STOR
F. P. Ramsey
The Economic Journal, Volume 37, Issue 145 (Mar., 1927), 47-61.
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Tue Jan 5 11:53:12 GMT 1999
A CONTRIBUTION TO THE THEORY OF TAXATION
THE problem I propose to tackle is this: a given revenue
is to be raised by proportionate taxes on some or all uses of income,
the taxes on different uses being possibly at different rates; how
should these rates be adjusted in order that the decrement of
utility may be a minimum? I propose to neglect altogether
questions of distribution and considerations arising from the
differences in the marginal utility of money to different people;
and I shall deal only with a purely competitive system with no
foreign trade. Further I shall suppose that, in Professor Pigou's
terminology, private and social net products are always equal
or have been made so by State interference not included in the
taxation we are considering. I thus exclude the case discussed
in Marshall's Principles in which a bounty on increasing-return
commodities is advisable. Nevertheless we shall find that the
obvious solution that there should be no differentiation is entirely
erroneous.
The effect of taxation is to transfer income in the first place
from individuals to the State and then, part, back again to
rentiers and pensioners. These transfers will slightly alter the
demand schedules in a way depending on the incidence of the
taxes and the manner of their expenditure. I neglect these
alterations; ¹ and I also suppose that "a given revenue means
a given money revenue, money being so adjusted that its
marginal utility is constant.
""
(
""
This problem was suggested to me by Professor Pigou, to
whom I am also indebted for help and encouragement in its
solution.
In the first part I deal with the perfectly general utility
function and establish a result which is valid for a sufficiently
small revenue, and takes a peculiarly simple form if we can
treat the revenue as an infinitesimal. I prove, in fact, that
in raising an infinitesimal revenue by proportionate taxes on
given commodities the taxes should be such as to diminish in
the same proportion the production of each commodity taxed.
In the second part I assume that the utility function is
quadratic, which means roughly that the supply and demand
1 The outline of a more general treatment is given in the Appendix.
48
THE ECONOMIC JOURNAL
[MARCH
curves are straight lines, but does not exclude the most general
possibilities of joint supply and joint demand. With this assump-
tion we can show that the rule given above for an infinitesimal
revenue is valid for any revenue which can be raised at all.
In the third part I give certain important special cases of
these general theorems; and in part four indicate certain practical
applications.
PART I
(1) I suppose there to be altogether n commodities on which
incomes are spent and denote the quantities of them which are
produced in a unit of time by x₁, x2... xn. Some of these
commodities may be identical, save for the place or manner of
their production or consumption; e.g., we can regard sugar
used in tea as a different commodity from sugar used in coffee,
and corn grown in Norfolk as different from that grown in Suffolk.
In order to avoid double reckoning we suppose that these com-
modities are all either consumed or saved; e.g., we include
household coal, but not industrial coal except in so far as an
increase in the stock of industrial coal is a form of saving, so that
this rate of increase can form one of our quantities x. The
quantities ₁, 2 ・・・ can be measured in any convenient different
units.
(2) We denote by u = F(x₁... n) the net utility of pro-
ducing and consuming (or saving) these quantities of commodities.
This is usually regarded as the difference of two functions, one of
which represents the utility of consuming, the other the disutility
of producing. But so to regard it is to make unnecessary
assumption of independence between consumption and pro-
duction; to assume, for instance, that the utility of a hot bath is
the same whether one does or does not work in a coal mine.
This assumption we do not require to make.
(3) If there is no taxation stable equilibrium will occur for
values of the x's which make u a maximum. Let us call these
values ₁, 2... n or collectively the point P. Then at P
we have
ди
0
r = 1,
... n.
əxr
J²u
d²u =
dxdx, is a negative definite form.
əxrəxs
Suppose now taxes are levied on the different commodities
49
1927]
A CONTRIBUTION TO THE THEORY OF TAXATION
at the rates A₁, A₂... An per unit in money whose marginal
utility is unity. Then the new equilibrium is determined by
du
=), 1
r = 1, . . . n
(1)
əxr
In virtue of these equations we can regard the λ's as functions
of the x's, which vanish at P, and satisfy identically
Əx Əs
Ju
(2)
=
dxs
дост
dxdxs
Also the revenue R = Σλαγ.
We shall always suppose R to be positive, but there is no
a priori reason why some of the A's should not be negative; they
will then, of course, represent bounties.
(4) Our first problem is this: given R, how should the X's
be chosen in order that the values of the x's given by equations (1)
shall make u a maximum.
du
I.e., u is to be a maximum subject to ‚¤ = R (where λ, is ;
дост
We must have
0 = du = dx, for any values of dx,
subject to
Əλs
dor,
0 = dR = Σλλα. + ΣΣ.α.
dxr
and so we have
A
Əs
Əx
Σxs
Xxs
Xxs
dxn
3
R
0 (say).
ΣΣ
XrXs
dx
(5) These equations determine values of the x's which are
critical for u, and it remains to discuss the possibility of a plurality
of solutions and to determine conditions under which they give
a true maximum. We shall show that if R is small enough
they will have a unique solution x₁, x . . . Xn, which tends to
X₁, Xq... n as R→0, and that this solution will make u a true
maximum.
¹ E.g., if u = u₁ — ₂ (consumers' utility - producers' disutility)
Juz
Ju ди1
əx
= demand price of rth commodity-supply price = tax.
Əx
Əx
E
No. 145.-VOL. XXXVII.
əxs
dxz
An
[MARCH
For, since
d²u = E
dxdx, is negative definite at P,
əxs
An)
(-)" (x1, x2
0(11, 22
is positive at, and therefore near, P. Hence
Xn)
we can express the x's as functions of the λ's. The equations (3)
then become
dr Rr(₁,
• An)
r = 1, 2, ... n.
Əλs
əxr
For the denominator ΣΣ xrxs is a negative definite form with
du and so cannot vanish near P (and therefore also > 0). The
Jacobian of these last equations with regard to the X's will tend
to 1 as R tends to 0, and they will therefore have a unique solution
λι, An which tends to 0... 0 as R tends to 0. Hence
the equations (3) have a unique solution tending to P as R→0.
We have now to consider the conditions for a maximum
which are obtained most simply by Lagrange's multipliers.
If we consider u + KR
ди
ƏR
we should have
+ K = 0
dxr
dxr
K
or
1+ K-
= 0 if
has the meaning it has in equations (3).
0
+0
or
K =
1-0
0
Then
d²u = d² (u +₁R)
d(૪
= d²u +
0
1-0
d²R
(calculated as if the variables x were independent ¹), and in a
sufficiently small neighbourhood of P we shall have 0 < any
assigned positive constant and so du + d²R negative
definite with d²u. This establishes the desired result.²
0
1-0
(6) Suppose now R and the λ's can be regarded as infini-
tesimals; then putting
„Əλ₂
dr
dxs
=
Σ
dxs
8
equations (3) give us, using (2),
1 See, e.g., de la Vallée Poussin, Cours d'Analyse, 4th ed., t. 1, p. 149.
2 Clearly also we shall get a maximum at any point for which d²R is negative
and 0 <1; i.e., if d²R is everywhere negative (3) will give a maximum for all values
of up to 0 = 1, which gives a maximum of R. This covers the case treated in
Part II and so also any case approximating to that.
50
THE ECONOMIC JOURNAL
1927]
A CONTRIBUTION TO THE THEORY OF TAXATION
x
dx₂
R
dxs
2
= 0
<0,
Əx
Σ -Xs
ΣΣ...
dxs
and their solution is evidently given by
dx₁
dx₂
dxn
= -0 <0.
(4)
.
Xn
x1
X₂
i.e., the production of each commodity should be diminished in
the same proportion.
(7) It is interesting to extend these results to the case of a
given revenue to be raised by taxing certain commodities only.
If the utility were the sum of two functions, one of the taxed and
the other of the untaxed commodities, it is obvious that our
conclusions would be the same as before. But in the general
case the question is by no means so simple.
Let us denote the quantities of the commodities to be taxed
by x₁ . . . Xn, and those not to be taxed by y₁ ... Yn.
du
If
λ =
then λ, is the tax per unit on Xr,
𐐀х,
du
and if
fp =
r=0 ('s and u's functions of x's and y's),
dy,
also as before
а),
Əx
дра
ало днее _ дняя
dx dys
dyr'
=
and
(5)
əxs
dys
𐐀хт
and we have to maximise u subject to
Σηλής =
R, μ = 0, t= 1, . .
T=1
We have
0
du
λ.dx.
T
Əλs
Əs
0 = dR = Edx + x ・dxr + Ex-
dy:
dy:
↑
Σόμι dar + Σ
dμe dyu
0= = dµ::
t = 1, . . . m.
=
əxr
dyu
↑
16
Solving these last equations (du: = 0) for the dy's we obtain
dy: Exirdxr
(6)
.
дре
, дне
[r= 1,
:m).
(7)
where
-Xur=0
𐐀х,
1 yu
(t = 1,
(The possibility of solution is guaranteed by the discriminants of
du not vanishing.)
Whence 0 = d.R = Edx-(dr + Ex
Ex. 2 + 1 Zz, 34 xv).
Σ Σχ.
(X4).
dy:
#
=
+ ['m
-
m.
Əx
əxs
E 2
51
[MARCH
.. instead of equations (3) we have
dr
(3¹)
(ads
для
Jxr
+ m₂ Xtr
t=1 dyt
8=1
It can be shown that these give a maximum of u with the
same sort of limitations as equations (3) do.
(8) And if the λ's are infinitesimal
Əλ
Əλ₂
dr => dxs + E dy
dxs
t dy
Əλs
+ I'm Xis dir
дли
Endxs
by (5), (6).
-
əxr
8=1
дрее
But
ΣXIS JxT
-
ΣΣΟΜ
tudy Xis Xur by (7)
t
дре
Σχιδούς
дрее
(by symmetry since
=
t
Əλs
• Eda. ( + Extra),
dr
since
όλο
Σdas(
dxr
So
=
dyt
dxs
8
and so equations (3′) are satisfied by
dxn
dx₁
-
Xn
X1
i.e., as before the taxes should be such as to reduce in the same
proportion the production of each taxed commodity.
(9) Further than this it is difficult to go without making some
new assumption. The assumption I propose is perhaps un-
necessarily restrictive, but it still allows scope for all possible
first-order relations between commodities in respect of joint
supply or joint demand, and it has the great merit of rendering
the problem completely soluble.
I shall assume that the utility is a non-homogeneous quadratic
function of the x's, or that the A's are linear. This assumption
simplifies the problem in precisely the same way as we have
previously simplified it by supposing the taxes to be infinitesimal.
We shall, however, make this new assumption the occasion for
exhibiting a method of interpreting our formulae geometrically
in a manner which makes their meaning and mutual relations
considerably clearer.
It is not, of course, necessary, nor would it be sensible to
suppose the utility function quadratic for all values of the
variables; we need only suppose it so for a certain range of
values round the point P, such that there is no question of imposing
taxes large enough to move the production point (values of the
52
THE ECONOMIC JOURNAL
dyu
дни
dyt
ape
53
1927] A CONTRIBUTION TO THE THEORY OF TAXATION
x's) outside this range. If we were concerned with independent
commodities, this assumption would mean that the taxes were
small enough for us to treat the supply and demand curves as
straight lines.
PART II
(10) Let u constant + Zarr + EΣBrsXrXs, (Brs= Bsr), and
let us regard the x's as rectangular Cartesian co-ordinates of
points in n-dimensional space.
du
The point P(₁,
n) is given by
0,
and at that point
d²u 2EEßrsdxdxs is a negative definite form.
:. EEßrsrs is a negative definite form,
and the loci u = constant are hyper-ellipsoids with the point
P for centre.
du
Since
=
= ar +22ßraxs
(8)
dxr
R
Σλα.
=
Σαγκ, + 2 ΣΣβrstras
(9)
=
and the loci R constant are hyper-ellipsoids with the point
Q, whose co-ordinates are ₁, 2..., for centre.
(The equations for Q are those for P with their first degree
terms doubled and their constant terms unaltered.)
Moreover, the hyper-ellipsoids u = constant, R= constant
are all similar and similarly situated. The figure shows these
relations for the case of two commodities only.
u=c'
Q
u=C
R=P
X₁
(11) If we are to raise a revenue p we must depress production
= p.¹
to some point on the hyper-ellipsoid R =
1
¹ We can depress production to any point we please because the connection
between the x's and λ's is one-one.
54
THE ECONOMIC JOURNAL
[MARCH
To do this so as to make u a maximum we must choose a
point on this hyper-ellipsoid at which it touches an ellipsoid of
the family u = constant. There will be two such points which
will lie on the line PQ: one between Q and P making u a
maximum, the other between 0 and making u a minimum.
For the point of contact of two similar and similarly situated
hyper-ellipsoids must lie on the line joining their centres. Since
the maximum of u is given by a point on OP we have as before
that
The taxes should be such as to diminish the production of all
commodities in the same proportion.
And this result is now valid not merely for an infinitesimal
revenue but for any revenue which it is possible to raise at all.
The maximum revenue will be obtained by diminishing the
production of each commodity to one-half of its previous amount,
i.e., to the point Q.
(12) If in accordance with this rule we impose taxes reducing
production from ₁, ₂. ...n to (1 k)₁, (1 k)₂.
(1 k)n.
We get from (8) λr = ar + 2(1 — k) Σßrsxr,
8
but at P
λ = 0, so that 0= ar + 2Σrsr;
T
therefore
>r = kar
(10)
i.e., the taxes should be in the fixed proportions λ₁:₂:
:An:: α₁:a₂
: an independent of the revenue to be raised.
Also R = Σλrr = k(1 - k)Σαrār,
4k(1 k) x the maximum revenue (got by putting
k = 1).
(13) Since k is positive it follows from (10) that the sign of
A, is the same as that of ar, and unless the ar are all positive some
of the λ, will be negative, and the most expedient way of raising a
revenue will be by placing bounties on some commodities and
taxes on others.
The sort of case in which this might occur is that of sugar
and particularly sour fruits, e.g. damsons. A tax on sugar
might reduce the consumption of damsons more than in pro-
portion to the reduction in the total consumption of sugar and so
require to be offset by a bounty on damsons.
(14) We can now consider the more general problem: a given
revenue is to be raised by means of fixed taxes μ₁... μm on
Im commodities and by taxes to be chosen at discretion on the
remainder. How should they be chosen in order that utility
may be a maximum ?
[MARCH
.. instead of equations (3) we have
dr
(3¹)
(ads
для
Jxr
+ m₂ Xtr
t=1 dyt
8=1
It can be shown that these give a maximum of u with the
same sort of limitations as equations (3) do.
(8) And if the λ's are infinitesimal
Əλ
Əλ₂
dr => dxs + E dy
dxs
t dy
Əλs
+ I'm Xis dir
дли
Endxs
by (5), (6).
-
əxr
8=1
дрее
But
ΣXIS JxT
-
ΣΣΟΜ
tudy Xis Xur by (7)
t
дре
Σχιδούς
дрее
(by symmetry since
=
t
Əλs
• Eda. ( + Extra),
dr
since
όλο
Σdas(
dxr
So
=
dyt
dxs
8
and so equations (3′) are satisfied by
dxn
dx₁
-
Xn
X1
i.e., as before the taxes should be such as to reduce in the same
proportion the production of each taxed commodity.
(9) Further than this it is difficult to go without making some
new assumption. The assumption I propose is perhaps un-
necessarily restrictive, but it still allows scope for all possible
first-order relations between commodities in respect of joint
supply or joint demand, and it has the great merit of rendering
the problem completely soluble.
I shall assume that the utility is a non-homogeneous quadratic
function of the x's, or that the A's are linear. This assumption
simplifies the problem in precisely the same way as we have
previously simplified it by supposing the taxes to be infinitesimal.
We shall, however, make this new assumption the occasion for
exhibiting a method of interpreting our formulae geometrically
in a manner which makes their meaning and mutual relations
considerably clearer.
It is not, of course, necessary, nor would it be sensible to
suppose the utility function quadratic for all values of the
variables; we need only suppose it so for a certain range of
values round the point P, such that there is no question of imposing
taxes large enough to move the production point (values of the
52
THE ECONOMIC JOURNAL
dyu
дни
dyt
ape
1927]
A CONTRIBUTION TO THE THEORY OF TAXATION
55
We have λ = μη, . . . λη
m
= μlm,
hyperplanes (n - 1
folds) whose intersection is a plane n m fold which we will
call S. S will cut the hyper-ellipsoids u = constant, R = con-
stant in hyper-ellipsoids which are similar and similarly situated
and whose centres are the points P', and Q' in which S is met by
the m-folds through P and Q conjugate to S in u = c or R = c.
As before the required maximum is given by the point of contact
of two of these hyper-ellipsoids in S, which must lie upon the
line P'Q'.
du
Now the hyperplane λ = μ1 ΟΙ
= μ₁ is conjugate in
дх1
uc to the diameter
x₂ = X2, X3 = X3,
xnxn.
Hence S is conjugate to the m-fold
Xm+1= m+, • • . Xn =
Xn,
and the co-ordinates of P' satisfy these equations, since they lie
on this m-fold.
Similarly the co-ordinates of Q' satisfy
Xm+1= m+1, ... Xn= n.
And so the desired production point lying on the line P'Q' satisfies
+1
Xm+2
Xn
Xm+1
Xm+2
In'
i.e., the whole system of taxes must be such as to reduce in the
same proportion the production of the commodities taxed at
discretion.
PART III
(15) I propose now to explain what our results reduce to in
certain special cases. First suppose that all the commodities
are independent and have their own supply and demand equations,
i.e., we have for the rth commodity the demand price
Pr=
pr(xr)
and the supply price
qr =
fr(xr).
::
Ar= Pr― gr = $r(xr) — fr(xr),
Əλr
and equations (3) become, since
0, r+s,
əxs
0.
x₁1'(x₁)-f1'(x₁)}¯¯¯ x₂{₂' (x₂) — ƒ₂'(x₂)}
These equations we can express in terms of elasticities in the
following way.
56
THE ECONOMIC JOURNAL
[MARCH
Suppose the tax ad valorem (reckoned on the price got by the
producer) on the rth commodity is pr, then
λ₂ = µgr = prfr(xr),
and
r(x) = f(xr) + dr = (1 + µr) fr(xr).
+ fr
..
xr{pr' (xr)-fr' (x)}
ft (x₂)
Xr
fr(xr)
(1 + r)
r(r)
pr(xr)
now xr fr(xr)
f(x) is the reciprocal of the elasticity of supply of the
commodity reckoned positive for diminishing returns, and
() is the reciprocal of the elasticity of demand, reckoned
-X pr(xr)
positive in the normal case.
Hence if we denote by pr and er the elasticities of demand and
supply,
1 + μ₂),
pr=0(=
Er
Pr
=+ =\0
Pr
Er
or
flr=
(11)
1
Pr
(valid provided the revenue is small enough, see § 5).
For infinitesimal taxes is infinitesimal and
My
pln
-
(12)
1
+
+
€1
P1
En
Pn
i.e., the tax ad valorem on each commodity should be proportional
to the sum of the reciprocals of its supply and demand
elasticities.
(16) It is easy to see
(1) that the same rule (12) applies if the revenue is to be
collected off certain commodities only, which have supply
and demand schedules independent of each other and all
other commodities, even when the other commodities are
not independent of one another.
(2) The rule does not justify any bounties; for in stable
1
1 1
equilibrium, although may be negative, + must be
positive.
Er
Pr Er
(3) If any one commodity is absolutely inelastic, either
for supply or for demand, the whole of the revenue should be
+
56
THE ECONOMIC JOURNAL
[MARCH
Suppose the tax ad valorem (reckoned on the price got by the
producer) on the rth commodity is pr, then
λ₂ = µgr = prfr(xr),
and
r(x) = f(xr) + dr = (1 + µr) fr(xr).
+ fr
..
xr{pr' (xr)-fr' (x)}
ft (x₂)
Xr
fr(xr)
(1 + r)
r(r)
pr(xr)
now xr fr(xr)
f(x) is the reciprocal of the elasticity of supply of the
commodity reckoned positive for diminishing returns, and
() is the reciprocal of the elasticity of demand, reckoned
-X pr(xr)
positive in the normal case.
Hence if we denote by pr and er the elasticities of demand and
supply,
1 + μ₂),
pr=0(=
Er
Pr
=+ =\0
Pr
Er
or
flr=
(11)
1
Pr
(valid provided the revenue is small enough, see § 5).
For infinitesimal taxes is infinitesimal and
My
pln
-
(12)
1
+
+
€1
P1
En
Pn
i.e., the tax ad valorem on each commodity should be proportional
to the sum of the reciprocals of its supply and demand
elasticities.
(16) It is easy to see
(1) that the same rule (12) applies if the revenue is to be
collected off certain commodities only, which have supply
and demand schedules independent of each other and all
other commodities, even when the other commodities are
not independent of one another.
(2) The rule does not justify any bounties; for in stable
1
1 1
equilibrium, although may be negative, + must be
positive.
Er
Pr Er
(3) If any one commodity is absolutely inelastic, either
for supply or for demand, the whole of the revenue should be
+
1927]
A CONTRIBUTION TO THE THEORY OF TAXATION
57
collected off it. This is independently obvious, for taxing
such a commodity does not diminish utility at all. If there
are several such commodities the whole revenue should be
collected off them, it does not matter in what proportions.
(17) Let us next take the case in which all the commodities
have independent demand schedules but are complete substitutes
for supply; i.e., with appropriate units the demand price
the supply price
Pr =
= $r(xr),
qr = f(x₁+
+ xn).
Let us put
2 = x₁ +
+ xn.
We can imagine this case as that of a country in which
all commodities are produced at constant returns by the applica-
tion of one kind of labour only, the increase in the supply price
arising solely from the increasing marginal disutility of labour,
and the commodities satisfying independent needs. Then z
will represent the amount of labour.
Equations (3) give us
dr
-0=
xror' (xr) - zf'(2)*
Or if ur represents the tax ad valorem and pr the elasticity of
demand for the rth commodity and the elasticity of supply of
things in general, we get, by a similar process to that of § 15,
€
+ 1)0
pr
flr=
(13)
0
1
-
Pr
If the taxes are infinitesimal we have
flr
=
(14)
+
Pr
In this case we see that if the supply of labour is fixed (abso-
lutely inelastic, →0) the taxes should be at the same ad valorem
rate on all commodities.
(19) If some commodities only are to be taxed it is easier to
work from the result proved in § 8 for an infinitesimal revenue,
that the production of the commodities taxed should be diminished
in the same ratio.
Suppose, then, X₁, . . . xm are to be taxed, m+1 • • . Xn
untaxed.
Let
dx₁ kx1,
dxm = - kxm.
...
58
Let
now
also
..
THE ECONOMIC JOURNAL
2² = x₁ + x₂ +.
+xm
Z" = xm+1 + . . . + xn.
A₁ = ₁(x + dx₁) — zf(z + dz)
= $₁'(x₁)dx₁ — f'(z)dz.
k
k
dz
μ1
• flm =
P1 ZE
pm
dz
dz' + dz"
kz' + dz",
dxm+1
Pm+1xm+1
dxn
dz"
Pnin
dz
€2
dz
€2
€2²
m+1
[MARCH
0=
dxm+1
dxm+2
kz'
=
procr
Ez + por
Pm+1xm+1 Pm+2xm+2
m+1
Σmxr
1
P1 = k
= K (²=²+2 +
+
etc.
€
+
m+1
As before we see that of two commodities that should be
taxed most which has the least elasticity of demand, but that if
the supply of labour is absolutely inelastic all the commodities
should be taxed equally.
PART IV
(20) We come now to applications of our theory; these cannot
be made at all exactly without data which I, at any rate, do not
possess. The simplest result is the one which we have proved
in the general case for an infinitesimal revenue (§ 8); this means
that it is approximately true for small revenues, and that the
approximation approaches perfection as the revenue approaches
zero. It is thus logically similar to the theorem that the period
of oscillation of a pendulum is independent of the amplitude.
We have also extended the result to any revenue which does not
take the production point outside a region in which the utility
may be taken to be quadratic, i.e., the supply and demand
schedules linear.
The sort of cases in which our theory may be useful are the
following:
(21) (a) If a commodity is produced by several different
methods or in several different places between which there is no
mobility of resources, it is shown that it will be advantageous to
discriminate between them and tax most the source of supply
which is least elastic. For this will be necessary if we are to
maintain unchanged the proportion of production between the
two sources (result analogous to § 19 with supply and demand
interchanged).
1927] A CONTRIBUTION TO THE THEORY OF TAXATION
59
(b) If several commodities which are independent for demand
require precisely the same resources for their production, that
should be taxed most for which the elasticity of demand is least
(§ 19).
(c) In taxing commodities which are rivals for demand, like
wine, beer and spirits, or complementary like tea and sugar,
the rule to be observed is that the taxes should be such as to
leave unaltered the proportions in which they are consumed
(§ 14). Whether the present taxes satisfy this criterion I do not
know.
(d) In the case of the motor taxes we must separate off so
much of the taxation as is offset by damage to the roads. This
part should be so far as possible equal to the damage done.
The remainder is a genuine tax and should be distributed
according to our theory; that is to say, it should be placed
partly on petrol and partly on motor-cars, so as to preserve
unchanged the proportion between their consumption, and should
be distributed between Fords and Morrises, so as to reduce their
output in the same ratio. The present system fails in both these
respects.
(22) (e) Another possible application of our theory is to the
question of exempting savings from income-tax.¹ We may con-
sider two uses of income only, saving and spending, and sup-
posing them independent we may use the result (13) in § 17.
We must suppose the taxes imposed only for a very short time ²
and that they raise no expectation of similar taxation in the
future; since otherwise we require a mathematical theory con-
siderably more difficult than anything in this paper.
On these assumptions, since the amount of saving in the very
short time cannot be sufficient to alter appreciably the marginal
utility of capital, the elasticity of demand for saving will be
infinite, and we have
1
+
P1
€
μ₁ (tax on spending)
P1
₂ (tax on saving)
0,
€
and we see that income-tax should be partially but not wholly
remitted on savings. The case for remission would, however,
1 No account is taken of graduation in this.
2 Strictly, we consider the limit as this time tends to zero,
[MARCH
THE ECONOMIC JOURNAL
60
be strengthened enormously by taking into account the expecta-
tion of taxation in the future.
(23) It should be emphasized in conclusion that the results
about "infinitesimal" taxes can only claim to be approximately
true for small taxes, how small depending on data which are not
obtainable. It is perfectly possible that a tax of 500% on whisky
could for the present purpose be regarded as small. The unknown
factors are the curvatures of the supply and demand curves; if
these are zero our results will be true for any revenue whatever,
but the greater the curvatures the narrower the range of "small'
taxes.
On the other hand, the more complicated results contained
in equations (3), (3′), (11), (13) may well be valid under still
wider conditions. But these are, in the general case, too com-
plicated to be worth setting down in the absence of practical
data to compare with them.
APPENDIX
We can also say something about the more general problem
in which the State wishes to raise a revenue for two purposes;
first, as before, a fixed money revenue, R₁, which is transferred
to rentiers or otherwise without effect on the demand schedules ;
and secondly, an additional revenue, R₂, sufficient to purchase
fixed quantities, a₁, a2,... an of each commodity.
Let us denote by pr, qr, as before, the demand and supply
prices of the rth commodity, and the tax on it by λ. Then if
xr is the amount of the rth commodity consumed by the public
(or by the State out of R₁), xr + ar is the amount produced, and
we have
ди
= √₂
ƏxT
Pr(x1, x2,. , Xn) — qr(x₁ + α₁, x2 + A2, ..., Xn + An),
R1 + R2 = Σλαr, Rg = Σarqr,
so that u is to be a maximum subject to
Arxr Zarqr = R₁ constant,
whence
λ₂
0
əxs
aqs
Zxzx
Σασσαν
3
λ₂
0, which replace equations (3).
=
or
das
Σ(as + as)gari
aps
Σxx
1927] A CONTRIBUTION TO THE THEORY OF TAXATION
61
Although these equations do not give such simple results as
we previously obtained for an infinitesimal revenue or a quadratic
utility function, in the cases considered in § 15 and § 17 they lead
us again to the equations (11) and (13).
For, taking the case of § 15, in which the commodities are
independent both for demand and supply, and, as before, denoting
by the rate of tax ad valorem on the rth commodity and by
Pr, er its elasticities of demand and supply for the amounts Xr,
Xr+ar respectively consumed and produced by the public, we
have
fly
= 0
Xr + Ar
dqr
Xr
qr
d(xr+ar)
qr dx,
fr
or
= 0
1 + fr
Pr
Er
whence fr
which is equation (11) again.
0
1
Pr
And we can similarly derive equation (13) from the assumption of independence for demand and equivalence for supply.
F. P. RAMSEY
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