lecture25_slides

lecture25_slides - Econ 121. Intermediate Microeconomics....

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Econ 121. Intermediate Microeconomics. Eduardo Faingold Yale University Lecture 25 Outline of the course I. Introduction II. Individual choice III. Competitive markets IV. Market failure Outline of the course I. Introduction II. Individual choice III. Competitive markets IV. Market failure Monopoly (Ch. 24, 25) Oligopoly (Ch. 27) Externalities (Ch. 34) Public Goods (Ch. 36) Externalities and Property Rights Consider two roommates, Alice and Bob. Alice is a smoker, Bob hates smoking. They care about money and smoke / clean air. Utilities are: uA.mA; s A/ and uB .mB ; c B / Of course, the amount of clean air that Bob consumes, c B , and the amount of smoke that Alice consumes, s A, sum to a constant K s A C c B D K: We may as well write uA.mA; s A/ and uB .mB ; K s A/ Externalities and Property Rights Now suppose we have well-defined property rights. This means we have well-defined endowment points A A .!m ; !s /; B B .!m ; !c /: with A B !s C !c D K: A The meaning is that A is entitled to an amount !s of smoke. Property rights for the externality good are no different from property rights over any other good. To say that Alice owns three pair of shoes is has the same meaning as Alice has the right to smoke 30 cigarettes a day. Alice may very well be willing to trade away from her endowment point and smoke less/more. This means that she may be willing to be "bribed" to smoke less, or maybe Bob is willing to be bribed to put up with more smoke. As long as property rights are well defined, we can consider a competitive market for smoke. Externalities and Property Rights Figure enters here. Externalities and Property Rights The competitive equilibrium will definitely depend on the endowment point, that is, the initial allocation of property rights. The initial allocation of property rights may not be Pareto efficient. In fact, the whole point of assigning property rights is precisely that people can trade away from it. Despite the likely inefficiency of the endowment allocation, the competitive equilibrium will be Pareto efficient. This is just the First Welfare Theorem at force here. Externality problems arise when property rights are not well-defined. "My neighboor thinks he can play the trumpet at 3 o'clock in the morning; I am sure he cannot. I cannot negotiate a price with him, for he thinks he is entitled to playing the trumpet loud at 3am. The only way is to take him to court." Public goods Examples: public defense, sidewalks, bridges, etc. Common feature: everyone must consume the same amount. No individual can decide how much to consume of military tanks! Society must decide on a common amount. When the public good is provided, everyone benefits from it, even though people may value it differently. Public goods are a tricky kind of externality, for one cannot exclude individuals from benefiting from it. Potential for Pareto inefficiency. Public goods 2 roommates, Alice and Bob, want to buy a TV. - Initial wealth levels are mA and mB ; t - their contributions to buy the TV are g A and g B ; - and x A and x B denote the amount of money left for private consumption, so that x i C g i D mi for i D A, B: Utility functions are uA.x A; G/ and uB .x B ; G/ where G 2 f0; 1g is a binary variable that indicates whether or not they have the TV. Finally, the TV costs c , hence GD1 () gA C gB c: Public goods First question we should ask is: Should they buy the TV? To answer this question, consider the notion of reservation price. The reservation price of Alice, r A, is the amount of money that she has to contribute to get the TV that makes her just indifferent between having a TV and not having it: uA.mA r A; 1/ D uA.mA; 0/: The reservation price of Bob is defined similarly. Public goods Thus, buying the TV is part of a Pareto efficient allocation if and only if rA C rB c: Indeed, if the condition is satisfied, then consider any contribution scheme .g A; g B / such that g A C g B D c and r A gA; rB gB : No way to improve Alice without hurting Bob and vice-versa. Exercise: prove the converse. That is, show that if the condition above is violated, then providing the public good cannot be part of a Pareto efficient allocation. Public goods How much of the public good should be provided, in general? Consider now the case where G is not a binary choice, but a continuous choice. That is, G is any number in the interval OE0; 1. Cost of provision C.G/. Utilities: uA.x A; G/ and uB .x B ; G/ Public goods Efficiency requires: x A ;x B ;G max uA.x A; G/ subject to uB .x B ; G/ D uB x A C x B C c.G/ D mA C mB m Public goods Lagrangean: L D uA.x A; G/ C .uB .x B ; G/ uB / C .x A C x B C c.G/ m/ FOC: A M Ux C D 0; B M Ux C D 0; A M UG C M U B C c 0.G/ D 0 Hence, A B M UG = C M UG = Cc 0.G/ D 0 ) jMRS AjCjMRS B j D M C.G/: Public goods Compare with voluntary contribution: max uA.mA gA gA; gA C gB / FOC: A A M Ux C M UG D 0 ) MRS A D 1 Likewise, for B . Thus, jMRS Aj C jMRS B j M C in general. Free rider problem. Public goods Back to binary provision problem: G D 0; 1, cost of provision c . Utilities: ui .G; t i / D vi G ti ; i D 1; : : : ; N; where ti is how much agent i is taxed. Idea that does not work: charge each individual a tax that increases with his valuation of the public good. Problem: ppl will want to pretend they value the public good less than they do, generating inefficient level of provision. Another idea: let us share the cost equally among us; we build the bridge iff the sum of our valuations exceeds the cost of the public good. Problem: incentive to overstate preference for the public good, generating inefficiency again. Vickrey-Clarke-Groves Mechanism Consider the framework of the binary provision model of the previous slide. Here is a "solution" to the public goods problem, known as the VCG mechanism. O Each agent i reports a valuation vi to a trustworthy central agency, henceforth the mechanism. Such report, vi , may or may not be equal to agent i 's actual valuation, vi . O Each agent is free to send any report he wants. In fact, it is infeasible to coerce agents to be truthful, because valuations are private information. So, if agents are to report their valuations truthfully, it is because they will be given incentives to do so. Vickrey-Clarke-Groves Mechanism Based on the reports v1; : : : ; vN , the mechanism decides O O - whether or not to provide the public good, G D 0 or 1; - how much each agent should contribute. The rule is as follows: G D 1 if and only if and X i vi O c: ti .v1 ; : : : ; vN / O O 8 c P O ^ N j i vj ^ ^ c < N P D c ^ j i vj N O ^ ^ : 0 c N P if v O Pj i j if O j i vj P if O j i vj P if O j i vj c N c N c N c N PN < 0; j D1 PN 0; j D1 PN 0; j D1 PN < 0; j D1 vj O vj O vj O vj O c N c N c N c N 0 0 <0 <0 The idea behind the VCG mechanism It looks complicated, but it is not. Let us examine case by case. c O Suppose j D1 vj N 0, so that, according to the agents reports, it is efficient to provide the public good. PN The idea is to split the cost in a way that pivotal agents pay more than the c c even share N , whereas the non-pivotal agents pay exactly N . Who are the pivotal agents in this case? These are the agents without whose report society would have found it efficient to not provide the public good. Thus, if PN j D1 vj O c N 0 then if and only if agent i is pivotal X j i vj O c < 0: N The idea behind the VCG mechanism Hence, when it is efficient (according to the reports) to provide the good, the VCG mechanism taxes each individual the even share c=N plus, on top of that, the pivotal agents (and only them) pay an amount that equals the externality that they impose on the other agents, i.e. X j i vj O c > 0: N Likewise, suppose that it is inefficient (according to the reports) to provide the good, i.e. N X vj O j D1 c < 0: N Who are the pivotal agents in this case? Those are the agents without whose report society would have found it efficient to provide the good, i.e. agent i is pivotal if and only if X j i vj O c N 0: The idea behind the VCG mechanism In this case, the mechanism does not provide the good and the non-pivotal agents are not taxed. But, the pivotal agents--those that revert the decision from provision to no provision--are taxed a positive amount, which equals (again) the externality that they impose on the others: X j i vj O c >0 N The idea behind the VCG mechanism The fundamental property of the VCG mechanism is that it is strategy proof, i.e. every agent finds it in his best interest to report his valuation truthfully. Let us see why. Consider agent i , who has valuation vi . He is going to consider two scenarios: (i) when the reports of the other agents satisfy X j i vj O c < 0: N (i) when the reports of the other agents satisfy X j i vj O c N 0: The idea behind the VCG mechanism Consider the first scenario (i), i.e. Two subcases are possible: c ** First, suppose vi C j i vj N O 0. Then, if agent i reports truthfully, the public good is provided and agent i is considered pivotal; thus, agent i 's utility is c N P j i vj O c N < 0. P vi ti D vi X c C vj O N j i c ; N which is non-negative. Therefore, the agent does not have any incentive to lie and report his value PN c to be some vi vi such that j D1 vj N O O 0, since doing so is going to yield the same provision decision and is not going to affect ti either. The idea behind the VCG mechanism Does he have an incentive to lie and report that his value is some vi vi O PN c O such that j D1 vj N < 0? Well, if he does that, then the public good is not provided and the agent is not considered pivotal, so he gets utility 0 (no public good/no tax), which is less than the utility that he gets when he reports his type truthfully. The idea behind the VCG mechanism c ** Second, suppose vi C j i vj N < 0. Then, if agent i reports O truthfully, the public good is not provided and agent i is considered non-pivotal; thus, agent i 's utility is zero. c N P Therefore, the agent does not have any incentive to lie and report his value PN c O to be some other vi vi such that j D1 vj N < 0, since doing so is O going to yield the same provision decision (i.e. no good) and is not going to affect ti either (i.e. still zero tax). The idea behind the VCG mechanism Does he have an incentive to lie and report that his value is some other PN c O vi vi such that j D1 vj N O 0? Well, if he does that, then the public good is provided and the agent is considered pivotal, so he gets utility vi ti D vi X c C vj O N j i c ; N which is strictly less than 0 in the current case. Thus, the agent has no incentive to misreport his valuation to the mechanism. The idea behind the VCG mechanism To finalize the proof, we still need to consider scenario (ii), i.e. when P c 0. The analysis is similar to the above and is therefore O j i vj N omitted (do it as an exercise). Thus, the VCG mechanism is strategy proof and implements the efficient provision rule (i.e. it provides the good if and only if the sum of the valuations exceeds the cost). Clearly, the amount of revenue raised by the mechanism exceeds the cost c when the good is provided, since everyone is contributing with at least c=N . The "problem" with the VCG mechanism is that it raises too much revenue: to provide incentives, the pivotal agents pay strictly more than c=N when the good is provided; and strictly more than 0 when the good is not provided. It turns out that this budget surplus is an inefficiency that cannot be avoided. That is, there is no mechanism that implements the efficient provision rule and is both strategy proof and budget-balanced. This is an important drawback of the VCG mechanism. Although not used in practice, the ideas behind the VCG mechanism have had tremendous impact in Economic Theory, particularly in problems concerning the allocation of private goods (auction theory). ...
View Full Document

This note was uploaded on 03/10/2012 for the course ECON 121 at Yale.

Ask a homework question - tutors are online