SNF Project No Markeder for helsetjenester. The project is financed by the Research Council of Norway

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1 :RUNLQJ3DSHUR $QDOWHUQDWLYHZD\WRPRGHO PHULWJRRGDUJXPHQWV E\ )UHG6FKUR\HQ SNF Project No Markeder for elsetjenester Te project is financed by te Researc Council of Norway INSTITUTE FOR RESEARCH IN ECONOMICS AND BUSINESS ADMINISTRATION Bergen, November 2003 ISSN Dette eksemplar er fremstilt etter avtale med KOPINOR, Stenergate, 0050 Oslo. Ytterligere eksemplarfremstilling uten avtale og i strid med åndsverkloven er straffbart og kan medføre erstatningsansvar.

2 An alternative way to model merit good arguments Fred Scroyen First version: September 2002 Tis version: October 2003 Abstract Besley (988) uses a scaling approac to model merit good arguments in commodity tax policy. In tis paper, I question tis approac on te grounds tat it produces wrong recommendations taxation (subsidisation) of merit (demerit) goods wenever te demand for te (de)merit good is inelastic. I propose an alternative approac tat does not suffer from tis deficiency, and derive te ensuing first and second best tax rules, as well as te marginal cost expressions to perform tax reform analysis. JEL code: H2 Keywords: merits goods, commodity taxation, tax reform analysis Tis version of te paper was written wile visiting te Dept. of Economics, Universitat Autònoma de Barcelona, wic provided a very friendly and ospitable work environment. Financial support by te Programme for Healt Economics in Bergen (HEB) troug an SNF grant is gratefully acknowledged. I sould like to tank Kjetil Bjorvatn, Bart Capéau, Agnar Sandmo, Bertil Tungodden and two anonymous referees for elpful comments and discussions. Remaining flaws are entirely mine. Dept. of Economics, Norwegian Scool of Economics & Business Administration, Helleveien 30, N-5045 Bergen (Norway). fred.scroyen@n.no.

3 Introduction Empirical tax reform analysis for European countries almost witout exception reports on te very ig welfare costs involved in marginally raising total tax revenue troug an increase in te VAT or excise rate on alcoolic beverages and tobacco. Te coexistence of suc a ig marginal welfare cost wit muc lower values for oter commodities and services seemingly points at te possibility of significant welfare increases by reducing of taxes on alcool and tobacco, and raising tose on, say, transport. Still, autors of suc studies asten to remark tat teir analysis does not take into account te demerit good arguments tat probably motivated tese ig excise tax levels in te first place. Altoug te (de)merit good argument dates back to Musgrave (959), its introduction in te optimal taxation literature did not come before Besley s (988) analysis. 2 Te reason for tis delay is tat standard second-best analysis of economic policy takes place in a welfaristic framework, wile (de)merit good arguments drive a wedge between te evaluations of citizens and tose of policy makers. Te strengt of Besley s (988) model is twofold. First, it incorporates merit good arguments in an oterwise very standard welfaristic setting, and terefore allows to demonstrate clearly ow putting one foot into non-welfaristic terrain affects te familiar first- and second-best policy rules. Second, Besley relates te (de)merit good considerations to were we would expect tem to ave teir root: te consumption pattern of te individual agent, rater tan te aggregate level of consumption of particular commodities. 3 Unfortunately, Besley s first-best analysis suffers from a deficiency tat puts te framework into question. As I will sow below, is model in fact prescribes to subsidise demerit goods and to tax merit goods wenever te demand for tose goods is inelastic. Keeping in mind te fact tat abit formation is often responsible for an inelastic demand for cigarettes and alcoolic beverages, one is ten led to te paradox tat suc demand sould be encouraged rater tan discouraged. Altoug a normative model cannot be subjected to te same kind of falsification tests as positive models, if its policy prescriptions go exactly against one s gut-feeling, ten one sould ask weter te framework witin wic tose prescriptions were derived is an appropriate one. I tink one can question Besley s Decoster & Scokkaert (989) for Belgium, Madden (995) for Ireland, Kaplanoglou & Newbery (2002) for Greece, Scroyen & Aasness (2002) for Norway. 2 Several years earlier, Sandmo (983) outlined for different degrees of market completeness te policy implications of a divergence between te agent s beliefs about te future states of te world, and tose of te social planner. Besley s (988) model, and te present one, concern te case of (de)merit wants: te divergence between te agent s preferences and tose of te social planner. 3 See e.g. Pazner (972).

4 framework,butattesametimetatitcanbeeasilyremediedinawaytat preserves its twofold strengt. To sow my arguments in te sarpest way, I start by explaining tem in a first best setting. Tis, I do in te next two sections. In section 4, I offer an alternative way of modeling (de)merit good arguments and derive te ensuing first best policy rules. Second-best rules are presented in section 5, and in section 6, I derive te marginal cost expressions for tax reform analysis. Section 7concludes. 2 Consumer beaviour and government opinion Consider a representative consumer wose preferences over te numéraire commodity (z R + ), a (de)merit good (y R + ), and a standard commodity (x R + ) can be represented by te monotonic and strongly quasi-concave utility function u(z,x,y). Let te consumer prices for tese commodities be given by (,q x,q y ) and te consumer s exogenous disposable income equal to m T,were T is te lump sum tax and m is pre-tax income. Tis consumer ten solves te problem Te first order conditions are 4 max u(z, x,y) () z,x,y s.t. z + q x x + q y y = m T. u 2 = q x, u 3 = q y, (2) and, togeter wit te budget constraint, tese are satisfied by te optimal commodity demands z(q x,q y,m T ), x(q x,q y,m T ), and y(q x,q y,m T ). (3) Te government evaluates te allocation of resources according to te modified utility function u g (z, x,y). (4) It takes individual beaviour (3) as given and is concerned wit solving: max t x,t y,t ug (z(q x,q y,m T ),x(q x,q y,m T ),y(q x,q y,m T )) (5) s.t. t x x(q x,q y,m T )+t y y(q x,q y,m T )+T R (λ) were q x = p x + t x and q y = p y + t y. 4 Subscript i wit a function denotes a partial derivates w.r.t. te it argument. 2

5 Te first order conditions are sown in te appendix to provide us wit te following tax rules: t x = u 2 (z,x,y) ug 2 (z,x,y) u g t y = u 3 (z,x,y) ug 3 (z,x,y) u g (6a) (6b) wo ave te intuitive interpretation of driving a wedge between te consumer s marginal willingness to pay (MWP) for eac good, and tat of te government. 3 Besley s scaling approac Besley (988) proposes te following specification for te government s evaluation function: u g (z,x,y) =u(z, x,θy). (7) and defines te tird commodity as a merit (demerit) good wenever θ > (<). Tis scaling approac dates back to Fiser & Sell (967) wo used it to construct an index for te true cost of living wen people s tastes cange or wen products cange in quality. In te present context, te government converts te quantity of te (de)merit good into efficiency units, but oterwise fully respects individual preferences. Wit tis specification, te earlier derived tax rules become 5 t x = u 2 (z,x,y) u 2 (z, x,θy), (8a) t y = u 3 (z,x,y) θ u 3 (z,x,θy). (8b) To see wat tese rules imply, consider te preference ordering represented by te CES utility function u(z, x,y) =(α z ρ + α 2 x ρ + α 3 y ρ ) /ρ,witα i > 0 (i =, 2, 3), and < ρ <. Wit suc preferences, u 3 = α 3 y ρ α, ug 3 = z u g α 3 α θ θy ρ z and u 2 = ug 2 = α u g 2 x ρ α. No tax sould be levied on te standard z commodity,wileforte(de)meritgood,weave t y = ( θρ ). (9) q y θ Wenever te elasticity of substitution is below one ( < ρ < 0), we get te paradoxical result tat a merit good sould be taxed wile a demerit good sould be subsidised! 5 In deriving te first best rule for te tax on te (de)merit good (8b), Besley (988) made a mistake. Te correct rule was provided in a comment by Feean (990). 3

6 Wat is going on? Wy does a seemingly natural way of modelling (de)merit good arguments lead to results wose intuitive appeal depends on te elasticity of substitution? Let us consider te limiting case of Leontief preferences, and ignore for te sake of grapical representation te standard non-numéraire good. Suppose tat a person s preferences can be represented by min{z, y}. Tis person as L-saped indifference curves wit te corners lying on te 45 o line, as sown in figure. z z=y z=½y Leontief preferences and a demerit good Te government, on te oter and, tinks of good y as a demerit good and subscribes to te preference ordering represented by min{z, y}. Te associated 2 indifference curves are te dased lines. Clearly, te government s preferences are more favourable to commodity y tan te agent s preferences are! Wit Leontief preferences, no finitesubsidywillbeabletodistortteagent s budget allocation away from te laissez-faire solution, but once te degree of substitutability becomes positive, it will. Te reason wy discounting a commodity leads to subsidisation sould now be clear. Te government respects tat red wine is complementary to a pasta meal. But it regards one bottle of wine only alf as good as you do. In order to get te maximal utility out of a pasta dis, it wants you to drink more wine, not less. Tis paradox olds true more generally. Notice tat te government s MWP for te (de)merit good (θ u 3 (z,x,θy)) as te following elasticity w.r.t. θ: Ã log( ug 3 ) u g = log! u y u z (z,x,θy). (0) log θ log(θy) Tus, weter te government s MWP exceeds or falls sort of tat of te consumer depends on weter te (own) demand price elasticity for te (de)merit good exceeds or falls sort of in absolute value. If te demand for tobacco, say, is inelastic, te demand price elasticity is likely to be large, and under te scaling y 4

7 approac te government s MWP for tobacco will exceed tat of te smoking agent. 6 4 An alternative way of modeling merit good arguments Te previous analysis indicates te need for an approac tat ties down in a more robust way te relationsip between te government s MWP and tat of te consumer. I propose to model tis relationsip by means of te total willingnessto-pay (TWP) function in terms of te numéraire commodity: F (x, y, u). () Tis function gives te amount of te numéraire required to bring te consumer at te utility level u wen consuming x and y units of te oter two commodities; its grap is te indifference surface. Te marginal willingness to pay (MWP) for te (de)merit good is ten F (x, y, u) = F 2 (x, y, u). (2) y If te government is of te opinion tat consumers appreciate te (de)merit good too (muc) little, ten a natural way of proceeding is to attribute to te government te MWP function F g 2 (x, y, u) = F 2 (x, y, u)+µ(y), (3) were µ(y) is (negative) positive for a (de)merit good. 7 Notice tat (3) does not directly impose any single crossing in te commodity space because it is conditional on te utility level u: te amount of z te consumer believes is necessary to reac tat utility level need not be te same as te amount te government believes is required. Indeed, integrating tis MWP function to a TWP function gives F g (x, y, u) =F (x, y, u) y g µ(χ)dχ, (4) were y g can be tougt of as te level of consumption above wic te government s marginal evaluation starts to deviate from te consumer s. 6 Wit additive preferences quasi-linear in te numéraire, te (own) elasticity of te inverse demand scedule is exactly te reciprocal of te (own) Marsallian price elasticity. 7 More complicated modifications of te MWP function are possible, but one sould make sure tat for a corresponding TWP function to exist, te cross partial derivatives sould be symmetric: F g 2 = F g 2 (Frobenius teorem). 5

8 Teinterpretationof(4)isasfollows. Supposeteagentas(x, y) available. Ten in order to reac a utility level u, sebelievesf (x, y, u) units of te numéraire are required, wile te government, convinced of te merit properties of te tird commodity, believes F g (x, y, u) units are sufficient. Wat (3) ten tells is tat te slope of te government s indifference curve troug (F g (x, y, u),x,y) differs from te slope of te consumer s indifference curve troug (F (x, y, u), x, y) wit a factor µ(y). To tis TWP function corresponds a utility function u g ( ), defined as It is easy to sow tat à u g (z,x,y) =u z + u g (F g (x, y, ν),x,y,ν) ν (all ν). (5)! µ(χ)dχ,x,y. (6) y g (see appendix). A sufficient condition for u g ( ) to be strongly quasi-concave is tat µ 0 ( ) 0. Te two MRS expressions necessary to compute te optimal tax rates are terefore u g 2 u g (z, x, y) = u 2 (z + µ(χ)dχ,x,y) (7a) y g u g 3 u g (z, x, y) = u 3 (z + µ(χ)dχ,x,y)+µ(y). (7b) y g Again, te reason (7b) is not exactly µ(y) is tat te evaluation ere is at te same bundle, not at te same utility level as in (3). A sufficient condition for single crossing of indifference curves (in te sense tat sign(mrsyz g MRS yz )= sign(µ(y)))istatte(de)meritgoodisanormalgood(intesensetat ( u 3 z ) > 0). Inserting (7) in (6) and using a Taylor expansion of u i (z + R y µ(χ)dχ,x,y) y around u i (z, x, y) gives µ u 2 t x ' µ u 3 t y ' (z, x, y) z (z, x, y) z µ(χ)dχ y g µ(χ)dχ µ(y) y g (8a) (8b) Te round bracket terms on te rs denotes te uncompensated effect of a marginal increase in z on te demand price for eac of te non-numéraire commodities. Te signs of tese effects are related to te normality of tese 6

9 commodities (altoug wit 3 or more commodities, te income effects are more involved). To fix ideas, suppose tat te government considers y as a merit good, µ(χ) > 0, χ 0, andtatbotx and y are normal goods. Ten bot non-numéraire commodities sould be subsidised. First, te merit good sould be subsidised wit a rate at wic te government s MWP exceeds tat of te consumer (µ). But in addition, bot non-numéraire goods sould be subsidised for te difference in willingness to pay for all inframarginal units. Te intuition is as follows. Te government is convinced tat te consumer is better off tan se erself believes se is (because se ignores te merit properties of te tird good). To teextenttattenon-numéraire commodities are normal goods, it considers bot commodities as underconsumed (relative to te numéraire) andwantsto stimulate teir consumption by subsidisation. 5 Second-best rules Suppose now tat tere are many consumers, indexed by, wit not necessarily te same preferences or income level, and tat te government cannot differentiate commodity taxes across consumers. Let social preferences be given by W = X λ u g z,x,y = X Z! y λ u Ãz + µ(χ)dχ,x,y. (9) y g To focus on te efficiency arguments, I assume tat differentiated lump sum taxes are available. Te government s problem is ten 8 X max λ u g z (q x,q y,m T ),x (q x,q y,m T ),y (q x,q y,m T ) t x,t y,{t } (20) X s.t. t x x (q x,q y,m T )+ X t y y (q x,q y,m T )+ X T R (γ) = λ u g Defining β def and proceeding in te same way as in te first best case, γ we get: X µ bx bx µ µp β q y t bx bx x q t = x P µ q y tx (2) y t y by by q y P by P by q y were a ^denotes a compensated price effect and t x def = u 2 u ug 2 u g and t y def = u 3 u ug 3 u g 8 I agree wit Capéau and Ooge (2003) tat te reaction functions entering te social welfare function sould be tose of te real agents, and not, as Besley (988) proceeded, tose of pantom agents tat are endowed wit te rigt preferences. 7

10 are te tax rates tat person would face under first best. 9 By premultiplying (2) troug by te inverse of te aggregate Slutsky matrix, we could isolate te vector of second best tax rates t x t y, but it is only under Hicksian independence between te two non-numéraire commodities ( bx q y =0)tattisbecomes illuminating: t y = X β y bε yy y a bε a t y (22) yy wit bε def yy = by q y q y,y a def = P y y and bε a def yy = P by q y q y. Expression (22) reflects y a te second best nature of te policy. As te government cannot individualise commodity tax rates, it cooses a uniform tax rate on te (de)merit commodity wic is a weigted average of te individual first best rates, were te weigt depend on te social marginal utility of income (β ), te sare in aggregate consumption ( y ) and te sensitivity of individual relative to aggregate demand y a ( bε yy ). A similar expression is true for t bε a x. yy Under first best, te social marginal utilities of income, β, sould all be equal to unity. Under uniform commodity taxation, tis is no longer necessarily te case. Te first order condition for te lump sum tax on consumer can now be written as: β µt x x m + y t y m x = µt x m + t y y m (23) sowing tat even at an optimal income distribution te social marginal utilities of income will deviate from one to te extent tat (i) te individual first best tax rates differ form te uniform ones, and (ii) te income effects are different from zero. Te latter case occurs wit quasi linear preferences in te numéraire. I will now give an example were all individual firstbesttaxratesareidentical and used to implement te optimal allocation. Suppose preferences are quasi-omotetic, meaning tat for eac agent tere are some survival quantities z,x, and y suc tat u (z,x,y) =F [v (z z,x x,y y )] wit F 0 ( ) > 0 and v ( ) omogenous of degree in (z z,x x,y y ). We ten ave te following Lemma Suppose tat consumers ave (i) quasi omotetic preferences tat are (ii) identical, wit (iii) bot x and y positive. Suppose tat (iv) µ( ) is m m constant and (v) y g = y. Ten te vectors of individual first best tax rates are identical and te first best allocation can be implemented by tis common vector andasetoflumpsumtaxestatsetallβ equal to unity. 9 More correctly would be to say tat tey caracterise te firstbesttaxrates,aste differences in MRS are all evaluated at te second best solution. 8

11 Te proof, of wic te details are spelled out in te appendix, goes rougly as follows. (i) means tat te MWPs are omogenous of degree zero in its arguments. (ii), (iv) and (v) ten imply tat te individual (first best) tax rates are identical across consumers. Wit (i), all substitution effects are proportional in income, wic makes te (common) vector of first best tax rates proportional to te vector te government uses, te factor of proportionality being te average β plus te covariance of te βs wit supernumerary incomes. But wit (i), income effects areindependentofincomeandby(iii)and(23)β =(all ), wic means te factor of proportionality is as well. Implementation of te first best allocation is tus not incompatible wit an unequal income distribution. But te combination of assumptions for tis to appen is quite stringent and in general second best tax rates will be a genuine weigted average of te firstbestones. 6 Tax reform analysis Te tax rules derived in te previous sections caracterise te optimal solution under first and second best. For a policy maker, tese rules may not be of primary interest (i) because income distribution policy is not necessarily on te same agenda as commodity tax policy, and (ii) because te existing commodity tax structure puts a straigtjacket on wat can be acieved troug a reform. More interesting are ten te rules tat indicate in wic direction marginal tax canges sould occur, and tat can easily be expressed in terms of accounting and statistical data. For tis purpose, one is interested in te marginal cost in terms of social welfare, W, of raising government revenue, R, wit one Euro by canging te tax on commodities x and y: MC x = W/ t x,mc y = W/ t y. (24) R/ t x R/ t y Expressions of tis kind ave been discussed in detail by Amad & Stern (984), wo sow tat a neat parameterisation is obtained by multiplying nominator and denominator by te respective after tax prices. Since te (de)merit good arguments only affect te nominators, I limit myself to tis part of te MC-expressions. Let me for simplicity assume tat µ(y) µ and y g =0. For commodity x, wetenavetat W t x = X λ eu ½ z + eu 2 eu x eu + 3 eu y + µ ¾, (25) were a ~ above a marginal utility denotes tat it is evaluated at te bundle (z + µy,x,y ). Using te first order Taylor approximation eu i ' q eu i + 9

12 u i z u (z,x,y) y µ (i =2, 3)andteaddingupcondition z x +q y x +q y = x, tis expression can be rewritten as λ eu g ½x x u2 (z, x, y) z y µ µ y + u 2 (z, x,y) z ¾ y µ. W ' X t x (26) Multiplying troug by q x, and turning derivatives into elasticities, we finally get q x W t x ' X µ ¾ λ eu g ½q x x µy ε xxδ q x x xz z + ε yx +δ q y y yz, (27) z were te Marsallian price elasticities and te demand price elasticities are defined as ε xx = x q x x, ε yx = y q x y, δ xz = u2 z z u 2, δ yz = u3 z Similar operations lead to an analogue expression for q y W t y : q y W t y ' X z u 3. (28) µ ¾ λ eu g ½q y y µy ε xyδ q x x xz z + ε yy +δ q y y yz, (29) z wit similar definitions for ε xy and ε yy. Notice ow te existence of a (de)merit good argument corrects bot MC expressions in a similar way. Writing µy = ηq y y, te curly bracket terms can in principle be calculated using expenditure data (expenditures on x and on y, and expenditures on x and y relative to z) and uncompensated price and quantity elasticities. If tese elasticities are not available at te ouseold (or income decile) level, tey can as an approximation be replaced by te aggregate elasticities. One can ten produce different rankings of te marginal costs, for different values of te (de)merit parameter η. 7 Discussion and concluding remarks In tis paper, I ave questioned te scaling approac proposed by Besley (988) to model merit good arguments on te ground tat it often leads to counterintuitive policy prescriptions. I ave proposed a different approac wic directly interferes wit te marginal willingness to pay for a (de)merit good. Wic approac to coose? If one is convinced tat a (de)merit good argument sould at least in a first best world precribe te subsidisation (taxation) of tat good, ten te approac presented ere meets tis criterion better tan 0

13 te scaling approac: only wen te merit good is a very inferior one will te policy rule (8b) prescribe taxation; te scaling approac would do so wenever te good is price inelastic. But even if te present approac prescribes taxation, ten a rationale can be provided: insofar te merit good considerations also apply to te inframarginal units consumed, te government believes te agent is ricer tan se tinks se is. Respecting te strong inferiority of te good, it wants te agent to consume less of it. Te analysis as been performed for tree commodities. However, its generalisation to n commodities sould be straigtforward. References Besley T (988) A simple model for merit good arguments, Journal of Public Economics 35, Capéau B & E Ooge (2003) Merit goods and pantom agents, Economics Bulletin 8(8), -5. Decoster A & E Scokkaert (989) Equity and efficiency of a reform of Belgian indirect taxes, Recerces Économiques de Louvain 55, Feean J (990) A simple model for merit good arguments a comment, Journal of Public Economics 43, Fiser F A & K Sell (967) Taste and quality cange in te pure teory of te true cost-of-living index, in J N Wolfe (ed) Value, Capital and Growt: Papers in Honour of Sir Jon Hicks, (Edinburg: Edinburg UP), Kaplanoglou G & D Newbery (2002) Indirect taxation in Greece: evaluation and possible reform, CES working paper 66, Munic. Madden D (995) Labour supply, commodity demand and marginal tax reform, Economic Journal 05, Musgrave R (959) Te Teory of Public Finance (New York: McGraw-Hill). Pazner E (972) Merit wants and te teory of taxation, Public Finance 27, Sandmo A (983) Ex post welfare economics and te teory of merit goods, Economica 50, 9-33 Scroyen F & J Aasness (2002) Indirect marginal tax reform analysis for Norway, mimeo, Norwegian Scool of Economics.

14 Appendix Derivation of te firstbesttaxrules. Te first order conditions for an interior maximum are given by: ½ ¾ ½ ¾ L z = u g + ug 2 x t x u g + ug 3 y x y u g + γ x + t x + t y =0 (30a) q ½ ¾ ½ x ¾ L z = u g + ug 2 x t y q y u g + ug 3 y x y q y u g + γ y + t x + t y =0 (30b) q y q y q ½ ¾ ½ y ¾ L z T = ug m + ug 2 x u g m + ug 3 y x u g γ +t x m m + t y y =0(30c) m Performing L t x L L x and T t y Ly gives: T ½ ¾ ½ ¾ bz + ug 2 bx u g + ug 3 by bx by u g + eγ t x + t y = 0 (3a) q ½ ¾ ½ x ¾ bz + ug 2 bx q y u g + ug 3 by bx by q y u g + eγ t x + t y = 0 (3b) q y q y q y were abdenotes a compensated price effect and eγ def = γ. Using te omogeneity u g condition on te compensated price effects, (3) may be rewritten as # # ug 2 u g u g 3 u g i q y 0 0 q x " bx by Because te substitution matrix bx qx by qx bx q y by q y bx qy by qy = eγ " bx q t x t x y by bx q y by q y. (32) is negative definite, (32) reduces to eγt x = q x ug 2 u g, and eγt y = q y ug 3 u g. (33a) Inserting tese conditions back in te FOC for T,sowstateγ =. Since q x = u 2 and q y = u 3,tefirst best tax rates are as in te text. Derivation of u g (z, x,y) By definition of F ( ) " Ã z + µ(χ)dχ y g = F x, y, u z + m " Ã z = F x, y, u z + 2!# µ(χ)dχ,x,y y g!# µ(χ)dχ,x,y y g (34) y g µ(χ)dχ (35)

15 By definition of F g ( ), F g [x, y, u g (z, x,y)] = z, wic means te above expression canbewrittenas " à F g [x, y, u g (z, x,y)] = F x, y, u z +!# µ(χ)dχ,x,y y g y g µ(χ)dχ (36) From (4) à F g [x, y, u g (z,x,y)] = F "x, g y, u z + y g µ(χ)dχ,x,y!# (37) Terefore à u g (z,x,y) =u z + y g µ(χ)dχ,x,y!. (38) Proofoflemma. ³ u g Expression 2 u can be written as v2 v (z z + y g µ(χ)dχ,x x,y y ). (39) (³ u 2 u g looks similar but as te extra term µ(y )). For tese preferences, it can be sown tat te Marsallian demands for commodity x is of te form b (q) x x (q, m )=x + b(q) m (p) [m m (p)]. (40) were b(q) is minimal expenditure necessary to generate one unit of utility and m (p) def = z + q x x + q y y denotes survival income. Similar expressions old for te oter two commodities. Under te lemma s assumptions, expression (39) reduces for every agent to v i ( b(q) x + µ[ b(q) y], b(q) x, b(q) y), (4) v q z q y q y since te expression is omogenous of degree 0 in its arguments and te term tus drops out. Every agent will ten ave te same vector of (first best) m m(p) b(q) m(p) 3

16 tax rates, t c, say. Since all substitution effects are proportional to supernumerary income, (2) reduces to P β (m m(p)) P (m m(p)) tc = t, or (42) [β av + cov(β, m m(p) m av m(p)) )]tc = t (43) On te oter and, because te income effects are independent of income, (23) can be written as β r(t c )=r(t), werer( ) denotes te net increase in government revenue from a marginal increase in te lump sum grant to a consumer. Combining bot results gives: r(t) r(t c ) tc = t (44) If ten r(t) > (<) r(t c ) ten t c À ( ) t. Because bot income effects are positive, tis implies tat r(t c ) > (<) r(t), a contradiction. Terefore t c = t and β =(all ). 4