A Complete Example of an Optimal. Two-Bracket Income Tax

Transcription

1 A Coplete Exaple of an Optial Two-Bracket Incoe Tax Jean-François Wen Departent of Econoics University of Calgary March 6, 2014 Abstract I provide a siple odel that is solved analytically to yield tidy expressions for the Pareto e cient tax structures and the optial twobacket arginal tax rates. It is for the special case of equally-sized groups of two skill types and no exogenous spending requireents of the governent. The results and the exposition give a self-contained treatent of the central ideas of optial incoe taxation. JEL codes: A22, A23, H21 Keywords: econoic education, optial taxation 1

2 1 Introduction The theory of optial incoe taxation exaines the tradeo between equity and e ciency in designing the personal incoe tax schedule. It fors a cornerstone of odern public nance and garnered a Nobel Prize for Jaes Mirrlees and Willia Vickery in The subject is very uch alive today in both public discourse and acadeic research. For exaple, French president Francois Hollande kept to his capaign pledge by iposing a 75 percent tax on earnings over 1 illion euros at the end of 2013, while a study by Diaond and Saez (2011) suggests that the top optial incoe tax rate in the United States should be 73 percent, which is uch higher than the current top rate. The theory of optial incoe taxation appears in undergraduate textbooks (e.g. Rosen et al. 2012) and in ore advanced textbooks (e.g. Salanié 2011). However, the atheatical treatent of optial incoe taxation relies on the theory of optial control, which is often not taught in econoics curriculla. A siple pedagogical exaple can therefore be useful for deonstrating soe key ideas of the theory. In this spirit, Slerod et al. (1994) use indi erence curve diagras to construct Pareto e cient tax structures and they provide a nuerical siulation of the social welfare axiizing tax rates, when only two tax brackets are peritted and there are two (equally-sized) classes of workers ( high and low skill); and there are no exogenous spending requireents. 1 This is an instructive exaple, but 1 The authors also provide optial tax siulations for the case of a large nuber of skill classes. The restriction to two tax brackets is advocated by Slerod et al. as a eans 2

3 the reliance on nuerical solutions ay be less satisfying or convincing for students than an algebraic result. Hence, in this article, I construct a siilar two-bracket exaple but with a coplete analytical solution. In deriving the social optiu two-bracket tax structure, I use and explain Slerod et al. s proposition regarding Pareto e cient tax structures. I derive the Pareto e cient tax schedules for the increasing arginal tax rate case and the decreasing arginal tax rate case, and then I show that the social welfare axiizing tax schedule has decreasing arginal tax rates, regardless of the relative weight attached to the low-skilled group. The expressions for the optial tax rates are very siple. My exaple odel follows the typical optial tax setup. A worker is endowed with a given skill level which is re ected in the wage she receives. The governent would like to engage in redistributive taxation fro the high-skilled to the low-skilled, but is assued to be unable to condition tax liabilities directly on skill level. Hence, the second fundaental welfare theore is o the table. Taxing a worker s observed incoe con ates the endowent of skill (wage) with the choice of hours of labor e ort, thereby distorting the labor-leisure argin and inducing excess burden. In this context, the literature has produced several general insights on the structure of the optial tax schedule. In particular, there is the faous zero arginal tax rate at the top result and the soewhat less known zero arginal tax rate at the botto result. 2 My exaple focuses on these two insights. of econoizing on taxpayer copliance costs. 2 These results necessarily hold only when certain boundedness conditions apply on the 3

4 2 Policy Insights I brie y discuss the two ain insights before turning to the illustrative odel. 2.1 Zero at the Top If no one in the population has an incoe (or ore precisely a wage rate) higher than a certain level, then the arginal tax rate should be zero at the top of the incoe scale (Mirrlees 1971, Seade, 1977). Suppose, on the contrary, that the top incoe tax rate is positive. Lowering the rate a little bit increases the top earner s reward, which will induce hi to work a little ore. He will be better o (by revealed preference) and ore tax revenue will be generated, which can be used to redistribute incoe to lower earners. This arguent can be repeated until the top arginal tax rate is zero. A key assuption for the result is that the highest attainable skill level (wage) is known. When this is not the case, siulations by Mirrlees (1971) suggest that, while the optial top tax rate ay not be zero, the optial tax structure is characterized by decreasing arginal tax rates near the top. A arginal tax rate schedule with this degressive feature was introduced in the Swiss cantons of Scha hausen in 2004 and Obwalden in 2006, although degressive incoe taxation was rejected by the Swiss federal court as unconstitutional in incoe distribution. This will be clari ed below. The practical iportance of these two results is disputed since they provide little guidance on the structure of optial taxation outside the top and botto extrees. However, students of public nance ust at least understand the results. 4

5 2.2 Zero at the Botto If no one in the population earns zero incoe (i.e., no one is idle) in the optial arrangeent, then the arginal tax rate should be zero at the botto of the incoe scale (Seade 1977). Suppose, on the contrary, that the botto rate is positive. The revenue raised by this arginal tax rate cannot be used to nance redistribution downwards because there is no one further down the scale. Hence, the revenue loss has no equity iplication, so that only e ciency atters for the arginal tax rate at the botto. This iplies the botto tax rate should be zero. The key assuption here is that everyone earns positive incoe under the optial tax schee, which in turn requires the wage rate at the botto of the incoe scale not to be too low. Otherwise, it ay be optial to tolerate idleness aong the least productive workers in order to generate higher tax revenues fro the ore productive workers by setting a positive the arginal tax rate at the botto Pareto E cient Taxation When there are only two arginal tax rates allowed in the syste, then obviously zero at the top and zero at the botto cannot both hold siultaneously, as there would be no tax revenue. As we shall see, Pareto e cient tax structures in a two-bracket tax syste will feature one or the other of these two end of the scale results. A Pareto e cient tax structure eans that there is no change in the tax syste that preserves budgetary balance while aking at least one person better o without haring others. 5

6 Any tax structure that is not Pareto e cient obviously cannot be social welfare axiizing. The speci c social welfare function and the distribution of skill levels then deterine which of the Pareto optial tax structures is socially best. 3 A Siple Model Preferences Suppose the utility function is quasi-linear in consuption (c) and labor (L): 3 u(c; L) = ln c "L: (1) The budget constraint of an individual requires consuption expenditures to equal after-tax incoe: c = T () (2) where = wl and T () is the tax liability at the incoe level and the price index for consuption goods has been noralized to equal one. A useful device for optial incoe tax analysis is to rewrite the utility function in ters of consuption and before-tax incoe (instead of labor e ort) by 3 This is equivalent to the utility function u(c; l) = ln c "(1 l), where l is leisure, after substituting for l using the tie constraint l + L = 1. For the utility function (1), the elasticity of substitution between labor and consuption is 1. The uncopensated elasticity of labor supply is 0, while the copensated elasticity of labor supply is 1. The utility function (1) di ers fro the one used by Slerod et al. (1994). 6

7 substituting =w for L into (1) to obtain u(c; =w) = ln c "=w (3) where the wage rate w is treated as a paraeter of the utility function and is a choice variable. Note that utility is decreasing in before-tax incoe because a higher level of requires ore labor e ort. On a diagra with c on the vertical axis and on the horizontal axis the direction of increasing utility is toward the north-west, where consuption is highest and before-tax incoe i.e., labor e ort is lowest. Each indi erence curve slopes upward at an increasing rate. That is, the arginal rate of substitution between c and is given by MRS ;c = dc d = "=w 1=c = "c w > 0 and the total derivative of the MRS ;c with respect to is d (dc=d) d = " w dc " 2 d = c > 0: w Single-Crossing Property An iportant property of the transfored utility function (3) is the socalled single-crossing property, which will enable the social planner to design the tax schedule in a anner that separates the di erent skill types in the equilibriu. The single-crossing property states that the indi erence curves 7

8 are less steep the higher the d = "c < 0: (4) w2 Figure 1 copares indi erence curves for a high- versus low-skilled worker that pass through a given point (c 0 ; 0 ) Two-Bracket Tax Schedule Suppose the tax syste is restricted to having two arginal tax rates: 1 on incoe below a threshold of and 2 on incoes above the threshold. To allow for redistribution a deogrant (i.e. a lup-su transfer) B is paid to everyone. Thus the two-bracket tax syste is described by T () = B + 1, if < (5) T () = B ( ), if. (6) Figure 2 illustrates the after-tax budget constraint (2) for an individual in the space of c and. Its slope is dc=d = 1 T 0 (), where T 0 () is the arginal tax rate, which equals 1 for < and equals 2 for. In the absence of taxation the budget constraint would be given by the 45 degree ray fro the origin. For exaple, earning B would enable a consuption level c B as indicated by the point B 00. However, as a result of taxation, the incoe B buys the lesser aount ^c B as shown by point B. Hence, the vertical di erence 8

9 BB 00 is the tax revenue collected fro an individual earning B. Notice that this is equivalent to the horizontal distance to the 45 degree line indicated by BB 0. Now consider a di erent incoe level, A, which in Figure 2 lies to the left of where the after-tax budget constraint crosses the 45 degree line. An individual earning A has a negative tax liability. That is, he receives a positive net transfer (B 1 A > 0) fro the governent, equal to the horizontal distance AA Social Welfare Maxiization For any given tax schedule an individual axiizes utility by nding the point in the space (c; ) that puts her on the highest feasible indi erence curve. I shall assue that there are only two skill levels, high (H) and low (L), characterized by their wages w H > w L. Given a tax schedule, denote the corresponding utility axiizing incoe levels of the high- and lowskilled workers as H and L and the optial consuption levels as c H and c L. If there are n H high-skilled workers and n L low skilled workers and the governent has an exogenous revenue requireent R then the equation for budgetary balance is n H T ( H) + n L T ( L) = R: (7) 9

10 Assue that social welfare is given by the function! = n H u(c H; H=w H ) + n L u(c L; L=w L ) (8) where is a welfare weight on the utility of low-skilled individuals. The objective of optial incoe taxation in this exaple is to choose the vector ={ 1, 2,, B} to axiize (8) subject to (7). I will ipose further that R = 0 and n H = n L. The rst assuption eans that the only otive for taxation is redistribution. This assuption cobined with the second one allows e to easily depict budgetary balance in a diagra. 4 Pareto E cient Tax Structures To characterize the optial two-bracket incoe tax syste, I begin by identifying the characteristics of Pareto e cient incoe tax structures. I shall only consider progressive tax structures with non-negative arginal tax rates ( 1 0, 2 0) and a positive deogrant (B > 0). 4 A progressive tax structure is one where the share of incoe paid in taxes is rising with incoe: d[t ()=] d > 0. The two-bracket syste is progressive if ( 1 2 ) < B. A basic proposition of Slerod, Yitzhaki, Mayshar, and Lundhol (1994), which tailors the analysis of Sadka (1976) and Stiglitz (1982) to a two-bracket tax syste, is the following. 4 It is possible for an optial tax structure to consist of negative arginal tax rates and a negative deogrant. These are not seen in reality and I ignore the possibility. 10

11 Let L denote the before-tax incoe of the low-skilled at the low-skilled consuer s optiu point, and let u H denote the indi erence curve of the high-skilled corresponding to the high-skilled s consuer optiu point. Then the socially optial two-bracket incoe tax syste = ( 1 ; 2 ; ; B) can be restricted to one of the following two cases: (1) Decreasing arginal tax rates (1) 1 > 2 = 0, with u H touching both branches of the budget constraint. (2) Increasing arginal tax rates (2) 0 = 1 < 2, with L = Part (1) of the proposition is a case of a progressive decreasing arginal tax rate schedule ( 1 > 2 ). Part (2) is a case of a progressive increasing arginal tax rate schedule ( 1 < 2 ). Both cases are progressive if the low-skilled are net transfer recipients and the high-skilled are net taxpayers (assuing there is no governent spending except for the deogrant). Each part of the proposition contains two stateents: a stateent about which tax rate is optially set to zero; and a stateent about tangency conditions between an indi erence curve and the budget constraint. Figure 3 illustrates Part (1) of the proposition and gure 4 illustrates Part (2). Each part of the proposition can be proven with diagras siilar to 11

12 gures 3 and 4. In order to ake this exposition self-contained, I provide the geoetric proofs in the appendix, although they are available in Slerod et al. (1994). I now use the proposition to derive analytical expressions for the Pareto e cient tax structures and the optial two-bracket arginal tax rates. 5 An Algebraic Solution I now characterize algebraically the two Pareto e cient tax structures for the odel and then I will deterine which one generates the highest social welfare. 5.1 Decreasing Marginal Tax Rates ( 1 > 2 = 0) Fro Part (1) of the proposition for Pareto e cient tax rates, we can ipose 2 = 0. Then the low-skilled s utility axiization proble is to nd u L = ax L ln [(1 1 ) L + B] " L w L (9) giving the solution 5 L = w L " B 1 1 : (10) Substituting (10) back into the utility function (9) gives an expression for the indirect utility function. To siplify the notation, fro herein I shall 5 We can verify later that the optial tax structure indeed iplies that L > 0. 12

13 noralize the low-skilled wage to unity: w L = 1. Hence u L = ln(1 + ") + ln (1 1 ) + "B=(1 1 ): (11) Recall that for the high-skilled we need to nd two tangency points between an indi erence curve and the budget constraint. One tangency point is on the segent with the arginal tax rate 1 > 0; the other tangency point is along the segent with 2 = First Segent Tangency Analogous to the low-skilled s optiization proble (9) the high-skilled worker solves u 1 H = ax H ln [(1 1) H + B] " H w H ; (12) yielding 1 H = w H " B 1 1 (13) and u 1 B H = ln(1 + ") + ln(1 1 ) + ln w H + " ; (14) (1 1 )w H where I have replaced the superscript asterisk on u H and H with 1 to indicate that these are the levels of utility and before-tax incoe corresponding to the rst segent of the budget constraint. 13

14 5.1.2 Second Segent Tangency Along the second segent of the budget constraint 2 = 0 and the optial incoe 2 H exceeds the threshold. The utility axiization proble is u 2 H = ax ln ( H 1 + B) " H w H : (15) The solution is 2 H = w H " + 1 B; (16) where again the superscript 2 on H indicates optiality on the second segent of the budget constraint. After substituting (16) into (15) and siplifying the expression, the indirect utility function is u 2 H = ln(1 + ") + ln w H + " B 1 w H : (17) Double Tangency Condition The next step is to equate the utilities u 1 H and u2 H since they are associated with the sae indi erence curve. After soe rearrangeent of ters, I obtain one of the key equations of the solution to the e cient decreasing arginal tax rates. ln(1 1 ) = " w H 1 B + 1 : (18)

15 5.1.4 Budget Balance The second key equation for the solution is the requireent that governent expenditures equal tax revenues, when the low-skill have incoe L given by (10) (with w L = 1) and the high-skill have incoe 2 H given by (16). Thus, T ( L ) + T (2 H ) = 0 can be written, after soe rearrangeent of ters, as B = 1 (1=" + ) (1 1) (2 1 ). (19) Pareto E cient Decreasing Tax Rates I can now use the equations (19) and (18) to derive expressions for B and in ters of 1 and the paraeters of the odel (w H and "). After soe tedious algebra, I obtain = 1 2" w H 2" B = 1 (1 1 ) " (2 1 ) (2 1 ) ln(1 1 ) (20) 1 ( 1 ) 2 (1 1 ) (1 1 ) w H ln(1 1 ): (21) 2" (2 1 ) 2" Finally, the solution for the Pareto e cient value of 1 > 0 can be copleted by choosing 1 to axiize the indirect utility function of a representative low-skilled individual u L given by (11), after substituting the right-hand side of (21) for B. This is equivalent to u L = ax 1 (2 w H ) ln(1 1 ) + 1 (22) 15

16 yielding the elegant solution 1 = w H 1 (23) which is positive as w H > 1 by assuption. To ensure that 1 < 1, an upper bound w H < 2 is required on the high-skilled wage. Hence, 1 = w L < w H < 2 is iposed on the odel. Notice that as w H approaches 1 the optial tax rate 1 goes to zero. That is, as inequality vanishes, so does the need for redistribution. The solutions for and B follow fro substituting 1 into (20) and (21). 1 = w H (3 w H) 2" (w H 1) ln(2 w H) + (w H 1) (24) B = (2 w H) ((w H 1) w H ln(2 w H )) : (25) 2" Low-Skilled Labor Supply Before proceding to the welfare calculation, we ust verify whether the lowskilled workers engage in non-negative labor supply at the proposed solution. That is, we ust check that L 0 in (10) after setting w L = 1, 1 = w H 1, and substituting for B using (25). Low-skilled workers supply non-negative labor until the high-skilled wage w H reaches approxiately When w H exceeds this aount, L = 0, since negative labor supply is eaningless. This eans that our solution is only applicable for high-skilled wages below 6 The critical value for w H is given by the equation w H (1 ln(2 w H )) = 3: 16

17 the critical point. For higher values of w H, the deogrant and optial rstbracket tax rate 1 ust be recalculated based only on the incoe H of the high-skilled. I shall ignore this discontinuity in the solution by further restricting the high-skilled wage to be below the critical level, call it wh c, such that low-skilled workers supply positive labor. This iplies a critical axiu value for 1 = wh c Social Welfare with Decreasing Tax Rates The optiized utility levels are obtained by substituting the solution for 1 into (22) to obtain u L gives and the solutions for 1,, and B into (17). This u L = (1 + ln ") + (w H 1) + (2 w H) ln(2 w H ) (26) 2 2 u H = u L + ln w H : (27) Social welfare under a Pareto e cient decreasing tax structure is then! = u H + u L = (1 + )u L + ln w H (28) = (1 + ) (1 + ln ") + (w H 1) + (2 w H) ln(2 w H ) (29) ln w H : 17

18 5.2 Increasing Marginal Tax Rates (0 = 1 < 2 ) Fro Part (2) of the proposition for Pareto e cient tax rates, we can ipose 1 = 0. The low-skilled s utility axiization proble is u L = ax L ln ( L + B) " L w L ; (30) which, after inserting w L = 1, yields, L = 1 " B (31) and u L = (1 + ln ") + "B (32) The high-skilled worker pays a tax rate of 2 only on incoe exceeding. Her objective is u H = ax H ln ((1 2 ) H B) " H w H ; (33) resulting in and H = w H " 2 B (34) u H = (1 + ln ") + ln w H + ln(1 2 ) + " 2 "B + : (35) (1 2 )w H (1 2 )w H 18

19 5.2.1 Budgetary Balance and Tangency Condition Budgetary balance requires T ( L )+T ( H ) = 0 where L and H are given by (31) and (34). This iplies 2 H 2 = 2B: (36) Fro Part (2) of the proposition, it is required that L =. Cobining this requireent with (36), and using the expression for H, gives 2 wh " 2 B " B = 2B; (37) which can be rearranged to obtain a second expression for : = (w H 1)(1 2 ) " 2 B ( 2 ) ( 2 ) 2 : (38) The expression (38) for can be equated to L, given the tangency condition of Part (2), to deliver a solution for B in ters of 2 : B = (w H 1) 2 2" ( 2 ) 2 2"(1 2 ) : (39) Our nal task for solving the optial tax rate is to axiize the uility of the low-skilled, u L given by equation (32), after substituting for B using (39).7 7 Note that L > 0 when (39) is used to substitute for B in (31). Hence, the low-skilled supply positive labor e ort in equilibriu. 19

20 The axiization proble is u L = ax 2 2 (w H 1) ( 2 ) : (40) The solution to the rst-order condition is 2 = 1 p 1=w H : (41) Only the negative root is adissable as otherwise 2 > 1. Hence, the optial value of 2 is 2 = 1 (w H ) 1=2 : (42) Observe again that if inequality vanishes (w H! 1) then 2 approaches zero, since the otive for redistribution disappears. By substituting 2 into (39) and anipulating the ters, I obtain the solution for B : B = 2 w 1=2 H 1 : (43) 2" In turn, B is substituted into (31), to nd the optial threshold, using the fact that = L : = 2 (w H) 1=2 1 2 : (44) 2" 20

21 Finally, the optiized values of utility and social welfare are as follows: u L = (1 + ln ") + (w H) 1=2 1 2 (45) 2 u H = (1 + ln ") + (1=2) ln w H + (w H 1) (46) 2w H! = (1 + )(1 + ln ") + (1=2) ln w H + (w H 1) 2w H (47) + 2 (w H) 1=2 1 2 : 5.3 Welfare Optiu: Decreasing or Increasing Marginal Tax Rates? The Pareto e cient tax structures have been solved for, but which one yields the highest level of social welfare? The answer requires a coparison of the social welfare values for the decreasing arginal tax rate (DMRT) case, given by (29), and the increasing arginal tax rate case (IMRT), given by (47). Thus, =! (DMRT)-! (IMRT) (48) (wh 1) = (1 + ) + (2 w H) ln(2 w H ) 2 2 (w H 1) +(1=2) ln w H 2w H 2 (w H) 1=2 1 2 = (1=2) (w H 1) 2 =w H + (2 w H ) ln(2 w H ) + ln w H (49) h +(=2) (w H 1) (w H ) 1=2 1 i 2 + (2 wh ) ln(2 w H ) : 21

22 An inspection of (49) reveals that every ter in it is positive. Hence, the optial decreasing arginal tax rate structure generates the highest social welfare, regardless of the size of the welfare weight. The zero arginal tax rate at the top case prevails in this exaple, which corroborates the observed tendency often reported by researchers using nuerical siulations, that the welfare optiu features declining arginal tax rates near the top of the incoe distribution. 6 Conclusion In this article, I have provided a siple odel and deonstrated step-by-step how to solve it analytically to yield Pareto e cient tax structures and the social welfare axiizing tax policy. It is for the special case of two arginal tax rates, equally-sized groups of two skill types, and no exogenous spending requireents of the governent. The results and the exposition give a selfcontained treatent of the central ideas of optial incoe taxation. The presentation builds on the analysis of Slerod et al. (1994). In contrast to that paper, however, I present closed for solutions for the optial tax rates. Closed for solutions to optial incoe tax probles are rare in the literature, which usually deals with ore coplicated versions of the odel. The optial tax structure that arises fro the analysis con rs the zero at the botto result that is coonly found in optial tax odels. The current edia focus on incoe inequality and taxation shows that the 22

23 optial incoe tax proble is very uch alive in public debate. 23

24 References [1] Diaond, P. and Eanuel S The case for a progressive tax: fro basic research to policy recoendations. Journal of Econoic Perspectives 25(4): [2] Mirrlees, J. A An exploration in the theory of optial incoe taxation. Review of Econoic Studies 38(2): [3] Rosen, H.S., Wen, J.-F., Snoddon, T Public nance in Canada. Toronto: McGraw-Hill Ryerson. [4] Sadka, E On incoe distribution, incentive e ects and optial incoe taxation. Review of Econoic Studies 43(2): [5] Salanié, B. 2011, The econoics of taxation. Cabridge, MA: MIT Press [6] Seade, J. K On the shape of optial tax schedules. Journal of Public Econoics 7(1): [7] Slerod, J., Yitzhaki, S., Mayshar, J. and Lundhol, M The optial two-bracket linear incoe tax. Journal of Public Econoics 53(2): [8] Stern, N.H On the speci cation of odels of optial incoe taxation. Journal of Public Econoics 6: [9] Stiglitz, J Self-selection pareto e cient taxation. Journal of Public Econoics 17:

25 7 Appendix 7.1 Pareto E cient Decreasing Marginal Tax Rates Zero at the Top: Why 2 = 0? I will rst show that if 1 > 2, then 2 ust equal zero, as otherwise there is a Pareto-iproving change to the incoe tax syste. In gure 5 the initial budget constraint is given by OD BE and has a kink at point D (at incoe 0 ), such that 1 2 > 0. Thus both arginal tax rates are positive along the segents foring OD BE, contrary to the requireent of Part (1) for a Pareto e cient structure. The low-skilled locate at point A and the high-skilled locate at B, that is, where the respective indi erence curves are tangent to the budget constraint. With our assuption of equal nubers high- and low-skilled workers, the net transfer to each low-skilled worker, shown as AA, ust equal the net tax payent of each high-skilled worker, shown as BB. Note that the single-crossing property is what ensures that a two-bracket schedule can be designed such that only the high-skilled locate on the second segent of the budget constraint, earning ore before-tax incoe than do the low-skilled. Now increase the cuto for the rst tax bracket 0 to and reduce 2 to zero. The budget constraint becoes ODBE, with the segent DBE cutting through the previous tangency point B. The lower ability individuals reain at the sae equilibriu point as before (point A) but the higher 25

26 ability individuals are better o soewhere along the segent DBE than they are at their previous optiu at B. In the illustration the high-skilled now choose point F on the segent DBE where they earn ore incoe and they attain higher utility. This change in 2 has no tax revenue iplications: the horizontal distance between the 45 degree line and point F is, by construction, the sae as the horizontal distance BB. Thus it could not have been optial to have a positive 2 whenever it is optial to have 2 1. Notice that, even though 2 = 0, the tax schedule ebodied in the budget constraint ODBE reains redistributive fro the high-skilled toward the low-skilled. In the construction of gure 5, I noted that the low-skilled are no worse o by reducing 2 to zero while increasing the cuto, but the high-skilled are better o. However, the tax syste can also be changed in such a way that both skill types are strictly better o when 2 is reduced to zero. Consider gure 6, where points B and F correspond to the points shown previously in gure 5. Suppose the cuto is increased all the way to 00 while setting 2 = 0. The high-skilled locate at point G on the segent D GE, where they are clearly better o than at point B. However, at G there is ore tax revenue than at B, since G lies further to the right of the 45 degree line than point B. The additional tax revenue can be distributed to both skill types in the for of a larger deogrant, B. Geoetrically, a larger B eans the entire budget constraint is shifted horizontally to the left. The low- and high-skilled will adjust their labour supply along the new budget constraint (i.e., their 26

27 optial incoes, L and H ). The process stops when budgetary balance is restored. In the nal equilibriu, everyone is on a higher indi erence curve than they began at (i.e. when 2 > 0). The iportant point, however, is that Pareto iproveents are possible until 2 is reduced to zero Tangency Condition We can now focus on the tangency condition in Part (1) to fully characterize the optial two-bracket incoe tax syste with decreasing arginal tax rates. Figure 7 shows a tax schedule ODD E, where 2 = 0 but the indifference curve u H does not touch both branches of the budget constraint, as required by the tangency condition of Part (1). The low-ability types locate at point A and the high-ability types locate at point C. It is then possible to increase the cuto fro 0 to and at the sae tie to lower 1 such that: (i) the new segent O AD cuts through point A (i.e. where the low-skilled s indi erence curve u L is tangent to the line OAD) and (ii) the indi erence curve u H is now tangent to the lower branch O AD at point B and tangent to the upper branch D CE at point C. Assuing the high-ability individuals continue to choose point C, they are unperturbed by the tax refor; but the low-ability individuals are strictly better o at soe points on the branch O AD than they are at A. In particular, they face a lower arginal tax rate and they work ore than before, for exaple at point F, where indi erence curve u 0 L is tangent to the line O AD. The aount of net transfers they receive declines, because point F ust lie to the right of point A, and, since 27

28 the net taxes paid by the high-skilled has not changed, the governent has extra oney on its hands, which it can disburse as a higher deogrant. The higher deogrant in turn would shift the entire budget line O FD CE to the left in parallel way, so that all individuals will be strictly better o copared to the original tax schedule. Thus any degressive optial two-bracket tax structure with two ability types requires u H to touch both branches of the budget constraint, as required by Part (1). It is not optial to ake the line segent O AD any steeper than what is depicted in the gure, that is, to reduce 1 any further, as this would induce the high-skilled workers to abandon point C on the second budget segent, in favour of a point on the rst budget segent, where their tax payents would be lower and the governent could no longer a ord its existing level of redistribution. Notice, too, the role of the single-crossing property in this construction. It is the relative shallowness of the highskilled s indi erence curves that leads to a tangency on the segent O FD at point B while the low-skilled achieve a tangency along the sae segent at point F. Consequently, in the equilibriu, the high-skilled earn ore incoe than the low-skilled; and redistribution fro those endowed with w H toward those with w L can occur through the incoe tax syste even without the governent being able to identify beforehand the skill levels of the di erent individuals. 28

29 7.2 Pareto E cient Increasing Marginal Tax Rates Zero at the Botto: Why 1 = 0? Now let us see why the proposition requires 1 = 0 when the arginal tax schedule is increasing. Figure 8 shows a situation where 2 1 > 0. The initial tax schedule is depicted as O AD E (with a kink at D ). In this case, one can lower the incoe threshold fro 0 to and reduce 1 to zero, such that the budget constraint becoes OADE, with the segent OAD being parallel to the 45 degree line, re ecting 1 = 0. This tax change will leave the high-skill individuals unperturbed at B while raising the utility of the low-skill individuals, without changing their tax bene ts. The low-skilled now locate at point F on the line OAD, where their utility is higher than at point A. Thus Figure 8 establishes that an increasing two-bracket arginal tax rate schedule cannot be optial unless the botto rate is zero Tangency Condition Pareto e ciency also requires that the low-skilled indi erence curve be tangent to the budget constraint precisely at the cuto incoe level. In gure 9 the schedule ODBE features 1 = 0 but L 6= thus violating the tangency condition of Part (2). As shown in gure 9, it is then possible to reduce 0 to such that the tax schedule now has a kink at point A. Low-ability individuals continue to locate at point A, where their indi erence curve u L is the highest achievable along OABE. Thus they get the sae 29

30 net transfer as before. At the sae tie, high-ability individuals becoe strictly better o by choosing a point such as B along the budget constraint segent ABB E copared to their utility at point B. In particular, they face a lower arginal tax rate, they work ore, pay ore in net taxes, and they consue ore. The additional tax payents can be used to raise the deogrant, which would shift OABB E to the left in a parallel fashion and ake everyone better o. Thus L = is a necessary condition for Pareto e cient tax structures, as required by Part (2). 30

31 c Figure 1 u L u H c The indifference curves exhibit the single-crossing property.

32 Figure 2 c c B B B Tax B Slope = 1 τ 2 A A Slope = 1 τ o A B An illustration of the budget constraint.

33 Figure 3 c 45 o line u H u L 0 * L * H An illustration of a Case (1) optial tax: τ 1 > 0, τ 2 = 0.

34 Figure 4 c 45 o line u H u L 0 * L * H An illustration of a Case (2) optial tax: τ 1 = 0, τ 2 > 0.

35 Figure 5 c 45 o line E F E B u H B u L D D O A A 0 ' A progressive decreasing tax schedule: if τ 1 * > 0, then τ 2 * = 0 is optial.

36 Figure 6 c 45 o line E E F G E u H B u L D D D A '' 0 ' Points B and F are the sae as in figure 5. The cut-off has been increased '' to. The high-skilled now choose point G. They are better off than at point B and they pay ore taxes.

37 c Figure 7 45 o line u H C B D O O A F u L u L D 0 ' A progressive decreasing tax schedule. The indifference curve of the high-skilled touches both branches of the budget constraint and τ 2 = 0.

38 c Figure 8 45 o line u H E B u L D D F O O A 0 ' A progressive increasing tax schedule: if τ 1 > 0, then τ 2 = 0 is optial.

39 c Figure 9 45 o line u H E B E B u L D A O 0 * = L ' * A progressive increasing tax schedule: u L is tangent to the * budget constraint where L =, and τ 1 = 0.