Market Failure: Externalities

Market Failure: Externalities

Ram Singh

Lecture 21

November 10, 2015

Questions

What is externality?

What is implication of externality for efficiency of competitive equilibrium?

What are the corrective measures available?

What are the relative merits of the corrective measures?

Can the market itself take care of externality?

WE with FOPs I

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 FOP 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}

WE with FOPs II

Let,

p= (¯p₁, ...,¯p_J)be the given the output price vector, consider any arbitrary allocation of FOPs across firms, say

w^∗= (w₁^∗, ...w_L^∗)be the given input price vector

(y^∗1, ...,y^∗J)be the profit maximizing output production level of FOPs for firmsj =1, ...,J.

(z^∗1, ...,z^∗J)be the profit maximizing demand of FOPs for firms j =1, ...,J.

WE with FOPs III

Recall, Proposition

The Competitive equilibrium (WE) is Pareto optimum.

Proposition

The equilibrium factor allocation,(z^∗1, ...,z^∗J), is Pareto optimum.

Proposition

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

WE with FOPs IV

Proposition

The production plan(y¹, ...,y^J)maximizes the aggregate/total profit for the economy if and only if it is a Pareto optimal plan.

However, in presence of externality, all these results breakdown

in fact, the existence of WE cannot be guaranteed anymore Government intervention is needed - generally but not always

A simple illustration I

Assume

There are two ‘competitive’ firms

Firm 2 uses only one FOP, sayl, to produce one marketable output Firm 1 also uses the FOPl but it produces a marketable output along with another ‘non-marketable’ output/inpute

There is no market ine

Firm 1 decides on the level ofe; firm 2 has no direct control over choice ofe

The profit functions areπ₂(y₁,l₁,e,p,w)andπ₂(y₂,l₂,e,p,w), respectively

A simple illustration II

Note for givenpandw, we have

φ_i(e,p,w) = maxπ_i(y_i,l_i,e,p,w)

≡ φ_i(e) Note: You can think of

φ₁(e)as the maximum profit for 1 given the level ofeopted by firm 1.

φ₂(e)as the maximum profit for 2 given the level ofeopted by firm 1.

Assume

φ⁰₁(e)>0, φ⁰⁰₁(e)<0, φ⁰₂(e)<0, φ⁰⁰₂(e)>0,i.e.,

eis good for firm 1 but bad for firm 2.

A simple illustration III

Moreover, there existse, such that¯ φ⁰₁(¯e)<0.

Question

Which firm is the cause behind the externality?

Firm 1 will solve maxe{φ₁(e)}. It will choosee^pthat solves the following FOCs:

φ⁰₁(e) =0 (1)

That is,φ⁰₁(e^p) =0. However, the total profit maximization problem is

maxe {φ₁(e) +φ₂(e)} (2)

For this OP, the FOCs is:

φ⁰₁(e) +φ⁰₂(e) =0 (3) Lete^∗solve (3).

A simple illustration IV

That is,φ⁰₁(e^∗) +φ⁰₂(e^∗) =0. Clearly, e^p>e^∗.

Question

What is wealth maximizing level of externality - e^por e^∗? What is Kaldor efficient level of externality - e^por e^∗?

WE with Externality I

Assume

There areJ firms;j =1,2, ...,J

FOPs arel =1, ...,L,e. Set of FOPs isL={1, ..,L,e}

There is no market ine

Firm 1 decides how much ofeto use/produce

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}

Individual Production Levels I

For simplicity, assume there are only two firms.

Take any price vectorp¯= (¯p₁, ...,p¯_J)for outputs andw¯ = ( ¯w₁, ...,w¯_L)for inputs. Firm 1 will chooseeandy¯¹that solves

= max

y¹,e

(

p¯1f¹(z¹,e)−

L

X

k=1

w¯k.z_k¹ )

wheref¹(z¹,e)is the output level, when input vector used isz¹= (z₁¹, ...,z_L¹) along withe.

Whenf¹(.)is strictly increasing and strictly concave, the demanded(z¹,e) will solve the following FOCs:

Individual Production Levels II

z_l¹solves : ¯p₁∂f¹(z¹,e)

∂z_l¹ =w_l^∗ for alll =1, ...,L, (4) esolves : ¯p1

∂f¹(z¹,e)

∂e =0. (5)

The 2nd firm will solve:

max

z²

(

¯p₂f²(z²,e)−

L

X

k=1

w_k^∗.z_k², )

The demanded(z²)will solve the following FOCs:

z_l²solves : p¯₂∂f²(z²,e)

∂z_l² =w_l^∗ for alll=1, ...,L, (6) So, the WE factor allocation is characterized by (4), (5) and (6).

Pareto Optimal Levels

The total profit maximization problem is max

e,z¹,z²







2

X

j=1

(¯p_jf^j(z^j,e)−w^∗.z^j)







,i.e.,

max

e,z¹,z²

p¯₁f¹(z¹,e)−w^∗.z¹+ ¯p₂f²(z²,e)−w^∗.z² (7) For this OP, the FOCs are:

z_l¹solves : ¯p₁∂f¹(z¹,e)

∂z_l¹ =w_l^∗ for alll =1, ...,L, (8) esolves : ¯p1

∂f¹(z¹,e)

∂e + ¯p2

∂f²(z²,e)

∂e =0. (9)

z_l²solves : p¯₂∂f²(z²,e)

∂z_l² =w_l^∗ for alll=1, ...,L, (10) So, the PO allocation of factors is characterized by (8), (9) and (10).

Corrective Measure: Quantity Regulation

Quantity Regulation:

Firm is allowed to produce up toe^∗and not beyond Sever penalty for production beyonde^∗

In equi, Firm 1 will choosee=e^∗ The outcome is Pareto efficient.

Corrective Measure: Pigouvian Tax (Price Regulation)

Let us go back to the simple case.

There are two firms

Firm 1 causes negative externality for Firm 2 Suppose,

Govt imposes tax on externality ‘creator’

Firm 1 pays a per unit tax¯t=φ⁰₂(e^∗).

Now, 1 will choosee^pthat solves:

maxe {φ₁(e)−te},i.e.,max

e {φ₁(e)−φ⁰₂(e^∗).e}

φ⁰₁(e)−¯t = 0,i.e., φ⁰₁(e)−φ⁰₂(e^∗) = 0,i.e., e^p=e^∗.

Corrective Measures: Subsidy

Suppose,

Govt offers subsidy to the externality creator for a reduction in externality level belowe^∗

Gross subsidy is:s(e) = (e^∗−e)φ⁰₂(e^∗).

Now, 1 will choosee^pthat solves:

maxe {φ₁(e) +s(e)},i.e.,max

e {φ₁(e) + (e^∗−e)φ⁰₂(e^∗)}

φ⁰₁(e) +s⁰(e) = 0,i.e., φ⁰₁(e)−φ⁰₂(e^∗) = 0,i.e., Again,e^p=e^∗.

Corrective Measures: Liability

Let

φ¯₂be the profit in the absence of externality, i.e.,φ¯₂=φ₂(e=0) Suppose,

The externality creator is required to compensate the ‘victim’ of externality

Firm 1 pays a compensation equal to loss;l(e) = ¯φ₂−φ₂(e).

Now, 1 will choosee^pthat solves:

maxe {φ₁(e)−l(e)},i.e.,max

e {φ₁(e)−[ ¯φ₂−φ₂(e)]}

φ⁰₁(e)−l⁰(e) = 0,i.e., φ⁰₁(e)−φ⁰₂(e) = 0,i.e., e^p=e^∗.