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Frank Cowell: Microeconomics Public Goods MICROECONOMICS Principles and Analysis Frank Cowell Almost essential Welfare and Efficiency Almost essential Welfare and Efficiency Prerequisites August 2006

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Frank Cowell: Microeconomics Overview... The basics Efficiency Contribution schemes The Lindahl approach Public Goods Characteristics of public goods Alternative mechanisms

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Frank Cowell: Microeconomics Characteristics of public goods Two key properties that we need to distinguish: Two key properties that we need to distinguish: Excludability Excludability You are producing a good. You are producing a good. A consumer wants some. A consumer wants some. Can you prevent him from getting it if he does not pay? Can you prevent him from getting it if he does not pay? Rivalness Rivalness Consider a population of people all consuming 1 unit of commodity i. Consider a population of people all consuming 1 unit of commodity i. Another person comes along, also consuming 1 unit of i. Another person comes along, also consuming 1 unit of i. Will more resources be needed for the ? Will more resources be needed for the ? These properties are mutually independent These properties are mutually independent They interact in an interesting way They interact in an interesting way

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Frank Cowell: Microeconomics Typology of goods: classic definitions Rival? [ Yes ][ No ] pure private [??] pure public [ Yes ] [ No ] Excludable?

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Frank Cowell: Microeconomics How the characteristics interact Private goods are both rival and excludable Private goods are both rival and excludable Public goods are nonrival and nonexcludable Public goods are nonrival and nonexcludable Consumption externalities are non- excludable but rival Consumption externalities are non- excludable but rival Non-rival but excludable goods often characterise large-scale projects. Non-rival but excludable goods often characterise large-scale projects. Example: defence Example: defence Example: National defence (E) you can't charge for units of 'defence (R) more population doesn't always require more missiles Example: National defence (E) you can't charge for units of 'defence (R) more population doesn't always require more missiles Example: bread Example: bread Example: Bread (E) you can charge a price for bread (R) an extra loaf costs more labour and flour Example: Bread (E) you can charge a price for bread (R) an extra loaf costs more labour and flour Example: bridge Example: bridge Example: Wide Bridge (E) you can charge a toll for the bridge (R) an extra journey has zero cost Example: Wide Bridge (E) you can charge a toll for the bridge (R) an extra journey has zero cost Example: flowers Example: flowers Example: Scent from Fresh Flowers (E) you can't charge for the scent (R) more scent requires more flowers Example: Scent from Fresh Flowers (E) you can't charge for the scent (R) more scent requires more flowers

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Frank Cowell: Microeconomics Private goods n h x i x i h h=1 Non-optional public goods x i x i 1 x i 2 Aggregating consumption: Pure rivalness means that you add up each persons consumption of any good i. Pure nonrivalness means that if one person consumes good i then all do so. Optional public goods x i max h ( x i h ) Pure nonrivalness means that if you provide good i for one person it is available for all. How consumption is aggregated over agents depends on rivalness characteristic How consumption is aggregated over agents depends on rivalness characteristic Also depends on whether the good is optional or not Also depends on whether the good is optional or not

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

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Frank Cowell: Microeconomics Public goods and efficiency Take the problem of efficient allocation with public goods. Take the problem of efficient allocation with public goods. The two principal subproblems will be treated separately... The two principal subproblems will be treated separately... Characterisation Characterisation Implementation Implementation Implementation will be treated later Implementation will be treated later Characterisation can be treated by introducing public-goods characteristics into standard efficiency model Characterisation can be treated by introducing public-goods characteristics into standard efficiency model Jump to Welfare: efficiency

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Frank Cowell: Microeconomics Efficiency with public goods: an approach Use the standard definition of Pareto efficiency Use the standard definition of Pareto efficiency Use the standard maximisation procedure to characterise PE outcomes... Use the standard maximisation procedure to characterise PE outcomes... Specify technical and resource constraints Specify technical and resource constraints These fix utility possibilities These fix utility possibilities Fix all persons but one at an arbitrary utility level Fix all persons but one at an arbitrary utility level Then max utility of remaining person Then max utility of remaining person Repeat for another person if necessary Repeat for another person if necessary Use FOCs from maximum to characterise the allocation Use FOCs from maximum to characterise the allocation

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Frank Cowell: Microeconomics Efficiency: the model Let good 1 be a public good, goods 2,...,n private goods Let good 1 be a public good, goods 2,...,n private goods Then agent hs consumption vector is Then agent hs consumption vector is (x 1 h, x 2 h, x 3 h,..., x n h ) where is the same for all agents h. where x 1 is the same for all agents h. and hs consumption of good 2,3,...n and x 2 h, x 3 h,..., x n h is hs consumption of good 2,3,...n Agents 2,…,n h are on fixed utility levels Agents 2,…,n h are on fixed utility levels h Differentiating with respect to involves a collection of n terms Differentiating with respect to x 1 involves a collection of n h terms good 1 enters everyones utility function. good 1 enters everyones utility function.

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Frank Cowell: Microeconomics Efficiency: the model Let good 1 be a public good, goods 2,...,n private goods Let good 1 be a public good, goods 2,...,n private goods Then agent hs consumption vector is Then agent hs consumption vector is (x 1 h, x 2 h, x 3 h,..., x n h ) where is the same for all agents h. where x 1 is the same for all agents h. and hs consumption of good 2,3,...n and x 2 h, x 3 h,..., x n h is hs consumption of good 2,3,...n Agents 2,…,n h are on fixed utility levels Agents 2,…,n h are on fixed utility levels h Problem is to maximise U 1 (x 1, x 2 1, x 3 1,..., x n 1 ) subject to: Problem is to maximise U 1 (x 1, x 2 1, x 3 1,..., x n 1 ) subject to: U(x 1, x 2 h, x 3 h,..., x n h ), h = 2, …, n h U h (x 1, x 2 h, x 3 h,..., x n h ) h, h = 2, …, n h f = 1, …, n f f (q f ) 0, f = 1, …, n f i= 1, …, n x i q i + R i, i= 1, …, n Use all this to form a Lagrangean in the usual way… Use all this to form a Lagrangean in the usual way…

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Frank Cowell: Microeconomics Finding an efficient allocation max L ( [x ], [q], ) := U 1 (x 1 ) + h h [U h (x h ) h ] f f f (q f ) + i i [q i + R i x i ] where x 1, x 2 h, x 3 h,..., x n h x h = (x 1, x 2 h, x 3 h,..., x n h ) x i = h x i h, i = 2,...,n q i = f q i f Lagrange multiplier for each utility constraint Lagrange multiplier for each firms technology Lagrange multiplier for materials balance, good i

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Frank Cowell: Microeconomics FOCs For any good i=2,…,n differentiate Lagrangean w.r.t x i h. If x i h is positive at the optimum then: x 1, x 2 h, x 3 h,..., x n h h U i h (x 1, x 2 h, x 3 h,..., x n h ) = i But good 1 enters everyones utility function. So, differentiating w.r.t x 1 : n h x 1, x 2 h, x 3 h,..., x n h h U j h (x 1, x 2 h, x 3 h,..., x n h ) = 1 h= Differentiate Lagrangean w.r.t q i f. If q i f is nonzero at the optimum then: f i f (q f ) = i Likewise for good j: f j f (q f ) = j MU to household h of good i shadow price of good i Sum, because all are benefited shadow price of good 1

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Frank Cowell: Microeconomics Another look at the FOC... U 1 h (x h ) 1 = U i h (x h ) i An important rule for public goods: An important rule for public goods: For private goods i, j = 2,3,..., n : For private goods i, j = 2,3,..., n : n h h= Condition when good 1 is public and good i is private Condition when good 1 is public and good i is private Sum over households of marginal willingness to pay = shadow price ratio of goods = MRT Sum of marginal willingness to pay U j h (x h ) j j f (q f ) = = U i h (x h ) i i f (q f )

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Frank Cowell: Microeconomics Overview... The basics Efficiency Contribution schemes The Lindahl approach Public Goods Private provision of public goods? Alternative mechanisms

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Frank Cowell: Microeconomics The implementation problem Why is the implementation part of the problem likely to be difficult in the case of pure public goods? 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 In the general version of the problem private provision will be inefficient We have an extreme form of the externality issue We have an extreme form of the externality issue We run into the Gibbard-Satterthwaite result We run into the Gibbard-Satterthwaite result

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Frank Cowell: Microeconomics Example Good 1 - a pure public good Good 1 - a pure public good Good 2 - a pure private good Good 2 - a pure private good Two persons: A and B Two persons: A and B Each person has an endowment of good 2 Each person has an endowment of good 2 Each contributes to production of good 1 Each contributes to production of good 1 Production organised in a single firm Production organised in a single firm

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Frank Cowell: Microeconomics [–][–][–][–] 1,13,0 0,32,2[+]Alf Bill [+] [–][–][–][–] Public goods: strategic view (1) If Alf reneges [–] then Bills best response is [–]. If Bill reneges [–] then Alfs best response is [–]. Nash equilibrium

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Frank Cowell: Microeconomics [–][–][–][–] 0,03,1 1,32,2[+]Alf bill [+] [–][–][–][–] Public goods: strategic view (2) If 1 plays [–] then 2s best response is [+]. If 2 plays [+] then 1s best response is [–]. A Nash equilibrium By symmetry, another Nash equilibrium

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Frank Cowell: Microeconomics Which paradigm? Clearly the two simplified +/– models lead to rather different outcomes. Clearly the two simplified +/– models lead to rather different outcomes. Which is appropriate? Will we inevitably end up at an inefficient outcome? Which is appropriate? Will we inevitably end up at an inefficient outcome? The answer depends on the technology of production. The answer depends on the technology of production. Also on the number of individuals involved in the community. Also on the number of individuals involved in the community.

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Frank Cowell: Microeconomics A Voluntary Approach (1) Consider in detail the implementation problem for public goods Consider in detail the implementation problem for public goods Logical to view the way individual action would work in connection with public goods Logical to view the way individual action would work in connection with public goods Begin with a simple contribution model Begin with a simple contribution model Take the case with n h persons. Take the case with n h persons. Then see what the classic solution would look like Then see what the classic solution would look like

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Frank Cowell: Microeconomics A Voluntary Approach (2) Each person has a fixed endowment of (private) good 2: Each person has a fixed endowment of (private) good 2: R 2 h R 2 h And makes a voluntary contribution of some of this toward the production of (public) good 1: And makes a voluntary contribution of some of this toward the production of (public) good 1: z h = R 2 h – x 2 h z h = R 2 h – x 2 h This is equivalent to saying that he chooses to consume this amount of good 2: This is equivalent to saying that he chooses to consume this amount of good 2: x 2 h x 2 h

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Frank Cowell: Microeconomics A Voluntary Approach (3) Contribution of all households of good 2 is: Contribution of all households of good 2 is: n h n h z = z h h=1 h=1 This produces the following amount of good 1: This produces the following amount of good 1: x 1 = z x 1 = z So the utility payoff to a typical household is: So the utility payoff to a typical household is: U h x 1, x 2 h U h x 1, x 2 h

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Frank Cowell: Microeconomics A Voluntary Approach (4) Suppose every household makes a Cournot assumption: Suppose every household makes a Cournot assumption: n h n h z k = z (constant) z k = z (constant) k=1 k=1 k h k h Given this and the production function agent h perceives its optimisation problem to be: Given this and the production function agent h perceives its optimisation problem to be: max U h z + R 2 h – x 2 h, x 2 h max U h z + R 2 h – x 2 h, x 2 h This problem has the first-order condition: This problem has the first-order condition: U 1 h x 1, x 2 h z z + R 2 h – x 2 h – U 2 h x 1, x 2 h U 1 h x 1, x 2 h z z + R 2 h – x 2 h – U 2 h x 1, x 2 h

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Frank Cowell: Microeconomics A Voluntary Approach (5) The FOC yields the condition: The FOC yields the condition: 1 U 1 h x 1, x 2 h 1 U 1 h x 1, x 2 h z h z h U 2 h x 1, x 2 h z h z h U 2 h x 1, x 2 h MRT = MRS h MRT = MRS h However, for efficiency we should have: However, for efficiency we should have: 1 U 1 h x 1, x 2 h 1 U 1 h x 1, x 2 h h h z h z h U 2 h x 1, x 2 h z h z h U 2 h x 1, x 2 h MRT = h MRS h MRT = h MRS h Each person fails to take into account the externality component of the public good provision problem Each person fails to take into account the externality component of the public good provision problem

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Frank Cowell: Microeconomics Outcomes with public goods Production possibilities 0 Contribution equilibrium Efficiency with public goods MRT = MRS Myopic rationality underprovides public good... ll ll x* x1x1 x2x2 ll ll x ^ MRT = MRS

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Frank Cowell: Microeconomics Graphical illustrations We can use two of the graphical devices that have already proved helpful. We can use two of the graphical devices that have already proved helpful. The contribution diagram: The contribution diagram: Nash outcomes Nash outcomes PE outcomes PE outcomes The production possibility curve The production possibility curve

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Frank Cowell: Microeconomics Outcomes of contribution game l l l l zbzb b (·) a (·) ll ll ll ll ll ll ll ll ll ll ll ll ll ll Alfs ICs in contribution space Alfs reaction function Alf assumes Bills contribution is fixed Bills ICs in contribution space Bills reaction function Likewise Bill Cournot-Nash equilibrium Efficient contributions Cournot-Nash outcome results in inefficient shortfall of contributions. zaza

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Frank Cowell: Microeconomics Overview... The basics Efficiency Contribution schemes The Lindahl approach Public Goods Personalised taxes? Alternative mechanisms

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Frank Cowell: Microeconomics A solution? Take the standard efficiency result for public goods: Take the standard efficiency result for public goods: j MRS j = MRT This aggregation rule has been used to suggest an allocation mechanism This aggregation rule has been used to suggest an allocation mechanism The Lindahl solution is tax-based approach. The Lindahl solution is tax-based approach. However, it is a little unconventional. However, it is a little unconventional. It suggests that people pay should taxes according to their willingness to pay 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. The sum of the taxes covers the marginal cost of providing the public good.

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Frank Cowell: Microeconomics An example Good 1 - a pure public good Good 1 - a pure public good Good 2 - a pure private good Good 2 - a pure private good Two persons: Alf and Bill Two persons: Alf and Bill Simple organisation of production: A single firm Simple organisation of production: A single firm

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Frank Cowell: Microeconomics Willingness-to-pay for good 1 x1x1 x1x1 U a ()/U a () 1 2 U b ()/U b () Plot Alfs MRS as function of x 1 WTP by Alf for x 1 the more there is of good 1 the less Alf wants to pay for extra units Bills MRS as function of x 1 x1x1 MRS 21 (x 1 ) a b WTP by Bill for x 1 x1x1 Bill is less willing to pay for good 1 than Alf Use this to derive efficiency condition

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Frank Cowell: Microeconomics Efficiency 1/ z x1x1 x1x1 * x1x1 x1x1 U a ()/U a () 1 2 U b ()/U b () 1 2 h U h ()/U h () MRS for Alf and for Bill Sum of their MRS as function of x 1 MRT as function of x 1 Efficient amount of x 1 Consider these as demand curves for good 1 For a public good we aggregate demand vertically MRS at efficient allocation. MRS 21 (x 1 ) a * MRS 21 (x 1 ) b * h MRS 21 (x 1 ) h * Can we use these WTP values to derive an allocation mechanism?

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Frank Cowell: Microeconomics Lindahl solution 1/ z x1x1 x1x1 * x1x1 x1x1 U a ()/U a () 1 2 U b ()/U b () 1 2 h U h ()/U h () Efficient allocation of public good Willingness-to-pay at efficient allocation. The Lindahl solution suggests that people pay should taxes according to their willingness to pay Combined tax prices p a + p b just cover marginal cost of producing the amount x 1 * of the public good Charge these WTPs as tax prices papa pbpb p a + p b But what of individual rationality?

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Frank Cowell: Microeconomics The Lindahl Approach let h, set by the government. let p h is the tax-price of good 1 for person h, set by the government. The FOC for the households problem is: The FOC for the households problem is: U 1 h (x 1, x 2 h ) U 1 h (x 1, x 2 h ) 1. = p h U 2 h (x 1, x 2 h ) U 2 h (x 1, x 2 h ) For an efficient outcome in terms of the allocation of the two goods: For an efficient outcome in terms of the allocation of the two goods: n h 1 n h 1 p h = p h = h=1 z z h=1 z z Conditions 1,2 determine the set of household-specific prices { } Conditions 1,2 determine the set of household-specific prices { p h } h MRS h = MRT

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Frank Cowell: Microeconomics The Lindahl Approach (1) But where does the information come from for this personalised tax-price setting to be implemented? But where does the information come from for this personalised tax-price setting to be implemented? Presumably from the households themselves Presumably from the households themselves In which case households may view the determination of the personalised prices strategically. In which case households may view the determination of the personalised prices strategically. In other words h may try to manipulate (and thus the allocation) by revealing false information about his MRS In other words h may try to manipulate p h (and thus the allocation) by revealing false information about his MRS

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Frank Cowell: Microeconomics The Lindahl Approach (2) Take into account this strategic possibility Take into account this strategic possibility Then h solves the utility-maximisation problem: Then h solves the utility-maximisation problem: choose (x 1, x 2 h ) to max U h (x 1, x 2 h ) subject to choose (x 1, x 2 h ) to max U h (x 1, x 2 h ) subject to 1. the budget constraint: p h x 1 + x 2 h R 2 h p h x 1 + x 2 h R 2 h 2. the following perceived relationship: x 1 = (c + p h x 1 ) x 1 = (c + p h x 1 ) But here p h is endogenous: But here p h is endogenous: So this becomes exactly the problem of voluntary contribution So this becomes exactly the problem of voluntary contribution

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Frank Cowell: Microeconomics The Way Forward Given that the Lindahl problem results in the same suboptimal outcome as voluntary contribution (subscription) what can be done? Given that the Lindahl problem results in the same suboptimal outcome as voluntary contribution (subscription) what can be done? Public provision through regular taxation Public provision through regular taxation Change the problem Change the problem Change perception of the problem Change perception of the problem

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Frank Cowell: Microeconomics Overview... The basics Efficiency Contribution schemes The Lindahl approach Public Goods Truth-revealing devices Alternative mechanisms

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Frank Cowell: Microeconomics A restricted problem One of the reasons for the implementation problem is that one invites selection of a social state, where is large. One of the reasons for the implementation problem is that one invites selection of a social state, where is large. Sidestep the problem by restricting. Sidestep the problem by restricting. We would be changing the problem We would be changing the problem But in a way that is relevant to many situations But in a way that is relevant to many situations Suppose that there is an all-or nothing choice. Suppose that there is an all-or nothing choice. Replace the problem of choosing a specific amount of good 1 from a continuum … Replace the problem of choosing a specific amount of good 1 from a continuum … …by substituting the choice problem select from {NO-PROJECT, PROJECT} …by substituting the choice problem select from {NO-PROJECT, PROJECT}

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Frank Cowell: Microeconomics The Clark-Groves approach Imagine a project completely characterised by Imagine a project completely characterised by the status-quo utility, the status-quo utility, the payment required from each member of the community if the project goes ahead the payment required from each member of the community if the project goes ahead the utility to each person if it goes ahead. the utility to each person if it goes ahead. For all individuals For all individuals utility is separable and utility is separable and income effect of good 1 is zero: income effect of good 1 is zero: U h x 1, x 2 h x 1 + x 2 h U h x 1, x 2 h x 1 + x 2 h

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Frank Cowell: Microeconomics The C-G method (2) Person h has endowment of of private good 2. Person h has endowment of R 2 h of private good 2. The project specifies a payment for each person conditional on the project going ahead. The project specifies a payment z h for each person conditional on the project going ahead. Total production of good 1 is () where Total production of good 1 is (z) where := z := h z h Social states states = {, } where Social states states = { 0, 1 } where : (0)= 0 0 : (0) = 0 : ()= 1 1 : (z) = 1 Measure the welfare benefit to each person by the compensating variation CV. Measure the welfare benefit to each person by the compensating variation CV h.

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Frank Cowell: Microeconomics R 2 – z b Project payoffs x2x2 a R 2 – z a 0 1 R2R2 a Bill x2x2 b R2R2 b 0 1 Alf Consumption space for Alf and Bill x1x1 x1x1 Endowments and preferences Outcomes if project goes ahead Compensating variation for Alf, Bill ° ° Alf would like the project to go ahead. Bill would prefer the opposite. The elements of CV is positive for Alf......negative for Bill But sum is positive Should project go ahead?

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Frank Cowell: Microeconomics A criterion for the project Let CV h be the compensating variation for household h if the project is to go ahead. Let CV h be the compensating variation for household h if the project is to go ahead. Then clearly an appropriate criterion overall is Then clearly an appropriate criterion overall is n h n h CV h > 0 CV h > 0 h=1 h=1 Gainers could compensate losers Gainers could compensate losers But how do we get the right information on CVs? But how do we get the right information on CVs? Introduce a simple, powerful concept Introduce a simple, powerful concept

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Frank Cowell: Microeconomics Use announced information Approve the project only if this is positive Approve the project only if this is positive n h n h CV h > 0 CV h > 0 h=1 h=1 If person k is pivotal, then impose a penalty of this size If person k is pivotal, then impose a penalty of this size n h n h CV h CV h h=1 h=1 h k h k Theorem: a scheme which Theorem: a scheme which approves a project if and only if announced CVs is non-negative, and approves a project if and only if announced CVs is non-negative, and imposes the above penalty on any pivotal household imposes the above penalty on any pivotal household will guarantee that truthful revelation of CVs is a dominant strategy.

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Frank Cowell: Microeconomics 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

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Frank Cowell: Microeconomics [ No ] Nil forgone gains of others costs imposed on others Nil[Yes]Decision Everyone else says: [Yes] [No] Public goods: revelation Two possible states Payoff table Agent h decision An example

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

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Frank Cowell: Microeconomics Example: choices If agent 1 declares… If agent 1 declares… v = v 1 then outcome is no project v = v 1 then outcome is no project reverses sign of willingness to pay – so must pay penalty reverses sign of willingness to pay – so must pay penalty gets payoff of –z 1 gets payoff of –z 1 v < v 1 then outcome and payoff are as above v < v 1 then outcome and payoff are as above v > v 1 then v > v 1 then if v v 1 is small, outcome and payoff are as above if v v 1 is small, outcome and payoff are as above if v v 1 is large, project goes ahead and payoff is v 1 if v v 1 is large, project goes ahead and payoff is v 1 If agent 2 declares… If agent 2 declares… v = v 2 then outcome is no project and gets payoff of 0 v = v 2 then outcome is no project and gets payoff of 0 v < v 2 then outcome and payoff are as above v < v 2 then outcome and payoff are as above v > v 2 then v > v 2 then if v v 2 is small, outcome and payoff are as above if v v 2 is small, outcome and payoff are as above if v v 2 is large, outcome reversed and payoff is v 2 + z 2 if v v 2 is large, outcome reversed and payoff is v 2 + z 2

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Frank Cowell: Microeconomics Example: outcomes Payoff to agent 1 is… Payoff to agent 1 is… –z 1 if declares v 1 –z 1 if declares v 1 –z 1 or v 1 otherwise –z 1 or v 1 otherwise but z 1 + v 1 = ½N[a + b] v 1 … but z 1 + v 1 = ½N[a + b] v 1 … … so declaring v 1 is optimal … so declaring v 1 is optimal Payoff to agent 2 is… Payoff to agent 2 is… 0 if declares v 2 0 if declares v 2 0 or v 2 or v 2 + z 2 otherwise 0 or v 2 or v 2 + z 2 otherwise but v 2 < 0 and z 2 < 0 … but v 2 < 0 and z 2 < 0 … … so declaring v 2 is optimal … so declaring v 2 is optimal Overall outcome Overall outcome Each has incentive to report truthfully Each has incentive to report truthfully More resources are paid (in penalties) than are necessary to produce the public good More resources are paid (in penalties) than are necessary to produce the public good

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Frank Cowell: Microeconomics The C-G model – assessment Strengths: Strengths: only uses announced information only uses announced information elicits truth-telling elicits truth-telling Drawbacks: Drawbacks: Restriction to Ziff preferences Restriction to Ziff preferences Does not ensure budgetary balance Does not ensure budgetary balance The tipping mechanism can be used as the foundation for more interesting design problems. The tipping mechanism can be used as the foundation for more interesting design problems.

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Frank Cowell: Microeconomics Summary A big subject. A few simple questions to pull thoughts together: A big subject. A few simple questions to pull thoughts together: What is the meaning of market failure? What is the meaning of market failure? Why do markets fail? Why do markets fail? Whats special about public goods? Whats special about public goods?

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Frank Cowell: Microeconomics Public goods: summary Characterisation problem: Characterisation problem: replace the MRS = MRT rule by MRS = MRT Implementation problem: Implementation problem: The Lindahl "solution" may not be a solution at all if people can manipulate the system.

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Frank Cowell: Microeconomics Public goods The externality feature of public goods makes it easy to solve the characterisation problem The externality feature of public goods makes it easy to solve the characterisation problem Implementation problems are much harder. Implementation problems are much harder. Intimately associated with the information problem. 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. Mechanism design depends crucially on the type of public good and the economic environment within which provision is made.

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