Microeconomics: Barter Economy and Core

Microeconomics: Barter Economy and Core

Ram Singh

Lecture 1

September 19, 2016

Questions

What is the best possible outcome under a command and control (i.e., socialist) economy?

What is the best possible outcome under a decentralized economy with following features:

Rule of Law,

Complete freedom to trade

Individuals have complete information about each other

Trade/exchange of goods is costless and individuals are willing to cooperate with each other

What is the best possible outcome under a decentralized economy with following features:

Rule of Law,

Complete freedom to trade

Individuals may not have full information about one another However, the market is competitive

Exchange Economy: Basics I

To start with, 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

For a 2×2 economy, let initial endowments be(e¹,e²), where

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

Exchange Economy: Basics II

Definition

Allocation: An allocation is a distribution of available goods among the individuals.

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

Exchange Economy: Basics 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 with non-wasteful allocations?

Exchange Economy: Basics 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, thetotal endowmentof 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

Exchange Economy: Basics 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

Set of (non-wasteful) allocations is the set

X={x= (x¹, ...,x^N)}

wherexⁱ is such that

x_jⁱ ≥0 for alliandj, for alli =1, ...,N, for allj=1, ...,M; 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 ???

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

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

u^j(x^j) > u^j(e^j)holds for at least onej.

Remark

If a feasible allocation(x¹,x²)is Pareto superior to(e¹,e²), then (e¹,e²)is Pareto inferior to(x¹,x²)

So,(e¹,e²)CANNOT be Pareto Optimum

However,(x¹,x²)may or may not be Pareto Optimum

Pareto Optimal Allocations II

Definition

Allocation(x¹,x²)is Pareto Optimum, if there does not exist 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.

Outcome under Barter 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.

An Example

Example

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

Endowments: e¹= (1,9), ande²= (9,1)

Preferences:uⁱ(x,y) =x.y. That is,u¹(x₁¹.x₂¹) =x₁¹.x₂¹and u²(x₁².x₂²) =x₁².x₂²

Allocation:x¹= (3,3), andx²= (7,7)

Clearly,x= (x¹,x²)is Pareto superior toe= (e¹,e²).

That is,

e= (e¹,e²)will be rejected in favour ofx= (x¹,x²).

Formally speaking,e= (e¹,e²)will be blocked by allocationx= (x¹,x²).

Rejectable Allocations I

Assume all exchanges are voluntary.

For a two-person two-goods economy: Allocationy= (y¹,y²)will be blocked/rejected, if any of the following holds:

1 u¹(e¹)>u¹(y¹); or

2 u²(e²)>u²(y²); or

3 There exists a feasible allocation(x¹,x²)that is Pareto superior to (y¹,y²), i.e., for some(x¹,x²)

uⁱ(xⁱ) ≥ uⁱ(yⁱ).fori=1,2.And u^j(x^j) > u^j(y^j)

holds for at least onei.

Rejectable Allocations II

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?

Core Allocations: Properties I

For a two-person two-good economy, an allocationx= (x¹,x²)belongs to the Core, only if

Everyiprefersxⁱ at least as much aseⁱ,i =1,2 Allocationx= (x¹,x²)is Pareto Optimum

Question

For the above example, describe the set of Core allocations?