An Exploration in the Theory of Optimum Income Taxation
Author(s): J. A. Mirrlees
Source: The Review of Economic Studies, Vol. 38, No. 2, (Apr., 1971), pp. 175-208
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An
Exploration in
Optimum
the
Income
Theory
of
2axa on12
J. A. MIRRLEES
Nuffield College, Oxford
1. INTRODUCTION
One would supposethat in any economicsystemwhereequalityis valued,progressive
income taxationwould be an importantinstrumentof policy. Even in a highly socialist
economy,whereall who workare employedby the State,the shadowpriceof highlyskilled
labourshouldsurelybe considerablygreaterthan the disposableincomeactuallyavailable
to the labourer. In WesternEuropeand America,tax rateson both high and low incomes
are widelyand lengthilydiscussed3:but thereis virtuallyno relevanteconomictheoryto
appealto, despitethe importanceof the tax.
Redistributiveprogressivetaxationis usuallyrelatedto a man'sincome(or, rather,his
estimatedincome). Onemightobtaininformationabout a man'sincome-earning
potential
from his apparentI.Q., the numberof his degrees,his address,age or colour: but the
natural,and one wouldsupposethe most reliable,indicatorof his income-earning
potential
is his income. As a result of using men's economic performanceas evidence of their
economicpotentialities,completeequalityof social marginalutilitiesof income ceases to
be desirable,for the tax systemthat would bring about that resultwould completelydiscourageunpleasantwork. The questionsthereforearisewhat principlesshouldgovernan
optimumincome tax; what such a tax schedulewould look like; and what degree of
inequalitywould remainonce it was established.
The problemseemsto be a ratherdifficultone evenin the simplestcases. In this paper,
I make the followingsimplifyingassumptions:
(1) Intertemporalproblemsare ignored. It is usual to levy income tax upon each
year'sincome,with only limitedpossibilitiesof transferringone year'sincometo another
for tax purposes. In an optimumsystem,one would no doubtwish to relatetax payments
to the whole life patternof income,4and to initialwealth; and in schedulingpaymentsone
would wish to pay attentionto imperfectpersonalcapitalmarketsand imperfectforesight.
The economydiscussedbelowis timeless. Thusthe effectsof taxationon savingareignored.
One mightperhapsregardthe theorypresentedas a theoryof " earnedincome" taxation
(i.e. non-propertyincome).
(2) Differencesin tastes, in familysize and composition,and in voluntarytransfers,
are ignored. These raise ratherdifferentkinds of problems,and it is naturalto assume
them away.
I
2
First versionreceivedAug. 1970;final versionreceivedOctober1970 (Eds.).
Work on this paperand its continuationwas begun duringa stimulatingand pleasurablevisit to the
Departmentof Economics,M.I.T. The influenceof PeterDiamond is particularlygreat, and his comments
have been very useful. Earlierversionswere presentedat the Cowles Foundation, to the Economic Study
Society, at the London School of Economics,and to CORE. I am gratefulto the membersof these seminars
and to A. B. Atkinson for valuablecomments. I am also greatlyindebtedto P. G. Hare and J. R. Broome
for the computations.
3 Discussions on (usually) orthodox lines, including many important points neglected in the present
paper, can be found in [7], [1], [5, Chapters5, 7, 8], and [6, Chapters11 and 12]. [2] is close in spirit to
what is attemptedhere.
4 Cf. [7, Chapter6].
175
176
REVIEWOF ECONOMICSTUDIES
(3) Individualsare supposedto determinethe quantityandkindof labourtheyprovide
by rationalcalculation,correspondingto the maximizationof a utilityfunction,and social
welfareis supposedto be a functionof individualutilitylevels. It is also supposedthat the
quantityof labour a man offers may be varied within wide limits without affectingthe
pricepaidfor it. Thefirstassumptionmaywell be seriouslyunrealistic,especiallyat higher
incomelevels,whereit does sometimesappearthat thereis consumptionsatiationand that
workis done for reasonsbarelyconnectedwith the incomeit providesto the " labourer".
(4) Migrationis supposedto be impossible. Sincethe threatof migrationis a major
influenceon the degreeof progressionin actualtax systems,at any rate outsidethe United
States,this is anotherassumptionone would rathernot make.'
(5) The State is supposedto have perfect informationabout the individualsin the
economy,their utilitiesand, consequently,their actions. In practice,this is certainlynot
the case for certainkinds of incomefrom self-employment,
in particularwork done for the
workerhimself and his family; and in some countries,the extent of uncertaintyabout
incomesis very great. Yet it seems doubtfulwhetherthe neglectof this uncertaintyis a
simplificationof much significance.
(6) Variousformalsimplificationsare madeto renderthe mathematicsmoremanageable: thereis supposedto be one kind of labour(in a specialsenseto be explainedbelow);
there is one consumergood; welfareis separablein terms of the differentindividualsof
the economy,and symmetric-i.e. it canbe expressedas the sumof the utilitiesof individuals
when the individualutilityfunction(the samefor all) is suitablychosen).
(7) The costs of administeringthe optimumtax scheduleare assumedto be negligible.
In sections 2-5, the more generalpropertiesof the optimumincome-taxschedule,
and the rules governingit, are discussed. The treatmentis not rigorous. Neverthelessa
readerwho wantsto avoid mathematicaldetailscan omit the last page or two of section3,
and will probablywant to glancethroughsection4 ratherrapidly. In section 6, I begin
the discussionof specialcases. The mathematicalargumentsin sections6-8 are frequently
complicated. If the readergoes straightto section9, wherenumericalresultsarepresented
and discussed,he should not find the omission of the previous sections any handicap.
He may, nevertheless,find it interestingto look at the resultsand conjecturespresentedat
the beginningof section7, and at the diagramsfor the two cases discussedin section8.
Rigorousproofs of the maintheoremswill be givenin a subsequentpaper,[4].
2. MODEL AND PROBLEM
Individualshave identical preferences. We shall suppose that consumptionand
workingtime enterthe individual'sutilityfunction. Whenconsumptionis x and the time
workedy, utilityis
u(x, y).
x andy both haveto be non-negative,andthereis an upperlimitto y, whichis takento be 1.
In fact, it is assumedthat: u is a strictlyconcave, continuouslydifferentiable,function
(strictly)increasingin x, (strictly)decreasingin y, definedfor x> 0 and 0 < y< 1. u tends
to - oo as x tends to 0 from above or y tends to 1 from below.
The usefulnessof a man's time, from the point of view of production,is assumedto
vary-from person to person. To each individualcorrespondsa numbern such that the
quantityof labourprovided,per unit of his time, is n. If he worksfor timey, he provides
a quantityof labourny. Thereis a knowndistributionof skills,measuredby the parameter
n, in the population. The numberof personswith labourparametern or less is F(n). It
1 The relationof optimumtax schedulesto propensitiesto migrateis discussedin anotherpaper under
preparation.
MIRRLEES
177
OPTIMUM INCOME TAXATION
will be assumedthat F is differentiable,so that there is a density function for ability,
is n an n-man.
f(n) = F'(n). Call an individualwhose ability-parameter
The consumptionchoice of an n-manis denotedby (xn,yn). Writezn = nyn for the
labourhe provides. Then the total labouravailablefor use in productionin the economy
is
Z=
. . .(1)
{ 00 z,,f(n)dn,
and the aggregatedemandfor consumergoods is
X
=
... (2)
xnf(n)dn.
0
In orderto avoid the possibilityof infinitelaboursupply,I assumethat
nf(n)dn< oo
S
.(3)
.
Eachindividualmakeshis choiceof (xn,yn)in the lightof his budgetconstraint. Using
an income tax, the governmentcan arrangethat a man who suppliesa quantityof labour
z can consume no more than c(z) after tax: the governmentcan choose the function c
arbitrarily. It makessenseto imposethe restrictionon the government'schoice of c, that
c be upper semi-continuous,for then all individualshave availableto them consumption
choicesthat maximizetheirutility,subjectto the budgetconstraint1:
... (4)
(xn,yn) maximizes u(x, y) subject to x ? c(ny).
Notice that (xn,yn) may not be uniquelydeterminedfor everyn.2 I write:
..
Un = u(Xn, Yn))
(5)
Proposition 1. There exists a numberno > 0 such that
Yn= O (n
yn>O
Proof.
_
no),
...(6)
(n>no).
If m<n, and ym>0, u[c(mym), ym]<u [c (n.
m)
m
<u.
Conse-
quently, ym = 0 if yn = 0, since then Ym= 0 gives the utility unto n-man. Thus
has the desiredproperties.|
n
=
inf [n Iy?O]
Proposition 2. Any function3 of n, (xn, yn), that satisfies (4) for some upper semicontinuousfunction c also satisfies (4) for some non-decreasing,right-continuousfunction c'.
1
To say that c is upper semi-continuousmeans that
lim sup c(zi) = c(z) when lim zi = z.
i-~o
If
we can supposethat xi-x
of c,
un = sup {u(x, y) I x _ c(ny)}, and u(xi, yi)->un, xi _ c(nyj)
and yi->y (since {yi}and therefore{xi} is bounded). By the uppersemi-continuity
x ? limsup c(nyj) = c(ny);
and by the continuityof u, u(x, y) = lim u(xi, yi) = u,. Thereforethe supremumis attained.
2 In other words, we have a correspondence,
providinga set of utility maximizingchoices for n-men.
It arises when the consumptionfunction c coincideswith the indifferencecurve for part of its length. It is
convenient neverthelessto use the notation of the text, despite its suggestion that we are dealing with a
function.
3 It is easy to see that the result is true for a correspondencealso.
-178
REVIEWOF ECONOMICSTUDIES
If x' < c'(ny'), then, for any e>0, there exists
Proof. Define c'(z) = sup c(z').
z' < z
y Y
y'such that x - , ?c(ny'). Thus u(x - B,Yn) <un which implies, since u is a decreasu(x, yn)
u,. It follows that
ing function in y, that u(xn- 8, Yn)? Unn
Un Letting
n
(xn, yn) maximizes u subject to x < c'(ny).
c' is clearlya non-decreasingfunctionof z. To prove that it is right-continuous,take
a decreasingsequencezi-z. c'(zi) is a non-increasingsequence,and thereforetends to a
limit,whichis not less than c'(z). If it is equalto c'(z), thereis no moreto prove. Suppose
it is greater. Then for some i>0 each c'(z')>c'(z)+s.
Therefore,thereexists a sequence
(fi) such that 2' ? z' and c'(zi) > c(')> c'(z) +B. The second inequalityimplies that
zi>z. Thus z'-iz. Yet lim supc() > c(z), which contradictsupper semi-continuity.
Thus in fact, c is right-continuous.
11
This propositionsays that the marginaltax rate may as well be not greaterthan 100
per cent. We shall considerlaterwhetherit shouldbe positive.
The governmentchooses the functionc so as to maximizea welfarefunction
w
=
0
G(un)f(n)dn.
...
(7)
I use the functionG here,ratherthan writingunalone, becauseI shalllaterwant to devote
specialattentionto the case uXy= 0 (whenu can be writtenas the sumof a functiondepending only on x and a functiondependingonly on y). In maximizingwelfare,the government
is constrainedby productionpossibilities:it mustbe possibleto producethe consumption
demands,X, arisingfromits choiceof c, with labourinputno greaterthanZ. Theproduction constraintis written
X < H(Z).
... (8)
We have not yet fully specifiedthe possibilitiesavailableto the government,since, if
(xn,yn) is not uniquelydefined,it is not clearwhetherthe governmentor the consumeris
allowed to choose the particularutility-maximizingpoint. Perhapsit is reasonableto
supposethat the governmentcan choose, and that the necessityfor market-clearingwill
make its choices actual. But it will turn out that the issue is of no significancewhen we
make the followingassumption,as we shall:
(A) Ynis uniquelydefinedfor all n exceptfor a set of measure0.
Thus the class of functionsc from whichthe governmentchooses is furtherrestricted
by the requirementthat the functionlead to choices satisfying(A). It will appearin due
coursethat (A) is satisfiedfor all functionsc in the particularcases we shall be most concernedwith.
3. NECESSARYCONDITIONSFOR THE OPTIMUM
On the assumptionthat an optimumfor our problem exists, we shall now obtain
conditionsthat it must satisfy. The mathematicalargumentwill not be rigorous. To do
the analysisproperly,one must attend to a numberof rathertrickypoints. Since these
technicaldetails tend to obscurethe main lines of the argument,rigorousproofs will be
presentedseparately,in the continuationof this paper. The nature of these neglected
difficultieswill be discussedbrieflyin the next section.
The key to a reasonablyneat solutionof the problemis to find a convenientexpression
of the conditionthat each man maximizeshis utilitysubjectto the imposed" consumption
the derivativeof u[c(ny), y] with respect
function" c. If we supposethat c is differentiable,
to y mustbe zero. Denotingthe derivativeof u withrespectto its firstand secondarguments
by ul and u2, respectively,we have
ulnc'(ny)+u2
= 0.
OPTIMUM INCOME TAXATION
MIRRLEES
179
Recollect that unis the utility of n-man. Then a straightforwardcalculation,using the
first-ordercondition(9), yields
- u1yc'
dun
_
...(10)
n
dn
(Theexpressionson the rightare, of course,alternativeexpressionsfor thepartialderivative
of u with respectto n, evaluatedat the maximum. The case wheren entersu in a more
generalmannercan be analyzedby using this more generalequation. We shall returnto
this point later.)
Our problemis to maximizew subjectto the constraintof the productionfunction,
X < H(Z), the differentialequation(10), and the definitionun = u(xn,yn). Thosewho are
familiarwith the PontriyaginMaximumPrinciplewill see that this is a form of problem
fairly suitablefor treatmentby it. Shadowpricesp and w have to be introducedfor X
and Z. Then we would like to maximize
W-pX+wZ
= f[G(un)-pxn +wynn]f(n)dn
. . .(11)
subjectto (10). unis to be regardedas the state variable,yn (say) as the controlvariable,
while xn is determinedas a function of un and Ynfrom the equationUn= u(xn,yn). The
Hamiltonianis
M=
G[(un) - pxn + wynn]f(n) -4)nYnU2
n
whereOn is a functionof n satisfyingthe differentialequation
=~
dn
__
Du
fl+
-f[G(n)-
...(12)
Ynshouldthen be chosen so as to maximizeM:
= 0~
[wn+ pu2] f(n) + O9nRIY
....(13)
wherethe functionqi(u,y) is definedby
f(u, y) = -yu2(x, y),
u = u(x, y),
... (14)
and y is its partialderivativewith respectto y. (Notice, at the same time, that
fu- =
-YU12/ul.)
Equation(12) can now be integratedto obtain an expressionfor in; which, when
substitutedin (13), providesus with an equationto be satisfiedby the optimumwe seek.
Beforegoing on to use this equation,however,we shall deriveit in a differentway, by a
more explicituse of the methodsof the calculusof variations. The use of the Maximum
Principlehas a numberof seriousdisadvantages.It does not showus how to obtaincertain
importantsupplementaryconditionson the optimum. The analysisprovidesno hint as
to how it couldbe maderigorous. It does not provideanyinsightinto the kindof maximization that is going on. Whenwe have done a more explicitvariationalanalysis,we shall
be betterable to see wherethe logical holes are, and to understandwhy things come out
the way they do.
REVIEW OF ECONOMIC STUDIES
180
For this purpose, I prefer to write (10) in integrated form:
n
Un
~dm
YmU2(Xm, Y.)
-
m
o
+ u(c(O), 0),
5
= JY(U., Y.) -+ Uos...
m
o
using the notation ql introduced above, and denoting the utility allowed to a man who does
no work by u0. Suppose first that ql is independent of u (corresponding to the sepcial case
0). If we consider a variation from the optimum which changes the functions un
U12=
and Yn by " small " variations bun and bYn,we deduce from (15) that these variations must
be related by
dm
Jn
.. .(16)
fryYm m +L8oU
m
o
This variation will bring about changes in W, X, and Z. As before, introduce shadow
prices (in terms of welfare) for X and Z. Then the variation must leave (11) stationary:
bUn =
0
=
XJ[G(un)-Pxn +wYnn]f(n)dn
= {[G'(Un)un-P
(ubun
-u 2 6n
+ WbYn jf(n)dn,
... (17)
where the variation in x is calculated as follows:
bUn= 3u(xn, Y)
=
...(18)
UlbXn +u23yn
It remains to substitute (16) in (17), yielding,
Jp [
=
I:
{:
un)
y Ymdm
+ buo] + [wn+p
[G'(um)- -P]f(m)dm. Y + (wn+pp
82]
y
n}f(n)dn
)ff(n)} yndn
ffP](n)dn. buo.
+ f[G'(un)-
... (19)
The second equation is obtained by inverting the order of integration in the double integral.'
(19) is to be satisfied for all possible variations of the function yn, and the number u0.
Since u0 can be either increased or decreased at the optimum (if, as is to be expected in
general, some people will do no work at the optimum),
f
G'([u - P-j f(n)dn = 0
...(20)
at the optimum.
1 The double integralis
X G'(u)-P f(n)
mdn.
-
The region over which the integrationtakes place is definedby 0 ? m _ n. Thus, when the orderof integration is inverted,n ranges betweenm and oo for given m. The integralcan thereforebe written
77
[G'-Pl-]
f(n)dn. y8ym
which is seen to justify (19) on permutingthe symbols m and n.
'
MIRRLEES
OPTIMUM INCOME TAXATION
181
If all variationsin yn werepossible-and this is a questionwe shall take up shortlywe could also claim that the expressionwithincurlybracketsought to be zero:
(wn+p
[I
l{:
=
)f(n)
n
ul
G'(um)]f(m)dm.
-
U1
n
...(21)
It should be noticed that this equation will only be valid for n > nO: it does not apply to
n for which yn = 0 (exceptno) because, there, not all variationsof the functionyn are
possible,sinceYncannotbe negative.
Finallywe know that the marginalproductof labourshould be equal to the shadow
wage:
pH'(Z) = w.
...(22)
These equations,(20) and (21), have been workedout underthe special assumption
that ql is independentof u. In the more generalcase, we have to replace(16) by
bun=
f
o
+ 5u0,
Tmnly6ymm
...(23)
where
Tmn=expf4(u
J
m
' .
. .
. (24)
(4
To show this, we can go back to the differentialequation(10). Applyingthe variation,
we obtainfrom it,
d bun = -1 /luUn +
ly6Yn-
dn
n
n
... (25)
This is a firstorderlinearequation,and can thereforebe solvedby the standardmethodto
give the solution(23).
Having replaced(16) by (23), we can now go throughthe rest of the calculationas
before. We find that (20) is generalizedinto
co
T
[G'(u.)-p/u1]T0nf(n)dn= 0;
... (26)
while (21) becomes
(wn+pu2fu1)f(n)
=
J[p/u1-
G'(um)]Tnmf(m)dm.
...
(27)
Notice that we have Tnmhere, although it was Tmnthat appeared in (23).
If theseequationsare correct,the two integralequations,(15) and (27) may be thought
of as determiningthe two functionsunandyn, giventhe threeparametersu0,w,andp. The
values of these parametersare fixed by the three equations(26), (22), and (8). We have
enoughrelationsto determinethe optimumtax schedule,since the functionc can be determined once we know unand yn.
4. NECESSARYCONDITIONS: A COMPLETESTATEMENT
The argumentused to derive these conditionsfor the optimumtax schedulehad a
numberof weak points. It is indeedunlikelythat the relationshipsderivedabove hold in
general. Among the weak points of the argument,notice that
(i) the existenceof the shadowpricesp and w was assumedwithoutproof;
(ii) the optimumtax schedule,and the resultingfunctionsxn,Yn,and unwereassumed
to be differentiable;
(iii) the applicationof the variationwas quite heuristic; and
(iv) no justificationwas providedfor assumingthat the functionYncould be varied
arbitrarily (for n >no).
M
182
REVIEWOF ECONOMICSTUDIES
I shall not commenton (i) and (iii), which,thoughimportant,are technicalmatters: they
can be justified. (ii) is not satisfiedin general: there was no reason to supposethat it
would be. When (ii) is not satisfied,the first-ordercondition, (9), for maximizationof
utility ceasesto be meaningful. Finally,(iv) is neverjustified. The functionyn is derived
from the impositionof the consumptionfunctionc, and we have no a prioriinformation
about it. We must expect that some conceivablefunctionsyn can never arise from the
impositionof a consumptionfunction. The class of possibley-functionsis no doubt quite
complicatedin certaincases. Fortunatelyit is possibleto specifythat class quitesimplyin
the realisticcases, and it is then possibleto use the variationalargumentrigorously.
Problem (ii) is dealt with in the rigorous analysis by dependingon equation (15)
insteadof the differentialfirst-ordercondition(9). It is a remarkablefact thatthis condition
holds if and only if the variousfunctionsarisefromutility-maximization
underan imposed
consumptionfunction,evenwhenthat functionis not differentiable.For proof, the reader
is referredto [4].
To deal with problem(iv), we have to restrictthe class of utilityfunctionsconsidered.
We assumethat
(B) V(x, y) = -yu2/u, is an increasingfunctionof y for each x>0 (and boundedin
0 < x < x, 0 < y < y for any < oo and y< 1).
It will be noticedthat this is an assumptionabout preferences,not just about the form of
the utility function used to representpreferences. The second part of the assumptionis
readilyacceptable. The first, and main part of the assumptionholds if and only if, for a
given level of consumptionx, a one per cent increasein the amountof work done requires
a largerincreasein consumptionto maintainthe sameutilitylevel,the greateris the amount
of work beingdone. It is equivalentto assumingthat (in the absenceof taxation)the consumer'sdemandfor goods is an increasingfunction of the real wage rate (at any given
non-wageincome.' Fewindividualsappearto havepreferencesviolating(B), andintuitively
it is ratherplausible. We shalllateruse the fact that (B) holds if preferencescan be representedby an additiveutility function. (It will be noticed that, as y-> 1, V-> + 00, so that
the assumptionmust hold for some rangesof y.) If the assumptiondoes not hold, the
theory of optimumtaxationis more complicated.
The point of the assumptionis indicatedin
Theorem 1. Under Assumption(B), zn = nyn maximizes utility for every n under some
consumptionfunction c if and only if
(i) zn is a non-decreasingfunction definedfor n >0;
(ii) 0 < zn<nfor all n>O.
1 This equivalenceis fairly obvious from an indifferencecurve diagram. For a formal proof thait(B)
impliesthat consumptionis an increasingfunctionof the wage rate,let w be the wage rate, and m non-labour
income (both measuredin terms of goods). (B) states that wy, regardedas a function of x and y, is an
increasingfunction of y. Write x and y as functions of w and m, putting x = x(w, m), y = y(w, m) and
x' = x(w', m), y' = y(w', m) where w'> w. I shall show that x'> x.
that x" = x(w", m") = x, and
y
To do this, choose w" and m" such
w
= y(w", m") = W y
Since x"-w'y" = m, (x', y') is preferredto (x",y"); and therefore
x'-x
> w"(y'-y")
e
L(wtytwty)
W
=
=
!! (w'y'-wy)
W
w' (X'-X),
since x'- wy' = m = x - wy. This implies, with our assumptionw'< w', that x'> x.
The conversepropositioncan be proved by reversingthe steps.
MIRRLEES
183
OPTIMUM INCOME TAXATION
For a rigorousproof of this theorem,the readeris referredto [4]. For a heuristicjustification,supposethat znis differentiable,and that c is twice differentiable.The first order
condition,(9), can be written
-
Oz
u(c(z), z/n) =
I [zc'(z)-V(c(z),
z
z/n)] = 0.
...(28)
Furthermore,we have the second-ordercondition, that the derivativeis non-increasing
at zn. Since it is zero there, this is also true when we drop the positivefactor ul/z. In
otherwords,
Oz
[ZC'(z) - V(c(z), z/n)]
_ 0, at z =
Zn.
... (29)
Now differentiate the equation znc'(zn)- V(c(zn)zn/n)= 0 with respect to n:
azc8-v]lz
Oz
= Zn
~dn
zn/n)z/n2.
V(c(zn),
. . .(30)
It follows from (29) and assumption(B) that
dzn >0
...(31)
dn
unlesszn = 0. In fact znis strictlyincreasingwhenn > nO and c is differentiable;a corner
in c causes zn to be constantfor a range of values of n. (An indifferencecurve diagram
makes this clear.) Condition(ii) of the theoremclearlyhas to be satisfiedby the utility
maximizing choice.
To prove that a suitableconsumptionfunctionexists for a given z-functionsatisfying
the two conditions,one definesc by the first-ordercondition(28). (30) then shows(nearly)
that the second-orderconditionfor a maximumis satisfied. This does not yet proveglobal
maximizationof utility,but that also is true.
It shouldbe noticedthat, as a corollaryof Theorem1, condition(A) holds whencondition (B) holds, for zn is shown to be non-decreasingeven if it is a correspondence. It
thereforetakesa singlevaluefor all but a countableset of valuesof n. Afortiori,condition
(A) is satisfiedin this case.
Theorem1 at once impliesthat zn and thereforealso xn are non-decreasingfunctions
when the optimumtax scheduleis imposed. Furthermore,it shows us quite straightforwardlywhat changesin the functionyn we are allowedto contemplatewhen applying
the variationalargumentthat allowablesmall changesshould make only a second-order
differenceto the maximand. The rigorousargumentis still complicated,in part because
one has to allowfor the possibilitythatznis constantoversomeintervals,and discontinuous
at somevaluesof n. Thefull statementof the result,whichis provedin [4],is as follows:
Theorem 2. If preferences satisfy assumption (B) and (un,x", yn) arise from optimum
income taxation, then
(i) zn = nyn is a non-decreasingfunction of n;
(ii) Un=
n
Uo-3 [YmU2(Xm,
ym)/m]dm
(n > 0);
...
(32)
(iii) at all points of increase of zn (i.e., where zn>zn, for all n'<n, or zn<zn for all
n'>n)
An
=
[w+u82I/nul]f(n)-
K
[
G-A
(u)
Tnmf(m)dm=0,
...
(33)
184
REVIEW OF ECONOMIC STUDIES
wheresuperscripts" (n) ", etc. indicatethat the function is evaluatedat n-man
(etc.)'sutility-maximizing
choice,and
(n) + ynU(n)U(n)/U(n)
= -u(-n)
~
m-
). dm'
-
exp
...(34)
=
(iv) If n e [nl, n2], where z is constant on [nl, n2], and [nl,
n2]
Ymyu12(xm',Ym')/U(Xm,
of constancyfor z,
rn
{Amdm
ni
2,
rn2
is a maximal interval
];*(35)
Of,JAmdm < 0;
...(36)
n
(v) If z is discontinuous
at n, Ynis definedto be lim ym,Xn is definedby
m*n-
(9n, Y7n)= Un = U(Xn,Yn),
andui1,etc., denoteu1 evaluatedat n,,5n,whileu1, etc., denoteevaluationat x",yn,
- ,nln) w+ U2/nul
(Wyn-xnln) - (w57n
YnU2
,YnU2
-
w + ii2/ni1
y
(37
jy
If qlyis a non-decreasing
functionof y for constantu, znis continuous
for all n.
(vi)
{G
[-)iG'(um)]
o
(vii) X=
w
T0mf(m)dm= 0,
...(38)
_ui
H(Z),
...(39)
H'(Z).
...(40)
It will be noticedthat in this statementw is the commodityshadowwage rate (wlpin the
earliernotation),whileA (i/p in the previousnotation)is the inverseof the marginalsocial
utility of commodities(nationalincome). The second part of (v) should be particularly
noted, sincewe are quite likelyto be willingto assumethat y is a non-decreasingfunction
of y, and it is a great advantagenot to have to worryabout possiblediscontinuitiesin Zn.
It does not seempossible,unfortunately,to delimita class of casesin whichone can be sure
that [0, no] will be the only intervalof constancyfor z. It shouldbe mentionedthat, when
4fyis not non-decreasing,and the equations(37) may possibly apply, the conditionsof
Theorem1 may definemore than one candidatefor optimality,and then only directcomparisonof the welfaregeneratedby the alternativepaths so definedwill solve the problem.
5. INTERPRETATION
If n is not in an intervalof constancyfor z, and c(.) is thereforea differentiable
function
at zn,the first-ordercondition(9) applies. It can be written
.. .(41)
-u2/nu1 = c'(z).
If we denotethe marginaltax rate, dd[wz
-
c(z)], by 0, we have
=
w+u2/nul
d(wz)
wO= d [wz-c(z)]
dz
= A-
a
Tnmf(m)dm,
...(42)
MIRRLEES
OPTIMUMINCOME TAXATION
185
by (33). (42)suggeststhe considerationsthatshouldinfluencethe magnitudeof the marginal
tax rate. First,it can tell us somethingaboutthe sign of 0: we alreadyknowthat 0 will not
be greaterthan 1, but we werenot previouslyableto say anythingaboutits sign. Of course,
we expectthat it will not usuallybe negative. Using (42) and the conditionsin Theorem1,
we can establishthis rigorously.
Note firstthat 1- AG'u1is a non-decreasingfunctionof n, sincexnis a non-decreasing
functionof n, and a G = G'u1a decreasingfunctionof x. If 1-AG'u1werealwayspositive
Ox
or alwaysnegative,Equation(38) could not be satisfied. Therefore
00 l
f
(1-AG'u1)Tn,f(m)dm
Jn ul
is increasingin n for n less than some n, and decreasingfor n> n; but in any case positive
for n >n. (Here we use the propertiesu1>0, Tmn>O.) Since the integralis zero when
n = 0, it is non-negativefor all n. Consequentlythe marginaltax rate is non-negativeat
all points of increaseof z. If n is not a point of increaseof z, c is not differentiableat Zn.
It is easily seen that, if [n1, n2] is a maximalintervalof constancyof z, - u2/nu1 is equal
to the left derivativeof c at n1, and the right derivativeat n2. Thus both the " right" and
"left" marginaltax ratesare non-negativein this case. Summarizing:
Proposition 3V1 If assumption(B) is satisfied, wz- c(z) (the " tax function ") is a nondecreasingfunction for all z that actually occur (and may therefore be taken to be a nondecreasingfunction for all z).
Havingestablishedthat the integralin Equation(42) is non-negativefor all n, we can
see that the marginaltax ratewill be greaterif therearerelativelyfew n-menthanotherwise;
or if the utility-valueof work, -yuy, is more sensitiveto work done (utilitybeing held
constant); or if n is closer to ni,the value of n at which 1 = AG'u1(and the integralis
thereforea maximum). Iff is a single-peakeddistribution,the firstconsiderationsuggests
that marginaltax rates should be greatestfor the richest and the poorest; but the last
considerationtells the otherway.
In any case, it is importantto note than no, the largestn for whichyn = 0, may be
quite large: if the numberwho do not work in the optimumregimeis large,the marginal
tax rate may not be high at zero income. Explicitly,we can rewriteEquation(38) in the
form
[U1(X~,
0) -i'G'(uo)] F(no)+
A[--lG']
Tnomf(m)dm= 0
...(43)
which,when combinedwith Equation(33) (for n = no) gives
w+
u2(Xo,
nOlxO
?)
)
= t;e (uo, 0) F('no)[AG'(uo) - 1I]
0O(O
ul(xo,
... (44)
?)-
Unfortunately,one cannot get much informationfrom these " local" conditions,at least
for smalln. For any detail,and in particularfor numericalresults,one must examinethe
whole system of equations. It is easierto do that for particularexamplesof the general
problem,and that is what we shall do in succeedingsections. It may be noted, however,
that Equation(44) does provideus with someinformationabout no and x0. For example,
it is clearthat no can be zero only if F/nf tends to 0 as n tends to 0; indeed,since the left
hand side of Equation (44) is bounded, no = 0 only if x0 = 0, and therefore 1/u1 = 0.
It follows that no = 0 only if F/(n2f) is boundedas n-*0, whichmeansthat F tendsto zero
faster than exp (- 1/n). This excludesthe cases usually consideredby economists. We
1 The analysis and result can be generalizedto the utility function u(x, z, n) where the parametern
can indicate-variationsin tastes as well as skill. The extension is fairly routine and will not be discussed
here.
186
REVIEWOF ECONOMICSTUDIES
mayconcludeat this stagethatit will be optimal,in the mostinterestingcases,to encourage
some of the populationto be idle.
A numberof conclusionshave been obtained,but they are fairlyweak: the marginal
tax ratelies betweenzero and one; in a largeclassof cases,consumptionand laboursupply
vary continuouslywith the skill of the individual; therewill usuallybe a group of people
who oughtto workonly if theyenjoyit. The mainfeatureof the resultsis thatthe optimum
tax scheduledependsupon the distributionof skillswithinthe population,and the labourconsumptionpreferencesof the population,in sucha complicatedway thatit is not possible
to say in generalwhethermarginaltax ratesshouldbe higherfor high-income,low-income,
or intermediate-income
groups. The two integralequationsthat characterisethe optimum
tax scheduleare, however,of a reasonablymanageableform. One expectsto be able to
calculatethe schedulein particularcases withoutgreat difficulty. In the next sectionsof
the paper,we shall show how this can be done in certainspecialcases, and obtainfurther
propertiesof the optimumtax in these cases.
6. ADDITIVE UTILITY
An interestingcase ariseswhen, for all x and y,
Ul2 =
0.
...
(45)
Thus ul dependsonly on x, and u2 only on y.
Proposition4. If assumption(45) is satisfied,V(x,y) is an increasingfunction of y,
boundedfor smallx andy.
Proof. V = -yu2(y)/u1(x),and V2 = (-u2-yu22)/u1>0.
Corollary. Underassumption(45), Theorem1 applies.
Boundednessis obvious.11
In particularwe know,from statement(v) of that Theoremthat yn is continuousprovided that fryis non-decreasing. In the presentcase, this conditionis equivalentto the
requirementthat
-yu2(y) is convex.
...(46)
Thereis no reasonwhy this assumptionshouldhold in general,but it is easilycheckedfor
any particularcase. We shall now restrictattentionto cases for which(46) holds.'
If we restrictattentionalso to caseswherez is strictlyincreasingwhenn >no, the optimum situationwill be a solutionof the equations
W+
8
nu,
Un
=
Uo-J
= VI,
n2fi(n)
)
YmU2
o
(
n
Ul
-AG')f(m)dm,
... (47)
...(48)
d.
m
We shall furtherassumethatf is continuouslydifferentiable. Since xn,yn are continuous
in this case, it follows that unand
+ _!2 -)liy are differentiablefunctionsof n. Write
nu,.
w+
v=-
U2
nu,
4iy
...(49)
1 In [4] a theoremis provedwhich states that the conditions of Theorem2 are in fact sufficient(as well
as necessary)for an optimum in the special case now being considered.
MIRRLEES
OPTIMUM INCOME TAXATION
187
u and v are continuouslydifferentiablefunctionsof x and y. Since au >0, au <0, and,
Ox
ay
as can easilybe seen, eV <0 eV <0, the Jacobiana(u, v) is alwaysnegative. Consequently
Ox
ay
o(x,
y)
x and y can be expressedas continuouslydifferentiable
functionsof u and v, and are therefore themselvesdifferentiablefunctionsof n.
We can now write Equations(47) and (48) as differentialequations:
dv
n
du
1
nf
f
Vm_ v (2+
I
dn
2G'
_
n2ul
Y2
dn
... (50)
n2
...(51)
n
which, as we havejust shown,can be thoughtof as equationsin u and v. The particular
solution we seek, and the particularvalue of A,,are definedby the boundaryconditions,
Equations(39), (40),
Vno=
2f(n)
...(52)
UAG'(un))]
whichis the form (38) takes here,and
vnn f(n)-0
....(53)
(n- oo),,
which is apparent from Equation (47). Provided that zn is strictly increasing for n
>
no,
a
solutionthat satisfiesall those conditionswill, by Theorem2 of [4],providethe optimum.
Equations (39) and (40), the production function and the marginal productivity
equation,may be ignoredin the calculations. Correspondingto the particularvalues of
w and 2 usedin the calculation,one obtainsvaluesfor Xand Z. Thuswe knowthe optimum
tax schedulewhen the marginalproductis w and the averageproductis X/Z. In this way
one could obtaina rangeof tax schedulescorrespondingto differentaverageproductsand
marginalproducts-which is what one wants. Of course, it is desirableto choose 2 so
that the averageproductwill be relatedto the marginalproduct,w, in a reasonableway.
This shouldnot presentany greatdifficulty.
To determinethe sign of
dv
dn
_
(22 _
nu,
_
\)
dy
I dn
nu,
1(
dy
dn
y
(u22 _
n nu,
Yn+ ndy we calculate,from Equation(49),
dn
=
dn
_Vly
y+
u2u11 dx
nu2 dn
u2
n2ul
u2
-
+
n2ul
U2UI1)
3
nu
dz
dn
dy
u2211
_
nu3 dn
_Y
n
U2U11
nu3
y
v(
nu,
du
dn
-
*(54)
22
n2ul
substitutingfrom (51). Therefore,using (50)
Lu22 _vr* +u2u11 dz = YU22 _yy
-nu,
nu, _ dn
nu,
=-yl(2~+n
+ U2 -
nu
+YL//
)+
+
)
+ fn2t(
V+G$
+
n
... (55)
REVIEWOF ECONOMICSTUDIES
188
We may thereforecheckthe assumptiondz > 0 by examiningthe solutionto see whether
dn
+ YYY
(2+ -
2
-
-G'> 0.
...(56)
Equation(56) is equivalentto dz >0 becausethe expressionin squarebracketsin Equation
dn
(55) is negative,termby term.
In computation,one can proceedas follows:[1] A value of
|u
A
is chosen. To get the right order of magnitude,one can calculate
{
G'fdn (cf. (38))for someparticularfeasible,and aprioriplausible,
'fdn/
allocationof consumptionand labour.
[2] A trial value of no> 0 is chosen. (It shouldbe bornein mind that the inequality
vno > 0 may, with (52), restrictthe rangeof possibleno.)
[3] Bearingin mind that yno= 0, the values of vnoand uno are obtainedfrom (49)
and (52).
[4] The solution of equations(50) and (51) is calculatedfor increasingn until either
(56) fails to be satisfied,or it becomesapparentthat (53) will not be satisfied(see
[6] below).
[5] If (56) fails to be satisfied,zn is kept constant,un(and v")being calculatedfrom
(49) until (56) is satisfiedagain, when zn is allowed to increaseand the solution
pursuedas in [4].
[6] The attemptedsolutionshouldbe stoppedif unor xn beginsto decrease,or vn or
Ynfall to zero, or xn, yn cannot be calculated(e.g. becauseunexceedsthe upper
bound of u, if there is one). Other stopping rules can be given for particular
examples,dependingon the structureof the solutionsof the equations.
[7] A rangeof trialvaluesof no mustbe usedto findthe one that most nearlyprovides
a solution satisfying (53). Efficientrules for iteration might be obtained in
particularcases.
7. FEATURES OF SOLUTIONS
Solutionsmay,for all I know,be verydiversein theircharacteristics;but examination
of the equationssuggestsa numberof comments. First we note that v"will always lie
between 0 and
1
,
since
qy(O),
1+
0
<
U2
nu
l+
U2
nu, <
...(57)
We are thereforeled to expectthat v tendsto a limit as n->oo. (It mightcycle for certain
forms off, of a kind one would perhapsbe unlikelyto use.) y is also bounded,by 0 and
1, and is thereforelikelyto tend to a limit. Oneis thenled to certainconjecturesaboutthe
limits,which ought to hold for sufficientlyregularf and u.
MIRRLEES
OPTIMUM INCOME TAXATION
189
Let
- +y+2<oo.
...(58)
b00
nfdn < oo, y > 0: otherwisen2f is increasingfor large n, thereforebounded
(Since
0
below.) Further,suppose
aex
u,
..>(59)
)*
as x-+ oo. Thenthereappearto be threecases; in each of which one expectsthe following
resultsto hold.
(i) u<1. Asn -+oo,
and
Yn-+1
v .-.+.(61)
0.
..
.(60)
The marginaltax rate,
0-+ 1.
...(62)
(ii) u = 1. As n-+oo,
Yn Y~
wherey is defined(uniquely)by
and
.(63)
**
YU2(Y)= -a,
Vn-[-(1 + )u2(j))-YU22(0)]1.
. . .(64)
...(65)
Furthermore,
0-> 1j+
...(66)
v
...(67)
where
YU22(Y)
UY)
(iii) 1u> 1. As n-+oo,
*.*(68)
and
v-[-(1
+y)u2(0)]1.
...(69)
1
...(70)
1+y
(It may be noted that, in a naturalsense,(66) holds for all cases.)
Beforeindicatingthe reasonsfor these conjectures,a few words of interpretationmay
be in place. On the whole, the distributionof incomefrom employmentappearsto be of
Paretianform at the upper tail1: Equation(58) holds with y between 1 and 2, roughly
speaking. It is not improbable,however,that marginalproductivityper workingyear is
distributeddifferentlyfromactualincomes: the lognormaldistributionis the mostplausible
simpledistribution. For this, y = oo, and
nf _ logn
f
for largen;
(a2
a2
is the varianceof the distributionof logarithmof incomes).
1 See the generalassessmentby Lydall [3].
...(71)
REVIEWOF ECONOMICSTUDIES
190
The realismof alternativeassumptionsabout utility may be assessedby calculating
the responseof the consumerto a linearbudgetconstraint,x = wy+a. It is easy to see
that utility-maximization
requires(sinceul2 0)
u1(x)
1
x=wy+a.
=-,
U2(y)
W
... (72)
If u1 = cxx, we have to solve
caw = -(a+wWY)4U2(Y).
(If aw < &-au2(0), y
=
O.)
...(73)
Clearlythe solutionhas the followingproperties:
y-+ I as w- oo if < 1,.
y-+Oas w-*oo if j> 1.J
(Cf. (61) and (68).) Also
x-a+w
aw
(y<1),
13(5
(
)t
These asymptoticpropertiessuggestthat the case yu 1 is particularlyinteresting.
Whenp = 1, since, by (73)
a = - --yax
w
U2
-Yu2-+c
as w-*oo;
i.e.
.
. . (76)
Y- ,
wherey is definedby (64). (Cf. (63).) If in addition,
u2(y) = -(-y)A
(3>0),
.. .(77)
we have
90(1 y)
=
y(1-9)
=
a,
v.
The choiceof a maybe influencedby consideringthaty = 0 whenwla < 1/a. It is interesting to note that, if
ac=2, 3 = 1, y =2,
v=2,
y=2/3,
and, if our conjecturesare correct,
0-+60 per cent.
Thiscaseis perhapsnot completelyunrealistic;but it shouldbe remembered
thatthe homogeneousformfor u meansthat the decisionnot to workdependsonly on the ratio of earned
to unearnedincome,whichis not a veryrealisticassumption.
It will be noticed that, in this case, the asymptoticmarginaltax rate is very sensitive
to the value of ,u(in the neighbourhoodof 1).
The reasonsfor the conjecturesEquations(60)-(70)(in fact, I can providea proof of
(iii) and will do so below)are as follows. One expectsthat, as n-* oo,the relevantsolution
of the differentialequationswill tend towardsa singularityof the equations: not only will
y and v tend to limits,but n dy and n dvwill tend to zero. Denote the postulatedlimit of
dn
dn
y,, by y. Considerfirstthe case ul = oax-(4u< 1).
MIRRLEES
OPTIMUM INCOME TAXATION
191
In this case utilityis unbounded. I shall show that y = 1. If not, u2 and ify tend to
finitelimits,and, from (51), we have
nu dx
Therefore,since u1 dx= o
dn
dy\ -
_u(y+n
[
1 x -Ja
x
X1
U2(-
...(78)
dn I1- y
-
=-9u2(y
log n[1+o(1)].
...(79)
I
This impliesthat
nu1 = 0[n(log n)f 1i]
...(80)
o00.
..(81)
Therefore
1
('x
n2f(n)J
[i
--A jf(m)dm=
1>0,
IF
1+
[1 + 2]
...(82)
which is readilyseen to be inconsistentwith (80) if the distributionis either Paretianor
lognormal.
We must thereforeexpect that y = 1. Supposenow that 1+ 2, the marginaltax
flu
rate, tends to a limit t < 1. Then
dx
dn
_
U2
(y
nu,
dy\
dn
...(83)
+nI->1+1,1
and consequently
X
...(84)
n
This impliesthat
-=
U1
(1- )0n'1 + o(1)],
-
...(85)
OC
from whichwe can deducethe behaviourof
I
{
oo. In the Paretiancase,f-
n-2-y,
as n->
U(I-
)f(m)dm
...(86)
it is easily seen that
...(87)
I--(2+y-u)-1>O.
Since 1-I =
limn-v
u2
. u2
J-,
and y
nu,
1_ d logfu2
tends to -oo as y-*1 (if it tends
dy
u2
to a limit at all), we musthave U2 -+O,whichis inconsistentwith the assumption < 1. In
nu,
the lognormalcase, one obtains
1-l = lim
u2 constant
u2
If 'y
u2
nu1
log
n
1 tendedto a finitelimit, since
log n
log I u2 # log(1-l) +log (nu,) #(1-ji) log n,
Y
...(88)
192
REVIEWOF ECONOMICSTUDIES
1
d log U2
u I would tend to a finite limit as y-+1; which is clearly impossible.
log I U21 dy
_
Thus in the lognormalcase too, we expect that t = 1. This explainsthe conjecturesin
the case u< 1.
If,u=
1,
dx
ocd(log x) = nu,
=
dn
d(log n)
yn(
dy
*dy(89)
dn
which thereforecannot tend to oo, since in that case uj1 =
finiteM, so that
21
f
1
>nM
n eventuallyfor any
f(m)dmbecomesunboundedas n- oo.
< 1 and
We can expect,therefore,that y-
logx
...(90)
Y-yU2(y).
log n
It is easily seen that the only plausiblevalue of y is that for whichlog x/log n- 1, i.e.
Then if 1+
U2
nu1
... (91)
-a.
YU2(Y) =
-4, we shall have
nocx
--1
-+
-U2(Y)
U-,
1-t
and
VIy
f( m)Pfdm-+
n2f(n) Ju
(I)'r
-U2(Y)
Y)
Y'
whichsuggeststhat
t=frt) G') -U2(Y)
Y
...(92)
(1 - t)(1+ v)/y,
in the notation(57). This is equivalentto (56). In particular,we expectthat I = 0 in the
lognormalcase.
When u> 1, the utility functionis boundedabove, and a more generaland rigorous
treatmentis easy. un is an increasingfunction,and beingnow boundedtendsto a finiteii.
We shall write
u(x,y) = x(x)+p(y).
...(93)
Since x is an increasingfunction,X(x)also tends to a finite limit j. Thus p(y) tends to a
limit, and so does y. The limit of y must be zero, since otherwise(32) implies u-+co,
whichis now impossible.
Now
=
qy
nu,
ip
nul
... (94)
-+
-U2(0)
OPTIMUM INCOME TAXATION
MIRRLEES
in this case (since
-,
being <
,
nu,
193
is bounded . Therefore Equation (50) becomes
U2
n
dv1
dn(Y+1+o(1))v+
+o(l)
~~~~U2(0)
...(95)
in the Paretiancase. From (95) one deduces,by the usual methodof solvinga first-order
lineardifferentialequation,that
1
_ U2(0)(Y
...(96)
+ 1)
from which it follows at once that the marginaltax rate tends to (y+ 1)-. It is easily
checkedthat in the lognormalcase the marginaltax rate tends to zero.
In the next section,a particularcase is examinedin detail, and providesconfirmation
for some of our conjectures.
8. AN EXAMPLE
CaseI. Let us, by way of illustration,analyzethe followingcase:
u = oclog x+]og(1-y)
G(u)= f(n)
1.
(lPog
= 1exp
n+1)2
(log
[
J
2
L
(The last assumesa lognormaldistributionof skills: the averageof n is 0_= O607...).
We put w = 1. With these assumptions,Equations(50) and (51) become V
dv
dn
logn
n
du
dn
y
n(l-y)'
x
A
n2
n2
where
x
1-
V= 01+
/V/
nu,
],*=
- y)
-y)2
1/cc(l
n(
=
(1
Y)(
1y
x
e
and
eu= x (1-y).
For simplicity,we considerthe case ,B= 0 first,and put
1-y,
s=
t
=
log n.
The equations become, since u = a log (an) + a log
s-
+ log s,
dv = v t+ 1 -s+2e-t,
ds
= [1-a_
dt
-(1
1
(1 + a)s](s2-v)+ocs(vt+2e-t)
+a)s2-(1-a)v
In the case of 9 = 0, we define G = u.
...(98)
(99)
REVIEWOF ECONOMICSTUDIES
194
Solutionsof these equationsare depictedin Fig. 1. We now establishtheirproperties.
We rememberthat, in the optimumsolution,0< v<s2 (for the marginaltax rate, v/s2, is
between0 and 1). Using this fact, we can deducefrom the firstequationthat
v-+O (to so).
Supposethat, for some t, vt > 1. Then
d- v
dt
>vt+V
v-s2
s
>vt-1_O,0
'I
v~~~~~~~~~~~~~~~~
2-t~~
1
5
FIGuVR 1
since v>0, and s < 1. Therefore v is increasing at an increasingrate, contradicting
v<s2 < 1. Thisshows that, in fact,
0<v< l/t.
...(100)
The two equationstogetherimply that
I1-(1 +oc)s I-a2
d '-a _
...(101)
s a (S -V),
[S 'a(S2 V)] s
dt
as one may see if one multipliesthe firstby as, and the secondby [(1 +oc)s2-(1- cx)v],and
subtracts. Write
-
r =sa(s2_v).
... (102)
MIRRLEES
OPTIMUM INCOME TAXATION
195
so that
dr
1-(1 +ex)s r
_
dt
When s <
,
...(103)
s
r increases; when s>
,
r decreases. For this reason s cannot tend to a
limit otherthan l/( +oc): we shall show more, that s-?1/(1 +oc). (Cf. Fig. 1.)
Since v-+0, given s>0, thereexists to such that 0<v,<e for all t _ to. Then
1+a
s
If
rt>(l+o)-
a
a -
1-a
cs a
1+a
<r<s
rS
...(104)
(t_ to).
, the right hand inequality implies that
1
St>-~~.
l+oc
Thereforer is decreasing. If
1++a
rt<(l
+oa)-
... (105)
1-a
max[l, (1 + oc) ccJ.
a -e
. ..(106)
we obtain from the left hand inequality(104),
St a <(1+x)-
a
-e{max[1,
(l+
O)~
a } <(1O)-
]-St
... (107)
a
if, either oc? 1 (in which case {...} ? 0 since s _ 1), or a> I and st>
.
Thus, in
fact
St <1+cx
and, by (98), rt is increasing. Combiningthese two results,we deducethat
...(108)
l+cc
a
-+<(l+o()
whichin turnimplies,since v>0, that
St-1+
. ...(109)
1+c
Our demonstrationthat v and s tend to limits 0 and
,
respectively,confirmsthe
conjecturesfor the specialcase. It is readilycheckedthatexactlythe sameargumentsapply
to the case /3>0. As we have noted previously,the marginaltax rate is v/s2. Thus, as
t ->oo
o0-.
... (1 10)
It is a strikingresult; but we shouldnote at once that 0 is a poor approximationto V/S2
even for larget. This becomesapparentwhenwe demonstratethat vt-* 1
I1+o
>s > 0 for an unboundedset of values of t.
Suppose the contrary,that vtIf Vt>
1+e
+ s, and t is large enough to imply that st<
dv > ...
1
1+
(Ili)
REVIEW OF ECONOMICSTUDIES
196
Thus vt continuesgreaterthan 1 +6, and - > is for all largert: but this impliesthat
dt
1+0e
v-* c, which we have alreadyshown to be false. If on the otherhand vt<
l+cx
and
-?,
t is greaterthan 2 and is largeenoughto imply
vt<
4
A
2tet
+
<
4
1
Stx
-
~~~~1+0e
then
d-(vt) = v + t(Vt-S) + Vt + Ate-t
s
dt
-8
I__
1+
< 1+
2
+
+
1
This impliesthat vt becomesnegative,which is impossible. Therefore vt-
I
for
<6
all largeenought:
Vt-*
I
.
(113)
I+o
Thus
o=
2_
v/s=...
14)
=..(1
t
Only 1 per cent of our populationhave t > 1P7(one in a thousandhave t > 2*4).
Since one mightwant to have acas low as 1, the above approximationis clearlyratherbad
even at t = 2., 2 How bad will becomeapparentin the next section.
CaseII. It is also of interestto examinethe case of a skill-distribution
with Paretian
tail:
nf'
f
... (115)
y+2,y>?
The equationsfor the optimumbecome(with,B= 0),
dv_
dt
-
vy(t)+
v
-s++
...(116)
et,
dtt
ds =[I1-oec-(I
+ O)S](S2-v) + os(vy(t)+ et)..(17
dt
(I + Oe)S2 Oe-)V
1 In this example, r2= 1: that is, the standarddeviationof log n is 1. This is done merely for convenience in manipulations. A preciselysimilartheory holds for a generallognormaldistribution.
It can be shown, by continuing the methods of the text, that vt-
l
I
-t
while s =
I
+0 (
The fact that the optimum path is tangential to the vertical at (s, v) = (11 , O)implies that s<
for large t, since otherwiser would be decreasing,and that, as can be seen from the diagram,is inconsistent
Thus we have the situationportrayedin Fig. 1.
with dv ??
2
The case /> 0 can be treatedin a preciselysimilarway, to obtain the same qualitativeresults.
MIRRLEES
OPTIMUM INCOME TAXATION
197
and, exactlyas before,one has the equation
dr _1-(1+ao)s
dt
s
where r = s
(
(S2-v).
(118)
The situation is portrayed in Fig. 2. The broken curves have
equations
s
a(s _v) ri i
2,2 3) v= s
...(119)
V~~~~~~~
v
1/
/ //
S2~~~~~~~~~~~~S
~ ~ ~~~~~
vs +1
FIGuRE 2
with 0 < r1< r2< r3. It will be noted that such a curve,with equation
v= s2-rs
ac(r constant),
...(120)
alwayscuts from below the curve
dVI
v2=
= sv - rs3+ p_
constant)
(p
r cntant),>
...
...(121)
(120)
that passesthroughthe samepoint. This follows from the calculation,
ds
dN1
N
_
ds
dv2
_
ds \ys+
d vs3 ~rs
>0.
...(122)
198
REVIEW OF ECONOMIC STUDIES
This remark will prove very useful; but first we want to establish that, for large t,
the sign of dt is nearly the same as the sign of v_-YS
dt
vs +1
Let e' be a positive number, and let t1 be so large that Ivy(t)+Ae-t - vy I>6' when
t ? t1. Since s = 1 at to = log no, s< 1 at t only if s = 1 for some previous t1;
1+o
1+o
if (for the given t) t1 is the greatest such, we have from Equation (118)
rt> t =(1
-a)-
a(nf
+
2-Vtl
t St
|
{(1+a)2
1+oc dt
since as t- oo, 0>
d-
dt
5
...(123)
=A/\>0,
implies
-t
Vt<
s2
+
o(1)
VSt + 1
< S2_
... (124)
yS t3+o(1).
Therefore st is positively bounded below, say
st > A'>0.
... (125)
Hence, when_t _ t1,
dt
= vy(t)+ v
s
-s+Ae-t
>,
... (126)
if
2s'
s2
S+ +
ys+
... (127)
11
y
Similarly, we can show that
dv<,
dt
... (128)
if
v<
s
YS+1
Now write ?= 8'/( + 1).
2
Y+
...(129)
1
It is clear that, if, for some t
v s-I +s
YSl
and s >
=1+aoc
t,
MIRRLEES
199
OPTIMUM INCOME TAXATION
then dv >0 and also dr <0. Therefore,by the propertiesof the two sets of curves(cf.
dt
dt
Fig. 3), v
s2_is increasing. Thusfor all subsequentt,~~~~~~~~dv
>e', and v-?oo. Sucha path
dt
Ys+1
cannot be optimum. Consequentlyon the optimumpath, if t _ tl,
1 or v?
either s<
l+oc
S2
ys+1
+C.
...(130)
-?.
...(131)
Similarly, for t > tl,
either s> 1 or v> s
1+oc
-ys+1
X~~~~~~
+
pi
/~~
FIGuRE 3
Suppose that at t1, s> 1
(An exactly similar argumentapplies if s<
1*)
Then r is decreasing,and continuesto do so until
2(
) N
r =r' =(1+Soc)~ aE ( 1+oe)2
1 (1+oe+y) +6,.
(Cf. Fig. 4.) Once s <
Only then can s becomeless than 1
1+x ~
~
~~+
Thereforeat no time is
r<r" = (1+Oc)
1,
r increasesagain.
a G +a)2(1 +a
Nor can we have r>r' at any later time. Thus we have found t2 such that, when t > t2,
LMPQ in Fig. 4, which containsX, and can be
(st, v) lies in the curvilinearparallelogram
made as small as we pleaseby suitablechoice of 6'. Thereforeas t-+oo,
1+o
(t
oc)(l+oc+y)32)
REVIEWOF ECONOMICSTUDIES
200
The optimumpath is indicatedby XZ in Fig. 2. On it, the marginaltax rate,
v2
.(133)
'
~~~~~..
l +a+Y
s2 1+ocy
whichconfirmsour conjecturein this specialcase.' 2
It shouldbe noted that we have not shown,in eitherof these cases, that s diminishes
(nor even that z = ny = et(1 - s) increases)all along the path: the possibilitythat z is
constantfor somerangeof n, in the optimumregime,remainsin both the exampleswe have
discussed. Calculationof specificcases is requiredto settlethis issue. Suchcalculationis
not difficultwith the informationabout the solutionthat we now have.
/
/
sZ
/
/
//
~~/
/
X~~
-M+T
w~~~~~~~~~~~~~~
Ls
FIGu1
4
9. A NUMERICAL ILLUSTRATION
The computationswhose results are presentedin the tables below were carriedout
for the first case examined above, with oc= 1, but with a more realisticvalue for c2.
Computationshave also beencarriedout for the case U2 = 1, andtheseprovidean interesting contrastto the mainset of calculations. In all cases,we take w = 1; and for computationalconvenience,the averageof log n is -1. Thismeansthatthe averagemarginalproduct
of a full day'swork is ef272, but it amountsonlyto a choice of unitsfor the consumption
good. The results show, for particularvalues of the averageproduct of labour, X/Z,
what is the optimumtax schedule,and what is the distributionof consumptionand labour
in the population.
1 The case 3>0 can be treatedin a preciselysimilarway, to obtain the same qualitativeresults.
2
It is possible to calculate optimum tax schedulesexplicitlyfor a uniform (rectangular)distribution
of skills; but since that distributionis of no great interest in the presentcontext, the analysis is omitted.
OPTIMUMINCOME TAXATION
MIRRLEES
201
For purposesof comparison,one naturallywants to know what would have been the
optimum position if it had been possible to use lump-sumtaxation (or, equivalently,
directionof labour). Let us considerthis first for the case ,B= 0. We shall assume a
linearproductionfunction
...(134)
X=Z+a
(whichone thinksof as applyingonly overa certainrangeof valuesof Z, includingall those
that are to be considered). In the full optimum,we maximize
[log x + log (1- y)] f(n)dn
T
subject to
.(135)
..
fxf(n)dn = fnyf(n)dn +a.
It is clearthat x will be the samefor everyone:
...(136)
x = x?,
and that yn must maximize
...(137)
log (1-y)+Jny/x0,
for otherwisewe could improvemattersby changingyn(for a set of n of positive measure,
of course)and changingthe constantx correspondingly.Maximizationof (137)yields
where the notation
[...]+
.. .(138)
[1-x?/n]+,
y=
means max (0,
...).
It is worthnoticingthat in the full optimum,only men for whomn > xo actuallywork,
and an interestingcuriositythat, with the particularwelfarefunction specifiedin (135),
utilitywill be less for morehighlyskilledindividuals. This is, as we have seen, impossible
underthe income-tax. The value of xo is determinedby the productionconstraint:
(n-xo)f(n)dn+a,
x?
...(139)
xo
where,for convenience,we havetaken
f(n)dn = 1. In the case of the speciallognormal
distributionused here,it can be shownthat this equationreducesto
...(140)
a.
2x0- x?F(x) - e-ff2 [-F(e
2X0)]=
Solution of this equationgives the consumptionlevel in the full optimum,and also the
skill-levelbelow which no work is requiredof a man, namelythat at which a full day's
labourwould providea wage equal to the consumptionlevel.
Whenf,>0, a similartheoryholds. In that case, x>xo for men with n>xo, but it is
still the case that such men are made to have a lower utility level than their less skilled
neighbours. The equationcorrespondingto (140)is a little more complicatedand will not
be reproduced. For n> xo, consumptionand labourare
Xn = (x0)(1+P)/(1+2P)n/(l+2P)
Yn=
..2.(141)
-(X01n)(1
In the tables, certainfeaturesof the optimalregimeunderincometaxationare given,
along with xo for the full optimumfor the samelinearproductionfunction. In TablesI-X,
the lognormaldistributionhas parametersa = 0 39. This figureis derivedfrom Lydall's
figuresfor the distributionof incomefrom employmentfor variouscountries([3], p. 153).
It is intendedto representa realisticdistributionof skills within the population. In each
202
REVIEW OF ECONOMIC STUDIES
TABLE I
(Case 1)
0-39, mean n = 0 40, XIZ = 0 93.
ot= 1, =0, c
Full optimum for X
Z-0-013:
xO = 0419, F(xO) = 0-045.
Partialoptimum (income-tax): xo = 0-03, no = 0-04, F(no) = 0-000.
F(n)
x
y
0
010
050
0 90
0.99
0-03
010
0-16
025
0-38
0
0-42
045
048
0-49
Population average
0417
x(l -y)
0 03
005
0-08
0-13
0419
z
full
optimum
x
0
009
0 17
0-29
0 45
0419
019
0 19
0419
0419
0418
0419
TABLE II
Same case as TableL
z
x
0
005
0.10
0-20
0 30
040
050
0 03
0*07
0*10
0418
0-26
0*34
043
o=1,
=,
Average
tax rate
per cent
-34
-5
9
13
14
15
Marginal
tax rate
per cent
23
26
24
21
19
18
16
TABLE HI
(Case 2)
cr = 039, mean n = 0 40, X/Z= 140.
Full optimum for X = Z+0-017:
xO = 0-21, F(xO) = 0-075.
Partialoptimum(income-tax): xo = 005, no = 0-06, F(no) = 0-000.
F(n)
x
y
x(l -y)
z
Full
optimum
x
0
0410
0 50
0 90
0 99
0*05
0411
0-17
0-27
0 40
0
0-36
0*42
0-45
0-47
0.05
0 07
0.10
0-15
0-21
0
0-08
0415
0-28
043
0-21
0-21
0-21
0-21
021
Population average
0418
0417
0-21
TABLE IV
Same case as TableIII.
z
0
005
040
1013
0-20
0 30
040
0 50
x
005
0'09
0*21
0-29
0*37
0-46
Average
tax rate
per cent
-80
-30
-5
3
6
8
Marginal
tax rate
per cent
21
20
19
17
16
15
MIRRLEES
203
OPTIMUM INCOME TAXATION
TABLE V
(Case 3)
= 1, P - 1, o = 0 39, mean n = 0 40, X/Z = 1V20.
Full optimum for X = Z+0-030: xO = 0 16, F(xO) = 0 016.
Partialoptimum(income-tax): xo- 007, no = 009, F(no) = 0000.
F(n)
x
0
0 10
0 50
0 90
0.99
0*07
0*12
0-17
0-26
0*39
Populationaverage
0 18
x(l -y)
y
z
Full
optimum
x
0
0-28
037
043
0*46
0-07
0*08
0*11
015
0-21
0
0-07
0-14
0-26
0-42
0416
0-18
0-21
0 25
0 29
0 15
0 21
TABLE VI
Same case as Table V.
z
Average
tax rate
per cent
x
0-07
0 11
0414
022
0-29
0 37
0A45
0
0 05
0'10
0-20
0*30
0 40
0 50
-113
-42
-8
2
7
10
Marginal
tax rate
per cent
23
28
27
25
23
21
19
TABLE VII
(Case 4)
= 1, ,3 = 1, o = 0 39, mean n = 0 40, X/Z= 0-98.
Full optimumfor X = Z-0 003: xO= 0414,F(xO)= 0 007.
Partial optimum (income-tax): xO = 0 05, no = 0-07, F(no) = 0 000.
F(n)
x
0
0410
0 50
0 90
0.99
0-05
0410
0415
0*24
0*37
Populationaverage
0416
y
0
0 33
0-41
0*46
0*48
x(l -y)
z
x
005
0 07
0 09
0*13
0.19
0
0-08
0415
0-28
0 44
0-14
0417
0-20
0-23
0 26
0417
0419
TABLE VIII
Same case as Table VII.
z
x
0
0 05
010
020
0 30
0 40
0 50
005
0 08
012
019
0-26
0 34
0-41
Full
optimum
Average
tax rate
per cent
-66
-34
7
13
16
17
Marginal
tax rate
per cent
30
34
32
28
25
22
20
204
REVIEW OF ECONOMIC STUDIES
TABLE IX
(Case 5)
039, mean n = 0 40, X/Z = 0-88.
o=
, ,B= 1, o =
Full optimum for X = Z-0-021;
xO = 0-13, F(xO) = 0 004.
Partialoptimum(income-tax): xo = 004, no = 0 06, F(no) = 0 000.
x
F(n)
0
0 10
0 50
0 90
0.99
004
0 09
0O14
0O23
0O36
Population average
0415
y
0
0O36
043
0-48
0 50
x(1 -y)
004
0-06
0-08
0O12
0418
z
Full
optimum
x
0
0-08
0-16
0-29
0 45
0-13
0-15
0-18
0-22
0-25
0417
0.19
TABLE X
Same case as TableIX.
Average
z
x
0
005
010
0-20
0 30
0 40
0 50
004
007
0.10
0-17
0-24
0-31
0 39
tax rate
per cent
per cent
-43
-3
15
20
22
21
TABLE Xl
(Case 6)
1, mean n = 0-61, XIZ
= 1,o =
oc= 1,
Full optimum for X = Z-0-013:
Marginal
tax rate
35
39
36
31
27
24
21
0 93.
xO = 0-25, F(xO) = 0 35.
Partialoptimum (income-tax): xO= 0410,no = 0-20, F(no) = 0-27.
F(n)
x
y
0
0*10
0 50
0 90
0.99
0*10
0*10
0-14
0-32
0*90
0
0
0*15
0-41
0*49
Populationaverage
0418
x(l -y)
0*10
0*10
0.11
0.19
0*46
z
Full
optimum
x
0
0
0-06
0 54
1-84
0-25
0-25
0-28
0 44
0-62
0-20
0-32
TABLE XII
Same case as TableXI.
z
x
0
0 10
0-25
050
1P00
1P50
2-00
3 00
0.10
0415
0-20
030
0-52
0 73
0*97
1P47
Average
tax rate
per cent
-50
20
40
48
51
51
51
Marginal
tax rate
per cent
50
58
60
59
57
54
52
49
MIRRLEES
OPTIMUM INCOME TAXATION
205
.4
.3
s 1-4-
*cXs
1
I0
*2
.4
3
FIGURE
5
z
5
Optimum
Consumnptio
Function
.3 i Distribution of
4
Consumption
after Tax
.2
CaSe 5:
X
1c=,
;nea
mean
Distributionof Income
k/;///,;;;t -efore Tax
/7<SX/B/
10
I=,6-.39,
X/Z=_____.88,
-
/
//777rr-.
12
.3
FIGuRE
.4
6
.5
z
t7.40,
206
REVIEW OF ECONOMIC STUDIES
case, x0, no, and the values of x, y and x(I -y) (which measures utility) at the 10 per cent,
50 per cent, 90 per cent and 99 per cent points of the skill-distribution are given. In separate
tables, the average and marginal tax rates are given for a representative range of values of
z. Graphs of the optimal consumption schedule (x = c(z)) are given in Figs. 1 and 2.
In Fig. 2, the distributions of xn and zn are displayed in case 5.
It will be noticed at once that, under the optimum regime, practically the whole
population chooses to work in each of these cases: this contrasts, in some cases, with the
full optimum, where sometimes a substantial proportion of the population is allowed to be
idle. In most cases, a significant number work for less than a third of the time. It is also
somewhat surprising that tax rates are so low. This means, in effect, that the income tax
x
1*5
.5
Case 6:
?
.5
1
FIGURE
*5
c=I
2
z
7
is not as effective a weapon for redistributingincome, under the assumptions we have made,
as one might have expected. It is not surprising that tax rates are higher when,B = 1.
When objectives are more egalitarian, more output is sacrificed for the sake of the poorer
groups. Nevertheless, the difference between the optimum when only an income tax is
available, and the full optimum, is rather large.
The examples have been chosen for X/Z fairly large: this corresponds to economies
in which the requirements of government expenditure are largely met from the profits of
public production, or taxation of private profits and commodity transactions. Tax rates
are, as one might expect, fairly sensitive to changes in X/Z (i.e. to the production possibilities
in the economy, and the extent to which income taxation is used to finance government
expenditure as well as for " redistribution "). Tax rates are mildly sensitive to the choice
of,f. (When c = 4i,the main features are unchanged).
Perhaps the most striking feature of the results is the closeness to linearity of the tax
schedules. Since a linear tax schedule, which may be regarded as a proportional income
tax in association with a poll subsidy, is particularly easy to administer, it cannot be said
that the neglect of administrative costs in the analysis is of any importance, except that
MIRRLEES
OPTIMUM INCOME TAXATION
207
considerations of administration might well lead an optimizing government to choose a
perfectly linear tax schedule. The optimum tax schedule is certainly not exactly linear,
however, and we have not explored the welfare loss that would arise from restriction to
linear schedules: nevertheless, one may conjecture that the loss would be quite small.
It is interesting, though, that in the cases for which we have calculated optimum schedules,
the maximum marginal tax rate occurs at a rather low income level, and falls steadily
thereafter.
This conclusion would not necessarily hold if the distribution of skills in the population
had a substantially greater variance. The sixth case presented has a = 1. So great a
dispersion of known labouring ability does not seem to be at all realistic at present, but it
is just conceivable if a great deal more were known to employers about the abilities of
individual members of the population. The optimum is in almost all respects very different.
Tax rates are high: a large proportion of the population is allowed to abstain from productive labour. The results seem to say that, in an economy where there is more intrinsic
inequality in economic skill, the income tax is a more important weapon of public control
than it is in an economy where the dispersion of innate skills is less. The reason is, presumably, that the labour-discouraging effects of the tax are more important, relative to the
redistributive benefits, in the latter case.
10. CONCLUSIONS
The examples discussed confirm, as one would expect, that the shape of the optimum
earned-income tax schedule is rather sensitive to the distribution of skills within the population, and to the income-leisure preferences postulated. Neither is easy to estimate for real
economies. The simple consumption-leisure utility function is a heroic abstraction from
a much more complicated situation, so that it is quite hard to guess what a satisfactory
method of estimating it would be. Many objections to using observed income distributions
as a means of estimating the distribution of skills will spring to mind. Yet the assumptions
used in the numerical illustrations seem to fit observation fairly well, and are not in themselves implausible. It is not probable that work decisions are entirely, or even, in the long
run, mainly, determined by social convention, psychological need, or the imperatives of
cooperative behaviour: an analysis of the kind presented is therefore likely to be relevant
to the construction and reform of actual income taxes.
Being aware that many of the arguments used to argue in favour of low marginal tax
rates for the rich are, at best, premissed on the odd assumption that any means of raising
the national income is good, even if it diverts part of that income from poor to rich, I must
confess that I had expected the rigorous analysis of income-taxation in the utilitarian
manner to provide an argument for high tax rates. It has not done so. I had also expected
to be able to show that there was no great need to strive for low marginal tax rates on low
incomes when constructing negative-income-tax proposals. This feeling has been to some
extent confirmed. But my expectation that the minimum consumption level would be
rather high has not been confirmed. Instead, virtually everyone is brought into the workforce. Since this conclusion is based on the analysis of an economy in which a man who
chooses to work can work, I should not wish to see it applied in real economies. So long
as there are periods when employment offered is less than the labour force available, one
would perhaps wish to see the minimum income-level, assured to those who are not working,
set at such a level that the number who choose not to work is as great as the excess of the
labour force over the employment available. A rigorous analysis of this situation has still
to be attempted. The results above do at least suggest that we should allow the least skilled
to work for a substantially shorter period than the highly skilled.
I would also hesitate to apply the conclusions regarding individuals of high skill: for
many of them, their work is, up to a point, quite attractive, and the supply of their labour
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REVIEW OF ECONOMIC STUDIES
may be rather inelastic (apart from the possibilities of migration). There is scope for further
theoretical work on this problem too. I conclude, for the present, that:
(1) An approximately linear income-tax schedule, with all the administrative advantages it would bring, is desirable (unless the supply of highly skilled labour is much more
inelastic than our utility function assumed); and in particular (optimal!) negative incometax proposals are strongly supported.'
(2) The income-tax is a much less effective tool for reducing inequalities than has
often been thought; and therefore
(3) It would be good to devise taxes complementary to the income-tax, designed to
avoid the difficulties that tax is faced with. In the model we have been studying, this could
be achieved by introducing a tax schedule that depends upon time worked (y) as well as
upon labour-income (z): with such a schedule, one can obtain the full optimum, since one
can, in effect, construct a different z-schedule for each n.A Such a tax would not be fully
practicable, but we have other means of estimating a man's skill-level-such as the notorious
I.Q. test: high values of skill-indexes may be sought after so much for prestige that they
would not often be misrepresented. With any such method of taxation, the risks of evasion
are, of course, quite great: but if it is true, as our results suggest, that the income tax is not
a very satisfactory alternative, this objection must be weighed against the great desirability
of finding some effective method of offsetting the unmerited favours that some of us receive
from our genes and family advantages.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
Blum, W. J. and Kalven, H. Jr. The Uneasy Casefor Progressive Taxation (University
of Chicago Press, 1953).
Diamond, P. A. " Negative Income Taxes and the Poverty Problem-a Review
Article ", National Tax Journal (September 1968).
Lydall, H. F. The Structure of Earnings (Oxford, 1968).
Mirrlees, J. A. " Characterization of the Optimum Income Tax " (unpublished).
Musgrave, R. A. The Theory of Public Finance (McGraw-Hill, 1959).
Shoup, C. S. Public Finance (Weidenfeld and Nicolson, 1969).
Vickrey, W. Agendafor Progressive Taxation (Ronald Press, N.Y., 1947).
1 The essentialpoint of these proposalsis that the marginaltax rate (as representedby rulesfor deductions from social securitybenefits)should be significantlyless than 100 per cent. Proposals of this kind
have sometimes been put forward in terms that suggest-quite wrongly of course-that any plausiblesoundingnegativeincome-taxproposalis betterthan a systemin which all earningsare deductedfrom social
securitybenefits. It was a majorintention of the presentstudy to providemethods for estimatingdesirable
tax rates at the lowest income levels, and a surprisethat these tax rates are the most difficultto determine,
in a sense. They cannot be determinedwithout at the same time determiningthe whole optimum incometax schedule. To put things anotherway, no such proposal can be valid out of the context of the rest of the
income-taxschedule.
2 J am indebtedto FrankHahn for pointingthis out. It would seem to be true that lump-sumtaxation
is possible in any formal model whereuncertaintyis not introducedexplicitly.