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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
Piacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring 2018. 2/26
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.
Piacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring 2018. 3/26
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
Piacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring 2018. 4/26
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.
Piacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring 2018. 5/26
Resolving Externalities
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...
Piacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring 2018. 6/26
Resolving Externalities
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
Piacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring 2018. 7/26
Resolving Externalities
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)
Piacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring 2018. 8/26
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
Piacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring 2018. 9/26
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
Piacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring 2018. 10/26
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
Piacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring 2018. 11/26
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!
Piacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring 2018. 12/26
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
Piacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring 2018. 13/26
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!
Piacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring 2018. 14/26
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.
Piacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring 2018. 15/26
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.
Piacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring 2018. 16/26
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
Piacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring 2018. 17/26
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
Piacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring 2018. 18/26
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
Piacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring 2018. 19/26
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!
Piacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring 2018. 20/26
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
Piacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring 2018. 21/26
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...).
Piacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring 2018. 22/26
The Paradox of Voting
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
Piacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring 2018. 23/26
The Paradox of Voting
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.
Piacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring 2018. 24/26
The Paradox of Voting
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)
Piacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring 2018. 25/26
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
Piacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring 2018. 26/26
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