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## Public Goods

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**Prerequisites**Almost essential Welfare and Efficiency Public Goods MICROECONOMICS Principles and Analysis Frank Cowell August 2006**Public Goods**Overview... The basics Efficiency Characteristics of public goods Contribution schemes The Lindahl approach Alternative mechanisms**Characteristics of public goods**• Two key properties that we need to distinguish: • Excludability • You are producing a good. • A consumer wants some. • Can you prevent him from getting it if he does not pay? • Rivalness • Consider a population of 999 999 people all consuming 1 unit of commodity i. • Another person comes along, also consuming 1 unit of i. • Will more resources be needed for the 1 000 000? • These properties are mutually independent • They interact in an interesting way**Typology of goods: classic definitions**Rival? [ Yes ] [ No ] [??] [ Yes ] pure private Excludable? pure public [??] [ No ]**How the characteristics interact**Example: Bread (E) you can charge a price for bread (R) an extra loaf costs more labour and flour • Private goods are both rival and excludable • Public goods are nonrival and nonexcludable • Consumption externalities are non-excludable but rival • Non-rival but excludable goods often characterise large-scale projects. Example: bread Example: National defence (E) you can't charge for units of 'defence‘ (R) more population doesn't always require more missiles Example: defence Example: Scent from Fresh Flowers (E) you can't charge for the scent (R) more scent requires more flowers Example: flowers Example: Wide Bridge (E) you can charge a toll for the bridge (R) an extra journey has zero cost Example: bridge**Aggregating consumption:**• How consumption is aggregated over agents depends on rivalness characteristic • Also depends on whether the good is optional or not Private goods nh xi =Sxih h=1 Pure rivalness means that you add up each person’s consumption of any good i. Pure nonrivalness means that if you provide good i for one person it is available for all. Optional public goods xi =max h( xih) Pure nonrivalness means that if one person consumes good i then all do so. Non-optional public goods xi =xi1 = xi2 =...**Public Goods**Overview... The basics Efficiency Extending the results that characterise efficient allocations Contribution schemes The Lindahl approach Alternative mechanisms**Public goods and efficiency**• Take the problem of efficient allocation with public goods. • The two principal subproblems will be treated separately... • Characterisation • Implementation • Implementation will be treated later • Characterisation can be treated by introducing public-goods characteristics into standard efficiency model Jump to “Welfare: efficiency”**Efficiency with public goods: an approach**• Use the standard definition of Pareto efficiency • Use the standard maximisation procedure to characterise PE outcomes... • Specify technical and resource constraints • These fix utility possibilities • Fix all persons but one at an arbitrary utility level • Then max utility of remaining person • Repeat for another person if necessary • Use FOCs from maximum to characterise the allocation**Efficiency: the model**• Let good 1 be a public good, goods 2,...,n private goods • Then agent h’s consumption vector is (x1h, x2h ,x3h,...,xnh) where x1is the same for all agents h. and x2h ,x3h,...,xnh is h’s consumption of good 2,3,...n • Agents 2,…,nhare on fixed utility levels uh • Differentiating with respect to x1involves a collection of nh terms • good 1 enters everyone’s utility function.**Efficiency: the model**• Let good 1 be a public good, goods 2,...,n private goods • Then agent h’s consumption vector is (x1h, x2h ,x3h,...,xnh) where x1is the same for all agents h. and x2h ,x3h,...,xnh is h’s consumption of good 2,3,...n • Agents 2,…,nhare on fixed utility levels uh • Problem is to maximise U1(x1,x21,x31,...,xn1) subject to: • Uh(x1,x2h,x3h,...,xnh) ≥ uh, h = 2, …, nh • Ff(qf) ≤ 0,f = 1, …, nf Technological feasibility • qfi is net output of good i by firm f • xi ≤qi + Ri ,i= 1, …, n Materials Balance**Finding an efficient allocation**Lagrange multiplier for each utility constraint max L( [x], [q], l, m, k) := U1(x1) + åhlh[Uh(xh) uh] åf mfFf (q f) + åi ki[qi + Ri xi] where xh = (x1,x2h,x3h,...,xnh) xi = åh xih , i = 2,...,n qi = åf qi f Lagrange multiplier for each firm’s technology Lagrange multiplier for materials balance, good i**FOCs**MU to household h of good i shadow price of good i • For any good i=2,…,n differentiate Lagrangean w.r.t xih. • If xih is positive at the optimum then: lhUih (x1,x2h,x3h,...,xnh) = ki • But good 1 enters everyone’s utility function. So, differentiating w.r.t x1: nh • å lhUjh (x1,x2h,x3h,...,xnh) = k1 • h=1 • Differentiate Lagrangean w.r.t qif. If qif is nonzero at the optimum then: mfFif(qf) = ki • Likewise for good j: mfFjf(qf) = kj Sum, because all are benefited shadow price of good 1**Another look at the FOC...**• For private goods i, j = 2,3,..., n : Ujh(xh) kj Fjf(qf) ——— =— =—— Uih(xh) ki Fif(qf) • Condition when good 1 is public and good i is private Sum of marginal willingness to pay nh åh=1 U1h(xh) k1 ——— =— Uih(xh) ki • An important rule for public goods: Sum over households of marginal willingness to pay = shadow price ratio of goods = MRT**Public Goods**Overview... The basics Efficiency Private provision of public goods? Contribution schemes The Lindahl approach Alternative mechanisms**The implementation problem**• Why is the implementation part of the problem likely to be difficult in the case of pure public goods? • In the general version of the problem private provision will be inefficient • We have an extreme form of the externality issue • We run into the Gibbard-Satterthwaite result**Example**• Good 1 - a pure public good • Good 2 - a pure private good • Two persons: A and B • Each person has an endowment of good 2 • Each contributes to production of good 1 • Production organised in a single firm**Public goods: strategic view (1)**• If Alf reneges [–] then Bill’s best response is [–]. • If Bill reneges [–] then Alf’s best response is [–]. Alf [+] 2,2 0,3 • Nash equilibrium [–] 3,0 1,1 [+] [–] Bill**Public goods: strategic view (2)**• If 1 plays [–] then 2’s best response is [+]. • If 2 plays [+] then 1’s best response is [–]. Alf [+] 2,2 1,3 • A Nash equilibrium • By symmetry, another Nash equilibrium [–] 3,1 0,0 [+] [–] bill**Which paradigm?**• Clearly the two simplified +/– models lead to rather different outcomes. • Which is appropriate? Will we inevitably end up at an inefficient outcome? • The answer depends on the technology of production. • Also on the number of individuals involved in the community.**A Voluntary Approach (1)**• Consider in detail the implementation problem for public goods • Logical to view the way individual action would work in connection with public goods • Begin with a simple contribution model • Take the case with nh persons. • Then see what the “classic” solution would look like**A Voluntary Approach (2)**• Each person has a fixed endowment of (private) good 2: • R2h • And makes a voluntary contribution of some of this toward the production of (public) good 1: • zh=R2h –x2h • This is equivalent to saying that he chooses to consume this amount of good 2: • x2h**A Voluntary Approach (3)**• Contribution of all households of good 2 is: nh z = S zh h=1 • This produces the following amount of good 1: x1= f(z) • So the utility payoff to a typical household is: Uh(x1 , x2h)**A Voluntary Approach (4)**• Suppose every household makes a “Cournot” assumption: nh S zk =`z (constant) k=1 kh • Given this and the production function agent h perceives its optimisation problem to be: • max Uh(f(`z + R2h –x2h) , x2h) • This problem has the first-order condition: • U1h(x1 , x2h) fz(`z + R2h –x2h) – U2h(x1 , x2h) = 0**A Voluntary Approach (5)**• The FOC yields the condition: 1 U1h(x1 , x2h) • ———— = ————— fz(Shzh) U2h(x1 , x2h) • MRT = MRSh • However, for efficiency we should have: 1 U1h(x1 , x2h) • ———— = Sh ————— fz(Shzh) U2h(x1 , x2h) • MRT = Sh MRSh • Each person fails to take into account the “externality” component of the public good provision problem**^**x* x Outcomes with public goods x2 • Production possibilities • Efficiency with public goods • Contribution equilibrium MRT = MRS • Myopic rationality underprovides public good... MRT = SMRS x1 0**Graphical illustrations**• We can use two of the graphical devices that have already proved helpful. • The contribution diagram: • Nash outcomes • PE outcomes • The production possibility curve**Outcomes of contribution game**• Alf’s ICs in contribution space zb • Alf’s reaction function • Bill’s ICs in contribution space • Bill’s reaction function ca(·) • Cournot-Nash equilibrium • Efficient contributions • Alf assumes Bill’s contribution is fixed • Likewise Bill’ cb(·) • Cournot-Nash outcome results in inefficient shortfall of contributions. za**Public Goods**Overview... The basics Efficiency “Personalised” taxes? Contribution schemes The Lindahl approach Alternative mechanisms**A solution?**• Take the standard efficiency result for public goods: SjMRSj = MRT • This aggregation rule has been used to suggest an allocation mechanism • The “Lindahl solution” is tax-based approach. • However, it is a little unconventional. • It suggests that people pay should taxes according to their willingness to pay • The sum of the taxes covers the marginal cost of providing the public good.**An example**• Good 1 - a pure public good • Good 2 - a pure private good • Two persons: Alf and Bill • Simple organisation of production: A single firm**Ua(•)/Ua(•)**12 b a MRS21(x1) MRS21(x1) Ub(•)/Ub(•) 12 Willingness-to-pay for good 1 • Plot Alf’s MRS as function of x1 • WTP by Alf for x1 • Bill’s MRS as function of x1 • WTP by Bill for x1 • the more there is of good 1 the less Alf wants to pay for extra units x1 x1 • Bill is less willing to pay for good 1 than Alf • Use this to derive efficiency condition x1 x1**Ua(•)/Ua(•)**12 x1 Ub(•)/Ub(•) b * a * 12 MRS21(x1) MRS21(x1) x1 ShUh(•)/Uh(•) 12 h * ShMRS21(x1) * x1 Efficiency • MRS for Alf and for Bill • Sum of their MRS as function of x1 • MRT as function of x1 • Efficient amount of x1 • MRS at efficient allocation. • Consider these as demand curves for good 1 • For a public good we aggregate demand “vertically” 1/fz • Can we use these WTP values to derive an allocation mechanism? x1**Ua(•)/Ua(•)**12 x1 Ub(•)/Ub(•) 12 x1 ShUh(•)/Uh(•) 12 * x1 Lindahl solution pa • Efficient allocation of public good • Willingness-to-pay at efficient allocation. • Charge these WTPs as “tax prices “ • The “ Lindahl solution” suggests that people pay should taxes according to their willingness to pay • Combined “tax prices” pa + pb just cover marginal cost of producing the amount x1* of the public good pb 1/fz But what of individual rationality? pa + pb x1**The Lindahl Approach**• let ph is the “tax-price” of good 1 for personh, set by the government. • The FOC for the household’s problem is: U1h(x1, x2h) • ———— = ph U2h(x1, x2h) • For an efficient outcome in terms of the allocation of the two goods: nh1 • S ph = —— h=1fz(z) • Conditions 1,2 determine the set of household-specific prices { ph} ShMRSh = MRT**The Lindahl Approach (1)**• But where does the information come from for this personalised tax-price setting to be implemented? • Presumably from the households themselves • In which case households may view the determination of the personalised prices strategically. • In other words h may try to manipulate ph (and thus the allocation) by revealing false information about his MRS**The Lindahl Approach (2)**• Take into account this strategic possibility • Then h solves the utility-maximisation problem: • choose (x1, x2h) to max Uh(x1, x2h) subject to • the budget constraint: • phx1 + x2hR2h • the following perceived relationship: • x1 = f(z,phx1) • But here ph is endogenous • So this becomes exactly the problem of voluntary contribution**The Way Forward**• Given that the Lindahl problem results in the same suboptimal outcome as voluntary contribution (subscription) what can be done? • Public provision through regular taxation • Change the problem • Change perception of the problem**Public Goods**Overview... The basics Efficiency Truth-revealing devices Contribution schemes The Lindahl approach Alternative mechanisms**A restricted problem**• One of the reasons for the implementation problem is that one invites selection of a social state qQ, where Q is large. • Sidestep the problem by restricting Q. • We would be changing the problem • But in a way that is relevant to many situations • Suppose that there is an all-or nothing choice. • Replace the problem of choosing a specific amount of good 1 from a continuum … • …by substituting the choice problem “select from {NO-PROJECT, PROJECT} ”**The Clark-Groves approach**• Imagine a project completely characterised by • the status-quo utility, • the payment required from each member of the community if the project goes ahead • the utility to each person if it goes ahead. • For all individuals • utility is separable and • income effect of good 1 is zero: • Uh(x1 , x2h) = y(x1) + x2h**The C-G method (2)**• Person h has endowment ofR2h of private good 2. • The project specifies a payment zh for each person conditional on the project going ahead. • Total production of good 1 is f(z) where • z := Sh zh • Social states states Q = {q0 , q1} where • q0 : f(0)= 0 • q1 : f(z)= 1 • Measure the welfare benefit to each person by the compensating variation CVh .**a**b x2 x2 b a R2 R2 b b a a R2 – z R2 – z Project payoffs • Consumption space for Alf and Bill • Endowments and preferences • q° • Outcomes if project goes ahead Alf • The elements of Q • q′ • Compensating variation for Alf, Bill • Alf would like the project to go ahead. • Bill would prefer the opposite. x1 0 1 Bill • q° • CV is positive for Alf... • ...negative for Bill • But sum is positive • q′ Should project go ahead? x1 0 1**A criterion for the project**• Let CVh be the compensating variation for household h if the project is to go ahead. • Then clearly an appropriate criterion overall is nh • S CVh > 0 h=1 • Gainers could compensate losers • But how do we get the right information on CVs? • Introduce a simple, powerful concept**Use announced information**• Approve the project only if this is positive nh S CVh > 0 h=1 • If person k is pivotal, then impose a penalty of this size nh S CVh h=1 hk • Theorem: a scheme which • approves a project if and only if announced CVs is non-negative, and • imposes the above penalty on any pivotal household will guarantee that truthful revelation of CVs is a dominant strategy.**The pivotal person**• Pick an arbitrary person h. • What would be the sum of the announced CVs if he were eliminated from the population? • If this sum has the opposite sign from that of the full sum of the CVs, then h is pivotal. Adding him swings the result. • We use this to construct a mechanism. • Consider the following table**Public goods: revelation**• Two possible states Everyone else says: • Agent h decision [Yes] [No] • Payoff table Decision [Yes] Nil S costs imposed on others [No] S forgone gains of others Nil An example**Summary**• A big subject. A few simple questions to pull thoughts together: • What is the meaning of “market failure”? • Why do markets “fail”? • What’s special about public goods?**Public goods: summary**replace the MRS = MRT rule by S MRS = MRT • Characterisation problem: The Lindahl "solution" may not be a solution at all if people can manipulate the system. • Implementation problem:**Public goods**• The externality feature of public goods makes it easy to solve the characterisation problem • Implementation problems are much harder. • Intimately associated with the information problem. • Mechanism design depends crucially on the type of public good and the economic environment within which provision is made.

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