Design: Public Goodsdarp.lse.ac.uk/pdf/EC426/EC426_16_04_H.pdf · Design Public Goods PG...

Design Public Goods PG Mechanisms Conclusions References

Design: Public Goods

Frank Cowell

EC426http://darp.lse.ac.uk/ec426

17 October 2016

http://darp.lse.ac.uk/ec426


Outline

DesignFundamentalsResult

Public GoodsCharacterisationPublic Goods: voluntarism

PG MechanismsPG: Restricted problemSecond-best schemes

Conclusions


An approach to design

• Start from same point as in Lecture 1• Arrow (1951) insight is fundamental to Public Economics

• helps understand concepts of social welfare (Lecture 1)• but also connects to an “information+incentives” problem

• Why can’t the government just do what it likes?• maybe exogenous constraints• more basic: the problem of “manipulability”

• Begin by making this idea more rigorous• connect back to the SWF approach

• Then formalise this in a typical PE application• game theoretic approach• incomplete information• hidden characteristics


Social welfare and individual values

• A social state: θ ∈Θ

• Individual i’s evaluation of the state vi (θ) , i = 1, ...,n• vi: member of some general class U• U: evaluation or utility functions Θ→ R

• A profile: [v1, . . . ,vi, . . .vn]

• ordered list of functions vi• set of all profiles: V

• SWF problem: find a constitution Σ : V→ U satisfying• Unrestricted domain• Pareto unanimity• Independence of Irrelevant Alternatives

• IF U,P, I hold then Σ must be dictatorial (Arrow 1951)

• except where there are fewer than three social states• “dictator” i∗: if vi∗ (θ)> vi∗ (θ

′) then society prefers θ to θ ′


Social choice and manipulation

• Use the same framework as previous slide• social state: θ ∈Θ

• individual preferences vi (·) ∈ U, i = 1, ...,n• profile: [v1, . . . ,vi, . . .vn] ∈ V

• A social choice function Γ : V→Θ

• compare this with the constitution Σ

• same domain, but different kind of “output”

• Does an individual i have power in the SCF?• if all tell the truth about preferences: θ = Γ(v1, . . . ,vi, . . .vn)• if i misrepresents preferences: θ = Γ(v1, . . . , vi, . . .vn)

• This reveals a fundamental problem• if vi

(θ)> vi (θ) then there is an incentive to misrepresent

• the social-choice function Γ is manipulable


Implementation

• Is the SCF Γ consistent with private economic behaviour?• yes if the θ picked out by Γ is also the equilibrium of an

appropriate economic game

• A mechanism is a partially specified game:• rules of game are fixed• strategy sets are specified• preferences not yet specified

• Plug preferences into the mechanism:• does the mechanism have an equilibrium?• does the equilibrium correspond to the desired θ?• if so, θ is implementable

• Wide range of possible and interesting mechanisms• Example: the market as a mechanism• Implementation problem: find/design an appropriate mechanism


Design result

• Result on the SCF, Γ (Gibbard 1973, Satterthwaite 1975, Ninjbat 2012)

If the set of social states Θ contains at least threeelements; and Γ is defined for all logically possiblepreference profiles and Γ is truthfully implementable indominant strategies, then Γ must be dictatorial

• Closely related to the Arrow theorem

• Has profound implications for public economics• misinformation may be endemic to the design problem• may only get truth-telling mechanisms in special cases

• Interested in two types of solution:1. “Full information” (“first best”) solutions

• needs an information-revealing mechanism

2. Second-best solutions• built-in constraints to prevent misrepresentation


Typology of goods

Rival Non-rivalExcludable pure private [?]

Non-excludable [?] pure public

• Excludable: is there a way of making people pay for the good?

• Rival / nonrival: need extra resources to supply an extra person?

• For a private good, aggregate consumption is found by summingthe consumption of n individuals

• For a public good, aggregate consumption equals theconsumption of each of the n individuals


Efficiency: model

• Let xi be vector of goods consumed by person i, x is aggregatevector of goods

• Consumer i has utility function ui• To find an efficient allocation:

• max utility of any one person i• keeping the n−1 others on a fixed utility level u` (x`) = υ`

• satisfying production constraint Φ(x) = 0

• Lagrangean is ui (xi)+∑ 6=i λ` [u` (x`)−υ`]−µΦ(x)

• For a private good j consumed by person i we have the FOC

λi∂ui (xi)

∂xij= µ

∂Φ(x)∂xj

• Public good j = 1 consumed by everyone equally. The FOC:n

∑`=1

λ`∂u` (x`)

∂x`1= µ

∂Φ(x)∂x1


Efficiency: result

• Use FOC for a max to characterise efficiency conditions :

• If goods j and k are both private

∂ui (xi)

∂xik

/∂ui (xi)

∂xij=

∂Φ(x)∂xk

/∂Φ(x)

∂xj

for every agent i : MRSi = MRT

• If good j is private and good 1 is public

n

∑i=1

∂ui (xi)

∂xi1

/∂ui (xi)

∂xij=

∂Φ(x)∂x1

/∂Φ(x)

∂xj

∑ni=1MRSi = MRT


Efficiency conditions

• Derived from the FOCs• If “wrong condition” applied to PGs – get under-provision


Strategic view

• Consider two types of Public-good game. In each case:

• players (Greek, Roman)• actions ([+], [−]): (contribute, not-contribute) to public good• payoffs denoted by letters for each player

Game 1Roman

[+] [−]

Gre

ek [+] β ,B δ ,A

[−] α,D γ,C

• Three efficient outcomes• But none of these is a NE

Game 2Roman

[+] [−]

Gre

ek [+] β ,B γ,A

[−] α,C δ ,D

• Three efficient outcomes• Maybe implementable?


Voluntary provision

• Two-good model: i’s utility is ψ(g)+ xi

• i is endowed with income yi, contributes an amount zi

• so private consumption is xi = yi− zi

• g produced from total contributions g = φ (z), z := ∑n`=1 z`

• Suppose every agent i makes a Cournot assumption:

• assumes that ∑n`=1 z`− zi is a constant, z−i

• perceives problem as “choose zi to max ψ (φ (z−i + zi))+ yi− zi”

• FOC for perceived problem is ψ ′ (g)φ ′ (z)−1 = 0• So we get ψ ′ (g) = 1

φ ′(z) ; in other words MRSi = MRT

• But for efficiency we need ∑ni=1 MRSi = MRT


Outcomes of contribution game

Each party makes Cournot assumption

• Nash equilibrium is at intersection of the reaction functions• Efficient outcomes given by locus of common tangencies


Willingness To Pay and the Public Good

• pi reflects person i’s Willingness To Pay: MRSi = WTPi

• Sum of WTP equal MC of producing public good g∗


Lindahl

• Introduce concept of “tax price” for funding public goods• pi: “tax price” for i of PG, set by government (Lindahl 1919)

• These “prices” must satisfy pi = WTPi and ∑ni=1 pi = MRT

• What if i realises that “price” depends on announced WTP?• WTP announced strategically: i announces WTPi knowing that

• pi = WTPi• amount of public good is g = φ (const+pig)• private consumption is xi = yi−pig

• Suppose i announces WTPi to maximise utility ψ(g)+ xi

• then this becomes exactly the problem of voluntary contribution!


Way forward

• Lindahl results in the same suboptimal outcome as voluntarism

• What can be done?• public provision through regular taxation• change the problem• change perception of the problem

• Alternatives to elementary model of individual rationality:• truthful revelation as a social norm (Johansen 1977)• reciprocity motives in the utility function (Guttman 1987)• co-operative outcome in a repeated game (Pecorino 1999)

• More promising: alternative institutional mechanisms• Pivot mechanisms in a restricted choice problem• “Forced” reciprocity• Provision-point mechanisms• Lotteries


A binary project

• An all-or-nothing choice (Clarke 1971, Groves and Ledyard 1977)

• Θ = {0,1}. A project in Θ completely characterised by

• each person i’s endowment yi of private good• payment zi by i if project goes ahead (∑i zi = z, φ(z) = 1)• a system of penalties

• All have zero income effect (ziff) utility: τiψ(g)+ xi

• where ψ is increasing, concave, τi is a taste parameter• τi reflects Willingness To Pay


Preferences for a binary project

v1 (θ◦)> v1 (θ

′) ; v2 (θ◦)< v2 (θ

′)


A criterion for the project

• Let CVi be compensating variation for i if project goes ahead• If S := ∑

n`=1 CV` then appropriate criterion seems to be S > 0

• gainers could compensate losers• but S is unobservable (information on preferences is private)

• So, use instead the announced CV, CVi, and define

S :=n

∑`=1

CV`, S−i := S− CVi, i = 1, ...,n

• If S and S−i have opposite signs then person i is pivotal

• Now consider the following criterion1. Approve (reject) the project if S≥ 0 (S < 0 )2. If i is pivotal, then impose a penalty of S−i on person i

• Mechanism guarantees that truth-telling is a dominant strategy


Payoff to i under the mechanism

S < 0 S≥ 0(θ ◦ chosen) (θ ′ chosen)

S−i < 0 vi (θ◦) vi (θ

′)− S−i

S−i ≥ 0 vi (θ◦)− S−i vi (θ

′)

• Person i ’s true valuations are vi (θ◦) ,vi (θ

′)

• Person i announces vi (θ◦) , vi (θ

′)

• Limitations:

• the amounts S−i have to be computed for all n persons• mechanism applies only to binary projects


A binary choice

• Tax/subsidy to persuade people to reciprocate? (Gradstein 1998)

• All have same ziff utility, endowed with 1 unit of private good• i’s contribution choice is represented as zi ∈ {0,1}• i has unobservable cost of contribution ci• utiity: ψ(g)+1− zici

• Public good production : g = φ(q)• where q is proportion of contributors

• For an efficient outcome want low-cost agents to contribute• for some cut-off value c,

zi =

{1 if ci ≤ c,0 otherwise

• If provision is left to private action there will be underprovision


Binary choice: tax/subsidy

• Government knows the distribution F(·) of contribution costs

• can condition on the threshold value c

• Design tax/subsidy scheme based on observables:

• subsidy s > 0 if you contribute• tax t > 0 if you don’t contribute

• person with critical cost c gets utility:

• ψ(g)+1− c+ s = ψ(g)+1− t• implies s+ t = c

• Those with costs ci < c will contribute; those with ci > c will not• Breakeven achieved if sF(c) = t[1−F(c)]

• the necessary tax to achieve this is t = cF(c)


Provision-point mechanism

• Voluntary contribution plus target value z∗ plus refund scheme

• target z∗ exceeded: a rebate in proportion to your contribution• if z∗ is not reached, all contributions refunded

• If total contributions are z and agent i’s share is πi, utility is

ψ(φ(z∗))+πi[z− z∗]+ yi− zi, if z≥ z∗

yi otherwise

• Each agent appears to have an incentive to report truthfully

• Issues arising:

• z∗ must be exogenous• but how is z∗ determined?• better than voluntarism in practice? (Rondeau et al. 2005)


Lottery mechanism

• If it is a fair lottery with fixed prize P then amount of publicgood is g = φ(z−P)

• probability of winning is πi = zi/z• expected utility is ψi(g)+πiP+ yi− zi

• Again i makes the Cournot assumption when maximising

• FOC gives ψ ′i (g) = β (P)/(zi−P) where β (P) := 1− zP/z2i < 1

• A higher P results in more public good being provided

• Fixed-prize lottery introduces an offsetting externality

• each time you buy a lottery ticket you affect others’ chances ofwinning (Morgan 2000, Morgan and Sefton 2000)


Summary

• Design principles associated with social-choice problem

• Arrow and Gibbard-Satterthwaite theorems connected• associated with an imperfect-information problem

• Public goods combine special properties

• more than one “cause for market failure”• easy to solve the characterisation problem• implementation problems are much harder

• Mechanism design depends on:

• the type of public good• the economic environment (Morgan 2000, Rondeau et al. 2005)


Bibliography I

Arrow, K. J. (1951). Social Choice and Individual Values. New York: John Wiley.

Clarke, E. H. (1971). Multi-part pricing of public goods. Public Choice 11, 17–33.

Gibbard, A. (1973). Manipulation of voting schemes: a general result. Econometrica 41, 587–601.

Gradstein, M. (1998). Provision of public goods in a large economy. Economics Letters 61(2), 229–234.

Groves, T. and J. Ledyard (1977). Optimal allocation of public goods: A solution to the free rider problem.Econometrica 45, 783–809.

Guttman, J. M. (1987). A non-Cournot model of voluntary collective action. Economica 54, 1–20.

Johansen, L. (1977). The theory of public goods: misplaced emphasis? Journal of Public Economics 7, 147–152.

Lindahl, E. (1919). Positive Lösung, die Gerechtigkeit der Besteuerung, reprinted as "just taxation - a positive solution". InR. A. Musgrave and A. T. Peacock (Eds.), Classics in the Theory of Public Finance. Macmillan, London.

Morgan, J. (2000). Public goods and lotteries. Review of Economic Studies 67, 761–784.

Morgan, J. and M. Sefton (2000). Funding public goods with lotteries: Experimental evidence. Review of EconomicStudies 67, 785–810.

Ninjbat, U. (2012). Another direct proof for the Gibbard-Satterthwaite theorem. Economic Letters 116, 418–421.

Pecorino, P. (1999). The effect of group size on public good provision in a repeated game setting. Journal of PublicEconomics 72(1), 121–134.

Rondeau, D., G. Poe, and W. Schulze (2005). VCM or PPM? a comparison of the performance of two voluntary publicgoods mechanisms. Journal of Public Economics 89, 1581–1592.

Satterthwaite, M. A. (1975). Strategy-proofness and Arrow’s conditions. Journal of Economic Theory 10, 187–217.

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