Externalities and Public Goods - Universitetet i oslo · Piacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring 2018. 1/26 Externalities and Public Goods ECON 4240

Piacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring 2018. 1/26

Externalities and Public Goods

ECON 4240 Spring 2018

Snyder et al. (2015), chapter 17

University of Oslo

1.3.2018


Defining Externalities

Definition 1

An externality occurs whenever the activities of one economic actoraffect the activities of another actor in ways that are not reflectedin market transactions.

Examples of externalities

◮ Traffic: congesting the road

◮ Transmittable diseases: infecting others

◮ Pollution: damage to health or environment paid for

Not an externality

◮ A new hotel opens in a small town. All other hotels’ revenuesdecrease.


Defining Externalities

◮ Main issue with externalities:there is no market price providing correct incentives(correct from a societal point of view)

◮ Externalities can be among firms, among consumers, orbetween consumers and firms.

◮ Special case of externality: utility of individual i depends onthe utility of individual j (friends, relatives, colleagues?)

utility = US(x1, ...,xn;UJ)

∂US

∂UJ> 0 altruism; ∂US

∂UJ< 0 envy


Resolving Externalities

Coasian bargaining & property rights:

◮ Coase Theorem: In a competitive economy with complete

information and zero transaction costs, the allocation of

resources will be Pareto-efficient if all property rights are

assigned.

Examples (somewhat rare):

◮ Cheshire, a small village in Ohio, has been polluted and finallybought up in 2002 by American Electric Power for roughly 20million USD

◮ French producer of mineral water Vittel who bargained withsurrounding farms in order to maintain the water qualitynecessary for its product.

◮ Compensation paid by wildlife protection groups to ranchersfor sheep lost to wolf predation.



Coasian bargaining & property rights: The stated examples

◮ Tend to be Pareto improvements

◮ Difficult to say whether the allocations reached in examplesare actually Pareto optimal

Practical problem

◮ transaction costs of such negotiations

◮ e.g. think about climate change and who has to sit at thetable...



Command and control: Prescribe

◮ pollution level

◮ pollution intensity of production

◮ technology (e.g. BATNEEC = Best Available Technology Not Entailing

Excessive Costs)

Issue:

◮ might require targeting different firms differently

◮ prescribing a technology does not incentivize development ofnew technologies

A more market-based approach:

◮ Make firms liable for damages



Market-based approaches:

◮ Pigovian taxation

◮ Tradable permits

What’s nice:

◮ Both achieve that costs are equalized across firms, a necessarycondition for cost efficiency.

◮ Both usually imply some incentives for R&D of bettertechnologies (though not necessarily first best)


Externalities and Allocative Efficiency: Analytical Example

General equilibrium model with goods x and y , one consumer

◮ with utility U(xc ,yc)

and two firms producing with technologies

◮ xo = f (yi ), yo = g(xi ,xo) with g1 > 0, g2 < 0.

Note that to keep things “simple”

◮ x is an input for y , y is an input for x

Externality:

◮ the output yo depends on the output of other firm’s xo◮ e.g. firm 2 produces y downstream from the polluting firm 1

who creates pollution proportional to x output.

Initial stocks:

◮ x and y (owned by the consumer)

→ Constraints: xc + xi = xo + x and yc + yi = yo + y


Efficient allocation

Finding the socially efficient allocation: maximizeL= U(xc ,yc)+λ1[f (yi )− xo ]+λ2[g(xi ,xo)− yo ]+

+λ3(xc + xi − x0− x)+λ4(yc + yi − yo − y)

1. ∂L∂xc = U1+λ3 = 0

2. ∂L∂yc = U2+λ4 = 0

3. ∂L∂xi = λ2g1+λ3 = 0

4. ∂L∂yi = λ1fy +λ4 = 0

5. ∂L∂xo =−λ1+λ2g2−λ3 = 0

6. ∂L∂yo =−λ2−λ4 = 0

Where, g1 :=∂g(xi ;xo)

∂xi , g2 :=∂g(xi ;xo)

∂xo , and fy :=df (yi )dyi


Efficient allocation

Requirements on the marginal exchange between x and y :

◮ From (1) and (2): MRS ≡λ3λ4

= U1U2

◮ From (3) and (6): MRS = λ3λ4

= λ2g1

λ2= g1

(productivity of x in firm 2’s y production)

◮ From (4), (5) and (6):

MRS = λ3λ4

= −λ1+λ2g2

λ4= −λ1

λ4+ λ2g2

λ4= 1

fy−g2

◮1

fyexchange ratio of y and x in firm 1’s production

◮ −g2 externality of x production on firm 2’s y production

So in particular efficient production requires:

◮ fy =1

g1+g2where g2 < 0 increases the required productivity of

f to make up for the externality


Competitive Allocation

Competitive allocation:

◮ The 2 firms buy inputs from the consumer and sell output tothe consumer at equilibrium prices px and py

◮ Firms maximize profits (profits go to the owner=consumer)

◮ Consumer maximizes utility given the equilibrium prices

results in

◮ Consumer’s utility maximization: MRS = pxpy

◮ Profit max firm y w.r.t. to input x : px = pyg1

◮ Profit max firm x : py = px fy

→ In particularpypx

= fy =1g1

inefficient!


Solutions to the Externality Problem: Taxation

Tax firm 1’s output x :

◮ Firm will choose (px−t)fy = py ⇔1fy= px

py−

tpy

◮ Using that firm 2 still produces such that pxpy

= g1

◮ We find that t =−pyg2 (> 0 for neg. externality) delivers

MRS = pxpy

= g1 =1fy−g2 as required for the efficient solution

◮ the tax −pyg2 values the loss from externality x ’s effect on they production

Note:

◮ Size of externality can depend on level of output (see fig 17.1)

◮ The optimal tax is based on the value of the externality at theoptimal solution


Solutions to the Externality Problem: Pollution Rights

Alternative to taxation: a market for pollution rights

◮ Firm x has to buy pollution rights XT

◮ Firm y sells pollution rights XT

◮ endogenous price of pollution right: r

◮ Firm y sells rights as long as positive marginal profit of sale:Profits πy = pyg(xi ,x

T )+ rxT −pxxi

⇒∂πy

∂xT = pyg2+ r = 0 ⇒ r =−pyg2

◮ Firm x produces until the marginal profit is positive.Same condition as under the tax with t = r =−pyg2

→ efficient solution!

◮ Difference to tax:Here the price for the pollution rights forms endogenously!


Public goods

Definition 2

A good is non-rival if the consumption by one individual does notdiminish the consumption benefits for other individuals.

Definition 3

A good is exclusive if it is relatively easy to exclude individuals frombenefiting from the good once it is produced. A good isnon-exclusive if it is impossible (or costly) to exclude individualsfrom benefiting from the good.

Definition 4

A good that is non-rival and non-exclusive is called a public good.


Public Goods and Resource Allocations

◮ Examples of non-exclusive goods: national defense, mosquitocontrol, vaccination, a piece of public art

◮ Examples of non-rival goods: a road when traffic is low, TV orradio signal, a piece of public art, national defense, mosquitocontrol, vaccination.


Public Goods and Resource Allocations

Consider a simple general equilibrium model:

◮ Two individuals (A and B) and two goods (x and y)

◮ Good x is a public good.

◮ Initial allocation of Y : yA and yB

◮ Good y can be consumed or used for the production of good x:x = f (yAs + yBs )

◮ Utilities: UA(x ,yA− yAs ) and UB(x ,yB − yBs )

◮ Non-rivalry: the good x enters in two utility functions at thesame time.

◮ Non-exclusivity: each individual enjoys the entire amount ofthe public good regardless of her contribution to theproduction of x


Efficient Resource Allocation

Conditions for a Pareto efficient allocation

◮ Maximize utility of individual A,ensuring a given level of K utils to individual B:

L(x ,yAs ,yBs ,λ ) = UA(x , yA− yAs )+λ (UB(x , yB − yBs )−K )

◮ Inserting x = f (yAs + yBs ) we find FOC:∂L

∂yAs= UA

1 f′−UA

2 +λUB1 f ′ = 0

∂L∂yB

s= UA

1 f′−λUB

2 +λUB1 f ′ = 0

◮ Eliminating λ yields efficiency requirement:UA

1

UA2+

UB1

UB2= 1

f ′,

that is: MRSA+MRSB = 1f ′

(=MRT )or:Sum of Marg. Willingnesses to Pay = Marg. Production Cost

Samuelson Condition


Market Equilibrium

Market equilibrium

◮ Two utility-maximizing individuals choosing how much tocontribute to the production of the public good x

◮ We consider a game in which the two agents choosesimultaneously how much to contribute


Market Equilibrium

◮ Individual A chooses a contribution sA as to maximize utility:

maxsA

UA[f (sA+ seB), yA− sA]

◮ FOC:UA

1

UA2

=1

f ′so MRSA =

1

f ′

◮ seB is the belief of individual A about B’s contribution. Inequilibrium everyone holds correct beliefs, so seB = sB

◮ Similarly the FOC from individual B maximization gives:MRSB = 1

f ′

◮ So efficiency (Samuelson condition) is violated!


Voting and Resources Allocation

What now?

◮ One solution to the public good problem is for the state tocollect resources - say with taxes - and use those to providepublic goods

◮ Say individuals have different preferences over a public good:how to decide how much/ what kind of public good to provide?

◮ That’s a question dealt with in public choice theory

◮ One way: direct voting (clubs, cooperatives, referendums)

We end this part of the lecture with a brief teaser on public choicetheory


The Paradox of Voting

We now look at properties of majority voting

◮ The issue is not specific to voting on public goods

◮ Condorcet Paradox: the social preference among alternativesmight not be transitive even if every individual has transitivepreferences

◮ transitivity is considered a desirable rationality constraint onpreferences/choice

The issue is not specific to majority voting. In fact, majority votingis one of the “better” voting systems (can be formalized...).



We take 3 voters & 3 alternatives

◮ Anne, Bob and Carlo (A,B,C)

◮ Alternatives x ,y ,z

◮ Preferences: Anne: x ≻A y ≻A z ,Bob: z ≻B x ≻B y ,Carlo: y ≻C z ≻C x

◮ Voting between x and y : social preference x ≻ y

Anne and Bob vote x , whereas Carlos votes y

◮ Similarly: y ≻ z and z ≻ x

→ social preferences are not transitive

Note: Majority voting is always pairwise about just two candidates



Alternative: Plurality vote, vote on all candidates simultaneously.The one with the most votes wins.

◮ Particularly susceptible to spoilers:The availability of a third candidate taking votes from thesecond (but not the first) can lead a first candidate to victoryeven though most voters preferred the second.

◮ Likely example:Florida decided national US 2000 elections. Florida outcome:

◮ G.W. Bush 2,912,790 Republican◮ Al Gore 2,912,253 Democratic◮ R. Nader 97,488 Green

Assuming green voters would have otherwise voted Democraticrather than Republican:Nader spoiled the elections for Gore and the Democrats

Note: Most voting schemes susceptible to strategic voting, soresults not necessarily presenting true preferences.



A positive result: Black (1948) has shown

◮ if alternatives are unidimensional

◮ and every individual has single-peaked preferences, i.e., a bestlevel and above and below it’s getting continuously worseReasonable for vote on the level of a public good

→ then pairwise majority voting results in a transitive social

ordering/preference

◮ If we order voters by their peak preference, the alternative thatwins every pairwise vote is the alternative that is preferred bythe median voter (median voter theorem)


Summary

What have we learned?

◮ Market failures occur if there are◮ externalities not paid for on the market◮ public goods (non-rival and non-exclusive)

◮ Fixing market failures:◮ Coase (?)◮ Command & Control◮ Tax◮ Tradable Permits

◮ Samuelson condition for optimal provision of public good

◮ Some basics regarding voting on public good provisions


End of First Part of Lecture

What have we learned?

◮ Partial equilibrium (short-run, long-run)

elasticities & tax incidence/welfare

◮ Walrasian equilibrium (firms non-strategic)

Welfare properties (including two theorems)

◮ Inperfect competition (strategic firms, game theory)

Strategy space very important for the outcome:

Betrand vs Cournot competion (& extensions)

◮ Externalities & Public goods

Policy instruments to “fix” the market

Do not forget “term paper” (multiple choice midterm):out: Thursday March 15, in: Monday March 19

Documents

Externalities and Public Goods - Universitetet i oslo · Piacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring 2018. 1/26 Externalities and Public Goods ECON 4240