Allowing Firms to Choose Between Formula Apportionment. and Separate Accounting Taxation 1. Thomas A. Gresik. University of Notre Dame.

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1 Allowng Frms to Choose Between Formula Apportonment and Separate Accountng Taxaton Thomas A. Gresk Unversty of Notre Dame August 03 Please do not cte wthout permsson Abstract: Ths paper analyzes the effect on frm behavor and natonal tax revenues when multnatonals are allowed to choose whether to be taxed under ether separate accountng rules (transfer prces) or an apportonment formula n a two-country model. Whle separate accountng s shown to generate output and condtonal labor demand dstortons that lead frms to prefer separate accountng, some frms may prefer formula apportonment when the formula creates suffcently strong proft-shftng dstortons. Country preferences are only partally algned wth frm preferences due to the dfferent tax base dstortons each method generates. In general, frm choce cannot make both countres better off. I thank Andreas Haufler, Ka Konrad, Wolgang Schön, Drk Schndler, and partcpants of the 03 Skatteforum (Tax Forum) n Norway and the research semnar at the Max Planck Insttute for Tax Law and Publc Fnance n Munch, Germany for all ther comments.

2 . Introducton. Formula apportonment and separate accountng represent two common ways for a frm to apporton ts taxable ncome among the tax jursdctons n whch the frm operates. Formula apportonment apportons taxable ncome based on a frm's relatve shares of observable economc varables n each jursdcton such as sales, wages, employment, and captal. Separate accountng apportons taxable ncome by relyng on transfer prces set by the frm and audted by the tax authortes n each locaton. Both methods dstort frm behavor. Over the last decade nternatonal tax economsts have studed the producton, sales, and tax revenue dfferences between these two methods motvated n part by EU ntatves to develop a coherent tax polcy. In recent years, the European Commsson and the OECD have taken opposng postons on whch approach s preferred wth the EC advocatng for the adopton of a specfc apportonment formula and the OECD favorng a tradtonal separate accountng approach. However, the EC (0) proposal does not recommend the exclusve adopton of an apportonment formula. Rather t recommends a tax apportonment system n whch frms operatng wthn the EU can choose between usng a specfed apportonment formula or usng standard transfer prcng methods (see page 5 and Chapter III, EC (0)). In ths regard, the EC proposal s smlar to the current Canadan polcy that offers mult-provnce frms the opton of allocatng taxable ncome va an apportonment formula or va transfer prces. 3 The purpose of ths paper s to study the economc mplcatons of hybrd taxaton systems that allow each frm to choose the system under whch ts tax labltes wll be calculated. The most common approach n the extant lterature to comparng formula apportonment wth separate accountng s to compare the equlbrum allocatons that result from a specfc European Commsson (0) and OECD (00) descrbe each organzaton's poston. 3 Canadan frms make ths choce by organzng ther mult-provnce operatons accordng to ether a subsdary-type structure (transfer prcng) or a branch-type structure (formula apportonment). Mntz and Smart (004) use the choces made by frms to estmate the elastcty of taxable ncome wth respect to tax rates for frms who can engage n ncome shftng va transfer prces and for those frms that opt to be taxed va an apportonment formula.

3 apportonment formula to those that result from a specfc model of transfer prce regulaton. Key examples of such papers nclude Nelsen, Ramondos-Møller, and Schjelderup (003 and 00) and Echner and Runkel (008 and 0). 4 NRS (003) shows that a shft from separate accountng to formula apportonment can actually exacerbate ncome shftng va transfer prces f the frm operates n olgopoly markets whle NRS (00) shows that tax revenues can ether rse or fall from a shft to formula apportonment dependng on the cost of ncome shftng and the magntude of pure frm profts. ER (008) provdes suffcent condtons for a sales only formula to ncrease tax revenues relatve to separate accountng. ER (0) endogenzes nterest rates and then shows that formula apportonment wll generate hgher tax revenues than separate accountng f the elastcty of substtuton between captal and labor s suffcently large. All of these papers as well as the bulk of the rest of ths lterature study the trade-offs between separate accountng and formula apportonment n a complete nformaton model wth a representatve frm or wth dentcal frms. Whle we have learned a good deal about the economc trade-offs assocated wth a shft from a separate accountng system of taxaton to an apportonment formula system from complete nformaton models wth a sngle, representatve frm, such models have two key lmtatons. Frst, the fundamental challenge tax authortes face when dealng wth frms that operate n multple jursdctons s that the authortes lack suffcent nformaton to properly apporton the frm's taxable ncome across jursdctons. It s ths lack of nformaton that provdes frms wth the scope to engage n tax-nduced proft and producton shftng. Most exstng proft apportonment models are unable to address the effect of lmted government nformaton on the economc outcomes generated by separate accountng and formula 4 One excepton s Redel and Runkel (007) n whch a multnatonal s requred to apporton ts taxable ncome generated wthn a unon of countres usng an apportonment formula whle t uses transfer prces to apporton global taxable ncome between unon countres and a non-unon country. Ther model stll conssts of a sngle representatve frm and does not allow the frm to choose among several tax systems for apportonng unon ncome. 3

4 apportonment models. For example, an ncomplete nformaton ssue such models cannot address s the crtque Hellersten and McLure (004) rase about the practcal dffcultes of usng apportonment formulas that rely on factors (such as captal) whose valuatons are dffcult for tax authortes to assess due to lmted nformaton about a frm's nvestment opportuntes. Second, representatve frm models are at odds wth studes such as Mntz and Smart (004) that explot dfferences n the choces frms make between beng taxed under separate accountng or formula apportonment rules as these models admt no scope for hybrd tax systems of the sort Canadan provnces use and that s proposed by the European Commsson. One paper that has ncorporated both prvate and heterogeneous frm nformaton s Gresk (00). To the best of my knowledge, ths s the only paper to nclude both features n a model of multnatonal taxaton. 5 In Gresk (00), I analyze a model wth heterogeneous frms that dffer based on prvate productvty nformaton and n whch the countres have mperfect audtng technologes. Both countres collect tax revenue from domestc frms and foregn frms. In that paper, I show that a sales-only formula wll generate greater expected tax revenues than separate accountng as long as the audtng procedures are suffcently accurate. Whle ths paper does not allow for each frm to choose ts method of ncome allocaton, t does show that tax revenue dfferences from an apportonment formula versus a separate accountng system vary not only wth dfferences n frm productvty but also wth the locaton of a frm's ntermedate good producton and ts fnal good sales. In addton, the dfferences n tax revenues to each country need not be monotonc wth respect to frm productvty. Ths earler paper of mne then rases the queston of whether countres can earn hgher tax revenues by explotng frm selecton patterns. The purpose of ths paper s to develop a model of heterogeneous multnatonal frms, 5 Burbrdge, Cuff, and Leach (006) study proft tax systems for heterogeneous natonal frms. Although they do not address the ssue of multnatonal taxaton, they do fnd "substantal dfferences" n economc outcomes relatve to the predctons of complete nformaton studes. 4

5 gve them the choce between usng ether an apportonment formula or transfer prces to calculate natonal tax labltes, and then analyze the pattern of choces made by the frms. The model s presented n secton. Each frm can produce and sell a fnal good n ether or both of two countres. Fnal good producton requres a frm to employ captal and labor n the destnaton country usng a constant returns to scale technology and an ntermedate good that s produced by the frm exclusvely n country (the upstream country). In general, a frm's choce of a tax system wll depend on the net effect of an output dstorton, a (per unt) proft-shftng dstorton, and a condtonal factor demand dstorton. A key queston addressed by ths paper s how these dstortons together nfluence frm choce. In secton 3, the analyss focuses on the tax method choce of a frm on both extensve and ntensve margns. A frm's extensve margns determne whether t should sell ts fnal good n each country whle the ntensve margns determne how much a frm should sell n each country. The analyss of the extensve margns dentfes a lnkage effect n whch a frm produces predomnately for country (the downstream country) sales n order to engage n proft-shftng under. Ths effect arses when country has the hgher tax rate and the frm has hgh margnal costs of captal or an neffcent fnal good producton technology. Thus, near the country extensve margn the lnkage effect allows the frm to ncrease country sales n order to shft profts nto country under whle a small (or zero) amount of country sales provdes almost no opportunty for proft-shftng ablty under. The analyss of the ntensve margns focuses on the trade-offs between and when the countres are symmetrc n order to dentfy the role of tax rate dfferentals. Theorem 3 presents a choce formula that explans how output, proft-shftng, and condtonal labor demand dstortons together determne whch tax system generates hgher post-tax proft for a frm for a general class of output-based apportonment formulas. The choce formula reveals that the output dstorton effect always favors as the beneft to a frm from the strategc use of ts 5

6 transfer prces s ncreasng n ts country sales. Gven the constant returns to scale assumpton, also generates a larger, and postve, condtonal labor demand dstorton due to the fact that an ncrease n a frm's effectve tax leads frms to substtute labor for captal. Under, country 's tax rate s sole determnant of a frm's effectve tax rate appled to fnal good producton n country whle, under, a frm's effectve tax rate appled to fnal good producton n country s an average of the tax rates n countres and. As a result, the effect of a change n country 's tax rate on ts captal-labor mx n country s moderated under. Snce labor expenses are tax deductble whle captal expenses are not, t s the effect of tax rates on a frm's condtonal labor demand that nfluences a frm's choce between and. Fnally, the proft-shftng effect can favor ether method. Thus, n order for a frm to prefer, must nduce much larger proft-shftng benefts so that t overcomes the output and condtonal labor demand dstorton advantage of. Whle the extensve margn analyss suggests that wll be preferred by hgh cost frms when the downstream country has the lower tax rate, the ntensve margn analyss suggests that frms wth low producton costs may favor ether method especally when the two countres have smlar tax rates. Ths ambguous frm rankng arses because as a frm's margnal cost of captal goes to zero, the output dstorton dsappears at equal tax rates as both and become pure proft tax systems and the condtonal demand for labor goes to zero. A frm's rankng then depends on the relatve magntudes of the proft-shftng dstortons under and and the magntude of these dstortons depends on the apportonment formula one uses. The choce formula n Theorem 3 reveals that the relevant characterstc of an apportonment formula that determnes the magntude of ts proft-shftng dstorton s the rate at whch the apportonment formula shfts taxable ncome nto country as country sales ncrease. Through an example, I show how ths rate vares across two dfferent formulas and leads to dfferent choces by lowcost frms. 6

7 Dependng on the magntudes of both the extensve margn and the ntensve margn effects, t s possble to construct examples n whch all frms prefer and n whch n both hgh-cost and low-cost frms choose and medum-cost frms choose. However, t s also possble to construct examples n whch all frms prefer as hgh cost frms can favor when the upstream country has the lower tax rate. In ths case, the extensve margn effect s neutralzed snce a frm would cease sales n the downstream market before dong so n the upstream market thus elmnatng the lnkage effect for hgh cost frms. Low cost frms can also favor dependng on the relatve magntude of the margnal proft-shftng dstortons under. In secton 4, the analyss focuses on the effect of a frm's choce on each country's tax revenues. Theorem 4 assocates the dfference n country 's tax revenues from a frm under and wth the dfference n that frm's proft on the ntensve margn. For small tax rate dfferences, preferences of the low tax rate country and the frm are algned whle preferences of the hgh tax rate country and the frm are msalgned. Thus, Theorem 4 mples that allowng frms to choose ther tax method ncreases tax revenues n the low tax rate country and decreases tax revenues n the hgh tax rate country. Ths suggests that absent sgnfcant dfferences between countres, gvng frms the choce of tax method benefts one country at the expense of the other. In secton 5, I dscuss the mpact of wage rate and demand dfferences between the countres. Sgnfcant dfferences n wage rates or demand can result n most frms preferrng the same method. Examples show that ths can happen even wth farly small wage or demand dfferences. However, wage rate dfferences tend to push a frm's rankng n the opposte drecton as demand dfferences do. Thus, to the extent that wage rates and demand are postvely correlated, one would expect a frm's choce to be prmarly drven by the tax rate effects descrbed n Theorem 3. Fnally, Theorem 5 extends Theorem 4 to descrbe the 7

8 relatonshp between each country's rankng of and and each frm's rankng. Wth country asymmetres a country's rankng depends on the frm's rankng and the tax rate dfferences of Theorem 3. For a country, the frst-order effect of a change n the effectve tax rate on the dfference n tax revenues between the two methods s equal to the change n the tax base. For a frm, the frst-order effect of a change n the effectve tax rate on frm proft under each method s equal to the sze of the tax base. For and, the tax bases are equal at equal tax rates whle the change n the tax base wth respect to the effectve tax rate s not. Thus, the change n the tax base creates a second-order effect on frm proft but a frst-order effect on a country's tax revenues. Ths dfference between the effect of the margnal tax base on frm proft and on tax revenues creates a wedge between frm preferences and country preferences. As a result, ths wedge also creates some msalgnment between the preferences of the low tax rate country and the frm and t creates some algnment of preferences of the hgh tax rate country and the frm. Secton 6 then examnes the potental for frm choce to be Pareto mprovng relatve to a polcy of requrng all frms to use ether or. In general, f frm choce ncreases one country's tax revenues, t wll decrease the other country's tax revenues. I offer concludng remarks n secton 7.. Model. In ths secton, I descrbe a smple model n whch heterogeneous frms can have dfferental preferences over beng taxed under a formula apportonment system or a separate accountng system. There are two countres ( and ) and a contnuum of multnatonal frms. Each multnatonal operates a subsdary n each country. Denote country j's rate by t j. Gven the tax rates t and t, each multnatonal chooses the tax system under whch t wll operate, f needed t sets a transfer prce for any ntermedate goods produced by one subsdary and transferred to the other, and t makes producton and sales decsons n each country. Every multnatonal frm has a subsdary n each country whch s responsble for 8

9 producng and sellng a fnal good usng labor, captal, and an ntermedate good. All fnal good producton s sold locally. Intermedate good producton occurs only n country and uses both f f labor and captal. The fnal good producton functon n each country s mn{ zfk, (, l )} where z denotes the quantty of the ntermedate good used, country j for fnal good producton, and f k j denotes the quantty of captal used n f l j denotes the quantty of labor used n county j for fnal good producton. Note that one unt of the ntermedate good s requred to produce one unt of the fnal good whle the frm can engage n some substtuton between captal and labor. The sub-producton functon, F, exhbts constant returns to scale. The ntermedate good producton functon s Gk (, l ; ) mn{ k, l / } where k denotes the quantty of captal used n ntermedate good producton, l denotes the quantty of labor used n ntermedate good producton, and s a parameter that determnes f ntermedate good producton s more ( > ) or less ( < ) labor ntensve than fnal good producton. Denote the margnal economc cost of captal for a frm by. Varaton among frms n or, reflectng ether potental dfferences n productvty or potental dfferences n the nternal projects avalable to each frm, wll consttute the two possble sources of frm heterogenety n ths model and wll account for varaton frm preferences of and. In fact, whle the Leontef producton functon for the ntermedate good s admttedly smple n nature, the producton structure n the model s rch enough to generate sophstcated frm choce patterns whle also allowng one to dentfy the key economc channels through whch each tax method nfluences the choce patterns. 6 Denote the country j wage rate by w j. Labor n each country s mmoble and each frm takes the wage rate n each country as gven. Snce the focus n ths paper s on the choce made by each ndvdual frm and the effect of each frm's choce on tax revenues n each country, t s not necessary to specfy the cumulatve dstrbuton functons of margnal captal costs and productvtes n the economy. However, such nformaton would be mportant when 6 Most of the examples wll focus on varaton n μ for convenence except when the results wth regard to θ dffer. 9

10 consderng factor prce determnaton and n studyng tax competton ssues. Gven the novelty of studyng frm choce even n a partal equlbrum settng, I leave the study of these general equlbrum effects to future work. Each frm serves market j wth nverse demand functon p ( q ). Assume for each country j that p j (0) s fnte and there exsts a quantty q j such that for all qj qj, pj( qj) 0. Defne a frm's revenue functon n each country j by r ( q ) where q j denotes the quantty of the fnal good produced and sold n country j. rj (0) 0 and r j (0) pj(0) 0. For each j, assume that rj() s strctly concave and that p j (0) s fnte. A multnatonal's pre-tax operatng proft n country equals r ( q ) w ( l f l ) ( k f f k k ) and ts pre-tax operatng proft n country f f equals r ( q ) wl k. To emphasze the role of tax rate dfferences, t wll help to analyze the case of symmetrc countres. When country symmetry s nvoked t wll mean that the revenue functons and the wage rates are dentcal, r ( q) r ( q) and w = w. 3. Multnatonal decsons. Each multnatonal maxmzes ts global post-tax proft takng as gven the tax rates, the wage rates, ts margnal economc cost of captal, and ts technology for producng the ntermedate good. 3. Separate accountng. Wth a separate accountng tax system (), each frm sets a transfer prce that determnes how much the subsdary n country pays the subsdary n country for the unts of the ntermedate good t uses. Denote ths transfer prce by. The countres jontly audt each frm's transfer prce to lmt the amount of proft-shftng nduced by tax rate dfferentals. Perunt audt costs ncurred by a multnatonal are denoted by the strctly convex functon C( w) for whch C() 0 and C(0) = 0. 7 Postve audt costs can reflect the costs ncurred j j j j 7 Ths specfcaton abstracts from actual transfer prce regulatons under whch the audtng costs vary wth the tax rates but shares the same qualtatve propertes mentoned n the text. See 0

11 by the frm to defend ts transfer prce durng an audt as well as expected penaltes mposed by the tax authortes. Wrtng C() as a functon of - w mples an arm's-length prce for the ntermedate good of w snce devatons from ths prce reflect taxable-proft-shftng by the multnatonal. If < w, the multnatonal s shftng taxable proft nto country and f > w, the multnatonal s shftng taxable proft nto country. An alternatve defnton of the arm's-length prce would be w +. Snce captal costs are generally not 00% taxdeductble, usng ths alternatve defnton of an arm's-length prce would effectvely allow frms to use non-tax-deductble costs to shft some taxable profts at no (audtng) cost. For a frm that chooses to be taxed under a separate accountng system, ts global post-tax economc proft equals ( q, q,, l, l, l, k, k, k ) ( t )( r( q ) w( l l ) q ) f f f f f ( t )( r ( q ) w l q ) C( w) q ( k k k ). f f f () Note that total audt costs equal C()q. Defnng total audt costs n ths way emphaszes the fact that snce proft-shftng s proportonal to the volume of ntermedate good shpments so too are potental penaltes for excessve proft-shftng. Most countres do not permt the full taxdeductblty of captal costs. For smplcty, I assume that 00% of captal costs are not taxdeductble. f f The problem each multnatonal solves s to maxmze () subject to q F( k, l ) for j j j j{,} and ( q q G k, l ; ). Holdng q, q, and fxed, a frm's condtonal factor demands, k q q K and, l q q L, are solutons to standard after-tax, cost-mnmzaton problems so that f f ( ), and K ( q, t ) and L ( q, t ) satsfy Fk (, l ) q and K L F / F /( t ) w where the "K" and "L" subscrpts denote partal dervatves of the subproducton functon. Note that K and L do not depend on q j or t j. Gresk and Osmundsen (008) and Nelson, Ramondos-Møller, and Schjelderup (00) for models that ncorporate ths level of detal n ther models.

12 Dfferentatng () mples that a frm's optmal transfer prce, condtonal on q, satsfes t t C( w) or w ( C) ( t t ). If ths optmal transfer prce s ether negatve or too large, the frm's problem can be unbounded due to the ablty of the frm to produce arbtrarly large quanttes n country to shft arbtrarly large proft nto country ( < 0) or nto country ( r( q)/ q w). Many countres have a "commensurate wth ncome" provson n ther tax codes that lmt such extreme proft-shftng. However, as long as C() s suffcently convex, such extreme values of wll not be proft maxmzng. Thus, we can defne *( q ) w ( C) ( t t ) to be a frm's proft-maxmzng transfer prce gven country producton q. Substtutng these condtonal values nto () mples that a frm solves ts proftmaxmzaton problem by choosng q and q to maxmze ( q, q ) ( t )( r( q ) w ( L ( q, t ) q )) ( t )( r ( q ) w L ( q, t ) wq ) ( t, t ) q ( K ( q, t ) K ( q, t ) q q ) () where ( t, t ) ( t t )( C ) ( t t ) C(( C) ( t t )). ( t, t) 0for all t and for all t t, ( t, t ) 0 by the strct convexty of C(). ( q, q) s strctly concave n q and q. The market bounds, q and q, and a suffcently convex audt cost functon wll mply that a soluton to the frm's problem exsts. Denote the proft-maxmzng quanttes by q ( t, t, w, w,, ) and proft by ( t, t, w, w,, ). By the Envelope Theorem, q ( t, t, w, w,, ) and defne a frm's ndrect post-tax economc / ( K ( q ) K ( q ) q q ) 0. (3) Drect comparatve statcs calculatons further mply for {,} that q / 0. Thus, multnatonals wth hgher economc costs of captal earn lower post-tax proft and produce less n each country. Note also that frm proft s lnear n f one holds output or nput levels constant. Any change n output and nput levels from a change n must yeld hgher proft than

13 ths passve benchmark. Thus, must also be convex n. For suffcently large values of μ, a frm wll no longer fnd t proftable to sell n country and/or country. Defne so that, for all, q 0. Lemma explans how to rank and n order to determne n whch country a frm wll cease producton frst as margnal captal costs ncrease. Lemma. If the countres are symmetrc and demand s suffcently large, then ( )( t t ) 0for all t t. Lemma states that a frm operatng under wll shut down sales n the hgh-tax country before t shuts down sales n the low-tax country. The demand requrement s needed to prove that whenever t < t as q can be non-monotonc n t for suffcently low μ as a frm may wsh to produce n country smply to create some capacty for proft-shftng. Wth suffcently large demand, a frm consders shuttng down sales n ether market only when μ s large enough so that the margnal captal costs outwegh any ncentve to produce n country solely to engage n proft-shftng. No such requrement s needed to prove that when t > t. Wth regard to, the Envelope Theorem mples that less productve frms earn less ndrect proft as / ( t ) wq ( t ) wq 0. (4) Less productve frms also sell less n each country as q / s negatve for {,}. By the same argument used n the precedng paragraph, s also convex n. An analog exsts to Lemma for the values of θ above whch a frm ceases to produce n each country. 3. Formula apportonment. Wth a formula apportonment system (), each frm reports ts total taxable proft to each country. An apportonment formula s then used to dvde the total taxable proft between the two countres. In ths paper, I restrct attenton to apportonment formulas defned by 3

14 ( q, q ) 0 ( q, q ),, whch determnes the weght put on country 's tax rate, such that ( q, q ) 0 (wth equalty only f λ=), and ( q, q ) 0(wth equalty only f λ=0). In addton, ( q,0) must equal one as the frm s not operatng n country. Note, however, that (0, q ) need not equal zero as the frm s stll producng the ntermedate good n country. Gven ( q, q), the effectve tax rate for a frm wll be T ( q, q ) t ( ( q, q )) t. (5) Two examples of apportonment formulas wth ths structure are the output formula, q /( q q ) r( q )/( r( q ) r ( q )). 8 Ths specfcaton, and the revenue/sales formula, does not nclude formulas that depend on nputs such as wages, employment, and/or captal (see footnote 7). An apportonment formula wll be called symmetrc f there s a non-negatve, nondecreasng functon, ( q ) wth (0) 0 so that ( q, q) ( q)/( ( q) ( q)). For symmetrc apportonment formulas, ( q, q ) /, ( q, q ) ( q, q ) 0, (, ) 0 q q, and ( q, q ) ( q, q ) 0. An apportonment formula wll be symmetrc f t does have any bult-n bas that favors operatons n one country over the other, as n the case of the output formula, and f the frms operate n symmetrc countres. Gven (5), a frm's post-tax economc proft s ( q, q, k, k, k, l, l, l ) ( T)( r( q ) r ( q ) w( l l ) wl ) ( k k k ). (6) f f f f f f f f f f Each frm's problem s to maxmze (6) subject to q F( k, l ) for j{,} and j j j qq G( k, l ; ). Holdng q and q fxed, a frm's condtonal factor demands, are solutons to standard after-tax, cost-mnmzaton problems so that k q q, l q q ( ), and K and L, K ( q, q ) and f f L ( q, q ) satsfy Fk (, l ) q and FK / FL / ( T) w. Note that, because both output quanttes nfluence the effectve tax rate, the condtonal factor 8 If margnal revenue can be negatve, the revenue rule can volate ether 0 or 0. However, both of these cases can be shown not to be proft-maxmzng. 4

15 demands depend on both quanttes. 9 Substtutng these condtonal demands nto (6) mples that a frm solves ts proftmaxmzaton problem by choosng q and q to maxmze ( q, q ) ( T) ( q, q ) ( K K q q ) (7) 0 where 0 ( q, q ) r( q ) r ( q ) w ( L ( q, q ) ( q q )) w L ( q, q ) denotes a frm's global pre-tax taxable proft. Unlke n the case, 0, as now both q and q affect T. Moreover, need not be globally concave n q and q. However, the market bounds, q and q, mply that proft-maxmzng quanttes exst. Denote these quanttes by q ( t, t, w, w,, ) and proft by ( t, t, w, w,, ). By the Envelope Theorem, q ( t, t, w, w,, ). Defne a frm's ndrect post-tax economc / ( K K q q ) 0 (8) and snce the equlbrum quanttes must be ncentve compatble, ( K / q ) q / ( K / q ) q / 0. (9) Note that (8) and (9) hold for all apportonment formulas and not just quantty-based formulas. The ablty of a frm to adjust ts output and nput levels n response to a change n mples that s also convex n. Remark. Each system s equvalent to an system but not necessarly one defned by a formula ( q, q). Lemma. If the countres are symmetrc, then ( )( tt) 0for all t t. Lemma s the analog to Lemma for. The proof shows that a frm wll sell more n the low-tax country than n the hgh-tax country, and as a result, t wll frst shut down sales n the j j j 9 If one were to consder apportonment formulas that use nput shares or nput-cost shares, t need no longer be the case that q F( K, L ) as some frms mght have an ncentve to overemploy captal or labor n order to nfluence T. 5

16 hgh-tax country. For frms wth dfferent productvty parameters, / ( T) w ( q q ) 0, (0) and snce the equlbrum quanttes must be ncentve compatble, (( T) w ( t t ) ) q / (( T) w ( t t ) ) q / 0. () s convex n. Analogs to Lemmas and exst wth regard to varaton n θ. 3.3 Multnatonal selecton. The analyss n sectons 3. and 3. reveal that the global post-tax ndrect proft functons under and are decreasng convex functons of and of. Each frm's choce of a tax system wll be based on the dfference n these ndrect proft functons,. Each system provdes a frm wth dstnct proft-shftng channels: transfer prcng wth and producton and sales shftng wth. However, both systems rely on the lnkage of upstream (country ) ntermedate good producton wth downstream (country ) sales n order to actually shft profts. To hghlght the mportance of ths lnkage effect on a frm's preference between and, we focus on a frm's preferences when a preponderance of ts sales occur n one country (secton 3.3.) as ths s where extensve margn effects are evdent and when the frm operates n symmetrc countres wth a focus on the ntensve margn effects (secton 3.3.) Extensve margn effects (Domnant country sales). A frm wll choose to sell almost exclusvely n one country f economc condtons mply lmted proft-earnng opportuntes n the other country. Holdng fxed any dfferences n revenue functons and wage rates, such near exclusvty wll occur f a frm's margnal cost of captal s suffcently hgh wth the country dfferences determnng whch country receves almost all of a frm's sales. Theorem consders the case n whch a frm's margnal cost of captal s suffcently hgh so that most of ts sales occur n country. Theorem. Assume that max{, } mn{, }. There exsts ˆ such 6

17 that, for all ( ˆ, ), a frm wth margnal cost of captal strctly prefers to be taxed under than under. The nterval, [, ), corresponds to margnal captal costs for whch a frm would sell nothng n country and strctly postve quanttes n country under and. A necessary condton for ths range to be well-defned under country symmetry s thatt t. For these values of μ, a frm strctly prefers over because affords no proftable opportuntes for proft-shftng as the frm always faces an effectve tax rate of t whle stll allows the frm to shft profts va transfer prcng. By contnuty, ths preference order extends to frms wth margnal captal costs slghtly below so that country s the predomnant but not exclusve sales locaton. Wth neglgble sales n country, the frm's ablty to shft profts under s also very lmted whle ts country producton affords greater potental to shft profts under. Fgure llustrates the proft dfferences between and dentfed n Theorem when t =.5, t =.3, w =w = =, r( q ) (0 q ) q, Fk ( f, l f ) mn{ k f, l f }, C( w ) ( w ), and ( q, q ) r( q )/( r( q ) r ( q )). The fgure graphs ( ) ( ) so postve values ndcate hgher post-tax proft under whle negatve values ndcate hgher post-tax proft under. The crcle dentfes approxmately the values of μ referenced n the theorem. The peak n Fgure around μ = 4.5 concdes wth, the value of μ beyond whch a frm sells ts fnal good only n country. For μ just above, affords no proft-shftng as q = 0 so T = t. However, a frm can stll shft profts under whch favors. By contnuty of Π, a frm's preferences for perssts for some μ just below even though a frm would now be sellng postve amounts n both countres. Thus, t s the mportance of usng sales volume to shft profts that favors for hgh cost frms when the downstream country has the lower tax rate. I wll dscuss the rest of the fgure n the next secton. 7

18 Theorem consders the mrror-mage case n whch almost all of a frm's sales occur n country. Now t s the case that max{, } mn{, }. (Wth symmetrc countres, ths case arses when t t.) For μ between and, a frm sells only n country under and and earns equal post-tax proft under each system. Now a frm's choce when μ s just below depends on whether t prefers to sell nothng n country under before t sells nothng n country under. Theorem. Assume that max{, } mn{, }. There exsts ˆ such that, for all ( ˆ, ), a frm wth margnal cost of captal strctly prefers to be taxed under f, and t strctly prefers to be taxed under f. Fgure : () - () (w =, t =.5, t =.3) The economc condtons that generate ths case lmt the frm's ablty to shft profts under ether system as the magntude of shfted profts s ted to country sales. Theorem dentfes the preferred tax system for frms that predomnately sell n country wth the system that supports hgher country sales as that system offers the greater capacty for proft-shftng. For example, when, a frm wth would stll prefer to sell postve quanttes n both countres under but sell only n country under. However, f ths frm chose but elected to sell only n country, t would earn the same proft as t could by choosng and only sellng n country. Snce t prefers to sell postve quanttes n both countres under, the strct concavty of pre-tax profts mples that ts profts must be strctly hgher under. An 8

19 analogous argument shows why a frm's preferred tax system swtches when. Fgures and 3 llustrate both of the scenaros descrbed n Theorem. Fgures and 3 are generated under the same assumptons and parameter values used for Fgure except that for Fgure, w =, t =.3, and t =.5, whle for Fgure 3, w =.5, t =.3, and t =.5. The knk n Fgure around μ = 4.6 concdes wth, the value of μ beyond whch a frm sells only n country. Fgure corresponds to the case n Theorem for whch. As a result, a frm wth a value of μ slghtly below strctly prefers. Fgure 3 corresponds to the case n Theorem for whch. Now a frm wth a value of μ slghtly below at around 4.67 strctly prefers. The effect of the revenue functons and the wage rates on the orderng of and be seen by comparng a frm's margnal profts wth respect to country sales under each system where q ( t, t) ( t t)( r (0) w) ( t t) ( r( q ) w( q L ( q )). () q The margnal proft-shftng dstortons unque to are captured by the transfer prcng term, Δ, whle the latter two terms n () descrbe the margnal proft-shftng dstortons unque to. The frst -term reflects the ablty of the frm under to have ts country profts taxed at t by sellng nothng n country whle under ts country profts are always taxed at t. When t > t, ths ablty favors. The second -term reflects the ablty of the frm to nfluence ts margnal tax rate. Snce λ s negatve (an ncrease n q decreases the share of a frm's global proft taxed at t ), ths term favors when t > t. A margnal decrease n w strengthens both of the -dstorton terms but the change n the second term s proportonal to country sales under whch also ncreases as w decreases. Thus, a sgnfcantly lower wage rate n country (relatve to country 's wage rate) can shft the preference of a frm that sells almost exclusvely n country from to. can 9

20 Fgure : () - () (w =, t =.3, t =.5) Fgure 3: () - () (w =.5, t =.3, t =.5) Wth these baselne results establshed for frms that operate near ts frst extensve margn where t shuts down sales n one country, I now consder the selecton patterns drven by a frm's ntensve margns under country symmetry Intensve margn effects wth symmetrc countres. A frm's ndrect profts under and are dentcal when the countres have equal tax rates. Thus, some dfference n tax rates s needed to generate a strct preference for one method over the other. Wth dentcal wage rates, revenue functons, and tax rates, a frm's factor demands and output choces n both countres wll also be dentcal wth ether or. It s also the case that f the countres are too dfferent, all frms on the ntensve margn wll prefer the same method. To emphasze the role of tax rate dfferences n a frm's tax method choce n a settng n whch dfferent frms are most lkely to choose dfferent methods, I assume n ths secton that the frms operate n symmetrc countres and denote ndrect frm profts by ( t, t, ) and t t (,, ). Theorem 3. Assume that at t = t, a frm wth margnal cost of captal and productvty parameter produces strctly postve output n both countres. For t suffcently close to t, country symmetry and apportonment formula symmetry mply that 0

21 0 ( t t) q w L ( t, t, ) ( t, t, ) ( K 3 / q ) ( t (0) ) r C ( t) r t O t 3 (( t) ) (3) where all output and nput quanttes are calculated at t t. Eq. (3) s the second-order Taylor seres expanson of near equal tax rates (t,t ). At equal tax rates, not only are the frm's ndrect profts equal, so too are ts margnal ndrect profts wth respect to t,.e., ( t, t)/ t ( t, t)/ t. However, ( t, t )/ t and ( t, t )/ t, whle both non-negatve, need not be equal for all t meanng that dfferences n the convexty of the frm's ndrect proft functons wth respect to t wll account for a frm's choce between and when the countres' tax rates are smlar. Snce the margnal ndrect proft functons are equal at equal tax rates, one can measure the relatve convexty of the ndrect proft functons by comparng ( t, t )/ t and ( t, t )/ t each of whch conssts of three terms: an output dstorton (due to the ndrect effect of a change of t on margnal ndrect proft va q and q ), a proft-shftng dstorton (due to the drect effect of a change n t ), and a condtonal factor demand dstorton (due to the ndrect effect of a change of t va the condtonal factor demands). The bracketed term n (3) equals the dfference between these dstortons under and. Denote ths term by (, ; t). The frst term n s postve and represents the dfference n country output dstortons between and. For both and, a hgher tax rate n country lowers country output whch n turn ncreases margnal proft. However, the output dstorton under s twce as large as the output dstorton under as the amount of proftshftng under depends only on country producton whle under the frm can use both country and country output to shft profts. The second and thrd terms of represent the dfference n the proft-shftng effects under and. Under, the proft-shftng effect s postve. Recall from () that s the

22 after-tax proft earned from proft-shftng per unt of output. In ths case, / t / C(0) at equal tax rates. As a result, the optmal amount of proft shfted per unt of output s ncreasng at an ncreasng rate as t t ncreases. The proft-shftng effect under s also postve (but s subtracted from the effect) as the proft-shftng opportuntes under ncrease wth t t just as they do under. Note that ts magntude depends on how the apportonment formula vares wth output n each country. The fourth term of Γ represents the dfference n the condtonal labor demand dstortons under and. The condtonal labor demand dstortons are postve, snce an ncrease n t gves frms the ncentve to substtute labor for captal, and under the assumpton of constant returns to scale n fnal good producton they are larger under. The larger condtonal labor demand dstorton under arses for three reasons. Frst, the effect of t on a frm's effectve tax rate under s reduced by λ relatve to due to the apportonment formula. Ths effect s mportant because the condtonal labor demand dstortons measure the after-tax effect of a change n t. Second, wth t also effects a frm's captal-labor mx n both countres as t changes the after-tax wage rate. But the effect n each country s reduced by λ snce what matters to a frm s the after-tax wage rate, w( T), whch s only partally determned by t. Thrd, under t only affects taxable ncome through country labor costs whle under t affects taxable ncome, now π 0, through country and country labor costs. The proof of Theorem 3 n the appendx shows under country and formula symmetry and equal tax rates that L / t L / t L / t. Thus, the aggregate condtonal labor demand dstortons on the tax base under equal the condtonal labor demand dstortons on country taxable ncome under whle the effect on the tax rate under s stll reduced by λ (whch equals / at equal tax rates). Snce the output and condtonal labor demand dstortons created by a change n t always favor, the only way for a change n t to generate hgher proft under s f the

23 apportonment formula generates a larger proft-shftng effect. As μ goes to zero, the output dstorton goes to zero because n the lmt both tax methods are pure proft taxes at equal tax rates, and the condtonal demand for labor also decreases. For example, wth a Cobb-Douglas producton functon, L / ts proportonal to L whch n turn goes to zero as μ goes to zero. Thus, for the frms wth very low margnal costs of captal, ther preferred tax method wll be determned prmarly by the proft-shftng dstorton. Notce that the magntude of the proftshftng dstorton from s proportonal to λ, the rate at whch a frm's global operatng proft s taxable n country as country sales ncrease. Consder how λ dffers between an output formula and a sales formula. Wth symmetrc countres and equal tax rates, /4q under an output formula whle ( / 4 q) ( r / p) / 4qunder a sales formula snce margnal revenue s less than prce at strctly postve quanttes. 0 Thus, (3) would suggest that frms wth a low margnal cost of captal have a stronger ncentve to prefer over a sales only formula than over a quantty formula. Fgure 4 llustrates the net margnal tax ncentves a frm faces gven a small tax dfferental wth a sales apportonment formula under the same assumptons for Fgures and as a functon of μ. (Note that the fnal good producton functon used n these examples generates no condtonal labor demand dstorton.) Snce postve values n Fgure 4 favor, frms wth margnal costs of captal below approxmately.4 prefer whle frms wth margnal costs of captal above.4 prefer. Wth an output formula nstead of a sales formula, the analog to Fgure 4 would show that all frms prefer. Ths change s consstent wth the fact that an output formula generates stronger proft-shftng dstortons when μ s small. Comparng Fgures and 4 also llustrates the mportance of analyzng a frm's preferences near ts extensve margns separately from the effect of ts ntensve margns. When t = t, so the extensve margn effect n favor of dentfed n Theorem 0 One could also wrte the expressons for λ n terms of q as all sales quanttes are equal under country symmetry and equal tax rates. 3

24 cannot be dentfed at equal tax rates. That s, the extensve margn effect only materalzes when t t. However, by usng Theorems and 3 together one can see n Fgure the part of the selecton pattern nduced by the ntensve margn ncentves and the part nduced by the extensve margnal ncentves. Thus, n Fgure 4 the ntensve margn ncentves result n low-μ frms preferrng and the extensve margn ncentves result n hgh-μ frms preferrng. Fgure 4: () - () for small tax dfferentals Fgure 5: (θ) - (θ) for small tax dfferentals Fgure 5 llustrates the ntensve margn ncentves a frm faces gven a small tax dfferental wth a sales apportonment formula under the same assumptons used for Fgures and but now as a functon of θ and wth μ =.. I wsh to note two features. Frst, the net output dstorton does not vary wth ntermedate good productvty when revenues n each country are quadratc. Thus, the only ntensve margn varaton n frm profts under and (at equal tax rates) must come from the proft-shftng dstorton and the condtonal labor demand dstorton terms. Second, one can see that the non-monotonc preferences of over can be generated solely by ntensve margn ncentves. Introducng the extensve margn effects when t t would further ncrease frm proft under for values of μ close to the country extensve margn. Eq. (3) also suggests that f the ntensve margn ncentves support, they should do so for frms wth low margnal costs of captal. Wth quadratc revenue functons and the 4

25 producton functons used n the runnng example, 8 ( ) /. (4) ( t ) ( t ) ( t ) ( t ) The last term n (4) reveals that how a frm's choce between and changes wth respect to μ can depend on whether (,) s concave or convex n q. (Recall that for a symmetrc formula, f (,) s concave n q t must be convex n q and vce versa.) When s close to zero, wll be decreasng n as long as (,) s not too convex n q. For both the output rule and the revenue rule, λ(,) s strctly concave whch means any proft advantage has at = 0 wll dmnsh at ntally ncreases. 4. Tax revenues. I now turn to the ssue of how the optmal choce behavor of each frm (descrbed n secton 3) affects tax revenues n each country under the assumptons of country symmetry and formula symmetry. The tax revenues collected by country by a frm that elects equal TR ( t, t ) t [ r ( q ) w ( L ( q, t ) q ) ( C) ( t t ) q ] (5) whle the tax revenues collected by country by a frm that elects equal TR ( t, t ) ( q, q ) t ( q, q ). (6) 0 The formulas for country 's tax revenues are analogous and can be found n the proof of Theorem 4 as (A.5) and (A.6). Eqs. (5) and (6) ndcate that all three dstortons, output dstortons, proft-shftng dstortons, and factor demand dstortons, can nfluence country 's tax revenues through the tax system choce each frm makes. To determne how each of these dstortons nfluence a country's tax revenue rankng of and on the ntensve margn, I adopt a smlar approach to that found n secton 3. Begnnng wth equal tax rates, note that a country's tax revenues under both methods are dentcal. Snce t does not affect q or L, TR ( t, t)/ t t q / C(0). That s, on the margn, a change n t only affects country 's tax revenues through ts effect on a frm's transfer 5

26 prce. Snce ths proft-shftng effect s only one of three dstortons that nfluence a frm's rankng of and, usng Theorem 3 one can also wrte ( K / q ) L TR ( t, t )/ t t [ w ] 3 t ( t) r t (7) where the bracketed term n (7) represents the proft-shftng dstorton as the total second-order dstorton from a change n t wth the output and condtonal labor demand dstortons taken out. In a smlar fashon, note that (6) mples that TR ( t, t )/ t d( )/ dt whle one can also 0 wrte / t d( )/ dt d / dt (see (A.8) n the proof of Theorem 3). Thus, the 0 0 effect of a change n t on country 's tax revenues under equals the second-order effect on frm proft plus the effect on a frm's tax base, π 0, whch ncludes both an output dstorton and a condtonal labor demand dstorton. In the proof of Theorem 4, I show that the tax base dstorton exactly offsets the output and condtonal labor demand dstortons so that country 's rankng of and depend only on a frm's rankng and the sgn of t t. Recall from Theorem 3 that the dfference n frm profts has the same sgn as the dfference n the second dervatve terms n (3). As a result, for small tax rate dfferences country wll prefer the same method that a frm prefers when country has the lower tax rate and t wll prefer the opposte method a frm prefers when country has the hgher tax rate. Country 's preference s the exact opposte of country 's for any set of tax rates snce when country has the lower tax rate, country has the hgher tax rate. Theorem 4. Assume that at t = t, a frm wth margnal cost of captal and productvty parameter produces strctly postve output n both countres. Then country symmetry and apportonment formula symmetry mply that ( t, t) ( t, t) TR ( t, t) TR ( t, t) t( t t) O(( t t) ) t t and (8) 6

27 ( t, t) ( t, t) TR ( t, t) TR ( t, t) t( tt) O(( t t) ). (9) t t Fgures 6 and 7 llustrate the relatonshp descrbed n Theorem 4 between the proft dfferental that drves each frm's choce and the tax revenue dfferental that defnes the effect of frm choce on each country's tax revenues when t =.3 and t =.3. Fgure 6 focuses on country tax revenues whle Fgure 7 focuses on country tax revenues. The thn lnes correspond to the proft dfferentals and the thck lnes correspond to the tax revenue dfferentals (scaled by.0 to allow for easer comparsons). Notce that the sgns of the proft dfferental and the tax revenue dfferental are the same n Fgure 6 for all μ whle the dfferentals have opposte sgns n Fgure 7. The same sgn patterns would emerge when lookng at varaton n θ. When country has the lower tax rate, the graphs n Fgures 6 and 7 would be swtched. Thus, f one country collects more tax revenues from a frm wth the ablty to choose between and, the other country wll collect fewer tax revenues. Perhaps not surprsngly, the extensve margn effects of choce whch encourage frms wth hgh margnal costs of captal to select when country s the low-tax country also ncrease country 's tax revenues from these frms under relatve to. For a hgh-tax country, t prefers that these frms choose. Snce the extensve margn effect creates a strct preference by country for, there wll exst a set of frms (below the extensve margn) who also margnally prefer. Thus, the extensve margn can create some scope for algnng frm preferences wth the hgh-tax country but the effect has been small n all the examples I have computed. Ths effect of the extensve margn n partally algnng the preferences of the hgh tax rate country wth a frm's choce can be seen n the crcled regon of Fgure 8 for μ between approxmately 5.6 and 5.8. What looks lke a dscontnuous jump n TR TR for μ at 5.8 s n fact not dscontnuous. The very steep segment corresponds to a small range of values of μ n whch a frm sells nothng n country under but sells a small amount n country under. To the rght of the knk n both functons, one sees the dfference n frm profts and the dfference n country tax revenues when a frm sells ts fnal good only n country under and. It s mportant to note that for values of μ close to the extensve margn value of 5.8, country tax revenues are negatve snce the frm s generatng zero revenues n country and s shftng some ncome to country. In practce, frms often have other domestc operatons that can mask proft-shftng that may not be commensurate wth ts operatng proft n a country. I allow for negatve tax revenues for ths reason. However, f one were to lmt a frm's proft-shftng by the operatng proft generated n a country, an extensve margn effect would stll exst although ts magntude would be reduced and the locaton of the extensve margn would shft to the left. 7