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1 CESifo Woking Pape Seies OPTIMAL FACTOR INCOME TAXATION IN THE PRESENCE OF UNEMPLOYMENT Ekki Koskela Ronnie Schöb Woking Pape No. 279 Apil 2 CESifo Poschinges Munich Gemany Phone: +49 (89) /1425 Fax: +49 (89) hp://.cesifo.de

2 CESifo Woking Pape No. 279 Apil 2 OPTIMAL FACTOR INCOME TAXATION IN THE PRESENCE OF UNEMPLOYMENT Absac Accoding o convenional isdom inenaionally mobile capial should no be axed o should be axed a a loe ae han labou. An impoan undelying assumpion behind his vie is ha hee ae no make impefecions, in paicula ha labou makes clea compeiively. A leas fo Euope, hich has been suffeing fom high unemploymen fo a long ime, his assumpion does no seem appopiae. This pape sudies he opimal faco axaion in he pesence of unemploymen hich esuls fom he union-fim age bagaining boh ih opimal and esiced pofi axaion hen capial is inenaionally mobile and labou immobile. In seing ax aes he govenmen is assumed o behave as a Sackelbeg leade oads he pivae seco playing a Nash game. The main conclusion is ha in he pesence of unemploymen, he convenional isdom uns on is head; capial should geneally be axed a a highe ae han labou. Keyods: Opimal faco axes, union age bagaining, Sackelbeg and Nash games JEL Classificaion: H21, J51, C7 Ekki Koskela Univesiy of Helsinki Depamen of Economics P.O. Box 54 (Unioninkau 37) 14 Univesiy of Helsinki Finland ekki.koskela@helsinki.fi Ronnie Schöb Univesiy of Munich Cene fo Economic Sudies Schacks Munich Gemany onnie.schoeb@ces.vl.uni-muenchen.de

3 1 1. Inoducion In he lieaue on he opimal axaion he condiions fo poducion efficiency ee idenified in a seminal pape by Diamond and Milees (1971). Accoding o hei esul, i is no desiable o diso poducion decisions as a pa of he opimal ax package if (i) pue pofis o pue ens ae fully axed aay, and (ii) a full se of commodiy axes exiss so ha hee ill be one ax insumen fo each magin of subsiuion in he consume's uiliy funcion. Poducion efficiency is achieved if axes on facos of poducion ae no diffeeniaed. Fo vaious easons, hoeve, such as impefec obsevabiliy, legal consains, ec. i may no be possible o fully ax pue ens, in hich case he govenmen is foced o ely on disoionay axes. In he heoy of opimal axaion, he Ramsey ule and is special case, he 'invese elasiciy ule', ell ho disoionay axes should hen be designed so as o minimize he excess buden of he ax sysem. Accoding o he invese elasiciy ule he govenmen should levy he highes ax on he mos inelasic aciviy. This agumen lies behind he convenional isdom concening he axaion of capial income: inenaionally mobile capial should no be axed o should be axed a a loe ae han labou because capial is moe sensiive han labou o changes in is on ax aes. 1 The applicaion of he Ramsey ule o he invese elasiciy ule usually assumes ha hee ae no make impefecions, in paicula ha he labou make cleas compeiively. A leas in he case of Euope, hich has been suffeing fom high unemploymen fo a long ime, his assumpion does no seem appopiae. Hence he quesion aises: should capial sill be exemped fom axaion o be axed a a loe ae han labou in he pesence of unemploymen? This pape shos ha hee ae seveal easons o ague ha his convenional isdom migh no hold in he pesence of unemploymen. Fisly, unde involunay unemploymen he supply of labou is locally infiniely elasic. Accoding o he invese elasiciy ule his ould sugges ha labou should no be axed a a highe ae han capial. Secondly, involunay unemploymen due o he age ae being highe han he compeiive age ae means ha he pivae maginal cos of labou exceeds he social maginal cos of labou. A ay o incease employmen and hence elfae is o 1 Cf. e.g. Egge and Haufle (1999) fo a ecen discussion of his agumen.

4 2 subsidize labou inpu elaive o capial inpu, fo hich social maginal cos equals he old inees ae. We use he fameok hich has been developed by Koskela and Schöb (1998) o analyse he employmen and elfae effecs of a evenue-neual faco ax efom, hich inceases he souce-based capial ax and educes he labou ax, o analyze he condiions fo an opimal faco ax sysem. The model consides a small open economy, hee he domesic poducion is epesened by a single fim facing monopolisic compeiion fom aboad. Capial is assumed o be pefecly mobile acoss counies, hile labou is inenaionally immobile. Wage and heeby unemploymen deeminaion is modelled by he 'igh-o-manage' appoach, accoding o hich he age ae is negoiaed in a bagaining pocess beeen he epesenaive ade union and he fim and he fim hen unilaeally deemines employmen. The govenmen levies axes subjec o vaious consains so as o maximize oal suplus, hich is linea in okes' ne-of-ax age income, he money-meic uiliy hich he unemployed deive fom leisue and unemploymen benefi paymens, and he ne-of-ax pofis. We sudy he ules fo opimal faco axes in he pesence of unemploymen by saing fom he benchmak case hee he govenmen is no esiced in axing pue pofis. Then e move on o he moe ealisic case of esiced pofi axaion and exploe is implicaions, boh fo individual faco axes and fo he sucue of faco axaion. In he analysis of he opimal axaion e conside fis a model ih a Sackelbeg game sucue hee he govenmen chooses ax aes fis, and he labou make oganizaions hen deemine he age ae in a age negoiaion, aking he ax aes as given. The esuls deived fom he Sackelbeg game ill hen be compaed ih he esuls deived fom a Nash game beeen he govenmen and he labou oganizaions. The pape is oganized as follos: Secion 2 pesens he basic model and some qualiaive esuls, hile Secion 3 ses up he social elfae maximizaion poblem unde he appopiae consains. Welfae maximizing faco axes ih opimal pofi axaion ae deived in Secion 4, hile he popeies of opimal faco axes ih esiced pofi axaion ae pesened in Secion 5. In Secion 6 e elae ou esuls o he exising lieaue. Finally, Secion 7 concludes.

5 3 2. The model We apply a fameok hich has been used by Koskela and Schöb (1998) o analyze he employmen and elfae effecs of a evenue-neual ax efom hich inceases he souce-based capial income ax and educes labou axes. We conside a small open economy, hee domesic poducion is epesened by a single monopolisic fim hich poduces good Y ih capial K and labou L as inpus. Capial is assumed o be pefecly mobile beeen counies hile labou is inenaionally immobile. Technology is assumed o be linea-homogeneous and is epesened by a consan elasiciy of subsiuion poducion funcion Y = f ( L, K). (1) The monopolisic fim faces oupu demand D( p), hich is deceasing in he oupu pice p and is assumed o be isoelasic, i.e. Y = D( p) = p ε (2) ih ε ( ) D( p) p p Y denoing he oupu demand elasiciy. The close subsiues fo good Y on he old make ae, he moe elasic oupu demand becomes. The fim maximizes pofis, hich ae given by = p ( Y) Y K L. (3) The fim consides he goss inees ae as given. I is given by he ne-of-ax inees ae plus a souce-based capial ax, i.e. = (1 + ) ih denoing he capial ax ae. The goss age % consiss of he ne-of-ax age, hich is negoiaed beeen a ade union and he fim, plus he labou ax, i.e. labou axes and social secuiy conibuions, so ha = (1 + ). To guaanee a pofi maximum, he oupu demand elasiciy mus exceed uniy, i.e. ε >1, in hich case pofi maximizaion implies ha he fim ill se a pice hich exceeds he consan maginal cos c( %, %) by a consan mak-up faco ε ( ε 1) > 1. All N okes of he economy ae epesened by a ade union hich maximizes is N membes' ne-of-ax income. Each membe supplies one uni of labou if employed, o zeo labou if unemployed. The ne-of-ax income of a oking membe hence equals he ne-of-ax age ae.

6 4 Being unemployed a ade union membe has an ouside opion b hich depends on he uiliy deived fom leisue and he unemploymen benefi ansfes fom he govenmen. The objecive funcion of he ade union can hus be ien as V * = L + b( N L). 2 (4) The age ae is deemined in a bagaining pocess beeen he ade union and he fim and he fim hen unilaeally deemines employmen. This is modelled by using a 'igh-o manage' model hich epesens he oucome of he bagaining by an asymmeic Nash bagaining. 3 The fall-back posiion of he ade union is given by V = bn, i.e. if he negoiaions beak don, all membes eceive hei esevaion age equal o he ouside opion. The fall-back posiion of he fim is given by zeo pofis, i.e. =. Using V V * V, he Nash bagaining maximand can be ien as = β 1 V β, (5) ih β epesening he bagaining poe of he ade union. The fis-ode condiion ih espec o he ne-of-ax age ae is V = + ( 1 β) =. (6) V β Using a CES poducion echnology e ill apply he explici fomulaion of he age elasiciy of labou demand, η L L = σ + s( σ ), ih σ being he elasiciy of subsiuion beeen L, ε labou and capial and s = L cy being he cos shae of labou (cf. Koskela and Schöb 1998) o fuhe develop condiion (6), ( βη + (1 β) s(1 ε) ) + β = ( b) L, =. (7) Equaion (7) implicily deemines he negoiaed ne-of-ax age fom Nash bagaining as a funcion of he ax policy paamees and so ha e have =, ). ( 2 The assumpion of a linea objecive funcion is fo analyical and exposioy convenience. All qualiaive esuls can be shon o hold fo isoelasic concave objecive funcions of he ade union. 3 This appoach can be jusified eihe axiomaically (cf. Nash 195), o saegically (cf. Binmoe, Rubinsein and Wolinsky 1986).

7 5 To deive he opimal ax fomulas e have fis o kno ho age negoiaions ae affeced by he ax sysem. We heefoe povide some compaaive saics esuls e ill use lae on. The effec of a change in he labou ax ae on he ne-of-ax age ae is ( b) z = =, (8) y + ( b) z(1 + ) ih y = β( 1+ ηl, ) + (1 β)(1 ε) s and z = [ β( σ ε) + (1 β)(1 ε) ] s. As he second-ode condiion is assumed o hold houghou, i.e. = y + ( b) z(1 + ) <, e can infe ha sign ( ) = sign ( z) = sign ( s ) if labou and capial ae pice complemens σ < ε, as e ill assume in ha follos. (Noe ha ε > 1). Fo a CES poducion echnology, he paial deivaive of he cos shae of labou ih espec o he goss age ae is given by so ha e have s > < s = (1 s)(1 σ) σ 1 = =, < > < as σ < 1 = as σ = 1. (9) > as σ > 1 If he elasiciy of subsiuion is less han one, an incease in he labou ax ae ill lead o an incease in he cos shae of labou s. A lage shae s implies ha a one pecen change in he age ae induces a lage incease in oal and maginal cos and esuls in a loe oupu. This ill lead fims o lay off moe okes. Hence, he ade union benefis less fom demanding highe ages. By conas, if he cos shae of labou s inceased due o highe labou axes, pofis ould fall a a highe ae if he ade union succeeded in inceasing he ne-of-ax age. Theefoe he fim ill oppose age inceases moe songly and demand loe ages. An incease in he labou ax ae eakens he ade union's bagaining posiion and senghens he fim s bagaining posiion a he same ime. Boh effecs ok in he same diecion and he ne-of-ax age ill fall. If, on he conay, subsiuabiliy is high, e.g. σ > 1, he ne-of-ax age ae ill ise.

8 6 An exogenous incease in he capial ax ae has an effec on he cos shae of labou opposie o ha of he incease in he labou ax ae. 4 Hence, depending on he elasiciy of subsiuion, he oal effec of an incease in is: > as σ < 1 = as σ = 1. (1) < as σ > 1 Finally, e conside he govenmen budge. The govenmen equies a fixed amoun of ax evenues o finance he public good G and, in addiion, i has o pay unemploymen benefis b o all unemployed okes. Denoing he oal numbe of okes by N, he numbe of unemployed okes is given by N L. The govenmen levies he labou ax on age income and a soucebased ax on domesic capial inpu. In addiion hee is a pofi ax on domesic pofis so ha he govenmen budge consain is given by L + K + = G + b ( N L). (11) To focus on efficiency aspecs of he opimal ax sucue only, e conside he oal suplus as an appopiae social planne s objecive funcion (cf. Summes, Gube and Vegaa 1993). (Disibuional consideaions ene only in ha e allo fo an unemploymen benefi paymen o susain he exisence minimum of unemployed okes.) Toal suplus consiss of he age income equal o L, hich accues o okes, b( N L), he money meic-uiliy unemployed deive fom leisue and unemploymen benefi paymens, and he ne-of-ax pofi income ( 1 ). As e hold G consan e suppess he em G in he elfae funcion. Fuhemoe, he income fom he domesic capial sock is also assumed o be consan and heefoe is no explicily consideed in he elfae funcion eihe. All domesic pofis go o domesic capialiss. 5 Hence, he elfae funcion is given by S = L + b( N L) + (1 ). (12) 4 This can be seen fom deiving he cos shae of capial ( 1 s) ih espec o he capial ax ae (cf. Koskela and Schöb 1998). 5 Fo an analysis hen foeignes eceive a facion of domesic pofis, see Huizinga and Nielsen (1997).

9 7 3. Welfae maximizaion: basic seup We sa by consideing a model ih a Sackelbeg game sucue, hee he govenmen chooses ax aes fis and he labou oganizaions hen deemine he age ae in a age negoiaion, aking he ax aes as given. Lae on, e examine he implicaions of a Nash game. Hence, he govenmen maximizes elfae (12) subjec o he budge consain of he govenmen (11), he oucome of he age negoiaion, hich is implicily given by he fis-ode condiion of he Nash bagaining (7), and he consain on he pofi ax ae (14): max { S,,, = L + b( N L) + (1 ), s.. L + K + = G + b ( N L). (11) ( βη + (1 β) s(1 ε) ) + β = ( b) L, =. (7) The Lagangian fo he elfae maximizaion is (14) L ( G + b ( N L) L K ) µ + ϕ( ) = L + b( N L) + (1 ) λ,(15) hee λ, µ and ϕ descibe he shado pices of he consains (11), (7) and (14), especively. The fis-ode condiions ih espec o he pofi ax ae, he o faco ax aes and he ne-ofax age ae can be expessed (afe some manipulaions) as follos: L = ( λ 1) = ϕ, (16a) L L [ ( b λb ) + λ ] Lη ( 1)(1 ) + λ Kη + λ L µ (1 + ) = = L, K, ( ( b λb ) + λ ) Lη ( 1)(1 ) + λ Kη + λ K µ (1 + ) = = L, K,, (16b), (16c) L ( ( b λb ) + λ ) Lη ( 1) ( (1 )) + λ Kη λ + L µ = = L, K,,(16d)

10 8 using he folloing expessions of he faco demand elasiiciies: η = K K = s( σ ) K, ε, η L, = L L = (1 s)( σ ε) and ηk, = σ + (1 s)( σ ε). By inspecing he complemenay slackness condiion ( ), ϕ, ϕ =, e can disinguish o cases. If ϕ =, he pofi ax consain is no binding and he govenmen can choose he pofi ax ae opimally. This case ill be analysed in Secion (4). The ohe case, hee he consain is binding, i.e. ϕ >, ill be discussed Secion (5). 4. Welfae maximizaion ih opimal pofi axaion If ϕ =, he fis-ode condiion ih espec o he pofi ax ae (16a) educes o λ = 1. (17) The shado pice λ epesens he maginal cos of public funds. A maginal cos of public funds equal o one indicaes ha he govenmen can aise axes o mee is evenue equiemen ihou imposing any cos on sociey. Hence, he economy faces no ax disoion bu is lef ih labou make disoions. To analyse ho hese labou make disoions affec elfae, e subac (16d) fom (16b), using λ = 1. This yields µ (1 + ) + 1 =. (18) As e kno fom he second-ode condiion, <, he shado pice µ mus be equal o zeo if he ems in backes ae non-zeo. The fis em in backes epesens he ne-of-ax age elasiciy ih espec o he labou ax. As can be seen fom equaions (8) and (1), his elasiciy has a non-negaive value fo σ 1. Hoeve, i akes a negaive value fo σ < 1, bu as long as he labou ax ae is no fully shifed ono okes, i.e. as long as an incease in he labou ax ae inceases he goss age ae, he absolue value of he elasiciy is belo one. I is shon fomally in Appendix 1 ha he ne-of-ax age elasiciy ih espec o he labou ax is indeed alays lage han 1, hich is also in confomiy ih empiical sudies (cf. e.g. Lockood and Manning 1993

11 9 and Holm, Honkapohja and Koskela 1994). Theefoe, he ems in backes mus alays be negaive and condiion (13) holds only if he shado pice µ =. This esul suggess ha if he govenmen can use pofi axaion ihou any esicion, i.e. apply non-disoive axaion, he Nash bagaining consain is no binding. Inuiively, haeve neof-ax age ae is fixed in he age negoiaion beeen he ade union and he fim, he govenmen can choose an appopiae age ax o subsidy o obain he goss age hich opimizes social elfae. Solving he equaion sysem (16b) and (16c) ih espec o he faco ax aes and making use of ϕ =, λ = 1 and µ =, e obain: and =, (19) ( b b ) =. (2) These o ax aes ensue ha boh goss faco pices equal hei social oppouniy cos. Fom equaion (1) i follos ha he maginal poduciviy of capial, hich equals he goss inees ae, also equals he old inees ae, =, hich is he ue oppouniy cos of domesic capial consumpion. Any ax on capial ould be bone by labou and pofi income. As he capial one alays eceives, less capial inpu in domesic poducion due o a capial ax ae ould hus educe labou income and pofis and hence elfae. I should be noed (ih espec o he lae analysis) ha he poducion echnology has no impac on his esul. PROPOSITION 1: If he govenmen can se he pofi ax opimally, i should no levy any capial ax. This poposiion has also been deived in he special case of he monopoly union model by Boees and Schneide (1999) and by Riche and Schneide (2). I is also a ell-knon popey of opimal axaion fo economies ih compeiive labou makes (see e.g. Bucovesky and Wilson 1991).

12 1 Subsiuing ino equaion (2) he definiion of he goss age ae yields = b b, i.e. he goss age equals he disuiliy of labou, in ohe ods he social cos of labou. Whaeve he oucome in he age negoiaion is, he govenmen can inoduce a age subsidy hich guaanees ha he maginal poduciviy of labou equals he maginal social cos of labou and hus maximizes social elfae. The age subsidy is equal o he mak-up beeen he ne-of-ax age ae and he maginal evenue poduc of labou he age negoiaion yields, given his subsidy. This esablishes full employmen in he sense ha hee is no involunay unemploymen anymoe. Hence, e have PROPOSITION 2: If he govenmen can se he pofi ax opimally, i should levy a age subsidy hich compleely offses he mak-up beeen goss and ne-of-ax age ae as deemined in he age negoiaions. Poposiion 2 confims fo a unionized labou make he esul by Guesneie and Laffon (1978) ha in a fis-bes old, he oupu of a pice make should be subsidized such ha he make pice equals he maginal cos. Hee, he ade union execises monopoly poe in he labou make by inceasing he ne-of-ax age ae above he disuiliy of labou. Whaeve mak-up is deemined in he age negoiaions, he govenmen can alays subsidize labou so ha he maginal poduc value of labou equals he maginal oppouniy cos of labou. Hence, ih unesiced pofi axaion, he govenmen can alays achieve poducion efficiency and hus esoe he fis-bes allocaion. 5. Welfae maximizaion ih esiced pofi axaion So fa e have focused on unesiced pofis. In pacice, hoeve, his is fo seveal easons he excepional ahe han he nomal case. Fisly, ax auhoiies may have difficulies in disinguishing beeen pue pofis and eun o capial invesmens. Secondly, opimal pofi axaion may be impossible if hee ae insiuional o legal consains. Hence, e no un o he moe elevan case hee ϕ >, i.e. he pofi ax consain is binding and he pofi ax ae is se a he uppe bound fo he pofi ax ae.

13 11 In his case, as pofis ae alays posiive, i can be seen diecly fom equaion (16a) ha λ > 1, i.e. he maginal cos of public funds exceeds uniy. This means ha he govenmen has o apply disoing axes o aise evenues fo he finance of public goods. Bu his is no he only disoion he economy faces. As can be shon, he labou make consain also becomes binding, i.e. he govenmen canno offse he inefficiency caused by seing he ne-of-ax age ae above he social cos of oking, b b. Inuiively, he govenmen has o apply disoing axes o finance he age subsidy. While a maginal mak-up due o he age negoiaion has only a second-ode effec on elfae, he loe ax evenue equiemen geneaes a fis-ode elfae gain. This is a sandad second-bes esul accoding o hich, in he pesence of moe han one disoion, i is no opimal o esablish he fis-bes soluion in only one seco. Fomally, he shado pice µ, hich epesens he social cos of labou make impefecion, can be signed by subacing (16d) fom (16b): (1 ) + µ + 1 = ( λ 1) L <. (21) As has been discussed in Secion 3, he ems in backes on he lef-hand side ae posiive. Hence, condiion (21) can hold only if µ >, i.e. educing he labou make disoion due o age negoiaions is alays elfae impoving (cf. Appendix 2). The loe he ne-of-ax age ae as a esul of he age negoiaion, he loe he elfae loss of disoive axes ill be. This ill be ue iespecive of he quesion of hehe he ne-of-ax age ae changes as a consequence of a ax ae change. 5.1 Opimal faco axes hen he ne-of-ax age ae emains unchanged If he elasiciy of subsiuion is uniy, hich is he case fo a Cobb-Douglas poducion funcion, he ne-of-ax age ae is independen of he ax aes (cf. Secion 2), so e have =. Using he calculaions given in Appendix 3, he opimal faco ax fomulas fo his case ae 1+ σ= = 1 ε λ ( 1 ), (22)

14 12 and 1 ( b λb = λ ) ε λ o 1 1+ σ= 1 ( ). (23) Equaion (22) shos ha he capial ax becomes sicly posiive. The posiive capial ax esuls fom he esicive pofi axaion, hich foces he govenmen o ely on disoionay axaion. The capial ax ae is highe, he loe he feasible pofi ax ae and he highe he maginal cos of public funds λ. A compaison of equaion (23) ih he opimal labou ax fomula fo unesiced pofi axaion, equaion (2), shos ha he labou ax ae is also highe. The fis em on he igh-hand side, hich epesens he subsidy componen of he ax ae, is inceasing in he maginal cos of public funds λ. The eason fo his effec is ha he inenalizaion has o be financed by disoing axes and becomes moe cosly, he highe he maginal cos of public funds is. Thee is a second posiive em, hich one migh efe o as he Ramsey componen of he labou ax ae. This ax componen epesens he opimal ax one should levy on labou o minimize he excess buden of axaion. The age subsidy pa is a leas paially offse by he Ramsey componen. Hence, in he case of Nash age bagaining ih esiced pofi axaion, a posiive labou ax is possible as a pa of he opimal ax eamen of facos of poducion. These esuls can be summaized in he folloing o poposiions. PROPOSITION 3: If he govenmen canno se he pofi ax opimally and faco axes ill have no effec on age negoiaion, he govenmen should levy a posiive capial ax. PROPOSITION 4: If he govenmen canno se he pofi ax opimally and faco axes ill have no effec on age negoiaion, he opimal ax eamen of labou ill consis of a subsidy componen and a Ramsey ax componen. As a compaison ih he capial ax fomula shos ha he Ramsey componen of he labou ax equals he capial ax ae. Hence, as long as he ne-of-ax age is no affeced by he ax sysem

15 13 he only diffeence in faco axes sems fom he age subsidy paid o educe he labou make disoion: o 1 ( ) b λb = > 1, (24) + λ 1+ σ= 1 σ= 1 This is oh emphasizing in an addiional poposiion. PROPOSITION 5: If he govenmen canno se he pofi ax opimally and he faco axes ill have no effec on age negoiaion, he capial ax ae should be highe han he labou ax ae. Poposiion 5 implies ha in he absence of any labou make disoions, faco ax aes should be equal. The eason fo equipopoional Ramsey componens can be seen fom applying he so-called 'invese elasiciy ule', accoding o hich he govenmen should levy he highes ax ae on he mos inelasic aciviy. In he sandad lieaue on axing mobile capial (see e.g. Bucovesky and Wilson 1991, Egge and Haufle 1999), his agumen has been pu foad o jusify a zeo ax on capial, hich is infiniely elasic in supply, and a posiive ax on labou, hose supply elasiciy is finie. Hoeve, in he pesence of unemploymen he esul no longe holds. Fisly, unde involunay unemploymen he supply of labou is locally infiniely elasic, hich suggess accoding o he invese elasiciy ule ha labou should no be axed a a highe ae han capial. Secondly, hee is a disoion in he labou make and he ne-of-ax age ae exceeds he maginal disuiliy of labou. This is an agumen fo he govenmen o subsidize labou elaive o capial. 5.2 Opimal faco axes hen he ne-of-ax age ae changes No e un o he moe geneal case hee he elasiciy of subsiuion beeen facos of poducion diffes fom one. In his case he oucome of he age negoiaion is affeced by changes in faco axaion as e shoed in Secion 3. Solving he sysem of equaions (16b)-(16c) fo he CES poducion funcion case ih espec o he ax aes and making use of λ > 1 and µ >, e obain he geneal opimal faco ax fomulas

16 = 1 ε λ ( 1 ) µ (1 + ) + λ (1 s) cyσ (25a) and o λ µ + ( ) 1 ( b b 1 1 (1 ) = λ ε λ λ scyσ (25b) hee ( ) s = b 1+ (1 s)(1 σ) < > > < [ βσ ε + 1 β] = σ = 1. (26) Compaed ih he fomulas pesened in Secion 5.1, hee is a ne em he second and hid ems on he igh-hand side in (25a) and (25b) especively hich capues he effec ha changes in he age ae ill have on he opimal faco axes. Since e have aleady discussed he ohe ems, e ill focus on hese ne ems only. Fom equaion (25a) e can deduc PROPOSITION 6: If he govenmen canno se he pofi ax opimally and faco axes ill affec he age negoiaion, he opimal capial ax should fall sho of (exceed) he Ramsey componen if he elasiciy of subsiuion beeen capial and labou is smalle (geae) han one. This esul has a naual inepeaion. If he elasiciy of subsiuion beeen capial and labou is less han one, a fall in he capial ax ae deceases he ne-of-ax age ae as he oucome of Nash bagaining. The labou make disoion due o he diffeence beeen he ne-of-ax age and he social maginal cos of labou becomes smalle. On he conay, if he elasiciy of subsiuion exceeds one, hen a ise in he capial ax ae ill decease he ne-of-ax age ae and heeby educe he labou make disoion. Wih espec o he labou ax ae, e obain PROPOSITION 7: If he govenmen canno se he pofi ax opimally and faco axes affec he age negoiaion, he opimal labou ax should exceed (fall sho of) he Ramsey componen plus

17 15 he age subsidy if he elasiciy of subsiuion beeen capial and labou is smalle (geae) han one. Poposiion 7 has an inepeaion analogous o Poposiion 6. Wih he elasiciy of subsiuion being less han one, a ise in he labou ax ae decease, he ne-of-ax age ae so ha he labou make disoion becomes smalle and vice vesa. Then he labou make disoion can be deceased by aising he labou ax ae. While he capial ax unambiguously exceeds he labou ax ae boh in he pesence of unesiced pofi axaion and in he case of a Cobb-Douglas poducion funcion, he geneal case needs fuhe elaboaion. Subacing equaion (25b) fom equaion (25a), e obain o 1 ( λ ) µ (1 + ) b b = + λ λ cyσs(1 s) (27) hee he second em on he igh-hand side depends on he elasiciy of subsiuion. Poposiion 5 shos ha hen he faco axes have no effec on he age negoiaion, he capial ax ae should be highe han he labou ax ae in he pesence of unemploymen. If he age negoiaion is affeced by he faco axes, hen one should incease he capial ax and decease he labou ax even fuhe if he elasiciy of subsiuion exceeds one. If σ is less han one, i is opimal o incease he labou ax ae and decease he capial ax ae o alleviae he labou make disoion. Fo he lae case, i canno be uled ou ha he labou ax ae exceeds he capial ax ae. These findings ae summaized in PROPOSITION 8: If he govenmen canno se he pofi ax opimally, he capial ax ae should be highe han he labou ax ae if he elasiciy of subsiuion is geae han o equal o one. If he elasiciy of subsiuion is less han one, hen he elaive size of opimal faco axes emains ambiguous a pioi.

18 A Nash game So fa e have assumed ha he govenmen acs as a Sackelbeg leade. In his secion e ill biefly illuminae he alenaive Nash assumpion, hee he govenmen axes he ne-of-ax age ae deemined in age negoiaions as given and he labou oganizaions in un ake he ax aes as given (cf. Hesoug 1984). The Nash equilibium condiions fo he deeminaion of he labou ax ae by he govenmen and of he age ae as he oucome he age bagaining can be easily deived. Fo he Nash bagaining soluion e have aleady assumed ha he labou oganizaions ake he ax aes as given. Diffeen o he maximizaion poblem in Secion 3, hoeve, e no have o maximize elfae ih espec o condiions (11) and (7) only because he govenmen akes he ne-of-ax age ae as given. As his is equivalen o he maximizaion poblem hee he labou make disoion consain is no binding, he opimal ax fomulas ae: 1+ Nash 1 1 = 1 ε λ ( 1 ), (28) and 1+ 1 ( b λb = λ ) ε λ o 1 Nash This can be summaized in a poposiion. ( ). (29) PROPOSITION 9: If he govenmen canno se he pofi ax opimally and he govenmen and he labou oganizaions play Nash, capial axes should alays be non-negaive and exceed he labou ax ae. Noe ha if he pofi ax ae can be se opimally, he capial ax ae should be zeo and labou should be subsidized. 6 In his case, he maginal cos of public funds is uniy and he ax fomulas ae he same as in he Sackelbeg game [equaions (19) and (2)]. 6 I is shon in Appendix 4 ha unde he assumpions made he Nash equilibium is unique and sable.

19 17 6. Relaed lieaue Thee is ecen lieaue hich deals ih he opimal faco axaion in he pesence of unemploymen. One of he fis papes dealing ih opimal axaion in he pesence of involunay unemploymen is Bovenbeg and van de Ploeg (1996). Alhough hey do no conside capial axaion bu sudy opimal axaion, opimal povision of public goods and envionmenal policy, hei analysis ih espec o he opimal labou ax ae can easily be applied o models ih inenaionally mobile capial as consideed hee. They sho ha (in he absence of any exenaliies) he opimal labou ax ae sikes a balance beeen o objecives. Fisly, he labou ax seves he pupose of aising ax evenues. Theefoe, he labou ax ae should be highe (he labou subsidy loe), he highe he maginal cos of public funds and he loe he pofi ax ae, ceeis paibus. Secondly, he subsidy componen is used o offse he labou make aioning. As hei esuls ae deived fo he case of igid ne-of-ax age aes, ou Poposiions 1 and 4 genealize hei esul o he case of endogenous age deeminaion hee he ax sysem migh affec he neof-ax age ae. Using a monopoly union model, Riche and Schneide (2) sho ha if pofi axaion is esiced, he capial ax may be used as an indiec ool o educe he labou make disoion, i.e. he union s abiliy o aise he ne-of-ax age above he maginal cos of labou, if i affecs he labou demand elasiciy and hence he monopoly poe of he ade union. This esul (see hei Poposiion 7(ii)) is in line ih ou Poposiion 6 and shos ha non-zeo capial ax aes ae in geneal desiable (i) o minimize he excess buden of axaion if pofi axaion is esiced and (ii) o educe he labou make disoion due o monopoly union poe if he ne-of-ax age ae is affeced by he capial ax ae. Appaenly, hoeve, a esul deived by Boees and Schneide (1999) and Riche and Schneide (2) fo he monopoly union case and Fues and Hube (1999) fo he Nash bagaining fo he opimal capial ax ae in he case of unesiced pofi axaion seems o be in shap conas o ou Poposiion 1. The eason fo ha is ha hey assume ha pofis ae fully axed aay, i.e. ha he pofi ax ae is fixed a 1%. Hence, hey acually conside a special case of esiced pofi axaion ahe han opimal pofi axaion. This esicion applies if he ax evenue equiemen

20 18 exceeds he pofi income in he economy (hich is only possible if faco axes affec he ne-of-ax age ae). In his case he maginal cos of public funds is above uniy and he labou make disoion consain becomes binding. By conas, in deiving Poposiions 1 and 2, e allo he govenmen o adjus he pofi ax ae opimally, i.e. e assume ha pofi incomes exceed he govenmen s ax evenue equiemen. In his case he Nash bagaining consain is no binding and he fis-bes soluion can be esoed. If one ould allo he govenmen o se he pofi ax opimally, ou Poposiion 2 could be confimed in hei models as ell. 7 As long as pofi axaion is no esiced, labou make disoions do no jusify a posiive ax on mobile capial. Theefoe, he ell-knon esuls of opimal axaion in economies ih compeiive labou makes (see e.g. Bucovesky and Wilson 1991, Razin and Sadka 1991 and fo a ecen discussion Egge and Haufle 1999) can be genealized if pofis can be fixed opimally. Boees and Schneide (1999) also compae he model hee he govenmen is a Sackelbeg leade ih he model hee hee is a Nash game beeen he govenmen hich ses he ax aes, and he monopoly union hich ses he ne-of-ax age ae. 8 They sho ha unde he Nash assumpion he capial income should no be axed and labou should be subsidized. This is a special case of ou Poposiion 9 as hey assume ha = 1. The moe geneal esul of Poposiion 9 shos, hoeve, ha heneve pofi canno fully be axed aay, he capial ax ae should be posiive. This confims he esuls deived by Buce (1992), Minz and Tulkens (1996) and Huizinga and Nielsen (1997) fo he case of compeiive labou makes, namely, ha if pofi income canno be fully axed, a souce-based capial ax seves as a ool o ax pofi indiecly. 7 A poof is available upon eques. 8 Fues and Hube (1999) also analyze he Nash game beeen he govenmen and he labou oganizaions. Hoeve, hey assume ha he govenmen akes he goss age as given. Alhough i does no mae hehe one assumes ha he ne-of-ax age o he goss age is deemined in age negoiaions fo he Sackelbeg game, he Nash oucome cucially depends on ha he govenmen consides o be unaffeced by is on acions.

21 19 7. Conclusions I is ell knon ha if i is no possible o ax pue pofis fully, he govenmen is foced o ely on disoionay axes. In he heoy of opimal axaion, he Ramsey ule and is special case, he 'invese elasiciy' ule, ell ho he disoionay axes should be hen designed so as o minimize he excess buden of he ax sysem. The invese elasiciy ule equies ha he govenmen levies he highes ax ae on he mos inelasic aciviy. This agumen lies behind he convenional isdom ha inenaionally mobile capial should no be axed o should be axed as a loe ae han labou. Applicaions of he Ramsey ule o of he invese elasiciy ule usually assume ha hee ae no ohe make impefecions, in paicula ha labou makes clea compeiively. A leas fo Euope, hich has been suffeing fom high unemploymen fo a long ime, his assumpion does no seem appopiae. Hence, i is impoan o ask hehe he convenional isdom, accoding o hich capial should be axed a a loe ae han labou, sill holds in he pesence of unemploymen. In his pape e have sudied he opimal faco axaion in he pesence of unemploymen hich esuls fom he union-fim age bagaining boh ih opimal pofi axaion and ih esiced pofi axaion hen capial is inenaionally mobile and labou immobile. Ou main conclusion is ha in he pesence of unemploymen he convenional isdom uns on is head; capial should geneally be axed a a highe ae han labou. The opimal levels of faco axes depend on specific feaues of he siuaion, like he game sucue beeen he govenmen and he pivae seco, he popeies of poducion echnology and he quesion of hehe he unesiced pofi axaion is feasible o no. Counies ih igid labou makes should heefoe be vey caeful in adoping ax policies hich ae appopiae fo counies hee labou makes ae sufficienly flexible.

22 2 Appendix 1: Ne-of-ax age elasiciy Using he explici fomulaions fom he CES poducion funcion fo he second deivaives, = y + ( b) z(1 + ) < and = y + ( b z ih y = β( 1+ η ) + (1 β)(1 s z β( σ ε) + (1 β)(1 ε) s and [ ] ) L, ε) = so ha he change in he ne-of-ax age ae due o a change in he labou ax ae is given by: ( b) z = =. (A1) y + ( b) z(1 + ) Subsiuing his ino he definiion of he ne-of-ax age elasiciy yields ω S (1 ) xz =. 1 y(1 + ) + xz The condiion > 1 holds if y <. Calculaing he ne-of-ax-age ae fom condiion (7) ω yields 1 = y( y β) b (A2) As ω > b i follos immediaely fom inspecion of (A2) ha β > implies y <. Hence, > 1. Q.E.D. Appendix 2: m > ih esiced pofi axaion Subacing (16d) fom (16b) implies ha he folloing condiion mus hold: [ ( 1+ )] = ( λ 1) < µ L (A3) so ha he lef-hand side of (A3) mus be negaive as ell. Is backe em can be eien as follos: (1 + ) (1 + ) = + 1 hee he fis em in backes on he igh-hand side is he ne-of-ax age elasiciy ih espec o he labou ax ae. As has been poved in Appendix 1, i is lage han 1. Hence (A4) is negaive and he condiion (A3) holds only if µ >. Q.E.D. (A4) Appendix 3: Deivaion of he opimal faco ax fomulas Fo he case ϕ = and hence =, eaanging he equaions (16b) and (16c) yields

23 21 Lη Lη L, L, (A5) Kη Kη K, K, λ λ (1 λ)(1 = (1 λ)(1 ( λ ) b b L ( λ ) b b K ) Lη ) Lη L, L, + µ µ (1 + (1 + ih = 1+ ) (1 ) (cf. Koskela and Schöb 1998). Applying Came s ule and ( + using he fac ha he deeminan of he lef-hand side maix is equal o = LK σε yields λ ( b λb ) (1 λ)(1 ) = + Lσε [ L η Kη ] K, K, µ λ L η L, L, + λ K ηl, L K, L, (1 + ) K η ), ) K,, (A6) (1 λ)(1 ) (1 ) [ K L + µ λ = + η η ]. (A7) Kσε η Using he explici elasiciy fomulas, e have ( sη (1 s η ) = cyσs Lη K η = cy. (A8a) K, K, K, ) K, ((1 s) η sη ) = cyσ(1 ) Kη Lη = cy. (A8b) L, L, L, L, s Hence, e end up ih condiions (25a) and (25b). Appendix 4: Sabiliy and uniqueness of he Nash game equilibium The opimal faco ax aes ih esiced pofi axaion unde he Nash game beeen he govenmen and he panes of he age negoiaion ae given in equaions (28) and (29). This appendix shos ha unde he assumpions made, he equilibium of his game is unique and sable. We fis subsiue he expession fo he opimal capial ax ino he equaion fo he opimal ax eamen of labou o deive afe some manipulaion he eacion funcion of he govenmen given he age ae and he opimal capial ax * L ( = ) = λ ( b λb ) =. (A11) 1+ Fo given ax aes, he age ae as he oucome of bagaining is [ βη + (1 β) s(1 ε) ] + β = ( b) L, =. (A12) The slopes of he eacion cuves of he govenmen and he age bagaining on he, ) space can be expessed, especively, as (

24 22 d d = 1+ L = < (A13) and d d = = < => < as σ = 1. (A14) > The sabiliy of he equilibium of his Nash game equies ha he eacion cuve fo he age bagaining mus be flae han ha fo he govenmen. Moe pecisely, he sabiliy condiion can be ien as = L L = (1 ) ( 1 ) = 1 > (A15) and i holds in ou model iespecive of he size of he elasiciy of subsiuion and he elaive bagaining poe of he ade union and he fim. I is easy o see ha he sabiliy condiion is also independen of hehe e have opimal o esiced pofi axaion. Refeences Binmoe, Kenneh G., Aiel Rubinsein and Ashe Wolinsky (1986): "The Nash Bagaining Soluion in Economic Modelling," Rand Jounal of Economics 17, pp Boees, Sefan and Kesin Schneide (1999): "Govenmen vesus Union: The Sucue of Opimal Taxaion in a Unionized Labo Make," Finanzachiv 56, pp Bovenbeg A. Lans and Fedeick van de Ploeg (1996): "Opimal Taxaion, Public Goods and Envionmenal Policy ih Involunay Unemploymen," Jounal of Public Economics 62, pp Buce, Neil (1992): "A Noe on he Taxaion of Inenaional Capial Flos," The Economic Recod 68, pp Bucovesky, Sam and John D. Wilson (1991): "Tax Compeiion ih To Tax Insumens", Regional Science and Uban Economics, 21, pp Diamond, Pee A. and James A. Milees (1971): "Opimal Taxaion and Public Poducion I: Poducion Efficiency", Ameican Economic Revie, 61, pp Egge, Wolfgang and Andeas Haufle (1999): "Capial Taxaion and Poducion Efficiency in an Open Economy", Economics Lees, 62, pp Fues, Clemens and Bend Hube (1999): "Tax Coodinaion and Unemploymen", Inenaional Tax and Public Finance, 6, pp Guesneie, Roge and Jean-Jacques Laffon (1978): Taxing Pice Makes, Jounal of Economic Theoy 19, pp Hesoug, To (1984): "Union Wage Responses o Tax Changes," Oxfod Economic Papes 36, pp Holm, Pasi, Seppo Honkapohja and Ekki Koskela (1994): "A Monopoly Union Model of Wage Deeminaion ih Capial and Taxes: An Empiical Applicaion o he Finnish Manufacuing," Euopean Economic Revie 38, pp

25 23 Huizinga, Hay and Søen Bo Nielsen (1997): "Capial Income and Pofi Taxaion ih Foeign Oneship of Fims," Jounal of Inenaional Economics 42, pp Koskela, Ekki and Ronnie Schöb (1998): Why Govenmens Should Tax Mobile Capial in he Pesence of Unemploymen, Univesiy of Munich, CES Woking Pape No.175, Novembe. Koskela, Ekki, Schöb, Ronnie and Hans-Wene Sinn (1998): "Polluion, Faco Taxaion and Unemploymen", Inenaional Tax and Public Finance 5, pp Lockood, Ben and Alan Manning (1993): "Wage Seing and he Tax Sysem," Jounal of Public Economics 52, pp Minz, Jack und Heni Tulkens (1996): "Opimaliy Popeies of Alenaive Sysems of Taxaion of Foeign Capial Income," Jounal of Public Economics 6, pp Nash, John (195): "The Bagaining Poblem," Economeica 18, pp Razin, Assaf and Efaim Sadka (1991): "Inenaional Tax Compeiion and Gains fom Tax Hamonizaion," Economic Lees 37, pp Riche, Wolfam and Kesin Schneide (2): Taxing Mobile Capial ih Labo Make Impefecions, Univesiy of Domund, Depamen of Economics, fohcoming in: Inenaional Tax and Public Finance. Summes, Laence, Jonahan Gube and Rodigo Vegaa (1993): "Taxaion and he Sucue of Labo Makes: The Case of Copoaism," Quaely Jounal of Economics 58, pp