Discussion Papers in Economics

Transcription

1 Discussion Papers in Economics No No 00/1 000/ Dynamics Correcting of Maret Output Failure Growt, Due Consumption to Interdependent and Pysical Preferences: Capital in Two-Sector Wen Is Piecemeal Models of Policy Endogenous Possible? Growt by by E Randon Farad & P Nili Simmons Department of Economics and Related Studies University of Yor Heslington Yor, YO10 DD

2 Correcting Maret Failure Due to Interdependent Preferences: Wen Is Piecemeal Policy Possible? E Randon & P Simmons y July, 00 Abstract Generally, implementation of Pigovian taxes to correct for maret failure requires an enormous set of information For eac commodity-person combination a di erent tax is required to correct te resulting maret ine ciency In tis paper, we analyse interdependent preferences and ine ciency of te maret solution wit te aim of nding conditions justifying simple rules for suc taxes We examine te utility possibility curve and Scitovsy community indi erence curve, allowing for general utility interdependence and agent eterogeneity In particular we sow te equivalence of taxes derived from te Marsallian and compensated demand approaces We move on to analyse te welfare cost of consumption externalities and sow tat it decomposes into part due to individuals coosing suboptimal quantities and part due to individuals using valuations tat are not socially optimal We sow wat forms of externality can justify simple policy corrections In particular, we analyse te conditions wic are required for te maret failure to be corrected by: 1) speci c indirect ad valorem taxes on commodities, ) te same proportional tax rate on every commodity, ) a proportional income tax rate on eac individual Te conditions are related to te restrictions necessary to ave H syntetic consumers witout externalities wo replicate beaviour of individuals wit externalities An example wit two individuals and tree goods concludes te paper Keywords: Consumption externalities, Piecemeal policy JEL classi cation: D, D11 1 Introduction Usually some taxation metod is suggested for te correction of externalities Tese ideas are based on Pigovian transfers and Lindal pricing in wic te taxes serve to replace private marginal rates of substitution by social marginal rates of substitution (Lindal (1919), Bergstrom (190), Milleron (19), ) However in general di erent commodity and person speci c taxes are necessary in eac maret and te appropriate tax rate depends on te particular Pareto optimal allocation of commodities wic is under consideration Te tax autority needs full information on preferences and tecnology to implement Pigovian taxes, Lindal distributive mecanisms do not require tis but require restricted forms of externality to be able to acieve e ciency (Bergstrom (19), Tian (00)) Hence tere is interest in nding conditions under wic simple taxes can be used In reality we ave a mixture of personal income taxation, general sales taxes at a more or less uniform rate (VAT) and speci c commodity indirect taxes eg on alcool and tobacco Because tese tax rates are not di erentiated by Dipartimento di Scienze Economice, Strada Maggiore, 01 Bologna, Italy y Department of Economics, University of Yor, Yor YO1 DD UK 1

3 personal spending patterns, tey are feasible to administer Some researc tries to de ne conditions under wic piecemeal policy is possible, in wic te appropriate correction in one maret is independent of activity in oter marets Tere is some looseness in tis idea-for example if only one maret is subject to externalities, it means tat taxation is directed only to transactions in tat maret If several marets are subject to consumption externalities it migt mean tat te taxes necessary to restore Pareto optimality in one maret are una ected by exogenous socs in oter marets eg supply socs In te context of a distorted sector in wic consumer and producer prices di er, Jewitt (1981) and Blacorby, Donaldson & Scworm (1991) nd tat piecemeal policy is only justi ed if te set of e cient points can be described by a relation between two aggregates of commodities corresponding to te distorted and non-distorted sectors: However te reasons for te distortion are not modelled In particular, te distortion is not an endogenous function of quantities as it is wit consumer externalities Caracterising corrective tax rules wit general interdependent preferences and nding wat sort of preference restrictions must be imposed if te necessary taxes are to tae simple forms are still open researc issues Kooreman & Scoonbee (00) analyse consumption externalities in a setting wit a xed income distribution After sowing tat Pareto improvements are generally possible starting from a maret solution wit consumption externalities 1, tey consider an example in wic individual preferences ave te linear expenditure form and te consumption externalities enter troug te subsistence terms Imposing particular assumptions (identical preferences, equal incomes, te subsistence term for commodity i depends only on te total consumption of tat commodity i by all oter consumers) tey compute te welfare losses due to externalities as te deviation from an equal utility distribution Pareto optimum and ten compute te commodity taxes tat will mae te maret solution and Pareto optimum coincide In tis paper we explore te implications of general interdependent preferences and te teoretical properties wic are required if piecemeal policy is to be possible Wit general preferences, we start by sowing tat a Pareto optimum corresponding to a given utility distribution can be reaced troug a decentralised maret system using Pigovian taxes on compensated demands wit consumption externalities We caracterise te system of Pigovian taxes required for tis tas Ten we sow tat a Pareto optimum corresponding to a given income distribution can also be acieved as a maret solution from Marsallian demands wit Pigovian taxes We ten sow tat te compensated and Marsallian approaces are equivalent wic gives us a measure of te welfare cost of te externalities de ned in terms of te expenditure functions of te 1 So long as tere is at least one good suc tat all individuals value te consumption of tat good by eac oter individual at less tan its maret value Following up a suggestion of Polla(190) and Kapteyn et al (199)

4 individuals Tis cost decomposes into part due to te wrong pricing of goods in te maret solution, and part due to individuals consuming te wrong quantities to implement te Pareto optimum Te equivalence of te compensated and Marsallian approaces is also of interest because te maret solution is a Nas equilibrium in mutual best responses but te strategic interpretation of te two is di erent since in one case te individuals environment as prices and income, and in te oter prices and a given utility level Next we try to analyse wen piecemeal policy is justi ed and proceed by de ning necessary conditions on preferences under wic te individual compensated demand for any good under externalities depends only on prices, individual utility and te consumption by oter individuals of tat particular good Tis is te scenario in wic it most liely tat simple taxes will succeed It turns out tat te externalities must enter individual preferences and expenditure functions as a form of subsistence level/cost In a sense te linear expenditure type system wit te form of externalities Kooreman & Scoonbee use is one of te forms tat must prevail if piecemeal policy is to be justi ed Given tis form of preferences we nd furter restrictions under wic simple taxes will wor In particular if externalities tae te form of varying in a linear way wit te total consumption of all oter individuals of eac good (we call tis case linear popular no spillover externalities) ten wen: - Only one good as an external e ect, te Pigovian taxes on any individual are identical for oter goods However for te externality inducing good te tax on any individual depends on te relative strengt of externalities between oter individuals - For every individual te strengt of externality is equal for eac good ten for any pair of goods te ratio of te tax rates of te two goods is equal for all individuals (and it is equal to te strengt of te externality on te two goods) If te social and private marginal valuations of individuals coincides ten for eac good every individual faces te same Pigovian tax rate, it is as if tere are speci c indirect ad valorem taxes on commodities - For eac individual tere is a common strengt of externality for every good wic di ers by individual: eac individual pays te same proportional tax rate on every commodity so tis is equivalent to a proportional income tax rate - For every individual and every commodity tere is a common strengt of externality Here te proportional income tax rate of eac individual is actually at te same rate for all individuals Interestingly in tis case te compensated Nas equilibrium aggregate demands will also tend to satisfy te usual Slutsy symmetry and negative semide niteness restrictions so tat in te aggregate it may be as

5 if tere are syntetic consumers wose beaviour witout externalities replicates aggregate demand Finally we illustrate te results by computing te compensated and Marsallian Pareto optima, Nas equilibria, Pigovian taxes and te social cost of te externalities troug an example wit two individuals and tree commodities wic is designed to sow te relations between te general concepts explicitly Te plan of te paper is to review te dual of te Pareto optimality problem witout externalities (Section ) We ten use tis dual formulation in te presence of consumption externalities to de ne te appropriate taxes in general and ten sow ow individuals facing tese taxes interact to produce a Nas equilibrium wic is Pareto optimal First we compute te Pigovian taxes required to ensure tat te maret solution (a compensated Nas equilibrium) yields a Pareto optimum wit a given utility distribution (Section ) Ten we consider Pigovian taxation wit interdependent preferences and xed welfare weigts (Section ) We establis lins between te compensated and te Marsallian approac (Section ) We compute te welfare cost of te externalities (Section ) Next we discuss te piecemeal policy issue (Section ) An example wit goods and individuals concludes te paper Te Setting We wor wit H individuals indexed ; as preferences given by u (x ) in te absence of externalities were x is a consumption allocation of n commodities Wit externalities we write u (x ; x ~ ) were x ~ is an ordered list of te consumption allocation of all individuals oter tan : We represent te resource constraint of te system by a linear function x = y i We can interpret tis as a linear transformation locus for te economy, in tis case represents bot te maret price and average cost of te it commodity wic is in perfectly elastic supply Alternatively it may represent a maret budget constraint for a group of H individuals, so is te maret price for good i and y represents te disposable resources of te group For example te group may be a family or a team witin an organisation In te family case we tin of a family model in wic te H individuals are eac family members and te family as exogenous resources y wic can be allocated to purcase consumption goods for di erent family members Pareto Optimality Witout Externalities Wen tere are no externalities, te problem of nding a Pareto optimal allocation is related to te ideas of utility possibility curves and Scitovsy community preference elds Gorman (19) Given a xed utility

6 distribution (u 1 ; ::u H ), te Scitovsy community indi erence curve is: (u 1 ; ::u H ) = min[ 1 ; :: n ju (x ) u ; x ]: (1) giving te minimum amounts of te aggregate quantities of goods tat place eac individual on is prescribed indi erence curve (Here min and max operators are understood in a vector sense) For eac set of aggregate quantities a distribution of commodities between consumers is implicitly de ned wic just allows attainment of te utility distribution De ne te utility possibility curve for xed aggregate quantities of goods ( 1 ; :: m ) by: U( 1 ; :: m ) = max[u 1 ; ::u n ju (x ) u ; x ] () Te utility possibility curve indicates te maximum level of utility tat an individual can acieve given te utility level of te oters Gorman(19) sows tat = ( 1 ; :: m ) (u 1 ; ::u H ) i (u 1 ; ::u H ) U( 1 ; :: m ): Wit te linear resource constraint () becomes U(p; y) = max[u 1 ; ::u n ju (x ) u ; x ; i y] () Under regularity conditions a point u 1 ; ::u H is in U(p; y) i te group resources are just su cient to reac tis utility distribution Analogously to (1) we ave y = G(p; u 1 ; ::u H ) = min[ i ju (x ) u ; x i ] = min[ x ju (x ) u ; = 1::H] = g (p; u ) () were g (p; u ) is te expenditure function of individual : Tis implies tat te group can decentralise te tas of attaining a particular Pareto optimal utility distribution by allocating y of te group resources to and leaving to mae teir own coices From a version of Hotelling s lemma we can de ne te aggregate compensated demand F i (p; u 1 ; ::u H ) u 1 ; ::u H )=@ g (p; u )=@ and so te aggregate compensated demands inerit te properties of te individual compensated demands; in particular tey ave a symmetric and negative semide nite Jacobian wit respect to p and are omogeneous of degree zero in p: u(:) is continuous, strictly quasiconcave and locally nonsatiated

7 Pareto Optimality wit Externalities 1 Wit a xed utility distribution Wit general externalities u (x ; x ~ ) te problem of attaining a Pareto optimum as te form min[ x ju (x ; x ~ ) u ; = 1::H] Te rst order conditions are = u (x ; x ~ ) = u ; 1::H () were c is te marginal social cost of individual : Tat is all marginal e ects of an individuals consumption must be taen into account so tat in particular () is lost 11 Implementation of a compensated Pareto optimum If eac individual were set te tas of reacing a xed utility level at minimum cost min[ x ju (x ; x ~ ) u ] () tey would set = (x ; x ~ ) U (x ; x ~ ) = u and would ignore te e ect of teir consumption on oters Solving (8) for one individual gives te compensated reaction curves x c = (p; c u ; x ~ ) and solving tese equations in turn yields te compensated Nas equilibrium (CNE) demands : x NEc = NEc (p; u 1 ; ::u H ) (9) However we could introduce individual and commodity speci c Pigovian taxes so tat te cost to of a unit of good i becomes and (8) becomes c = U (x ; x ~ ) = u : We continue to assume u () is strictly quasi concave and nonsatiated in x ; and continuous in all variables Given our assumptions tese are also su cient

8 or c c n U (x ; x ~ ) = u : = i = 1::n 1 (11) Te compensated Pigovian taxes must be selected to yield () wic leads to (see Appendix A1): c c n = = n = = c n c i (1) wic in turn can be written for good i as c i1 =c n1 c ih =c nh = P i 1 = [ c 1 =@x n1 ] P i H = [ c H =@x nh ] (1) were i is te t element of te inverse of te HxH matrix =@x i1 =@u =@x i H =@x i1 =@u H =@x 1 =@x i =@u 1 =@x i1 1 ::: ::: H =@x ih 1 =@u H =@x 1 =@x ih =@u 1 =@x =@x ih =@u =@x i ::: 1 Tis matrix will be nonsingular if at te Pareto optimal point in question te marginal externalities for good i are independent-tere is su cient diversity between individuals for no one individual to be a ected by externalities in a way wic is a linear combination of te external e ects for oter individuals It is evident tat for eac individual ; te Pigovian taxes are only determined up to a factor of proportionality, tey serve to equate te social and private marginal rates of substitution Substituting (1) into (11) gives () Tis means tat if we solve all te (n + 1)H equations in (11) for te unnowns x and c wit c de ned by (1) ten it is equivalent to solving te system of equations () In oter words a Nas equilibrium in wic eac individual faces te corrected prices, given consumption of te oter individuals and a given utility level replicates te socially optimal way of acieving te same utility distribution If tese taxes are used as xed numbers ten decentralised coice will lead to individuals coosing te Pareto optimal consumption bundle so long as te decentralised coice problem remains well de ned One di culty is tat te tax could be non-positive wic would mae te e ective maret price of te good in question negative Tis could occur if x as suc a strong positive e ect on te utility of oter individuals tat it is e cient to pay per unit of consumption of i: But ten since is nonsatiated, tere will be no decentralised solution- will coose an in nite consumption of good i: We rule tis out Apart from So tey can be scaled to yield zero tax revenue in aggregate

9 tis case, te decentralised coice problem is well beaved under Pigovian taxation given te regularity assumptions-te rst order conditions will caracterise te individuals best coice Wit xed welfare weigts Instead of obtaining a Pareto optimum troug minimising te aggregate cost of nancing te utility distribution, te Pareto optimum could be de ned by maximising a linear combination of individual utilities subject to te aggregate budget constraint max U (x ; x ~ ) (1) s:t: : pi x = y leading to m = x = y Te corresponding individual problem for some income distribution (y 1 ; ::::y H ) is (1) max U (x ; x ~ ) (1) s:t: : pi x = y Solving (1) for one individual gives te Marsallian reaction curves (Kooreman & Scoonbee) x m = m (p; y ; x ~ ) and solving tese equations in turn yields te Marsallian Nas equilibrium (MNE) demands: x NEm = NEm (p; y 1 :::y H ) (1) Tere are ten two reasons for lac of equivalence between te Pareto optimum and te maret solution Firstly private and social marginal rates of substitution (mrs) di er so in te maret solution individuals ignore te e ects of teir consumption on oters; secondly te income distribution may not align wit te particular welfare weigts tat are being used 8

10 1 Implementation of a Marsallian Pareto optimum If we introduced Pigovian taxes on eac person and good m ; te rst order conditions in te maret solution would become m m =@x i = 1::n 1 (18) = x = y i and if we want tese to replicate (1) te taxes must satisfy (see Appendix A) m m @u n = m i = = (19) i wic for good i can be written m i1 =m n1 m ih =m nh = m P i 1 = 1 =@x n1 ] m P i H = H =@x nh ] (0) were te 0 s ave te same de nition as previously Tus te Marsallian Pigovian taxes coincide wit te compensated Pigovian prices if = m = c In addition te income distribution must be cosen to matc te welfare weigts ie if we de ne te individual utility levels acieved in te Marsallian Pareto optimum and in te maret solution wit Marsallian Pigovian taxes by v P O(p; y; 1; ::: H ) and v m(m p; y 1 ; ::y H ) respectively ten y 1 ::y H must be cosen so tat v P O (p; y; 1 ; ::: H ) = v m ( m p; y 1 ; ::y H ) = 1::H (1) giving y ( m ; p; y; 1 ; ::: H ): To ensure tat all resources are exactly consumed, te taxes must be scaled to give zero tax revenue Equivalence between Marsallian and Compensated Nas Equilibria Tere are lins between te compensated and Marsallian Nas equilibria From standard teory we now tat if g (p; u ; x ~ ) = minfp x u (x ; x ~ ) = u g x v (p; y ; x ~ ) = maxfu(x ; x ~ )jpx = y g x 9

11 ten as identities in p; u ; y ; x ~ g (p; v (p; y ; x ~ ); x ~ ) = y v (p; g (p; u ; x ~ ); x ~ ) = u Tus if y gives a maximal utility level of u ten y is te minimal cost incurred to reac te utility level u : Setting x ~ to be its Nas equilibrium demand function g (p; v (p; y ; NEm ~ (p; y 1 ::y H )); NEm ~ (p; y 1 ::y H )) = y = p F (p; y 1 ::y H ) v (p; g (p; u ; NEc ~ (p; u 1 :::u H )); NEc ~ (p; u 1 :::u H )) = u We also now from standard teory tat te Marsallian and compensated demands satisfy associated identities: c (p; u ; x ~ ) = m (p; g (p; u ; x ~ ); x ~ ) m (p; y ; x ~ ) = c (p; v (p; y ; x ~ ); x ~ ) so evaluated at te relevant Nas equilibrium demands c (p; u ; x ~ ) = m (p; g (p; u ; NEm ~ (p; g 1 ::g H )); NEm ~ (p; g 1 ::g H )) m (p; y ; x ~ ) = c (p; v (p; y ; NEc ~ (p; v 1 :::v H )); NEc ~ (p; v 1 :::v H )) From tis we deduce tat (i) te Nas equilibrium demands for eac good and individual in te CNE wit p; 1 ::: H ave identical values to te MNE demands wit p and individual incomes y set to te costs of eac individual of purcasing te goods in te CNE (ie g ) (ii) te Nas equilibrium demands for eac good and individual in te MNE wit p; y 1 :::y H ave identical values to te CNE demands wit p and individual utilities u set to te utility levels of eac individual in te MNE Tus for any CNE tere is an associated income distribution generating a MNE wic yields te utility distribution of te CNE wit identical demands for eac individual and good From tis we can deduce equivalence of te Marsallian and compensated Pareto optima since tese can be represented as Nas equilibria wit Pigovian taxes Comparing(11) and (18) te solutions will coincide if tere are suitable lins between te utility distribution, te aggregate resources, te income distribution and te welfare weigts Tere is a consumption allocation x wic solves (11) and attains te utility 10

12 distribution u 1 :::u H at minimum aggregate cost Setting y = i x te allocation x also solves (18) Ten wit y = y tere are values of for wic te allocation x also solves:(1): Tis re ects Gorman s result tat te consumption allocation is in te Scitovsy community indi erence curve if and only if te utility distribution de ning te Scitovsy community indi erence curve is attainable from tat consumption allocation Tis result extends to te case of externalities Tere is also a form of () G(p; u 1 ; ::u H ) = = c NEc ( c p; u 1 ; ::u H ) g ( c p; u ; NEc ~ ( c p; u 1 ; ::u H )) In te sequel we concentrate on te compensated demand scenario because of te equivalence between tis and te Marsallian scenario outlined above Te Pigovian taxes are endogenous in tat tey depend on all quantities consumed by eac individual in te particular Pareto optimum Tey vary by bot commodity and individual Tere are some general properties of te Pigovian taxes: using c c n = = c suppose tat tere are no external e ects in good n c c c = = = 0 = Ten and c c n = = n c n = = = i Second suppose tat in addition good i as no external e ects so tat = () = 0 = Te Pigovian taxes become c c n = 1 so no correction is required in te it maret 11

13 Te Welfare Cost Of te Externality We can use te framewor of consumer surplus to compute te welfare cost of consumption externalities Private coices ave a cost to of g (p; u ; x ~ ) = min[ x ju (x ; x ~ ) u ] wic in te Nas equilibrium is g (p; u ; x NE ~ ) = eg (p; u 1 ; :::u H ): Te aggregate cost of attaining u 1 ; :::u H wit decentralised decisions is ten eg (p; u 1 ; :::u H ) = x NE Wit Pigovian taxes, te e cient way of attaining u 1 ; :::u H as a cost of Hence te welfare cost can be measured by g (p; u ; x P O ~ ) = C = g (p; u ; x P O ~ ) = = x P O (x P O x NE ( ) x NE ( ) eg (p; u 1 ; :::u H ) () x NE ( )) + (x NE ( ) x NE ( )) Te welfare cost can be decomposed into part corresponding to te resource misallocation arising from te incorrect coice of quantities and part arising from te misvaluation of commodities In Figure 1 te two parts of te welfare cost are outlined in bold p,mv Πp p MSV MPV x P0 x NE Figure 1: Welfare cost decomposition 1

14 Piecemeal Policy 1 Compensated Nas Equilibrium wit No Spillovers Piecemeal policy involves te idea of being able to correct for maret failure in one maret in a way wic is independent of conditions in oter marets For example, if tere is only one commodity wic exibits externalities ten piecemeal policy is possible if te Pigovian taxes in all te oter marets are equal to zero More generally, piecemeal policy is valid if te Pigovian taxes in one maret are invariant to canges in conditions (in eiter preferences or prices) in oter marets, or in wealt Bot tese ideas involve some notion of independence of te external e ect in di erent marets A natural place to start is to consider te case in wic any individual demand for any good only as external e ects corresponding to consumption of te same good by oter individuals For example one consumers spending on say mobile pones is only in uenced by te beaviour of oters troug teir spending on mobile pones Tis is close to a necessary condition for piecemeal policy to be possible From (1) for = n to be independent of quantities consumed of goods oter tan i; n requires tat (@u =@x )=(@u =@x i ) be independent of x j ; j = i; n meaning tat u () in Nas equilibrium is separable in commodities De ne externalities wit no commodity spillovers to exist if eac individuals compensated reaction curve for any good i depends only on prices, utility and te consumption of oter individuals of tat good x c = c (p; u ; x i1 ; ::x 1 ; x +1 ; ::x ih ) () Te rst question is ten wat are te individual preferences tat generate tis form of compensated demand? Since c () must be omogeneous of degree zero in p, it follows tat we must be able to write te expenditure function as g (p; u ; x ~ ) = c () and moreover by di erentiating () and appealing to Hotelling s rule, j Tis is impossible in te case of Marsallian demands: it would mae eac individuals demand for any good depend on prices, individual income and te demand for tat good by eac oter individual bx = f (p; y ; x i1 ; ::x 1 ; x +1 ; ::x ih ) However tis is inconsistent wit te budget constraint; to old for all p; M needs te identity pi f (p; y ; x i1 ; ::x 1 ; x +1 ; ::x ih ) = y and di erentiating troug wrt x i implies =@x i = 0 identically or tat tere can actually be no externality Essentially all goods compete for consumer income 1

15 so tat for i = c =@p j must be independent of te externality e ects x i1 ; ::x 1 ; x +1 ; ::x ih because te LHS must be independent of x i1 ::x ih and te RHS independent of x j1 ::x jh : Tat c =@p j@x i = 0 so tat c () = ( ; u ; x i1 ; ::x 1 ; x +1 ; ::x ih ) + (p; u ) Integrating tis over leads to an individual expenditure function of te form g (p; u ; x ~ ) = i A ( ; u ; x i1 ; ::x 1 ; x +1 ; ::x ih ) + B (p; u ) () so tat 0 s compensated demands ave te form x ( ; u ; x i1 ::x ih (p; u () Tis as te implication tat in te compensated demands tere are no spillovers of externalities between commodities We need A i () to be omogeneous of degree one in and so e ectively it is linear in ;and B() also to be omogeneous of degree one in p: Imposing tis leads to g (p; u ; x) = i e A (u ; x i1 ; ::x 1 ; x +1 ; ::x ih ) + B (p; u ) () Tis as a relation to a Klein-Rubin linear expenditure system of preferences; in fact it extends tis by allowing te subsistence level to vary wit te standard of living of te consumer u : Interestingly Polla (19), Kapteyn et al (199) and Kooreman & Scoonbee (00) focus on interdependence in te Klein- Rubin utility function, interpreting te subsistence parameter as a linear function of te quantities consumed by oter individuals 11 Te Linear No Spillover Case A special form of () maes te externalities wor only troug a linear combination of te consumption of oters (Kapteyn et al (199)) u (x ; x ~ ) = u (x 1 + = w 1 x 1 ; ::; x n + = w n x n ) Ten g (p; u ; x) = w i x i + B (p; u ) i = Note tat B () itself can be interpreted as an expenditure function: it must be concave and omogeneous of degree one in p and represents an arbitrary form of base utility Here te compensated demands are x = = w i x i (p; u 1

16 Te strengt of te external e ect is independent of prices and of te standard of living A furter specialisation arises if 0 s preferences react only to te total consumption of oters of eac good and in a way tat is independent of te standard of living Empirically examples would be congestion goods (public transport) and networ goods (mobile telepones) In tis case wen marginal external e ects are constants, ; individual preferences ave te form u (x ; x ~ ) = u (x x 1 ; ::; x n + n x n ) and te compensated demands become = = x = = x i (p; u (8) so it is te total consumption of oters wic a ects an individual s compensated demand We call tis linear popular no spillover externalities Kooreman & Scoonbee use a linear expenditure system wic as tis form Te Pigovian Taxes For linear no spillover externalities from (1) te taxes are given by P 1 i = [ c 1 =@x n1 ] = c i1 =c n1 c ih =c nh P i H = [ c H =@x nh ] (9) were [ i ] is a matrix of constants being te inverse of 1 w i1 ::: w ih1 w i1 1 : w ihh 1 w i1h w ih 1 Te e ect of linearity of externalities is tat [ i ] is independent of prices or te level of individual income or utility In te popular case te matrix simpli es to A = 1 i ::: ih i1 1 : ih i1 i 1 and ten it can be sown tat (see appendix A) i = =( Y i 1)= det(a) (0) For eac commodity-person, te tax factors into a product of a person speci c term, common to all commodities, and a person-commodity speci c term Te person speci c tax ( =( = )) re ects te di erence between te maret and social marginal valuations of individual at te Pareto optimum as 1

17 measured troug te marginal utilities of te last good Te commodity speci c part depends only on te strengt of te various externalities and it is independent of prices or te level of utility Tis is due to te linearity of te externality e ect Moreover for any commodity te ratio of te Pigovain taxes on any two individuals ; 0 is independent of te externalities imposed by any oter individual ( c 0=c n 0) = (c = c n) = [( 0 1)=( 0 1)] = [ c 0@u 0= 0=c 0@u = ] so tere is an independence of irrelevant externalities property 8 Restricted types of popular externalities will generate commonly observed taxation regimes, for example a personal income tax system or a system of speci c indirect commodity taxes In particular if - Only good 1 as an external e ect ( = 0; i > 1) Te Pigovian taxes on any individual are identical for goods i >, owever for good 1 te tax on individual depends on te relative strengt of externalities between oter individuals - For every individual te strengt of externality is equal for eac good ( = i ) For any pair of goods te ratio of te tax rates of te two goods is equal for all individuals (and it is equal to te strengt of te externality on te two goods) If te social and private marginal valuations of individuals coincide ten for eac good every individual faces te same tax rate, it is as if tere are speci c indirect ad valorem taxes on commodities - For eac individual tere is a common strengt of externality for every good ( = ) Eac individual pays te same proportional tax rate on every commodity so tis is equivalent to a proportional income tax rate - For every individual and every commodity tere is a common strengt of externality ( = ) Te proportional income tax rate of eac individual is actually at te same rate for all individuals Te Compensated Nas Equilibrium Te compensated reaction curves (8) can be solved commodity by commodity for te compensated Nas equilibrium 1(p;u 1) 1 i1 ::: i1 x i 1 i i : = ih ih 1 x H (p;u H 8 Te structure of te taxes in (0) and tis property would also old if u = U (x 1 ; :::x n ; = x 1 ; ::: = x n ): 1

18 leading to x NE = ( (p; u and aggregate Nas equilibrium compensated demands NE i = ( ( (p; u (1) (1) is of special interest since eac term B (p; u ) as all te properties of an expenditure function In particular it as a negative semide nite Jacobian, so if ( ( )) > 0 te Jacobian of te aggregate compensated Nas equilibrium demands will also satisfy te sign restrictions of negative semide niteness Note tat te aggregate demand may fail to ave a negative semide nite Jacobian if some or all of te coe cients are negative Te aggregate compensated Nas equilibrium demand ave a symmetric Jacobian if ( ( B (p; j = ( ( j B (p; u A clear case wen tis olds is if = for all i: te case of common externality e ects across commodities Ten we can tin of tese aggregate demands as coming from H syntetic consumers were te t consumer as an expenditure function ( ( ))B (p; u ) exibiting no externalities (recall eac B () as te properties of an expenditure function) and wose compensated demand is z i = ( ( (p; u () A Commodity, Individual Example 1 Te utilities As in Polla, Kapteyn and Kooreman & Scoonbee, we tae an LES utility function 9 for = 1; : u (x 1 ; x ) = log(x + x i ); = ; = 1 () were x = (x 1 ; x ; x ): Te individual expenditure function is g (p; u ; x ~ ) = i x i + exp(u ) Y ; = 9 Detailed calculations are available on request 1

19 Demands in a Compensated Pareto Optimum For i = 1; ; and ; = 1; = x P Oc = c (1 i ) i c (1 ) () were: Tus: x P Oc = c = exp(u ) Y ( 1 i ) 1 i i (1 ) (exp(u ) Y Y 1 i 1 i Y 1 + (1 i ) (exp(u i ) 1 i Te cost to eac individual of attaining te utility level u is! y c Y 1 i = (exp(u ) (1 i i ) 1 i! i Y 1 i (exp(u ) (1 ) 1 i i Compensated Demands in a Nas Equilibrium i () Y i p i Y Y i Y i i ) () ) ) i ) On te oter and in te maret solution, te compensated Nas equilibrium demands for eac individual and good are x NEc = exp(u ) Y + exp(u ) i (1 i ) Y i i () Compensated Pigovian Prices Applying equation (1) te Pigovian taxes are given by c = (1 i)(1 ) (1 i )(1 ) ; c = 1 (8) for eac = : Te terms in good represent te e ects of te marginal valuation of te individual for good n For positive Pigovian prices we require eiter 0 i < 1 for all i; or i > 1 for all i; : Compensated Demands wit Pigovian Pricing Replacing te prices in (8) wit te tax corrected prices c te Nas equilibrium demands become x NEc = exp(u ) Y Y ( c p ) 1 ( c ) + (9) i exp(u ) i ( c p i) 1 i (1 i ) 18 Y ( c i) i Y i i

20 Putting (8) in (9) we get te Pareto optimal demand stated in () so tese price corrections do eliminate te maret failure Te tax revenue is de ned by ( 1) x NEc Demands in a Marsallian Pareto Optimum For given aggregate resources y and wit a welfare weigts ; 1 respectively for individuals 1 and te Pareto optimal demands are x P Om = y (1 i ) i y (1 ) were 1 = ; = 1 : In a particular Pareto optimum te costs to te two individuals are " y 1 = y i1 (1 ) # i i1 (1 i ) (1 i1 ) " y = y (1 ) i # i1 i (1 i1 ) (1 i1 ) wic de nes te income distribution necessary to sustain te Pareto optimum in marets Note tat te sum of tese is equal to y: Marsallian Demands in a Nas Equilibrium wit Fixed Income Distribution We derive tese from te more general equations (1) below were all m = 1: 8 Marsallian Pigovian Prices Using (0) tese are given by m = (1 i)(1 ) (1 i )(1 ) (0) Notice tat te Marsallian Pigovian prices are equal to te compensated Pigovian prices 9 Marsallian Demands in a Nas Equilibrium wit Fixed Income Distribution and Pigovian Pricing Te Marsallian Nas equilibrium demands wit Pigovian pricing ave te form x NEm 1 = [(y + y ) 1 A 1 ( y + y )A 1 (y + y ) 1 A + ( y + y )A + A y ]=(p 1 D) (1) x NEm = [( 1 y + y ) A 1 (y + 1 y )A 1 (y + y )A ( y + y ) A + A y ]=(p D) x NEm = [( 1 y + y )A (y + 1 y )A (y + y ) A + ( y + y )A + A y ]=(p D) 19

21 were for = ; ; = 1; A 1 = 1 (1 ) M 1 M A = 1 (1 ) M 1 M M A = (1 1 1 ) M 1 M 1 M A = 1 1 (1 )(1 ) M M A = (1 1 1 )(1 ) M 1 M 1 A = (1 )(1 1 1 ) M 1 M 1 M M and D = (1 1 )( ) (1 1 )(1 1 1 ) 1 m (1 1 )(1 11 ) 1 m (1 1 )(1 1 ) m ( )(1 1 ) m 11 m ( )(1 1 ) m 11 m 1 + ( )(1 1 ) 1 m (1 1 1 )(1 1 ) m 1 m ( )(1 1 ) m 11 m 1) To see te equivalence between tese demands and tose in a Pareto optimum for given aggregate resources y and relative welfare weigts on te two individuals, de ne individual incomes by te expenditure of eac individual in te particular Marsallian Pareto optimum: y 1 = y = = y = y " " x P Om i1 x P Om i (1 ) i1 (1 i ) i (1 i1 ) (1 ) i i1 (1 i1 ) i1 i (1 i1 ) # # Substitute tese into te Marsallian Nas equilibrium demands wit Pigovian pricing to get x NEm = ( p; y; y ) (wit 1 = ; = 1 ) in te form x NEm = N yy + N y y D () 0

22 Detailed calculation sows tat N y = A + B N y = A Y were A 11 = ( 1 1) ( 1 + )( 1 1)( 11 ( 11 1) ( 1 1)) ( 1)( )( 1 1) A 1 = ( 1 1) ( 1 + )( 1 1)( 1 ( 1 1) ( 11 1)) ( 1)( )( 1 1) A 1 = ( 1 1) (1 1 )( 11 1)( 1 ( 1 1) + 1 ( 1)) ( 1)(1 1 )( 1 1) A = A 1 = ( 1 1) ( )( 11 1)( ( 1) + 1 ( 1 1)) ( 1)( )( 1 1) ( 1 1) ((1 1 ) ( ))(1 )( 1 1)( 1 1)(1 11 ) ( )( 1) (1 1 )( 1 1) A = ( 1 1) ((1 1 ) + ( ))(1 )( 1 1)( 1 1)(1 11 ) ( )( 1) (1 1 )( 1 1) = (1 11 ) (1 1 ) (1 1 ) Y = Te functions B are dependent on and and do not justify te space to display ere 10 Te ratio Y =D is relatively simple: Y=D = ( 1)( )( 1 1)( ) ( 1 1) ( 1 1)( 1 + )( 1 1)( ) so tat N y =D reduces to te coe cient of y in Marsallian Pareto optimal demand It is more tedious to sow tat te coe cient N y =D also reduces to te coe cient of y in te Marsallian Pareto optimal demand but in fact it does 10 Equivalence of te Compensated and Marsallian Pareto optimum Here we sow tat for any CPO wit a utility distribution 1 ; tere is an aggregate resource level and welfare weigts in wic te Marsallian Pareto optimal demands coincide wit te compensated Pareto optimal demands 10 Details are available on request 1

23 Given a compensated Pareto optimum we can compute te aggregate resources tat it requires by summing te values of te individuals expenditure functions at te CPO: y = g (p; u 1 ; u ) = Y 1 i exp(u ) 1 i Y ; = () Notice tat witout externalities te feasible utility distribution from given income y tat maes say u 1 is maximal gives 1 all te aggregate income y and as noting But wit externalities tis may not be true-1 may be better o from aving some of te income if tere are positive externalitiesin te Marsallian demands at te Pareto optimum corresponding to y; ; replace y by te expression () and ten tae te compensated and Marsallian demands for te rst good by te rst individual, equate tem and solve for giving = exp(u 1 ) Y 1 i 1 i i1 i1 Y i1 i1 = Y 1 i exp(u ) 1 i Y wic as te interpretation of te sare of te total cost of attaining te Pareto optimum attributable to individual 1: Wit tese values of y and ; x P Oc 11 Social Cost = x P Om for eac pair of values i; : Te social cost of te externality SC is te sum of losses incurred by eac individual, in turn te individual losses are te sum of te losses on eac commodity wic depend on te strengt of preference for te commodity and on te strengt of te externality " Y " ( i 1) SC = exp(u ) ( i 1) + ( 1) ( 1) =1 Conclusions # 1 Y (i 1) # ( i 1) In tis paper we use te ideas of a utility possibility curve and a Scitovsy community indi erence curve to implement a Pareto optimum wen tere are consumption externalities We use te long establised idea of Pigovian taxation to analyse tis, focussing on situations in wic piecemeal policy is possible in te sense tat corrective tax policy in one maret is largely independent of tax policy in oter marets We sow tat for tis to be possible individual preferences must ave a form in wic te externality enters as an adjustment to a subsistence term in individual utility or cost Tis is interesting since for oter reasons te literature as suggested modelling interdependent preferences in tis way We sow tat if te

24 externality as tis form and also enters only troug te sum of te consumption of oter individuals and linearly ten various simple tax systems can be used to correct for te externalities In particular te correct taxes on one good can be computed independently of oter goods, and for cases were tere is furter restriction on te form of te externality (especially across individuals or across goods) te commonly used taxes suc as speci c excise taxes or a personal income tax can implement a Pareto optimum Tis provides a justi cation for concentrating on tese forms of preferences, of course te oter justi cation is empirical-suc preferences are liely to arise wit networ of congestion goods Our results generalise tose in te literature on caracterising Pigovian taxes and provide te lin to piecemeal policy We also give a decomposition of te welfare cost of externalities On te positive economics side we nd conditions under wic te maret solution for compensated demands (wic as te form of a Nas equilibrium due to te consumption interdependence) will eiter be downward sloping or ave a symmetric Jacobian wit respect to prices Finally we examine a two individual, tree commodity example to see ow te taxes loo and te features of te welfare cost of te externalities References [1] Bergstrom, T (190), A Scandinavian Consensus Solution for E cient Income Distribution Among Non-malevolent Consumers, Journal of Economic Teory, 8-98 [] Bergstrom, T (19), Collective Coices and te Lindal Allocation Metod in Economics of Externalities by S Y Lin, Eds, Academic Press: New Yor, [] Blacorby, C, Davidson, R and W Scworm, (1991), Te Validity of Piecemeal Second-Best Policy, Journal of Public Economics, -90 [] Gorman, WM (19), Community Preference Fields, Econometrica 1, -80 [] Kapteyn, A, Van de Geer, S A, Van de Stadt, H and T Scoonbee, (199), Interdependent Preferences: an Econometric Analysis, Journal of Applied Econometrics 1, -8 [] Kooreman, P & Scoonbee, L, (00), Caracterizing Pareto Improvements in an Interdependent Demand System, Journal of Public Economic Teory, - [] Jewitt, I, (1981), Preference Structure and Piecemeal Second Best Policy, Journal of Public Economics 1, 1-1

25 [8] Lindal, E (1919), Die Gerectigeit der Besteuring, Lund: Gleerup (englis translation: Just taxation-a Positive Solution In Classics in te Teory of Public Finance, by R A Musgrave and A T Peacoc, Eds, Macmillan: London, 198) [9] Milleron, J C (19), Teory of Value wit Public Goods: a Survey Article, Journal of Economic Teory 9, 19- [10] Polla, R (190), Habit Formation and Dynamic Demand Functions, Journal of Political Economy, - [11] Polla, R A, (19), Interdependent Preferences, American Economic Review, 09-0 [1] Scitovsy, T (191), A Note on Welfare Propositions in Economics, Review of Economic Studies 9, pp-88 [1] Scitovsy, T (19), Some Consequences of te Habit of Juding te Quality by Price, Review of Economic Studies 1, 100- [1] Scitovsy, T (19), Two Concepts of External Economies, Journal of Political Economy, 1-11 [1] Tian, G (00), A Unique Informationally E cient Allocation Mecanism in Economies wit Consumption Preferences, International Economic Review, A Appendix A1 Deriving Pigovian Taxes wit xed utility distribution From () Solve =@x : and put tis in (x ; x ~ c = = 1 = = (x ; x ~ ) c

26 c c n = (x ;x ~ ) c = = = n c P (x ;x ~ n c @u n @x n Note tat in te linear no spillover case tis becomes c c n = = i c n = = = Rearranging () and writing te system of equations for good i c c i = i i i (A1) c c + n n i @x c @x i i1 1 i1 ih ::: @x i ::: 1 i1 = n1 ih = nh 1 ih = c 1@u 1 =@x n1 ::: ::: 0 =@x n ::: 0 ::: ::: c H@u H =@x nh n1 nh = i1 = n1 ih = i i1 1 @x i ::: 1 c 1@u 1 =@x n1 ::: ::: 0 = =@x n 0 ::: ::: c H@u H =@x i1 1 ih ih 1 = ih nh pn P i 1 = c 1 =@x n1 1 pn P i H = c H =@x nh were i 1 are te elements of te inverse of te matrix of marginal externality e ects wit 10 s on te diagonal A Deriving Pigovian Taxes wit welfare weigts and xed income distribution TePareto optimum as P = = (A)

27 n = =@x = = (A) Solve (A) =@x : Use = = i =@x =@x i for any (A) = = 1 1 = Solve for = n m m n = = n i = = i (A) Rearrange (A) c c @x = i (A) In matrix notation (A) is =@x i1 =@u =@x i H =@x i1 =@u H =@x 1 =@x i =@u 1 =@x i1 =@x ih 1 =@u H =@x 1 =@x ih =@u 1 =@x =@x ih =@u =@x i ::: 1 1 =@x n1 ::: ::: 0 i1 = =@x n = ih = nh 0 ::: ::: H =@x nh from (1) =@x n1 =@x nh = Te solution is ten i1 = n1 ih = nh = P i 1 = 1 =@x n1 ] P i H = H =@x nh ] (A)

28 A Pigovian Taxes Wit Popular No Spillover Externalities Let B = We assert tat te inverse of A is C : e 1 1 ::: 1 1 e ::: ::: e H c ij = =i;j (e 1)= det(b) i = j () c jj = e i c ij =i;j c j = [e i =i;j (e 1) + =i;j l=;j (e l 1)] = det(b) for any i = j; eac j () det(b) = e j c jj + i=j c ij for any j = (e j e i 1) =i;j (e 1) + (e j 1) =i;j l=;j (e l 1) To verify tis note tat e 1 c 11 + =1 c 1 e 1 c 1 + =1 c ::: e 1 c 1H + =1 c H e c 1 + = c 1 e c + = c ::: e c H + = c H BC = e H c H1 + =H c 1 e H c H + =H c ::: e H c HH + =H c H A typical o diagonal term of BC as te form e i c ij + =i c j = e i c ij + =i;j c j + c jj = e i =i;j (e 1)= det(b) =i;j l=;j (e l 1)= det(b) + [e i =i;j (e 1) + =i;j l=;j (e l 1)] = det(b) = 0 wilst a typical diagonal term as te form e j c jj + =j c j = e j [e i =i;j (e 1) + =i;j l=;j (e l 1)] = det(b) =j l=;j (e l 1)= det(b) = [e j e i =i;j (e 1) + e j =i;j l=;j (e l 1) =j l=;j (e l 1)] = det(b) = [e j e i =i;j (e 1) + e j =i;j l=;j (e l 1) =j;i l=;j (e l 1) l=i;j (e l 1)] = det(b) = [e j e i =i;j (e 1) + (e j 1) =i;j l=;j (e l 1) l=i;j (e l 1)] = det(b) = [(e j e i 1) =i;j (e 1) + (e j 1) =i;j l=;j (e l 1)] = det(b) = 1

29 Hence te inverse of B is indeed given by C: Te matrix we are actually interested in is 1 ::: H 1= 1 1 ::: 1 A = 1 1 : H = 1 1= ::: ::: 1= H 1 0 ::: 0 0 ::: ::: H derived from B by setting e i = 1= i A 1 = = = From () and () 1 0 ::: 0 0 ::: 0 A = B 0 0 ::: H ::: 0 0 ::: 0 C 0 0 ::: H 1= 1 1 ::: 1 1 1= ::: 1 C 1 1 ::: 1= H c 11 = 1 c 1 = 1 ::: c 1H = 1 c H1 = H c H = H ::: c HH = H c ij = i = =i;j(( 1)= i det(b) (1= i ) =i;j (( 1)= =i;j l=;j (( l 1= l ) c jj = j = j det(b) Te Pigovian tax term is te sum of te terms in any row of To evaluate [c ii + =i c i ] = i consider c 11 = 1 c 1 = 1 ::: c 1H = 1 c H1 = H c H = H ::: c HH = H c ii + =i c i = e c i l=;i c li + l=i c il = [e l=;i (e l 1) + l=;i j=l;i (e j 1) l=i j=l;i (e j 1)] = det(b) = [e l=;i (e l 1) + l=;i j=l;i (e j 1) l=i; j=l;i (e j 1) j=;i (e j 1)] = det(b) = [e j=;i (e j 1) + l=;i j=l;i (e j 1) l=i; j=l;i (e j 1) j=;i (e j 1)] = det(b) = [(e 1) j=;i (e j 1)] = det(b) = [ j=i (e j 1)] = det(b) 8

30 Replacing e j by 1= j [c ii + =i c i ] = i = j=i ( j 1 )=( i det(b)) j = j=i( j 1) det(a) 9