Download SOME ANALYTICS OF TI{E LAFFER CURVE James M

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.
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