Welfare Economics: Lecture 12

Welfare Economics: Lecture 12

Ram Singh

Course 001

October 20, 2014

Fair Vs Efficient

Question

1 What is a fair allocation?

2 Is a ‘fair’ allocation also an efficient allocation?

3 Is a ‘fair’ allocation Pareto efficient?

4 Does a competitive market lead to ‘fair’ allocations?

Fairness: Two Definitions

Consider aN×M pure exchange economy. Let, (e¹, ...,e^N)be the vector of initial endowments.

Definition

Allocationx= (x¹, ...,xⁿ)is ‘Fair’ if it is equal division of endowments. That is, if

(∀i,j ∈N)

"

xⁱ =x^j = PN

i=1eⁱ N

# .

Definition

Allocationx= (x¹, ...,xⁿ)is ‘Fair’ if it is Non-envious/envy-free. That is, if (∀i,j∈N)[xⁱ Ri x^j], i.e.,

(∀i,j ∈N)[uⁱ(xⁱ)≥uⁱ(x^j)].

Are these definitions equivalent to each other?

Pareto Optimal Allocations I

Theorem

An allocationˆx= (ˆx¹, ...,ˆx^I)is PO iff: For some utility levelsU¯², ...,U¯^I, xˆ= (ˆx¹, ...,ˆx^I)is a solution of the following

max

x¹,...,x^I{u¹(x¹)}

s.t.xⁱ ≥0, and

uⁱ(xⁱ) ≥ U¯ⁱ, for all i>1

I

X

i=1

xⁱ =

I

X

i=1

eⁱ

See MWG, Section 16.F

Fair Vs Pareto Efficient

Question

1 Is an ‘Equal division’ allocation ‘Non-envious’ ?

2 Is a Non-envious allocation also an Equal division allocation?

3 Is an ‘Equal division’ allocation Pareto Efficient?

4 Is a Non-envious allocation Pareto Efficient?

Fair but Not Efficient

Consider a 3×3 exchange economy. Let u¹=3x1+2y1+z1

u²=2x₂+y₂+3z₂ u³=x₁+3y₁+2z₁ e¹=e²=e³= (1,1,1)

Consider an allocationy= (y¹,y²,y³), where y¹= (3,²₃,0),y²= (0,0,2)andy³= (0,⁷₃,1)

Can Fair be Efficient? I

Consider a 2×2 pure exchange economy:

The goods are;x andy.

u¹(.) =x^α.y^1−αandu²(.) =x^β.y^1−β, andα=β =¹₂ the total initial endowment vector is(¯x,¯y)>>(0,0).

Let

x1andx2denote the amounts of goodx allocated to individuals 1 and 2, resp.

y₁andy₂denote the amounts of goody allocated to individuals 1 and 2, resp.

Can Fair be Efficient? II

Clearly

MRS¹ = y₁ x₁ MRS² = y₂ x2

= y¯−y₁ x¯−x1

So the set of PO allocations is solution to y₁

x1

=y₂ x2

= y¯−y₁ x¯−x1

But,

y₁ x1

= ¯y−y₁

¯x−x1

⇒ y₁

x1

=y¯ x¯

Can Fair be Efficient? III

Question

Is fair (equal) division of endowments a P.O allocation?

Consider a 2×2 exchange economy:

The goods are;x andy.

u¹(x₁,y₁) =x₁.y₁andu²(x₂,y₂) =x₂.y₂

the total initial endowment vector is(¯x,¯y)>>(0,0).

So the set of PO allocations is solution to

MRS¹ = MRS²,i.e.,

∂u(.)

∂x1

∂u(.)

∂y1

=

∂u(.)

∂x2

∂u(.)

∂y2

Can Fair be Efficient? IV

Under ‘equal division’ allocation, i.e., whenx1=x2andy1=y2, we have

∂u(.)

∂x1

∂u(.)

∂y1

=

∂u(.)

∂x2

∂u(.)

∂y2

Fairness under Markets

Question

Is competitive equilibrium allocation fair?

Proposition

When preferences are strongly monotonic and initial allocation is ‘Equal’, the competitive equilibrium is fair (non-envious)

Suppose,e¹=e²=...=eⁿ, and(x^∗1, ...,x^∗n)is a Walrasian equilibrium allocation. Suppose,

(∃i,j∈ {1, ...,n})[uⁱ(x^∗i)<uⁱ(x^∗j)].

Then,(x^∗1, ...,x^∗n)not a WEA. So, the following holds.

(∀i,j∈ {1, ...,n})[uⁱ(x^∗i)≥uⁱ(x^∗j)].

Effect of Endowments I

Question

Does an increase in endowment make the person better off?

Answer is ‘Yes’, for well behaved utilities. Consider a 2×2 pure exchange economy:

The goods are;x andy.

u¹(.) =x^α.y^1−αandu²(.) =x^β.y^1−β,α, β∈(0,1)andα6=β the initial endowments aree¹(.) = (¯x1,¯y1)>>(0,0)and e²(.) = (¯x2,y¯2)>>(0,0).

Effect of Endowments II

Question

Find out the competitive equilibrium prices and allocations.

How do the competitive equilibrium allocations change with initial endowments

For any given price vectorp= (p1,p2)>>(0,0), you can check that demands are:

x1(p) = α(p1x¯1+p2y¯1) p1

y₁(p) = (1−α)(p₁x¯₁+p₂y¯₁) p2

x₂(p) = β(p1x¯2+p2¯y2) p₁

y2(p) = (1−β)(p1¯x2+p2y¯2) p₂

Effect of Endowments III

Market Clearance for 1st good implies:

x1(p) +x2(p) = ¯x1+ ¯x2

α(p1¯x1+p2¯y1)

p₁ +β(p1x¯2+p2¯y2)

p₁ = ¯x1+ ¯x2

αp1¯x1+αp2y¯1+βp1¯x2+βp2y¯2 = p1¯x1+p1¯x2

Re-arranging the last equality, we get

p2(α¯y1+βy¯2) = [(1−α)¯x1+ (1−β)¯x2]p1,i.e., p₁^∗

p₂^∗ = (α¯y₁+βy¯₂)

(1−α)¯x₁+ (1−β)¯x₂ (1)

Effect of Endowments IV

Letp^∗₂=1. In view of (1), the equilibrium demands can be written as

x₁^∗(p^∗) = α

¯x₁+ ¯y₁(1−α)¯x₁+ (1−β)¯x₂ (α¯y₁+βy¯₂)

y₁^∗(p^∗) = (1−α)

¯x₁ (α¯y1+βy¯2)

(1−α)¯x₁+ (1−β)¯x₂+ ¯y₁

x₂^∗(p^∗) = β

x¯2+ ¯y2

(1−α)¯x1+ (1−β)¯x2

(α¯y₁+β¯y₂)

y₂^∗(p^∗) = (1−β)

x¯2

(α¯y1+β¯y2) (1−α)¯x1+ (1−β)¯x2

+ ¯y2

From (1)

∂(^p_p^∗1∗ 2

)

∂¯x1

=− (α¯y₁+βy¯₂)(1−α) [(1−α)¯x1+ (1−β)¯x2]² <0.

Effect of Endowments V

Now, you can verify the following:

∂x₁^∗

∂x¯₁ = α+ ¯y1

(1−α) [(α¯y₁+βy¯₂)] >0

∂y₁^∗

∂x¯1

= (1−α) (α¯y₁+βy¯₂)(1−β)¯x₂ [(1−α)¯x1+ (1−β)¯x2]² >0

∂x₂^∗

∂x¯1

= βy¯2

(1−α) (α¯y1+βy¯2) >0

∂y₂^∗

∂x¯₁ = −¯x2(1−β) (1−α)(α¯y1+βy¯2) [(1−α)¯x₁+ (1−β)¯x₂]² <0