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Lecture Notes: Public Economics

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Lecture Notes: Public Economics Lecture Notes: Public Economics Fall 2018 Contents I Public Goods 6 0.1 Introduction . 7 0.1.1 What are Public Goods? . 7 0.2 The Model . 7 0.3 Optimal Provision of Pure Public Good . 8 0.4 Can the Optimal Allocation be Decentralized? . 12 0.5 Lindahl Equilibria . 14 0.5.1 Is Lindahl Equilibrium a Reasonable Market Mechanism? . 16 0.5.2 Nash Implementation . 20 0.6 Positive Models of Private Provision of Public Goods . 23 0.6.1 A Static Model of Private Contributions . 24 0.6.2 Comparative Statics Regarding Changes in the Wealth Distribution 30 0.6.3 Multiple Public Goods and Spheres of Influence . 32 0.7 Does Public Provision Crowd Out Private Provision? . 33 0.7.1 Guide to the Recent Literature . 34 0.8 Dynamic Voluntary Provision of Public Goods . 40 0.8.1 Admati and Perry (ReStud, 1991) . 40 0.8.2 Marx and Matthews (ReStud 2000) . 44 0.9 Provision of Public Goods with Private Information . 46 0.9.1 An Illustrative Example: two agent case . 46 0.9.2 Impossibility Result for Large Economies (Mailath and Postlewaite, REStud (1990) . 48 0.10 Local Public Goods . 51 0.10.1 A Class of Tiebout Models (Bewley Econometrica 1981) . 52 2 CONTENTS 3 0.10.2 Are Tiebout Equilibria Pareto Effi cient? . 54 0.10.3 A Tiebout Model with Effi cient Equilibrium . 57 0.10.4 Empirical Studies Related to Tiebout Hypothesis . 60 0.11 Public Goods Bibliography . 62 II Social Arrangements 63 0.12 A model incorporating social arrangements: . 64 0.12.1 Model . 64 0.12.2 Related empirical work . 70 0.13 Applications of concern for rank . 71 0.13.1 Effort choice . 71 0.13.2 Conspicuous consumption . 72 0.13.3 Investment: “Investment and concern for relative position,”(with H. Cole and G. Mailath) Review of Economic Design 6, 241—261 (2001). 75 0.14 Social Assets Mailath and Postlewaite IER (2006) . 77 0.14.1 Model:.................................. 77 0.14.2 Social Arrangements Bibliography . 82 III Public Choice (Silverman) 90 0.15 Introduction . 91 0.16 Models of Political Competition . 91 0.16.1 The Downsian Model of Political Competition . 91 0.16.2 Criticisms and Weaknesses with the Downsian Model . 94 0.16.3 Probabilistic Voting Models (minor footnote) . 94 0.16.4 Supermajorities (interesting footnote) . 96 0.16.5 Citizen Candidates (important class of models) . 97 0.16.6 Equilibrium in Citizen Candidate Model . 98 0.17 Comparing Political Institutions . 101 0.17.1 Lizzeri and Persico (AER 2001) . 102 0.17.2 Diermeier, Eraslan & Merlo (Econometrica 2003) . 104 0.17.3 Literature . 105 4 CONTENTS IV Discrimination and Affi rmative Action 106 0.18 Theoretical Models of Discrimination . 107 0.18.1 Introduction .............................. 107 0.19 Taste-Based Discrimination . 107 0.20 Statistical Discrimination . 108 0.20.1 Phelps (AER, 1972) . 109 0.20.2 Arrow (1973) . 110 0.20.3 Coate and Loury (AER, 1993) . 113 0.21 Discrimination Due to Inter-Group Interactions . 118 0.21.1 Moro and Norman J. Public Econ (2003) . 118 0.21.2 Mailath, Samuelson and Shaked (AER, 2000) . 119 0.21.3 Eeckhout (REStud, 2006) . 121 1 Affi rmative Action 123 1.1 Origins of Affi rmative Action . 123 1.2 Theoretical Studies of the Effect of Affi rmative Action . 125 1.2.1 Coate and Loury’sPatronizing Equilibrium . 125 1.2.2 Moro and Norman (2001) . 129 1.2.3 Fang and Norman (2001) . 130 1.2.4 Fryer (2010) . 132 1.3 Evaluation of the Effects of Affi rmative Action . 132 1.3.1 Donohue and Heckman (1991) . 132 1.3.2 Moro (2001) . 133 V Welfare Reform and Social Security 135 2 Welfare Reform 136 2.1 Some Institutional Details . 136 2.2 Welfare Reform Bill of 1996 . 139 2.3 Incentive Effects of the Welfare System (Moffi tt 1992) . 140 2.4 Workfare versus Welfare: Besley and Coate (1992) . 140 2.5 A Proposal for Welfare Reform (Keane 1995) . 146 2.6 Is Time-limited Welfare Compassionate? (Fang and Silverman 2001) . 147 CONTENTS 5 Acknowledgement These notes began from notes written by Hanming Fang and Dan Silverman. I have edited them substantially in several places and added entirely new sections. They should not be held responsible for errors or omissions. Part I Public Goods 6 0.1. INTRODUCTION 7 0.1 Introduction 0.1.1 What are Public Goods? A good is called a pure public good if “each individual’sconsumption of such a good leads to no subtraction from any other individual’s consumption” (Samuelson 1954, p387) This is commonly referred to as non-rivalry in use. There are two important areas of economics in which public goods play an important role. The first is the case of spending on things such as national defense: the cost of providing a missile defense system is independent of the number of people inhabiting the protected area, and it is impossible to defend some but not all the inhabitants. This is the prototypical example used to motivate the role of government in providing such public goods. Historically, this was the set of problems that motivated the interest in public goods. While interesting, more recently a second class of problems in which public goods play an important role is family economics. A married couple jointly consumes many goods and shares the cost of those goods. How a couple decides on the amount of time spent on child rearing and how much each contributes to that amount is central to understanding child development, labor force participation, the degree of matching assortivity and probabilities of divorce. Chapters 2 and 3 in Browning, Chiappori and Weiss (2014) are a good reference to this area. We will primarily consider the simplest case with a single private good and a single public good. 0.2 The Model n consumers, indexed by i = 1, ..., n • xi : agent i’sconsumption of private good and denote x = (x1, ..., xn) as the vector of • private consumption G: the (common) consumption of public good • Agent i’spreference described by the utility function • ui (xi,G) 8 which is differentiable and increasing in both arguments, quasi-concave and satisfies Inada Condition wi : agent i’sendowment of private good and • n W = wi i=1 X is the total endowment of private good; and public good endowment is taken to be zero Public good may be produced from the private good according to a production func- • tion f : R+ R+ where f > 0 and f < 0. That is, if z is the total units of private ! 0 00 goods that are used as inputs to produce the public good, the level of public good produced will be G = f (z) . 0.3 Optimal Provision of Pure Public Good We first ask the normative question of what is the optimal level of pure public good. We assume that the government of a fully controlled economy chooses the level of G, and the allocation of private goods x = (x1, ..., xn) to agents according to the Pareto criterion. Definition 1 An allocation (x,G) Rn+1 is feasible if there exists some z 0 s.t. 2 + ≥ n xi + z W ; • i=1 ≤ GP f (z) . • ≤ Alternatively we could define that an allocation (x,G) Rn+1 is feasible if 2 + n 1 xi + f (G) W. ≤ i=1 X Definition 2 A feasible allocation (x,G) is Pareto optimal if there exists no other feasible allocation (x0,G0) s.t. ui x0 ,G0 ui (xi,G) i = 1, ..., n i ≥ 8 and for some i 1, ..., n , 2 f g ui xi0 ,G0 > ui (xi,G) . 0.3. OPTIMAL PROVISION OF PURE PUBLIC GOOD 9 That is, a feasible allocation (x,G) is Pareto optimal if there is no way of making an agent strictly better off without making someone else worse off. Now we can characterize the set of Pareto optimal allocations. It is the solution to the following problem: max u1 (x1,G) x,G,z f g s.t. ui (xi,G) ui 0 for i = 2, 3, ..., n, (multiplier i) ≥ n W xi z 0 (multiplier ) ≥ i=1 X f (z) G 0 (multiplier ) ≥ G 0, z 0 and xi 0 for all i = 1, ..., n ≥ ≥ ≥ where ui are treated as parameters of the problem. Inada conditions on the utility function implies that the non-negativity constraints can be ignored. The necessary and suffi cient (suffi ciency due to quasi-concavity assumption on u and f) Kuhn-Tucker conditions are: @ui (xi,G) (xi : ) i = 0 (1) @xi n @u (x ,G) (G : ) i i = 0 i @G i=1 X (z : ) + f 0 (z) = 0 where we have set 1 = 1 by convention. From the fist n equalities, we obtain i = . @ui (xi,G) /∂xi From the last equality, we obtain = f 0 (z) Plugging these n + 1 equalities into the the middle condition regarding G, we get n @u (x ,G) /∂G 1 i i = . (2) @u (x ,G) /∂x f (z) i=1 i i i 0 X This condition is referred to as the Samuelson condition, the Lindahl-Samuelson condition, or sometimes even the Bowen-Lindahl-Samuelson Condition and is probably familiar to anyone who have taken an intermediate course in public economics. 10 Interpretations: The left hand of equation (2) is the sum of the marginal rates of sub- stitutions of the n agents. To see this, note that from agent i’s indifference curve, the term @ui (xi,G) /∂G @ui (xi,G) /∂xi denotes the quantity of private good agent i is willing to give up for a small unit increase in the level of the public good.
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