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Public Goods

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.

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Public Goods

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  1. Prerequisites Almost essential Welfare and Efficiency Public Goods MICROECONOMICS Principles and Analysis Frank Cowell August 2006

  2. Public Goods Overview... The basics Efficiency Characteristics of public goods Contribution schemes The Lindahl approach Alternative mechanisms

  3. 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

  4. Typology of goods: classic definitions Rival? [ Yes ] [ No ] [??] [ Yes ] pure private Excludable? pure public [??] [ No ]

  5. 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

  6. 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 =...

  7. Public Goods Overview... The basics Efficiency Extending the results that characterise efficient allocations Contribution schemes The Lindahl approach Alternative mechanisms

  8. 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”

  9. 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

  10. 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.

  11. 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 • xi ≤qi + Ri ,i= 1, …, n • Use all this to form a Lagrangean in the usual way…

  12. 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

  13. 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

  14. 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

  15. Public Goods Overview... The basics Efficiency Private provision of public goods? Contribution schemes The Lindahl approach Alternative mechanisms

  16. 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

  17. 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

  18. 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

  19. 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

  20. 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.

  21. 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

  22. 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

  23. 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)

  24. A Voluntary Approach (4) • Suppose every household makes a “Cournot” assumption: nh S zk =`z (constant) k=1 kh • 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

  25. 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

  26. ^ 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

  27. 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

  28. 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

  29. Public Goods Overview... The basics Efficiency “Personalised” taxes? Contribution schemes The Lindahl approach Alternative mechanisms

  30. 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.

  31. 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

  32. 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

  33. 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

  34. 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

  35. 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

  36. 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

  37. 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 + x2hR2h • the following perceived relationship: • x1 = f(c + phx1) • But here ph is endogenous: • So this becomes exactly the problem of voluntary contribution

  38. 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

  39. Public Goods Overview... The basics Efficiency Truth-revealing devices Contribution schemes The Lindahl approach Alternative mechanisms

  40. A restricted problem • One of the reasons for the implementation problem is that one invites selection of a social state qQ, 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} ”

  41. 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

  42. 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 .

  43. 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

  44. 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

  45. 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 hk • 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.

  46. 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

  47. 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

  48. Example: model • Amount of public good is 0 or 1 • if public good is produced cost is shared equally • population of size N each pay 1/ N of total • Agents’ valuations differ • valuation of his net of contribution to public good • vh = a +[ h − 1][b – a ] / [N − 1] ,h = 1,2,…,N • assume b > 0 > a • Mean valuation is ½[a + b] • project should go ahead if a + b> 0 • assume, however, that a + b < 0 • Define zh:= ½N[a + b] − vh • measures the sum-of-valuations if h is excluded. • Suppose v1 < v2 < 0 and that z1 > 0, z2 < 0 • both agents 1 and 2 would prefer no project • agent 1 is pivotal if reports truthfully (z1 is opposite sign to a + b ) • agent 2 is not pivotal if reports truthfully (z2 is same sign as a + b )

  49. Example: choices • If agent 1 declares… • v = v1 then outcome is no project • reverses sign of willingness to pay – so must pay penalty • gets payoff of –z1 • v < v1 then outcome and payoff are as above • v > v1 then • if v −v1 is small, outcome and payoff are as above • if v −v1 is large, project goes ahead and payoff is v1 • If agent 2 declares… • v = v2 then outcome is no project and gets payoff of 0 • v < v2 then outcome and payoff are as above • v > v2 then • if v −v2 is small, outcome and payoff are as above • if v −v2 is large, outcome reversed and payoff is v2+z2

  50. Example: outcomes • Payoff to agent 1 is… • –z1 if declares v1 • –z1 or v1otherwise • but z1 + v1 = ½N[a + b] < 0 so that –z1 > v1 … • … so declaring v1 is optimal • Payoff to agent 2 is… • 0 if declares v2 • 0 or v2 or v2+z2otherwise • but v2 < 0 and z2 < 0 … • … so declaring v2 is optimal • Overall outcome • Each has incentive to report truthfully • More resources are paid (in penalties) than are necessary to produce the public good

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