Journal of Public Economics 29 (1986) 263-279. North-Holland

SOME ANALYTICS OF TI{E LAFFER CURVE

James M. MALCOMSON*Uniaersity of Southampton, Southampton SO9 !NH, UK

Received May 1983, revised version received January 1986

It is shown that, in a general equilibrium model with one private good, one public good, labourand an income tax, certain widely-assumed properties of the Laller curve do not necessarilyhold. For well-behaved functional forms it may not be continuous and may not have an interiormaximum. Its slope depends on technology as well as on the tax elasticity of labour supply. Forcerlain technologies, a more negative elasticity may imply a more positive slope. Moreover, therelevant tax elasticity is a general equilibrium one which may differ in sigrr from the widely-quoted partial equilibrium one.

l. Introduction

Since its reincarnation in the guise of the Laffer curve, the idea thatincreases in average tax rates lead first to an increase and then to a decreasein tax yields has played a large role in the popular discussion about the sizeof the government sector. Discussion of the Laffer curve has also filtered intothe more academic literature. See, for example, Beenstock (1979), Buchananand Lee (1982), Canto, Joines and Laffer (1978), Feige and McGee (1983),Fullerton (1982) and Lambert (1985), who have made use of the concept, andAtkinson and Stern (1980), Hemming and Kay (1980), Hutton and Lambert(1980) and Mirowski (1982), who have responded to, and criticized, the usethat has been made of it.

Even the more academic literature, however, has taken many of theproperties of the relationship between average tax rates and tax revenues forgranted. Although the relationship represented by the Laffer curve is, inprinciple, a general equilibrium one, much of the discussion of its form hasbeen within a partial equilibrium context. And Canto, Joines and Laffer(1978), Feige and McGee (1983) and Fullerton (1982), who use generalequilibrium models, choose particular functional forms for technology, factorsupplies, and product demands without serious discussion of the conse-quences 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 valuablecomments on earlier versions of this paper. Thcy are not, of course, responsible for any errorsthat remain.

C047-2721186153.50 @) 1986, Elsevier Science Publishers B.V. (North-Holland)

zffi J.M. Malcomson, Analytics of the Infier curue

purpose of the present paper, therefore, is to investigate the properties of theLaffer curve within a simple general equilibrium framework without impos-ing particular functional forms for technology and preferences that may ruleout a priori certain possibilities.

The essence of the Laffer curve is simple enough. The curve represents therelationship between the average tax rate imposed by a government and thetotal tax revenue. Tax revenue is the product of the ave(age tax rate and thetax base. A common argument for the shape of the Laffer curve runs asfollows. If the average tax rate is zero, then so, obviously, is the revenue fromthat tax. If the average tax rate is 100 percent, the tax revenue is also zerosince no rational agent would generate a base for a 100 percent tax. Inbetween, therefore, as the average tax rate is increased from zero to 100percent, the tax revenue must first increase, then reach a maximum, andhnally decrease, so giving the Laffer curve the general shape shown in fig. l.

to ta ltaxrevenue

O I averagetax

Fie. I rate

That shape has been used as the basis for a number of results. In aneconomy with a single tax, it implies that increasing the average tax rate toincrease the size of the government sector has diminishing returns and musteventually lead to a fall in government revenue. By estimating for the UnitedKingdom a curve on which this shape is imposed, Beenstock (1979) hasargued that, 'even in terms of a narrowly conceived revenue objective, the taxsystem is more or less at its limit'. Also on the basis of this shape, Buchananand Lee (1982) argue that a revenue maximizing government with a timehorizon limited to the next election will tend to set too high an average taxrate even in terms of its own objective. More generally, it is the existence of a

J.M. Malcomson, Analytics of the Lafler curue

downward sloping portion of the curve which has led to the 'supply side'argument that cutting the average tax rate would generate a sufficientincrease in the tax base actually to increase total tax revenue.

These conclusions depend crucially on the Laffer curve having the shapeassumed. But, even given the endpoints of zero tax revenue with average taxrates of zero and 100 percent, the general shape in fig. I depends, asAtkinson and Stern (1980) and Hemming and Kay (1980) have noted, on thecurve being a continuous function. One might also add (though no more willbe said about this here) that the assumption of zero tax revenue at anaverage tax rate of 100 percent may be too strong. Economic activity mayjust go underground to join the black economy as long as the probability ofdetection is less than one. The government will then collect revenue fromthat part that is detected. Feige and McGee (1983) have explored further therole of the black economy in the context of the Laffer curve.

In discussing what it is that causes the curve to turn down, the generalpresumption in both the theoretical and the empirical literature seems to bethat it is the disincentive effect of higher taxation on labour supply that islikely to be the dominating influence. To isolate the role of the tax elasticityof labour supply, the present paper, therefore, uses a simple generalequilibrium model, indeed, about the simplest model that allows the generalequilibrium implications to be analysed while still retaining a genuine rolefor government policy.r For this model, it turns out that the nature oftechnology is an equally important determinant of the slope of the Laffercurve. Indeed, for certain technologies, a more negative tax elasticity oflabour supply may imply a more positive slope because of general equilib-rium effects on wages and profits. Moreover, the relevant tax elasticity is ageneral equilibrium one taking account of consequential changes in wagesand profits, not the more usually quoted partial equilibrium one from whichit may even differ in sign. Finally, the taffer curve may not be continuousand may not have an interior maximum. It could slope upwards at everyaverage tax rate less than 100 percent with a discontinuity at 100 percent.

2. The basic model

To maintain the spirit of previous discussions of the Lalfer curve, thepresent 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 generalequilibrium models have provided no useful role for the additional revenue raised-by-a taxincrease. In Canto, Joines and Laffer (1978), tax revenue is merely passed back to the (implicitlyidentical) 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 asIump-sum transfers and any deficit financed by lump-sum taxation. For obvious reasons, modelsin which additional tax revenue has no real use, or in which lump-sum taxes and subsidies arepermitted, are not ideal for analysing the consequences of increases in distortionary taxation.

265

266 J.14. Malcomson, Analytics of the Lafer curoe

importance of income taxation in most developed countries, this tax will betaken to be an income tax. There is no formal diffrculty in extending themodel to include many types of taxes - the Laffer curve then just depictstotal government expenditure as a function of the average tax rates for eachtype of tax - but for the points to be made here no additional insights seemto be offered by this, For simplicity, individuals will be taken to be identicaland, to provide a genuine role for government, there will be one public goodas well as one private good. The former will be treated as a pure public goodwhich is produced by the private sector but purchased, and provided toconsumers, only by the government. Again, there is little formal difhculty inextending the number of such goods. Since with identical individuals thepurely redistributive policies of governments become pointless, these will beignored here. All economic agents are assumed to be price takers.

Let there be N identical individuals, each with a twice differentiable utilityfunction U(ho -h,x,!), where h0 is the maximum possible hours of laboursupply, h the actual hours of labour supply (so lro-lr is hours of leisure), xthe amount of the private good consumed, and y the amount of the publicgood provided by the government. The individual's budget constraints aregiven by

for wh *nZA;otherwise,

where x0 is each individual's endowment of the private good, r the marginalrate of income tax, A the amount of income exempt from tax, w the hourlywage rate, and n each individual's share of the profits of firms. [In (1), anynet sales of the private good are treated as a disposal of assets and so notsubject to income tax.l Let (t(1-r)w, (l -t)n,y,tA) denote each individual'ssupply function for labour hours obtained by maximizing the utility functionsubject to the constraint (1), with f treated as zero if no tax is payable. Formost of what follows, this supply function will be taken to be differentiablefor wh+T =,4, that is, as long as taxes are actually being paid.

The production side of the economy is represented by the profit functionof the private sector iirms. This is denoted [I(p,w), where p is the price atwhich the public good is sold to the government, the price of the privategood being normalized at 1. II{p,w) is assumed to be twice differentiable.Then by standard duality theory, IIr(') is the firms' supply function for thepublic good and -II*(') their demand function for labour hours. Theproperties that this profit function must satisfy in order to represent a validtechnology are that Ir(') be non-negative for positive p and w and strictlyconvex. It is therefore assumed that

fxo +(1 -t)(wh*z- A\+ A,* : l *o

+wh+n ,(1)

(2)i l ro>0; [ '*]0; i l ooi l nn- n'*or},

J.M. Malcomson, Analytics of the La/fer curue

where the arguments p and w of these second derivatives are dropped fornotational simplicity. Each individual's share of the profits of firms is thengiven by n=n(YN.

This paper considers the only situation of practical interest, namely that inwhich tax allowances do not exceed total income so that some taxes areactually paid. Then the government's tax revenue is given by

tN(wh * n - A) : tlLl(p, w) - wII "(p, w) - N.4)1, (3)

the equality following from the properties of the profit function mentionedabove. Its expenditure is

py: pn e(p,w). (4)

Budget balance requires that these two be equal. Equilibrium requires thatsupply and demand be equal in the markets for the private good and forlabour hours but, because of Walras' Law, only one of these needs to beconsidered explicitly. Hence, the general equilibrium prices of the economyare 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].

267

The Laffer curve plots total tax revenue as a function of the tax rate rwhen allowance is made for the general equilibrium effects on prices andquantities traded.z Denote this by R(f). The concern here is with the shape ofthis curve as r changes and the simplest way to investigate that is byconsidering its derivative. Since government revenue is given by the right-hand side of (6), the slope of the Lafler curve is given by the derivative of theleft-hand side of (6), namely

R'(t):p[I r*(p,w)dwIdt +UI r(p,w)+ p[I or(p,w))dpIdt.

(6)

(7)

To evaluate this requires the comparative statics of the system (5) and (6) inorder to obtain expressions for dw/dr and dpldt. These can be obtained in

2Traditionally, the Lalfer curve is treated as a relationship between total tax revenue and theaverage taxrate. Here f is the marginal tax rate, which equals the average tax 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 convenienthere to treat thc Laffer curve as a relationship between total tax revenue and the marginal taxrate. Provided u{ is held constant, as it is here, the average tax rate increases with the marginaltax rate so the slopes of the two dillerent versions of the Laffer curve always have the sanesigns.

268 J.M. Malcomson, Analytics of the Laffer cunte

the usual way by total differentiation. To simplify the notation, the argu-ments of the profit function will be dropped where this causes no ambiguityand dhldt will be used to denote the total derivative (that is, taking accountof the consequontial changes in the price of the public good, the wage rateand, hence, pro{-rts) of the supply function of labour hours with respect to thetax rate. Then the total differential of 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.

Let / denote the determinant of the matrix of coefficientssides of (8) and (9), that is

(e)

on the left-hand

^ -!il** Dn, I" - lt*n nn* pil np $ - tlII o+ twII .,r+ pil ool

: plfl nnn oo- il',oJ + ( I - t)n eII nn

>0, (10)

the sign following from the restrictions (2), i.e. the strict convexity of theprofit function, and the assumption that the public good cannot be a netinput so that I/o must be non-negative. The solution of (8) and (9) for dpldtand dddr is given by

dpldt:ln**(n -wiln- N,4) + N(rwll*,u +pII*)dhldt)lA, (11)

dw I dt :{ - N(l - t)n e+ Mn *e+ pn rofdhldt- i l *o(n--wnn-NA)\ lA.

Substitution of these into (7) gives the slope of the Laffer curve as

R'(r): t(II- wII.- NA\lp(fr,oil**-nt*r)+n en*nf

(12)

+ rN[wp(II ,oil n*- rl,*r1 + n ,1pn_o * wlr,n*)f dhldt\ I a. ( 1 3)

Since A>0, the strict convexity of the profit function and the fact that thepublic good cannot be an input ensure that the first term on the right-handside of (13) is necessarily positive whenever tax allowances do not exceedtotal income so that some taxes are actually paid. Hence, a necessary (but

J.M. Malcomson, Analytics of the Lffir curae 269

not, of course, suffrcient) condition for the Laffer curve to have a negativeslope is that the second term be negative, which requires that either dhldt orits coeflicient (which depends on the profit function and, hence, on thetechnology of the economy), but not both, be negative.

It has been widely recognized, in the literature, indeed it is the basis ofmost of the discussion of the Laffer curve, that what happens to workinghours as the average tax rate is increased is crucial for its slope - though notall the discussion has adequately reflected the fact that it is the generalequilibrium effect taking account of consequential changes in prices and wages,rather than the partial equilibrium effect holding prices and wages constant,which is the relevant effect. More will be said about this in section 4 below.What, as far as I am aware, has never been questioned is the sign of thecoefficient of dhldt in (13). The presumption implicit in the literature is thatthis coeffrcient is always positive so that the greater the reduction (smallerthe increase) in hours of work as the result of an increase in the marginal taxtate t, the smaller the gain (or the greater the loss) in the government's taxrevenue. As will be shown in the next section, however, that is by no meansnecessarily the case.

3. Technology anrl the Laffer curve

The purpose of this section is to investigate the sign of the coeffrcient ofdh/dt in eq. (13) above. To do this it is instructive lirst to consider separatelythe different terms in that coeflicient. The term wdIIoJInn-II3,o) is neces-sarily positive because of the strict convexity of the profit function. The termIIrwIInn is non-negative since il*n70 by the strict convexity of the profitfunction and IIo, the output of the public good, cannot be negative. ilno,however, is the effect of a change in the wage rate on the supply of the publicgood which will be negative as long as labour hours are the only input.Hence, the overall sign of the coefficient is a balance of positive and negativeterms.

To provide some further feel for what is involved, it is useful to express thecoefiicient of dhldt in terms of elasticities. Define

eyP=pn eef n e;

eto: p[I nof II n.

ern=wII.of II o; ELn?w[Innf IIn;

(14)

These are, respectively, the elasticity of the supply of the public good withrespect to its own price and with respect to the wage rate, and the elasticityof demand for total labour hours (denoted t) with respect to the wage rateand 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",]. (1 5)

Note that If.*<O so that the term in square brackets on the left-hand side of(15) has the opposite sign to the term in square brackets on the right-handside. The term in parentheses on the right-hand side must be negative for theprofit function to be strictly convex, as must er-. The only positive term onthe right-hand side of (15), therefore, is e1o. Hence, for the left-hand side of(15) to be negative requires Ea, to be sufliciently large but, of course, toolarge a value would violate the condition for convexity of the profit function.A smaller absolute value for €r", would make violation of this convexitycondition less likely but these two elasticities are not independent since theyboth involve the same cross partial derivative of the profit function. As aresult, it is not easy to sign the overall expression at this level of generality.

Some examples, however, will serve to illustrate that the assumption thatthe coeflicient of dhldt in (13) is positive, which is implicit in the discussionin the literature on the Laffer curve, is not necessarily true. In view of thefact that this coeflicient can be expressed conveniently in terms of elasticitiesas in (15), it will be no surprise that a profit function based on the Cobb-Douglas form (suitably modihed to fit the details of the model) both makes iteasy to evaluate and results in it having an unambiguous sign. Consider,therefore, the profit function

I I (p ,w) : Apow-b* (B-Cw) , where A, B ,C,b>0,a-b> l . (16)

The restrictions on the signs of the coeflicients .4, a, b, and C are to ensurethat the public good is always an output and labour always an input. Theadditional requiremett, a-b> 1, is then necessary and suflicient to ensurethat the profit function is locally convex. It is, in fact, then also globallyconvex for all positive linite p and w. The term (B-Cw) is used to modifythe CobfDouglas function in order to allow the private good to be anoutput. Everything that follows holds for B:C:0 but in that case therequirements that the profit function be convex and linearly homogeneous inall prices are suflicient to ensure that the private good is an input for everypositive p and w. That would not actually be inconsistent with the modelused here since allowance has been made for the possibility of a strictlypositive endowment of the private good but it seems more in the spirit of thediscussion to assume B and C strictly positive so that both the private goodand the public good can be outputs. The implication of this is to give theeconomy, in effect, an additional endowment B of the private good at a cost

J.M. Malcomson, Analytics of the Lalfer curue

Cw. ln this way, (16) provides a tractable form which satisfies globally theconditions for a valid profit function and which allows the private and publicgoods to be outputs and labour hours to be an input. For this function,straightforward differentiation and manipulation establishes that the ex-pression in (15), which is the coe{Iicient of dhldt in (13), is always zero,independent of the values of p and w. That is to say, the slope of the Laffercurve is completely independent of how hours of work adjust in response toa change in the rate of income tax. That slope is then necessarily positive atall tax rates giving a positive tax yield.

A natural way to generate more general examples is to use the standardflexible functional forms which provide second-order approximations to anyarbitrary twice dilferentiable profit function. Typically, these give less clear-cut results because the sign of the coeflicient of dhldt in (13) then depends onthe actual values of 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 positive or anegative slope for any given tax rate r one therefore needs to specify a utilityfunction and solve the whole model for p and w as functions of r. Toillustrate the kind of results that can be obtained on the basis of the profitfunction alone, however, consider the translog form:

ln II(P, w) : aot a, ln P * urln w

+ +lP L tlt d' * 2F t tln p ln w + B 22Qn w)2f.

One problem with this form is that, short of a drastic simplification such asassuming fq:0, all i, i, and so reducing it to a CobFDouglas profitfunction, no restrictions on the parameters will ensure that it is globally.convex, but this is a disadvantage it shares with all the other standardflexible functional forms for profit functions which have variable, as well asfixed, inputs. Sse Diewert (1974) on this. For this reason, it can be consideredonly as a local approximation.

With the translog form it is convenient to work with the input and outputshares defined by

sy: pn p(p,w\lII(p,w); s"=wII n(p,w)[I(p,w). (1 7)

These shares are, in general, functions of p and w but, for notationalsimplicity, the arguments are suppressed. Linear homogeneity of the profitfunction in all prices ensures that the shares of all inputs and outputs(including the private good which is left implicit in the functional form) mustsum to unity but, unlike in the case of translog cost functions, they do notnecessarily lie in the interval [0,1]. The shares of inputs are negative and theshares of outputs may be greater than l. In terms of these shares, the

271

J.M. Malcomson, Analytics of the Lffir curue

elasticities in (1a) are given by

€yp:sy- 7+Brrlsr; rrw:sr *|nlsr"

sLn:sz-1 * |zr ls i l rLp:sy*Frzlsr .

Then the condition for local convexity of the prolit function is

(sy- 1 *811/sr)(s"- 1+ firrlrr)*(sr* prrlsl)@1* f nlsr)<0. (18)

The sign of the coefficient of dhldt in (13), evaluated using the elasticities in(15), is given by

- sign [(s, - 1 + f , r/sr)(sr - | + B rrl s") - (s, * F, zl s ")(s r* fl nl sn)

+ sy+ f ,rl s r*sr. - 1 + fr ,rl s"l.

(1e)

Clearly, the signs of these in general depend on the values taken by theshares and I have found no restrictions, apart from the Cobb-Douglasspecial case with constant shares, which sign them independently of thosevalues. Consider, however, the point at which sy: -sr-1, which implies thatthe private good is an output with share 1 also. In view of what was saidabove, these are legitimate values for the shares. Then the profit function isstrictly convex provided

- f , r (2+ f lzz)+( l - f , r ) t <0,

which holds for

- JlTr'(z+ Fzr\f<(1- Frr)< + Jlfrtr(z+ Frr)).

Since one would normally expect frn,fzr>0 and frz50 and these con-ditions are sufficient to ensure that the elasticities above have the appropriatesigns, let

$ * Fd: + r/l0rr(2+ Fr)f - 6,

fr">U(2+ P2),

where

o<6< Jrp, r (2+Br) ) - r

(20)

(21)

J.M. Malcomson, Analytics of the Laller curae 273

which ensure that the profit function is strictly convex at this point and B*has the expected sign. The expression in (19) then reduces to

- sign {6, - 6(1 + 2\/ lp r r(2 * f , r)l) + t / tfl r rQ + fl , )f - (2 + F 1,,7} .

For d close to zero, this takes the opposite sign t" {r/l|rr(2+ f ,r)l-(2+ flrz)|,which is positive for p1r>2*F* and negative for ftr<Z*fzr, neither ofwhich is inconsistent with (21). Hence, the coeffrcient of dhldt in (13) can beeither positive or negative at the point sr: -sr:l depending on the relativesizes of pt, and prr.

These examples serve to show that theoretical grounds alone are notsullicient to establish that a decrease in working hours as a result of a taxincrease necessarily reduces the revenue gain from that tax increase. Whetheror not it does is an empirical question, the answer to which cannot simply beassumed.

4. Taxation and working hours

The purpose of this section is to consider the relationship between themarginal tax rate t and working hours. Three main points will be madeabout this. The first is that, even with a twice differentiable utility function,working hours need not be a continuous function of the marginal tax rate,which, in itsell is sufficient to undermine the continuity of the LafIer curve.Although the possibility of discontinuity in labour supply has been recog-nized in the theoretical literature, the point is emphasized here because ithas been neglected in much of the discussion of the Laffer curve and anumber of the arguments made [for example, the necessity of a downwardsloping portion and the arguments of Buchanan and Lee (1982)] depend onthe continuity of that curve. The second is that the partial equilibrium effecton working hours of a change in the marginal tax rate is not necessarilyopposite in sign to the effect of a change in the wage rate net of the marginaltax rate. This is well known in the labour supply literature [see, for example,Hausman (1981)] but is reiterated here because it too has been neglected insome of the literature on the Laffer curve. Feige and McGee (1983), forexample, adopt a labour supply function which depends only on themarginal wage net of tax even though the income tax in their model is not aproportional one and, in citing empirical evidence, some authors quote theeffect of changes in wages after tax without mentioning that this may differfrom the effect of changes in the marginal tax rate.

The third point is that the general equilibrium effect on working hours ofa change in the marginal tax rate taking account of consequential changes inprices, wages and the proaision of public goods, which is the relevant effect fordiscussions of the shape of the Laffer curve [see eq. (13)], is different from,

J.M. Malcomson, Analytics of the LoJler curue

and may indeed have a different sign from, the partial equilibrium effectholding prices, wages and the provision of public goods constant. The latterinvolves only the labour supply function, the former the comparative staticsof an equilibrium at which demand equals supply and, hence, depends on thelabour demand curve as well. The distinction is important in considering theimplications of empirical studies for the shape of the Laffer curve. Empiricalstudies of labour supply measure the partial equilibrium effect, yet these arewidely quoted lsee, for example, Beenstock (1979)l in discussions of theLaffer curve without this distinction being made.

To see these points, consider the utility maximizing choices of theindividuals in the model of section 2. Substitution for x in the utility functionfrom the budget constraint (1) for the case considered in the previous sectionin which wh+n>,4 (which must hold for there to be any tax revenue),allows the maximization to be written

max U(f to -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 themultiplier attached to the constraint h<ho and the inequalities bracketedtogether are complementary.

* u,(.)+w(l _t)uzo_l:ll, ,._l=3}(23\

Denoting by U^. the second derivative of U(.) with respect to h, the second-order condition for an interior maximum can be written

(J m=U r r(. ) * 2w(I- r) U, e(.) + w2 1t - 4z Lt 2 2() 50. (24)

In what follows it will be assumed that the inequality in (2a) is strict.To illustrate the first point, let the endowment xo be zero and the utility

function have the Stone-Geary form:

ln (ha - h) + aln [( 1 - t\(wh * n - A) + A- x*l + b ln (/ - y*), (2s)

for some constants a,b,x*,y*>0.Then, if the solution for /r has O<h<ho,ittakes 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 increase or decrease as the tax rate increases. But itis immediately clear from (25) that for t:l the optimal value for h is zero;hence, in ths case where x*>A,therc must bo a discontinuity at /:1. Thenthe Laffer curve too will be discontinuous at t=l and may have no interiormaximum.

That the effect on the supply of hours of an increase in the tax rate r is notthe opposite of that of an increase in the wege net of tax is also obviousfrom (26) since w andt do not appear solely in the form (1 -t)w. There are,however, some more general points to make about this. Standard manipu-lations give the partial comparative static effect of a change in f on hoursworked for interior ft as

)hl 0t : - {wU 2(.) - (wh * n - A)lU, zO- }o( I - t)u z r(. )l V( - U il. Q7)

The partial comparative static effect of a change in the wage after tax isgiven by

a hl ALwQ - t)f : {u z() - hlU, r(')- }r( I * t)U zzOf} l( - U r ).

If z happens to equal A (for example, if they were both zero), then

0hl0t: -wAhl7lw(1-41.

But, as long as unearned income is included in the tax base or the incometax has an exemption level (or, more generally, is progressive in some otherway), this simple relationship will not hold in general.

One can say more about the relationship between the two. Note that

Ahl A{wQ- r)} : U r()l(- U rr) * h 1hl 1xo, (28)

where the first term is the substitution elfect (necessarily non-negative) andthe second the income effect. Also.

)hl 0t : - w{U z(') lF U ̂ ^) + lh + (n - A) lwlAfl 0*o\. (2el

Since positive tax revenue requires wh+n- A>0, it is clear that, with apositive income effect in (28) (that is, leisure is an inferior good), the signs of(28) and (29) must be opposite. With a negative income effect, however, thatis not necessarily the case. With n<A,(28) positive implies (29) negative butboth could be negative. With n) A, (28) negative implies (29) positive but

276 J.M. Malcomson, Analytics of the L$fer curue

both could be positive, The economic rationale for this is straightforward.Consider n> A. A change in the tax rate t then has a bigger income effectthan a change in the wage rate after tax because it affects all sources ofincome, not just earned income, so that when the income and substitutioneffects are different in sign the income effect may outweigh the substitutioneffect in one case even though it does not in the other. This warns againstassuming that a positively sloped labour supply curve necessarily implies anegative response of working hours to an increase in the tax rate. It alsowarns against assuming [as is done by Feige and McGee (1983)] a functionalform for the supply of labour hours which depends only on the marginal net-of-tax wage when the tax structure is progressive or suppliers of labout haveunearned income.

The final point to be considered in this section is the relationship betweenthe partial equilibrium effect of a change in the tax rate t on working hoursfor giuen profits and wage rate and the general equilibrium effect takingaccount of consequential changes in profits and the wage rate. Again consideran interior solution for h, Then, from total differentiation of (23) with w, nand y treated as functions of r and with (11) and (12) used to substitute fordpldt and d{dt, we get that the general equilibrium effect dhldt has theform:

dhldt: [( - U ila)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 (tw II *. * pil n)+ N{( I - t\U z- [,,o[ U r s - ]e( 1 - r) Uzrl]

x {( t - iln e+ twII no I pil oo},

fl : - (I - wil n - N,4) {II**[ (t - t)n olu r z - w( 1 - t)U 2] I N

* II rolU n- lv(l - r)Uz.ll

+ II.e( I - t)U r- n *nlU rt- w(l - r)Uzsll).

A and ( - Ur,,) are positive from the strict convexity of the profit function andthe second-order condition for utility maximization, respectively, but a and fcan, for different profit and utility functions, take on a variety of valueswhich make it impossible to determine the relative signs and magnitudes ofdhldt 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 separable in 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 leisure be a.normal good, Then

[U' . ( ' ) - t { , (1- t )U23( ' ) ] :0; lU . . .O-w(l - t )U22Of >0. (31)

Moreover, let the substitution effect dominate the income effect so that0hl0t<0, as is assumed in most discussions of the Laffer curve, and let thetechnology be represented by the modified Cobb-Douglas profit function ofthe previous section, eq. (16). Then

a:(1 - t )yNfa-t( t+b) l

the sign following from the fact that 0<r<1 and a-b> 1, which ensure thatthe first term in square brackets on the right-hand side is positive and,together with (27) and 0hl6t<0, that the term in braces is positive as well.AIso, under these assumptions,

+ p2by3(a - b - t)Ur n(.) - w(l - t)u zzgJ/(aN twz),

which, in view of (31) and (a-b-1)>0, is certainly positive for fS+. Thisand a>0 inserted into (30) ensure that dhldt>0 for any tS} even though0h/0t<0. of course, with this technology we already know from the previoussection that the sign of dhldt is irrelevant for the slope of the Laffer curve. Itnevertheless provides a neat and tractable way of illustrating that the generalequilibrium effect of a change in the tax rate t on working hours may havethe opposite sign to the partial equilibrium effect. Hence, empirical estimatesof the sign of the partial equilibrium effect obtained from estimates of thesupply function for hours cannot simply be assumed to give the sign of thegeneral equilibrium effect.

5. Summary and conclusion

This paper has investigated the properties of the Laffer curve for an

x{wu "(.1-!4*n*n- A)tu,z(.)-w(t -t)u zzgf\ f *ro,

(-u illaht\t + p: prnl+-|+!t + u]fu,1.11w

J.M. Malcomson, Awlytics of the Laffer cunse

economy with one private good, one public good and labour under anincome tax. It has been shown that, under perfectly reasonable assumptions,certain properties of that curve which are widely assumed to hold may not,in fact, hold for that economy.

The Laffer curve is a general equilibrium relationship between the averagerate of tax and total tax revenue. As such, even when labour is the onlyfactor and the only tax is an income tax, its shape depends on more than thelabour supply function. It is equilibrium labour hours, determined by boththe demand for and supply of labour, which is the relevant quantity variablefor the part of the income tax arising from earned income and the responseof this to a change in the average rate of tax may have the opposite sign tothe partial elasticity of labour supply with respect to that average tax rate.But what happens to the wage rate and to profits as the average tax ratechanges is also important for the tax base. For perfectly reasonable tech-nologies, the interplay of all these influences may result in the response ofworking hours to an increase in the average tax rate having no effect oq oreven an inverse effect on, the revenue gain from that tax increase. Even forwell-behaved utility functions and technologies, the Laffer curve may not becontinuous and may have no interior maximum.

These results highlight the limitations of the assumption that the Laffercurve has the general shape illustrated in fig. 1. The shape of that curve forany real economy must be determined from empirical evidence, not fromtheory. But one conclusion of the present paper is that one must be careful,more careful than much of the existing literature, in interpreting thesignificance for the shape of the Laffer curve of empirical evidence about, forexample, the labour supply function.

Another conclusion is that empirical work aimed at determining the shapeof the Laffer curve needs to use a model sufficiently rich to allow for theeffects discussed here. For example, the labour supply function must allowthe sign of the effect of an increase in the marginal tax rate to differ fromthat of a reduction in the wage net of tax if the economy has a progressiveincome tax or wage earners have unearned income. The technology assumedmust be sufliciently rich to allow the general equilibrium effect of a taxchange on labour hours to have either the same sign as, or the opposite signfrom, the partial equilibrium effect and to allow income tax revenue torespond positively or negatively to changes in equilibrium hours. Imposing apriori restrictions on these effects, as can easily be done by apparentlyinnocent choices of functional form, may lead to incorrect conclusions aboutthe shape of the Laffer curve.

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