externalities - university of chicagohome.uchicago.edu/bdm/pepp/externalities.pdf · public goods:...
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Externalities
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Outline
Introduction
Public Goods: Positive Externalities
Policy ResponsesPersuasionPigovian Subsidies and Taxes
The Second Best
Take Aways
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Key Ideas
What is an externality?
Externalities create opportunities for Pareto improvingpolicy
Externalities require “active” and ongoing policyinterventions
The optimal (second best) policy intervention involvestrade-offs
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Externalities
Person A’s action imposes an externality on person B if A’saction affects B’s payoffs
Suppose Person A can take action a or not take action a
Person A’s action a imposes a negative externality onPerson B if B’s payoff is always lower when A does a thanwhen A does not do a
I Positive externalities are defined analogously
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The General Principle
People do not internalize their externalities
I Externalities do not influence decision making
Relative to the utilitarian optimum
I Too little of actions with positive externalities
I Too much of actions with negative externalities
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The Policy Challenge
Get people to internalize their externalities
I Persuasion
I Subsidies/taxes
I Regulation
I Decentralized mechanisms
Weight the benefits of internalizing externalities against thecosts of the policy intervention
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Outline
Introduction
Public Goods: Positive Externalities
Policy ResponsesPersuasionPigovian Subsidies and Taxes
The Second Best
Take Aways
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What is a Public Good?
Non-excludable: If I have access to the good, so do you
Non-rival: My use of the good does not diminish youraccess to it
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Examples
Clean air
National defense
Protection from asteroids
Knowledge
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The Model
N > 1 people each decide how much effort to contribute
Player i’s contribution is ei
Public good production technology is additive
G = e1 + e2 + . . .+ eN
i’s cost of effort is e2i
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Best ResponsesPlayer i′s best response to e−i is found by solving:
maxei
Public Goods︷ ︸︸ ︷e1 + e2 + . . .+ ei + . . .+ eN −
Costs︷︸︸︷e2i
First-order condition
1 − 2e∗i = 0
BRi(e−i) =1
2
Because of the additive technology, player i’s best responseactually doesn’t depend on what anyone else is doing
I This wouldn’t be true if efforts were complements11 / 40
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Nash Equilibrium
The unique equilibrium is all players choose effort 12
Player i’s equilibrium payoff is:
N × 1
2−(
1
2
)2
=2N − 1
4
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The First Best
The profile of efforts that maximize the utilitarian welfareis called the first best
The utilitarian welfare is
N ×N∑i=1
ei −N∑i=1
e2i
The equilibrium efforts are not the first best because peopledo not internalize their positive externalities
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Finding the First BestIf each player chooses the same effort (e), each individual’spayoff is:
Public Goods︷ ︸︸ ︷N × e −
Costs︷︸︸︷e2
The first best solves:
maxeN(N × e− e2
)
N(N − 2eFB
)= 0
eFB =N
2
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Finding the First BestIf each player chooses the same effort (e), each individual’spayoff is:
Public Goods︷ ︸︸ ︷N × e −
Costs︷︸︸︷e2
The first best solves:
maxeN(N × e− e2
)
N(N − 2eFB
)= 0
eFB =N
2
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Finding the First BestIf each player chooses the same effort (e), each individual’spayoff is:
Public Goods︷ ︸︸ ︷N × e −
Costs︷︸︸︷e2
The first best solves:
maxeN(N × e− e2
)
N(N − 2eFB
)= 0
eFB =N
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Social Dilemma
Equilibrium effort by each individual is 12
I Individual payoff is 2N−14
The first best is for each individual to exert effort N2
I Payoff would be N × N2−(N2
)2= N2
4
Each individual is better off if all individuals exert moreeffort. This is especially true as society gets big.
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20 40 60 80 100
500
1000
2000
2500
N
Utility
1500
Individual payoff if everyone does equilibrium effort
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Outline
Introduction
Public Goods: Positive Externalities
Policy ResponsesPersuasionPigovian Subsidies and Taxes
The Second Best
Take Aways
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Policy Responses
The fundamental problem is the failure to internalizeexternalities
Policy makers must look for instruments to get people tointernalize externalities
They must also pay attention to any unintendedconsequences of using those instruments
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Outline
Introduction
Public Goods: Positive Externalities
Policy ResponsesPersuasionPigovian Subsidies and Taxes
The Second Best
Take Aways
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Free Riding
Even though payoffs are higher if everyone does first-besteffort, no individual has an incentive to behave in thissocially optimal way
Each individual has an incentive to free ride, even ifeveryone else does social optimum
(N − 1) × N
2+
1
2−(
1
2
)2
=2N2 − 2N + 1
4>N2
4
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20 40 60 80 100
1000
2000
4000
5000
N
Utility
3000
Individual payoff if everyone does equilibrium effort
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Outline
Introduction
Public Goods: Positive Externalities
Policy ResponsesPersuasionPigovian Subsidies and Taxes
The Second Best
Take Aways
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Pigovian Subsidies in Public
Goods Game
Suppose provide a subsidy, σ, for each unit of effort
Subsidy must be funded with inefficient taxation
It costs tax payers τ > 1 dollars for each dollar of revenuethe government collects
Each member of society pays an equal share of the taxburden to fund subsidies for everyone other than herself
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Best Response to Subsidy
maxei
e1 + e2 + . . .+ ei(1 + σ) + . . .+ eN − e2i −τσ∑
j 6=i ej
N − 1
First-order condition
1 + σ − 2e∗i = 0
BRi(e−i, σ) =1 + σ
2
To achieve the first-best level of effort choose σ = N − 1
I People fully internalize externalities
I This is not the socially optimal subsidy24 / 40
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Direct Regulation vs. Pigovian
Intervention
Informational requirements
Monitoring
Flexibility
In either case, requires ongoing intervention
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Outline
Introduction
Public Goods: Positive Externalities
Policy ResponsesPersuasionPigovian Subsidies and Taxes
The Second Best
Take Aways
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The Idea
Any policy lever government pulls to address one issue hasspillover effects in another domain
Good policy balances these benefits and costs
The second best policy is the socially optimal intervention,given all the constraints and effects
When policy effects other domains, it is rarely sociallyoptimal to achieve the first best in the given domain
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An Example
Monopolist in a polluting industry
Break monopoly to improve consumer surplus
Also increases production, which imposes externalities
Optimal policy with respect to market power may not beperfect competition because of constraint from thepollution domain
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Second-Best Defined
The second-best policy is the policy that maximizes theutilitarian social welfare, taking into consideration all thevarious effects of the policy.
The second-best action is the action players are induced totake when the second-best policy is implemented.
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Finding the Second-Best Subsidy
Take the equilibrium effort as a function of the subsidy, σ,as given
Given the equilibrium effort induced by a given subsidy, σ,calculate the utilitarian social welfare as a function of σ
Find the σ that maximizes that social welfare
I The σ that does so is the second-best subsidy and theeffort it induces is the second-best effort.
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Individual Welfare under Subsidy
Recall, given a subsidy σ, each individual will choose effort
BRi(e−i, σ) =1 + σ
2≡ e∗,σ
To fund the subsidy, each individual must pay a taxes of
(N − 1) × e∗,σ × σ × τ
N − 1= e∗,σ × σ × τ
ui(σ) =
Public Goods︷ ︸︸ ︷N × e∗,σ +
i’s subsidy︷ ︸︸ ︷e∗,σ × σ −
i’s cost︷ ︸︸ ︷(e∗,σ)2 −
i’s tax burden︷ ︸︸ ︷e∗,σ × σ × τ
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Utilitarian Welfare
Utilitarian welfare from subsidy σ is
U(σ) = N × ui(σ)
Substituting for e∗,σ = 1+σ2
U(σ) = N
[Public Goods︷ ︸︸ ︷N
(1 + σ
2
)+
i’s subsidy︷ ︸︸ ︷(1 + σ
2
)σ−
i’s cost︷ ︸︸ ︷(1 + σ
2
)2
−
i’s tax burden︷ ︸︸ ︷(1 + σ
2
)στ
]
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The Second Best
The second-best subsidy solves
maxσ
U(σ)
σSB =N − τ
2τ − 1
e∗,σSB
=1 + N−τ
2τ−1
2=N − 1 + τ
2(2τ − 1)
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Four Points
The more people, the larger is the second-best subsidybecause externalities are more important
The larger is τ , the smaller is the second-best subsidybecause the inefficiencies of funding it loom large
Second-best subsidy is less than the subsidy that inducesthe first-best effort
N − τ
2τ − 1< N − 1
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Four Points
The more people, the larger is the second-best subsidybecause externalities are more important
The larger is τ , the smaller is the second-best subsidybecause the inefficiencies of funding it loom large
Second-best subsidy is less than the subsidy that inducesthe first-best effort
N − τ
2τ − 1< N − 1
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Four Points
The more people, the larger is the second-best subsidybecause externalities are more important
The larger is τ , the smaller is the second-best subsidybecause the inefficiencies of funding it loom large
Second-best subsidy is less than the subsidy that inducesthe first-best effort
N − τ
2τ − 1< N − 1
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1 ! 4!
200 !
400 !
600 !
800 !
τ!
Subsidy that implements first best effort σ = N − 1
Second best subsidy as a function of τ
σSB =N − τ2τ − 1
1000 !
0!2! 3! 5! 8!6! 7!
1 ! 4!
100 !
200 !
300 !
400 !
τ!
First best effort: eFB =N2
Second best effort as a function of τ :
e*,SB =N − 1+ τ
2τ
500 !
0!2! 3! 5! 8!6! 7!
Effort with no subsidy: e* =12
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Four Points
The more people, the larger is the second-best subsidybecause externalities are more important
The larger is τ , the smaller is the second-best subsidybecause the inefficiencies of funding it loom large
Second-best subsidy is less than the subsidy that inducesthe first-best effort
N − τ
2τ − 1< N − 1
For any τ > 1, welfare is higher under second-best subsidythan under either no subsidy or subsidy that inducesfirst-best effort
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10 20 30 40 50
100
200
300
400
N
Utility
τ = 1.25
Welfare under σ
Welfare under σSB
Welfare with no subsidy
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10 20 30 40 50
50
100
150
N
Utility
τ = 1.5
Welfare under σ
Welfare under σSB
Welfare with no subsidy
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Outline
Introduction
Public Goods: Positive Externalities
Policy ResponsesPersuasionPigovian Subsidies and Taxes
The Second Best
Take Aways
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Take Aways
Externalities create Pareto inefficiency
Policy can improve situation by incentivizing people to(partially) internalize their externalities
Policy intervention must be active and ongoing
If policy has other costs, the socially optimal policy (secondbest) does not achieve complete internalization ofexternalities
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