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Journal of Public Economics29 (1986)263-279.North-Holland SOME ANALYTICS OF TI{E LAFFER CURVE JamesM. MALCOMSON* SO9 !NH, UK Uniaersityof Southampton, Southampton ReceivedMay 1983,revisedversionreceivedJanuary1986 It is shown that, in a generalequilibriummodel with one privategood, one public good, labour propertiesof the Laller curve do not necessarily and an income tax, certain widely-assumed hold. For well-behavedfunctionalforms it may not be continuousand may not havean interior maximum.Its slopedependson technologyas well as on the tax elasticityof labour supply.For cerlain technologies,a more negativeelasticitymay imply a more positiveslope.Moreover,the relevant tax elasticityis a generalequilibrium one which may differ in sigrr from the widelyquotedpartial equilibriumone. l. Introduction Since its reincarnation in the guise of the Laffer curve, the idea that increasesin averagetax rates lead first to an increaseand then to a decrease in tax yields has played a large role in the popular discussionabout the size of the governmentsector.Discussionof the Laffer curve has also filtered into the more academicliterature.See,for example,Beenstock(1979),Buchanan and Lee (1982),Canto, Joines and Laffer (1978),Feige and McGee (1983), Fullerton(1982)and Lambert(1985),who havemadeuseof the concept,and Atkinson and Stern(1980),Hemmingand Kay (1980),Hutton and Lambert (1980)and Mirowski (1982),who have respondedto, and criticized,the use that has beenmade of it. Even the more academic literature, however, has taken many of the propertiesof the relationshipbetweenaveragetax rates and tax revenuesfor granted. Although the relationship representedby the Laffer curve is, in principle, a general equilibrium one, much of the discussionof its form has been within a partial equilibrium context. And Canto, Joines and Laffer (1978), Feige and McGee (1983) and Fullerton (1982), who use general equilibrium models,chooseparticular functionalforms for technology,factor supplies, and product demands without serious discussionof the consequences of those choices for the implied shape of the Laffer curve. The fI am grateful to John Hutton, Peter Lambert and two anonymous referees for valuable comments on earlier versions of this paper. Thcy are not, of course, responsible for any errors that remain. C047-2721186153.50 @) 1986, Elsevier SciencePublishers B.V. (North-Holland) zffi J.M. Malcomson,Analyticsof the Infier curue purposeof the presentpaper,therefore,is to investigatethe propertiesof the Laffer curve within a simple generalequilibrium framework without imposing particular functional forms for technologyand preferences that may rule out a priori certainpossibilities. The essenceof the Laffer curve is simpleenough.The curve representsthe relationshipbetweenthe averagetax rate imposedby a governmentand the total tax revenue.Tax revenueis the product of the ave(agetax rate and the tax base. A common argument for the shape of the Laffer curve runs as follows.If the averagetax rate is zero,then so, obviously,is the revenuefrom that tax. If the averagetax rate is 100 percent,the tax revenueis also zero since no rational agent would generatea base for a 100 percent tax. In between, therefore, as the average tax rate is increasedfrom zero to 100 percent, the tax revenuemust first increase,then reach a maximum, and hnally decrease, so giving the Laffer curve the generalshapeshownin fig. l. total tax revenue I O average tax rate Fie. I That shape has been used as the basis for a number of results. In an economywith a singletax, it implies that increasingthe averagetax rate to increasethe size of the governmentsectorhas diminishingreturns and must eventuallylead to a fall in governmentrevenue.By estimatingfor the United Kingdom a curve on which this shape is imposed, Beenstock(1979) has arguedthat, 'evenin terms of a narrowly conceivedrevenueobjective,the tax systemis more or lessat its limit'. Also on the basisof this shape,Buchanan and Lee (1982) argue that a revenuemaximizing governmentwith a time horizon limited to the next electionwill tend to set too high an averagetax rate evenin terms of its own objective.More generally,it is the existenceof a J.M. Malcomson, Analytics of the Lafler curue 265 downward sloping portion of the curve which has led to the 'supply side' argument that cutting the average tax rate would generate a sufficient increasein the tax baseactually to increasetotal tax revenue. These conclusionsdependcrucially on the Laffer curve having the shape assumed.But, even given the endpointsof zero tax revenuewith averagetax rates of zero and 100 percent, the general shape in fig. I depends,as Atkinson and Stern(1980)and Hemmingand Kay (1980)have noted,on the curve being a continuousfunction.One might also add (though no more will be said about this here) that the assumptionof zero tax revenue at an averagetax rate of 100 percentmay be too strong. Economic activity may just go undergroundto join the black economyas long as the probability of detection is less than one. The governmentwill then collect revenuefrom that part that is detected.Feigeand McGee (1983)have exploredfurther the role of the black economyin the contextof the Laffer curve. In discussingwhat it is that causesthe curve to turn down, the general presumptionin both the theoreticaland the empirical literature seemsto be that it is the disincentiveeffect of higher taxation on labour supply that is likely to be the dominating influence.To isolatethe role of the tax elasticity of labour supply, the present paper, therefore, uses a simple general equilibrium model, indeed,about the simplestmodel that allows the general equilibrium implications to be analysedwhile still retaining a genuinerole for governmentpolicy.r For this model, it turns out that the nature of technologyis an equally important determinantof the slope of the Laffer curve. Indeed, for certain technologies,a more negative tax elasticity of labour supply may imply a more positive slope becauseof generalequilibrium effectson wagesand profits. Moreover,the relevanttax elasticityis a general equilibrium one taking account of consequentialchangesin wages and profits, not the more usually quoted partial equilibrium one from which it may even differ in sign. Finally, the taffer curve may not be continuous and may not have an interior maximum. It could slope upwards at every averagetax rate lessthan 100percentwith a discontinuityat 100percent. 2. The basicmodel To maintain the spirit of previous discussionsof the Lalfer curve, the present analysis will use a model with just a single tax and, given the lln addition to using restrictive functional forms, some of the studies using general equilibrium models have provided no useful role for the additional revenue raised-by-a tax increase. In Canto, Joines and Laffer (1978), tax revenue is merely passedback to the (implicitly identical) individuals as lumpsum transfers, thus eliminating income effects.In Fullerton'(1982), any government surplus ov€r actual 1973 expenditure is also passed back to individuals as Iump-sum transfersand any deficit financed by lump-sum taxation. For obvious reasons,models in which additional tax revenue has no real use, or in which lump-sum taxes and subsidiesare permitted, are not ideal for analysingthe consequences of increasesin distortionary taxation. 266 J.14.Malcomson,Analyticsof the Lafer curoe importance of income taxation in most developedcountries,this tax will be taken to be an income tax. There is no formal diffrculty in extending the model to include many types of taxes - the Laffer curve then just depicts total government expenditureas a function of the averagetax rates for each type of tax - but for the points to be made here no additional insights seem to be offeredby this, For simplicity,individualswill be taken to be identical and, to provide a genuinerole for government,there will be one public good as well as one private good. The former will be treatedas a pure public good which is produced by the private sector but purchased,and provided to consumers,only by the government.Again, there is little formal difhculty in extending the number of such goods. Since with identical individuals the purely redistributivepoliciesof governmentsbecomepointless,thesewill be ignored here. All economicagentsare assumedto be price takers. Let there be N identical individuals, each with a twice differentiableutility function U(ho-h,x,!), where h0 is the maximum possiblehours of labour supply, h the actual hours of labour supply (so lro-lr is hours of leisure),x the amount of the private good consumed,and y the amount of the public good provided by the government.The individual's budget constraintsare given by +(1 -t)(wh*z- A\+ A, for wh*nZA; * : l *fxo o otherwise, +wh+n, (1) where x0 is each individual'sendowmentof the private good, r the marginal rate of income tax, A the amount of income exemptfrom tax, w the hourly wage rate, and n each individual's shareof the profits of firms. [In (1), any net salesof the private good are treated as a disposalof assetsand so not subjectto incometax.l Let (t(1-r)w, (l -t)n,y,tA) denoteeachindividual's supply function for labour hours obtainedby maximizingthe utility function subjectto the constraint(1), with f treatedas zero if no tax is payable.For most of what follows, this supply function will be taken to be differentiable for wh+T =,4, that is, as long as taxesare actuallybeing paid. The production side of the economyis representedby the profit function of the private sector iirms. This is denoted[I(p,w), where p is the price at which the public good is sold to the government,the price of the private good being normalized at 1. II{p,w) is assumedto be twice differentiable. Then by standard duality theory, IIr(') is the firms' supply function for the public good and -II*(') their demand function for labour hours. The propertiesthat this profit function must satisfyin order to representa valid technologyare that Ir(') be non-negativefor positive p and w and strictly convex.It is thereforeassumedthat il ro>0; ['*]0; il ooil nn- n'*or}, (2) J.M. Malcomson, Analytics of the La/fer curue 267 where the argumentsp and w of these secondderivativesare dropped for notational simplicity. Each individual's share of the profits of firms is then givenby n=n(YN. This paper considersthe only situation of practicalinterest,namely that in which tax allowancesdo not exceedtotal income so that some taxes are actuallypaid. Then the government'stax revenueis given by tN(wh* n - A) : tlLl(p, w)- wII "(p,w)- N.4)1, (3) the equality following from the propertiesof the profit function mentioned above.Its expenditureis py: pn e(p,w). (4) Budget balancerequiresthat thesetwo be equal. Equilibrium requiresthat supply and demand be equal in the markets for the private good and for labour hours but, becauseof Walras' Law, only one of these needsto be consideredexplicitly. Hence, the generalequilibrium prices of the economy are given by - il n(p,w): Nf [( I - t)w,(l - t)nfu,w)lN,Il o(p,w),tA], (5) p[I o(p,w):tLII(p, w)- wII *lp,w) - N A]. (6) The Laffer curve plots total tax revenueas a function of the tax rate r when allowanceis made for the generalequilibrium effectson prices and quantitiestraded.zDenote this by R(f). The concernhere is with the shapeof this curve as r changesand the simplest way to investigate that is by consideringits derivative.Since governmentrevenueis given by the righthand side of (6), the slopeof the Lafler curve is given by the derivativeof the left-handside of (6), namely R'(t):p[I r*(p,w)dwIdt+UI r(p,w)+p[I or(p,w))dpIdt. (7) To evaluatethis requiresthe comparativestaticsof the system(5) and (6) in order to obtain expressionsfor dw/dr and dpldt. Thesecan be obtained in 2Traditionally, the Lalfer curve is treated as a relationship between total tax revenue and the averagetaxrate. Here f is the marginal tax rate, which equals the averagetax rate only if l:0. Permitting A to be nonzero provides an increasc in generality with little increase in complexity. Since the average tax ratc is then not a parameter but an endogenous variable, it is convenient here to treat thc Laffer curve as a relationship between total tax revenue and the marginal tax rate. Provided u{ is held constant, as it is here, the average tax rate increaseswith the marginal tax rate so the slopes of the two dillerent versions of the Laffer curve always have the sane signs. 268 J.M. Malcomson, Analytics of the Laffer cunte the usual way by total differentiation.To simplify the notation, the arguments of the profit function will be dropped where this causesno ambiguity and dhldt will be usedto denotethe total derivative(that is, taking account of the consequontialchangesin the price of the public good, the wage rate and, hence,pro{-rts)of the supply function of labour hours with respectto the tax rate. Then the total differentialof the system(5) and (6) is If*,,,dw+[Inpdp: -N(dh/dr)dr, (8) ItwII nn+ pII .Jdw + [( 1- r)Ile + twfl no* pII on]dp :ln -wll* -N.,4ldr. (e) Let / denote the determinant of the matrix of coefficientson the left-hand sidesof (8) and (9), that is Dn, I "^ --!il** pil pil tlII twII np $ .,r+ lt*n nn* o+ ool : plfl nnn oo- il',oJ + ( I t)n eIInn >0 , ( 10) the sign following from the restrictions(2), i.e. the strict convexity of the profit function, and the assumptionthat the public good cannot be a net input so that I/o must be non-negative.The solution of (8) and (9) for dpldt and dddr is given by dpldt:ln**(n -wiln- N,4)+ N(rwll*,u+pII*)dhldt)lA, dwI dt :{ - N(l - t)n e+ Mn *e+ pn rofdhldt -i l *o (n --w n n -NA) \lA. (11) (12) Substitutionof theseinto (7) givesthe slopeof the Laffer curve as R'(r): t(II- wII.- NA\lp(fr,oil**-nt*r)+n en*nf + rN[wp(II ,oil n*- rl,*r1+ n ,1pn_o* wlr,n*)f dhldt\I a. ( 13) SinceA>0, the strict convexity of the profit function and the fact that the public good cannot be an input ensurethat the first term on the right-hand side of (13) is necessarilypositive whenevertax allowancesdo not exceed total income so that some taxes are actually paid. Hence,a necessary(but J.M. Malcomson,Analyticsof the Lffir curae 269 not, of course,suffrcient)condition for the Laffer curve to have a negative slopeis that the secondterm be negative,which requiresthat either dhldt or its coeflicient (which dependson the profit function and, hence, on the technologyof the economy),but not both, be negative. It has been widely recognized, in the literature, indeed it is the basis of most of the discussionof the Laffer curve, that what happensto working hours as the averagetax rate is increasedis crucial for its slope- though not all the discussionhas adequatelyreflectedthe fact that it is the general equilibrium effecttaking accountof consequential changesin pricesand wages, rather than the partial equilibrium effectholding pricesand wagesconstant, which is the relevanteffect.More will be said about this in section4 below. What, as far as I am aware,has never been questionedis the sign of the coefficientof dhldt in (13).The presumptionimplicit in the literature is that this coeffrcientis always positive so that the greater the reduction (smaller the increase)in hours of work as the result of an increasein the marginal tax tate t, the smaller the gain (or the greaterthe loss)in the government'stax revenue.As will be shown in the next section,however,that is by no means necessarilythe case. 3. Technologyanrl the Laffer curve The purposeof this sectionis to investigatethe sign of the coeffrcientof dh/dt in eq. (13) above.To do this it is instructivelirst to considerseparately the different terms in that coeflicient.The term wdIIoJInn-II3,o) is necessarily positive becauseof the strict convexityof the profit function.The term IIrwIInn is non-negativesince il*n70 by the strict convexity of the profit function and IIo, the output of the public good, cannot be negative.ilno, however,is the effectof a changein the wagerate on the supply of the public good which will be negativeas long as labour hours are the only input. Hence,the overall sign of the coefficientis a balanceof positiveand negative terms. To provide somefurther feelfor what is involved,it is usefulto expressthe coefiicientof dhldt in termsof elasticities.Define eyP=pneefn e; eto: p[I nofII n. ern=wII.of II o; ELn?w[Innf IIn; (14) These are, respectively, the elasticity of the supply of the public good with respect to its own price and with respect to the wage rate, and the elasticity of demand for total labour hours (denoted t) with respect to the wage rate and with respect to the price of. the public good. Then the coeflicient of dhldt J.M. Malcomson, Analytics of the Lafer curtse in (13)can be written tNlwp(n rJI *n- ilr*) + II o(pII*o* wII**11 : tN Il pII n[(eroe"r, - e ulyn) * elo * r1",]. (15) Note that If.*<O so that the term in squarebracketson the left-handside of (15) has the opposite sign to the term in square brackets on the right-hand side. The term in parentheseson the right-hand side must be negativefor the profit function to be strictly convex,as must er-. The only positive term on the right-hand side of (15), therefore,is e1o.Hence,for the left-hand side of (15) to be negativerequiresEa, to be suflicientlylarge but, of course,too large a value would violate the condition for convexityof the profit function. A smaller absolute value for €r", would make violation of this convexity condition less likely but thesetwo elasticitiesare not independentsince they both involve the same cross partial derivative of the profit function. As a result,it is not easyto sign the overallexpressionat this level of generality. Some examples,however,will serveto illustrate that the assumptionthat the coeflicientof dhldt in (13) is positive,which is implicit in the discussion in the literature on the Laffer curve, is not necessarilytrue. In view of the fact that this coeflicient can be expressedconvenientlyin terms of elasticities as in (15), it will be no surprisethat a profit function basedon the CobbDouglasform (suitablymodihedto fit the detailsof the model) both makesit easy to evaluate and results in it having an unambiguoussign. Consider, therefore,the profit function I I ( p , w ) : A p o w - b * ( B - C w ) , w h e r eA , B , C , b > 0 , a - b > l . (16) The restrictions on the signs of the coeflicients.4, a, b, and C are to ensure that the public good is always an output and labour always an input. The additional requiremett, a-b> 1, is then necessaryand suflicient to ensure that the profit function is locally convex. It is, in fact, then also globally convex for all positive linite p and w. The term (B-Cw) is used to modify the CobfDouglas function in order to allow the private good to be an output. Everything that follows holds for B:C:0 but in that case the requirementsthat the profit function be convex and linearly homogeneousin all prices are suflicient to ensurethat the private good is an input for every positive p and w. That would not actually be inconsistentwith the model used here since allowance has been made for the possibility of a strictly positiveendowmentof the private good but it seemsmore in the spirit of the discussionto assumeB and C strictly positiveso that both the private good and the public good can be outputs. The implication of this is to give the economy,in effect,an additional endowmentB of the private good at a cost J.M. Malcomson,Analyticsof the Lalfer curue 271 Cw. ln this way, (16) provides a tractableform which satisfiesglobally the conditionsfor a valid profit function and which allows the private and public goods to be outputs and labour hours to be an input. For this function, straightforward differentiation and manipulation establishesthat the expression in (15), which is the coe{Iicientof dhldt in (13), is always zero, independentof the valuesof p and w. That is to say, the slope of the Laffer curve is completelyindependentof how hours of work adjust in responseto a changein the rate of income tax. That slopeis then necessarilypositiveat all tax ratesgiving a positivetax yield. A natural way to generatemore generalexamplesis to use the standard flexible functional forms which provide second-orderapproximationsto any arbitrary twice dilferentiableprofit function. Typically, thesegive less clearcut resultsbecausethe sign of the coeflicientof dhldt in (13) then dependson the actual valuesof the price p and wage rate w which prevail in equilibrium. With such a profit function,to find out if the Laffer curve has a positiveor a negativeslopefor any given tax rate r one thereforeneedsto specifya utility function and solve the whole model for p and w as functions of r. To illustrate the kind of resultsthat can be obtained on the basisof the profit function alone,however,considerthe translogform: ln II(P,w): aot a, ln P* urln w w)2f. + +lPL tlt d' * 2Ft tln p ln w+ B 22Qn One problem with this form is that, short of a drastic simplificationsuch as assuming fq:0, all i, i, and so reducing it to a CobFDouglas profit function, no restrictionson the parameterswill ensure that it is globally. convex, but this is a disadvantageit shareswith all the other standard flexible functional forms for profit functions which have variable,as well as fixed,inputs. SseDiewert (1974)on this. For this reason,it can be considered only as a local approximation. With the translogform it is convenientto work with the input and output sharesdefinedby sy: pn p(p,w\lII(p,w); s"=wII n(p,w)[I(p,w). ( 17) These shares are, in general, functions of p and w but, for notational simplicity, the argumentsare suppressed.Linear homogeneityof the profit function in all prices ensuresthat the shares of all inputs and outputs (including the private good which is left implicit in the functional form) must sum to unity but, unlike in the caseof translogcost functions,they do not necessarilylie in the interval [0,1]. The sharesof inputs are negativeand the shares of outputs may be greater than l. In terms of these shares,the J.M. Malcomson,Analyticsof the Lffir curue elasticitiesin (1a) are givenby €yp:sy- 7+Brrlsr; sL n :sz-1 *|zrl si l rrw:sr *|nlsr" r Lp:sy*Fr zlsr . Then the condition for local convexityof the prolit function is (sy- 1*811/sr)(s"-1+ firrlrr)*(sr* prrlsl)@1*f nlsr)<0. (18) The sign of the coefficientof dhldt in (13),evaluatedusing the elasticitiesin (15),is givenby - sign [(s, - 1+ f , r/sr)(sr- | + Brrl s")- (s,* F, zls")(sr* fl sn) nl + sy+f ,rlsr*sr.- 1+ fr,rls"l. (1e) Clearly, the signs of thesein generaldependon the valuestaken by the shares and I have found no restrictions,apart from the Cobb-Douglas special case with constant shares,which sign them independentlyof those values.Consider,however,the point at which sy: -sr-1, which impliesthat the private good is an output with share 1 also. In view of what was said above, these are legitimate values for the shares.Then the profit function is strictly convex provided - f ,r(2 + fl zz)+(l-f ,r ) t <0, (20) which holds for - JlTr'(z+ Fzr\f<(1-Frr)<+ Jlfrtr(z+Frr)). Since one would normally expect frn,fzr>0 and frz50 and these conditions are sufficient to ensurethat the elasticitiesabove have the appropriate signs,let $ * Fd: + r/l0rr(2+Fr)f- 6, fr">U(2+P2), where o < 6 <J r p , r ( 2 + B r ) ) - r (21) J.M. Malcomson, Analytics of the Laller curae 273 which ensurethat the profit function is strictly convex at this point and B* has the expectedsign.The expressionin (19) then reducesto - sign{6,- 6(1+ 2\/ lpr r(2* f , r)l)+ t / tflr rQ+ fl, )f - (2+ F 1,,7} . For d closeto zero,this takesthe oppositesignt" {r/l|rr(2+ f ,r)l-(2+ flrz)|, which is positive for p1r>2*F* and negativefor ftr<Z*fzr, neitherof which is inconsistentwith (21). Hence,the coeffrcientof dhldt in (13) can be eitherpositiveor negativeat the point sr: -sr:l dependingon the relative sizesof pt, and prr. These examplesserve to show that theoretical grounds alone are not sullicient to establishthat a decreasein working hours as a result of a tax increasenecessarilyreducesthe revenuegainfrom that tax increase.Whether or not it doesis an empiricalquestion,the answerto which cannot simply be assumed. 4. Taxation and working hours The purpose of this section is to consider the relationship betweenthe marginal tax rate t and working hours. Three main points will be made about this. The first is that, even with a twice differentiableutility function, working hours need not be a continuousfunction of the marginal tax rate, which, in itsell is sufficientto underminethe continuity of the LafIer curve. Although the possibility of discontinuity in labour supply has been recognized in the theoretical literature, the point is emphasizedhere becauseit has been neglectedin much of the discussionof the Laffer curve and a number of the argumentsmade [for example,the necessityof a downward sloping portion and the argumentsof Buchananand Lee (1982)] dependon the continuity of that curve.The secondis that the partial equilibrium effect on working hours of a changein the marginal tax rate is not necessarily oppositein sign to the effectof a changein the wagerate net of the marginal tax rate. This is well known in the labour supply literature [see,for example, Hausman (1981)] but is reiteratedhere becauseit too has been neglectedin some of the literature on the Laffer curve. Feige and McGee (1983),for example, adopt a labour supply function which depends only on the marginal wagenet of tax even though the incometax in their model is not a proportional one and, in citing empirical evidence,some authors quote the effectof changesin wagesafter tax without mentioning that this may differ from the effectof changesin the marginaltax rate. The third point is that the generalequilibrium effecton working hours of a changein the marginal tax rate taking accountof consequential changesin prices,wagesand the proaisionof public goods,which is the relevant effect for discussionsof the shapeof the Laffer curve [see eq. (13)], is differentfrom, J.M. Malcomson,Analyticsof the LoJlercurue and may indeed have a different sign from, the partial equilibrium effect holding prices,wagesand the provision of public goods constant.The latter involves only the labour supply function, the former the comparativestatics of an equilibrium at which demand equalssupply and, hence,dependson the labour demandcurve as well. The distinction is important in consideringthe implications of empirical studiesfor the shapeof the Laffer curve. Empirical studies of labour supply measurethe partial equilibrium effect,yet these are widely quoted lsee, for example, Beenstock (1979)l in discussionsof the Laffer curve without this distinction being made. To see these points, consider the utility maximizing choices of the individualsin the model of section2. Substitutionfor x in the utility function from the budget constraint(1) for the caseconsideredin the previoussection in which wh+n>,4 (which must hold for there to be any tax revenue), allows the maximization to be written ma xU (fto-h ,xa +(1 -r) ( wh+ n- A\+ A,y\ h s.t. O<hsho. Q2) This gives rise to the following first-order conditions, in which ,l is the multiplier attached to the constraint h<ho and the inequalitiesbracketed togetherare complementary. _t)uzo_l:ll,,._l=3} * u,(.)+w(l (23\ Denoting by U^. the secondderivativeof U(.) with respectto h, the secondorder condition for an interior maximum can be written (Jm=U r r(.) * 2w(I- r)U, e(.)+ w21t- 4zLt22() 50. (24) In what follows it will be assumedthat the inequality in (2a) is strict. To illustrate the first point, let the endowmentxo be zero and the utility function have the Stone-Gearyform: ln (ha- h) + aln [( 1- t\(wh* n - A) + A- x*l + b ln (/ - y*), (2s) for someconstantsa,b,x*,y*>0.Then, if the solutionfor /r hasO<h<ho,it takes the form: 6 : {aho- (n - A)lw + {A- x*)/[(1- t)wf\l( + a), {26) J,M, Malcomson, Analytics of the Lafer cunte so that sign0hl0t:sign (x* - A). So, for interior h, h may increaseor decreaseas the tax rate increases.But it is immediatelyclear from (25) that for t:l the optimal value for h is zero; hence,in ths casewherex*>A,therc must bo a discontinuityat /:1. Then the Laffer curve too will be discontinuousat t=l and may have no interior maximum. That the effecton the supply of hours of an increasein the tax rate r is not the opposite of that of an increasein the wege net of tax is also obvious from (26) sincew andt do not appearsolely in the form (1 -t)w. There are, however,some more generalpoints to make about this. Standard manipulations give the partial comparative static effect of a change in f on hours worked for interior ft as )hl 0t : - {wU2(.)- (wh* n - A)lU, zO- }o(I - t)u zr(.)l V( - Uil. Q7) The partial comparativestatic effect of a changein the wage after tax is given by ahlALwQ- t)f : {u z() - hlU, r(')- }r(I * t)U zzOf}l( - Ur ). If z happensto equal A (for example,if they wereboth zero),then 0hl0t: -wAhl7lw(1-41. But, as long as unearnedincome is included in the tax base or the income tax has an exemptionlevel (or, more generally,is progressivein some other way), this simplerelationshipwill not hold in general. One can say more about the relationshipbetweenthe two. Note that AhlA{wQ- r)}: Ur()l(- U rr)* h 1hl1xo, (28) where the first term is the substitutionelfect (necessarilynon-negative)and the secondthe incomeeffect.Also. )hl0t : - w{Uz(')lF U^^)+ lh + (n- A)lwlAfl0*o\. (2el Since positive tax revenuerequires wh+n- A>0, it is clear that, with a positiveincome effectin (28) (that is, leisureis an inferior good), the signsof (28) and (29) must be opposite.With a negativeincome effect,however,that is not necessarilythe case.With n<A,(28) positiveimplies(29) negativebut both could be negative.With n) A, (28) negativeimplies (29) positive but 276 J.M. Malcomson,Analyticsof the L$fer curue both could be positive,The economicrationale for this is straightforward. Consider n> A. A changein the tax rate t then has a bigger income effect than a change in the wage rate after tax becauseit affects all sources of income, not just earnedincome,so that when the income and substitution effectsare different in sign the income effect may outweigh the substitution effectin one caseeven though it does not in the other. This warns against assumingthat a positively sloped labour supply curve necessarilyimplies a negativeresponseof working hours to an increasein the tax rate. It also warns againstassuming[as is done by Feigeand McGee (1983)]a functional form for the supply of labour hours which dependsonly on the marginal netof-tax wage when the tax structureis progressiveor suppliersof labout have unearnedincome. The final point to be consideredin this sectionis the relationshipbetween the partial equilibrium effectof a changein the tax rate t on working hours for giuen profits and wage rate and the general equilibrium effect taking accountof consequential changesin profits and the wagerate. Again consider an interior solution for h, Then, from total differentiationof (23) with w, n and y treated as functionsof r and with (11) and (12) used to substitutefor dpldt and d{dt, we get that the generalequilibrium effect dhldt has the form: dhldt: [( - Uila)U}t+ P]ll(-U h)a+a.), (30) where a : {( 1- t)n elu tz - w(I - t)U zz}+Nfee[ U r : - w(1- t)U rrf\ x (twII *. * pil n)+ N{( I - t\U z- [,,o[ U r s - ]e(1- r)Uzrl] x {( t - iln e+ twII noI pil oo}, fl : - (I - wil n - N,4){II**[ (t - t)n olu r z - w(1- t)U2] I N * II rolUn- lv(l - r)Uz.ll + II.e( I - t)Ur- n *nlU rt- w(l - r)Uzsll). A and ( - Ur,,) are positive from the strict convexity of the profit function and the second-ordercondition for utility maximization,respectively, but a and f can, for different profit and utility functions, take on a variety of values which make it impossibleto determinethe relative signs and magnitudesof dhldt and Ahldt at this level of generality. As an example to show that dh/dr can have the opposite sign to 0hl0t, consider the following case.Let the utility function be weakly separablein y, J.M. Malcomson, Analytics of the Lafer cun:e 277 that is, with the form: U(ho-h,x,y)=Il*LY(ho *h,x), yJ, and let leisurebe a.normal good, Then [U '.(')-t{,(1 -t)U 2 3 (') ]:0; lU ...O- w( l - t) U22Of >0. ( 31) Moreover, let the substitution effect dominate the income effect so that 0hl0t<0, as is assumedin most discussionsof the Laffer curve, and let the technologybe representedby the modified Cobb-Douglasprofit function of the previoussection,eq. (16).Then a :(1 -t)yN fa -t(t+b )l -t)u zzgf\*ro, x{wu"(.1-!4*n*n-A)tu,z(.)-w(t f the sign followingfrom the fact that 0<r<1 and a-b> 1, which ensurethat the first term in square brackets on the right-hand side is positive and, together with (27) and 0hl6t<0, that the term in bracesis positive as well. AIso, under theseassumptions, (-uillaht\t +p: prnl+-|+!t+ u]fu,1.11w - b - t)Ur n(.) - w(l - t)u + p2by3(a zzgJ/(aNtwz), which, in view of (31) and (a-b-1)>0, is certainlypositivefor fS+. This and a>0 insertedinto (30) ensurethat dhldt>0 for any tS} even though 0h/0t<0. of course,with this technologywe alreadyknow from the previous sectionthat the sign of dhldt is irrelevantfor the slopeof the Laffer curve.It providesa neat and tractableway of illustrating that the general nevertheless equilibrium effectof a changein the tax rate t on working hours may have the oppositesign to the partial equilibrium effect.Hence,empirical estimates of the sign of the partial equilibrium effect obtained from estimatesof the supply function for hours cannot simply be assumedto give the sign of the generalequilibrium effect. 5. Summary and conclusion This paper has investigated the properties of the Laffer curve for an J.M. Malcomson,Awlytics of the Laffer cunse economy with one private good, one public good and labour under an income tax. It has been shown that, under perfectlyreasonableassumptions, certain propertiesof that curve which are widely assumedto hold may not, in fact, hold for that economy. The Laffer curve is a generalequilibrium relationship betweenthe average rate of tax and total tax revenue.As such, even when labour is the only factor and the only tax is an incometax, its shapedependson more than the labour supply function. It is equilibrium labour hours, determinedby both the demandfor and supply of labour, which is the relevantquantity variable for the part of the income tax arising from earnedincome and the response of this to a changein the averagerate of tax may have the oppositesign to the partial elasticity of labour supply with respectto that averagetax rate. But what happensto the wage rate and to profits as the averagetax rate changesis also important for the tax base. For perfectly reasonabletechnologies,the interplay of all theseinfluencesmay result in the responseof working hours to an increasein the averagetax rate having no effectoq or even an inverse effect on, the revenuegain from that tax increase.Even for well-behavedutility functionsand technologies,the Laffer curve may not be continuousand may have no interior maximum. These results highlight the limitations of the assumption that the Laffer curve has the generalshapeillustratedin fig. 1. The shapeof that curve for any real economy must be determined from empirical evidence,not from theory. But one conclusionof the presentpaper is that one must be careful, more careful than much of the existing literature, in interpreting the significancefor the shapeof the Laffer curve of empirical evidenceabout, for example,the labour supply function. Another conclusionis that empiricalwork aimed at determiningthe shape of the Laffer curve needs to use a model sufficiently rich to allow for the effectsdiscussedhere. For example,the labour supply function must allow the sign of the effect of an increasein the marginal tax rate to differ from that of a reduction in the wage net of tax if the economyhas a progressive income tax or wage earnershave unearnedincome.The technologyassumed must be sufliciently rich to allow the general equilibrium effect of a tax changeon labour hours to have either the samesign as, or the oppositesign from, the partial equilibrium effect and to allow income tax revenue to respondpositivelyor negativelyto changesin equilibrium hours. Imposing a priori restrictions on these effects,as can easily be done by apparently innocent choicesof functional form, may lead to incorrect conclusionsabout the shapeof the Laffer curve. References Atkinson, Anthony B. and Nicholas H. 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