A Characterization of Optimal Feasible Tax Mechanism

Transcription

1 A Characterization of Otimal Feasible Tax Mechanism Byungchae Rhee Deartment of Economics Pennsylvania State University May, 24 Abstract In this aer, we study the following question: For a ublic good economy where the rovision of ublic goods is to be financed by roerty taxes collected from individuals, what is the otimal feasible tax mechanism when a social lanner is relatively uninformed of the roerties of the individuals? Using a Bayesian model, we rovide the full characterization of the otimal feasible tax mechanism with two agents and its roerties. We find that (i when the exected total endowment of the economy is relatively low enough or high enough, the incentive comatibility constraint does not bind so that first best taxation can be obtained; (ii the second best feasible tax mechanism requires a oor agent to ay relatively more than a rich agent, that is, it is regressive; and (iii the otimal feasible tax mechanism is increasing in the sense that the agent s tax ayment increases with his endowment. For the case of more than two agents, under certain mild assumtions we give some artial results similar to (i and (ii above. In addition, we find the otimal feasible tax mechanism for the corresonding infinitely large economy. Keywords: otimal taxation, feasibility, incentive comatibility, informational rent, second best JEL classification: H2, D7, D82 Introduction This aer is motivated by a ractical roerty or income taxation roblem: For a ublic good economy where the rovision of ublic goods is to be financed by roerty taxes collected from individuals, what is the otimal feasible tax mechanism when a social lanner is relatively uninformed of the roerties of the individuals? In this case, the roblem is that the social lanner has to take into account not only the individuals incentive to reort their wealth truthfully, but also the individual feasibility of the designed tax schedule in the sense that each individual s tax ayment I thank Professor Tomas Sjöström, my advisor, for his invaluable guidance and comments. I also thank Professors Kalyan Chatterjee, Steven Huddart, James Jordan, and Neil Wallace for their helful comments. Of course, all remaining errors are mine. Contact Information: 68 Kern Graduate Building, University Park, PA brhee@su.edu Some authors use the term interim efficient instead of otimal to emhasize the informational structure of their model. In this aer, we choose otimal and use second best if necessary to distinguish such a situation where the otimal tax mechanism is not first best. Also, throughout the aer, we will use the terms wealth, endowment, roerty or income of an agent interchangeably since they have the same meaning, the amount of resources the agent has initially.

2 should be consistent with their ability to ay. In articular, this kind of roblem, otimal rivate rovision of ublic goods, is frequently observed in a small economy such as a club or village. 2 Consider the following examle. Two thieves, Ali and Baba, want to build a door for their treasure cave. The quality of the door deends on the total contributions they make. Suose that Ali is relatively rich and has $2, and Baba is relatively oor and has $, but that none of them knows how much the other has. A social lanner, who does not know how much Ali and Baba have, asks them to reort their wealth in order to determine their contributions (taxes. What is the otimal feasible tax mechanism that maximizes the exected sum of utilities? If the social lanner wants to collect $3 for the door-building, she cannot imose $5 to each of the two thieves because it is not (individually feasible to Baba. The theory of otimal taxation has a long history. Since the seminal work by Mirrlees [97], the otimal taxation literature has studied the incentive asect of a tax mechanism and established many characterization results under a variety of economic situations. 3 Mirrlees [97] considers a labor income taxation roblem for an infinitely large economy and studies the otimality of redistributive taxation when each individual has rivate information about his own roductivity. He shows that the redistributive tax mechanism is subject to subotimality due to the informational asymmetry between the ublic olicy authority and the individuals. This situation is now well understood as a second best taxation. Following Mirrlees [97], many authors have analyzed a trade-off between efficiency and equity of otimal income taxation. Other authors have studied the otimal taxation roblem from the imlementation ersective (Guesnerie [995] and Piketty [993] among others. However, most of these works have assumed that there is a continuum of individuals and the tax schedule deends on an observable variable such as labor income so that there are no individual feasibility or bankrutcy roblems. Imlementation (or mechanism design theory, ioneered by Hurwicz [972] and Maskin [999], studies the imlementability of various social choice rules and the characterization of the imlementing mechanisms under different environments and informational assumtions. 4 Most of results in this literature, however, assume that the set of feasible outcomes is fixed and common knowledge so that this set does not deend on the realization of the economic environment. This is a quite restrictive assumtion, in articular, when agents have rivate information about their own endowments or roduction technologies. If a social lanner is relatively uninformed of the realization of agents endowments, she has to consider the feasibility roblem when designing an imlementing mechanism. The first study to exlicitly tackle this tye of feasibility roblem is Hurwicz, Maskin and Postlewaite [995]. 5 They considered the feasible imlementation roblem under comlete information in which a social lanner does not know the realization of agents endowments or roduction sets. Following them, there have been some extensions of their model to incomlete information cases. 6 However, those results have mainly focused on the imlementability 2 Fund-rasing is another examle of this roblem. See, for examle, Andreoni [998]. 3 For a survey of modern otimal taxation theory, see Stiglitz [985]. 4 For some recent surveys, see, e.g., Jackson [2a, b], Palfrey [22] and Maskin and Sjöström [22]. 5 The earlier version of this aer has been circulated since See for examle Hong [996, 998] and Tian [999]. See also Dagan, Serreno, and Volij [999], in which they study the feasible imlementation of a given taxation method which embodies the socially otimal tax level. However, their work considers the taxation roblem from the equity oint of view so that the total amount of taxes to be collected is exogenously given. In contrast, our model deals with the efficiency of a tax mechanism which endogenously determines the each agent s tax ayment as well as the total amount of taxes. 2

3 of a general social choice rule, but not on the efficiency of the imlementing mechanism. Such an efficiency roblem has been a major subject in otimal taxation theory. In this aer, we emloy the so-called endowment game created by Hurwicz, Maskin and Postlewaite [995] to model the otimal feasible taxation roblem of a ublic good economy with a finite number of agents. That is, using a Bayesian model, we set u the maximization roblem of a utilitarian social lanner who is relatively uninformed of the agents endowments. Since the number of agents in the economy is finite, each agent s tax ayment will be affected not only by his own endowment not also by the others endowments. We first consider the case of two agents and two otential tyes, and fully characterize its solution. Also, we can illustrate this solution grahically due to the low dimensionality of the roblem. The four main results of this aer are as follows. First, if the exected total endowment of the economy is relatively low enough or high enough, then first best feasible taxation can be obtained. This result is due to the fact that the incentive comatibility constraint does not bind at the corresonding first best feasible tax schedules when the economy is relatively oor or rich. Second, for the cases in which the incentive comatibility constraint does bind, the otimal feasible tax mechanism imoses a high tax rate on a oor agent when his neighbor is rich. The intuition behind this regressive taxation is that levying a tax on the oor agent does not cause an incentive roblem so that the social lanner, who does not mind which agent ays how much roortion of the total taxes, refers to imose as much tax as ossible on the oor agent rather than his rich neighbor who may request an informational rent as a reward for the revelation of his tye. Third, the otimal feasible tax schedule is increasing in the sense that the tax ayment of an agent is increasing in his endowment. Fourth, we conduct a comarative statics analysis on how the otimal feasible tax mechanism resonds to a change in the initial arameter values. We first study the resonses to a change in the robability distribution of endowment analytically, and then to a change in endowment arameters by means of simulation. In essence, these analyses show how each agent s tax ayment deends on the incentive comatibility constraint and the relative size of low endowment. As a natural extension, we consider the case of more than two agents. Although it is imossible to fully describe the otimal feasible tax mechanism for this case due to its high dimensionality and abundance of corner solutions, we obtain some artial results similar to those of the two-agent case under certain mild assumtions. In addition, we find the otimal feasible tax mechanism for the corresonding infinitely large economy, where the tax ayment of an agent is always equal to the amount of the low endowment. Finally, we would like to mention the two features of our model that distinguish it from the revious literature on ublic goods. Our model considers the continuous rovision of ublic goods under incomlete information. There is a huge literature on mechanism design and ublic economics which analyzes ublic good economies. However, most of the models in this literature have dealt with the discrete (in fact, binary rovision of ublic goods. 7 Although this discreteness makes the models mathematically simle and tractable, 8 it is a restrictive assumtion. In our model, the rovision of ublic goods is continuous because it is determined directly by the total amount 7 See, for examle, D Asremont and Gérard-Varet [979], Laffont and Maskin [979], and Gradstein [994]. See also Groves and Ledyard [987]. 8 One of many advantages that the discreteness assumtion brings about is to make the individual utility deend linearly on the rovision of ublic goods. 3

4 of taxes according to a constant returns to scale technology. 9 Another feature of our model is the direct linkage between taxes and level of rovision of ublic goods. For standard models in the imlementation literature, for examle D Asremont and Gérard-Varet [979] and Laffont and Maskin [979], there are little such a linkage, but mainly redistributive transfers among the agents, which are used to resolve incentive roblems. To this oint, our model is closely related the theory of rivate rovision of ublic goods where collected taxes and level of rovision of ublic goods are exlicitly related. The remainder of this aer is organized as follows. In Section 2, we resent the model for a ublic good economy. In Section 3, we fully characterize the otimal feasible tax schedule for the economy with two agents and two ossible tyes. Using the characterization results, in Section 4, we discuss the roerties of the otimal mechanism and rovide some comarative statics analyses. As an extension, we consider the case of more than two agents in Section 5. In Section 6, we give concluding remarks and future research agenda. 2 The Model 2. The Economy Consider a ublic good economy with n agents, 2 n <. Let N = {,...,n} denote the set of agents. There is one rivate good x R + and one ure ublic good y R +, where the rivate good can be used to roduce the ublic good according to a constant returns to scale technology. Without loss of generality, we normalize the roduction technology such that one unit of rivate good can be transformed into one unit of ublic good. Each agent i N has the same quasilinear von Neumann-Morgenstern utility function u on R 2 +, u(x i, y = log y + x i, where x i is the consumtion of rivate good by agent i. Initially, each agent i is endowed with rivate good ω i {, } only, where < <. Agent i is called oor when ω i = and rich when ω i =. Let Ω = {(, R 2 + : < } denote the set of all ossible airs of initial endowments. The information structure of this economy follows a standard incomlete information (Bayesian model. The rimitives of the economy are common knowledge, whereas each agent has rivate information about his own endowment. That is, agent i knows the realization of his own endowment ω i and the initial robability distribution of the other agents endowments, but does not know 9 Ledyard and Palfrey [999] emloy a model that allows continuous ublic goods rovision. However, under their assumtions of linear roduction and risk-neural references, it is equivalent to a discrete one. Also, Bergstrom, Blume, and Varian [986] study a continuous model, but their model assumes comlete information. The case where there is only one agent in the economy is trivial. The case where n = will be discussed in Section 5.3. In this aer, each agent s initial endowment may be interreted as a ortion of his total wealth above the subsistence level. Thus, it can be called the agent s taxable wealth for the rovision of ublic goods. This interretation will be made clear in Section 4.. 4

5 the realizations of the other agents endowments ω i. 2 Agents endowments are indeendently distributed according to Pr(ω i = = (, i N. Thus, an economic environment is equivalent to the realization of ω = (ω,...,ω n. 2.2 The Tax Mechanism A tax mechanism consists of message saces M i for each agent i N, and an outcome function f which mas each message rofile m M n i= M i into agents tax burdens t(m = (t (m,..., t n (m R n + and ublic good roduction y; f : m (t(m, y(m. The constant returns to scale technology imlies that y(m n i= t i(m for all m M, but without loss of generality, we can assume that the equality always holds since no taxes will be wasted. 3 Hence, we have the following simle definition. Definition 2. (Tax Mechanism and Schedule A tax mechanism Γ is defined as Γ = M, t, where t : M R n + is called a tax schedule. Given a tax mechanism Γ = M, t, let s i : {, } M i denote the strategy (reort of agent i. By the Revelation Princile (see Myerson [979], we are able to restrict our attention to a direct incentive comatible tax mechanism. Thus, we assume that M i = {, } for each i N. The exected utility of agent i when his endowment is ω i and he reorts s i, assuming the other agents are truthful, is n ωi U i (s i ω i, t = E ω i u i (ω i t i (s i, ω i, t j (s i, ω i j= n = E ω i log t j (s i, ω i + j= ( ωi ω i t i (s i, ω i. In this aer, we make two assumtions which a tax mechanism should satisfy. First, following Hurwicz, Maskin, and Postlewaite [995], we emloy the no exaggeration assumtion. Assumtion 2.2 (No Exaggeration For each i N, s i (ω i ω i. That is, no agent is allowed to overstate his endowment when reorting. 4 This assumtion artially relieves the informational disadvantage of the social lanner. Another assumtion is the anonymity of a tax mechanism: A tax schedule should not be affected by the change of agents names. 5 More secifically, this includes two conditions. First, an agent s tax ayment should not be affected by 2 Notational convention alies here, that is, given a vector a = (a,..., a n A = n i= Ai, a i = (a,..., a i, a i+,..., a n A i = j i A j, and a = (a i, a i. 3 This roerty may be viewed as a budget-balancedness. Comare with the Clarke-Groves mechanism where budget-balancedness is usually not satisfied, see Clarke [97] and Groves [973]. 4 It may be assumed that each agent is asked to ut his reort on the table. 5 This assumtion must hold for every society where taxation is based on a democratic rocess. 5

6 the change of order in the other agents reorts as long as the distribution of their reorts remains the same. Second, any two agents tax ayments should be the same if they reort the same endowment with other things being equal. Formally, Assumtion 2.3 (Anonymity For all i, j N, i. t i (s i, s i = t i (s i, σ(s i, ii. s i = s j = t i (s i, s = t j (s j, s s {, } n, where σ(s i is a ermutation of s i. Under the anonymity assumtion, let t L,(n kl,(k H denote an agent s tax ayment when he and (n k of the other agents reort and the remaining (k agents reort, k =,...,n. Define t H,(n kl,(k H similarly. Then, we can exress a tax schedule as t = ( (t L,(n kl,(k H n k=, (t H,(n kl,(k H n k=. Since we are considering a direct mechanism, we simly identify a (direct tax mechanism Γ = M, t with a tax schedule t in this aer. To state the social lanner s roblem, we need to look at three roerties that a tax mechanism should satisfy: Feasibility, Incentive Comatibility, and Individual Rationality. First, feasibility, one of the most imortant concets in this aer, imlies that no tax mechanism should imose more than the announced endowment. 6 That is, Definition 2.4 (Feasibility A tax mechanism t is feasible if for all k =,..., n, t L,(n kl,(k H and t H,(n kl,(k H. Throughout this aer, we require all tax mechanisms considered to be feasible. Second, by the Revelation Princile, we consider an incentive comatible tax mechanism only. Definition 2.5 (Incentive Comatibility: IC A tax mechanism t is (Bayesian incentive comatible if for all i N, U i (, t U i (, t. ( Note that due to the no exaggeration assumtion, the incentive comatibility of a tax mechanism for this economy is just one-directional; the inequality U i (, t U i (, t is meaningless. Third, to make the agents articiate in this ublic good economy, we need to make assumtions as to what will haen if an agent does not articiate. Notice that we have to distinguish between the situations in which an agent does not want to reort his endowment and in which an agent wants to leave the economy or denies to ay the imosed tax. In the former case, we assume that the social lanner can imose a tax on the agent as if he were to reort that his endowment is. 7 Under this assumtion, the exected utility of agent i who did not reort is U i (, t if his endowment is, or U i (, t if his endowment is. Since only incentive comatible 6 In this sense, the feasibility can also be called no-bankrutcy. 7 This kind of tax enforcement scheme seems well established in reality. Imlicit or exlicit membershi fee in a club is one of the examles. 6

7 tax mechanisms are considered, agent i who reorts his endowment will obtain U i (, t if his endowment is, or U i (, t if his endowment is. Thus, by inequality (, every agent will reort his endowment, which makes the individual rationality condition redundant in this model. In the latter case, we assume that the social lanner can revent the agent from enjoying ublic good by, for examle, exulsion from the economy. 8 Under this assumtion, the exected utility of the agent is u(x i, =, x i R +. Thus, the individual rationality constraint becomes redundant, too. As a result, we can ignore individual rationality by the above two assumtions. Finally, we add one more definition for a tax mechanism. Definition 2.6 (Increasingness A tax mechanism t is increasing if for all k =,..., n, t L,(n kl,(k H t H,(n kl,(k H. That is, a tax mechanism is increasing if an agent s tax ayment is increasing with his endowment. 2.3 The Social Planner s Problem The social lanner (or tax authority who does not know the true realization of the economic environment, but knows the robability distribution, wants to find an incentive comatible and feasible tax schedule t = ( (t L,(n kl,(k H n k=, (t H,(n kl,(k H k= n which maximizes the exected sum of agents utilities. Formally, given (, Ω and (,, the social lanner s roblem is 9 [ n ] (P n 8 9 (P n max t W(t; = E subject to i= Cancellation of club membershi may be an examle. Exlicitly, max t W(t; = n j= U i (ω i ω i, t (IC U i (, t U i (, t i N, (Feasibility t B(, [, ] n [, ] n. nc j j ( n j[ nlog (jt L,(j L,(n jh + (n jt H,(j L,(n jh ] (jt L,(j L,(n jh + (n jt H,(j L,(n jh + n ( + ( subject to (IC (Feasibility n n C j j ( n j[ ( log jt L,(j L,(n jh j= ] +(n jt H,jL,(n jh t H,jL,(n jh n n C j j ( n j[ ( log (j + t L,jL,(n jh j= +(n jt H,(j+L,(n 2 jh t L,jL,(n jh ], t L,(n kl,(k H, t H,(n kl,(k H, k =,..., n, where nc j is the number of ways of choosing j unordered outcomes from n ossibilities. That is, nc j = ( n j = n!. (n j!j! 7

8 Notice that only one (IC constraint is binding. For notational simlicity, given (,, define a function : R 2n + R R {, + } by (t; = U i (, t U i (, t. Then, a tax schedule t satisfies (IC if (t;. 3 Otimal Feasible Tax Mechanism for n = 2 In this section, we study the otimal feasible tax mechanism for the case of two agents. For n = 2, a tax mechanism t can be written as t = (t LL, t LH, t HL, t HH, where, for examle, t LH is the tax ayment of an agent when he reorts and the other agent reorts. The social lanner s roblem now becomes: Given (, Ω and (,, (P 2 max t W(t; = 2 [2 log(2t LL 2t LL ] + 2( [2 log(t LH + t HL (t LH + t HL ] + ( 2 [2 log(2t HH 2t HH ] + 2( + ( subject to [log(t LH + t HL t HL ] + ( [log(2t HH t HH ] (IC [log(2t LL t LL ] + ( [log(t LH + t HL t LH ], (Feasibility t LL, t LH, t HL, t HH. Note that for n = 2, (t; = [log (t LH + t HL 2 ] (2t LL (2t HH (t LH + t HL + (t LL + t HH 3. Possibility of First Best Taxation [ log t ] LH + t HL + (t HH t LH. 2t HH To begin with, we examine the ossibility of the first best tax schedule which is the solution to (P 2 without (IC constraint. If the social lanner were to know the realization of each agent s endowment, she could easily find the first best tax schedule. However, she does not have such an information, so the question is when the (IC constraint is not binding. First of all, to rule out the uninteresting cases, artition Ω (see Figure into Ω Ω 2 = {(, Ω : [, }, and = {(, Ω : [, }. When (, Ω 2, the social lanner can easily solve (P 2 by imosing a first best feasible tax schedule t {t B(, : t LL = t HH =, t LH + t HL = 2, and t LH }, 8

9 since t satisfies the (IC constraint; (t ; = ( t LH. If the social lanner insists that the tax schedule be increasing, then the unique solution to (P 2 is t = (,,,. Therefore, in the following we just focus on the case of (, Ω. According to the welfare function W(, it is easy to see that for (, Ω the first best feasible tax schedule is t F = ( t F LL, t F LH, t F HL, t F HH = ( ωl,, min{ +, 2}, min{, }. 2 To find the conditions under which t F is the solution to (P 2, consider the (IC constraint at t F : (t F ; = [log min{ +, 2} 2 (2 (2 min{, } min{ +, 2} + ( + min{, } ] [ log min{ +, 2} + ( ] min{, }. 2 min{, } Lemma 3. For (, Ω, (t F ; is strictly increasing in. Proof : Consider the two cases: (i (, [, (, ], and (ii (, [, (, ]. Case (i (, [, (, ]: In this case, it is clear that (t F ; Case (ii (, [, (, ]: If + < 2, Since (t F ; ( (t F ; = 2 + >, it follows that If + 2, then (t F ; Therefore, we have the result. = log ( + 2 (2 (2 >. = log ( (. (t F ; > lim = log ( + 2 >. 4 (t F ; = log >. For (, Ω, define ρ R by (t F ; ρ =, or equivalently, log min{+,2} 2min{ω ρ = H,} + ( min{, } log min{+,2} 2 (2 (2min{,} min{ +, 2} + ( + min{, }, and let ρ = min{, ρ}. Define also Ω F = {(, Ω : lim (t F ; }. 2 In fact, there is a continuum of first best feasible tax schedules if + > 2. However, given, t F is the solution to (P 2 while making (IC satisfied maximally, so we can assume without loss of generality that t F is the unique first best tax schedule. This oint will be made clear in the following. 9

10 45 Ω Ω 2 log 2.5 Ω F.5 Figure : Possibility of First Best Taxation Proosition 3.2 If ρ, then the first best feasible tax schedule t F is the unique solution to (P 2. In articular, if (, Ω F, then t F is the unique solution to (P 2 for all (,. Proof : By the definition of ρ and Lemma 3., if ρ, then (t F ;, which imlies that t F satisfies (IC. Since t F is feasible, the fact that t F is the unique first best feasible tax schedule roves the first result. It is obvious that lim (t F ; guarantees that (t F ; for all (,. Figure deicts the ossibility of first best feasible taxation. 3.2 Second Best Tax Schedule Assume that < ρ. To characterize the second best feasible tax schedule, we begin with three lemmas. The main urose of these lemmas is to lower the dimension of the social lanner s roblem. Lemma 3.3 Suose t is a solution to (P 2. Then, t HH = min{, }. Proof : There are two cases: (i, and (ii >. Case (i : Suose by way of contradiction that t HH <. Choose ε such that < ε t HH. Consider a new tax schedule t = (t LL, t LH, t HL, t HH + ε. Since log(2t HH t HH is strictly increasing in t HH (,, it follows that U i (, t > U i (, t U i (, t = U i (, t. Hence, t satisfies (IC. Also, t satisfies (Feasibility by the construction of ε. However, we have W(t ; > W(t ;, a contradiction to the hyothesis that t is a solution. Case (ii > : Suose by way of contradiction that t HH. If t HH <, choose ε such that < ε t HH. Then the same argument in Case (i induces a contradiction. If t HH >, choose

11 ε such that < ε t HH, and consider a new tax schedule t = (t LL, t LH, t HL, t HH ε. Then, the same argument in Case (i also gives a contradiction. Lemma 3.4 Suose t is a solution to (P 2. Then, t LH + 2. Proof : Suose by way of contradiction that t LH + t HL > 2. Note that this case is ossible only when + > 2. Choose ε such that < log(t LH + t HL log(t LH + t HL ε < ε 2. Such an ε is well defined since d dy (log y < 2 for y > 2. Consider a new tax schedule t = (t LL, t LH, t HL ε, t HH. Then, U i (, t = [log(t LH + ε + ε] + ( [log(2t HH t HH] [ > log(t LH + + ε ] + ( [log(2t 2 HH t HH] > [log(t LH + ] + ( [log(2t HH t HH] [log(2t LL t LL] + ( [log(t LH + t LH] > [log(2t LL t LL] + ( [log(t LH + ε t LH] = U i (, t, which imlies that t satisfies (IC. Also, t satisfies (Feasibility by construction. However, since 2 log(t LH + t HL (t LH + t HL = 2 log(t LH + ε (t LH + ε > 2 log(t LH + (t LH +, we have W(t ; > W(t ;, a contradiction to the hyothesis that t is a solution. Lemma 3.5 Suose t is a solution to (P 2. Then, t LH =, and. Proof : If =, then the claim is trivial. Hence, consider the case of >. Suose by way of contradiction that (i < ; or (ii t LH < and. Case (i < : Consider a new tax schedule t = (,,, min{, }. Then, (t ; = ( [ log min{, } + min{, } ] >, (2 which imlies that t satisfies (IC. Also, t satisfies (Feasibility. However, we have W(t ; > W(t ;, a contradiction to the hyothesis that t is a solution. Case (ii t LH < and : According to Lemma 3.4, we have two subcases: (a t LH +t HL < 2; or (b t LH + t HL = 2. Subcase (a t LH + t HL < 2: Choose ε such that < ε min{ t LH, 2 (t LH + t HL }. Consider a new tax schedule t = (t LL, t LH + ε, t HL ε, t HH. Then, U i (, t = U i (, t + ε > U i (, t = U i (, t + ( ε > U i (, t, (3

12 which imlies that (t ; >, that is, the (IC constraint is not tightly binding. Since (t; is continuous in t HL, we can choose δ (, ε such that t = t + (,, δ, still satisfies (IC and (Feasibility. Notice that t LH +t HL = t LH +t HL < t LH +t HL < 2, and t LL = t LL, and t HH = t HH. Hence, we have W(t ; > W(t ;, a contradiction to the hyothesis that t is a solution. Subcase (b t LH +t HL = 2: Notice that in this case > since = 2 t LH > 2 >. First, we want to show that t LL <. Suose not, that is, t LL =. Since t HH = by Lemma 3.3, it turns out that (t ; = ( log ( <, ω since < ρ = L log. This is a contradiction to the hyothesis that t satisfies (IC. So, t LL <. Consider a new tax schedule t = (t LL, t LH + ε, t HL ε, t HH where ε is chosen such that < ε t LH. Then, by (3, we have (t ; >, that is, the (IC constraint is not tightly binding. Since (t; is continuous in t LL and t LL <, we can choose δ (, t LL such that t = t +(δ,,, still satisfies (IC and (Feasibility. Since 2 log(2t LL 2t LL > 2 log(2t LL 2t LL, we have W(t ; > W(t ;, a contradiction to the hyothesis that t is a solution. Therefore, we conclude that t LH = and. By Lemmas , we can reduce the dimension of (P 2 from four to two. Let T = t LH +t HL. Lemmas imlies that we can restrict our attention to (T, t LL [2, min{ +, 2}] [, ], which now can be called a tax schedule. Define (IC-function z(, ; : [2, min{ +, 2}] [, ] R, by z(t, t LL ; = (t LL, t LH, t HL, t HH ; tlh =,t HH =min{,} [ T 2 = log (2t LL (2 min{, } T + ( t LL + min{, } ] [ ] T log 2 min{, } + (min{, }. Thus, a tax schedule (T, t LL satisfies (IC if z(t, t LL ;. Now, the social lanner s roblem (P 2 can be written as an equivalent but simlified version (P 2: Given (, Ω and (,, (P 2 max (T,t LL W(T, t LL ; = 2 [2 log(2t LL 2t LL ] + 2( [2 log T T] subject to (IC z(t, t LL ; (Feasibility (T, t LL [2, min{ +, 2}] [, ]. To find the second best tax schedule, first consider the shae of the (IC-curve z(t, t LL ; =. In fact, we can find a oint that satisfies z(t, t LL ; = for all (,. For (, Ω \ Ω F, let T = 2 min{, }e (min{,}, and { t LL = W ( ex log T } 2 T +, 2

13 where W is the rincial branch of Lambert W function. 2 By the definition of T, we can rewrite the (IC-curve as [ T 2 z(t, t LL ; = log (2t LL ( T T + ( ] [ t LL + log T T ], so, it is clear that z( T, t LL ; = for all (, ρ. That is, the (IC-curve z(t, t LL ; = always goes through the ivotal oint ( T, t LL. Furthermore, Lemma 3.6 i. ( T, t LL (2, min{ +, 2} [,. ii. If T, then t LL >. Proof : i. Since log(min{, } min{, } > log(, it is clear that T > 2. For (, Ω \ Ω F, log min{+,2} 2 min{,} + (min{, } = lim (t F ; >, so T < min{ +, 2}. To see that t LL [,, by Lambert W function, it suffices to show that ϕ(, log T 2 T + <. Note that since T > 2, ϕ(, = log(min{, } min{, } T + 2 < log(min{, } min{, }. ii. Since t LL <, the result is equivalent to t LL log t LL log, or log T 2 T +2. By 2 < T, we have the result. This lemma tells that if T, the ivotal oint ( T, t LL is above the feasible set [2, min{ +, 2}] [, ]. Another roerty of the (IC-curve is that it turns around the ivotal oint ( T, t LL counterclockwise as increases. Lemma 3.7 For all, (, such that <, if z(t, t LL ; =, then { z(t, t LL ; < if T < = T if T T. Proof : Since ( ( [ z(t, t LL ; = z(t, t LL ; + log T T ] ( [ = log T T ], we have the result easily. Figure 2 deicts the subsets of Ω that satisfy t LL and T >. 2 The Lambert W function is defined to be the function satisfying W(xe W(x = x. This function is defined on [ e,, and has a single real value on [, and two real values on [ e,. W, called the rincial branch, is the increasing art of W and W, called the ( th branch, is the decreasing art of W. The solution of the W(a log b. For more roerties on the Lambert W function, see Corless, et. al. [996]. (We reluctantly emloy the notational abuse, W, reviously used for the welfare function. Hoefully, it may not cause any confusion in the following. equation xb x = a is x = log b 3

14 t LL > T > log Figure 2: Relative size of ( T, t LL on Ω Now, consider the sloe of (IC-curve z(t, t LL ; =. Without loss of generality, we can restrict our attention to the domain of [, 2] [,, which includes all of the ossible (T, t LL. 22 Using the Imlicit Function Theorem, dt LL dt 2 T = ( z(t,tll ;= t LL z(t,t LL ;=. (4 Since we restrict t LL on [,, the denominator is negative. If 2, then the numerator is negative for all T [, 2]. If ( ( 2,, then the numerator is ositive for T, 2 and negative for ( T 2, 2. As a result, for (T, t LL [, 2] [,, dt LL dt z(t,tll ;= > if ( ( 2, and T = if ( 2, and T = 2 < otherwise, 2. Remark 3.8 According to the inequality (2, it turns out that for (, Ω, the curve defined by z(t, t LL ; = on (T, t LL [2, min{ +, 2}] [, ] has a negative sloe because the oint (T, t LL that has zero sloe cannot be in [2, min{ +, 2}] [, ]. The sloe of welfare-curve W(T, t LL ; = w, where w is a constant, is dt LL = ( ( 2 T dt W(T,tLL ;=w 2 ( t LL W(T,tLL ;=w <, (5 22 Note that z(t, t LL; = defines multile curves on R 2 while it defines a single curve on [, 2] [,. 4

15 for (T, t LL [, 2] [,. That is, the welfare-curve W(T, t LL ; = w has a negative sloe on [, 2] [,. To describe the second best feasible tax schedule, we need some definitions. First, for (, Ω such that + and (, ρ, define t LL (, by z( +, 2, t LL ; =. Second, for (, ρ, define T (2, min{ +, 2} by z(t, ; =. Third, define simly T o =. Finally, for T < and T <, define t o LL (, by z(, t o LL ; =.23 Now, we can state the main result of this aer. Proosition 3.9 For < ρ, the solution to (P 2 is (t LL,,, if + T o t = (t o LL,, T o, min{, } if T T o. (,, T, min{, } if T > T o Proof : Notice from (4 and (5 that the (IC-curve z(t, t LL ; = is tangent to the welfare-curve W(T, t LL ; = w at (T, t LL = (T o, t o LL. For the interior solution (the third case, we need to show that the tangent oint (T o, t o LL is maximizing the welfare function W( rather than minimizing. This can be done by showing that d 2 t LL dt 2 Differentiating (4 and (5, it follows that d 2 t LL dt 2 d2 t LL W(T o,t o LL ;=w dt 2 = z(t o,t o LL ;= > d2 t LL W(T o,t o LL ;=w dt 2. z(t o,t o LL ;= = >. [ ( tll T T( t LL T t LL t o LL ( t o LL For the corner solutions (the firs and second cases, we need to show that T = dt LL dt LL, dt dt resectively. From (4 and (5, we have dt LL dt LL dt dt W(T,tLL ;=w z(t,tll ;= W(T,tLL ;=w resectively, since t LL [,. Therefore, we have t as stated. z(t,tll ;= dtll dt ] T=T o,t LL =t o LL = ( ( tll T 2 for T, t LL T Table summarize the otimal feasible tax schedules and their relative size for each ossible case. 23 Using Lambert W function, we can exress t LL, T, and t o LL exlicitly as ( } t LL = W ( ex {log (+ 2 2 T log + ; T ( ( ( {( [ T = 2 W ex log ( (2ω 2 2 L( T ] ( 2 { ( } t o LL = W ( ex log 2 T + ωl log T. For T, W should be W if > 2 and W if 2. 2 log( T } ; 5

16 Cases t LL t LH t HH ρ (including Ω F = < min{ +, 2} min{, } < ρ + t LL < = + > T t o LL < T o < min{, } T > = < T min{, } Table : Otimal feasible tax schedules for n = Simulated Examles In this section, we illustrate some examles that show the secific otimal feasible tax schedules for different arameter values. Due to the low dimensionality of the social lanner s roblem, we can draw the results grahically. Examle 3. i. Suose first that (, = (.2,.5. In this case, ρ.28, so ρ for all (,. In articular, this is the examle of (, Ω F. Thus, the first best feasible tax schedule (t LL, t LH, t HL, t HH = (.2,.2,.5,.5 is obtained. ii. Suose that (, = (.,.8. In this case, ρ.3. (a If ρ, the first best tax schedule (t LL, t LH, t HL, t HH = (.,.,.8,.8 is obtained. Figure 3(a illustrates the case of =.3. (b If < ρ, by Proosition 3.9, the second best tax schedule t = (t LL,,, is obtained. Figure 3(b illustrates the case of =. in which the otimal tax schedule is (t LL, t LH, t HL, t HH = (.7,.,.8,.8. iii. Suose that (, = (.25,.8. In this case, ρ.4. (c If =.35 as illustrated in Figure 3(c, then T.2 >, so the second best tax schedule t = (,, T, = (.25,.25,.77,.8 is obtained. (d If =.2 illustrated in Figure 3(d, then T.97 <, so the second best tax schedule t = (t o LL,, T o, = (.2,.25,.75,.8 is obtained. iv. Figure 3(e (h show some other cases that have the second best tax schedule for different arameter values. 4 Proerties and Comarative Statics 4. Proerties of Otimal Feasible Tax Schedules First, consider the ossibility of first best feasible taxation. For a first best feasible tax schedule to be a solution to the social lanner s roblem, it should not give any incentive for an agent to misreort his endowment. Since the incentive comatibility constraint is unilateral in our model, 6

17 t LL W( = w z( = t LL z( = W( = w =. =. 2 =.2 + =.9 (a (, = (.,.8, =.3 T 2 =.2 + =.9 (b (, = (.,.8, =. T t LL t LL =.25 =.25 W( = w z( = W( = w z( = 2 =.5 + =.5 (c (, = (.25,.8, =.35 T 2 =.5 + =.5 (d (, = (.25,.8, =.2 T t LL =.35 t LL W( = w z( = =.6 W( = w z( = 2 =.7 + =.5 (e (, = (.35,.8, =.5 T 2 + =.2 =.4 (f (, = (.6,.8, =.5 T t LL t LL z( = W( = w =.6 W( = w =.2 z( = 2 =.4 + =.5 (g (, = (.2,.3, =.5 T 2 =.2 (h (, = (.6,.3, =.5 + =.9 T Figure 3: An examle of second best taxation 7

18 this requirement says that a rich agent should have no incentive to lie. According to the characterization results in the revious section, when (i ρ, or (ii (, Ω 2, the corresonding first best feasible tax schedules can be a solution to the social lanner s roblem (P 2. For the case of (ii, the endowment of a oor agent is large enough that the first best feasible tax schedule could imose the same amount of tax on each agent for any case. 24 Thus, a rich agent has no incentive to misreort his tye. On the other hand, for the case of (i, since the overall endowment level of the economy is small enough (the case of Ω F or the robability of low endowment is high enough, a rich agent worries mainly about that too low amount of ublic good would be rovided if he misreorts. Thus, he will not lie. Therefore, when the total endowment of the economy is relatively low enough or high enough, first best feasible taxation satisfies the incentive comatibility constraint so that it can be the solution to (P 2. Second, the otimal feasible tax schedule always imoses % tax rate on a oor agent when his neighbor is rich. 25 That is, t LH = for all (, Ω and all (,. This result reflects the effect of informational rent on the economy which ursues efficiency rather than equity as its objective. Due to the no exaggeration assumtion, the incentive comatibility constraint in this model is unilateral so that levying a tax on a oor agent does not create any incentive roblem as long as it is feasible. Thus, the social lanner, who does not mind which agent ays how much roortion of the total taxes, refers to imose as much tax as ossible on the oor agent rather than his rich neighbor who may request informational rent. Of course, the absolute amount of tax ayment of rich agent is strictly higher than that of oor agent. However, since there do exist some cases in which the tax rate imosed on the rich agent is strictly less than %, we can say that the otimal feasible tax mechanism is regressive. Third, the otimal feasible tax schedule is increasing. Proosition 4. For all (, Ω, t LL and t LH t HH. Proof : The first inequality is clear since Lemma 3.5 imlies that t LL. The second inequality is also clear since t LH = < min{, }. Thus, under the otimal feasible tax mechanism each agent s tax ayment is increasing with his endowment. Note, however, that this increasingness does not imly that marginal tax rate is increasing. Finally, we note the roerty that for all (, Ω, t LL t LH with strict inequality for some cases as can be seen in Table. That is, a oor agent may say, If my neighbor is rich, then I have to ay more! This can be interreted in a quite similar way used in the second roerty above; a oor agent should take some extra burden caused by his rich neighbor, which would not have been incurred had his neighbor been oor. In Rhee [24b], we tackle this roblem in detail by comaring the immigration incentives of an agent to the communities with different exected endowments. 24 Note that it is assumed that an increasing tax schedule is used on Ω As mentioned in Section 2., each agent s initial endowment is considered as his taxable wealth. Thus, the % tax rate is accetable in this sense. 8

19 4.2 Comarative Statics One of the interesting questions about the otimal feasible tax schedule is how the otimal feasible tax schedule t will resond to the change in the robability of low endowment. We are also interested in how t will change as or varies. In the following, we exclude the trivial case Ω 2 in which first best taxation is always ossible Resonses of t to Since both t LH and t HH are indeendent of, it suffices to analyze the resonses of t LL and t HL. Given (, Ω, if ρ, then t is indeendent of. Thus, suose < ρ. By Lemma 3.7, we have the following subcases. ( + In this case, only t LL = t LL deends on. Since the rincial branch W of the Lambert W function is strictly increasing, it follows that dt LL d >, that is, t LL is strictly increasing. (2 + > Case (i t LL : If T, then only t LL = to LL deends on. By the definition of to LL, it turns out that ( dt LL d = z(t o,t o LL ;= log T T t LL ( > z(t o,t o LL ;= since the nominator is negative by t o LL < and the denominator is negative by T < T = T o. Hence, t LL is strictly increasing. If T >, then only t HL = T deends on. Since T T and T > 2 by Remark 3.8, it follows that ( dt log T T d = z(t,ωl ;=, (6 2 T z(t,ωl ;= which imlies that is increasing.26 Case (ii t LL < : In this case, we claim that < T T. By Remark 3.8, T T is clear. To see that < T, note that [ ] [ z(, ; = log (2 ( T log T = log ] ( log 2ω T >. L That is, the (IC-curve is always above the oint (,, which imlies < T. Thus, only = T deends on. By (6, dt d <, so t HL is strictly decreasing. (3 Interretation Table 2 summarizes the resonses of t to for each ossible case. Figure 4 shows the examles for some different endowments values. 27 Roughly seaking, t LL is (weakly increasing for increases The equality holds only if t LL =. In Figure 4, β (, is defined such that z(, ; β =. 9

20 Cases dt LL d dt LH d d d dt HH d ρ (including Ω F < ρ > t LL T + T > + t LL < Table 2: A summary of the resonses of t to. However, is (weakly increasing for relatively low, but decreasing for large. In fact, these results show how the (IC constraint will change as the robability of low endowment increases. Suose first that the initial low endowment is small enough such that t LL (the areas of and 2 in Figure 2. In this case, the increase in makes the (IC constraint less tight for both t LL and t HL in the sense that the set of incentive comatible and feasible tax schedules becomes larger. 28 Thus, the social lanner can increase t LL or t HL as long as the feasibility constraint is binding. That is, for T (the interior solution case, t LL increases but t HL stays the same since the solution always occurs at T o =, and for T > (the corner solution case, increases but t LL is fixed at its feasible maximum. The economic intuition behind this result is as follows: The social lanner has to take into account the informational rent incurred by a rich agent. When is relatively small such that t LL, the social lanner can design an incentive comatible tax mechanism without much worrying about such an informational rent because the rich agent is reluctant to lie to avoid too small rovision of ublic good. Thus, as increases, the rich agent (IC constraint becomes less tight. (Figure 4(a(b(e(f. On the other hand, suose that the initial low endowment is relatively large such that t LL < (the area of 3 in Figure In this case, the increase in makes the (IC constraint less tight for t LL but tighter for t HL in the sense that the set of incentive comatible and feasible tax schedules becomes larger with resect to t LL, but smaller with resect to t HL. Thus, the social lanner would like to decrease and increase t LL as long as the feasibility constraint is binding. Since in this case t LL is already set at its maximum (the corner solution case, only should be decreased. This result can be interreted as follows: If is relatively large, then too small ublic good rovision is no longer a big roblem. Thus, as increases, the rich agent is more willing to lie, which imlies that the social lanner should decrease the rich agent s tax ayment thl to make him honest (Figure 4(c(d(g(h. 28 Recall by Lemma 3.7 that the (IC-curve z(t, t LL; = turns counterclockwise around the ivotal oint ( T, t LL as increases. 29 Note that in this case, T > by Lemma

21 t =.8 T o =.75 =.8.67 t =.25 t LL =.35 t LL β.29 ρ.4 ρ.72 (a (, = (.25,.8 (b (, = (.35,.8 t t =.8.68 =.8.7 =.6 t LL =.45 t LL (c (, = (.45,.8 (d (, = (.6,.8 t =.3 t =.3 T o = =.25 t LL =.35 t LL β.23 ρ.86 (e (, = (.25,.3 (f (, = (.35,.3 t =.3 t =.3.7 =.45 t LL.74 =.6 t LL (g (, = (.45,.3 (h (, = (.6,.3 Figure 4: Resonses of t LL and t HL to 2

22 t t =.8 =.8 t LL t LL =.8 =.8 (a =.8, =.5 (b =.8, =.5 t t =.3 =.3 t LL t LL (c =.3, =.5 (d =.3, =.5 t t.2 =.85 =.5 =.5 t LL =.85 =.5 =.5 t LL.2 (e =.5, =.5 (f =.5, =.5.7 t.7 t =.45 t LL =.45 t LL =.45 =.45 (g =.45, =.5 (h =.45, =.5 Figure 5: Resonses of t LL and t HL to and 22

23 2 =.6 E(y 2 =.6 E(y 2 =.2 2 =.7 ρ =.72 (a (, = (.35,.8 (b (, = (.6,.8 2 =.6 E(y E(y 2 min{,} = 2 =.3 =.8 (c =.8, =.5 =.8 (d =.3, =.5.23 E(y.46 2 =.2 E(y 2 =.7 =.35 =.65 =.6 (e =.35, =.5 (f =.6, =.5 Figure 6: Exected Total Provision of Public Good: E(y Resonses of t to or We have already studied rough resonses of t to or by its characterization for different cases on Ω. Now, we rovide some simulated examles for better understanding the otimal feasible tax mechanism. Figure 5(a (d show the resonses of t to and Figure 5(e (h to when =.5 or.5. Consider first the resonses to a change in. As increases, we can see that t LL is increasing, but is (weakly decreasing for lower values of and then increasing for larger values. These results can be interreted as follows. For lower, a first best solution like Figure 3(a is ossible so that t LL and t HL are set their maximum. As increases more, a corner solution like Figure 3(b could occur deending on the values of and. In this case, stays the same at its maximal but t LL would be less than. As increases furthermore, then an interior solution like Figure 3(d will haen. In this case, t LL will increase but t HL will decrease. As increases even further, a corner solution like Figure 3(e or (f is obtained. In this case, is decreasing initially and then 23

24 increasing, and t LL is increasing with. One of the interesting imlications from these results is that an increase in can decrease the tax burden of a rich agent if is relatively small. We can aly a similar interretation to the change of. That is, for lower, a first best solution like Figure 3(a is obtained, and then a corner solution like Figure 3(b and/or an interior solution like Figure 3(d is obtained deending on the values of and. Eventually, the otimal solution ends u an interior one like Figure 3(d or a corner solution like Figure 3(e or (f. Notice also that an increase in can decrease the tax burden of a oor agent if is relatively small and is relatively large (see Figure 3(e Exected Total Provision of Public Good Finally, we show how much ublic goods will be rovided as,, or varies. The exected total rovision of ublic good is exressed as E(y = 2 (2t LL + 2( (t LH + + ( (2t HH. The Figure 6 illustrates some examles of those resonses. Roughly seaking, E(y increases as or increase, and as decreases. 3 In articular, for large, the increase in may reduce E(y (Figure 6(d. This fact reflects the observation that the increase of may decrease so much. Thus, even if t LL and t LH increase with, the decrease of is still dominating, which results in the smaller E(y. 5 Otimal Feasible Tax Mechanism for n > 2 As the extension of the case n = 2, we now study the otimal feasible tax schedule for 2 < n <, t = ( (t L,(n kl,(k H n k=, (t H,(n kl,(k H k= n, which is the solution to (Pn. To begin with, consider the case of (, Ω 2. The social lanner can easily solve (P n by imosing a first best feasible tax schedule { t t [, ] n [, ] n : t L,(n L,H =, t H,L,(n H = ; since t satisfies the (IC constraint; 3 (n kt L,(n kl,kh + kt H,(n kl,(k H = n, and } t L,(n kl,kh, for k =,...,n. n (t F ; = n C j j ( n j( t F L,(j L,(n jh. (7 j= If the social lanner insists that the tax schedule be increasing, then the unique solution to (P n is t = (,...,;,...,. Therefore, in the following we just assume that (, Ω. 3 Although not rovided here, the case in which is quite small, say.5, shows the ossibility that E(y increases as increases. This is due to that fact that for a quite small the increase of t LL or as a small increases (see Figure 4(a(b(e or (f could increase E(y. 3 See Aendix A for derivation. 24

25 5. Possibility of First Best Taxation First of all, artition Ω into Ω A = {(, Ω : }, Ω Bi = {(, Ω \ Ω A : i + (n i < n (i + (n i + }, for i =,...,n. If (, Ω Bi, i =,...,n, the first best feasible tax schedule is { t F t [, ] n [, ] n : t L,(n L,H =, t H,L,(n H = ; t L,(n kl,kh =, t H,(n kl,(k H =, k =,...,n i; } (n kt L,(n kl,kh + kt H,(n kl,(k H = n, k = n i +,...,n. Since lim (tf ; = { log n +(n ( if i = ( t F L,L,(n otherwise <, 32 there is no (, Ω Bi for i =,...,n such that the first best feasible tax schedule t F can be the solution to (P n for every (,. Also, it turns out that the olynomial equation (t F ; = of may have multile roots so that it is imossible to define the unique ρ such that (t F ; for ρ. Now, suose that (, Ω A. In this case, the unique first best feasible tax schedule is j= t F = (,..., ;,...,. To find the condition under which t F is the solution to (P n, consider the (IC constraint at t : 33 [ n n j ] (t F ; = n C j n j ( k+mod(n j,2 n jc k log(k + (n k (8 (, k= where mod(x,2 is if x is even and if x is odd. Lemma 5. For all n 2 and all j =,...,n 2, (t F ; is strictly increasing in n j. Proof : See Aendix C. For (, Ω A, define ρ R by (t F ; ρ =, and let ρ = min{, ρ}. Define also Ω F = {(, Ω A : lim (t F n ; = log ( }. + (n Proosition 5.2 If ρ, then the first best feasible tax schedule t F is the unique solution to (P n. In articular, if (, Ω F, then t F is the unique solution to (P n for all (,. 32 n To see that log +(n ( < for (, Ω 2, note that d(lhs/d >. Since < n (n n in Ω B, it follows that LHS < lim log ( = (n ( <. n (n + (n 33 See Aendix B for derivation. 25

26 Proof : Same as the roof of Proosition 3.2. Corollary 5.3 lim n ΩF =. Proof : Since lim n lim (t F ; = ( <, we have the result. The intuition of this result is that as the number of agents increases, the incentive for a rich agent to misreort his tye increases because the ossibility that too little ublic good is rovided decreases. Thus, it becomes more difficult to satisfy the (IC constraint and finally the ossibility of first best feasible tax schedule gets to disaear. 5.2 Second Best Feasible Tax Schedule The same result as Lemma 3.3 holds for n > 2. Proosition 5.4 Suose that a tax schedule t is a solution to (P n. Then, Proof : Same as the roof of Lemma 3.3. t H,L,(n H = min{, }. On the contrary, the result like Lemmas 3.5 does not hold for n > 2. Nonetheless, we can find a similar result with a mild assumtion. Proosition 5.5 Suose that a tax schedule t is a solution to (P n. For each j {,...,n }, if t H,jL,(n jh >, then t L,(j L,(n jh =. Proof : If =, then the claim is obvious. Thus, assume that >. Suose by way of contradiction that t L,(j L,(n jh <. Choose ε such that < ε min{ t L,(j L,(n jh, t H,jL,(n jh }. Consider another tax schedule t which relaces t L,(j L,(n jh and t H,jL,(n jh in t by t L,(j L,(n jh = t L,(j L,(n jh + ε j, and t H,jL,(n jh = t H,jL,(n jh resectively. Then, it follows that ( ε U i (, t = U i (, t + n C j j ( n j n j > U i (, t = U i (, t + n C j j ( n j ( ε j > U i (, t, ε n j, which imlies that (t ; >. Since (t; is continuous in t H,jL,(n jh we can choose δ (, ε such that a new tax schedule t, which relaces t H,jL,(n jh in t by t H,jL,(n jh = 26