General Equilibrium with Production

General Equilibrium with Production

Ram Singh

Microeconomic Theory

Lecture 11

Producer Firms I

There areNindividuals;i =1, ...,N There areM goods;j =1, ...,M There areK firms;k =1, ...,K

Each firm has a set of production plans, i.e., production setY^k ⊂R^M The production setY^k is the set of feasible production plans for firmk Examples: LetM =3 andK =2. Suppose Firm 1 can produce six units of good 2 by using three units of good 1 and nine units of good 3, i.e.,

Firm 1: y¹ = (−3,6,−9) Firm 2: y² = (8,−31

2,−14) Economy: Y = (5,−21

,−23)

Producer Firms II

A typical production plan for firmk is

y^k = (y₁^k, ...,y_M^k), where

y_j^k >0 if goodj is an output produced by firmk y_j^k <0 if goodj is an input used by firmk.

Letp= (p₁, ...,p_M)be the given price vector. For anyy^k ∈Y^k, profit for firmk is

π(p,y^k) =p₁y₁^k +...+p_My_M^k =p.y^k. So, PMP for firmk is:

max

y^k∈Y^k

{p₁y₁^k+...+p_My_M^k},i.e., max

y^k∈Y^k

{p.y^k}

Producer Firms III

Let

Π^k(p) = max

y^k∈Y^k

π(p,y^k) Π^k(p)homogenous function of degree 1 inp.

Assume, for each firmk,

0∈Y^k ⊂R^M, i.e., firm can always earn Zero profit So, profit is non-negative

Y^k is closed and bounded and strictly convex

For any given price vectorp, the profit maximizing choice/production plan is unique

For different price vectorp⁰, profit maximizing choice/production plan will be different, in general.

Aggregate Production Plans I

Let

y^k = (y₁^k, ...,y_M^k)be a feasible production plan for firmk =1, ...,K. Corresponding to the above production plan, the aggregate production plan for the economy is given by

Y= (y¹, ...,y^K), wherey^k = (y₁^k, ...,y_M^k). (1)

Corresponding to the production plan (1), the total production of goodj is given by

K

X

k=1

y_j^k.

So, corresponding to the production plan (1), the total ‘output’ vector is:

(

K

X

k=1

y₁^k, ...,

K

X

k=1

y_M^k) =

K

X

k=1

y^k =Y

Aggregate Production Plans II

ifPK

k=1y_j^k >0, goodj is a net output for the economy ifPK

k=1y_j^k <0, goodj is a net input for the economy.

The aggregate production possibility set (for the entire economy) is

Y= (

Y|Y=

K

X

k=1

y^k, wherey^k ∈Y^k )

.

That is, ifYˆ ∈Y, then there exist production plansˆy¹, ...,ˆy^k, ...,ˆy^K such that yˆ^k ∈Yˆ^k, i.e.,ˆy^k is a feasible plan for firmk =1, ...,K; and

Yˆ =PK k=1ˆy^k

Aggregate Production Plans III

Proposition

Given the above assumptions onY^k and PMP for firms, 0∈Y⊂R^M

Yis closed and bounded and strictly convex

Question

WhyYis a bounded set?

Proposition

Given the above assumptions onY^k, for any given price vector p= (p₁, ..,p_M)>>(0, ...,0), the following PMP has a unique solution:

max

y^k∈Y^k

{p.y^k}for every k=1,...,K

Aggregate Production Plans IV

Proposition

Given the above assumptions onY, for any given price vector

p= (p1, ..,pM)>>(0, ...,0), the following PMP has a unique solution:

maxY∈Y{p.Y}

Efficient Production: Two definitions I

Definition

(Definition 1): Production PlanY= (y¹, ...,y^K)∈Yis ‘efficient’ if there is no other planZ= (z¹, ...,z^K)such that:Zis feasible, i.e.,Z∈Y, and

K

X

k=1

z_j^k ≥

K

X

k=1

y_j^k, for all goodsj

K

X

k=1

z_j^k >

K

X

k=1

y_j^k, for at least one goodj

Remark

Supposekth good is a net input. In that case,PK

k=1z_j^k >PK

k=1y_j^k implies that the production plan requires smaller quantity of this input.

Efficient Production: Two definitions II

SupposeY= (y¹,y²):

firm 1: y¹ = (−2,4,−6) firm 2: y² = (8,−4,−14) So,Y=P2

1yⁱ = (P2 1y1,P2

1y2,P2

1y3) = (6,0,−20) LetZ= (z¹,z²):

firm 1: z¹ = (−2,4,−6) firm 2: z² = (8,−31

2,−14) So,Z=P2

1zⁱ = (P2 1z1,P2

1z2,P2

1z3) = (6,¹₂,−20)

Efficient Production: Two definitions III

Definition

(Definition 2): For given price vectorp= (p₁, ...,p_M), production Plan Y= (y¹, ...,y^K)∈Yis efficient if it solves

maxY⁰∈Y{p.Y⁰},i.e.,

p.Y≥p.Y⁰ for allY⁰∈Y Question

Does (D1) imply (D2)? Does (D2) imply (D1)?

Production Equilibrium I

Total supply is given by

N

X

i=1

eⁱ +

K

X

i=k

y^k

Let

p¯= (¯p₁, ...,¯p_M)be a price vector.

Y¯ = (¯y¹, ...,y¯^K)be a production plan for the economy.

Definition

(¯Y,p)¯ is a competitive ‘production’ equilibrium only if: For allk =1, ...,K,

¯y^k solves max

y^k∈Y^k

{p.y¯ ^k}.

Production Equilibrium II

Proposition

Take any price vectorp¯= (¯p₁, ...,p¯_M). If¯y^k ∈Y^k solves max

y^k∈Y^k

{¯p.y^k}for k=1, ...,K.

LetY, where¯ Y¯ =PK

k=1¯y^k. Then, there is NO other planZ= (z¹, ...,z^K)such that: Zis feasible, i.e.,Z∈Y, and

K

X

k=1

z_j^k ≥

K

X

k=1

y¯_j^k, for all goods j

K

X

k=1

z_j^k >

K

X

k=1

y¯_j^k, for at least one good j

Production Equilibrium III

Proposition

Take any price vectorp¯= (¯p₁, ...,p¯_M). Suppose production plans¯y¹, ...,¯y^K are such that¯y^k ∈Y^k solves

max

y^k∈Y^k

{¯p.y^k}for all k =1, ...,K

. Then,Y¯ =PK

k=1¯y^k solves

maxY∈Y{¯p.Y}

That is, individual profit maximization leads to total profit maximization. Why?

Question

What assumption is made about Externality?

Production Equilibrium IV

Proposition

Take any price vectorp¯= (¯p₁, ...,p¯_M). IfY¯ solves maxY∈Y

{¯p.Y}

then there exist production plans¯y¹, ...,¯y^K such thaty¯^k ∈Y^k,Y¯ =PK k=1¯y^k, andy¯^k solves

max

y^k∈Y^k

{¯p.y^k}for all k =1, ...,K Proof: LetY¯solve maxY∈Y{¯p.Y}. That is,

(∀y∈Y)[¯p.Y¯ ≥p.Y]¯ Lety¯¹, ...,¯y^K are such thaty¯=PK

k=1y¯^k and¯y^k ∈Y^k. If possible, suppose for somek,y¯^k does not solve

max

y^k∈Y^k

{p.y¯ ^k}

Production Equilibrium V

So, there exists someyˆ^k ∈Y^k such that

¯p.ˆy^k >p.¯¯y^k Now, consider the production planZsuch that

Z = (z¹, ...,z^k, ...,z^K)

= (¯y¹, ...,yˆ^k, ...,¯y^K) Clearly,Z∈Y. It is easy to show that

p.Z¯ >¯p.Y,¯ a contradiction.

Privatized Economy I

Privatized Economy:(uⁱ(.),eⁱ(.),Y^k, θ^ik)_i∈{1,..,N},j∈{1,...,M},k∈{1,...,K}

All firms are privately owned

θ^ikis the (ownership) share ofkth firm owned byithe individual;

0≤θ^ik≤1 for alli=1, ...,Nandk =1, ...,K Pn

i=1θ^ik =1 for allk =1, ...,K

Now, for any given price vector,p= (p₁, ...,p_M), individuali solves max

xⁱ∈R₊^M

uⁱ(x), s.t. p.xⁱ ≤Iⁱ(p), where

Iⁱ(p) =p.eⁱ+

K

X

k=1

θ^ikπ^k(p).

Privatized Economy II

Remark

In view of the above assumptions on Production sets, π(p)≥0 is bounded above and continuous inp.

Moreover,Iⁱ(p)is a continuous function ofp

In view of the assumption onuⁱ(.), the solution of the above UMP is unique.

Definition

The excess demand function for goodjis

z_j(p) =

N

X

i=1

x_jⁱ(p,Iⁱ(p))−

K

X

k=1

y_j^k(p)−

N

X

i=1

eⁱ_j.

So, excess demand vector isz(p) = (z (p), ...,z (p)).

Privatized Economy III

Definition

Feasible Allocation: An allocation(X,Y), whereY= (y¹, ...,x^K)is feasible, if:

For allk,y^k ∈Y^k

N

X

i=1

xⁱ =

N

X

i=1

eⁱ+

K

X

i=k

y^k.

Existence of WE I

As before, in view of the assumption onuⁱ(.)andY^k,

the excess demand function is homogeneous function ofpof degree 0.

For equilibrium to existY+PN

i=1eⁱ >>0must hold for someY∈Y. Aspj →0 for somej, the excess demand becomes unbounded for one of such commodities.

Theorem

Consider an economy(uⁱ(.),eⁱ, θ^ik,Y^j), where i =1, ...,N, j=1, ...,M and k =1, ...,K . If uⁱ(.)and Y^j satisfy above assumptions, then there exists a price vectorp^∗>>0such thatz(p^∗) =0.

Efficiency of WE I

Definition

A feasible allocation(¯X,Y)¯ is Pareto optimum if there is no other allocation (X,Y)such that

N

X

i=1

xⁱ =

N

X

i=1

eⁱ +

K

X

i=k

y^k;

uⁱ(xⁱ) ≥ uⁱ(¯x) for alli ∈ {1, ...,N} and uⁱ(x^j) > u^j(¯x) for somej ∈ {1, ...,N}

Theorem

Consider an economy(uⁱ(.),eⁱ, θ^ik,y^k), where i =1, ...,N and k=1, ...,K . If uⁱ(.)is strictly increasing, then every WE is Pareto optimum.

Proof: Suppose(¯X,Y)¯ along withp^∗is WE. Therefore,

Efficiency of WE II

N

X

i=1

¯xⁱ =

N

X

i=1

eⁱ+

K

X

i=k

¯y^k

Suppose(¯X,Y)¯ is not PO. So, there is some(X,Y), such that

N

X

i=1

xⁱ =

N

X

i=1

eⁱ +

K

X

i=k

y^k;

uⁱ(xⁱ) ≥ uⁱ(¯xⁱ) for alli ∈ {1, ...,N} and uⁱ(xⁱ) > u^j(¯x^j) for somej ∈ {1, ...,N}

Therefore,

p^∗.xⁱ ≥ p^∗.x¯ⁱ for alli ∈ {1, ...,N} and

Efficiency of WE III

p^∗.

N

X

i=1

xⁱ > p^∗.

N

X

i=1

x¯ⁱ

p^∗.





M

X

j=1

y^j +

M

X

i=1

e^j



 > p^∗.





M

X

j=1

y¯^j+

M

X

j=1

e^j





p^∗.

M

X

j=1

y^j > p^∗.

M

X

j=1

y¯^j

a contradiction. Why?

Efficiency of WE IV

Theorem

Consider an economy(uⁱ(.),eⁱ, θ^ik,Y^j), where i =1, ...,N and k=1, ...,K . Suppose, uⁱ(.)and Y^j satisfy above assumptions, andy+PN

i=1eⁱ >>0for somey∈Y. If(¯x,¯y)is any feasible PO allocation, then there exist

’Cash’ transfers Tⁱ, i=1, ..,N such thatPN

i=1Tⁱ =0 A price vectorp¯ = (¯p1, ...,p¯M)

such that

1 x¯ⁱ maximizes uⁱ(xⁱ) s.t.p.x¯ ⁱ ≤p.e¯ ⁱ+PK

k=1θ^ikπ^k(¯p) +Tⁱ for all i =1, ...,N

2 y¯^k maximizesp.y¯ ^k for all k=1, ...,K

3 z^j(¯p) =0 for all j=1, ...,M