternatives is the set of (non-wasteful) allocations Individuals have (self-interested) preferences defined over the set of allocations

Each preference is/can be represented by a (self-interested) utility function

Each preference is complete, transitive and ????

Each utility function is monotone, and ???

Pareto Optimal Allocations I

Consider a two-person, two-good economy.

Definition

Allocation(x¹,x²)is Pareto superior to the endowment,(e¹,e²), if uⁱ(xⁱ) ≥ uⁱ(eⁱ)holds fori =1,2.And

uⁱ(xⁱ) > uⁱ(eⁱ)holds for at least onei.

Remark

If(x¹,x²)is Pareto superior to(e¹,e²), then (e¹,e²)is called Pareto inferior to(x¹,x²) (e¹,e²)CANNOT be Pareto Optimum (x¹,x²)may or may not be Pareto Optimum

Pareto Optimal Allocations II

Definition

Allocation(x¹,x²)is Pareto Optimum, if there does not exit another (feasible) allocation(y¹,y²)such that:

(y¹,y²)is Pareto superior to(x¹,x²).

Remark

In general, there can be several Pareto Optimum allocations.

Barter in an Ideal World I

Question

What is the best achievable outcome under Barter?

Assuming that:

there are no restriction on trade/exchange - their endowments with some other bundle

but individuals are free to trade - trade only if they wish to do so individuals have all the information needed for the trade.

Barter in an Ideal World II

Example

Consider the following two-person, two-good economy:

Endowments: e¹= (1,9), ande²= (9,1)

Preferences:uⁱ(x,y) =xy. That is,u¹(x₁¹x₂¹) =x₁¹x₂¹and u²(x₁²x₂²) =x₁²x₂²

Allocation:x¹= (3,3), andx²= (7,7)

Clearly, allocationx= (x¹,x²)is feasible. And,u¹(x¹)≥u¹(e¹), and u²(x²)≥u²(e²). In factu²(x²)>u²(e²).

That is,

allocationx= (x¹,x²)is Pareto superior toe= (e¹,e²) So, we say that: allocatione= (e¹,e²)is blocked by allocation x= (x¹,x²).

Barter in an Ideal World III

Definition

For a two-person two-good economy: Allocationx= (x¹,x²)will block y= (y¹,y²), if any of the following holds:

1 u¹(x¹)>u¹(y¹), and e¹≥x¹; or

2 u²(x²)>u²(y²) and e²≥x²; or

3 (x¹,x²)is Pareto superior to(y¹,y²), i.e.,

uⁱ(xⁱ) ≥ uⁱ(yⁱ).fori=1,2.And uⁱ(xⁱ) > uⁱ(yⁱ)

holds for at least onei.

Barter in an Ideal World IV

Remark

In the above example, recallx¹= (3,3)andx²= (7,7). You can verify that:

there is no other feasible allocationy= (y¹,y²)for which the following hold (with at least one inequality)

u¹(y¹)≥u¹(x¹), andu²(y²)≥u²(x²).

so, allocation(x¹,x²)cannot be blocked by any individual or both of them together.

Question

For the above example,

How many unblocked allocations are there?

What is the set of possible outcomes under Barter?

Blocking Coalition

Here is a general definition of Blocking Coalition.

Definition

LetS⊆ {1, ...,N}.Sis called a blocking coalition forx= (x¹,x², ...,x^N)if there is some vectorysuch that

X

i∈S

y_jⁱ = X

i∈S

e_jⁱ for allj =1, ...,M

uⁱ(yⁱ) =uⁱ(y₁ⁱ, ...,y_Mⁱ ) ≥ uⁱ(x₁ⁱ, ...,x_Mⁱ ) =uⁱ(xⁱ) for all i∈S uⁱ(yⁱ) =uⁱ(y₁ⁱ, ...,y_Mⁱ ) > uⁱ(x₁ⁱ, ...,x_Mⁱ ) =uⁱ(xⁱ) for somei ∈S Question

What will be the outcome (equilibrium) in a barter economy, where individuals have all the information?