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:
– 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 equilibrium
3
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
1 2
Ci (qi )  ci qi
2
1
2
Bi (Q)  bi Q  bi Q
2
Ci ' (qi )  ci qi
Bi ' (Q)  bi (1  Q)
9
Constrained Pareto efficiency
• Without side payments
max B1 (Q)  C1 (q1 )   k Wk  Bk (Q)  Ck (qk )
n
qi
k 2
• nFOCs
  j B j ' (Q)  i Ci ' (qi )  0 or
j 1
n
B j ' (Q)
 C ' (q )  1
j 1
j
j
C1 ' (q1 )
• Welfare weights λ1 = 1 and k 
Ck ' ( qk )
• λk and qi not determined
10
Unconstrained Pareto efficiency
• With side payments, agent i receives xi
n


max B1 (Q)  C1 (q1 )  x1    j W j  B j (Q)  C j (q j )  x j    x1   x j 
qi , xi
j 2
j 2



n

• FOC for xi: λj = μ = 1
• FOC for qi:
n
 B ' (Q)  C ' (q )  0
j 1
j
i
i
• All λj and qi determined, but xi not determined
11
Noncooperative Nash (NCN)
max Bi (Q)  Ci (qi )
qi
• FOCs
Bi ' (Q)  Ci ' (qi )  0
• Not Pareto-efficient (underprovision)
12
3. Nash bargaining
• With equal
bargaining
weights
(A
NCN
payoff)
j
n

max  log B j (Q)  C j (q j )  A j
qi
• FOCs
n

j 1
B j ' (Q)
Ci ' (qi )


Bi (Q)  Ci (qi )  Ai
j 1 B j (Q)  C j (q j )  A j
• Constrained Pareto optimal, generally unequal
welfare weights
1
i 
Bi (Q)  Ci (qi )  Ai
• Higher gain: Lower welfare weight, higher Ci’
13
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
max Wi  Bi (Qn  Yi )  Ci (qin  yi ) s.t. Yi  pi yi  0
yi
• FOCs
pi Bi ' (Qi )  Ci ' (qi )
• Agent i supplies yi, demands Yi
14
Equilibrium
• All agents demand the same amount, which is
the sum of all their supplies:
n
Yi  Y   y j
• Equilibrium prices Yi
Y
pi  
yi yi
yi 1

• Agent i’s supply share
Y
pi
j 1
• Constrained Pareto optimal:
n
Bi ' (Q) n 1 n yi
    1

i 1 Ci ' (qi )
i 1 pi
i 1 Y
15
Two agents, quadratic benefits and costs
Ci (qi )  ci qi
Bi ' (Q)  bi (1  Q)
bi
gi 
ci
• 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
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
max Wi  Bi (Q)  Ci (qsi )  P(qsi  qi )
qsi
• qsi = production, qi contribution
• P(Q) certificate price (perfect competition)
• FOC
P  Ci ' (qsi )  0
18
Nash bargaining
n
max J   logBi (Q)  Ci (qsi )  P(Q)(qsi  qi )  Ai 
qi
i 1
• FOC
n
B j ' (Q)  P' (Q)(qsj  q j )
j 1
W j  Aj

P(Q)

0
Wi  Ai
• All Wi – Ai must be the same
19
Unconstrained Pareto optimum
n
n
 B ' (Q)  P' (Q) (q
j 1
j
j 1
sj
 q j )  P(Q)
• Market clearing and perfect competition on
certificate market:
n
 B ' (Q)  P(Q)  C ' (q
j 1
j
i
si
)
• Outsourcing as a vehicle for side payments
20
Market exchange solution
maxWi  Bi (Qn  Yi )  Ci (qin  ysi )  P( ysi  yi )
yi
s.t. Yi  pi yi  0
• FOC
pi Bi ' (Q)  pi P' (Q)( ysi  yi )  P  0
yi
• In equilibrium: Bi ' (Q)  P' (Q)( ysi  yi )  P
Y
• Sum over i:
n
 B ' (Q)  P  C ' (q )
i 1
i
j
• Unconstrained Pareto optimum
j
21
Contributions
n
• Substituting P   Bi ' (Q) back into
i 1
yi
Bi ' (Q)  P' (Q)( ysi  yi )  P
Y
yields yi  Bi ' (Q)  P' (Q)( ysi  yi )
Y
 B j ' (Q)
• Every agent contributes in proportion to her
marginal benefits, adjusted by price
manipulation motive
• Remember with NBS: Every agent has the
same gain
22
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:
yi Bi ' (Q)  P' (Q)( ysi  yi )

Y
 B j ' (Q)
– Lindahl
– Producer’s price manipulation motive
23
Two agents, quadratic benefits and costs
Ci (qi )  ci qi
Bi ' (Q)  bi (1  Q)
• 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
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