General Equilibrium Analysis: Lecture 3

General Equilibrium Analysis: Lecture 3

Ram Singh

Course 001

September 19, 2014

Core when Utilities are ’Non-transferable’ I

Example

Consider the following three-person, two-goods 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

We have seen that:

ForN=2, allocationx= (x¹,x²), wherex¹= (3,3),x²= (7,7), belongs to the core

ForN=3, allocationx= (x¹,x²), wherex¹= (3,3),x²= (7,7), and x³= (5,5)does NOT belong to the core

Core when Utilities are ’Non-transferable’ II

Example Suppose,

There are four individuals and two goods as above.

Utility functions are: uⁱ(x₁ⁱ.x₂ⁱ) =x₁ⁱ.x₂ⁱ,i =1, ...,4.

Endowments aree¹= (1,9),e²= (9,1),e³= (1,9), ande⁴= (9,1).

Now, consider the allocationx= (x¹,x²,x³,x⁴), where x¹= (3,3) =x³andx²= (7,7) =x⁴.

1 Is allocationx= (x¹,x²,x³,x⁴)Pareto optimum?

2 Does it belong to the Core?

The Core when Utilities are Transferable I

Example

Production: Consider the following three firms/farmers producing one good:

Endowments of FOP:e¹= (e¹₁,e¹₂) = (1,9), e²= (e²₁,e²₂) = (9,1), ande³= (e¹₃,e₂³) = (5,5) Production Functions:

q¹(x₁¹.x₂¹) =x₁¹.x₂¹;q²(x₁².x₂²) =x₁².x₂²; andq³(x₁³.x₂³) =x₁³.x₂³ FOPs can be shifted/employed across farms/plants as desired

So, the maximum possible production level for Coalition,S, is Q^S = ¯e^S₁.¯e^S₂, where

e¯^S₁ =X

i∈S

eⁱ₁and ¯e^S₂ =X

i∈S

eⁱ₂

The Core when Utilities are Transferable II

1 Is output allocation(0,0,225)Pareto optimum?

2 Does output allocation(0,0,225)belong to the Core?

3 Is output allocation(45,45,135)Pareto optimum?

4 Does output allocation(45,45,135)belong to the Core?

You can show that the set of Core allocations is C(e,qi) ={(y₁,y2,y3)|yi ≥0& X

i∈S

yi ≥Q^S = ¯e^S₁.¯e^S₂},

where for eachS⊆ {1,2,3}(non-empty subsets). E.g., y1≥9; y2≥9; y3≥25; y1+y2≥100, etc.

An easy test

Let(uⁱ(.),eⁱ)_i∈Nbe a pure exchange economy, andx= (x¹,x², ...,x^N)be a feasible allocation. Consider the following statement.

Statement A:S⊆ {1, ...,N}is a blocking coalitions forx= (x¹,x², ...,x^N).

That is,Scan blockx.

Suppose, the statement A is true. In that case, which of the following claims is/are necessarily true?

1 Allocationx= (x¹,x², ...,x^N)is Pareto optimum

2 Allocationx= (x¹,x², ...,x^N)is not Pareto optimum

3 Allocatione= (e¹,e², ...,e^N)is Pareto optimum

4 Allocatione= (e¹,e², ...,e^N)is not Pareto optimum

Another easy test

Consider the following context:

There are three farmers - 1, 2 and 3

They can produce, and consume any one three goods;x= (x₁,x₂,x₃), y= (y₁,y₂,y₃)andz= (z₁,z₂,z₃);

Production decision is to be taken unanimously

In case of partial disagreement the majority will prevail - in case of total disagreement nothing will be produced

The preference relations are as follows

Table:(Preferences)

1 2 3

x1 y2 z3

y1 z2 y3

z1 x1 x3

The Core in Real World I

Remark

Exchange/cooperation and therefore the Core is an issue only if preferences are monotonic and ’super-additivity’ property holds in one sense or the other.

The initial endowment vectors aree¹= (1,0)ande²= (0,1). Utility functions are :u¹(x,y)andu²(x,y), wherex is the quantity of the first good andy is the quantity of the second good;x,y ≥0. Assuming:

1 uⁱ(x,y) =1,i=1,2

2 uⁱ(x,y) =x+y,i=1,2

3 u¹(x,y) =

1, whenx+y <1

x +y, whenx +y ≥1, andu²(x,y) =x+y. Find out:

The set of Pareto efficient allocation?

The set of Core allocation?

The Core in Real World II

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. ????

Remark

The above claims for co-operative ventures hold only for private goods.

In case of Common Property Resources, the opposite can hold