Homogeneous Utility Functions and Equality in "The Optimum Town". A Note
Author(s): Nicholas Stern
Source: The Swedish Journal of Economics, Vol. 75, No. 2 (Jun., 1973), pp. 204-207
Published by: Wiley on behalf of The Scandinavian Journal of Economics
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HOMOGENEOUS UTILITY FUNCTIONS AND
EQUALITY IN "THE OPTIMUM TOWN".
A NOTE'
NicholasStern
St. Catherine'sCollege,Oxford,England
In his recentpaper on "The OptimumTown" [2], Mirrleesshowedthat it
would be optimumto have equality of utilityin his model only when the
utilityfunctionof the individual(whereall individualsare identical)took a
special form.Mirrleesgave threeexamplesof classes of utilityfunctionsthat
would give equality at the optimum.The most importantof these classes
consistedof utilityfunctionshomogeneousin the consumptiongood (c) and
land occupied (a). It is rathercommonto use homogeneousutilityfunctions
in bothempiricalestimationand numericalexamples-see forexample,Solow
[5] and Dixit [1] fortheiruse in discussionsof equilibriumand optimality
in towns.It mightbe thought,therefore,
thatthiscase ofequality
(respectively)
was of some importance.The purposeof this note is to point out that most
simpleelaborationsof, or similarmodels to, the model Mirrleesused in the
initialpart of the paper forhis discussionof equalitywould not give the ho(HIE) result.The consequencesforthe HIE result
mogeneity-implies-equality
of two of the modificationsconsideredin this note are implicitin Mirrlees'
resultsbut our understanding
of the distribution
of utilityin these modelsis
helped by makingthese consequencesexplicit.The reasonswe shall usually
wantutilityto be unequal are discussedby Mirrlees[2] and Riley [4].
The two otherutilityfunctions
thatgiveequalitywhichwerenotedby Mirrlees are of less interest-forone of thesefunctions,
u--ekw(a, r), the equality
resultis robustto simplegeneralisations,
forthe other,u =v(c, az(r)), it is not
(r is the distancefromthe centreofthetownand k is a constant).
Mirrlees'problemin the initialpart of the paper was the maximisation
of
fu(c, a, r)rf(r)drwheref(r)is the populationdensity(so that whereall land is
occupied af= 1). The constraintswere a fixed population2nN and a fixed
amountof the consumptiongood 2r Y, i.e.,2
1 I am very gratefulto Avinash Dixit forhelpfuldiscussions. I am responsible forerrors.
I should like to thank the National Science Foundation for financial support and the
Department of Economics, M. I. T., forhospitality.
2 Mirrlees'
[2] numbersforequations (1)-(4) are (5), (6), (8), (9).
Swed.1. of Economics1973
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and equalityin "The OptimumTown" 205
Homogeneous
utilityfunctions
r/(r)dr= N
(1)
crf(r)dr= Y
(2)
Using Lagrangemultipliers
/z,A forthese constraintswe have the firstorder
conditionsw.r.t.to c and f
(3)
uc=A
and
u-cuc-aua=
(4)
/
Mirrleesexpressesthe firstordercondition(4), with respectto f,neatly as
follows:"The effectofaddingan extramanto thetownshouldbe thesame-in
ofthe place in whichspace is foundforhim.Suptermsofwelfare-regardless
c(r)at someone's
pose we put himat distancer. Thenwe givehimconsumption
we
the
area
to
at
reduce
r
and
distance
given
people
by a(r) in aggreexpense
The
HIE
is
result
immediate
from
since
u homogenousdegreea in c
(4)
gate."
is
a
to
thus
and impliescu,+aua equal
au
(1-a)u =
(5)
and u is independentof r provideda+ 1.1
There will be an analogue of condition(4) formost optimumtownmodels
and it is the modifications
to this conditionformost generalisations
of the
Mirrlees
that
the
model
cause
HIE
The
result
to
effect
of
simple
disappear.
extra
man
to
the
will
an
if
town
include
but
adding
(cue+au),
always
thereis any othercost to the remainingpopulationof addinga man at r that
dependson r (call this costq(r)) thenthe HIE resultdisappears.Equation (5)
becomes
(1 - a) u - 99(r)- /z
Then u is a monotonicincreasing(decreasing)functionof99(r)fora <l(ax> 1).
It willoftenbe easy to see how9, and thusutility,
dependson r. x= 1 implies9
is constant. > 1 is excludedif u is concavein (c, a), as assumedby Mirrlees
(see precedingfootnote).
We considerthreemodifications
to theMirrleesmodel-involvingrelaxation
1 The case a = 1 is rather peculiar, since it impliesy = 0. Thus, whereverwe put the extra
man we have zero net addition to the utility integral. This suggests that the solution for
this case may not be unique. Equal utilitymay well be one of the solutions of course.
Note that if we write (1) as the pair of constraints (la) rfdr?N and (Ib) rfdr> N
then (l a) "bites" fora < 1 and (i b) "bites" fora > 1. In otherwords maximizingsubject to
(l a) alone with a> 1 would lead us to solutions where (l a) did not bind. This is to be
expected since a > 1 correspondsto "increasing returns" in the utilityfunction(as a function of (c, a)) and should make us watchfulforsolutions with concentrationsof utility.
Swed.1. of Economics1973
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206 NicholasStern
of each of the main assumptions-tosee that in each case a functionenters.
Firstwe modifythe utilityfunctionforenvironmental
externalities;
second,we
that
land
is
needed
for
that
there
are
(or
suppose
transportation
congestion
costs),and thirdwe relax the assumptionthat workhoursare fixed.The first
two are consideredby Mirrleesalthoughtheireffecton the HIE resultis not.
1. Environmental
externalities.
The effectof adding an extra man now includes the diseconomieshe imposes on othersthroughincreasedpopulation
density.Thus,9 = --fu (noteu, <0 usually)whichwillin generaldependon r.
See Mirrlees[2] equation (48).
2. Land requiredforroads/congestion
costs.The distancer at whichwe place
theextramanwillaffectroadrequirements/congestion
costsat all pointsnearer
to the centrethanr. Thus 9 willbe an increasingfunctionofr. In the Mirrlees
case whereroad widthk is requiredper travellerat any point then (p(r)=
Srkuadrsinceua is themarginalutilityofthedisplacedland. Withmoregeneral
congestioncost models(see e.g., Dixit [1]) 9 will be the integralof the extra
congestioncost between0 and r.
3. Variableworkhours.Suppose that the utilityfunctiondependsonly on
consumptionand land (so that if homogeneousit satisfiestwo of Mirrlees'
conditionsforequalityat the optimum)but that leisurehoursare
sufficiency
fixedso that the further
a man lives out the less workhe does. Againwe shall
have 9 increasingwithr sincethefurther
out we place theextraman the more
is
thisassumptiontogetherwiththisutility
output lost. (Dixit [1] incorporates
functionin his model,and Riley [4] also considersvariableworkhours.)
Mirrleesshowsthatu =v(c, az(r)) willalso give equal utilitysince(3) and (4)
becometwo equationsin c and az whichare independentofr and thuscandaz
are equal foreveryone.This argumentdisappearsifa q9function
entersequation
(4). This case is of littleinterestwhen combinedwiththe fixedworkhours
assumptionsince it removesall aspects of requiredtravelto the centre.The
and land (thuswe shall
problemreducesto sharingout thegoodsconsumption
want z(r) to fallto zero at some pointto achievea finitetown,ifland area is
not constraineda priori).
The equalityresultwiththeutilityfunction
u = ekcw(a,r) is robustto changes
in the assumptionsof the model since equalityis a consequenceof equation
= kcu).'
(3) only
(A.=unote that the Rawlsian social welfarefunction2
We should
(wherewe maximize the welfareof the worstoff)willusuallygive equality.We can writethe
Rawlsian problem
maximize jfir dr
subject to u> i
1
2
(6)
To keep utilityconcave in (c, a) it is preferableto use = - ekCw(a, r).
u
See Rawls [3].
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and equalityin "The OptimumTown" 207
Homogeneous
utilityfunctions
forexample crfdr= Y
and any otherconstraints,
(2)
Taking a Lagrangemultiplierv(r) forconstraint(6) and maximizingw.r.t. c
(whichwe supposeforthe momentdoes not appear in constraintsotherthan
(2) and (6)) we have vu,- Arf= 0. Thus v willin generalbe positiveand u=if.
It is clear this resultis robustto modifications
of the model since v>0 will
usually resultfromconsideringvariationof any variable enteringthe utility
function.The Rawlsianproblemhas been consideredand the equalityresulted
notedby Dixit [1] and Riley [4].
References
1. Dixit, A. K.: The optimumfactorytown.
Mimeo, M.I.T., October 1972.
2. Mirrlees, J. A.: The optimum town.
Swedish Journal of Economics 74, No. 1,
March 1972.
3. Rawls, J.: A theoryof justice. Harvard
University Press, 1971.
4. Riley, J. G.: Optimal towns. Unpublished Ph D. Dissertation, M.I.T., June
1972.
5. Solow, R. M.: Congestion,density and
use of land in transportation. Swedish
Journal of Economics 74, No. 1, March
1972.
Swed.I. of Economics1973
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