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Massachusetts Institute of Technology
Department of Economics
Working Paper Series
OPTIMAL TAXATION AND
THE LE CHATELIER PRINCIPLE
Peter Diamond
James
Mirrlees
Working Paper 02-30
September 2002
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Abstract
Optimal Taxation and
the Le Chatelier Principle
A.
P.
Diamond and
J.
A. Mirriees
September, 2002
It is
when
a natural presumption that there should be less distorting taxation
there are
more
decisions based on the prices distorted by taxation.
This note shows the need for an additional assumption in order to reach
the conclusion.
We
good, labor, and
human
consider a competitive model with one consumption
capital.
decisions based on a price,
is
To represent a
we contrast the
chosen with that where the
human
difference in the
situation where
capital level
the Le Chatelier principle. Thus, the result
is
is
number
human
of
capital
a parameter, using
relevant for considering
a temporary tax, which just applies to one cohort already in mid-life,
with a permanent tax, which
lives.
will
apply to future cohorts over their entire
While the Le Chatelier principle signs the
effects
difference in substitution
between the two models, there are other terms that are relevant
as well.
The assumption
small relative to
its
of social welfare to
of the optimal
presumptive
that the income derivative of
substitution effect
wage taxation with
is
human
capital
is
sufficient to sign the response
fixed
human
capital at the value
wage tax with human capital chosen, thereby giving the
result.
Optimal Taxation and the Le Chatelier Principle
p. A.
Diamond and
J.
A. Mirrlees
September, 2002
Introduction
1
It is
a natural presumption that there should be
there are
more
decisions based
shows the need
for
less distorting
To
an additional aissumption
represent a difference in the
contrast the situation where
capital level
is
is
when
on the prices distorted by taxation. This note
in order to reach the conclusion.
We consider a competitive model with one consumption good,
capital.
taxation
human
number
capital
is
labor,
and human
of decisions based on a price,
chosen with that where the
we
human
a parameter, using the Le Chatelier principle. Thus, the result
relevant for considering a temporary tax, which just applies to one cohort
already in mid-life, with a permanent tax, which will apply to future cohorts
over their entire
lives.
in substitution effects
While the Le Chatelier principle signs the difference
between the two models, there are other terms that are
relevant as well.
The assumption that the income
small relative to
its
substitution effect
is
derivative of
sufficient to sign
human
capital
is
the response of social
welfare to
wage taxation with
wage tax with human
As
fixed
human
capital at the value of the optimal
capital chosen, thereby giving the presumptive result.
often the case with optimal tax analyses, the presence of income effects
is
complicates the analysis (contrast,
e.
g.,
Mirrlees, 1971 with
Diamond,
1998).^
Individual Choice
2
Assuming
linear taxation,
we can
Max
consider the choice problem'^
u{vjxh
^''»
where
w
is
+ b,x,h)
the net of tax wage, x the hours of labor supply, h the level of
human
capital investment,
that u
is
strictly
and
b
the level of lump
<
0,
sum income. We assume
concave and that the problem has an interior
relevant parameters. For this interpretation,
W3
(1)
x>0, h>l
we
naturally have uj
with the latter condition reflecting the cost of incresising
maximum
>
0,
human
uq
<
at
0,
capital
(earnings per hour of labor supply).'^ This choice problem yields optimal choice
For other comparative statics results on optimal taxation, see Helpman and Sadka (1978).
For a more general formulation of the interaction between human capital investments and
the return to labor supply, we could denote by E{x, h) the (gross of tax) level of earnings after
a human capital investment of h and a labor supply of x hours. Since human capital and the
return to working can interact in a rich varietj' of ways, it might be instructive to explore such
a model.
^For example, consider three cases where human capital investment has only financial costs,
only adds to the disutility of labor (for example by taking time) and only has an additive utility
'
^
cost.
u{wxh + b,x,h) = f {wxh
-\-
b
—
c {h)
= / {wxh +
6,
x
u{wxh + b,x,h) = f {wxh +
6,
x)
u{wxh
-\-
b,x,h)
-f-
,
x)
c (h))
—
c {h)
We
functions x{w,b) and h{w,b).
wage
derivatives by x^, Xb,
some
level,
and
denote the wage, income and compensated
x^;
h^,
and
hb,
the optimal choice of x depends on
particularly interested in x evaluated at h{w,
b).
ty, b,
h
If
he-
and
is
h,
We denote
held constant at
x{w,
b, h).
We
are
the derivatives of x
similarly to those of x.
It is useful
hy
z,
z
=
to rewrite the problem in terms of effective labor, which
xh.
Msixu{wz
z,h.
The
we denote
+
b,z/h,h)
(2)
2>0,/l>l
point of this version comes from the fact that
alone, that gives the
impact of a change
compensated own derivative of z
is
in the net
it
is 2,
not either x, or h
wage on
utility.
Thus the
positive, while there is not a necessary sign
condition for either x or h.
Given the structure of the individual choice problems, the Slutsky relations
take the form
As compensated own
he
=
hyj
— xhhb
(4)
Zc
=
Zu;
-
(5)
Xc
=
Xu,
— xhxb >
ZZb
derivatives of a bad,
with the assumptions c(l)
=
0„
c'
{h)
>
we
0, c" (fe)
>
>
also
0.
(6)
know
that Xc
>
0, Zc
>
or
Xch
+ xhc >
We
0.
denote the indirect
envelope condition,
v^{w,
b)
utility
functions as v{w,b) and v{w,b,h).
By
the
we have
=
v^{w,
h {w,
6,
b));
Vb(w,
=
b)
Vb{w,
b,
h {w,
b)).
(7)
Moreover, the demand derivatives of the two problems satisfy the relationships
Xb-
Note that these expressions can have
be (but
is
=
Xb
Xhhb
either sign.
<
not necessarily) a normal good [hb
increasing or decreasing in the wage.**
decreasing in
human
capital, just as
(9)
it
That
0)
is,
human
capital
may
and human capital may be
Also, labor supply can be increasing or
may be
increasing or decreasing in the
wage.
Given the structure of the individual choice problem, the Slutsky relation
gives us the condition
Xjjj
The
familiar
— x^ =
—
Xc
+ xh {xb —
Xb)
=
Xc
Le Chatelier Principle (Samuelson
''Note that there
derivative oh
Xc
human
is
no liquidity constraint
capital for
many
people.
in this
—
Xc
+ xhxhhb
(10)
(1947), exposited in Hatta
problem, a source of a positive income
(1987),
h
=
Milgrom and Roberts (1996), and Silberberg (1974))
tells
us that for
h{w,b),
zc{w,b)-zc{w,b,h)>0
or,
(11)
using the relationship between effective labor and hours of labor and
human
capital
h{xc-
Xc)
+ xhc>0
(12)
However, signing the difference in compensated derivatives
is
not sufficient in
general to sign the difference in Marshallian derivatives.
Ramsey Taxation
3
We turn now
to a formal analysis of the
Ramsey economy.
If
the revenue needs
of the government were fixed, then the level of tax to finance
the same in the two economies
we
are considering
supply derivatives would not change the result.
elasticities,
we assume that the government
is
-
them would be
the change in effective labor
In order to show the role of
using the revenue to finance a
public expenditure which has an additive impact on social welfare.
Thus we
are
considering the role of the elasticity on the level of public expenditure, related
to the question explored in Atkinson
We
h
is
assume a
linear tax
and Stern (1974).
on wage income.
We
are implicitly assuming that
not observable or not independently taxable for some other reason. For
simphcity,
we assume a
linear technology with a marginal product
m of effective
labor.
We
denote by g
expenditure
e,
the (additive) contribution to welfare of public good
(e)
with expenditure equal to net tax revenue.
human
case of a variable level of
we write
capital
Starting with the
social welfare as the
sum
of
individual utility (written as a function of the after-tax
wage and the
lump sum income, which
Ramsey problem) and
is
constrained to be zero for the
the social value of government expenditures.
where [w — m)x{w,b)h{w,b)
Differentiating
net-of-tax
V
and using Roy's
wage (and so the tax
V^{w*,0)
We now
fix
the
=
+b+
e
=
6
=
social welfare
V{w,b)
identity,
we have the
v{w,b)
+
is
written
as:
g{e)
(13)
condition for the optimal
rate):
=
-xhvb +
de
—g'
=
—xhvb —
g'
human
Then
level of
(14)
aw
[hx
+
{w
— m) (hx^ +
xh^)]
=
capital level at its optimal level, h{w*,0)
and define
social welfare as the restricted indirect utility function plus the value of public
good expenditure.
=
V{w,b,h{w*,0))
where [w — m)x('w,b,h{w* ,0))h{w* ,0)
We
want to calculate the derivative of
of-tax wage, evaluated at the optimal
human
+b+
e
=
5
=
v{w,b,h{w*,0))+g{e)
(15)
social welfare
wage
level for
with respect to the net-
the problem with variable
capital:
V^{w*,Q,h{w*,0))
de
=
v^{w,b,h{w*,0))
+
=
—xhvb — g'[hx
(w — m)(hx^)]
+
-^g'
aw
Subtracting (14) from (16) and using the Slutsky equation,
V-ui{w*,0,h{w'',Q))
We
(16)
we have
= g'{w-m){h{x^ - Xyj)+xh^)
=
g'{w
— m){h
=
g'{w
—
=
g'(w
— m){h
(17)
(xc
—
Xc)
+ hxh (x(, — x;,) + xhc +
rn){h (xc
—
x^)
+ hxhxhht, + xhc + xxhhb)
—
Xc)
+
[xc
xhc
+
^2:
[hx^
+
xxhhf,)
x) hi)
are looking for sufficient conditions for this derivative to be negative,
representing a local gain from increasing the tax (lowering the net-of-tax wage).
Since
we have a
positive tax,
w <
m
(assuming positive public expenditures)
and the
first
and second terms
to a positive from the
presumptive result
is
the last parenthesis on the right hand side add
in
Le Chateher Principle,
(hxh
+
x) hb
>
0.
However, we would expect that plausibly
effective labor supply in the constrained
(hxh
+ x) >
0, (just
as
we expect
human
capital
Thus a
sufficient condition
is
a normal good,
is
it
hi,
(12), a sufficient condition for the
problem
is
increasing in
human
to be increasing in the wage)
<
0.
capital,
and plausibly
This would result in a negative sign.
that the income derivative of
human
capital
be
small.
Intuitively, the issue is that
a large income
effect
changes the payoff from
taxation.
4
Optimal Taxation
The Ramsey
setting permits analysis of the presumption of less distorting taxa-
tion to finance public expenditures
when
there are fewer decisions. If distorting
taxes are used to finance both public expenditures and income redistribution,
then we can examine the impact on both of them. For this purpose, we consider
the
many
person optimal tax problem with linear taxation.
Starting with the case of a variable level of
human
we
write the
sum
level of
lump
g{e) the contribution to welfare of public
good
of individual utilities as a function of the after-tax
sum income. We denote by
capital
wage and the
expenditure
e,
which
is
equal to net tax revenue. Social welfare
v{w,b)
where
then
Y.{u\w,b))+m)
(w-m)^ (x'(w,b)ft'(t(;,6)) +6^(1) + e =
Differentiating
V^
=
is
=
V
and using Roy's
identity,
we have
-^(x'/,X) + ^5'
(19)
At the optimum (w*,6*), we have Vw{w* ,h*) and Vb{w* ,h*) equal to
We now
fix
the
human
zero.
capital level of each individual at its optimal level,
h'{w*,b*) and evaluate the derivatives of the
sum
of restricted indirect utility
functions plus the value of public goods expenditures.
We can write the
deriva-
tives as
K,
=
-J2i^''''<) +
=
-
^
{x'h'vl)
-
^9'
g'
[J2
10
(20)
{h'x')
+ {^-m)J2
{h'il)]
Subtracting (19) from (20) and using the Slutsky equation, we have
VUw\bn = g'(w-m)Y,{h'{xl-il) + x'hl)
=
g'{w
- m) J2ih'
{xl
-
This leads to the same analysis of the welfare
as with the
Ramsey
+
^l)
x'h\
(21)
+ h'x\h'xh +
effect of increased
x')hl)
wage taxation
case.
Turning to the guaranteed income provided,
similarly,
we have
= g'{w-m)Y,{h\xl-xl) +
V,[w\h*)
x^h^,)
(22)
= g'{w-m)Y,{{h^xl + x^W,).
Thus a
income
and Yl
is
sufficient condition for the
provided, Vb{w* ,h*)
((^'^/i
+
constrained problem
capital
sign.
is
<
^^)hb)
is
fixed,
<
would expect that
0.
we
still
human
effective labor
capital, {hxh
This would result
in
guaranteed
w <
m,
supply in the
+ x) >0,
and human
the presumed negative
the income derivative
get an increase in the
is
small but
wage tax when human
but the increased revenue from the increase in the wage tax
used to finance the additive public good, as
the lump
less
the presence of wage taxation,
less plausible case, if
of the implausible sign,
is
^®
is
increasing in
a normal good, hi
Considering a
capital
*-*
<
presumptive result that
sum income.
11
is
is
the saved revenue from lowering
5
References
p. A.
Diamond
(1998) 'Optimal income taxation:
An example
with a u-shaped
pattern of optimal marginal tax rates.' American Economic Review 88(1), 82'
95.
T. Hatta (1987), "Le Chatelier Principle," in The
of Economics, edited by
J.
Eatwell,
M.
New
Milgate, and P.
Palgrave:
A
Dictionary
Newman, London; The
Macmillan Press.
E.
Helpman and
E.
Sadka (1978), "The Optimal Income Tax: Some Compara-
tive Statics Results," Journal of Public
P.
Milgrom and
J.
9,
383-394.
Roberts (1996) "The LeChatelier Principle," American Eco-
nomic Review, March,
J.
Economics,
86, 173-179.
A. Mirrlees, (1971) 'Exploration in the theory of optimal income taxation.'
Review of Economic Studies 38(114), 175-208.
P. A.
Samuelson (1947), Foundations of Economic Analysis, Cambridge: Har-
vard University Press.
E. Silberberg (1974),
nomics, or.
nal of
How
to
"A Revision of Comparative
Do Comparative
Economic Theory,
7,
Statics
159-172.
12
Statics
Methodology
in
Eco-
on the Back of an Envelope," Jour-
Z003
Date
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