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