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**MICROECONOMICS Principles and Analysis Frank Cowell**

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

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**Overview... Characteristics of public goods Public Goods The basics**

Efficiency Characteristics of public goods Contribution schemes The Lindahl approach Alternative mechanisms

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**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 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 ? These properties are mutually independent They interact in an interesting way

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**Typology of goods: classic definitions**

Rival? [ Yes ] [ No ] [??] [ Yes ] pure private Excludable? [??] pure public [ No ]

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**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: flowers 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 Example: bridge

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**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 = S xih 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 =...

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

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**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”

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**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

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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 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.

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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 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 Ff(qf) ≤ 0, f = 1, …, nf xi ≤ qi + Ri , i= 1, …, n Use all this to form a Lagrangean in the usual way…

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**Finding an efficient allocation**

Lagrange multiplier for each utility constraint max L( [x ], [q], l, m, k) := U1(x1) + åhlh [Uh(xh) uh] åf mf F f (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

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**FOCs å lhUjh (x1, x2h, x3h, ..., xnh) = k1**

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 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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**Sum of marginal willingness to pay**

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

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**Overview... Private provision of public goods? Public Goods The basics**

Efficiency Private provision of public goods? Contribution schemes The Lindahl approach Alternative mechanisms

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**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

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**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

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**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

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**Public goods: strategic view (2)**

If 1 plays [–] then 2’s best response is [+]. Alf [+] 2,2 1,3 If 2 plays [+] then 1’s best response is [–]. A Nash equilibrium By symmetry, another Nash equilibrium [–] 3,1 0,0 [+] [–] bill

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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.

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**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

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**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

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**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)

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**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

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**A Voluntary Approach (5)**

The FOC yields the condition: U1h(x1 , x2h) ———— = ————— fz(Sh zh ) U2h(x1 , x2h) MRT = MRSh However, for efficiency we should have: U1h(x1 , x2h) ———— = Sh ————— fz(Sh zh ) U2h(x1 , x2h) MRT = Sh MRSh Each person fails to take into account the “externality” component of the public good provision problem

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**Outcomes with public goods**

x2 Production possibilities Efficiency with public goods Contribution equilibrium x ^ MRT = MRS Myopic rationality underprovides public good... x* MRT = SMRS x1

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**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

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**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

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**Overview... “Personalised” taxes? Public Goods The basics Efficiency**

Contribution schemes The Lindahl approach Alternative mechanisms

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A solution? 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.

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**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

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**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 Ua(•)/Ua(•) WTP by Bill for x1 MRS21(x1) a x1 the more there is of good 1 the less Alf wants to pay for extra units x1 Bill is less willing to pay for good 1 than Alf Ub(•)/Ub(•) MRS21(x1) b Use this to derive efficiency condition x1 x1

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**Efficiency 1/fz x1 Ua(•)/Ua(•) 1 2 x1 Ub(•)/Ub(•) 1 2**

MRS21(x1) a * MRS for Alf and for Bill Sum of their MRS as function of x1 x1 MRT as function of x1 Efficient amount of x1 Ub(•)/Ub(•) MRS at efficient allocation. MRS21(x1) b * Consider these as demand curves for good 1 For a public good we aggregate demand “vertically” 1/fz ShUh(•)/Uh(•) Can we use these WTP values to derive an allocation mechanism? ShMRS21(x1) h * * x1 x1

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**But what of individual rationality?**

x1 Lindahl solution Ua(•)/Ua(•) pa Efficient allocation of public good x1 Willingness-to-pay at efficient allocation. Charge these WTPs as “tax prices “ Ub(•)/Ub(•) pb 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 1/fz ShUh(•)/Uh(•) But what of individual rationality? pa + pb * x1 x1

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The Lindahl Approach let ph is the “tax-price” of good 1 for person h, 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: nh S ph = —— h= fz(z) Conditions 1,2 determine the set of household-specific prices { ph} Sh MRSh = MRT

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**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

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**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

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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

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**Overview... Truth-revealing devices Public Goods The basics Efficiency**

Contribution schemes The Lindahl approach Alternative mechanisms

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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} ”

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**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

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**The C-G method (2) 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 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 .

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**Should project go ahead?**

Project payoffs x2 a Consumption space for Alf and Bill Endowments and preferences q° R2 a Outcomes if project goes ahead Alf The elements of Q R2 – z a a q′ Compensating variation for Alf, Bill Alf would like the project to go ahead. Bill would prefer the opposite. x1 1 x2 b Bill R2 b q° CV is positive for Alf... ...negative for Bill But sum is positive R2 – z b b q′ Should project go ahead? x1 1

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**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

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**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 S CVh 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.

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**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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**Public goods: revelation**

Everyone else says: Two possible states Agent h decision [Yes] [No] Payoff table Decision [Yes] Nil S costs imposed on others [No] S forgone gains of others Nil An example

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**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 h is 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 )

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**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

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**Example: outcomes Payoff to agent 1 is… –z1 if declares v1**

–z1 or v1 otherwise 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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**The C-G model – assessment**

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

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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?

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**Public goods: summary Implementation problem:**

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:

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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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