Tax Policy and the Missing Middle: Optimal Tax Remittance with Firm-Level Administrative Costs*

Transcription

1 Tax Polcy and the Mssng Mddle: Optmal Tax Remttance wth Frm-Level Admnstratve Costs* Dhammka Dharmapala Unversty of Illnos at Urbana-Champagn Joel Slemrod Unversty of Mchgan John Douglas Wlson Mchgan State Unversty Ths verson: July 2010 Abstract: We analyze the optmal taxaton of frms when the government faces fxed (per-frm) admnstratve costs of tax collecton. The tax nstruments at the government s dsposal are a fxed (per-frm) fee and a lnear tax on output. If all frms n an ndustry are taxed, we show that t s optmal to mpose a postve fee to nternalze admnstratve costs. The output taxes satsfy the nverse elastcty rule for taxed ndustres, but ndustres wth suffcently hgh admnstratve costs should be exempted from taxaton. We also nvestgate the case where frms wth outputs below a cutoff level can be exempted from taxaton. It may be optmal to set the cutoff hgh enough to exempt a szable number of frms, even though some frms reduce ther outputs to the cutoff level, creatng a mssng mddle : small and large frms but not those of ntermedate sze exst. Thus, ths common phenomenon n developng countres may result from optmal polces. The paper also presents a modfed nverse-elastcty rule when output cutoffs are used, and t extends the analyss to nclude optmal nonlnear taxes on output. *We are grateful for comments on an earler draft receved from the Edtor (Emmanuel Saez), two anonymous referees, Alan Auerbach and attendees of semnar presentatons at Hunter College, the London School of Economcs, Queen s Unversty, Stanford Unversty, the Unversty of Calforna at Berkeley, Unversty College London, the World Bank, the Offce of Tax Polcy Research conference on The Role of Frms n Tax Systems, and the 2008 Centre for Busness Taxaton symposum at Oxford Unversty.

2 1. Introducton Some students of developng country economes have noted a phenomenon of the mssng mddle, n whch many small frms and a few large frms produce the bulk of value added. It forms part of a wder cluster of nterrelated characterstcs ncludng a large nformal sector and regulatory barrers to entry nto the formal sector that have often been explaned wth reference to the grabbng hand assocated wth bureaucratc neffcency and corrupton (e.g. Fredman et al., 2000; Djankov et al., 2002). In the development lterature, t has been suggested that the mssng mddle arses n part because taxes and regulatons are enforced only among large, formal-sector frms (e.g. Rausch, 1991). 1 Notably, though, the development lterature does not address under what condtons t s optmal for polcy to treat dfferentally frms of dfferent sze, perhaps n a way that generates a mssng mddle. Nor, ndeed, has the publc fnance lterature focused on ths ssue, largely because ts standard normatve framework does not allow for heterogeneous frm sze n a meanngful way. In ths paper, we provde a normatve framework for analyzng polces that apply dfferentally to small and large frms, and demonstrate that under some condtons optmal polces wll generate a mssng mddle a range of output that no frm produces. We address tax polcy, although some of the nsghts are also relevant to regulatory polcy. Our basc argument s that the government should economze on admnstratve costs by exemptng small frms from taxaton, even though dong so causes ntermedate-szed frms to reduce ther outputs to tax-exempt levels, thereby creatng a mssng mddle. 2 More broadly, a central am of our paper s to construct a model of optmal tax polcy that recognzes the central role that frms play n the remttance systems of all modern taxes, and the potental mportance of treatng frms dfferently accordng to ther sze, regardless of whether these taxes are leved n developng or developed countres. The mportance of frms becomes apparent once one recognzes that t s cost-effcent for the tax authorty to deal wth a small number of enttes wth relatvely sophstcated accountng and fnancal expertse, rather than a much larger number of 1 Tybout (2000) revews the emprcal evdence about ths phenomenon. 2 Our approach s consstent wth Gordon and L s (2009) basc contenton that the polces pursued by developng countres can potentally be explaned on normatve grounds, rather than through poltcal economy explanatons based on the self-nterest of non-benevolent polcymakers. 1

3 employees or provders of captal. 3 However, dealng wth small busnesses s not generally cost-effcent, and many tax systems partally or entrely exempt small busnesses from remttance responsblty. 4 Although specal tax treatment of small frms mght economze on complance and admnstratve costs, t also generally causes producton neffcency, n part because t provdes a tax-related ncentve for frms to be or stay small. The tradeoff between the costs of collecton and producton neffcency, and more generally the desgn of the remttance of taxes, has not been closely addressed by the optmal tax lterature. 5 The operaton of actual tax systems, however, requres consderable attenton to the remttance of mones to the tax authorty, ncludng both the admnstraton and enforcement of the tax rules, as well as the desgn of the tax rules wth the admnstratve and complance ssues n mnd. Ths s especally true n developng countres, where admnstratve constrants are often frst-order ssues n tax systems. In addton, the famous Damond and Mrrlees (1971) theorem on aggregate producton effcency posts that producton neffcences should not be tolerated f the government faces no constrants on ts ablty to levy optmal commodty taxes. But ther model of optmal taxaton gnores admnstratve costs and assumes that there are no untaxed profts, due ether to constant returns to scale or a 100 percent profts tax. Ytzhak (1979) and Wlson (1989) nvestgate optmal commodty taxaton when there s 3 The centralty of frms s llustrated by two recent studes that fnd that over 80 percent of all taxes are remtted by busness n the U.S. and the U.K. - see Chrstensen, Clne, and Neubg (2001) and Shaw, Slemrod, and Whtng (forthcomng). Anecdotal evdence suggests that collecton of taxes from busnesses s even more mportant n developng countres. 4 For an excellent revew of the sorts of specal regmes that countres apply to small busnesses, see Internatonal Tax Dalogue (2007). In many countres the exempton of small frms s de facto, due to lax enforcement. One mplcaton of these polces s that the collecton of taxes s hghly concentrated among relatvely large frms. A recent report asserts that the typcal dstrbuton of tax collectons by frm sze for Afrcan and Md-Eastern countres features less than one percent of taxpayers remttng over 70 percent of revenues, and the report gves specfc examples of hghly concentrated patterns: n Argentna, 0.1 percent of enterprse taxpayers remt 49 percent of revenues; and n Kenya, 0.4 percent remt 61 percent (Internatonal Tax Dalogue (2007). In contrast, most manufacturng employment n developng countres s n small frms (wth less than 10 employees); see e.g. Tybout (2000). Gauther and Gersovtz (1997) document the concentraton of tax payments for Cameroon. 5 In part, ths s because nearly all of modern tax theory s concerned wth what actons or states of the world trgger tax lablty, and vrtually none s concerned wth the system of remttance of funds to the government to cover that lablty. Indeed, elementary publc fnance textbooks often assert that the remttance detals such as whether the buyer or seller of a commodty remts the sales tax trggered by the sale are rrelevant to the consequences of a tax. The mportance of the remttance system s dscussed n Slemrod (2008). 2

4 costly tax admnstraton, but ther assumpton of constant returns to scale elmnates any role for frms. An earler paper by Heller and Shell (1971) presents a general framework for analyzng an optmal system of commodty taxes, lump-sum taxes, and frm-specfc lcensng fees and profts taxes when these tax nstruments are costly to admnster. In contrast, a major goal of the present paper s to nvestgate the optmal taxaton of ndvdual frms when frm-specfc taxes are not avalable and therefore exemptons from taxaton must be based on observed outputs. Keen and Mntz (2004) consder an output cutoff for exemptng frms from a value-added tax n the presence of admnstratve and complance costs. However, frms expect to earn untaxed profts n ther model, n whch case the Damond-Mrrlees theorem no longer holds and, n general, dfferent frms should be taxed at dfferent rates, even wthout admnstratve costs. 6 As we next explan, n our model admnstratve costs are solely responsble for the producton neffcences, and these costs requre an expanson n the set of tax nstruments. In our man model, the three avalable polcy nstruments are a constant tax rate on output, a fxed per-frm fee, and an output cutoff, below whch frms are not taxed. We show that when all frms n an ndustry are taxed, optmal polcy may nvolve the use of the fxed fee to nternalze the socal costs of tax admnstraton. 7,8 In our model, each ndustry s characterzed by constant returns to scale, because there s an effectvely unlmted number of ex ante dentcal potental producers. We assume that frms dscover heterogeneous productvtes after enterng the ndustry. In ths settng, the standard rules of optmal commodty taxaton hold f there are no admnstratve costs, enablng us to solate the mplcatons of ntroducng these costs. In partcular, the 6 Ths dscusson refers to the second of two models presented by Keen and Mntz (2004). In the frst model, frm sales are exogenous, so a cutoff rule does not dstort output decsons. (See also Zee (2005) for an extenson of ths model.) Both models dffer from our analyss by treatng net product prces as fxed (a small open economy), assumng an exogenous socal margnal value of government revenue, and not allowng a per-frm fxed fee as a polcy nstrument. 7 Internatonal Tax Dalogue (2007, p. 31) dscusses the fxed per-frm fee, also known as a patente system, as an example of a presumptve tax regme. The fxed per-frm fee could also be motvated by the entry fees and regstraton costs mposed by governments on frms, as measured and analyzed by Djankov et al. (2002). Aurol and Warlters (2005) also argue that entry barrers may be optmal n some crcumstances, but the reason s very dfferent: the entry barrers generate rents for ncumbents that the government can tax. 8 If, nstead of admnstratve costs borne n the frst nstance by the taxng authorty, frms ncurred fxed complance costs n the payment of taxes, then the fee would not be needed because these costs would already be nternal to the frm. Our other results are also easly modfed to account for complance costs. 3

5 Damond and Mrrlees (1971) theorem on aggregate producton effcency tells us that the tax system should not dscrmnate among frms n the same ndustry. Wth admnstratve costs, we dentfy cases n whch t s optmal to exempt small frms from taxaton. Ths creates producton neffcences that are never part of an optmal tax system n the Damond and Mrrlees framework. These neffcences occur because dfferent frms n the same ndustry sell output at dfferent prces, and also because some frms obtan the tax exempton by reducng ther outputs to neffcently low levels, creatng the mssng mddle descrbed above. It s mportant to emphasze that our clam s a theoretcal one that a mssng mddle can potentally be generated by optmal tax polces under certan crcumstances rather than an emprcal one (that observed mssng mddles correspond to exstng tax thresholds). However, there s an emergng body of emprcal evdence showng that frm sze can be affected by varous tax and regulatory thresholds. For nstance, Onj (2009) analyzes the ntroducton of a value-added tax (VAT) n Japan n The new VAT system ncorporated preferental treatment for small frms, wth a cutoff for elgblty of 500 mllon yen n sales. Onj (2009) fnds a clusterng of frms just below ths threshold followng the reform. Labor market regulatons also often vary by frm sze. In Italy, frms wth more than 15 employees face sgnfcantly more strngent employment protecton regulatons (and, n partcular, hgher frng costs). Usng dfferent emprcal approaches, Garbald, Pacell and Borgarello (2004) and Schvard and Torrn (2008) fnd sgnfcant effects of ths threshold, nvolvng slower frm growth and greater persstence close to the threshold. 9 Ths evdence provdes support for the emprcal mportance for the dstrbuton of frm sze of tax and regulatory thresholds of the type analyzed n our model. As a precursor to our model of heterogeneous frms, we frst nvestgate the optmal system of output taxes and fxed fees n an economy where frms dffer only across ndustres, not wthn ndustres. In partcular, n Secton 2 we characterze the 9 In addton, there s wdespread support for the more general noton that the dstrbuton of frm sze s nfluenced by taxaton and regulaton n developed as well as developng countres. Pagano and Schvard (2003) document sgnfcant dfferences n frm sze dstrbutons across European countres, and attrbute these n part to tax and regulatory polces. Henrekson and Johansson (1999) fnd that Sweden has a partcularly small share of medum-szed frms (wth employees), and explan ths wth reference to tax and regulatory polces favorng larger frms. 4

6 optmal system of output taxes and fxed fees, and we descrbe a rule to determne when admnstratve costs are suffcently hgh to justfy the exempton of an ndustry from taxaton. Secton 3 develops the heterogeneous-frm model on whch the rest of the paper reles. In Secton 4, we show that under certan condtons, cutoffs are part of the optmal tax structure. In Secton 5, we develop a modfed nverse-elastcty rule for the average net (of admnstratve costs) taxes on dfferent goods. In Secton 6, we ntroduce an optmal nonlnear tax on an ndustry s taxed output, emphaszng the use of the nonlnear structure to reduce taxes on frms that mght otherwse avod these taxes by neffcently reducng ther outputs to the cutoff level. Secton 7 concludes. 2. The Structure of Frm-Remtted Taxes across Homogeneous Industres We begn by focusng on dfferences across ndustres rather than on dfferences among frms wthn an ndustry. To do so, consder an economy wth many ndustres, each of whch has access to an unbounded mass of dentcal potental frms. Before makng ts entry decson, a frm knows that f t enters an ndustry, t wll face a strctly ncreasng, strctly convex cost functon gven by c ) + c e for a frm wth output y n ndustry, where the dervatve, c '(y ), s postve at all y, and c e s treated as an entry cost. When we later ntroduce heterogeneous frms, we wll assume an addtonal fxed producton cost ncurred by frms after enterng and choosng to produce. In ether case, the entry cost and ncreasng margnal cost mply U-shaped average cost curves. The tax nstruments consst of a constant margnal tax rate on output, t, and a fxed fee, b. The fxed fee wll turn out to be a crtcal component of the tax system when there are per-frm admnstratve costs n the collecton of taxes. Lettng p denote the producer prce for good, calculated net of the output tax, proft maxmzaton yelds a frm s output functon, y (p ), and ts proft functon, defned gnorng the fxed entry cost: ( y ( p, b ) p y ( p ) c ( y ( p )) b. (1) Free entry guarantees that p wll settle where p, b ) equals the entry cost. ( The demand for each good s obtaned from the utlty-maxmzaton problem for a representatve consumer wth utlty functon, U(X, L), where X s a vector of I goods 5

7 and L s the supply of an nput called labor. The representatve consumer supples ths labor to frms n each ndustry and receves all profts, whch are zero n equlbrum. Labor serves as the numerare, so the costs descrbed above are measured n unts of labor. So that we can work wth demand functons for each good that depend only on the good s own prce, we assume that the drect utlty functon s separable and quas-lnear n labor: U 1 (X 1 ) + + U I (X I ) L. Lettng q denote the prce that the consumer pays for good, utlty maxmzaton gves the demand functons, X (q ) for good, and the ndrect utlty functon, v (q ), where the wage s suppressed because t s fxed at one. In the presence of a unt tax t on good, the consumer prce q equals p + t. The number of frms producng good, M, s determned by the requrement that demand equals supply: X (q ) = M y p ). By Walras Law, these equlbrum condtons and ( the budget constrants for each agent (ncludng the government) mply that the labor market clears. In ths paper we focus on the mplcatons for optmal tax polcy of a fxed perfrm cost of collectng taxes. 10 Specfcally, the government ncurs a fxed per-frm admnstratve cost, A, when t collects taxes from a frm. Ths assumpton s ntended to capture the noton that there s a substantal fxed component to tax admnstraton and enforcement. For nstance, suppose that tax enforcement requres that (at least wth some postve probablty) the tax authorty must dspatch agents to audt the records of each frm. Although t may be the case that audtng a larger frm requres more resources, t s exceedngly unlkely that such dfferences, even f they exst, wll be proportonal n sze to the dfferences n frms output levels. 11 Although to smplfy the model we assume that admnstratve costs are fxed and hence ndependent of the sze of the frm, the qualtatve results apply as long as there s a relatvely large fxed component to these costs. The government s tax nstruments consst of the vectors of output taxes and fxed fees. The values of these control varables determne the market-clearng values of the consumer and producer prces, and the number of frms enterng each ndustry. 10 The model does not allow frms to splt up or combne for tax purposes. 11 See Slemrod (2006) for a revew of the emprcal evdence showng that the complance costs of busness taxes as a fracton of frm sze fall wth frm sze, suggestng a sgnfcant fxed aspect of ths part of tax collecton costs. 6

8 Followng the standard practce n optmal tax theory, t s convenent to take as control varables the consumer and producer prces, rather than the tax rates. 12 It s also convenent to treat the M s as control varables. The Lagrangan descrbng the government s problem s as follows: L = v ( (2) q ) X (q )- M y (p ) + B M (q -p )y (p ) b A E + ( p, b ) c. e The government seeks to maxmze the utlty of the representatve consumer, subject to three types of constrants. The constrant multplyng the Lagrange multpler λ s the requrement that demand equal supply n the market for good. The multpler λ B apples to the government budget constrant, where E s an exogenous revenue requrement. Fnally, each multpler pertans to the requrement that profts equal zero n ndustry. We frst show that the fxed fee should equal the per-frm admnstratve costs, leavng the output tax to fnance expendture needs. (All proofs appear n the Appendx.) Proposton 1. If frms producng good are taxed, then the optmal fxed fee equals admnstratve costs: b = A. The value of b can be thought of as a knd of Pgouvan tax: a taxed frm generates a socal cost n the form of admnstratve costs, and the fee should nternalze ths cost. 13 Ths reasonng clearly does not depend on the smplfyng assumpton that all frms are dentcal, so Proposton 1 carres over to the model of heterogeneous frms addressed begnnng n the next secton, assumng all frms are taxed. Note that, ceters parbus, the hgher s A (and therefore the optmal value of b ), the larger s the optmal output of a frm n ndustry, and the lower s the optmal number of such frms. Ths s true because a hgher b requres a hgher p to mantan the zero-proft condton, whch n turn mples a hgher y ; for gven q and thus total output X (q ), ths n turn mples a 12 In standard constant-cost models, p s fxed by the technology, but we shall see that ths s not the case here. 13 It s not a classc Pgouvan tax, as t addresses a fscal externalty that would not be present f no tax revenue were requred. 7

9 smaller M. Ths zero-proft condton mples that each frm produces at the bottom of ts average cost curve, nclusve of b. Because Proposton 1 tells us that b = A, ths mnmzaton of each frm s average cost mples that ts average producton plus admnstratve costs s mnmzed. Wth the fee addressng the external nature of the admnstratve costs, we should expect the output tax to satsfy the usual nverse-elastcty rule for an optmal tax system. 14 Ths turns out to be the case. Let denote the margnal utlty of ncome, and defne the prce elastcty of demand, measured postvely, as follows: X dx dq q X. (3) We then have: Proposton 2. If all frms producng good are taxed, then the optmal output tax structure satsfes the nverse-elastcty rule: t q 1 B X. (4) Proposton 2 asserts that, as b addresses the externalty, any remanng net revenue requrement ought to be collected as would otherwse be optmal, n ths case by followng the well-known nverse elastcty rule. In the case of heterogeneous frms consdered n the next secton, the set of producng frms and the set of taxed frms may be dfferentated by establshng a sze threshold for beng subject to tax. But under the current assumpton of homogeneous frms, producng frms can be dstngushed from taxed frms only by exemptng entre ndustres from taxaton. Exempton of an ndustry mght be optmal because, although admnstratve costs do not affect the optmal structure of output taxes on taxed goods (by Prop. 2), they may affect the optmal set of taxed goods. 14 The nverse-elastcty result apples because of our assumpton that the utlty functon s separable and quas-lnear n labor. See Auerbach and Hnes (2002) for a careful exposton of ths model. To derve (4), we use Roy s dentty, whch mples that = v (q )/X. 8

10 Suppose, n partcular, that for some reason admnstratve costs A ncurred n taxng frms n sector ncrease, whle the costs related to another sector j (or set of other sectors) decrease, so that the government budget stays balanced wth no change n taxes. The precedng argument tells us that the optmal structure of output taxes does not change, and therefore socal welfare stays constant, f we contnue to tax the same goods. Instead, the government should respond to the hgher A by rasng the fxed fee b, and lowerng b j. If, however, A gets suffcently hgh, then the government should reason that, f the ntal output taxes are low relatve to admnstratve costs, and a good s not such a large contrbutor to the government budget that exemptng t would requre much hgher taxes on the remanng goods, then t should be exempted from taxaton altogether n order to obtan the savngs n admnstratve costs. Indeed, one can derve a suffcent condton for ndustry exempton that obtans f the rato of admnstratve costs to tax revenue exceeds a lower bound that depends on the margnal cost of funds. 15 Ths argument has nterestng parallels to the work of Ytzhak (1979) and Wlson (1989), both of whch consder the optmal taxaton of a contnuum of goods that enter a representatve consumer s utlty functon symmetrcally. 16 In these models, taxng any good ncurs an admnstratve cost that vares across goods, whch suggests that taxng fewer goods s better than taxng many goods. But expandng the tax base reduces the standard deadweght loss from taxaton and, at the optmal number of taxed goods (tax base breadth), the margnal admnstratve cost equals the margnal savng n deadweght loss. In the Ytzhak and Wlson models, however, the reason that admnstratve costs vary across goods s exogenous. Nor could the source of varaton be related to admnstratve costs at the frm level, as these models adopt the standard assumpton that all frms exhbt constant returns to scale, renderng the sze of frms ndetermnate. In the present model, U-shaped average cost curves lmt the equlbrum sze of frms. 15 The margnal cost of funds s equal to MCPF 1 1 t q X. The suffcent condton s derved n Dharmapala, Slemrod, and Wlson (2009). 16 See also Slemrod and Kopczuk (2002), whch adds dstrbutonal concerns to the choce of how broad the tax base should be. 9

11 Heterogenety of producton technologes ntroduces heterogenety across sectors n the cost of collectng taxes, as t s more costly to collect taxes from ndustres whose technology favors small frms, even when the tax authorty optmally uses the polcy nstruments at ts dsposal (ncludng both the fxed fee that mrrors the fxed per-frm admnstratve costs and, as modeled below, a threshold sze for beng subject to tax). In ths manner, we endogenze nter-ndustry dfferences n admnstratve costs and obtan a tradeoff between these costs and the deadweght cost of taxaton that s smlar to the tradeoff studed by Ytzhak and Wlson: ndustres wth many small frms are lkely to exhbt relatvely hgh admnstratve costs n tax collecton, ncreasng the lkelhood that they should be exempted from taxaton. 3. A Model wth Heterogeneous Frms Wthn an ndustry, frms typcally dffer sgnfcantly n sze. For example, small grocery stores and large supermarkets both sell food. Gven our assumpton that admnstratve costs have an mportant fxed per-frm component, t follows that t mght be optmal to exempt small frms n an ndustry from taxaton. As prevously noted, the explct or de facto exempton of small frms s a wdespread phenomenon n tax systems. Thus t s especally mportant to have a theoretcal framework that wll enable the rgorous analyss of the optmal structure of such polces. The treatment of frm heterogenety n the model we develop s nspred by Hopenhayn (1992a, b) and Meltz (2003). The papers by Hopenhayn examne statonary equlbra for a stochastc model n whch frm-level productvty shocks follow a Markov process, generatng a pattern of entry and ext by compettve frms. Meltz examnes the steady-state equlbrum for a model n whch monopolstcally compettve frms learn ther productvtes mmedately after enterng the ndustry. Our model s a statc verson of the Meltz model, extended to nclude taxes, except that we follow Hopenhayn by assumng that frms behave compettvely. 17 In the model developed below, frms that are ex ante dentcal choose whether to enter an ndustry (takng nto account expected profts and taxes), competng away expected profts to zero. Followng Meltz (2003), however, ncurrng the cost of entry 17 An explct steady-state analyss s a straghtforward extenson of our model, assumng that the goal of tax polcy s to maxmze steady-state welfare. 10

12 enables a frm to ascertan ts productvty, and t chooses ts output accordngly. Thus, the model endogenously generates varaton n frm sze, and also allows frms to endogenously adjust both ther entry decsons and output choces n response to tax polces (ncludng any exemptons or cutoffs). These features of the model consttute sgnfcant advantages relatve to alternatve approaches to modelng frm heterogenety. For nstance, the tradtonal domnant frm model that has been extensvely used n the ndustral organzaton lterature typcally nvolves a sngle large frm that exercses prce leadershp, surrounded by a compettve frnge of small prce-takng frms. Whle ths model also entals heterogenety n frm sze, ths heterogenety s generated by the exogenously-mposed assumpton that one of the frms n the ndustry s domnant n the sense of choosng ts prce frst (see, e.g., Kydland (1979, p. 358)). Thus, t s unclear how the structure of frm sze n such a model would respond to varatons n tax polcy. 18 We frst descrbe the behavor of frms, and then turn to the government s optmal tax problem. Because we are nterested n how frms wthn a gven ndustry should be taxed, at frst we drop subscrpts dentfyng goods and focus on a sngle ndustry. In partcular, we assume that the government has chosen the consumer prce for ths ndustry, q, fxng demand at X(q), and we solve the sub-optmzaton problem of maxmzng net revenue, gven q. If revenue were not maxmzed, then t would be possble to move to a dfferent tax system that created a budget surplus that could be passed on to consumers through a welfare-mprovng reducton n q. The optmal vector of consumer prces s then nvestgated n Secton 6. In partcular, we derve a modfed nverse-elastcty rule for how the consumer prces on dfferent goods should be chosen. Buldng on our prevous model, assume agan that all frms ncur the same fxed cost to enter the ndustry, denoted c e, but now allow that the convex varable cost functon dffers across frms: c (y, ) for a type-φ frm, where φ s an ex ante unknown productvty parameter that takes on values over an nterval, [φ l, φ h ], wth a densty functon that s strctly postve at each value wthn ths range. 19 The value of φ cannot be dscovered (by the entrepreneur) unless the frm enters, although ts dstrbuton 18 In addton, an earler lterature on the effects of taxaton and regulaton wth an nformal sector allows for frm heterogenety (e.g. Fortn, Marceau and Savard, 1997), but typcally also nvolves ex ante exogenous dfferences among frms. Moreover, ths lterature does not derve optmal tax rules, as we do n ths paper. 19 Unless otherwse stated, φ l and φ h can be taken to be mnus nfnty and plus nfnty, respectvely. 11

13 (characterzed by cdf F(φ) and pdf f(φ)) s known ex ante, and there s an unbounded mass of dentcal potental entrants, each possessng ths dstrbuton. By assumng a contnuum of frms, we ensure that ndustry output s non-random, although (ex ante) any sngle frm s output s random. Each frm chooses ts output only after ncurrng the fxed cost, c e. We assume that a hgher value of φ decreases margnal costs: c ( y, ') c( y, ") y y for φ' > φ'' and all y. Thus, a frm s chosen output rses wth φ. In addton, we assume that c (y, ) s bounded from below by a postve number at all postve y, reflectng a fxed cost ncurred f any postve output s produced (n addton to the entry cost c e ), whch we assume for smplcty to be constant across frms. For a frm facng unt output prce p and fxed fee b, profts, calculated gnorng the fxed entry cost, are gven by ( p, b, ) py( p, ) c( y( p, ), ) b, (5) where the supply functon, y(p, φ), s obtaned by maxmzng profts. These profts are ncreasng n φ. Once a frm has entered the ndustry, t ncurs no addtonal costs f t exts the ndustry before producng (.e., c(y, φ) dscontnuously drops to zero at y = 0). Thus, t wll ext f the φ t draws s too small to yeld non-negatve profts; that s, f π(p, b, φ) < 0. Although ext entals no costs, the entry cost c e s sunk and cannot be recovered. For the subsequent analyss, we assume that there always exst some frms whose productvty s low enough that they ext. 20 Now consder ntroducng a tax-threshold level of output, y*, such that frms wth output equal to or less than y* are not taxed and thus receve the consumer prce q per unt of output. Frms wth output above y* pay the fxed fee b and are taxed at the rate t on (all) ther output, n whch case they receve the producer prce p (= q t) per unt of output. Thus, we nvestgate the optmalty of tax schedules that exhbt a dscontnuous 20 A frm that exts thus knows ts productvty. Ths knowledge, however, does not affect the results n any way, as frms are assumed not to have the opportunty to reenter (and because frms wth suffcently low productvtes that ext wll have no ncentve to reenter n any event). 12

14 jump n tax lablty as output rses above y*. 21 The assumpton here s that the government s unable to observe productvtes drectly, and must therefore base ts tax on observed output, whch we assume can be observed by the tax authorty wthout cost. 22 If y* s suffcently low, then no frm wll be wllng to produce untaxed output, because a frm cannot cover ther fxed costs f ts output s too low. As y* ncreases, however, eventually some frms producng above y* wll be tempted to lower ther output to y* to escape taxaton. Hgher levels of y* can nduce some frms to choose outputs below y*. But there wll stll be frms that produce at y*, but would produce above y* f the tax break at y* were not avalable. Fgure 1 llustrates the ncentves facng two such frms, whch dffer n ther productvtes. The optmally-taxed outputs for these hgh- and low-productvty frms are located where ther margnal cost curves, MC h and MC l, equal the producer prce p. The hgh-productvty frm ncurs a loss n producers surplus of area I+II+III from the drop n output from y h to y*, but ths drop s offset by the elmnaton of the tax burden once y* s reached. The low-productvty frm ncurs a smaller loss n producers surplus, gven by area I, because ts margnal cost curve s hgher by assumpton. Thus, t too reduces output to y*. There s then bunchng at output y* of frms wth productvtes wthn some nterval. Moreover, ths bunchng elmnates the producton of outputs between y* and some hgher output, gvng us the mssng mddle proposton dscussed n the ntroducton: Proposton 3 (the mssng mddle ). Under an optmal lnear tax system wth a cutoff y* that nduces some frms not to pay taxes, there exsts a y** > y* such that frms wth suffcently hgh productvtes produce outputs above y**, but no frms produce an output between y* and y**. 21 The case n whch there s a dscontnuous jump n tax lablty at the threshold s not necessarly unrealstc. Indeed, thresholds for VAT regstraton typcally operate n ths manner, although some governments mtgate the dscontnuty by applyng a lower rate over some range; see the dscusson n Keen and Mntz (2004, pp ). 22 In realty, observng output s not completely wthout cost, although we are confdent that observng output s less costly than observng, for example, profts. A more complete model of the process would consder an enforcement system that audts output and presumably deters understatement. We beleve that addng ths feature to the model would not fundamentally alter the paper s results. 13

15 q= p+t MC l p I III MC h II y* y l y h Fgure 1 Thus, the economy contans small frms and large frms, but s mssng frms wth ntermedate levels of output. Frms that would be producng these outputs nstead reduce ther outputs to y* to elmnate ther tax burdens. The lowest taxed output that frms are wllng to produce, y** n Proposton 3, leaves the margnal frm ndfferent between y* and y**: qy* - c(y*, φ) = py** - c(y**, φ) - b (p, b, φ). (6) Solvng ths equalty for φ gves us the lowest productvty possessed by frms producng taxable output, defned as a functon of prces, y*, and the fxed fee b: φ**(q, p, b, y*). Wth ths notaton, the total output of taxed frms s T Y M h y( p, ) f ( ) d. (7) **( q, p, b, y*) where M s the number of frms enterng the ndustry. Ths output and the total output of untaxed frms are non-random, because there s a contnuum of frms. Unlke the prevous model, M s no longer the number of frms that actually produce n the ndustry, because frms ext f there productvtes are too low. 14

16 Consder now the determnaton of untaxed output. Let φ m (q, y*) denote the lowest productvty of actve untaxed frms. If y* s not too hgh, there wll be no frms producng below y*. Then ths lowest productvty wll be determned by the zero-proft requrement, qy* - c(y*, φ) = 0, n whch case t declnes wth q and y*, notng that hgher y* reduces the neffcent reducton n output requred to acheve tax-exempt status. But at hgher values of y*, some frms maxmze profts at untaxed outputs below y*, n whch case small changes n y* have no effect on the value of φ m (q, y*). Instead, ths mnmum productvty s determned by the requrement that maxmzed profts n the absence of taxes equal zero: π(q, 0, φ) = 0. In addton, the mnmum productvty among bunched frms, denoted φ*(q, y*), s the productvty at whch y(q, φ) = y*, whereas φ*(q, y*) = φ m (q, y*) f no frms produce below y*. In ether case, total untaxed output may be wrtten U Y M *( q, y*) y( q, m ( q, y*) ) f ( ) d **( q, p, b, y*) y * f ( *( q, y*) ) d, (8) where the frst ntegral dsappears f y* s reduced to the pont where no frm desres to produce below y*. M s determned by the requrement that total supply equal demand: Y U + Y T = X(q). (9) To fully demonstrate a mssng mddle, we must set up the government s optmal tax problem and characterze cases where t s ndeed optmal for some frms to produce untaxed output. As dscussed above, we frst fx q and fnd the tax system that maxmzes government revenue. The Lagrangan for ths problem s: L = M h ( q p) y( p, ) b A f ( ) d (10) **( q, p, b, y*) 15

17 + X(q) - M *( q, y*) y( q, m ( q, y*) ) f ( ) d **( q, p, b, y*) y * f ( *( q, y*) ) d h y( p, **( q, p, b, y*) ) f ( ) d *( q, y*) h **( q, p, b, y*) + ( q,0, ) f ( ) d qy * c( y*, ) f ( ) d ( p, b, ) f ( ) d c. m ( q, y*) *( q, y*) **( q, p, b, y*) The Lagrange multpler,, multples the market-clearng constrant, and the Lagrange multpler,, multples the zero-proft constrant. These constrants account for the three types of frms that do not ext: those below the cutoff (f there are any), whch pay no taxes; those at y*, whch also pay no taxes; and those that produce n excess of the cutoff and are therefore taxed on ther output. The control varables are the producer prce p, whch determnes the tax t = q p; the fee, b; the number of entrants, M; and the cutoff, y*. The zero-proft constrant has an mportant mplcaton: because frms sales revenue qx(q) must equal producton costs plus tax payments, the maxmzaton of tax payments net of admnstratve costs s equvalent to the mnmzaton of total producton costs plus admnstratve costs. To depct ths mnmzaton problem, we can wrte total producton costs as a functon of the number of taxed frms, C(M T ), where the tax parameters are optmally chosen to acheve ths number M T ; then the optmal M T mnmzes C(M T ) + AM T. Ths s an unconstraned maxmzaton problem because the C(M T ) reflects not only the adjustment to the producer prce p to acheve zero profts (for the gven consumer prce q), but also the entry of frms to equate supply wth demand. If a cutoff s desrable, t should be set so that rasng t enough to reduce the number of taxed frms by a unt causes producton costs to rse by an amount equal to the savng n admnstraton costs: - dc/dm T = A. Note that producton plus admnstratve costs are mnmzed only n a secondbest sense, gven the lmtatons on the avalable tax nstruments. The second-best neffcences nherent n a cutoff rule wll be studed below. If we could dentfy frm productvtes and mplement a productvty cutoff rather than an output cutoff, then C+AM T could be lower. e 16

18 Ths mnmzaton problem suggests a defnton of deadweght loss from taxng any partcular good that generates frm-level admnstratve costs: t s the excess of C+AM T over the mnmum producton costs, where the latter would be acheved n ths compettve economy by settng y* equal to zero and taxng all frms at the same rate. An mportant nsght s that t s not generally optmal to smply mnmze producton costs when collectng taxes ncurs admnstratve costs. Returnng to the revenue-maxmzaton problem, the frst-order condton for M shows that the Lagrange multpler s the rato of revenue, net of admnstratve costs, to output, whch we denote T e : R X e T. (11) The basc dea s that f there s an exogenous unt ncrease n output X(q), then the number of frms n the ndustry can be ncreased to satsfy ths output expanson, and the resultng ncrease n tax revenue, T e, measures the socal gan. In the absence of a cutoff, T e would equal the output tax rate, t because the fxed fee b would equal admnstratve costs A, n ts role as a Pgouvan tax. We show n an addtonal onlne appendx, however, that the optmal t exceeds T e when frms do take advantage of a cutoff. 23 Intutvely, the hgher output tax, coupled wth a lower fxed fee, makes relatvely low levels of taxed output more attractve to frms, dscouragng some of them from bunchng at y*. In other words, the fee s now no longer solely servng the role of a Pgouvan tax, but s also beng used to control bunchng. Gven these conflctng consderatons, we are unable to sgn the dfference between b and A when there s an output cutoff, although the excess of t over T e lmts the extent to whch b can exceed A. Identfyng ths sgn s not, however, needed for our subsequent results. 4. Mssng Mddle or Isolated Bottom? The mssng mddle dentfed n Proposton 3 suggests an economy wth a szable number of small frms and large frms, but no ntermedate-szed frms. But 23 The onlne appendx s avalable at: See n partcular Proposton A.1, whch also shows that b = A wth heterogeneous frms and no output cutoff. Proposton A.2 states that t > T e under a suffcent condton that seems unlkely to be volated. 17

19 another possblty s that the level of the output cutoff used to exempt frms from taxaton s so low that only a small number of frms take advantage of t, generatng a sze dstrbuton of frms wth ntermedate- and large-szed frms and a few much smaller untaxed frms, representng an solated bottom of the sze dstrbuton. The next proposton shows that ths sze dstrbuton would never occur under an optmal output cutoff. Proposton 4. Startng from a welfare-maxmzng tax system wthout output cutoffs, ntroducng a cutoff for frms n a gven ndustry must lower welfare f the resultng set of untaxed frms s suffcently small. The basc dea here s that f only a few frms can gan by choosng to produce untaxed output, then they cannot beneft much. Start wth a cutoff y* that s so low that no frms produce y*, but then ncrease y* untl those actve frms wth the lowest output, y 1, are now ndfferent between contnung to produce y 1 and nstead reducng ther outputs to y*. The benefts of the tax exempton for these few frms must be offset by the proft losses they ncur to reduce ther outputs to untaxed levels. In terms of Fgure 1, f MC l s the margnal cost curve for one of these frms, then the proft loss from reducng output to y* approxmately equals the beneft of the tax reducton once y* s reached, II + ty*. Thus, the movement of frms to untaxed output s not generatng a rse n expected profts for the ndustry, and the government s therefore unable to rase taxes on frms above the cutoff wthout necesstatng a rse n the consumer prce to satsfy the zeroexpected-proft requrement. But wth some output now produced by untaxed frms and total output fxed at X(q), there must be a reducton n total taxed output, even after we account for the entry of frms nto the ndustry needed to keep supply equated wth demand after some exstng frms reduce ther outputs to y*. Ths declne n the tax base lowers tax revenue. 24 As prevously explaned, welfare cannot be maxmzed f revenue s not maxmzed, gven the consumer prce q. 24 Wth p and b ntally set to maxmze tax revenue n the absence of a cutoff (subject to the requrement that profts equal zero, gven q), the envelope theorem tells us that the welfare effect from adjustng y* to nduce a margnal number of frms to produce untaxed output does not depend on whether p and b also change by margnal amounts (but changes n p and b wll affect the lowest level of y* at whch some frms are wllng to produce untaxed output). 18

20 As the cutoff level s ncreased, frms ntally producng slghtly above y 1 now choose to reduce ther outputs to y*; and frms that were ndfferent between y* and y 1 now obtan hgher profts at the hgher y*, snce ther producton dsadvantage from producng at y* declnes as y* rses. 25 Thus, the cutoff starts to rase expected profts for the ndustry, enablng the government to ncrease ts taxes on frms above the cutoff wthout causng the consumer prce to rse. Whether the hgher taxes more than offset the loss n revenue from the ext of more frms from the tax base wll depend on ther tax payments relatve to ther admnstratve costs. If ther tax payments do not exceed admnstratve costs by much, then exemptng them from taxaton wll have lttle effect on the government budget, so the hgher taxes on frms above the cutoff wll result n a budget surplus, whch can be used to rase welfare. Thus, we are able to prove: Proposton 5. For a gven level of admnstratve costs, f the tax revenue collected from an ndustry n excess of these costs s suffcently small n the absence of an output cutoff, then the welfare-maxmzng tax system wll nvolve a cutoff that s hgh enough to nduce a postve measure of frms to produce untaxed output. Thus, the mportance of admnstratve costs relatve to tax collectons s the crtcal determnant of whether there should be an output cutoff. Increasng the cutoff generates addtonal profts for exstng untaxed frms, allowng the government to rase taxes on frms above the cutoff wthout necesstatng a rse n the consumer prce to keep expected profts equal to zero. But a hgher cutoff reduces taxable output, whch tends to reduce tax revenue. If the average net tax on output, T e, s small enough (but postve), however, then ths second effect wll be unmportant, so addtonal revenue can be generated by rasng the cutoff, at no cost to the consumer. Ths addtonal revenue can then be dstrbuted to the consumer through a reducton n the consumer prce. Note that ths welfare gan requres that the cutoff be hgh enough to nduce a szable number of frms to produce untaxed output. Consequently, Proposton 5 lends further support for the optmalty of the mssng mddle. In partcular, a welfaremprovng output cutoff nduces a szable number of frms to opt out of the tax system by lowerng ther tax rates, so that there are many small untaxed frms as well as some large 25 No frm wll choose to produce below y* untl y* s ncreased to a suffcently hgh level. 19

21 taxed frms, but there s a mssng mddle of ntermedate-szed frms. Although Proposton 5 places an upper bound on values of T e under whch a welfare-mprovng output cutoff must exst, ths upper bound s not necessarly small. 5. A Modfed Inverse-Elastcty Rule When there are no untaxed frms producng n any ndustry, the standard nverseelastcty rule derved n Proposton 2 holds. In that case, only the demand elastcty matters because rasng the consumer prce alone has no mpact on the producer prce receved by frms, the level of whch s fxed by the requrement that expected profts equal zero. Wth output cutoffs, n contrast, ths rule no longer apples. Untaxed frms receve the consumer prce, so supply elastctes matter. Thus, to state a new rule, we must frst defne the supply elastcty of untaxed output of good : U Y U q Y q U 0. (12) For ths defnton, we hold fxed the number of frms producng untaxed output and consder only the margnal mpact of the prce they receve (.e., the consumer prce) on ther output. A change n q, and any accompanyng changes n the other tax varables, wll also generally change the share of frms that produce untaxed output, denoted F **. We defne ths share elastcty as follows: F df dq ** F q ** 0 (13) The change n ths share drectly affects the net revenue obtaned from taxng good. Lettng R denote ths net revenue, we defne the revenue elastcty, R dr F **. (14) df ** R 20

22 The addtonal onlne appendx demonstrates that ths revenue elastcty s negatve (Proposton A.2). These three elastctes all enter our modfed nverse-elastcty rule. Agan let denote the consumer s margnal utlty of ncome, and B denote the margnal value of government revenue. Wth ths notaton, we now state the new rule as follows: Proposton 6. Assume that t and b are optmal for each good, gven exogenous (optmal or not) output cutoffs. Then the average net tax rate for each taxed good satsfes the followng modfed nverse-elastcty rule: 26 e T q X 1 U Y X U B R F. (15) Ths s a rule for the average net tax on output as a percentage of the consumer prce an average ad valorem net tax. Intutvely, t s the net tax rate that matters, not the gross rate, because a rse n output not only generates a socal gan n the form of addtonal gross tax revenue, but also a socal cost n the form of addtonal admnstratve costs. Recallng that t > T e (Proposton A.2 n the addtonal onlne appendx), ths rule places lower bounds on the output tax rates. The modfed nverse-elastcty rule tells us that not only should, ceters parbus, taxes be low on goods wth hgh demand elastctes, but, other thngs equal, they should also be low on goods wth hgh supply elastctes for untaxed output. In contrast, supply elastctes only matter n the Ramsey model when the standard assumpton of constant returns to scale s replaced wth decreasng returns, mplyng postve profts. In the current model, the assumpton of free entry mples constant returns to scale at the ndustry level, and zero expected profts. But taxng output at a hgher rate dstorts not only demand decsons, but t also dstorts supply decsons by ncreasng untaxed output at the expense of taxed output. The supply elastcty reflects ths latter dstorton, because the postve mpact of a hgher consumer prce on the supples of exstng untaxed frms 26 We lmt ths rule to taxed goods because we have seen that not all goods are necessarly taxed n the presence of admnstratve costs. 21

23 crowds out taxed output, through a reducton n the number of frms enterng the ndustry. But the hgher consumer prce reduces taxed output through a second avenue: frms producng taxed output fnd untaxed output more attractve, causng some of them to swtch. The share elastcty accounts for ths latter consderaton, and ts mportance depends on the net revenue elastcty, whch s shown n the addtonal onlne appendx to be negatve (Proposton A.2): the movement of a margnal frm from taxed to untaxed output lowers revenue. Thus, goods wth hgh share elastctes should, ceters parbus, be taxed at relatvely low net rates. Note, fnally, that a good wth a relatvely hgh share of output that s untaxed, gven by Y U X should have a relatvely low average tax, all else equal. The reason s that the supply elastcty for untaxed output becomes more mportant n the modfed nverseelastcty rule as the untaxed-output share rses. Now consder the case where the government uses a cutoff for some ndustres but not others. Wthout a cutoff, only demand elastctes enter the nverse-elastcty rule. Wth a cutoff, the supply-related elastctes also enter, and they all contrbute to a lower tax rate. Thus, we have shown: Proposton 7. Assume that t and b are optmal for each good, gven exogenous (optmal or not) output cutoffs. If taxed goods and j have the same demand elastcty, but the government uses a cutoff rule only for, then the optmal average net tax on output s lower for than for j: e T T j < q q j e. Presumably, a cutoff rule s more lkely to be used n ndustres wth relatvely hgh admnstratve costs. If ths s the case, Proposton 7 tells us that the ndustres wth the hgh admnstratve costs tend to have the lower average net taxes. In other words, any addtonal gross tax payments are not fully coverng the hgher admnstratve costs. The ntuton s that, although a cutoff rule lowers the total cost of taxng a sector s frms, t ncreases the margnal cost of so dong because a hgher tax rate causes both neffcent supply and demand responses. 22