Journal of Pubhc Economics 48 (1992) 361-375. North-Holland

Competitive tax theory in open economies Constrained inefficiency and a Pigovian remedy

R. Krelove*

Received February 19XY. revised version received hlarch IYY I

In an economy wth multiple governments. each is constrained to raise revenue for public expenditures ustnp dastortwnary tares on a directly mobile tax base. A povcrnment is said to bc competitive if the IOLII CO~I of IIS decisions, including the excess hurden. is perceived to be mternalired. The competitive euullibrium allocations are not constrained eflicient in wncral: there is another ce~ ,;f dlxtortickry t;lnes and associated public expenditures for which individuals in the wrnomy are better OIT. II is shown that the source of the Llilure can intcrpretcd as a m~s~np market. and the l’orm of the hesl decentralwcd remedy is derived.

alI be

1. Introduction

In this paper I invcstigatc an economy with multiple govcrnmcnts whose

economics arc linkd by trade in factors and goods, whcrc each is con-

straincd to raise rcvcnuc for public cxpcnditurcs employing distortionary

t3xcs on * ;I directly mohilc tax base. Specifically I consider ;I simple

intcrtanporal moclcl whcrc each jurisdiction taxes the capital cmploycd by

firms within its borders. A govcrnmcnt is said to bc competitive if the total

costs (and bcnclits) of its decisions. including the cxccss burden, is pcrccivcd

to bc intcrnalizcd. The compctitivc assumption rcquircs, among other

conditions, that govcrnmcnts arc net-of-tax price tukcrs for traded

C‘r~rrespo~nlo~~,~, IO: R. Krelove. Smlon Fraser University, Burnahy. I&C. VSA IS6. Canxlx *I am grateful to two anonymous referees , ~1s well as seminar participants ;II the University of

Toronto and Quwn’s University at Kingston for helpful comments and suggestions. An e:Irlicr version ws presented ;II the North American Summer Meetings of the Econometrw Society and appeared as University of Toronto Discussion Pap no. 881X. Wildasin (IYXY) has indepen- dently investigated the Jcs~gn of mterventions in a related model; he amrlyses the ekes of 3 subsidy tu 13x rate incrs;tses (abstracting from the issue of tin;mcing the subsidy program). Such il subsidy is not the bcs~ intervention in the tarpang sense. The views in this paper are mine. They do not neccswrily reflect the ofliwl position\ of the Intsrncltwnal Monetary Fund.

0047 -2727,YZ SM.00 I’ IYYZ --lilscvier Science Publishers I$.V. All rights reserved

commodities.’ In this context 1 mean to show two things. After describing the model and the equilibrium in section 2. I show first in section 3 that the

competitive equilibria are not constrained ellicient in general.’ This result is

counter-intuitive to some extent, since there is no obvious marginal incentive

problem. Analogizing to the fundamental theorem of welfare economics. the assumption that the burden of a tax, including the excess burden. is

internalized by a jurisdiction taking world prices as fixed appears to be

exactly what is needed to provide the correct marginal incentive to each government to make the appropriate second-best tradeoffs. This intuition breaks down. however, because governments’ decisions together affect world

prices, and these prices, through their effect on private maximizing behavior.

alter the possibilities for transferring resources between private budgets and

government budgets. Competitive communities ignore these additional

transfer possibilities in their decisions. One way to intcrprct the failure is that traded goods prices do not provide

:I sufficient set of signals to coordinate the decisions of independent competitive govcrnmcnts. and the second thing I do is ask what would constitute ;I just-sufkicnt set of signals. This question is taken up in section 4. where it is shown that at ~1 constrained eflkicnt allocntion :L tcrm cmcrgcs

that can be intcrprctcd as the shadow value of (traded) capital ;IS tax base to

the economy. It is then argued that the best remedy that prcsorvcs the

dc~cntrali/ation of decisions involves pricing govcrnmcnts net ‘consumption’

of the tas base. With lhis remedy thcrc arises ;I corrcspondcncc ~C~WL'CII

conipctitivc equilibria ;IIKI constrained cfkicnt :Illoc;itions. antI not (ncccssar-

ily) with full efficiency: there remains the cfficicncy loss that is ;I consequence

of the fact that revenue must be raised with distortionary taxes.

Many rcscarchcrs’ in local public finance hvc been conccrncd with

characterizing oulc0nics in environments where jurisdictions tax mobile commodities. The prcscnt paper represents a11 improvement over these

results in several respects. k’irst. the compctitivc case has not itlwitys been clearly distinguished from various noncompetitive environments where

governments have opportunities to mulct lhs foreigner, with the result that

‘An alloca[ion arwciatcd with ;L WI of (di\rortion;lry) I:II ralrs is non constrainsd eflicien~ if thcrc cxisrs anolher WI of pax r;Iks and aswciakd ullocrwn under which all individuals arc hctrcr off.

‘For cxxmplc. Gordon ( 19X.1). 0a1cs and Schwab ( IYX)o. Srarrell (19X0). WildaGn (IYXY). Wilson ( IYM). and %drow and M~csAowsk~ ( I9Xb).

sources of failure have been confounded.’ Second, simplifying assumptions

common in the literature obscure some important aspects of the problem,

leading to an incomplete understanding of the nature and extent of the

failure.5,b Third, the problem of the design of interventions for competitive

environments has not yet been addressed systematically.

2. The model and the competitive tas equilibrium

There are J jurisdictions (equivalently. communities) in the economy,

usually indexed j. Each community has a fixed population, oj; for simplicity assume II~= I. all j. There is one private good and two periods. labeled I and

7 Preferences are representable by the intertemporal utility function _.

I~j(_u,j..~2j.,~j). increasing in its arguments. The resident of j can either use his

endowment 1, in the first period for consumption xii. or lend it at an

after-tax price of savings, denoted r. Private firms in community j borrow an

amount kj in period I to produce output l;(kj) in the second period. The

marginal product of capital. denoted ,/‘;. is positive, and diminishing. J’; <O.

each j. Assume also that j;(O) =0 and that the marginal product bccomcs

Iarpc as the capital input approaches jrcro.

Communities ;Irc link4 by the costless mobility of capital and of the

private good in each period. NorIn;Ilix the producer price of the private

good to I in cxh period. Then lirm hohitvior in j is charoctsrizctl by the choice of k 10 maririiix l,/;(k) --q/i!. cxh j. whcrc q, is the known, pilraIllclriC producer price of capital in j. I.ot n,(q,) = IllilXk (/i(k) -q,ki tlcnotc

the Inxsiniizcd value of prolit in ,j.

In cxh community there is ;I govcrnmcnt that Icvics taxes and uses the

rcvcnuc to provide the public good. It is ilSSllIllCd for simplicity that the

comtnunity can commit in period I to ;I unit tax, I,. on capital input in the

community. Then the producer price of capital in j is given by q, = r + tj. as

the consumer price of capital is cclu;Ili/.cd across communities because of the

mcjhility assumption. If f, is the t:Ix rate, the community raises rcvcnuc r,k, in

period 1 when the capital input of domestic firms is k,. The public good is

produced from the private consumplion good in period 2 at a constant. unit,

364

marginal rate of transformation; community j’s budget constraint is then given by the inequality gjsr,kj+ ~~~ where T, is a lump-sum transfer to community j (which equals zero at the equilibrium).

Each community is assumed to be competitive. in the sense that it perceives that the total costs and benefits of its decisions are internalized. To make this operational, I make three assumptions. First, each community takes the global consumer price of capital, r, as fixed. independent of its own actions. Thus the community believes that private firms face a perfectly elastic supply of capital at the prevailing net-of-tax return. Second. the community government is benevolent in this its objective is the maximization of the level of utility of its residents. Third, since it is supposed that the government takes account of the effect of its decisions on n,(.), it is assumed that this surplus accrues entirely to the domestic residents. There are several equivalent institutional structures to rationalize this last assumption; for example, it can be assumed that the domestic firms are owned internally. Equivalently. it can be assumed that there is some fixed factor in each community. owned by the resident. that is the recipient of the surplus.’

Private consumer behavior in j is dcscribcd by the choice of lcvcls of private consumption in the bud@ sot to maximize utility, given prices and govcrnmcnt choices.’ Dcnotc the maximized lcvcl of utility by v,(I,.,~,: r. HI,);

that is.

r~,(lj,g,;r,rti,)~rll;Ix (ir,(.\.,,sL,,~,):r.~, +.~,~r!‘j+rr,(r+f,)+ttl,). (1) I I . I 1

whcrc ni, rcprcscnts period-_ 7 lunlp-SUIII incomc (which equals zero through- out the analysis). Let s,,(r, .qi, I,, m,) rcproscnt the demand function for

period- I consumption in j, and Ict s,(.) -r’j-.\:,,( .) rcprcscnt j’s saving function. With thcsc assumptions wc arc in a position to dsfinc the equilibrium concept, called a campctitivc tax equilibrium.

Ik/i’riifioti. A cottfpditii~c rus c*c/lri/ihriirrtl (CTli) is a list (.u, j, x2, kj. s,), rcprcscnling an allocation for the cconoiny, a list of capital tax rates (r,),, and a price r, such that

(a) C,SIj+Cjkj=Cj).‘j;

and, for each j,

(b) (r,,, .x2,) solves max,,~,x, ( ~c,(.u,,.u,;~,):r.u,+s~$r?.,+n,(r+r,)+rri~);

365

(c) (rj.gj) solves max t.p rj( t. g; r, mj) subject to

(i) gstkj+sj and

(ii) kj solves max, (f,(k)-(r+t)ki.

Capital market equilibrium is imposed implicitly through condition (a) of the definition. It is an implication of the conditions (b) and (c) that material balance in (produced) second-period consumption is satisfied. and so does not need to be entered explicitly. Note that each community takes r as parametric in its program given by condition (c).~

Consider the choice problem for some community j. Associate the multiplier ~~20 with constraint (i) above. Assuming enough differentiability and that the community’s problem is sufficiently well behaved, the first-order conditions are given by the constraints and by”

tj=$/‘; “ri_i’j , ( 1 i'j

whcrc urj rcprcscrits the maryinal utility of second-period consumption (irzj= I’,~ and tfUj= 11~~). The term i.J14~, is j’s pcrccivcd social marginal utility of govcrnmcnt rcvcnuc. norm:l~iizcd by the social marginal utility of private incomc. Obviously 7j/~~tjg I always at the CTE. From (3). ;),/u2,> I for positive tax rates. [Then from (2) the marginal rate of substitution bctwccn the public pooJ and period-2 consumption is greater than the marginal rate of tr;lnsk)rmatiorl (which cclu;~ls I).] A ncccssary and suflicicnt condition for ~hc constrained incff&ncy result of the next section is that this strict incqu;tlity holds for some j at the CTE, i.c. that the pcrceivcd marginal dcadwcight loss of the capital tax, (yj- uzj)/uzj as usually dctincd, is positive. That this is ;I gcncral characteristic of the equilibrium follows from the rcstrictcd possibilities in the model for transferring resources bctwccn the private budget constraint and the govsrnmcnt bud@” It is assumed

“It is being assumrd that the community has good knowltxlg:e of prefercnccs to bc used for some decisions but not fur othrrs; in parlicular the knowledge of prcfcrenccs is nol used IO levy (oprimal) lump-sum I;WS. This is 3 standard assumption in optimal IU theory. often juslilicd by the clssumption rh;:! Ihe govcrnmcnt may hare god information concerning the distribution of IASICS and endowments but cannot dcnrify the characteristics of a particular individual’.

“‘NOW that the j tirms’ upid demand function has slope tX,/Cr,= I//;. ’ 'l'osi~ivc public goods provision (positive 13x rates) will bc ;L characwristic of the CTII for il

larpc SCI of prrferences. One contrary cast arise when period-2 consumption and the public good arc perfec! substiiutcs for all individu:lls [so utility in j is given by, say. u,( ‘) =h,( t,,J + rz, +,g,. with /I,(.) mucasing and coruve].

throughout that a CTE exists, that the equilibrium choices can be character- ized by (2) and (3). and that the equilibrium world price of capital is positive and locally unique and varies smoothly (with not all derivatives zero) in government decisions in a neighborhood of the CTE.” It is said that the tax structure is incomplrtr if yji tlZi > I for some j at the equilibrium.

3. Strictly improving perturbations

In this section I examine the effect on utilities resulting from marginal changes in communities’ decisions around a competitive tax equilibrium. The feasible changes are constrained by private sector budgets and behavior, in particular by the maximizing behavior of firms and households, and material balance. The goal is to identify the circumstances under which there exists a perturbation that results in a strict Pareto improvement. It is shown that such an improvcmont exists in general; thus the CTE is not constrained efficient.

Consider a differential change (dt,. d,~,)~ in communities’ decisions around a conipctitivc tax equilibrium, along with changes (drj)j in community govcrnmcnts’ lump-sum incumc. Through private behavior the change inducts a change in the allocation pivcn by (d.~, i, d_~~,. dk,),, and a change in the net-of-tax intcrcst rate. dr. The perturbation is called fcasihlc when it satisfies 1, T, SO. and maintains povcrnnicnt budget balance for all j. allowing for private sector responses. LA !I’(‘, r,, IN,) dcnotc the equilibrium lcvcl of lhc ohjcclivc (utility) in community j. Since the change in taxes and public cxpcnditurc lcvcls is along the budgets starting from the CTli. and the chnngc in private hchavior is along private hudgcts starting from a consumer optimum, it follows from the cnvclopc thcorcm that the marginal change has no direct first-order cffccts on utility. so the lirst-order change in utility is given solely by the indirect cffccts, through dr. and through the rcvcnue cffcct from dr,. That is to say. the change in utility in j. JII,, is given by

dlij= (7/I’ t'r

dr + (‘d dTj. ?Tj

The goal is to find fcasiblc changes that satisfy &cj>O for all j. From the regularity assumption there exists a fcasihlc change yielding dr #O. Then. the lemma below follows directly from cq. (1). using cj risOO, and using (71~j/?s,=;,j (from the envelope thcorcm):

LOWFla. There exists u />asible marginal chungr from the CTE satisflving

du, > 0, all j. if‘und only if

7 ;‘, y f @ (5)

Using the envelope theorem and eqs. (3). the form of a typical elemen ,t in

the sum in eq. (5) is given by

using the fact that ~I,,,~=II~~ evaluated at the equilibrium. and where

sj=(_rj-.~,j) is saving in j. The first term on the RHS of (6) is the wealth

erect of the change in the consumer price of capital. the difference between

(6)

the change in value of savings and the change in profit income (from the

envelope theorem, c’nj/?r = -k,). The second term captures the induced effect

on utility through ;I change in tax revenue (and hence public expenditures); this offccf is nonzcro when the tax structure is incomplctc.

Summing cqs. (6) over j:

(7)

The sum on (he R tIS of (7) is a non-negative-wcightcd sum (with weights summing to I) of numbers, 11~~/;‘~, ;I II no larger lhan one and at least one of

which is less than one when the tax struclurc is incomplete. Hence the sum is

less than one, and Ihl: expression in brackets is negative. It follows

immcdia~cly from the lcmm;l that an improvement exists; formally, it has been proved that:

An important role in the failure of the proof is played by (6). in particular by the second term in that expression, which captures how ;L government’s choices afkct utilities through their indirect effect on market-determined variables (here. the equilibrium price of capital) when the tax structure is incomplete. This second term can be interpreted as follows. The fundamental allocation problem in the model is that in each community there arc restricted possibilities for transferring resources across the two budget

constraints. from the resident to the government. What can be called the ‘technology’ of the transfer possibilities depends on r, through its effect on private maximizing behavior. A change in r affects private behavior in a way that alters the marginal possibilities for transferring resources with the available imperfect tax instruments. Using (3). the term can be seen to be equal to tj?kj/c’r, which is the induced tax revenue in j from a change in r. evaluated at the equilibrium tax rate. Moreover, the direction of the effect is the same in all communities. so the sum does not vanish [unlike the wealth effects, captured in the first term in (6). the sign of which varies in general across j]. The failure arises because a competitive community ignores the effect of its decisions on the technology of transfer through its effect on world prices.

Since the RHS of (7) is negative when the tax structure is incomplete.. it follows from eq. (4) that any change that induces a fall in r is an improvement. The type of change that would accomplish this in the normal case would involve an increase in tax rates on capital. However, income effects on saving and interactions bctwecn saving and the public good can produce counter-intuitive rtsults. To briefly invcstignte this, total diffcrcntia- tion of the capital market equilibrium equation yields the comparative statics of the modol:

~:urthcrmorc. the change, dgi, is constrained to maintain the j-budget. i.c. d,~,=dr~+(r,/J‘;) dr +(k,+ f,.:j’;) dfj, with ~,rj~O. For simplicity. consider the case whcrc the incomc effect on savings is zero and the public good and saving arc ncithcr eomplcments nor substitutes. Then (%/c’f,= -

1 /_/‘y& ( I/j‘: - t?s:jCr)] < 0, each j, where ?.s:/i7r dcnotus the (non-nugativc) compensated cfkct on saving. Under these circumstances a uniform increase. dr, in aII tax rates will lower r, resulting in an improvement. Savings and the capital stock fall; hcncc thcrc is ovcrsaving at the CTI:. It can bc shown that total tax rcvcnue rises with the uniform incrcasc in taxes, so that expcnditurc on public services c;ln rise in each community. It is important to note that the indicated change actually increases the aggregate deadweight loss asso- ciatcd with the tax system alone (since the gap between every individual’s intertemporal marginal rata of substitution and the marginal product of capital in any community widens); Proposition 1 establishes that this loss is more than offset by an efkiency gain in public good provision.

In the special case, prominent in the previous literature, where the supply of savings is fixed, the first-order conditions characterizing the CTE, eqs. (2) and (3). continue to hold. because from any community’s point of view the

size of its tax base is independent of the domestic savings rate. Then eq. (6) continues to hold. without change. Eq. (8) becomes drxi( I:fy) = - xj(dt,/f;), so that any improvement involves raising taxes on average. Of course in this case a coordinated increase in taxes does not necessarily imply an increase in the excess burden of the tax system.

I close this section with two corollaries to Proposition I. First. an improvement exists satisfying dsj=O. all j. if any only if sj--(;‘j/rrri)kj <O. all j. This is more likely to be the case when communities’ capital account surpluses tend to be small and preferences for public goods are strong. in that ~j/u2j is large at the CTE.” Second. it follows from Proposition I that there exists an improvement [that is, the sum in (7) is nonzero] even if only one jurisdiction taxes capital at the CTE. Hence an improving change in that lone nonzcro tax rate ” does not generate an external benefit or cost for other communities by altering the size of their tax bases (since their tax rates are zero) - that is, the failure is not the consequence solely of a ‘fiscal externality’. It is an implication of Proposition I that the conventional wisdom on this matter. namely that the tiscal externality measures the gap bctwccn the private and the social incentive to tax a mobile base. is incomplctc.’ 5

4. A Pigovian rrrndy

l’hc demonstration of constraioncd incficicncy in the previous section is constructive in that it indicates the information that is nocdcd to idsntify an improving direction. With this information. communities may agree to some change. or in some contexts the changes may bc imposed. In this section I consider the possihilitics for improving intcrvcntions that prcscrvc the dcccntralization of decisions: in particular the goal is to find the best

‘-‘Note th;lt since the KtiS trf (7) is nonrero. o utilit;lri:m welfare improvement without translixs alw;~ys exists when Ihe 1;1x structure is incomplete. Ilence the prohlcm arises because the price ch;mgcs have advercc wcdth elTcc~s for some j. In il model with more tradrablcs (in the present model there is onl) one r&live world price), il is more likely mu be ;~ble IO manipul;lte world prices IO engineer benelici:d wealth CITCCIS. so thal it is more likely that an improvemrnr without transfers exists.

“‘This assumes th;lt. in the USC where the wealth rlTcct of the price change does not have the s;Lrnc sign for all nont;~ning communities. a trxder dr, IO nontaring community j has utility due for the resident equal tu the siLe or the transfer (IO make it possible IO OITS~I adverse wealth rll~ts. iT any). This complication does not detract from the point beins made here.

“The conventional argument is that a community. when it considers raising its tax rate. ignores thal a higher lit\ T;LIC creates a re;II benefit for other communities by incrrasing the sire of their Ian bases as domestic capital migrales in response IO the higher tan rate. The demonstrations have been verb4 and intuitive; the argument can be found in. for example, Gordon (19X3. p. 5X3). Oates and Schwab ( IWX. p, 343). Wildusin (19X9. p, lY6). Wilson (IYX6. p. 303; lYX7. p. 837). and Zodrow and Mieszkowski (I986, p. 369). Starrett (19X8. pp. 187-1Xx) is a tcxlhook treatment of Ihe analogous problem for local sales taxes. whcrc individuals can choose the community of purchase.

intervention in the sense of the minimal dimension set of signals necessary to decentralize a constrained efficient allocation.

The point of view is that the source of the failure of the CTE can be identified with a missing market for the mobile tax base. That is. at a constrained eflicient allocation a number, ci. emerges which is interpreted as the shadow value of capital as tax base. By completing the markets (that is, in our model, by adding an extra signal corresponding to this shadow value in guiding communities’ decisions), a correspondence is established between the competitive equilibrium of this modified economy and the constrained ellicient allocations. Thus, the economist’s standard response to esternalities in competitive private ownership economies carries over to the present context. which contains two novel aspects: first the mode) has competitive governments using distortionary taxes (these distortions are not usefully thought of as standard technological externalities); and second the compari- son allocations arc not fully (Pareto) efkient. but only second best.

The strategy of the approach in this section is, first. to define the set of constrained cflicicnt allocations prcciscly, and then to characterize those allocations by the first-order conditions for a constrained maximization problem. A modiftcd cconumy is then dclincd whcrc thcrc is a compctitivc niarkct for the tax base. and a compctitivc equilibrium for this modified economy is charactcrizcd by the first-order conditions for each community’s constrained maximk~tivn problem. A comparison of thcsc two sets of first- order conditions cstablishcs a corrcspondcncc bctwccn the critical points.

I begin with scvcral definitions. The economy is as dcfincd in section 2 above. A d~sj~~~r~ for the economy is dcfincd as an allocation (.ulj. _s~,.~~~, k,),, ;I set of capital tax rates (tj),. ;I net-of-tax price of capital, r, and a set of transfers among community govcrnmcnts ( r,)j. A design is Jiwsihl~~ if it satisfies:‘”

(i) Cj.\.I,‘+C,x,5;C,J’,;

(ii) Cj Sj~;O;

illld, for each j:

(iii) (.r,,, s2,) solves niax,,, ~) 1 ~r,(.u,,.\‘~;,~,):r.~, +~,~‘r)‘,+f[~(~+t~)+~)~~:;

(iv) g,sfjkj+r. .tnd I’ ’

(v) kj SOIVCS max) ( J;(k)-(r+lj)k);

where n,(.) denotes the maximized value of profit in community j. A feasible design is said to bc ~~n.s~roirtc~cl eJki~( if there dots not exist some other

fcasiblc design whcrc all individuals in the economy arc strictly bcttcr off.

Constrained efficient designs will be characterized by the first-order con-

ditions for the problem of choosing a design to maximize the welfare

function LV[( u,Jx, j, xzi. s~))~]. increasing in its arguments, subject to the

feasibility constraints (i)Hv). Thus. there exist numbers (Oj)j, 8jz0. and a

number SgO [the normalized multiplier associated with the constraint (i)]

such that a constrained efficient design solves. along with the constraints, for

each j:”

and

“ri = oj

[ 1

1 _ (j c’sj

l'lj c’gj ’

lj = ii - I;,.f‘;c I - p,,.

cs c 3 = c ( I - [jj),Vj. j(‘r ,

(9)

(10)

(11)

whcro for each j. /I, = [I/O, + S c’s,/?rr~~]. and whcrc, as before. c’sTI~‘r dcnotcs

the (Slutsky) compensated effect of r on saving. The term I/O, is the social

marginal utility of (period-2) income to j. normalized by the social marginal

wclfarc of govcrnmcnl rcvcnuc. The term /ji is the sum of this direct social

benefit of incomc IO ,j plus an indirect bcncfit gcncratcd by the extra savings

out of income. wcightcd by S; /j, is thus closely rclatcd to Diamond’s ( 1975)

net social marginal utility of income. The diffcroncc is that 5 appears rather

than the tax rate. The lcrni CS is the shadow tax ralc on capital (saving); it

can hc inlcrprctctl as the shadow value of capital as tax base to the

economy. l;rom cq. (0) the marginal rate of substiturion difkrs from the

m;irgin;il ralc of lr;Insformaliori (cqi~al to unity) for two reasons. First. the

conlplcmcnrnrity or substitutability bctwccn saving and the public good may

offset or roinforcc the cxccss burden of the system of taxes; this cffcct is

cvaluatcd in each community at the shadow tax rate. CC. The other Lcrm, (I,,

can lit on eilhcr side of I. From eq. (IO) the lax rate in community j cquals (5

minus an adjustment that dcpcnds on the elasticity of capital demand in j

(capluring the los( Oulpul tluc to the tax iIS rates depart from uniformity)

and the valuation of private income in j relative to government rcvcnue,

cxprcssing distributional concerns. [From cq. (I I), when the substitution

efkct is not zero. the savings-share-weighted sum of the /Ii is less than I.]

The form of (I I) is familiar from optimal commodity tax theory, where now

the reduction in tht: economy’s saving along the compensated supply curve is

evaluated as if there was a common tax rate equal to d. the shadow value of saving as tax base.‘“.”

I turn now to a consideration of the possibilities for decentralizing a

constrained efficient allocation. First. define the list of numbers (c;i)j by

cfij=O. The numbers (h,), become part of the data of a modified economy, where hj is interpreted as community i’s endowment of ‘rights to the tax vase’. Each community in the modified economy now has an additional

decision, the choice of a quantity of rights to tax base, hi. subject to the constraint that it hold rights at least equal to its capital account surplus,

allowing for an adjustment equal to I;,. It is assumed that these rights can be bought and sold at a price. denoted p. An equilibrium for the modified economy differs from a CTE defined in section 2 solely by the presence of

this extra ‘market’; formally:

Dcjinitifm. A compt~titirx~ t0.v c~fptifihriw ji)r rllc rr&j/kf ccortom~ rdtrfir.c lo

(fij)j is a list (.Y ,,, .~2j.sj. lij. /I~)~. a list of capital tax rates (fj)P and prices (r, p) such that:

(a) ~j.~,j+~jkj~‘cj~*,:

(b) CjjljsCjh,;

and, for each j.

(c) (Y,,. szj) solves max.,, _ll [ ~(,(.~,..\‘,,,~~):f_\‘, +s,~r!.j+nj(r+lj)+“~j);

(d) (<LJj, fj, hi) solves 111;1x U.,.h 1%,(l,g; r. ftr,) subject to

(i) .r 5 rk, t I’( I;, - It).

(ii) hzk,-s,. and

(iii) lij solves maxI, :./;(I,)-(r+f)k).

Clearly the (.Y,,, s2,,g,, k,. [,)j and r associated with ;I competitive t;ix equilibrium for the modified economy is ;I fcasiblc design, with T,=

p(hj-(k,-s,)). I‘hc term (k,-.sj) is j’s capital account surplus or deficit. Condition (ii) requires that the community hold rights at least as large as its

capital account. Then (fij-/I,) is j’s net trade in rights to the tax base. and p(I;,-11,) is the value of this net trade.

An institutional structure consistent with the equilibrium concept is as

R. Kreloce. Competitive tax theory 373

follows. A central clearing house (perhaps a federal government or, in the

international context, a supranational agency) assigns endowments (t;j)j of

rights to communities. It collects data on supplies and demands (the size of tax bases and savings rates) and attempts to establish an equilibrium price

for rights. A community is then charged or receives a payment equal to the

value of its capital account surplus after allowance for its endowment of rights.”

In the modified tax equilibrium concept, agents are decentralized with two signals, the prices r and p. Note that the signal p is relevant only to the decisions of the competitive governments; all private agents (households and firms) are decentralized at price r and, at equilibrium, the taxes (c~)~ chosen

by the communities. Consider the maximization problem for community j in the modified

competitive tax equilibrium. The first-order conditions are given by the

constraints and by:

!'~i=;.~ 7

[ 1 1 _,!>j "Zj 'Sj

a n d

(12)

(13)

whcrc yj is j’s pcrccivcd normalized marginal utility of govcrnmcnt rcvcnue at

the modified compctitivc cquillibrium. A comparison of eqs. (12) and (13)

with cqs. (!I)-( I I) characterizing constrained efliciency yields:

What Proposition 2 asserts is that the net-of-tax price of traded capital

incorrectly reflects the social scarcity value of saving to the economy. Thus, traded goods prices, while suffkient to guide the decisions of private agents

in the economy, arc not suflicicnt to coordinate the decentralized decisions of govcrnmcnts. They must in addition face the shadow value of tradeablcs as tax base. An attraction of the approach is that an indirect means, the market, may be usad to discover the shadow value of the tax base. Accompanying

‘“The anulogy IO mclrketablc permits for. say. pollution is immediate.

this informational parsimony is the remedy’s consistency with the trageting

principle, namely the principle that the best remedy for a failure works

directly on the relevant margin. This property is of value for the design of interventions in richer models where communities make decisions on a wider number of issues. some of them with solely domestic content.”

5. Concluding remarks

The focus of this paper has been an important class of multiple-jurisdiction

models. purged of spillover effects and of any exercise of monopoly power by

governments in world markets. When each jurisdiction must pursue its goals

using distortionary instruments. the competitive equilibrium is not con- strained efficient in general, even though each perceives that the total costs of its policies are internalized. The example analyzed here strongly suggests that this is a general phcnomsnon; broadly, the result cxtcnds to any economy whcrc the sacond-best instruments in agprogatc dctorminc world prices. An

implication of this is that much of normative tax theory. and more gcncrally the theory of optimal policy. dcvclopcd for closctl economics is in fact partial equilibrium theory in multi-country cnvironmcnts, cvcn when govcrnmcnts

arc compctitivc. In addition, the nalurc of the best dcccntralizcd intervention for the

economy has been analyzed. It is intuitively plcasing that this takes a

familiar form: the simple atlditivc structure dcrivcs from the fact that wc can

assign to tradcablc commodities a shadow value as tax hasc. While thcrc arc

large gaps, informational and othcrwisc. bctwccn concept and implcmcn-

tation that still need to bc bridged. the analysis points the way toward an

intcrvcntion policy potentially applicable to the design of intcrgovcrnmcntal grant schcmcs in fcdcral countries and to lax h~lrmoiiiz~ltioii efforts in the

international context.

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