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Essays on Optimal Feasible Taxation The Pennsylvania State University The Graduate School Department of Economics ESSAYS ON OPTIMAL FEASIBLE TAXATION A Thesis in Economics by Byungchae Rhee °c 2004 Byungchae Rhee Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2004 The thesis of Byungchae Rhee was reviewed and approved∗ by the following: J. Tomas Sj¨ostr¨om Professor of Economics Thesis Adviser Chair of Committee Kalyan Chatterjee Distinguished Professor of Economics and Management Science James S. Jordan Professor of Economics Steven J. Huddart Associate Professor of Accounting Robert C. Marshall Professor of Economics Head of the Department of Economics ∗ Signatures are on file in the Graduate School. Abstract In the first essay, we study the following question: If the provision of public good is financed by property (wealth or endowment) taxes, what is the optimal tax mechanism when agents have private information about his own endowment? Employing the state-dependent implementation model, under incentive compatibility and feasibility constraints, we provide the full characterization of an optimal tax mechanism with two agents and its properties. Some interesting comparative statics analyses are also provided. For the general n-agent case, some partial characterization results are obtained. In addition, we fully characterize the optimal tax mechanism for the corresponding infinitely large economy. The main goal of the second essay is to extend the model of the first essay to a heterogeneous economy in which agents have some different wealth characteristics that are publicly observable, e.g., race. Using the same analysis as in the first essay where a homogeneous economy is studied, we provide the full characterization of an optimal tax mechanism for such a heterogeneous economy with two agents and its properties. In addition to the similar results in the first essay, we find that if the level of low endowment is low enough, only the incentive compatibility constraint of a minority agent is binding. The third essay studies the optimal formation of local public good economies and the immigra- tion incentive of an agent. Using the characterizations of optimal tax mechanisms in the first and second essays, we compare the social welfare between homogeneous formation for which each local community consists of agents with same observable characteristics, e.g., race, and heterogeneous for- mation for which each local community consists of agents with different observable characteristics. The comparison shows that if the expected endowment of the economy is low enough, homoge- neous formation is optimal, while otherwise heterogeneous formation is optimal. We also study the immigration incentive and find that a minority (resp. majority) agent will choose a heterogeneous (resp. homogeneous) formation regardless of his endowment. iii Table of Contents List of Figures vii List of Tables viii Acknowledgements ix I A Characterization of Optimal Feasible Tax Mechanism 1 I.1 Introduction.................................... ..... 1 I.2 TheModel ........................................ 4 I.2.1 TheEconomy.................................... 4 I.2.2 TheTaxMechanism ............................... 5 I.2.3 TheSocialPlanner’sProblem. ...... 7 I.3 Optimal Tax Mechanism for n =2 ............................ 8 I.3.1 PossibilityofFirstBestTaxation . ......... 9 I.3.2 SecondBestTaxSchedule. 10 I.3.3 SimulatedExamples ............................. 17 I.4 PropertiesandComparativeStatics . .......... 18 I.4.1 PropertiesofOptimalTaxSchedules . ........ 18 I.4.2 ComparativeStatics . 20 I.4.2.1 Responses of t∗ to p ........................... 21 ∗ I.4.2.2 Responses of t to ωL or ωH ...................... 23 I.4.2.3 Expected Total Provision of Public Good . ..... 26 I.5 Optimal Tax Mechanism for n> 2 ............................ 27 I.5.1 PossibilityofFirstBestTaxation . ......... 27 I.5.2 SecondBestFeasibleTaxSchedule . ....... 29 I.5.3 Infinitely Large Economy: n = ∞ ......................... 30 iv I.6 ConcludingRemarks ............................... ..... 31 II Optimal Feasible Tax Mechanism for a Heterogeneous Economy 33 II.1 Introduction................................... ...... 33 II.2 TheModel ....................................... 34 II.3 optimalTaxMechanism . ...... 35 II.3.1 PossibilityofFirstBestTaxation . .......... 35 II.3.2 SecondBestTaxSchedule . ..... 35 II.3.3 SimulatedExamples . 36 II.4 PropertiesandComparativeStatics . ........... 39 II.4.1 PropertiesofoptimalTaxSchedules . ......... 39 II.4.2 ComparativeStatics . ..... 39 II.4.2.1 Responses of t∗ to p and q ....................... 39 ∗ II.4.2.2 Responses of t to ωL or ωH ...................... 44 II.4.2.3 Expected Total Provision of Public Good . ...... 44 II.5 ConcludingRemarks .............................. ...... 44 IIIOptimal Formation and Immigration of Local Public Goods Economies 45 III.1Introduction.................................. ....... 45 III.2TheModel ...................................... 47 III.2.1TheEconomy.................................. 47 III.2.2 Optimal Feasible Tax Mechanisms . ........ 48 III.2.2.1 HomogeneousFormation . 49 III.2.2.2 HeterogeneousFormation . 49 III.3WelfareComparison . ....... 50 III.4ImmigrationIncentives. .......... 55 III.5ConcludingRemarks . ....... 58 A Appendix 63 A.1 DerivationofEquation(I.7) . ......... 63 A.2 DerivationofEquation(I.8) . ......... 63 A.3 ProofofLemmaI.5.1............................... ..... 64 v B Appendix 65 B.1 Expressions of ∆W ..................................... 65 B.1.1 tLL ≧ ωL and [ωL + ωH ≦ 1] .......................... 65 B.1.2 £tLL ωL¤ and [ωL + ωH > 1] .......................... 66 e ≧ B.1.3 £tLL < ωL¤ ..................................... 66 e B.2 Expressions£ of ∆U¤ρ(ωk) for ρ = p, q and k = L,H .................... 67 e B.2.1 Immigration Incentives of q-type Agent with ωL ................ 67 B.2.1.1 tLL ≧ ωL and [ωL + ωH ≦ 1] ..................... 67 B.2.1.2 £tLL ωL¤ and [ωL + ωH > 1] ..................... 68 e ≧ B.2.1.3 £tLL < ωL¤ ................................ 68 e B.2.2 Immigration£ Incentives¤ of q-type Agent with ωH ................ 68 e B.2.2.1 tLL ≧ ωL and [ωL + ωH ≦ 1] ..................... 68 B.2.2.2 £tLL ωL¤ and [ωL + ωH > 1] ..................... 69 e ≧ B.2.2.3 £tLL < ωL¤ ................................ 70 e B.2.3 Immigration£ Incentives¤ of p-type Agent with ωL ................ 70 e B.2.3.1 tLL ≧ ωL and [ωL + ωH ≦ 1] ..................... 70 B.2.3.2 £tLL ωL¤ and [ωL + ωH > 1] ..................... 71 e ≧ B.2.3.3 £tLL < ωL¤ ................................ 71 e B.2.4 Immigration£ Incentives¤ of p-type Agent with ωH ................ 71 e B.2.4.1 tLL ≧ ωL and [ωL + ωH ≦ 1] ..................... 71 B.2.4.2 £tLL ωL¤ and [ωL + ωH > 1] ..................... 72 e ≧ B.2.4.3 £tLL < ωL¤ ................................ 72 e £ ¤ e vi List of Figures I.1 PossibilityofFirstBestTaxation . ........... 11 F I.2 Partition of Ω1 \ Ω by (T, tLL).............................. 15 I.3 ExamplesofOptimalTaxSchedules . ........ 19 e e ∗ ∗ I.4 Responses of tLL and tHL to p .............................. 24 ∗ ∗ I.5 Responses of tLL and tHL to ωL and ωH ......................... 25 I.6 Expected Total Provision of Public Good: E(y)..................... 26 II.1 Examplesofsecondbesttaxation . .......... 37 ∗ ∗ II.2 Responses of tLL and tHL to p and q ........................... 40 ∗ ∗ II.3 Responses of tLL and tHL to ωL and ωH ......................... 41 II.4 Expected Total Provision of Public Good: E(y)..................... 42 III.1 Welfare Comparisons: ωH = 0.8.............................. 52 III.2 Welfare Comparisons: ωH = 1.3.............................. 53 III.3 Welfare Comparisons: ωH = 2.5.............................. 54 III.4ImmigrationIncentives. .......... 57 vii List of Tables I.1 Optimal Tax Schedules for n =2.............................. 17 I.2 A summary of the responses of t∗ to p........................... 22 II.1 Optimal Tax Schedules for a Heterogeneous Economy. .............. 36 II.2 Responses of t∗ to p and q. ................................ 43 III.1CasesforWelfareComparisons . .......... 51 III.2 Cases for the Comparisons of Immigration Incentive . ................ 56 viii Acknowledgements First of all, I am very grateful to Professor Tomas Sj¨ostr¨om, my advisor, for his invaluable guidance, comments, and patience. I am indebted to my dissertation committee members, Professors Kalyan Chatterjee, Steven Huddart, and James Jordan. I also thank Neil Wallace for his encouragement. My special gratitude belongs to Professor Emeritus John Riew for his encouragement and guidance. Finally, I wish to express my deepest love to my wife, Sungran, and daughter, Jane. ix Chapter I A Characterization of Optimal Feasible Tax Mechanism I.1 Introduction This essay is motivated by a practical property (wealth or endowment) taxation problem:1 For an economy where the provision of public goods is to be financed by property taxes collected from agents, what is the optimal tax mechanism when a social planner does not know the agents’ property?2 What is important in this mechanism design problem is that we have to take into account not only the agents’ incentive to report their wealth truthfully, but also the feasibility of the designed tax schedule in the sense that each agent’s tax payment should be within their ability to pay. To see the problem of feasibility, consider the following simple example. Two thieves, Ali and Baba, want to build a door for their treasure cave. The quality of the door depends on the
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