Online Appendix: The Continuous-type Model of Competitive Nonlinear Taxation and Constitutional Choice by Massimo Morelli, Huanxing Yang, and Lixin Ye

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1 Online Appendix: The Continuous-type Model o Competitive Nonlinear Taxation and Constitutional Choice by Massimo Morelli, Huanxing Yang, and Lixin Ye For robustness check, in this section we extend our analysis to the continuous type model, which can be regarded as the limiting case o many inite types. As an overview, with a continuum o types, the tax schedule chosen under each regime is characterized by a second-order dierential equation with two boundary values. By ocusing on the case where the vertical types are distributed uniormly, we are able to show that under independent taxation, the higher the mobility, the higher the consumption or all but the highest and lowest types; the rich (types suiciently close to the highest type) pay lower tax, and the poor (types suiciently close to the lowest type) receive lower subsidy under competition; there exists a cuto type so that all types above are better o, and all types below are worse o with competition. Our computations conirm most o the indings rom the three type model regarding the preerences o the median type, who is responsible or the constitutional choice. Speciically, in the vertical dimension worker-consumers are distributed on [, ] with density unction (), where () is continuous, strictly positive everywhere in its support. All the other assumptions are the same as those in the previous discrete type model. As in the discrete type model, citizens can only be sorted in the vertical dimension. Thus, oering a tax schedule T (Q) is equivalent to oering a menu o consumption and production pairs {C(),Q()} [,].Deine the tax unction T () =Q() C(). In the autarkic economy (no tax), a citizen s optimal consumption is determined by u 0 (c )=1/. Again we will consider uniied and independent taxation rules. Under either the uniied or independent taxation rule, incentive compatibility has to hold or each type o citizen conditional on her State o residence. Deine V (, b ) = u(c( b )) Q(b ) to be the utility or a citizen with (vertical) type who accepts contract {C( b ),Q( b )}. Incentive compatibility requires that V (, ) V (, b ) (, b ) [, ] 2. Let v() denote the equilibrium rent provision to type- citizen: v() =V (, ). By the standard 1

2 Constraint Simpliication Theorem, the IC conditions are equivalent to the ollowing two conditions: v 0 () = Q() 2 = 1 [u(c()) v()] (1) Q 0 () 0 (2) Constraint (2) is the monotonicity requirement as in the three-type model. By (1), given v(), Q() is uniquely determined and so is C(). For convenience, we will work with the rent provision contract v(). 1 Itcanbeeasilyveriied that Q 0 = u 0 (C)C 0.Thus,asinthe three-type model, Q 0 () 0 iandonlyic 0 () 0. Given v() provided by the State in question and the other State s rent provision v i (), the type- market share or the State in question is given by x () =1+ v() v i(). (3) k For ease o analysis, rom now on we will work with the utility unction u(c) =2 C. 2 Uniied Taxation Under uniied taxation, the objective o the Federal authority is to maximize the weighted average utility o all the citizens in both States, where the weight unction w() =() (in the same spirit as in the three-type model). We ocus on the symmetric solution in which the same menu o contracts is applied to both States and the resulting market shares are symmetric (no citizen moves). We can thus drop the State index to write {C i (),Q i ()} = {C(),Q()}, i =1, 2. Mathematically, this can be ormulated as an optimal control problem: max Z v()()d s.t. v 0 () = 1 Q 0 () 0 Z h 2 p i C() v() [Q() C()]()d =0 The last constraint is the resource or budget constraint (RC). 1 This approach ollows the lead o Armstrong and Vickers (2001), who model irms as supplying utility directly to consumers. 2 Our main results should not be altered as long as we work with concave utility unctions. 2

3 To solve this optimal control problem, as is standard in the literature, we irst ignore the monotonicity constraint on Q() to consider the relaxed program (and this approach will be justiied i the solution o Q() is indeed monotone). To deal with the resource constraint, we deine the new state variable J() as ollows J() = Z [Q() C()]()d, hence J 0 () = [Q() C()](). Now(RC) is equivalent to J() =0and J() =0. The Hamiltonian o the problem is: H = v + λ 1 h 2 i C v + µ[(2 C v) C] Deine z = C, then the Hamiltonian can be rewritten as H = v + λ 1 [2z v]+µ[(2z v) z2 ] where λ and µ are the two costate variables. The optimality conditions are as ollows: H z = 2 λ + µ[2 2z] =0 (4) λ 0 = H v = + λ + µ (5) µ 0 = H J From (6), µ is a constant. From (4) and (5) we can get rid o λ to yield =0 (6) z z =2 1 µ + 0 (7) We can urther getting rid o µ by turning (7) into a second-order dierential equation: z 00 = 1 z 0 2+(z + z 0 2) 0 z() =, z() = µ (z ) (8) where the boundary conditions above are directly implied rom the transversality conditions λ() = λ() =0and (4). The above second-order (linear) dierential equation system has a closed-orm solution, which is given by 3

4 Independent Taxation z() = () Z µ (s) 2 1 () () µs + (s) s0 ds +, (9) (s) Z df (s) where µ =. s Under the independent taxation regime, each State i chooses its taxation schedule simultaneously and independently. Given v i (), the rent provision provided by the other State, State i will choose a rent provision v() to maximize the weighted average utility o the citizens residing in its own State. Again we ocus on symmetric equilibria, in which the two States choose the same taxation schedule. Suppose State 2 s rent provision contract is given by v (). Then i State 1 oers rent provision contract v(), by(3)thetype- market share or State 1 is given by η() =1+ 1 k [v() v ()]. Now State 1 s maximization problem can be ormulated as the ollowing optimal control problem: Z max v()()d s.t. v 0 () = 1 h 2 p i C() v() Q 0 () 0 J 0 () =[(2 p C() v()) C()]η()() J() =0,J() =0 where J() = R [(2 C v) C]η()()d is the state variable associated with the budget constraint. Note that the market share η() does not directly enter the State s objective unction. However, the States compete or high-type citizens as the market shares aect the resource constraints and hence the ability to redistribute. We again drop the monotonicity constraint Q 0 () 0 and deine the Hamiltonian (with z = C): H = v + λ (2z v)+µη[(2z v) z2 ]. The optimality conditions or a symmetric equilibrium are given by H z = 2 λ + µ[2 2z] =0 λ 0 () = H v = 2λ µ k µ 0 () = H J =0 µ is a constant 4 (2z v) z 2 + µ

5 Ater getting rid o λ, wehave: z 0 = 2 1 µ (z )0 v 0 = 1 (2z v) J 0 = (2z v) z 2 (2z v) z2 k Letting w =2z v, the above system becomes w 0 = 2z 0 v 0 =2z 0 w (10) J 0 = w z 2 (11) z 0 = 2 1 µ (z )0 w z2 k =2 1 µ (z )0 J0 k (12) From (11), we have w = 1 (J0 + z 2 ), (13) w 0 = 1 2 (J 00 +2zz 0 ) (J 0 + z 2 ) (14) Substituting (13) and (14) into (10), we have From (12), we have J 00 =2( z)z 0 (15) J 00 =2k k(z 00 + z 0 ) k(z + z 0 2) 0 Equating (15) and (16), and simpliying, we have µ 0 0 k( z) (16) z 00 = 1 z 0 2+(z + z 0 2) 0 z() =, z() = µ 0 + (z ) ( z)z0 k (17) where the boundary conditions above, as in the uniied taxation case, ollow rom the transversality conditions λ() =λ() =0. Note that this is again a second-order dierential equation system with 5

6 two boundary values. It is nonlinear, however, in this case. The complication is that a closed-orm solution is no longer available. The analysis can easily become intractable i we work with general distributions. For this reason in the next subsection we will ocus on the uniorm distribution case, where is distributed uniormly over,. The Uniorm Distribution Case Under uniied taxation, assuming that is uniormly distributed (i.e., 0 =0), (8) reduces to z 00 = 1 z 0 2 (18) z() =, z() = Substituting () =1/ into (9), we obtain the solution in the uniorm distribution case: log log z() =2 ( ) log log (19) It can be easily veriied that z 0 () > 0 i / 1 2log /,orequivalently, / γ 3.55 (20) Note that z 0 () > 0 implies that Q 0 () > 0. Given our ocus on perect sorting equilibria and to justiy our approach to solve the relaxed program by ignoring the monotonicity constraint, we maintain the sorting condition (20) throughout this section. 3 Intuitively, the higher the /, the more costly is sorting along the vertical dimension. When / is large enough, pooling at the lower end is optimal. It can be easily veriied that z >0 or, and z = or =,. Theresultoeiciency at the top is standard in the screening literature. Eiciency at the bottom, which is implied rom the transversality condition, however, is dierent rom what we have seen rom our base model with three types. 4 3 This is a similar condition to the one that Rochet and Stole (2002) impose to guarantee separating equilibrium in a nonlinear pricing setting with random participation. When this assumption ails, pooling occurs at the lower end. 4 A reconciliation is provided in the nonlinear pricing literature by Rochet and Stole (2002), who demonstrate that in a inite type model, the quality distortion or the lowest type disappears as the number o types goes to ininity. In the literature o optimal taxation, Seade (1977) provides a good intuition or the no-distortion-at-the-bottom result. 6

7 Since T 0 () =2( z)z 0, T 0 () > 0 or, under uniiedregime. Thatis,thetaxis increasing in the type. Given (RC), this also implies that the low types receive subsidies and the high types pay taxes. Under independent taxation, given that is uniormly distributed, (17) becomes: z 00 = 1 z 0 2k 2+ ( z)z0 (21) z() =, z() = Despite the lack o closed-orm solutions, we are able to explore some analytical properties o the equilibrium based on this ODE system. Our irst result is that under independent taxation, consumption is downward distorted or all but the top and bottom: Lemma 1 z I > 0 or (,). Proo. Deine y() = z I (). Theny() =y() =0, y 0 () =1 z 0 I (), andy00 () = 1 [1 + y0 2 k (1 y0 )]. It is equivalent to show that y never drops strictly below the zero line (y =0). First, we show that the curve is initially shooting above, i.e., y 0 () > 0. Suppose not, then there are two cases: Case 1: y 0 () < 0. Sincey() =0,inthiscasewehavey( + ) < 0. Thatis,they curve is initially shooting below. Given the endpoint condition y() =0, at some point the curve has to shoot back to the zero line. So there is ˆ (,), such that y 0 (ˆ) =0and y() < 0 or all (, ˆ]. Inthatcase, y 00 (ˆ) = 1ˆ [1 2 k y(ˆ)] < 0. This implies that y(ˆ + ) <y(ˆ) < 0, i.e., the curve keeps shooting below right ater ˆ. However, given the endpoint condition, the curve has to come back at some later point. But our preceding argument suggests that the curve can never come back to the zero line, contradicting the endpoint condition. Case 2: y 0 () =0.Inthiscase, y 00 () = 1 < 0. Thus y( + ) < 0. Now connecting our argument rom here with the argument in the irstcaseabove, we establish contradiction again. Thus we show that the curve is initially shooting above (y 0 () > 0). Given the endpoint condition, the curve will eventually drop back to the zero line. I it drops back to zero exactly at =, we 7

8 are done; otherwise, there is ˆ (,), such that y 0 (ˆ) =0and y(ˆ) < 0. Now ollowing the same argument above, y can never get back to zero, contradiction. This establishes that y() > 0 except =,. So as in the uniied taxation case, consumption is also distorted downward or all but the top and the bottom types or any k>0. Notethatthisisverydierent rom a result obtained in the duopoly case in Rochet and Stole (2002), who show that when competition is suiciently intense (k suiciently small), quality distortions disappear completely. The next lemma establishes that the equilibrium under independent taxation exhibits perect sorting. Lemma 2 Suppose condition (20) holds, then zi 0 () > 0 and hence T I 0 () > 0 or any [, ]. Proo. First, whenever z 0 I property. That is, z 0 I =0, z00 I = 2 > 0. By the single-crossing lemma, z0 I has the single crossing crosses zero line rom below at most once.5 What remains to be shown is that zi 0 () > 0. Now compare two dierential equation systems (18) and (21). Whenever zi 0 = z0 U (> 0), we have z00 I <z00 U (since z I > 0 by Lemma 1). By the single-crossing lemma, the curve zi 0 () z0 U () crosses zero line rom above at most once. Given the boundary conditions z I () z U () =z I () z U () =0, we conclude that zi 0 () z0 U () has to cross zero line exactly once. That is, there is a b (, ) such that zu 0 () <z0 I () or [, b ), and zu 0 () >z0 I () or (b, ]. Given that zu 0 () > 0, wehavez0 I () >z0 U () > 0. This completes the proo or zi 0 > 0. Givenz0 I > 0 and ( z I) > 0, wehaveti 0() =2( z I)zI 0 > 0 or (, ). The proo o Lemma 2 suggests that whenever the optimal solution under uniied taxation exhibits perect sorting, the equilibrium under independent taxation must exhibits perect sorting. On the other hand, it is possible that pooling occurs under uniied regime but the equilibrium under independent taxation exhibits perect sorting. 6 The implication is that sorting occurs more easily under a competition regime. The intuition is similar to that provided in Yang and Ye (2008): higher types receive higher rents under competition, which relaxes the IC constraint, making it easier to sort the agents. The next proposition displays interesting comparative statics with respect to the role o mobility: 5 Thereore, i there is pooling, it must happen at the low end. 6 Consider the ollowing example. is uniormly distributed on [1, 4], k =0.5. Under uniied taxation, the monotonicity constraint is violated and pooling occurs in the neighborhood o the low end. independent taxation exhibits perect sorting. However, the equilibrium under 8

9 Proposition 1 Let k 2 <k 1. Under independent taxation, (i) >z 2 >z 1 or all (, ); (ii) T 1 () >T 2 () and T 2 () <T 1 (); (iii) the tax schedule or (relatively) rich people is latter under k 2. Proo. (i) The two dierential equations under independent taxation are as ollows: z 00 1 = 1 [2 z0 1 2 k 1 ( z 1 )z 0 1], (22) z 00 2 = 1 [2 z0 2 2 k 2 ( z 2 )z 0 2]. Let y = z 2 z 1. We have y() =y() =0. We need to show that y() > 0 or all (, ). The proo idea resembles that o Lemma 1. First we show that y 0 () > 0. Suppose in negation, y 0 () 0. Case 1: y 0 () < 0. Given that y() =0, there exists b (, ) such that y 0 ( b )=0and y() < 0 or all (, b ]. But then it is easily veriied that y 00 ( b ) < 0. This implies that y will always remain strictly below zero ater initially shooting below, a contradiction. Case 2: y 0 () =0. It is easily veriied that in this case all higher derivatives at are zero: y (n) () =0or all n 2. This, combined with y() =0, implies that there exists b suiciently close to, such that y( b )=y 0 ( b )=y 00 ( b )=0. However, with notation z( b )=z 1 ( b )=z 2 ( b ) and z 0 ( b )=z1 0 (b )=z2 0 (b ), we can demonstrate that y 00 ( b )= 1 µ 2( b b z( b ))z 0 ( b 1 ) 1. k 1 k 2 Since z 0 ( b ) > 0 and b z( b ) > 0, the above expression implies that y 00 ( b ) < 0, a contradiction. So the y curve is initially shooting up. Given the endpoint condition, it will eventually come back to the zero line. I it comes back exactly at, we are done with the proo; otherwise it drops below zero beore reaching the end point. But then there is b (, ) such that y 0 ( b )=0and y() < 0 or all (, b ]. Applying the same argument to rule out Case 1 above, we can establish the contradiction. So y has to stay above zero except two boundary points. (ii) Similarly to the previous proo, that >z 2 >z 1 implies that v 2 cross v 1 at most once rom below. Again, the case that v 1 >v 2 or all can be ruled out. But so ar the case v 1 <v 2 or all cannot be ruled out. Thereore, we can only show T 2 () <T 1 (). (iii)notethatwehavez 1 () <z 2 () or any interior. This implies that at the neighborhood o, z 0 1 >z0 2.Asaresult,inthisneighborhood,T 0 1 >T0 2 as well. By continuity, we also have T 1 () >T 2 () or types suiciently close to, andt 2 () <T 1 () or types suiciently close to. As k goes down, the competition between two States becomes 9

10 more intense. Proposition 1 suggests that as mobility (or competition) increases, the consumption distortion is reduced, the rich (types suiciently close to the top) pay less taxes, and the poor (types suiciently close to the bottom) receive less subsidies. While these results are obtained computationally in our three type model, they are obtained analytically in this continuous type model. Thus the result that increased mobility leads to lower progressivity is a airly robust prediction. As in the three type model, as k 0, T () =0. The solution under uniied taxation, on the other hand, is independent o k, which can be regarded as the limiting case when k + (this can be seen rom comparing (8) and (17)). In Simula and Trannoy (2010), a curse o middle-skilled workers is identiied, in the sense that the marginal tax rate is negative at the top and the average tax rate is decreasing over some interval close to the top. Such a curse does not occur in our model. 7 The dierence arises or the ollowing reasons. In Simula and Trannoy, higher types have lower moving cost than lower cost types. This means that competition or top types is stronger than the competition or middle types, thus a negative marginal tax rate might occur at the top. In our model, all (vertical) types have the same moving cost given the same horizontal type. We have thus demonstrated that the curse o middle types may not arise in a model with outside options endogenously determined. We next turn to comparing the two taxation systems. This will be done by comparing the ODE systems (18) and (21). Using subscripts U and I to denote the uniied and independent taxation regimes, respectively, we can state the ollowing comparison results: Proposition 2 (i) There is a b (, ) such that z 0 I (b )=z 0 U (b ), z 0 I () >z0 U () or [, b ) and z 0 I () <z0 U () or (b, ]; (ii)z I () >z U () or any (, ); (iii) T 0 I () <T0 U () or (b, ). Proo. Part (i) is established in the proo o Lemma 2. Part (ii) ollows rom (i) given the boundary conditions z I () z U () =z I () z U () =0. For ( b, ], thatz U <z I and zu 0 >z0 I implies that T I 0() <T0 U (), ast 0 =2( z)z 0 under both taxation regimes. Thereore, under competition all types (, ) receive strictly higher consumption. Moreover, the tax schedule is latter or the rich (those with suiciently high types). Proposition 3 (i) There is a e (, ) such that v I ( e )=v U ( e ), v I () <v U () or [, e ) and v I () >v U () or ( e, ]; (ii) T I () >T U () and T I () >T U (). 7 Under independent taxation, T 0 =2( z)z 0 is always positive as ( z) 0 and z 0 > 0. 10

11 Proo. From the irst order conditions o the IC constraints, we have v 0 I v 0 U = 1 [2(z I z U ) (v I v U )]. (23) Over (, ), givenz I >z U, rom (23) we have v 0 I >v0 U whenever v I = v U. This implies that over (, ), v I and v U cross at most once, and at the intersection v I must cross v U rom below. Next we rule out the case that v I and v U never cross in the interior domain. Suppose v I () v U (). Thenv I () v U () or all and v I () >v U () or any >. This contradicts the act that v U () is the optimal solution under the uniied regime, while v I () is one o the easible schedules under the uniied regime. Thereore, v I () <v U (). Given that z I () =z U (), itmustbethecase that T I () >T U (). Next we rule out the case that v I () v U (). Suppose this is the case. Then v I () <v U () or all <. At, v I () <v U (), which implies that T I () >T U (). At, v I () v U (), which implies T I () T U (). For any interior (, ), v I () v U () = µ 2z I () z2 I () µ 2z U () z2 U () + T U() T I (). The irstterminthebracketispositivesince>z I () >z U (). I v I () <v U (), wemusthave T U () <T I () or all (, ). Thereore, R T I()d > R T U()d, violating the resource constraint R T I()d = R T U()d =0. Thus, v I crosses v U (rom below) exactly once at some interior (, ). Thisprovespart(i). Part (ii) ollows rom part (i) and the boundary conditions. So the rich (high-type citizens) are better o while the poor (low-type citizens) are worse o moving rom uniied to competitive taxation. The highest type (and the types suiciently close to the highest type) pay less tax and the lowest type (and the types suiciently close to the lowest type) get less subsidy under independent taxation. To illustrate, we consider the example with =1and =2. We can plot the tax schedules under both taxation regimes or any given value o k. Thecasewithk =0.5 isgiveninfigure3below.it is evident that or this case the tax schedule under independent regime is everywhere latter, which strengthens our analytical result given in Proposition 3. Generally speaking, higher types are taxed less and lower types get less subsidy under the independent system. 11

12 Uniied Regime T () Independent Regime Figure 3: Tax Schedule Comparison with Uniorm Distribution With these results at hand, we are now ready to examine the determinants o constitutional choice with a continuum o types. Constitutional Choice With continuous types the constitutional choice is determined by the median voter s preerence. As in the three-type model, the preerence o the median type can only be obtained using numerical computations. We thus go back to our model with general distributions or vertical types to characterize constitutional choice as a unction o the mobility parameter, the distribution o relative classes (the types), and the distribution o income. With any given distribution F (density unction ), our computations can be done based on (8) and (17). Since the Pareto distribution is commonly adopted to proxy real world income inequality in the taxation literature, we consider the ollowing truncated Pareto distribution amily: () = α α α and 1 F () = α 4 α 1 4 α, [1, 4].8 (24) Note that the uniorm distribution is a special case o the Pareto distribution amily (with α = 1). As α increases, the density becomes more tilted toward lower types (more poor people). The tax schedules under two taxation systems are compared in Figure 4 below (plotted or the case α =1and k =0.5), which exhibits the same pattern as in the case o uniorm distribution. 8 With the support o being [1, 4], the highest type s pre-tax income is 16 times that o the lowest type. 12

13 Uniied Regime 1 T ( ) Independent Regime Figure 4: Tax Schedule Comparison with Pareto Distribution Recall that with uniorm distribution we established that the utility schedule v I crosses v U once rom below. Our computation shows that this pattern o single crossing holds or truncated Pareto distributions as well. Let be the indierence type at which v I crosses v U. Then all the types below preer the uniied regime and all the types above preer the independent regime. The ollowing table shows how the indierence type shits as k changes (or the truncated Pareto distribution, the computations are done based on the case α = 0.15). Table 1: How shits as k changes k =1 k =0.5 k =0.3 k =0.2 k =0.1 k =0.03 Uniorm [1, 3] Pareto [1, 4], α = The above table indicates that is monotonically decreasing in k. ThisisconsistentwithResult?? in the three type model. Thereore, as the moving cost decreases, the measure o citizens who preer the uniied regime increases. As a result, the uniied regime is more likely to be chosen at the constitutional stage or a smaller moving cost, other things equal. The intuition or this result is analogous to that provided in the three type model. As k decreases, the previously indierent type (the median type) beneits less rom the presence o the rich (all the types above her), hence will switch her preerences toward the uniied regime, whose solution does not depend on k. For the range o mobility parameter k reported in the table, the uniied regime is always chosen in the uniorm distribution case (since the median type m =2). However, or the truncated Pareto distribution case, the median type is m = Hence the independent regime will be chosen or 13

14 cases k =0.3, 0.5, and 1, and uniied regime will be chosen or cases k =0.01, 0.1, and 0.2. We are also interested in how changes in the (type) income distribution aect the constitutional choice. Fix k =0.5, and consider the truncated Pareto distributions given in (24). The ollowing table reports how the indierence type and the median type m change as α varies: Table 2: How and m shit as α changes α m For all the cases we examined, the solutions exhibit perect sorting. Two observations are worth noting. First, as α increases (more poor around), the indierent type monotonically decreases. Again this is consistent with what we ound rom the three type model. This is intuitive: having more poor implies more taxes rom the higher types in the uniied regime, while in the independent regime the solution is closer to autarky. Thereore, the indierence type will decrease, as in Result??. However, i α is suiciently large (α > 0.15), the median type preers the uniied regime. Thus having more poor people in this continuous type case makes the choice o the uniied system more likely, which seems to be inconsistent with our inding in the three type model. This happens in this Pareto distribution case simply because the indierence type decreases slower than the median type: as the size o the poor increases, the median type becomes even poorer. This observation highlights a dierence between our three-type model and the continuous type model, that is, themediantypeis generically dierent rom the type who is indierent between the various constitutional choices, and they vary at dierent rates when the parameters change. Finally, we study how the degree o inequality aects constitutional choice by examining a distribution amily with mean preserving spread. Again, we ix k =0.5. Consider the ollowing distribution amily: 1 a () = a[10 a(2 )2 ], [1, 3] with a [0, 10). Thecasea =0corresponds to the uniorm distribution. As a increases, the distribution becomes more concentrated around the mean or median (which is 2 in this case), so inequality decreases. The computation results are reported in the ollowing table. ( is once again the cuto type who is indierent between the two tax regimes): Table 3: How shits as inequality parameter changes 14

15 a =0 a =3 a =5 a =7 a = The table shows that the relationship between inequality and the indierence type is not monotonic in this particular continuous type distribution case. 15