#### Transcript Extensive Form - London School of Economics

Prerequisites Almost essential Welfare and Efficiency Frank Cowell: Microeconomics August 2006 Public Goods MICROECONOMICS Principles and Analysis Frank Cowell Public Goods Overview... Frank Cowell: Microeconomics The basics Characteristics of public goods Efficiency Contribution schemes The Lindahl approach Alternative mechanisms Characteristics of public goods Frank Cowell: Microeconomics Two key properties that we need to distinguish: Excludability Rivalness You are producing a good. A consumer wants some. Can you prevent him from getting it if he does not pay? 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 Rival? Excludable? Frank Cowell: Microeconomics Typology of goods: classic definitions [ Yes ] [ No ] [ Yes ] [ No ] pure private [??] [??] pure public How the characteristics interact Frank Cowell: Microeconomics Example: bread Example: defence Example: flowers Example: bridge Private goods are both rival and excludable Public goods are nonrival and nonexcludable Consumption externalities are nonexcludable but rival Non-rival but excludable goods often characterise large-scale projects. Example: Bread (E) you can charge a price for bread (R) an extra loaf costs more labour and flour Example: National defence (E) you can't charge for units of 'defence‘ (R) more population doesn't always require more missiles Example: Scent from Fresh Flowers (E) you can't charge for the scent (R) more scent requires more flowers Example: Wide Bridge (E) you can charge a toll for the bridge (R) an extra journey has zero cost Aggregating consumption: Frank Cowell: Microeconomics How consumption is aggregated over agents depends on rivalness characteristic Also depends on whether the good is optional or not Private goods xi = S h=1x nh h i Optional public goods xi = max h ( xih ) Non-optional public goods xi = xi1 = xi2 =... 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. Pure nonrivalness means that if one person consumes good i then all do so. Public Goods Overview... Frank Cowell: Microeconomics The basics Extending the results that characterise efficient allocations Efficiency Contribution schemes The Lindahl approach Alternative mechanisms Public goods and efficiency Frank Cowell: Microeconomics Jump to “Welfare: 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 Frank Cowell: Microeconomics 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 Frank Cowell: Microeconomics Let good 1 be a public good, goods 2,...,n private goods Then agent h’s consumption vector is (x1h, x2h , x3h, ..., xnh) where x1 is the same for all agents h. and x2h , x3h, ..., xnh is h’s consumption of good 2,3,...n Agents 2,…,nh are on fixed utility levels uh Differentiating with respect to x1 involves a collection of nh terms good 1 enters everyone’s utility function. Efficiency: the model Frank Cowell: Microeconomics Let good 1 be a public good, goods 2,...,n private goods Then agent h’s consumption vector is (x1h, x2h , x3h, ..., xnh) where x1 is the same for all agents h. and x2h , x3h, ..., xnh is h’s consumption of good 2,3,...n Agents 2,…,nh are 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 f f F (q ) ≤ 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 Frank Cowell: Microeconomics max L( [x ], [q], l, Lagrange multiplier for m,each k)utility := constraint Lagrange multiplier h(xhfor h] U1(x1) + l [U ) u h h technology each firm’s Lagrange f (q f) multiplier for f mf Fmaterials balance, good i + i ki[qi + Ri xi] where xh = (x1, x2h, x3h, ..., xnh) xi = h xih , i = 2,...,n qi = f qi f FOCs Frank Cowell: Microeconomics For any good i=2,…,n differentiate Lagrangean w.r.t xih. MU to household h shadow price h of good i of good If xii 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: Sum, because all are benefited nh lU h shadow price of good 1 j h (x , x h, x h, ..., x h) 1 2 3 n = 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 Another look at the FOC... Frank Cowell: Microeconomics For private goods i, j = 2,3,..., n : Ujh(xh) kj Fjf(qf) ——— = — = —— Uih (xh) ki Fif(qf) Condition Sum of marginal willingness to pay nh h=1 when good 1 is public and good i is private 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... Frank Cowell: Microeconomics The basics Private provision of public goods? Efficiency Contribution schemes The Lindahl approach Alternative mechanisms The implementation problem Frank Cowell: Microeconomics 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 Frank Cowell: Microeconomics 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) [+] 2,2 0,3 [–] If Bill reneges [–] then Alf’s best response is [–]. 3,0 1,1 [+] [–] Alf Frank Cowell: Microeconomics If Alf reneges [–] then Bill’s best response is [–]. Bill Nash equilibrium Public goods: strategic view (2) [+] 2,2 1,3 Alf If 2 plays [+] then 1’s best response is [–]. A Nash equilibrium By symmetry, another Nash equilibrium [–] Frank Cowell: Microeconomics If 1 plays [–] then 2’s best response is [+]. 3,1 0,0 [+] [–] bill Which paradigm? Frank Cowell: Microeconomics 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) Frank Cowell: Microeconomics 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) Frank Cowell: Microeconomics Each person has a fixed endowment of (private) good 2: And makes a voluntary contribution of some of this toward the production of (public) good 1: R2h zh = R2h – x2h This is equivalent to saying that he chooses to consume this amount of good 2: x2 h A Voluntary Approach (3) Frank Cowell: Microeconomics Contribution of all households of good 2 is: nh z= Sz h 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) Frank Cowell: Microeconomics 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) Frank Cowell: Microeconomics The FOC yields the condition: 1 ———— fz(Sh zh ) MRT = MRSh However, for efficiency we should have: U1h(x1 , x2h) = ————— U2h(x1 , x2h) 1 ———— fz(Sh zh ) = S U1h(x1 , x2h) h ————— U2h(x1 , x2h) MRT = Sh MRSh Each person fails to take into account the “externality” component of the public good provision problem Outcomes with public goods Frank Cowell: Microeconomics x2 Production possibilities Efficiency with public goods Contribution equilibrium ^x MRT = MRS x* MRT = SMRS x1 0 Myopic rationality underprovides public good... Graphical illustrations Frank Cowell: Microeconomics 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 Frank Cowell: Microeconomics Alf’s ICs in contribution space Alf’s reaction function Bill’s ICs in contribution space Bill’s reaction function Cournot-Nash equilibrium Efficient contributions zb ca(·) Alf assumes Bill’s contribution is fixed Likewise Bill’ cb(·) za Cournot-Nash outcome results in inefficient shortfall of contributions. Public Goods Overview... Frank Cowell: Microeconomics The basics “Personalised” taxes? Efficiency Contribution schemes The Lindahl approach Alternative mechanisms A solution? Frank Cowell: Microeconomics Take the standard efficiency result for public goods: Sj MRSj = 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 Frank Cowell: Microeconomics Good 1 - a pure public good Good 2 - a pure private good Two persons: Alf and Bill Simple organisation of production: A single firm Willingness-to-pay for good 1 Frank Cowell: Microeconomics Plot Alf’s MRS as function of x1 WTP by Alf for x1 Bill’s MRS as function of x1 U1a(•)/U2a(•) WTP by Bill for x1 a MRS21(x1) x1 x1 the more there is of good 1 the less Alf wants to pay for extra units U1b(•)/U2b(•) b MRS21(x1) x1 x1 Bill is less willing to pay for good 1 than Alf Use this to derive efficiency condition Ua1(•)/Ua2(•) Efficiency Frank Cowell: Microeconomics a MRS21(x*1) x1 MRS for Alf and for Bill Sum of their MRS as function of x1 MRT as function of x1 Efficient amount of x1 Ub1(•)/Ub2(•) b MRS21(x*1) MRS at efficient allocation. x1 For a public good we aggregate demand “vertically” 1/fz ShUh1(•)/U2h(•) Can we use these WTP values to derive an allocation mechanism? h ShMRS21(x*1) x*1 Consider these as demand curves for good 1 x1 Ua1(•)/Ua2(•) pa Frank Cowell: Microeconomics x1 Lindahl solution Efficient allocation of public good Willingness-to-pay at efficient allocation. Charge these WTPs as “tax prices “ Ub1(•)/Ub2(•) pb x1 1/fz Combined “tax prices” pa + pb just cover marginal cost of producing the amount x1* of the public good ShUh1(•)/U2h(•) pa + But what of individual rationality? pb x*1 The “ Lindahl solution” suggests that people pay should taxes according to their willingness to pay x1 The Lindahl Approach Frank Cowell: Microeconomics let ph is the “tax-price” of good 1 for person h, set by the government. The FOC for the household’s problem is: 1. U1h(x1, x2h) ———— = ph U2h(x1, x2h) For an efficient outcome in terms of the allocation of the two goods: S MRSh = MRT nh 2. Sp h=1 1 h h = —— fz(z) Conditions 1,2 determine the set of household-specific prices { ph} The Lindahl Approach (1) Frank Cowell: Microeconomics 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) Frank Cowell: Microeconomics 1. 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: 2. the following perceived relationship: phx1 + x2h R2h x1 = f(z, phx1) But here ph is endogenous So this becomes exactly the problem of voluntary contribution The Way Forward Frank Cowell: Microeconomics 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... Frank Cowell: Microeconomics The basics Truth-revealing devices Efficiency Contribution schemes The Lindahl approach Alternative mechanisms A restricted problem Frank Cowell: Microeconomics 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 Frank Cowell: Microeconomics 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) Frank Cowell: Microeconomics Person h has endowment of R2h 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 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 . z := Sh zh Project payoffs x2a Consumption space for Alf and Bill Frank Cowell: Microeconomics Endowments and preferences R2a q° R2a – z a Outcomes if project goes ahead Alf The elements of Q q′ Compensating variation for Alf, Bill 0 x2b R2b x1 1 Bill Alf would like the project to go ahead. Bill would prefer the opposite. q° R2b – z b q′ CV is positive for Alf... ...negative for Bill But sum is positive Should project go ahead? 0 1 x1 A criterion for the project Frank Cowell: Microeconomics 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 Frank Cowell: Microeconomics 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 Frank Cowell: Microeconomics 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 [Yes] [Yes] [No] Decision Frank Cowell: Microeconomics Everyone else says: [No] Two possible states Agent h decision Payoff table S costs Nil imposed on others S forgone gains of others Nil An example Summary Frank Cowell: Microeconomics 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 Frank Cowell: Microeconomics Characterisation problem: Implementation problem: replace the MRS = MRT rule by S MRS = MRT The Lindahl "solution" may not be a solution at all if people can manipulate the system. Public goods Frank Cowell: Microeconomics 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.