General Equilibrium Analysis: Lecture 6

General Equilibrium Analysis: Lecture 6

Ram Singh

Course 001

September 26, 2014

First Fundamental Theorem I

The First Fundamental Theorem of Welfare Economics:

Consider an exchange economy(uⁱ,eⁱ)_i∈N. Theorem

If uⁱ is strictly increasing for all i =1, ..,N, then W((uⁱ,eⁱ)_i∈N)⊆C((uⁱ,eⁱ)_i∈N). That is,

Every WE/Competitive equilibrium is Pareto optimum;

Every WE/Competitive equilibrium is in the Core.

Question

What if the Core allocations are highly unequal Can markets lead to equitable outcomes?

First Fundamental Theorem II

Question

Supposey= (y¹, ...,y^N)is any feasible Pareto optimum allocation.

y= (y¹, ...,y^N)may or may not be equitable across individuals

If desired, cany= (y¹, ...,y^N)be achieved as a competitive equilibrium?

An Example I

Consider a 2×2 economy:

u¹(.) =x₁¹+2x₂¹, andu²(.) =x₁².x₂² Therefore,MRS1= ¹₂ andMRS2=^x22

x₁²

e¹(.) = (1,¹₂), ande²(.) = (0,¹₂) Individuals act as price-takers

Given any price vectors, in equi. person 2 will consume(x₁²,x₂²)such that:

MRS₂= ^p_p¹

2 and all income is spent.

That is, the demanded bundle(x₁²,x₂²)will be such that:

x₂² x₁² = ^p_p¹

2, i.e,

p2.x₂² = p1.x₁²and p₁x₁²+p₂x₂² = p₁.0+p₂ 2

An Example II

This gives us:

2p1x₁² = p2

2 i.e., x₁² = p2

4p₁.Moreover, x₂² = 1

4

For the 1st person, the demanded bundle(x₁¹,x₂¹)will be such that: if (x₁¹,x₂¹)>>(0,0).

MRS1 = p₁ p2

,i.e. 1 2 =p₁

p2

p₁x₁¹+p₂x₂¹ = p₁+p2

2. However,

An Example III

p1

p₂ > 1

2 ⇒ only 2nd good is demanded p₁

p₂ < 1

2 ⇒ only 1st good is demanded p₁

p2

= 1

2 ⇒ any(x₁¹,x₂¹) on the budget line can be demanded.

So, the only plausible equilibrium price vector will have: ^p_p¹

2 =¹₂. Let(p1,p2) = (1,2). At this price:

For 2nd person, the demanded bundlex²= (x₁²,x₂²) = (1/2,1/4) For 1st person, the bundlex¹= (x₁¹,x₂¹) = (1/2,3/4)lies on the budget line.

An Example IV

Therefore,

(x¹,x²= ((1/2,1/4),(1/2,3/4))along with(p₁,p₂) = (1,2)is a competitive equilibrium.

Remark

WE exists even though preferences are not strictly quasi-concave.

Question

For the above economy, suppose we are told that a WE exists. How can we find the WE?

Note

We know that WE is PO and is a Core allocation So, we can start with the set of PO points.

An Example V

The locus of tangencies of ICs, i.e, where MRS₁ = MRS₂,i.e. 1

2 =x₂² x₁² x₁² = 2x₂².

The only consistent point is

x¹= (1/2,3/4)andx²= (1/2,1/4).

Now, question is:

x¹= (1/2,3/4)andx²= (1/2,1/4)is a WE?

For what price vector, the utility maximizers persons 1 an 2 will choose x¹= (1/2,3/4)andx²= (1/2,1/4), respectively?

An Example VI

Given the nature of the preferences: Try anyp= (p1,p2)such that^p_p¹

2 =¹₂. Question

Consider, another PO point such asy¹= (1/4,5/8)andx²= (3/4,3/8). Can we induce it as WE?

Yes, try this by keepingp= (p1,p2)such that^p_p¹

2 = ¹₂, but by choosing T₁=−1

2 and T₂= 1 2

Now, in equi. person 2 will consume(x₁²,x₂²)such that: ^x22

x₁² =^p_p¹

2, i.e, p₂.x₂² = p₁.x₁²and

p1x₁²+p2x₂² = p1.0+p2.1 2+T2

You can check thatT2= ¹₂induces the 2nd person to buy 3/8.

An Example VII

WhenT₁=−¹₂and T₂= ¹

2, the only solution is (x₁²,x₂²) = (3/4,3/8).

Second Fundamental Theorem

The Second Fundamental Theorem of Welfare Economics:

Theorem

If uⁱ is continuous, strictly increasing, and strictly quasi-concave for all i =1, ..,N, thenanyPareto optimum allocation,y= (y¹, ...,y^N), such that yⁱ >>0,

can be achieved as competitive equilibrium with suitable transfers.

That is,y= (y¹, ...,y^N)is a WE with suitable transfer.

With suitable transfers, market can achieve any of the socially desirable allocation as competitive equilibrium.

2nd Theorem: Transfer of Goods

Suppose,y= (y¹, ...,y^N)is a feasible PO allocation, and we want to achieve allocation asy= (y¹, ...,y^N)a competitive equilibrium. There are two solutions.

Choose,´e= (´e¹, ...,´e^N)such that: For alli =1, ...,N yⁱ =eⁱ+ ´eⁱ.

It can be easily seen that there exists a price vector such thaty= (y¹, ...,y^N) a competitive equilibrium. Letp´= (p⁰₁, ...,p⁰_M)be such a price vector.

Remark

Choose any´e= (´e¹, ...,´e^N)such that the new endowment vectors (e¹+ ´e¹, ...,e^N+ ´e^N)lies on the budget line generated by the price vector p´ = (p⁰₁, ...,p_M⁰ ).

2nd Theorem: Cash Transfer I

Consider an exchange economy(uⁱ,eⁱ)_i∈N. Let,

x= (x¹, ...,x^N)be an the equilibrium without transfers. Clearly, for all i =1, ...,N

p.xⁱ =p.eⁱ Now, consider ‘cash’ transfers; individualigetsTi.

Let,y= (y¹, ...,y^N)be the equilibrium after ‘cash’ transfers. Now: For all i =1, ...,N

p.yⁱ =p.eⁱ+Tⁱ,i.e., for alli=1, ...,N

M

X

j=1

p_jy_jⁱ =

M

X

j=1

p_je_jⁱ+Tⁱ (1)

2nd Theorem: Cash Transfer II

That is,

N

X

i=1





M

X

j=1

pjy_jⁱ



=

N

X

i=1





M

X

j=1

pje_jⁱ



+

N

X

i=1

Tⁱ,i.e,

N

X

i=1

Tⁱ =

M

X

j=1 N

X

i=1

p_jy_jⁱ

!

−

M

X

j=1 N

X

i=1

p_jeⁱ_j

!

=

M

X

j=1

pj N

X

i=1

y_jⁱ−

N

X

i=1

eⁱ_j

!

= 0.