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1 Economics Department Discussion Papers Series ISSN Unit Versus Ad Valorem Taxes: Te Private Ownersip of Monopoly In General Equilibrium Carles Blackorby and Susama Murty Paper number 10/11 URL: ttp://business-scool.exeter.ac.uk/economics/papers/

2 Unit Versus Ad Valorem Taxes: Te Private Ownersip of Monopoly In General Equilibrium* Carles Blackorby and Susama Murty November 2010 Carles Blackorby: Emeritus Professor, Department of Economics, University of Warwick and GREQAM: Susama Murty: Department of Economics, University of Exeter: *We tank te worksop members of CRETA at te University of Warwick and CDE at te Deli Scool of Economics. November 13, 2010

3 Abstract Employing a general equilibrium framework, Blackorby and Murty [2007] prove tat, wit a monopoly and under one undred percent profit taxation and uniform lump-sum transfers, te utility possibility sets of economies wit unit and ad valorem taxes are identical. Tis welfare-equivalence is in contrast to most previous studies, wic demonstrate te superiority of te ad valorem tax in a partial equilibrium framework. In tis paper we relax te assumption of one undred percent profit taxation and allow te consumers to receive profit incomes from ownersip of sares in te monopoly firm. We find tat, under certain regularity conditions, for any fixed vector of profit sares, te utility possibility sets of economies wit unit and ad valorem taxes are not generally identical. But it does not imply tat one completely dominates te oter. Rater, te two utility possibility frontiers cross eac oter. Additionally, employing a standard partial equilibrium welfare analysis, we sow tat te Marsallian social surpluses resulting from te two tax structures are identical wen te government can implement unrestricted transfers. JEL classification: H21 Keywords: Ad Valorem taxes, unit taxes, monopoly, private ownersip economy, general equilibrium, second-best Pareto optimality. November 13, 2010

4 Unit Versus Ad Valorem Taxes: Te Private Ownersip of Monopoly In General Equilibrium by Carles Blackorby and Susama Murty 1. Introduction In a recent paper 1 we sowed, in te context of a general equilibrium model wit a monopoly sector, tat te utility possibility frontier in te face of ad valorem taxes is identical to te utility possibility frontier wit unit taxes. Tis result is contrary to almost all of te previous literature, wic demonstrates te welfare superiority of te ad valorem tax. 2 Te caracteristic of Blackorby and Murty [2007] model tat generates te contradictory result is te assumption tat te government levied profit taxes of one undred percent rebating any resulting surplus as a uniform lump-sum transfer (also called a demogrant), wic is a standard assumption in te general equilibrium literature on indirect taxes. 3 It could be argued tat te differences in te earlier results and Blackorby and Murty [2007] are because, in contrast to te general equilibrium approac of te latter paper, muc of te earlier literature employs a partial-equilibrium framework. Te models in tis literature are usually silent about te end use of te monopolist s profit and te government s revenue. For tis reason, peraps, tese analyses could be interpreted as being performed in an institutional structure were te government as control of and can do unrestricted redistribution of te available economic resources among consumers. 4 Suits and Musgrave [1955] sowed tat for every ad valorem tax, tere exists an equivalent unit tax tat can support te profit maximizing output of te monopolist under te ad valorem tax, and vice-versa. Te asymmetry between te unit and te ad valorem taxes arises because te monopolist s profits and te government s indirect tax revenues under an ad valorem tax and te equivalent unit tax are not equal. 5 Blackorby and Murty [2007] sowed, owever, tat te sum of te government s revenue and monopoly profit does not cange in te move from te ad valorem tax to te equivalent unit-tax. Tis must imply tat, even in a standard partial equilibrium welfare analysis conducted in an institutional setting were te government can implement personalized lumpsum transfers, te Marsallian social surpluses sould be te same across bot te tax systems. Tis result seems not to ave been demonstrated by te earlier literature. 1 Blackorby and Murty [2007]. 2 See Cournot [1838, 1960], Wicksell [1896, 1959], Suits and Musgrave [1955], Skeat and Trandel [1994], Keen [1998], and Delipalla and Keen [1992]. Lockwood [2004] is an exception. 3 See, for example, Guesnerie [1995] and Guesnerie and Laffont [1978]. 4 I.e., it can implement personalized lump-sum transfers. 5 If te ad valorem tax is positive, te government revenue (te monopoly profit) is iger (lower) under te ad valorem tax as compared to te equivalent unit tax. 1

5 Tis must also imply tat, in an institutional setting were te government s redistributive ability is constrained and te total governmental revenue from profit and indirect taxation is rebated to consumers as uniform lump-sum transfers, te ad valorem tax and te equivalent unit tax result in, not only identical monopoly output and consumer prices, but also identical consumer incomes and demands. Tus, every ad valorem-tax equilibrium also as a unit-tax equilibrium representation. Te converse is also true. Tis is te welfare equivalence of ad valorem and unit taxes demonstrated in te Blackorby and Murty model. In tis paper we relax te assumption tat government can tax te profits of firms, and allow consumers to benefit directly from profit incomes from teir ownersip of sares in firms. Te problem raised by private ownersip is tat, given tat te monopolist s profits and te government s indirect tax revenues under an ad valorem tax and te equivalent unit-tax are different, for a fixed vector of profit sares, te profit incomes and te demogrant incomes of te consumers cange wen moving from a system of ad valorem taxes to an equivalent system of unit taxes; ence, in general, a given ad valorem-tax equilibrium is not a unit-tax equilibrium of te same private ownersip economy. Tus, tere is no direct way to compare te set of unit-tax equilibria wit te set of ad valorem-tax equilibria for a given private ownersip economy. To circumvent tis problems, in tis paper, we ave resorted to an indirect and somewat novel procedure wic draws eavily on earlier work in second-best economies by Guesnerie [1980] and Quinzi [1992] in a somewat different context. Our metod exploits te Suits and Musgrave result and te continuity of te difference in te consumer incomes under unit (respectively, ad valorem) taxes and equivalent ad valorem (respectively, unit) taxes. In tis paper, we first conduct a standard partial equilibrium welfare analysis employing an example wit quasi-linear preference to demonstrate te equivalence of te Marsallian social surpluses tat result from unit and ad valorem taxation of a monopoly wen te government can do unrestricted personalized transfers. We also use tis simple example to demonstrate te central problem created by private-ownersip in te welfare comparison of unit and ad valorem taxes wen tere are restrictions of government s redistributive ability, namely, it can only implement a demogrant. Furter, we use tis example to outline te strategy tat we adopt in tis paper to facilitate suc a comparison and sow tat it works for our example. Te rest of te paper provides te proof of te above claim in a rater general model. Our main result is tat, under certain regularity assumptions, private ownersip of te monopoly firm implies tat te unit-tax utility possibility frontier and te ad valorem utility possibility frontier must cross eac oter. Tat is, tere is a region were unit taxes Pareto dominate ad valorem taxes and anoter were ad valorem taxes dominate unit taxes. 2. A Two-Good Example Consider a two-good economy, were te good indexed by zero is supplied by a monopoly, wile te unindexed good is a non-produced good tat is consumed by te consumers and used as an input by te monopoly. 2

6 2.1. Preferences and tecnology. Suppose tere are H consumers in te economy. Eac consumer as quasi-linear preferences tat are linear in te non-produced good, wic are represented by utility function u = x +φ ( x 0) := x +b x 0 (x 0 )2, 0 x 0 b 2, x 0. Ten te individual and aggregate Marsallian demands for te monopoly good are independent of consumer incomes and te aggregate demand for te non-produced good depends only upon te aggregate income and not upon its distribution. Tus, te individual and aggregate demand functions, as a function of consumer prices (q 0,q) and consumer incomes, w, are 6 x 0 (q 0,q) = b q 0 2 x 0 (q 0,q) = b Hq 0, and 2, x (q 0,q,w ) = 1 q [w b q 0 q 0 ], (2.1) 2 x (q 0,q,w ) = 1 q [ b Hq 0 w q 0 ], 2 (2.2) were b := b. Assume also tat te monopolist faces constant marginal costs, so tat is cost function is C(y0 u,q) = cqyu 0. His demand for input is tus cyu 0. Tis framework lends itself to a usual partial equilibrium welfare analysis Te case of unit taxation and personalized lump-sum transfers. Consider te case of an economy were (i) te government can implement a unit tax on te monopolist, (ii) te monopolist takes te price of te input as given, (iii) te initial endowment of te input is ω, (iv) tere is no initial endowment of te monopoly good, (v) tere is no tax on te competitive commodity, and (vi) government as full discretion to distribute, as personalized lump-sum transfers (w ), its revenue from taxation of te monopoly good, te monopolist s profits 8, and te endowment of te non-produced good tat is unused by te monopolist. If te monopolist is subject to a unit tax t 0 ten te net-of-tax (producer) price tat e receives is p u 0 = q 0 t 0. Te monopolist solves te problem Π mu (t 0,q) :=maxp u 0x 0 (p u p u 0 +t 0,q) cx 0 (p u 0 +t 0,q)q, (2.3) 0 wic yields te profit maximizing price, quantity, and profits of te monopolist and te consumer price as te functions p u 0 = P0(t u 0,q) = b Ht 0 +Hcq 2H Π mu (t 0,q) = (b Hcq Ht 0) 2 8H, y0 u = x 0 (q 0,q) = b Hcq Ht 0, 4, and q 0 = P0(t u 0,q)+t 0 = b+ht 0 +Hcq. 2H (2.4) 6 We use te notation w is used to denote a vector (w 1,...,w H ). 7 See, for example, Capter 10 in Mas-Colell et al [1995]. 8 I.e., it can implement one-undred percent taxation of profits 3

7 Note tat for te profit maximizing price and quantity to be non-negative, we require tat t 0 b Hcq b+hcq H H. (2.5) An equilibrium in tis economy wit taxation of monopoly profits, unit taxation of te monopoly good, and personalized lump-sum transfer is described below: x 0 (p u 0 +t 0,q) = y u 0 p u 0 = P0(t u 0,q) x (p u 0 +t 0,q,w ) = ω cy0. u (2.6) It follows from Walras law tat at any suc equilibrium, te government s budget is balanced: w = Π mu (t 0,q)+t 0 y0 u +qω = (b Hcq Ht 0)(b Hcq +Ht 0 ) 8H +qω. (2.7) (2.6) is a system of 3 equations in 4+H unknowns, namely, p u 0,yu 0,q,t 0,w 1,...,w H. Also it is omogeneous of degree zero in p u 0,q,t 0,w 1,...,w H. Hence it admits a armless normalization, say q = 1. Te non-produced good ten can be interpreted as a numeraire commodity. Wit tis normalization, tere are H degrees of freedom in coosing tax equilibria in tis economy. For example, an equilibrium specified by (2.6) is uniquely determined once we fix t 0 and w 0 for = 1,...,H 1 suc tat (2.5) olds and H 1 =1 w (b Hc Ht 0)(b Hc+Ht 0 ) 8H + ω, i.e., fixing tese variables fixes te equilibrium levels of all te remaining variables, i.e., under te normalization adopted, (2.6) is caracterized by y0 u = b Hc Ht 0, p u 0 = b Ht 0 +Hc 4 2H w H = (b Hc Ht 0)(b Hc+Ht 0 ) 8H +ω H 1 =1 w. (2.8) Tus,equilibriadescribedby(2.6)arefullyparametrizedbyteH variablest 0,w 1,...,w H 1. In particular, note tat if we fix t 0, we get a wole set of equilibria corresponding to different distributions of te total income. In eac of tese equilibria, since t 0 is fixed, (2.8) implies tat p u 0, te individual and aggregate demands for te monopoly good (and ence te individual and aggregate expenditure on te monopoly good), and te demands for input by te monopolist are te same as tese variables are independent of consumer incomes. Te total amount of te numeraire good tat is available for distribution to consumers is te initial endowment of tis good minus te amount used as input by te monopolist: ω cy0 u = ω c[b Hc Ht 0] 4. Hence, tese equilibria will differ only wit respect to consumptions of (and expenditures on) te numeraire good by different 4

8 consumers. Te quasi-linear structure of te preferences implies tat equilibrium allocation corresponding to a given t 0 are obtained by transferring te available amount of te numeraire good unit for unit between consumers. Depending on te amount of te numeraire good, x, tat e receives, te utility of consumer for a fixed t 0 is given by u = (2b H b Ht 0 Hc)(2b H+b+Ht 0 +Hc) +x. We define te frontier of equilibrium utility 16H 2 profiles of tis economy tat are made possible wen t 0 is fixed as U u (t 0 ) := { u 1,...,u H = 1,...,H, x 0 suc tat x = ω c[ b Hc Ht 0 ] 4 and u = (2b H b Ht 0 Hc)(2b H +b+ht 0 +Hc) 16H 2 +x }. For all u 1,...,u H U u (t 0 ), we ave u = φ ( ) x 0(p u 0 +t 0,q = 1 + x ( ) w,p u 0 +t 0,1 = φ ( ) x 0(p u 0 +t 0,1 + ω cy0 u (2.9) = (2b H b Ht 0 Hc)(2b H +b+ht 0 +Hc) 16H 2 +ω c[b Hc Ht 0], 4 (2.10) wic is a constant, so tat U u (t 0 ) is linear. For any level of te unit tax t 0 and as in any partial equilibrium welfare analysis, call te expression M u (t 0 ) = φ ( ) x 0 (p u 0 +t 0,1) cy0 u (2.11) te Marsallian surplus. We can rewrite M u (t 0 ) as M u (t 0 ) = φ ( ) x 0 (p u 0 +t 0,1) cx 0 (p u 0 +t 0,1)+q 0 x 0 (p u 0 +t 0,1) q 0 x 0 (p u 0 +t 0,1) =[ φ ( ) x 0 (p u 0 +t 0,1) q 0 x 0 (p u 0 +t 0,1)] + [p u 0x 0 (p u 0 +t 0,1) cx 0 (p u 0 +t 0,1)] + t 0 x 0 (p u 0 +t 0,1) =[ φ ( ) x 0 (p u 0 +t 0,1) q 0 x 0 (p u 0 +t 0,1)] + Π mu (t 0,1) + t 0 x 0 (p u 0 +t 0,1). (2.12) Given te quasi-linear structure of preferences, te first term in te last equation of (2.12) is te consumer surplus resulting from te monopolist s profit maximizing coice. Tus, te Marsallian surplus M u (t 0 ) is te sum of consumer surplus, te government s tax revenue, and te profit of te monopolist wen te monopoly is subject to a unit tax. 5

9 2.3. Te case of ad valorem taxation and personalized lump-sum transfers. Now suppose tat te monopoly commodity is taxed in an ad valorem manner, wit te ad valorem tax denoted by τ 0, so tat te price faced by te monopolist is p a 0 = q 0 1+τ 0. Te problem of te monopolist is Π ma (τ 0,q) :=maxx 0 (p a 0(1+τ 0 ),q)[p a p a 0 cq] (2.13) 0 yielding te solution p a 0 = Pa 0 (τ 0,q) = b+hcq(1+τ 0) 2H(1+τ 0 ). 9 Note tat for te profit maximizing price and quantity to be non-negative, we require tat b cqh cqh τ 0 (b+2cqh2 ) 2cqH 2. (2.14) Consider an exactly same economy as in te previous section wit government being able to do personalized lump-sum transfers, but wit te monopolist facing an ad valorem tax on is good. Let te incomes consumer receive in tis ad valorem economy be denoted by R. An equilibrium can be defined exactly as in te unit tax case. A armless normalization q = 1 can be adopted and te set of equilibria are fully parametrized by te H variables τ 0, R 1,...,R H 1. Exactly as in te above section, we can also define U a (τ 0 ), te frontier of equilibrium utility profiles of tis economy tat are made possible wen τ 0 is fixed and, M a (τ 0 ) = [ φ ( ) x 0 (p a 0(1+τ 0 ),1) q 0 x 0 ] + Π ma (τ 0,1) + p a 0τ 0 x 0 (p a 0(1+τ 0 ),1), te Marsallian surplus corresponding to τ 0. (2.15) 2.4. Unit vs ad valorem: te case of personalized lump-sum transfers. As demonstrated out by Suits and Musgrave [1955], for every unit tax t 0 tere exists an equivalent ad valorem tax tat results in te monopolist coosing te same level of te output (and conversely). Tis is obtained by solving te following for τ 0 : x 0 (P u 0(t 0,q)+t 0,q) = x 0 (P a 0(τ 0,q)(1+τ 0 ),q) (2.16) Wit q = 1 tis implies b Hc Ht 0 = b Hc(1+τ 0) 4 4 and we obtain te equivalent ad valorem tax rate as (2.17) τ 0 = t 0 c. (2.18) 9 Itiseasytocecktatteproblemyieldsteprofitmaximizingquantityandprofitsoftemonopolist as well as te consumer price as y0 a = x 0 (q 0,q) = b Hcq(1+τ0) 4, Π ma (t 0,q) = (b Hcq(1+τ0))2 8H(1+τ 0), and q 0 = b+hcq(1+τ 0) 2H. 6

10 Since te profit maximizing outputs of te monopolist are te same under bot t 0 and its equivalent ad valorem tax, te demands for te numeraire by te monopolist as an input are also equal in te two scenarios. Tis means tat te amounts of te numeraire left tat can be potentially distributed to consumers are also same under t 0 and te equivalent ad valorem tax. Clearly: wen te government can do unrestricted transfers of te numeraire between consumers, eac equilibrium allocation under a unit tax is also attainable as an equilibrium allocation under te equivalent ad valorem tax and ( ) U u (t 0 ) = U a t0. (2.19) c Furter, Blackorby and Murty [2007] result can be easily verified in tis example: Π mu (t 0,1) Π ma ( t 0 c,1) = [t 0y 0 t 0 c pa 0y 0 ] = [b Hc Ht 0] 2 > > t 0 < 0 t 0 < 0, 8H(c+t 0 ) = = (2.20) were y 0 is te output level cosen by te monopolist wen faced wit t 0 and p a 0 is te profit maximizing price for te equivalent ad valorem tax rate t 0 c. (2.20) says tat: te sums of monopoly profit and tax revenue are te same wen te monopolist faces a unit or an equivalent ad valorem tax, altoug for a positive tax on te monopolist, te monopoly profit under unit taxation is larger tan under an equivalent ad valorem tax, wile te tax revenue under unit taxation is smaller tan under an equivalent ad valorem tax. Te reverse is true if t 0 is negative. Since consumer surplus is te same under t 0 and its equivalent ad valorem tax, (2.20), (2.12), and (2.15) imply tat: te Marsallian surpluses under unit and ad valorem taxation are equal: M u (t 0 ) = M a ( t 0 ). (2.21) c Tis demonstrates te welfare equivalence of unit and ad valorem taxation of a monopoly in te case wen government can do personalized lump-sum transfers Te optimal tax on monopoly wit personalized lump-sum transfers. Te optimal tax on te monopoly good is obtained by coosing te tax rate tat maximizes te Marsallian surplus: φ (x 0(p u 0 +t 0,1)) cy0 u max t 0,p u 0,yu 0 subject to x 0 (p u 0 +t 0,1) = y u 0 and p u 0 = P u 0(t 0,1). (2.22) 7

11 From (2.10) tis is equivalent to (2b H b Ht 0 Hc)(2b H +b+ht 0 +Hc) max t 0 16H 2 c[b Hc Ht 0]. (2.23) 4 Te solution of tis problem is t 0 = ch b H. (2.24) Given (2.5) and (2.4), we find tat: te Pareto optimal unit tax on a monopolist wen te government can implement personalized lumpsum transfers is t 0 < 0. Furter, at tis tax rate, te consumer price q 0 of te monopoly good is equal to te marginal cost c of te monopolist. Tus, as in Guesnerie and Laffont [1978], te optimal tax on te monopolist wen te government can implement personalized lump-sum transfers is a subsidy tat leads im to coose a level of monopoly output tat corresponds to te level in a perfectly competitive economy: in tis economy, te distortion created by a monopoly can be fully corrected by subsidizing te monopolist. Te first-best utility possibility frontier is U FB U u ( t 0 ). (2.25) Since te ad valorem tax tat is equivalent to t 0 is t 0 c, we also ave U FB = U a t 0 ( ). (2.26) c 2.6. Reconciliation wit Skeat and Trandel [1994]. Skeat and Trandel [1994] argue tat for every unit tax on a monopolist tere exists an ad valorem tax tat yields a iger consumer surplus, a iger monopoly profits, and also a iger tax revenue to te government. Note te following wit respect to teir argument: (1) Teir proof for te existence of suc an ad valorem tax rate olds only if te government can implement positive rates of taxation. Te revealed preference argument employed will fail if te tax rate is negative. (2.) Tediscussionaboverevealstatifforeveryunittaxt 0 > 0tereexistsanadvalorem tax, say τ 0 > 0, tat yields a iger Marsallian surplus, ten tere is also a unit tax t 0 = cτ 0 > 0 tat is equivalent to τ 0 tat yields a iger Marsallian surplus tan t 0. We can continue te argument furter and establis tat tere exists also a unit tax and an equivalent ad valorem tat yield a iger Marsallian surplus tan t 0. Tis argument can go on for ever if we confine ourselves to positive rates of taxation. Tis is because in tis partial equilibrium framework based on Marsallian surplus, te optimal rate of taxation on te monopolist is negative. Tus, every positive unit tax 8

12 will always be Pareto dominated by anoter positive unit and an equivalent positive ad valorem tax. (3) At te Pareto optimal unit tax (wic is negative) tere exists no ad valorem tax tat can yield a iger Marsallian surplus. Tere owever exists an equivalent ad valorem tax tat yields te same Marsallian surplus. Tus, we reconcile te arguments of Skeat and Trandel [1994] wit our arguments above on te equivalence of unit and ad valorem taxation of a monopoly wit respect to te Marsallian surplus Te problem wit private ownersip wit no personalized lump-sum transfers. Now consider economies wit taxation of te monopoly good, were consumers receive a sare of te monopoly profits, te aggregate endowment, ω, of te numeraire commodity is eld by te consumers according to a distribution ω, and were government redistributes its tax revenue as uniform lumpsum transfers (Tat is, eac consumer receives 1/H of te government deficit or surplus.). Suppose te sare of consumer in te monopoly profit is θ [0,1] wit θ = 1. Consumer s income is composed of is profit income, te lump-sum transfer from te government, and is endowment income. A unit tax equilibrium in tis private ownersip economy wit profit sares θ = θ is given by x 0 (p u 0 +t 0,q) = y0, u p u 0 = P0(t u 0,q) (2.27) w = θ [p u 0y0 u cy0]+ u t 0y0 u H +qω,. Note tat, from Walras law, (2.27) implies tat te market for te non-produced input clears: x (p u 0 +t 0,q,w ) = ω cy0. u (2.28) (2.27)isasystemof2+H equationsin4+h unknownsp u 0,yu 0,q,t 0,w 1,...,w H. Tesystem is omogeneous of degree zero in p u 0,q,t 0,w 1,...,w H and so admits te normalization q = 1. Hence, effectively, tere is one degree of freedom in coosing equilibria, tat is te set of equilibria of tis θ ownersip economy can be parametrized by te variable t 0 suc tat (2.5) olds. Te equilibrium values of p u 0,yu 0,w 1,...,w H are (uniquely) determined once t 0 is fixed. Precisely, tey are p u 0 = b+hc Ht 0 2H, y0 u = b Hc Ht 0, and 2H w = (b Hc Ht 0) [θ (b Hc Ht 0 )+2t 0 ]+ω. 8H (2.29) Similarly too we can define an ad valorem tax equilibrium, were τ 0 is te ad valorem tax rate and p a 0 = q 0/(1+τ 0 ) is te producer price faced by te monopolist under te ad valorem tax. Te income of consumer under te ad valorem tax is R = θ [p a 0y a 0 c(y a 0)q]+ 1 H τ 0p a 0y a 0 +ω, (2.30) 9

13 wit p a 0 = Pa 0 (τ 0,1) and y0 a = 0( xa p a 0 (1+τ 0 ),1 ). As in Blackorby and Murty [2007], (2.20) and (2.18) can be used to sow tat, in an economy were monopoly profit is taxed at 100% and rebated back to te consumers as uniform lumpsum transfers, a unit tax equilibrium as an equivalent ad valorem tax representation and vice-versa. 10 Hence, te set of equilibrium allocations are te same under bot te tax systems, and tis implies tat te two taxes are equivalent in terms of individual well-being. However, wen monopoly profits are not taxed and tere is private ownersip of te monopoly, ten te switc from unit to an equivalent ad valorem tax (or vice-versa) implies tat te incomes of te consumers, in general, cange because of te difference in tecompositionofprofitincomeandteuniformlumpsumtransferfromtegovernment 11 : if t 0 0 and θ H 1 for all, ten we ave w = θ Π mu + 1 H t 0y 0 +ω (2.31) even toug R = θ Π ma + 1 H τ 0p a 0y 0 +ω, w = R. (2.32) In terms of consumer demands (given quasi-linear preferences wic are linear in te numeraire good), x (q 0,1,w ) x (q 0,1,R ),. (2.33) even toug x (q 0,1,w ) = x (q 0,1,R ) (2.34) and te consumer demands for te monopoly good, wic are not subject to income effects, also remain te same in tis switc. Te above implies tat altoug te aggregate demands remain uncanged in te switc from unit to te equivalent ad valorem tax, te individual demands for te numeraire good and, ence, te utilities of consumers cange as do te set of equilibrium allocations. 12 Tus, te issue of dominance cannot be studied directly. 10 Te income distribution acieved at a unit-tax equilibrium of suc an economy is also acieved by implementing te equivalent ad valorem tax. In particular, te income distribution acieved is te same as te one acieved in a private ownersip economy wit a demogrant, were te sare θ of any consumer is 1 H. (In suc private ownersip economies, at a unit tax and its equivalent ad valorem tax equilibria, we ave w = R.) 11 Unless θ = 1 H for all. Tis case is teoretically equivalent to te case of 100% taxation of profits. 12 For more general preference structures, te switc may not even result in an equilibrium allocation. 10

14 2.8. A sketc of te solution for te case of private ownersip wit no personalized lumpsum transfers. In order to be able to make a comparison of te two tax regimes in te general case (as well as for te example above) we proceed in an indirect manner wic ultimately yields results. Consider te move from unit-taxation to ad valorem taxation as te reverse is more or less te same. At every unit-tax equilibrium of a given private ownersip economy, te equivalent ad valorem tax leads to te same production decision by te monopolist. However, as discussed above, under tis ad valorem tax, te given allocation of profit sares results in different distributions of consumer incomes and ence different consumption decisions. We proceed in te following manner. First, for eac private ownersip economy θ = θ with consumers(tatis,foreacpossibleallocationofsarestoteconsumers), we define te unit-tax utility possibility set (UPS u (θ)) as te set of all utility profiles corresponding to all possible unit-tax equilibria of te given private ownersip economy. We construct te utility possibility frontier (UPF u (θ)) by maximizing te utility of one consumer olding te utilities of all oter consumers fixed and subject to equilibrium conditions for unit-taxation in te given private-ownersip economy. Next we construct te outer envelope of tese utility possibility frontiers. Tat is, for eac feasible fixed level of utilities for persons 2 troug H, we maximize, by coosing te allocation of private sares, te utility of consumer one. (See Figure 1, wic illustrates tis for H = 2). Figure 1: u 2 Unit Envelope UPF u ( θ) UPF u (ˆθ) Te unit-tax envelope u 1 11

15 Picking a particular fixed set of sares, say θ = θ, we ten searc along tis unit-tax envelope to see if tere is a point on it tat is also supported as an equilibrium of θ private-ownersip ad valorem economy. Under one set of regularity conditions we sow suc a point (a vector of consumers utilities), say ū = ū, exists by a fixed-point argument (see Figure 2.) Since ū lies on te unit envelope, tere exists a sare profile, say ψ = ψ, suc tat te Pareto frontier of te corresponding unit-tax economy is tangent to te unit envelope at ū. We sow tat under our regularity conditions, at ū, te consumer incomes and equilibrium prices and quantities in te ad valorem and unit economies are te same. However, we find tat ψ is not equal to θ and tat ū never belongs to te utility possibility set of te θ ownersip unit economy unless te sares in θ were all equal to 1/H (and ence equivalent to one undred per cent profit taxation problem tat was solved in Blackorby and Murty [2007]) or te optimal tax on te monopolist appened to be equal to zero. In tis way, we obtain a point in te utility possibility set of a θ private-ownersip economy wit ad valorem taxes wic is not present in te utility possibility set of a θ private-ownersip economy wit unit taxes, demonstrating tat unit taxation does not dominate ad valorem wen te monopoly is privately owned. (Figure 2 makes tis clear by indicating bot te utility possibility sets.) Figure 2: u 2 Unit Envelope UPF u ( ψ) ū UPF a ( θ) UPF u ( θ) ū UPS a ( θ), ū UPF u ( ψ), ū / UPS u ( θ) u 1 Te converse is proved in a similar way under anoter similar set of regularity constraints by searcing for a unit-tax equilibrium along te ad valorem-tax envelope for a given allocation of sares. If bot our sets of regularity constraints old simultaneously ten, taken togeter, tese results substantiate te claim tat neiter tax system Paretodominates te oter. 12

16 2.9. Implementing te solution in te case of quasi-linear preferences and constant marginal cost. For te quasi-linear example studied in Section 2.7, it was found tat set of equilibria corresponding to any θ = θ private ownersip economy wit unit taxation of te monopoly good and uniform lump-sum transfer is fully parametrized by t 0, were t 0 satisfies (2.5). Precisely, te parametrization was derived in (2.29). Tis means tat tere is a tax equilibrium in tis economy tat corresponds to te first-best optimal tax t 0 = ch b H. Te independence of te demands for te monopoly good from incomes of consumers under quasi-linear preference structures implies tat te demands for te monopoly good as well as te demand by te monopolist for te numeraire commodity as input at te privateownersip tax equilibrium will also be te demands at te first-best. Tis implies tat te tax equilibrium allocation of te θ private-ownersip economy corresponding to t 0 is also first-best Pareto optimal, in oter words, tere is a point on te utility possibility frontier UPF u (θ) tat is tangent to U FB. Tis is true for every θ private ownersip economy, were θ [0,1] for all and θ = 1. Clearly, from tis argument it follows tat: in te example wit quasi-linear preferences, te unit-tax envelope is a subset of U FB. Using exactly te same argument, we also claim tat: in te example wit quasi-linear preferences, te ad valorem-tax envelope is a subset of U FB. Furter, we also find tat: quasi-linear preferences imply tat te unit-tax envelope is a subset of te ad valorem-tax envelope. Tis can be verified by looking at te ranges of values tat te utility of eac consumer takes along te unit-tax envelope and te ad-valorem tax envelope. Consider te unit-tax envelope. Since te demands for te monopoly good, te input demand for te numeraire by te monopolist, and te unit tax are fixed at te first-best levels at every point on it, te monopoly profits and te demogrant are equal along all tese points. Te utility level of any consumer varies along tis envelope precisely because is income varies as is profit sare varies from zero to one along tis envelope. For every, letterangeof utilitylevelsalong tead-valoremenvelopebedenoted by u [u u,ū u]. 13 Similarly, for every, let te range of utility levels along te ad-valorem envelope be denoted by u [u a,ū a]. 14 Te upper-bounds correspond to te case were s profit sare is one, wile te lower bounds correspond to te case were s profit sare is zero. Te precise relation between te first-best frontier and te unit-tax and te ad valorem-tax envelopes is sown in Figure Straigtforward calculations yield u u = (2b H b H t 0 Hc)(2b H+b+H t 0+Hc) b (b+ht 0+Hc) 4H + (b+h t 0+Hc) 2 8H 2 16H + (b Hc H t 0) 2 4H t 0 + ω and ū u = (2b H b H t 0 Hc)(2b H+b+H t 0+Hc) 16H 2 + (b Hc H t 0) 4H [ (b Hc H t 0) 2 + t 0 ]+ω b (b+ht 0+Hc) 4H + (b+h t 0+Hc) 2 8H Straigtforwardcalculationsyieldu a = (2b H b Hc(1+τ 0))(2b H+b+Hc(1+τ 0)) 16H 2 ω b (b+hc(1+τ 0)) 4H + (b+hc(1+τ0))2 8H ]andū 2 a = (2b H b Hc(1+τ 0))(2b H+b+Hc(1+τ 0)) 16H 2 τ 0(b+Hc(1+τ 0)) H ]+ω b (b+hc(1+τ 0)) 4H + (b+hc(1+τ0))2 t 8H, were τ 2 0 = 0 c. + (b Hc(1+τ0)) 8H(1+τ 0) [ τ0(b+hc(1+τ0)) H ]+ + (b Hc(1+τ0)) 8H(1+τ 0) [ (b Hc(1+τ0))

17 Figure 3: u 2 FirstBestFrontier UnitEnvelope AdValoremEnvelope u 1 Te case of quasi linear preferences: te first-best frontier and te unit and ad valorem envelopes 14

18 If we now look at te differences between te lower and upper bounds of te intervals [u u,ū u] and [u a,ū a], ten we find tat 15 and ū u ū a = implying tat [u u,ū u] [u a,ū a] for all. t 0 (H 1) 16H 3 (c+ t 0 ) (b Hc H t 0 ) 2 < 0 (2.35) u u u t 0 a = 8H 2 (c+ t 0 ) (b Hc H t 0 ) 2 > 0 (2.36) Intuitively, tis is true because t 0 < 0, so tat from (2.20) it follows tat te demogrant under ad valorem taxation is smaller tan under unit taxation, wile te profits are larger under ad valorem taxation. Tus, for θ = 0 (respectively, θ = 1), wen te consumer receives only demogrant (respectively, profit) income, apart, of course, from is endowment income, is income and ence utility is larger (respectively, smaller) under te unit tax tan under te equivalent ad valorem tax. Precisely, tis implies tat te regularity condition (mentioned in te previous subsection and called Assumption 6 later in te general case), wic is required to prove tat ad valorem taxation does not dominate unit taxation, olds for tis example. 16 Te solution outlined in te previous section for welfare comparison of unit and ad valorem taxation under private ownersip can now be employed in tis example. Pick any profile of valid profit sare-profile θ. A searc along te ad valorem envelope will yield a utility profile ū tat also corresponds to an equilibrium in a θ private ownersip economy wit unit taxation. Tis is precisely because te unit envelope is a subset of te ad valorem envelope. Arguments made in te previous section wile explaining te solution in te general case follow to sow tat ad valorem taxation does not dominate unit taxation in private ownersip economies. Furter, a searc along te unit envelope for a utility profile tat corresponds to an equilibrium in an ad valorem private ownersip economy wit sare-profile θ may or may not be successful as te unit envelope is a subset of te ad valorem envelope. If successful, te arguments outlined in te solution provided in te previous section apply. If not, ten too te utility profile tat lies on te ad valorem envelope and corresponds to a private ownersip economy wit sare-profile θ is not attainable in a unit private ownersip economy wit sare-profile θ unless θ = H 1 for all. Tis demonstrates tat unit taxation does not dominate ad valorem taxation in private ownersip economies. 15 To sign tese expressions, note tat (2.18) implies 1+τ0 = c+t0 c, wic in turn implies tat c+t 0 > Foreveryconsumerandgivenanysareofprofitθ [0,1], teincomeassociatedwitanequivalent unit tax at any point on te ad valorem envelope lies between te incomes receives under ad valorem taxation at te two end points of te ad valorem frontier corresponding to sares zero and one. 15

19 2.10. Issues wit extension to more general economies. Te example studied above restricted focus to te case of quasi-linear preferences and did not allow for taxation of te numeraire good. In more general economies, te relations between te first-best frontier and te unit-tax and ad valorem-tax envelopes may be more general. Wit more general preferences, te demands for te monopoly good will not be independent of consumer incomes and ence will vary along te first-best frontier. Tis implies tat, even if tere are common instruments tat parametrize tax equilibria in bot economies wit personalized lump-sum transfers and economies wit private-ownersip 17, te utility profiles obtained at a private-ownersip tax equilibrium may be different from te one obtained at te first-best for te same set of values of tese instruments. Tis is because te income profiles associated wit tese instrument values may differ in te two equilibria in more general economies. 18 Tis may imply tat a second-best utility possibility frontier corresponding to a private-ownersip economy wit taxation of te monopoly good may not be tangent to te first-best frontier, i.e., te unit-tax (or te ad valorem-tax) envelope may not be a subset of te first-best frontier. Furter, it could also be te case tat neiter of te two envelopes is a subset of te oter Te general case: Description of te economy. Consider an economy were H is te index set of consumers wo are indexed by. Te cardinality of H is H. Tere are N +1 goods, of wic te good indexed by 0 is te monopoly good. Te remaining goods are produced by competitive firms. Te aggregate tecnology of te competitive sector is Y c, 20 te tecnology of te monopolist is Y 0 = {(y 0,y m ) y 0 F m (y m )}, were y m R N + is its vector of input demands and te tecnology of te public sector for producing g units of a public good is Y g (g) = {y g R N + F(y g ) g}. For all H, te gross consumption set is X R N+1. Te aggregate endowment is denoted by (ω 0,ω) R N+1 ++ and is distributed among consumersas ω0,ω. 21 Forall H, agrossconsumptionbundleisdenotedby(x 0,x ), andu denotesteutilityfunctiondefinedovertegrossconsumptionset. Teproduction bundle of te competitive sector is denoted by y c, of te public sector by y g, and of te monopolist by (y 0,y m ). 17 As we saw in te quasi-linear case, were te tax on te monopoly good, t0, was te common instrument. 18 For example, in a two-good case wit general preferences, unit-for-unit transfers between consumers of te amount of te non-monopoly good available at a first-best tat corresponds to tese instrument values may not be feasible due to income effects suc transfers generate. Tis is unlike in te example wit quasi-linear preferences. 19 See a working paper version of tis paper, Blackorby and Murty [2008], for te relative positions of te first-best frontier and te unit and ad valorem envelopes for te general case. 20 Aggregate profit maximization in tis sector is consistent wit individual profit maximization by many different firms, as we assume away production externalities. 21 Any H dimensional vector of variables pertaining to all H consumers suc as (u1,...,u H ) is denoted by u. 16

20 Te economy is summarized by E = ( ω 0,ω, X,u,Y 0,Y c,y g ). An allocation in tis economy is denoted by z = ( x 0,x,y 0,y m,y c,y g). A private ownersip economy isonewereteconsumersownsaresinteprofitsofbottecompetitiveandmonopoly firms. A profile of consumer sares in aggregate profits is given by θ H Te consumer price of te monopoly good is q 0 R ++, q R N ++ is te vector of consumer prices of te competitively supplied goods. Te wealt of consumer is given by w. Te producer price of te monopoly good is p 0 R ++, p R N + is te vector of producer prices of te competitively supplied goods. Te individual and aggregate consumer demands for te monopoly good are given by x 0 (q 0,q, w ) = x 0(q 0,q,w ), (3.1) and te individual and aggregate consumer demand vectors for te competitively supplied commodities are given by x(q 0,q, w ) = x (q 0,q,w ). (3.2) Te indirect utility function of consumer is denoted by V (q 0,q,w ). 23 We assume tat te monopolist is naive, in te sense tat it does not take into account te effect of its decision on consumer incomes. 24 Its cost and input demand functions are denoted by C(y 0,p)andy m (y 0,p), respectively. Teaggregatecompetitiveprofitandsupplyfunctions are denoted by Π c (p) and y c (p), respectively. We use te following general assumptions on preferences and tecnologies in our analysis. Assumption 1: For all H, te gross consumption set is X = R N+1 +, te utility function u is increasing, strictly quasi-concave, and twice continuously differentiable in te interior of its domain X. Tis, in turn, implies tat te indirect utility function V is twice continuously differentiable. 25 We also assume tat te demand functions (x 0 (),x ()) are twice continuously differentiable on te interior of teir domain. Assumption 2: Te tecnologies Y 0, Y c, and Y g (g) are closed, convex, satisfy free disposability. Y 0 and Y c contain te origin. Te public good production function F is strictly concave and twice continuously differentiable on te interior of its domain. Assumption 3: Te profit function of te competitive sector, Π c, is assumed to be differentially strongly convex and te cost function C(y 0,p) of te monopolist is assumed 22 H 1 is te H 1-dimensional unit simplex. Assuming tat consumers ave te same sares of monopoly and competitive sectors profits makes te notation considerably simpler witout any loss of generality. 23 Tere is also a public good g but, as it remains constant trougout te analysis, it is suppressed in te utility function. 24 Likewise we assume tat consumers are naive; tey do not anticipate canges in teirs incomes due to cange in te profits of te monopolist. 25 See Blackorby and Diewert [1979]. 17

21 to be differentially strongly concave in prices and increasing and convex in output. 26 Te competitivesupplyy c (p)isgivenbyhotelling slemmaas p Π c (p)andteinputdemands of te monopolist are given by y m (y 0,p) = p C(y 0,p). Te marginal cost y0 C(y 0,p) is positive on te interior of te domain of C A unit-tax private-ownersip equilibrium. Te monopolist s optimization problem, wen facing a unit tax t 0 R and wen te vector of unit taxes on te competitive goods is t R N, is P u 0 (p,t 0,t, w ) :=argmax p u 0 {p u 0 x 0 (p u 0 +t 0,p+t, w ) C(x 0 (p u 0 +t 0,p+t, w ),p) (3.3) As discussed in detail in Guesnerie and Laffont [1978] te profit function of te monopolist (te function over wic it optimizes) is not in general concave. Following tem we assume tat te solution to monopolist s profit maximization problem is locally unique and smoot. Under Assumptions 1, 2, and 3, te first-order condition for tis problem is q0 x 0 (p u 0 +t 0,p+t, w ) [p u 0 y0 C(y 0,p)]+x 0 (p u 0 +t 0,p+t, w ) = 0 (3.4) wic implicitly defines te solution p u 0 = Pu 0 (p,t 0,t, w ). Assumption 4: P0 u is single-valued and twice continuously differentiable function suc tat t0 P0(p,t u 0,t, w ) 1. (3.5) As discussed in Guesnerie and Laffont [1978], t0 P0 u 1 implies tat te government can control q 0 by controlling t 0. Since consumer demands are omogeneous of degree zero in consumer prices and incomes, q0 x 0 is omogeneous of degree minus one in tese variables. Also, te cost function C is omogeneous of degree one in p. Hence, it follows tat te left side of (3.4) is omogeneous of degree zero in p u 0,p,t 0,t, and w. Tis implies tat te function P0 u(p,t 0,t, w ) is omogeneous of degree one in p,t 0,t, and w. A unit-tax equilibrium in private-ownersip economy wit sares θ H 1 is given by 27 x(q 0,q, w )+y c (p) y m (y0,p) y u g +ω 0, (3.6) }. and x 0 (q 0,q, w )+y0 u +ω 0 0, (3.7) p u 0 P0 u (p,t 0,t, w ) = 0, (3.8) w = θ [p u 0y0 u C(y0,p)+Π u c (p)]+ 1 [t T (y c y m y g )+t 0 y0 u py g] H (3.9) +q T ω +q 0 ω0, H F(y g ) g 0, p u 0 0, p 0 N, q 0 = p u 0 +t 0 0,q = p+t 0 N. (3.10) 26 See Avriel, Diewert, Scaible, and Zang [1988]. 27 Te superscript notation T stands for transpose. A matrix wit subscript N is a square matrix wit dimension N. 18

22 3.2. An ad valorem-tax private-ownersip equilibrium. Te monopolist s profit maximization problem, wen confronted wit ad valorem taxes (τ 0,τ) is 28 P0(p,τ a 0,τ, R ) := { ) argmax p a 0 p a 0x 0 (p a 0(1+τ 0 ),p T (I N +τ), R (3.11) ( )} C x 0 (p a 0(1+τ 0 ),p T (I N +τ), R ),p, Assume tat P0 a is a single valued twice continuously differentiable function; Assumption 4 ten implies tat (1+τ 0 ) τ0 P0 a Pa 0. A monopoly ad-valorem tax equilibrium in a private ownersip economy wit sares θ H 1 satisfies and x(q 0,q, R )+y c (p) y m (p,y a 0) y g +ω 0, (3.12) x 0 (q 0,q, R )+y a 0 +ω 0 0, (3.13) p a 0 = P0 a (p,τ 0,τ, R ), (3.14) R = θ [p a 0y0 a C(y0,p)+Π a c (p)]+ 1 [τ 0 p a H 0y0 a +p T τ [y c y g y m ] p T y g] (3.15) +q T ω +q 0 ω0 H, F(y g ) g 0, (3.16) p a 0 0, p 0 N, q 0 = p a 0(1+τ 0 ) 0, q = (I N +τ)p 0 N. (3.17) As in te unit-tax case, te function P a 0 is omogeneous of degree one in its arguments. 4. Unit versus ad valorem taxes in private ownersip economies. Tis section consists of two subsections. In te first we set out te assumptions and notation tat we need and ten sow tat te set of unit-tax Pareto optima for a given private-ownersip economy does not contain te set of ad-valorem tax Pareto optima for te same private ownersip economy. Hence, unit taxation does not dominate ad valorem taxation. Te following subsection proves te converse. 28 Symbols in bold face suc as τ and p stand for diagonal matrices wit diagonal elements being te elements of vectors τ and p, respectively. 19

23 4.1. An ad valorem-tax private-ownersip equilibrium on te envelope of unit-tax utility possibility frontiers For eac possible profile of profit sares, θ H 1, we obtain a unit-tax Pareto frontier by solving te following problem for all utility profiles (u 2,...,u H ) for wic solution exists: U u (u 2,...,u H, θ ) := max V 1 (q 0,q,w 1 ) p u 0,p,t 0,t, w,q 0,q subject to V (q 0,q,w ) u, for = 2,...,H, and (3.6) to (3.10). (4.1) Te envelope for te Pareto manifolds of all possible private ownersip economies wit unit taxes (wic we will call te unit envelope) is obtained by solving te following problem for all utility profiles (u 2,...,u H ) for wic solutions exist: Û u (u 2,...,u H ) :=max θ Uu (u 2,...,u H, θ ) Denote te solution to tis problem by subject to θ = 1, and θ [0,1], H. (4.2) θ u = θ u (u 2,...,u ). (4.3) Tat is, for given utility levels, (u 2,...,u H ), θ u is te vector of sares tat maximizes te utility of consumer 1. Next we generate an algoritm tat identifies te ad valorem tax-equilibria tat lie on te unit envelope. Let A u be te set of all allocations corresponding to te utility profiles on te unit envelope Ûu (u 2,...,u ). Define te identity mapping I : A u A u wit image ( x 0 (z),x (z),y 0 (z),y m (z),y c (z),y g (z)) = z. 29 Let ρ u : A u R H wit image ρ u (z) = u (x 0 (z),x (z)) be a utility map of te allocations in A u. Tat is, for every z A u, te set of utility levels enjoyed by consumers at tat allocation is ρ u (z). Wit some abuse of notation, let θ u (z) = θ u (ρu (z)) for H be te solution of te problem (4.2) at te allocation z. Our strategy is based on a fixed point argument and requires te restriction tat all prices and taxes belong to a compact and convex set. A natural way to do so is to adopt 29 Tat is, for every z = ( x 0,x,y 0,y m,y c,y g) A u, te mapping I assigns x 0(z),x (z) = x 0,x, y c (z) = y c,y g (z) = y g,y 0 (z) = y 0, and y m (z) = y m. 20

24 a price normalization rule, wic te equilibrium system allows as it is omogeneous of degree zero in te variables. 30 Let b be suc a normalization rule suc tat te set S u b := {(p 0,t 0,p,t) R + R R N + R N b(p 0,t 0,p,t, w ) = 0 for any w } (4.4) is compact. Let ( p u 0, t 0, p, t and (p u 0,t 0,p,t) be te vectors of maximum and minimum values attained by p u 0,t 0,p, and t in S u b. For example, pu 0 solves max {p u 0 R + t 0,p,t, w suc tat b(p u 0,p,t 0,t, w ) = 0}. (4.5) Define te mapping ψ u : A u Sb u as ψ u (z) = (ψp(z),ψ u w(z)), u were ψp(z) u = (p u 0 (z),t 0(z),p(z),t(z)) is te vector of unit taxes and producer prices associated wit allocation z (a unit tax equilibrium), wile ψw(z) u = w (z) is te profile of consumer incomes associated wit allocation z. Under Assumptions 1 to 4, ψ u is continuous. For every z A u, q 0 (z) = p u 0 (z)+t 0(z) and q(z) = p(z)+t(z). Since Sb u is compact, tere exist ( q 0, q and (q 0,q) wic denote te vector of maximum and minimum possible consumer prices tat can be attained under te adopted price normalization rule. Given (i) an appropriate normalization rule and (ii) te fact tat, for a monopolist, p u 0 (z) y 0 C(y 0 (z),p(z)), we ave, for every z A u, p(z) [p, p], p u 0(z) [p u 0, pu 0], t(z) [t, t], t 0 (z) [t 0, t 0 ], q 0 (z) [q 0, q 0 ], q(z) [q, q], (4.6) and 31 C(y y0 0(z),p(z)) [p u 0, pu 0,]. (4.7) Next, we use Suits and Musgrave [1953] argument to sow tat we can separate q(z) and q 0 (z) into (equivalent) ad valorem taxes and producer prices defined by functions (τ 0 (z),τ(z)) and (p a 0 (z),p(z)), wic ensure tat (y 0(z),y(z)) and (p a 0 (z),p(z)) are te profit maximizing outputs and prices in te monopoly and te competitive sector wen te ad valorem taxes are (τ 0 (z),τ(z)), tat is, (τ 0 (z),τ(z)) and (p a 0 (z),p(z)) solve p a 0(z) = P a 0 q 0 (z) = p a 0(z)(1+τ 0 (z)) and ( ) τ 0 (z),p(z),t(z), w (z) > 0 q(z) = (τ(z)+i N )p(z). (4.8) From an argument in Suits and Musgrave [1953] 32, for every z A u, if t 0 (z) y0 C(y 0 (z),p(z)) > 1 (4.9) 30 Te rationale for tis including a discussion of valid normalization rules and a proof tat teir coice does not affect te solution (Lemma B3) is in Appendix B (Blackorby and Murty [2008]) of te working paper version. 31 Note, normalization rules suc as te unit emispere ensure tat y0 C(y 0 (z),p(z)) lies in a compact set wen te monopolist optimizes. Tis is because, under suc a normalizaton, p u = 0 and ence, 0 y0 C p u = See also Blackorby and Murty [2007]. 21

25 ten coosing τ 0 (z) = t 0 (z) y0 C(y 0 (z),p(z)), (4.10) τ(z) = p(z) 1 t(z) (4.11) p a 0(z) = q 0(z) 1+τ 0 (z) = q 0(z) y0 C(y 0 (z),p(z)) y0 C(y 0 (z),p(z)) + t 0 (z) > 0. (4.12) does te job 33 Define te mapping ψp(z) a := (p a 0 (z),τ 0(z),p(z),τ(z)). ψp a identifies te (equivalent) ad valorem taxes and prices associated wit an allocation z A u tat results in te same output decisions as in te unit tax equilibrium. Since τ 0 (z) and p a 0 (z) are continuous functions, (4.10) (4.12) imply tat tere exist compact intervals [τ 0, τ 0 ], [τ, τ], and [p a 0, pa 0 ] suc tat for every z Au, we ave τ 0 (z) [τ 0, τ 0 ], (4.13) and We define τ(z) [τ, τ] (4.14) p a 0(z) [ ] p a 0, pa 0. (4.15) ] S u b [p = a 0, pa 0 [τ 0, τ 0 ] [ p, p ] [τ, τ], (4.16) and for every z A u, we ave (p 0 (z),τ 0 (z),p(z),τ(z)) S u b, wic is a compact and convex set. Foreacallocationz A u weneedtobeabletoidentifyteincomesofteconsumers. Define an income map for consumer as te map r u : A u Sb u [0,1] R, wic for every z A u, π = (p u 0,t 0,p,t) Sb u, and θ [0,1] as image r u (z,π,θ ) =θ [p u 0y 0 (z) C(y 0 (z),p)+π c (p)]+ 1 H [ t 0 y 0 (z)+t T [y c (z) y m (z) y g (z))] p T y g (z) ] (4.17) +[p+t] T ω +[p u 0 +t 0 ]ω 0 33 If (4.9) is not satisfied ten tere is no ad valorem tax tat tat yields te same profit-maximizing output as te given unit tax t 0 (z), for as seen in te equation immediately below, violation of (4.9) would imply tat p a 0(z) is eiter less tan zero or does not exist. 22