pareto-efficient solutions for shared production of a public good work in progress
DESCRIPTION
Pareto-efficient solutions for shared production of a public good work in progress. Andries Nentjes , U of Groningen Bouwe Dijkstra , U of Nottingham Jan- Tjeerd Boom, Danish EPA Frans de Vries , U of Stirling. 1. Introduction. Private provision of a public good International examples: - PowerPoint PPT PresentationTRANSCRIPT
Pareto-efficient solutions for shared production of a public good
work in progress
Andries Nentjes, U of GroningenBouwe Dijkstra, U of Nottingham
Jan-Tjeerd Boom, Danish EPAFrans de Vries, U of Stirling
1. Introduction
• Private provision of a public good
• International examples:
– Greenhouse gas emission reduction
– Military alliances
• Nash equilibrium: Underprovision
2
A “new” solution: Market Exchange
• Nentjes (1990)
• How much yi of the public good would you be
willing to supply if you would get Yi = piyi from the
group in return?
• Equilibrium prices where all Yi = Σyj
– Unique stable equilibrium3
Comparison
• This paper: Nash bargaining
• Nentjes, Rübbelke, Dijkstra, De Vries:– Kaneko ratio equilibrium
– Guttman matching scheme
– Andreoni-Bergman tax-subsidy scheme
– Falkinger tax-subsidy scheme
– Roemer’s Kantian equilibrium
4
Nash bargaining
• Constructed to have desirable outcomes
• Bargaining process itself is black box
• Noncooperative implementation
– Binmore et al. ’86: 2 players, alternate offers
– Chae&Yang ’94, Krishna&Serrano ’96, Hart&Mas-Colell ’96:
n players, specific bargaining procedure, equilibrium concept
– Requires full information
5
Outsourcing
• E.g. emission trading• Each agent commits to a certain public good
contribution• Agent i who produces more than her
contribution earns certificates which she can sell to another agent j– Agent j can produce below contribution
6
Literature: International environmental policy
• Hoel (1991): Nash bargaining without emission trading
• Helm (2003): Noncooperative emission reduction with and without emission trading
• Boom (2006 thesis): Nash bargaining with and without emission trading
7
Outline
2. The model
3. Nash bargaining without outsourcing
4. Market exchange without outsourcing
5. Outsourcing
6. Conclusion
8
2. The model• n agents (i = 1,...,n) producing and consuming
a public good Q = Σqi
• Cost function Ci(qi) with Ci’, Ci’’ ≥ 0
• Benefit function Bi(Q) with Bi’ ≥ 0, Bi’’ ≤ 0
• Specific case: two agents, quadratic functions
9
iiiiiiii qcqCqcqC )('2
1)( 2
)1()('2
1)( 2 QbQBQbQbQB iiiii
Constrained Pareto efficiency
• Without side payments
• FOCsor
• Welfare weights λ1 = 1 and
• λk and qi not determined10
n
kkkkkk
qqCQBWqCQB
i 2111 )()()()(max
0)(')('1
iii
n
jjj qCQB 1
)('
)('
1
n
j jj
j
qC
QB
)('
)(' 11
kkk qC
qC
Unconstrained Pareto efficiency
• With side payments, agent i receives xi
• FOC for xi: λj = μ = 1
• FOC for qi:
• All λj and qi determined, but xi not determined11
n
j
n
jjjjjjjj
xqxxxqCQBWxqCQB
ii 2 211111
,)()()()(max
0)(')('1
ii
n
jj qCQB
Noncooperative Nash (NCN)
• FOCs
• Not Pareto-efficient (underprovision)
12
)()(max iiiq
qCQBi
0)(')(' iii qCQB
3. Nash bargaining
• With equal bargaining weights (Aj NCN payoff)
• FOCs
• Constrained Pareto optimal, generally unequal welfare weights
• Higher gain: Lower welfare weight, higher Ci’13
n
jjjjj
qAqCQB
i 1
)()(logmax
iiii
iin
j jjjj
j
AqCQB
qC
AqCQB
QB
)()(
)('
)()(
)('
1
iiiii AqCQB
)()(
1
4. Market Exchange Solution
• How much yi of the public good would you be willing to supply if you would get Yi = piyi from the group in return?– On top of the NCN amounts qin, Qn
• FOCs
• Agent i supplies yi, demands Yi
14
0..)()(max iiiiiniiniiy
ypYtsyqCYQBWi
)(')(' iiiii qCQBp
Equilibrium
• All agents demand the same amount, which is
the sum of all their supplies:
• Equilibrium prices
• Agent i’s supply share
• Constrained Pareto optimal:
15
n
jji yYY
1
ii
ii y
Y
y
Yp
i
i
pY
y 1
11
)('
)('
111
n
i
in
i i
n
i ii
i
Y
y
pqC
QB
Two agents, quadratic benefits and costs
• MES and NBS coincide– Probably not a general result
• Agent with highest gi has highest qi
• c1 = c2: High-benefit agent has highest Ci’
• b1 = b2: High-cost agent has highest Ci’
16
i
iiiiiiii c
bgQbQBqcqC )1()(')(
5. Outsourcing
• Stage 1: Each agent commits to a certain
public good contribution
• Stage 2: Agent i who produces more than her
contribution earns certificates which she can
sell to another agent j
– Agent j can produce below contribution
17
Stage two
• qsi = production, qi contribution
• P(Q) certificate price (perfect competition)
• FOC
18
)()()(max isisiiiiq
qqPqCQBWsi
0)(' sii qCP
Nash bargaining
• FOC
• All Wi – Ai must be the same
19
n
iiisisiii
qAqqQPqCQBJ
i 1
))(()()(logmax
0)())((')('
1
ii
n
j jj
jsjj
AW
QP
AW
qqQPQB
Unconstrained Pareto optimum
• Market clearing and perfect competition on
certificate market:
• Outsourcing as a vehicle for side payments20
)()()(')('11
QPqqQPQBn
jjsj
n
jj
)(')()('1
sii
n
jj qCQPQB
Market exchange solution
• FOC
• In equilibrium:• Sum over i:
• Unconstrained Pareto optimum21
0))((')(' PyyQPpQBp isiiii
Y
yPyyQPQB i
isii ))((')('
n
ijji qCPQB
1
)(')('
0..
)()()(max
iii
isisiiniiniiy
ypYts
yyPyqCYQBWi
Contributions• Substituting back into
yields
• Every agent contributes in proportion to her marginal benefits, adjusted by price manipulation motive
• Remember with NBS: Every agent has the same gain
22
n
ii QBP
1
)('
Y
yPyyQPQB i
isii ))((')('
)('
))((')('
QB
yyQPQB
Y
y
j
isiii
Lindahl pricing?
• Ask every public good consumer i how much he would demand at price pi
• Public good is supplied efficiently– Only with outsourcing
• MES contributions with outsourcing:
– Lindahl– Producer’s price manipulation motive 23
)('
))((')('
QB
yyQPQB
Y
y
j
isiii
Two agents, quadratic benefits and costs
• Comparing MES and NBS• Identical benefit functions:– High-cost agent pays low-cost agent
• Identical cost functions:– High-benefit agent pays low-benefit agent
• Payments lower in MES than in NBS– Attempts to manipulate the permit price
24
)1()(')( QbQBqcqC iiiiii
6. Conclusion
• Comparison of Nash bargaining and market exchange solutions for public good provision– Example: Two agents, quadratic benefits and costs
• Without outsourcing: both are constrained Pareto-optimal– MES and NBS coincide
• With outsourcing: both are unconstrained Pareto-optimal– Smaller transfers in MES
25
Extensions
• Other functional forms
• Asymmetric information
• Coalition formation
• Climate change policy simulations
• Experiments
26