General Equilibrium Analysis: Lecture 10

General Equilibrium Analysis: Lecture 10

Ram Singh

Course 001

October 15, 2014

Questions

What is the relationship between output prices and the wage rate for the relevant FOPs?

Does competitive market provide fair wage to all FOPs?

How does distribution of wealth affect the wage rates for different FOPs?

What market factors affect the wage rates for different FOPs?

What non-market factors affect the wage rates for different FOPs?

Firms and FOPs I

Readings: MWG^∗. Assume

There are no intermediate goods;

There are pure inputs (factors of production) and pure consumptions goods;

Pure inputs/FOPs arel =1, ...,L. Set of FOPs isL={1, ..,L}

total endowments of factors is¯z= (¯z₁, ...,z¯_L)>>0and is initially owned by consumers.

Consumers do not directly consume these endowments.

One firm produces only one good; goodj is produced by firmj. So, k =j, andj =1, ...,J. Set of firms and also the consumption goods is J={1, ..,J}

Firms and FOPs II

So production plan ofjth firm can be written asy^j ∈R^L+J y^j = (−z₁^j, ...,−z_L^j,0, ...,y_j^j, ...,0)

= (y₁^j, ...,y_k^j, ...,y_L+J^j )

where

y_k^j =







0, if k >Landk 6=j;

y_j^j, if k >Landk =j;

−z_k^j, ifk ≤L Assumez_k^j ≥0.

Firms and FOPs III

Take any price vectorp¯= (¯p1, ...,p¯J)for outputs andw¯ = ( ¯w1, ...,w¯L)for inputs. Firmjwill choose¯y^j ∈Y^L+J that solves

= max

y^j∈Y^L+J

(

−

L

X

k=1

w¯_k.z_k^j +

L+J

X

k=L+1

p¯_k.y_k^j )

= max

y^j∈Y^L+J

(

−

L

X

k=1

w¯k.z_k^j + ¯pj.y_j^j )

= max

y^j∈Y^L+J

(

¯pjf^j(z^j)−

L

X

k=1

w¯k.z_k^j )

wheref^j(z^j) =y_j^j, when input vector used isz^j = (z₁^j, ...,z_L^j).

Production Equilibrium I

Given output price vectorp, equilibrium in the factors market is a price vector¯ w^∗= (w^∗₁, ...,w^∗_L), and factor allocation for each firm

z^∗1 = (z^∗1₁, ...,z^∗1_L) ... = ...

z^∗J = (z^∗J₁, ...,z^∗J_L)

such that

J

X

j=1

z^∗j_l = ¯zl

andz^j∗solves

Production Equilibrium II

max

z^j

(

p¯_jf^j(z^j)−

L

X

k=1

w_k^∗.z_k^j, )

i.e.,

max

z^j

p¯_jf^j(z^j)−w^∗.z^j. (1) Assume

f^j(.)is strictly increasing and strictly concave for allj =1, ...,J Now, the equilibrium is characterized by the following FOCs:

p¯_j∂f^j(z^j)

∂z_l^j = w_l^∗ for alll =1, ...,L, &j=1, ...,J (2)

J

X

j=1

z^∗j_l =

J

X

j=1

∂c^j(.)

∂wl

= ¯zl for alll =1, ...,L. (3)

Production Equilibrium III

Remark

(w^∗1, ...,w^∗J)and(z^∗1, ...,z^∗J)depend onp= (p¹, ...,p^J)and

¯z= (z¹, ...,z^L).

Therefore, in a competitive equilibrium,(w^∗1, ...,w^∗J)and p= (p¹, ...,p^J)will be determined simultaneously.

Maximizing the Cake-size I

Theorem

The equilibrium factor allocation,(z^∗1, ...,z^∗J), maximizes the aggregate/total revenue for the economy.

From (1)z^∗j solves max_zj

np¯_jf^j(z^j)−PL

l=1w_l^∗.z_l^jo

. Therefore,(z^∗1, ...,z^∗J) solve

max

z¹,...,z^J





 X

j

p¯_jf^j(z^j)−w^∗.z^j





 Next, whenP

jz^j = ¯zmust hold, we have

Maximizing the Cake-size II

max

z¹,...,z^J





 X

j

(¯p_jf^j(z^j)−w^∗.z^j)







= max

z¹,...,z^J

X

j

p¯_jf^j(z^j)−w^∗.X

j

z^j

= max

z¹,...,z^J

X

j

p¯_jf^j(z^j)−w^∗.¯z

Therefore,(z^∗1, ...,z^∗J)solve

max

z¹,...,z^J





 X

j

¯p_jf^j(z^j),







(4)

such thatP

jz^j = ¯z.

Maximizing the Cake-size III

Remark

The Competitive equilibrium allocation is also Revenue maximizing allocation. Therefore,

The equilibrium allocation can be determined without determining factor prices.

Question

Should a county focus on Revenue (GDP) maximizing allocation of FOPs?

While deciding on allocation of FOPs, can we ignore the issue of equity in distribution of gains from growth?

Wage-rates I

Question

Are wage-rates fair under a competitive economy?

Let

z^∗ =

J

X

j

z^∗j and z=

J

X

j

z^j

f(z) = p¯1f¹(z¹) +...+ ¯pJf^J(z^J)

Given output price vector,¯p= (¯p1, ...,p¯J), we know thatz^∗solves:

maxz≥0{f(z)−w^∗.z}

s.t.PJ

z^j = ¯z.

Wage-rates II

This gives us the following FOCs:

w₁^∗ = ∂f(z)

∂z₁ = ¯p₁∂f¹(z)

∂z₁ +...+ ¯p_J∂f^J(z)

∂z₁ ... = ...

w_L^∗ = ∂f(z)

∂z_L = ¯p₁∂f¹(z)

∂z_L +...+ ¯p_J∂f^J(z)

∂z_L

Since, in equilibriumz= ¯z, we get (∀l∈L)

w_l^∗=∂f(¯z)

∂zl

= ¯p₁∂f¹(z)

∂zl

+...+ ¯p_J∂f^J(z)

∂zl

Wage-rates III

Remark

In a competitive setting

Each FOP is paid equal to its marginal social productivity (in money terms)

ACeteris Paribusincrease in supply of a FOP decrease its market price (wage). Why?

Question

How will distribution of wealth affects the market prices for FOPs?

How will an increase in supply of a FOP affect its market price?