Journal of Pubhc Economics 48 (1992) 361-375. North-Holland
Competitive tax theory in open economies Constrained inefficiency and a Pigovian remedy
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 lorm of the hesl decentralwcd remedy is derived.
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
Cr~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 Quwns 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. kirst. 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;
marginal rate of transformation; community js 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,(.) -rj-.\:,,( .) rcprcscnt js saving function. With thcsc assumptions wc arc in a position to dsfinc the equilibrium concept, called a campctitivc tax equilibrium.
Ik/iriifioti. 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
and, for each j,
(b) (r,,, .x2,) solves max,,~,x, ( ~c,(.u,,.u,;~,):r.u,+s~$r?.,+n,(r+r,)+rri~);
(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 communitys problem is sufficiently well behaved, the first-order conditions are given by the constraints and by
tj=$/; ri_ij , ( 1 i'j whcrc urj rcprcscrits the maryinal utility of second-period consumption (irzj= I,~ and tfUj= 11~~). The term i.J14~, is js 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. ifund 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
the change in value of savings and the change in profit income (from the
envelope theorem, cnj/?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:
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 governments
choices afkct utilities through their indirect effect on market-determined variables (here. the equilibrium price of capital) when the tax structure is...