Barter Exchange and Core: Lecture 2

Barter Exchange and Core: Lecture 2

Ram Singh

Course 001

September 16, 2015

The Question

Consider a N-person and M-good pure-exchange economy:

Let the endowment vectors be:e¹= (e¹₁,e¹₂, ...,e¹_M),..., eⁱ = (e₁ⁱ,eⁱ₂, ...,eⁱ_M),...,e^N= (e^N₁,e₂^N, ...,e^N_M), respectively.

An exchange/reallocation can lead to Pareto Superior outcome - the vector of initial endowments is not Pareto Efficient

Question

How can we redistribute the endowments such that:

Every individual prefers the reallocated bundle received over her initial endowment

No subset of individuals can do better for themselves using their own endowments

Every subset of individuals prefers the reallocation made to the subset as a whole over what they can do own their own.

An Example

Example

Consider the following two-person, two-goods 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 uⁱ(xⁱ) > uⁱ(yⁱ)

holds for at least onei.

Rejectable Allocations II

Since the preferences are monotonic, for a two-person two-good economy:

Allocationx= (x¹,x²)will blocky= (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.

Rejectable Allocations III

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, how many unblocked allocations are there?

Question

How to find the Core of a N×M economy?

The above statements are true forN×M economy as well.

Core Allocations: Properties II

That is, an allocationx= (x¹,x², ...,x^N)will belong to the Core, only if Everyiprefersxⁱ at least as much aseⁱ,i =1,2, ...,N- otherwisei will block it

Allocationx= (x¹,x², ...,x^N)is Pareto Optimum - otherwise all of them will block it together.

Remark

A Pareto Optimum allocation may not belong to the Core

will not belong to the Core if there exists a blocking coalition

Question

Does the set of Pareto optimum allocations depend on the initial endowments?

Blocking Coalition

Here is a general definition of Blocking Coalition.

Definition

LetS⊆ {1, ...,N}.Sis called a blocking coalitions 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

Core of Barter Exchange

Consider a pure exchange economy(uⁱ(.),eⁱ)_i∈N. For this economy, Definition

Core is a set of allocations,C(uⁱ(.)_i∈N,e), such that ifx∈C(uⁱ(.)_i∈N,e), then xCANNOT be blocked by any coalition.

Remark

The size of the Core, i.e., outcome of barter depends on the ‘nature’ of the economy:

the nature of individual preferences the initial endowment/wealth

the number of individuals in the economy

Size of the Core I

Example

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

Endowments: e¹= (1,9),e²= (9,1), ande³= (5,5) Preferences:u¹(x₁¹.x₂¹) =x₁¹.x₂¹,u²(x₁².x₂²) =x₁².x₂², and u³(x₁³.x₂³) =x₁³.x₂³

Now, consider the following allocation:

x¹= (3,3),x²= (7,7), andx³= (5,5).

Size of the Core II

When

x¹= (3,3),x²= (7,7), andx³= (5,5).

Question

Is the allocationx= (x¹,x²,x³)feasible?

Is allocationx= (x¹,x²,x³)Pareto-superior toe= (e¹,e²,e³)?

Is allocationx= (x¹,x²,x³)Pareto Optimum?

Does allocationx= (x¹,x²,x³)belong to the Core?

Consider a Coalition of 1 and 3, i.e.,S={1,3}. Letybe such that y¹= (2,5)andy³= (4,9).

Recall,e¹= (1,9)ande³= (5,5).

Size of the Core III

Question

How to find the Core allocations?

If there are 25 individuals, you have to check 2²⁵−1 as potential coalitions that may block an allocation.

Size of the Core shrinks with number of agents - Edgeworth (1881); Debreu and Scarf (1963); Aumann (1964), etc.

The Core in Real World

Question In real world,

Will bargaining among individuals always lead to one of the allocations in the Core?

Are there factors that can frustrate successful bargaining among individuals?

Question

Can market lead to the same set of outcomes as the Barter will in ideal world?

Can outcome under market be better than under the Barter?

Barter Vs Market I

1 Informational and logistical requirements Barter requires

Search costs - to identify suitable trading partners Successful negotiations

Market requires No search costs

No cooperation - only decision making at individual level

2 Relative Efficiency Barter

Pareto efficient outcome is unlikely, for large set of individuals Market

Pareto efficient outcome more likely, especially for large set of individuals

The claims are valid with or without production

Barter Vs Market II

3 Effect of Policy Interventions Barter

Policy intervention only through reallocation of endowments Market

Policy intervention through reallocation of endowments as well as direct transfers of ’purchasing power’

Remark

Neither Barter nor Market can guarantee the intended outcome Some endowments are not transferable - E.g. ????