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Tax reform from the gradient projection viewpoint
Journal
of Public Economics
TAX REFORM
15 (1981) 2755293.
FROM
North-Holland
Publishing
THE GRADIENT VIEWPOINT
Company
PROJECTION
J. TIROLE* Ecole Nationale des Pants et Chausstes and M.I.T, Cambridge, MA 02139, USA
R. GUESNERIE CEPREMAP, Received
75013 Paris, France
April 1979, revised version
received July 1980
This paper examines the problems of gradual tax reform. Desirable small moves of the tax system are determined using the principles of the well-known gradient projection algorithm. Three classical models of public finance are considered. For each of them a continuous time dynamic process is exhibited along which the social welfare function increases until convergence to local second best extrema. Although the selection of trajectories according to the gradient projection method has some arbitrariness, it leads to intuitively and economically appealing rules such as a tax reform analog of the classical optimal taxation result on production efficiency.
1. Introduction The recent literature on theoretical public finance has shown an increased concern for the problems of gradual tax reform, putting a particular emphasis on the local analysis of the feasible and desirable moves of a system from non-optimal points. Desirability can be evaluated according to different criteria. For example, in the context of the well-known Diamond-Mirrlees model (1971), which will be considered here, Guesnerie (1977) looks for directions of tax changes making every consumer better off, whereas Diewert (1979) and Dixit (1979) rely on a social welfare function to construct what they call respectively ‘optimal tax perturbations’ and ‘best small improvemements’.’ These procedures will remind the reader of the classical algorithm of search of the direction of steepest constrained ascent in applied mathematics. Such a direction is obtained by projecting the gradient of the objective *We are grateful to A. Dixit, H. Tulkens, and especially to the referee, whose helpful comments considerably improved the exposition. Part of the material presented here can be found in I irole (1978) who gratefully acknowledges CEPREMAP’s hospitality. ‘See also Weymark (1979) for a comparison of papers by Guesnerie and Diewert.
004772727/81/0000~000/$02.50
0 North-Holland
Publishing
Company
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J. Tirole und R. Guesnerie, Tax reform
function on the linear local approximation to the feasible set; this is the gradient projection method or the gradient projection algorithm.2S3 This paper applies the gradient projection method to three classical models in public finance. For each of them a continuous time dynamic process will be exhibited which leads to a steady increase in a given social welfare function until convergence to the set of points satisfying the (second best) optimality conditions is achieved. We should make at this point an important warning. The gradient projection algorithm that we single out here is certainly not the only one which could be used for our purpose (although it is certainly the simplest one). Furthermore, for a given algorithm, there exists different processes according to the choice of control variables; in other words, the trajectory given by the gradient projection method depends on the space in which the projection is made. This is the counterpart, here, of the problems faced by Dixit and Diewert in choosing the vectors, the norm of which is constrained. Hence, we do not claim to provide a unique solution to the problems of tax reform. Among the many monotone dynamic processes which can be expected to converge to local welfare optima, we single out one. This is in itself a desirable goal since tax reform theory might well content itself with an algorithm prescribing changes in taxes and public production, however arbitrary they may be. However, we do more here: the tax and public good production revision formulae (which rely on particularly natural principles) are both simple and economically appealing, as shown by the following prescriptions. The consumer prices revision can be decomposed into two moves, the first (socially desirable one) proportional to the marginal social surplus, and the second being the cheapest (in terms of private production) correction in order to ensure feasibility (section 4). Public production is guided at each instant by the private production prices; this rule can be considered as the reform analog of the Diamond-Mirrlees production efficiency rule at a second best optimum (section 5). Lastly, in a one consumer economy with lump sum transfer, consumer prices are revised according to their absolute index of discouragement (section 6). We also discuss the possibility of non-desirability of private production efficiency, and its effect on the above prescriptions. The rest of the paper is organized as follows: section 2 gives some background material on the gradient projection method, and section 3 presents the classical Diamond-Mirrlees model. The application of the gradient projection method to the model and the results mentioned above ’ ‘Notice that the use of gradient process in economics goes back at least to Arrow-Hurwicz. A systematic analysis of the usual planning processes of the first best theory in light of the gradient projection method has been recently attempted by d’Aspremont-Tulkens (1980). 3Gradient projection algorithms are usually defined for processes operating in discontinuous time [see Rosen (1961)]. We consider here their continuous time version.
J. Tit-ok and R. Guesnerie,
Tax reform
are derived in sections 4-6. Section 7 provides analysis and proves existence and convergence.
2. The gradient projection method -
a more careful
217
mathematical
A reminder
Let p,(x)
be an objective function, where x belongs to the Euclidean space that the feasible set is defined by M equality constraints {pi(x) = 0, i=l ,. . ., M}. All these functions are assumed to be continuously differentiable, and let VP,(X). V’pl (x), . . ., VpM(x) be their gradients (column vectors), S(x) the space spanned by Vpl(x),.. ., V’pM(x), and S(x) its orthogonal complement. S(x) is also the set of locally feasible moves at x. Let Q(x) be a matrix, derived from the matrix A(x) formed by Vpl(x), . . ., VpM(x), and composed of linearly independent vectors spanning S(x). The projection operator on S(x) is then RN. Assume
P(x)=I-Q(x)(Q’(x)Q(x))-‘Q’(x), where t denotes the transposition. The gradient projection of the objective
function
is
An alternative, but equivalent, development of this problem is as follows: Since RN= S(x)@s(x), any vector in RN can be decomposed unique/y as the sum of vectors in S(x) and S(x). This VpO(-l-) is the sum of P/)P(x) and some vector in S(x); that is, there exists a column vector a(x) such that
~PO(X)=~~p(x)+Q(x)&).
w
When the number of equality constraints is low, it is computationally easier to posit form (/l), and determine the components of a(x) by imposing the feasibility constraints (Vpi(x). Vpp(x) =O, Vi), than to use the general formula (u). The gradient projection is shown in fig. 1 for the case of one equality constraint, where Q(x)= (Vpl (x)). In the following, we shall work with two constraints (at most) and therefore shall systematically use this second method. Let us now discuss the link between the gradient projection problem and the problem of finding the ‘best small improvement associated with the Euclidean norm’. At a point x, the best small improvement associated with
278
the Euclidean
J. Tirole and R. Guesnerie,
norm,
where ll~xll= Jm
Tax reform
6x*, is given by
is the Euclidean
norm.
Fig. 1
It is easy to check that 6x* is proportional to the gradient projection V~,,(X).~ Thus the best small welfare improvement associated with the Euclidean norm (with equality constraints A’(x)Gx =0) or ‘optimal tax perturbation problem’ and the gradient projection problem are equivalent. The gradient projection method is just the algorithm corresponding to the one-time ‘best small improvement’. Note that if, in the best small improvement problem, one does not impose equality constraints, but only A’(x) 6x 5 0, the two methods may give different results. We shall discuss at length the implications of this observation for the models considered in sections 4. 5 and 6. ‘See for example Bazarra-Shetty (1979). Graphically, the best small improvement tries to minimize on s(x) the angle with VP,(X); clearly this leads to the projection of BP,(X) on s(x). The whole issue of the relationship between the gradient projection and the best small improvement was clarified thanks to conversations with A. Dixit.
219
J. Tirole and R. Guesnerie, Tax reform 3.
The economy There are 1 commodities
(h = 1,. . ., 1); the different
(all vectors in the paper will be column
price vectors belong
to
vectors).
(a) Consumers are indexed by i= 1,. . ., m; consumer i has ;I utility function Ui, and for consumer prices q and income Ii, solves the program: max Ui(xi) s.t. q .xisZi. The solution of this program, supposed unique, is denoted xi(q, Zi) (or xi(q) in sections 4 and 5, where Ii =O). We shall assume that xi is a C2 function on the interior of its domain. v(q,Zi)U,(x,(q, Zi)) is the indirect utility function and is assumed to be C2, and x=&xi is the total demand function. dX denotes the (1 x I) Jacobian matrix associated with the partial derivatives of x with respect to the consumer prices. For the economy with one consumer we shall call x1 the vector of derivatives of x with respect to income. (b) There is one private production sector with production set Y and production plan y. Given the production price system p, the private sector supplies y(p) which, we assume, is single valued and C2 on .P. 8Y denotes Jacobian matrix whose kernel contains, as is well known, the ray spanned p. Furthermore
we suppose that vectors 6u such that p .6u =0 and words, one can accommodate any demand) 6u such that p. 6u=O = dY _ ’ 6u (when small changes in production price production price to be taxed away.
its by
8Y is a one-to-one mapping between the the vectors 6p such that p ‘6~ = 0; in other desired small change in production (or net by a small production price change 6p
6Y - ’ is restricted to the appropriate &AS), and conversely net demand which have a zero value measured with the system are the only ones to be implementable through a change [see Guesnerie (1977)].5 Lastly, profits are assumed
(c) Finally, there is a public production sector controlled government, the production plan of which is denoted z. z satisfies where g is a C2 function. Vg denotes the gradient of g. For comments on the policy assumptions in this framework the reader is to Diamond-Mirrlees (197 1) or Guesnerie (1977).
‘Formally the assumption if the production function Guesnerie (1980)].
by the g(z)iO, general referred
requires that iY is of rank I- 1. This property holds, for example, 1s differentiable and has a non-zero Gaussian curvature 1 [see
280
J. Tirole and R. Guesnerie,
Tax reform
4. The gradient projection method in a model without a public sector In this section and the next we assume the system is in tight equilibrium:
Ii E 0, Vi. Suppose
that
at time t
(1) and let
(2) The following remark helps in interpreting C. Suppose that some policy change induces a change in aggregate consumption 6x. The production cost of the change is p .6x. Hence C, = xi pi(8xi/o?q,) is the production cost associated with a unit increase in the consumer price of commodity h, and C may be called the vector of production costs of consumer price changes. Following Guesnerie (1977) and section 3, the only directions of consumer price changes dq/dt which can be accommodated by a producer price change dp/dt are such that:
=04(p,q).dt=o.
dq
(3)
Thus the set of feasible consumer price changes can be represented by only one constraint, (3); the accommodating change in production prices will be given in a second stage by (4):
dp -
pY-
I,@ cxdc.
(4)
A change (dq/dt,dp/dt) satisfying (3) and (4) is said to be tight equilibrium preserving. For simplicity, the social welfare function will be assumed to be linear:
Let pi E ai(2K/ZIi)
be the marginal
social utility
of i’s income,
then
J. Tide
and R. Guesnerie,
281
Tux reform
Thus ( --&Pixih) is the marginal social surplus associated with a unit increase in the consumer price of h. We define a particular gradient projection algorithm in the consumer prices space by projecting ( -ci /&xi) on the hyperplane tangent to the set of feasible moves : {dq/dt 1C. (dq/dt)=O} (first stage), and then by accommodating dq/dt by the change in producer prices given by (4) (second stage). Proposition 1. Assume that C(p, q)#O. 7he gradient defined above leads to the following process:
4
dt=
projection
algorithm
(5)
(-FBlh)xi(Y))!-PIP,‘I)C(I’,q~,
where c(P2q)
’
-C (
Pi(qlxi(q) i
>
(6)
AP,4)= Ip3P~q)llz
’
(7)
Proof: From the analysis in section 2, the projection of the gradient of the objective function on the local approximation of the feasible set must have the form given by (5). To find p, we write the feasibility condition as C (dq/dt) =O or C (-xi/&xi) -~1) Cl)* =O. Eq. (7) is the accommodation formula, and recall that 11II is the Euclidean norm (see $2). n It is clear that the simplicity of the computations is due to our two stage procedure which allows us to project the gradient of the objective function on a single hyperplane in the consumer prices space. Another (one stage) procedure is to project the gradient of the objective function (-xi bixi, 0) in the (dq/dt,dp/dt) space on the intersection of the I hyperplanes defined by the market clearing conditions: ZX (dq/dt)-aY (dp/dt)=0.6 This problem is far more difficult, and would give results which are not easy to interpret; this remark vindicates our choice of the control variables (dq/dt) in the projection stage. But it is clear that it makes a difference into which space the projection is being made. In the local welfare optimization associated with our algorithm, the ‘optimal tax perturbation’ depends on whether only changes in consumer
282
J. Tirole and R. Guesnerie,
Tax reform
prices are constrained (according to the norm ((dq/dt\\, with corresponding accommodation by a move of production prices) or if both changes in consumer and production prices are constrained (according to the norm is important sensitivity problem is unavoidable in tax [[(dqldt), (dpldr)ll). Th’ reform theory. Of course we shall face it again when dealing with the other two models. Since we shall not systematically discuss this point again, the reader should keep in mind that the simplicity of our results is due to the particular choice of control variables in the projection stage. Let us now develop the relationship between the gradient projection technique, and Dixit’s (1979) ‘local welfare optimization’ and Diewert’s (1979) ‘optimal tax perturbation problem’. As explained in section 2, if only equality constraints are used in the definition of the feasible set, the three methods are equivalent. Dixit imposes only an inequality constraint: C(p,q) .(dq/dt) 5 0 (3’). Thus if in his problem the constraint is not binding, the ‘best change in consumer prices’ is proportional to the marginal social surplus vector and differs from our tight equilibrium preserving result. Actually Dixit shows that if ~(p, q) > 0, private production efficiency is desirable (while it is imposed in our model). Hence, in this case his result (eg. (29)) coincides with the one given by eqs. (5) and (6). It is only when p(p, q) < 0, that production efficiency is no longer desirable and that our tax revision differs from Dixit’s best small improvement. Let us comment now on proposition 1 and stress the economic meaning of our tax revision formulae.7 Let us suppose first that p(p,q)>O. The direction of change dq,/dt is the combination of two opposite changes: (1) In the first one ( -~ijlixi,), the consumption price of commodity h is ‘reduced’ in proportion to the sum of the quantities consumed by each consumer and weighted by the social marginal utilities of incomes. Hence, the price of a consumption good h should be reduced more if this good is consumed more or if it is demanded by people who have a high social marginal utility of income. (2) In a second step, corrections should be made in order to satisfy feasibility. These corrections consist in increasing (p>O) the price of commodity h in proportion of the production surplus generated by a unit increase in qh. Hence the price of a commodity whose price reduction is costly in terms of equivalent production should be increased more. The appropriate weights of these two opposite changes ~ the first one is the most socially desirable and the second one gives the cheapest correction ~ have to be inversely correlated with their production costs C ( --xi pixi) and C. (-C). Hence, the directions are defined by eqs. (5) and (6). ‘It is worth noticing here that the process given by proposition 1 is (as desirable) independent of the choice of units in which quantities are measured. Furthermore the direction defined by eqs. (5) and (6) - which is the only relevant information ~ is not modified if all pi are doubled, i.e. it does not depend upon the cardinal representation of W
J. Tirole and R. Guesnerie, Tax reform
283
p is negative if the first step price reduction induces a positive production surplus: C . (-xi pixi) ~0, i.e. if it releases, instead of consumes, resources. In this case, the second step, which is both costly and socially harmful, would not be desirable.’
5. The gradient projection method and the best investment firm
policy of a public
We now add a public firm to the preceding model. We impose public production efficiency and tightness of the equilibrium; that is, at time t the state of the economy is entirely defined by {p(t), q(t), z(t)} satisfying:
gMt))=O,
(8)
Cxi(q(t))=Y(p(t))+z(t).
(9)
L
Repeating the argument given in the preceding section we see that a tight equilibrium preserving direction of price and public production changes (dqldt, dpldt, dzldt) meets the following constraints:
C(p,q)
J&p+o,
(10)
Pg(z).$=O;
(11)
. We again use a two stage procedure. First in the (dq/dt,dz/dt) compute the projection of the gradient of the objective function
(12) space, we
(r > 1 /&Y; 0
on the intersection of the two hyperplanes defined by (10) and (11); we then accommodate the change in net demand for the private goods according to (12). ‘It can be argued that the desirability of production inefficiency (p(
284
J. Tirole and R. Guesnerie, Tax reform
Proposition the revisions
2. Assuring that either C # 0 or: p and Vg are not proportional, associated with the gradient projection are given by:
dq
(13)
z= dz -= dt
(14)
V(P, 4, z) proj,(,,p,
(15)
,
and (17)
is the whereprojHcz,p
production
H(z)={UI Proof:
following
projection
of p on the hyperplane
tangent
to the public
set:
According form:
Vg(z).U=O}.
to section
1 and p are determined
2 (eq. (/?)) the gradient
by the feasibility
constraints
projection
takes
the
(10) and (11):
(10’)
-IL11 vgll2+vp.
vg=o.
(11’)
J. Tirole and R. Guesnerie,
Thus 3.= v[(p . Vg)/ll Vg11’], and substituting dz
(14).
>, into (10’) gives (16). Finally,
(P. Vg) Vg IIBgl12
dt=vp-v
gives
28.5
Tax wforform
(Note
that
by
the
Schwarz
(/pII’
inequality,
W
2 (P. ~gY/ll ~gl12).
The most remarkable characteristic 2 is given by (14): supposing v>O, direction production
of
public
price
production
system
of the process exhibited one sees that ut euch
chunge
on the tangent
is
obtained
hyperplane
by
in proposition time
projecting
to the public
the
‘best’ p,
production
the set,
and this conclusion holds irrespective of the specific social weights pi. Hence, the production price system which is an adequate reference shadow price system in any second best optimum of this economy, is also an appropriate all Ilong a reference, in the sense just mentioned, outside the optimum process directed to the steepest ascent (see fig. 2).
Fig. 2
286
J. Tirole and R. Guesnerie, Tax reform
However, this latter condition would not hold if v(p, 4, z)
Let us now recapitulate eqs. (13t(17) when v>O. moves :
and give an intuitive economic understanding of The best move is obtained as the sum of three
(1) First, a move of public production j in the direction magnitude CI.This move generates a social surplus
=P . projH,,,p
= a
9We owe this remark
(
(P . VgJ2
-p+llPl12) II w
to the referee
(positive
if cc>O)
proj,,,,p,
with
J. Tide
(2) Second, -C(P,q),
reform
287
system social
in the direction of surplus generated is
consumption 6. The cost
price system in the associated is
and R. Guesnerie,
a move of the consumption with the same magnitude
Tax
price CL The
+l12. (3) Third, and finally, direction -cpixi,
a move of the with magnitude
Observing that the first and second step surpluses have to be integrally consumed in the third step, we conclude that 2 and 6 have to be inversely proportional respectively to C.(-CiBiXi) and (IIP112-[(P’~g)2/11~gl121 + IICJI”), which explains eg. (16). As in section 4, a ‘reduction’ of prices in the direction of the gradient of the social welfare function is combined with a correction in the direction of the vector of social costs of price changes. However, the weights of the two basic moves are modilied, as compared to the previous section: the weight of the correction is smaller thanks to the additional surplus generated by the public investment.
6. The gradient distortions
projection
method
in a one consumer
economy
with initial
We consider in this section a one consumer economy in which a price distortion exists and where income can be transferred from the government to the consumer in a lump sum way.” For such a model the desirability of a partial reduction of price distortion from the distorted initial situation is an issue which has often been considered in the literature. The problem actually has a reasonably simple structure and strong and appealing results have been obtained (see in particular Dixit (1975) for the derivation of the most important ones, and a more comprehensive set of references). It seems worth exhibiting the best infinitesimal change defined in this model by the gradient projection method. Dropping index i, we define as before a tight equilibrium preserving change (dqldt, dpldt, dlldt) as satisfying
dq
C(p,q,I).dt+r(PIq,l)dt=O.
where r is the production
dl
cost of a change
“We do not consider the public not alter the nature of the result.
firm any longer.
(18) in income: However,
r-p
introducing
.xr. a public
firm would
J. Tirole and R. Guesnerie, Tax reform
288
For a given change changes are now
(dqldt, dlldt),
the accommodating
production
dp d+y-’
The changes
(19)
in real income
dl’ dl -=--X’dt dt
price
I’ is given by
dq
(20)
dt ’
and is, by definition, proportional to the change in social welfare. We shall use real income as a control variable, and therefore we transform (18): dq dl’ (C+rx)~-++-=O~CU~-+r--~O, dt dt
C” has actually
dq
dl’
dt
dt
(21)
a simple expression:
CU=dX’p+p
‘X,X’
(axt+xx;)p
= 2x”p,
(22)
where 2X is the (symmetric) matrix of compensated derivatives. C” is thus the vector of production costs of compensated price changes. Another interpretation is given below. In the (dq/dt,dI’/dt) space, we project the gradient of the objective function (y) on the hyperplane defined by (21), and we then accommodate the change in production by (19). Proposition 3. The gradient prqjection income changes proportional to
dq
-= dt
method
-r(q,p,I)C”(q,p,I)=r(q,p,I)(~X”(q,I)T),
dl’ ~=llC”(q>PJl~“,
selects
prices
and
real
(23)
(24)
289
J. Tirole and R. Guesnerie, Tax rgform
dp z=aY
-; (p) -u ax
(25)
)
where T-q-p
(26)
is the tax vector. Proof:
In the (dq/dt, dZ’/dt) space, the gradient
projection
is given by
(27)
(26). Substituting Imposing constraint (21) gives -~~~C”~~2+r-~r2=0 into (27) and multiplying by 11cull2 + r2, we obtain (23) and (24). Note that q belongs to the kernel of 8X”‘, which implies dX”p=c?X”(q-T)= Lastly, (25) is the accommodating move in production prices. The main result is eq. (23). By analogy
the index of discouragement
of commodity
with Mirrlees
-2X”T.
n
(1976), who calls
h, we shall call
the absolute index of discouragement of h. C’, which is the vector of production costs of compensated consumer price changes, can thus be interpreted as (the opposite of) the vector of absolute indices of discouragement. Assuming 2~0 (that is, an increase in income induces a change of consumption which has a positive value at production prices) the consumption price of a good h should be reduced proportionally to its absolute index of discouragement. It is only in the case where these indices are
perfectly correlated with the distortions that the direction of steepest ascent actually coincides with a proportional reduction of distortions. Note also that, following a line of reasoning inspired by sections 4 and 5, one can argue
Tax reform
J. Tirole and R. Guesnerie,
290
that the case where r
7. Existence
of trajectories and convergence for gradient algorithms
In this section we will consider existence system of differential equations (13), (14) model with one public firm. Very similar specific problem of section 4, and similar process of section 6. First we have to check that a solution properties expected from its design. This is Proposition
4.
A
solution
p(t),
q(t),
and convergence problems for the and (15) of section 5, i.e. for the results would apply to the more methods would also apply to the of the process actually easily done in proposition
z(t)
of
the
system
of
has the 4.
differential
equations (13), (14) and (15) of section 5 defined on [0, T] and starting from P(O), q(O), z(0) which is u tight equilibrium, has, z$ it exists, the following Properties: (1) at each time t, {P(t), q(t), z(t)) is u tight equilibrium u_nd function oft. IlP(0ll= llP(O)ll> jlq(t)ll= 11q(O)ll; (2) W zs a non-decreusing Proof:
The proof of the first part of (1) is straightforward. The fact that the Euclidean norm of prices is preserved results from the integration of p. (dp/dt)=O, which [as in Guesnerie (1977)] is a consequence of the definition of 8Y - ’ and q (dq/dt) = 0, which is easily shown from (13). To prove the second part of proposition 4, let us call
~(~)=J~C~~2+~~P~~2--
(P . vgj2
IIvgl12 .
One then has
and thus d Wldt is strictly
positive
unless:
“Let us note, however, that the desirability of production inefficiency is much less likely than previously. When all commodities are not inferior r>O [see Dixit (1975, p. 107) for very closely related remarks].
J. Tirole and R. Guesnerie,
Tax reform
291
or (b) In the following
the corresponding
points
will be said to be of type (a) or
n
type (b).
The next question to be raised concerns the existence of solutions. Local existence is assured by the differentiability assumptions as soon as D > 0, but we need the existence of trajectories, i.e. of solutions defined (0, + a). For that we shall introduce the following assumptions. (HI). There does not exist p, q, z, and i, >O s.t. xi xi(q)= y(p)+ z, C(p, q) =o, p=i.Vg(z), g(z)=O.‘2 (Hl) ensures that D is always strictly positive. One can compare this assumption with assumption (H7) in Folgelman-Guesnerie-Quinzii (1978) and argue, as they do, that (Hl) is likely to be generically true. z, xicXi, (H2). Let A={x,,y,~I~~x~=y+ ‘physically feasible states’. A is compact. This is a standard
as’+.umption. It ensures
KEY,
g(z)=01
be the
that any trajectory
set of
will remain
in
the compact set K x projzA, where K = ip, q 1[(PJI= Ilp(O)ll and llq[\ = ~~q(O)~~>. To avoid problems at the boundary of 9, we add two other assumptions to analogous to assumptions (A) and (H3) in FogelmanGuesnerie-Quinzii, which the reader is referred for justifications. (H3). Let q” be a sequence of consumption prices converging to 4, such that for some (but not all) h:q,=O. Then either 11x(q”)1I++x or W(...,xi(q”),...)
to p, such that
Proposition 5. For any initial tight equilibrium p(O), q(O), z(O), the system defined by (13), (14) and (15) has a solution defined on (0, + cc). Proof: (H2), (H3) and (H4) imply that any trajectory stays subset of feasible states N contained in {K n 9 x P} x proj,A. “The reader will check that there is a close connection D >O and the constraints qualifications of the optimisation
between both literature.
in a compact
(Hl) and the fact that
J. Tirole and R. Guesnerie, Tax reform
292
Because of (Hl), there exists an open set 0 containing N and such that the second member of eqs. (13), (14) and (15) is continuously differentiable. Hirsch-Smale’s (1974) theorem gives the conclusion (p. 196). n Finally,
we have the following
Proposition 6. Every points of type (b). ProoJ: function
proposition.
trajectory
We use an extension
converges
either to points of type (a) or to
of Liapounov
theorem,
with
the Liapounov
I
V= max W(p,q,2) - WMt), 4(t), z(t)). i N V is non-increasing limit
Ps i
on trajectory
and being
bounded
above
from zero has a
max W - W(p(O), q(O), z(O)). N 1
Let I’+ be the limit set of the process, accumulation points of p(t,), q(tn), z(t,) One can see that on r+, V= P and dV/dt (1958), any trajectory (which is bounded) the set of points (a) and (b) which contains
i.e. the set of points which are for infinite sequences t,, t,+ +r;c. =O. Then, according to Hurewicz converges to I-+ and therefore to r+. n
Points of type (a) are local social welfare extrema in the sense that the necessary conditions of optimality are satisfied: they may be either global optima or local optima or saddle points. Points of type (b) are points at which an increase of social welfare is impossible if one is constrained, as supposed here, to have an efficient public production. Hence at such points temporary production inefficiencies in the public sector are desirable.
8. Conclusions It is clear there are generally many ways by which an objective function can be increased at a non-optimal point. This point makes tax reform somehow arbitrary, and has already been recognized by authors considering small moves of a system. The same point, though somewhat disguised, arises when choosing an algorithm and its control variables. Therefore the issue might well be the following. Does there exist a process which has given
J. Tirole und R. Guesnerie,
Tux reform
293
properties ~ monotonicity and convergence to the set of points satisfying the second best optimality conditions ~ and is easy to interpret and to implement? We show in this paper that the question has a positive answer. The use of the simplest algorithm ~ the gradient projection algorithm ~~~~ and a judicious choice of control variables leads to appealing rules in spite of their arbitrariness. In particular public production is guided by the private production prices. (See section 5. This is the tax reform equivalent of the Diamond-Mirrlees efficiency result). In a one consumer economy with lump sum transfers the change in consumer prices is proportional to the vector of absolute indices of discouragement (section 6). We feel that the economic insights and implementability of these rules vindicate the use of gradient projection methods in tax reform theory. References Arrow, K. and L. Hurwicz, 1960, Decentralization and computation in resource allocation, in: R.W. Pfouts, ed., Essays in economics and econometrics (University of North Carolina Press, Chapel Hill) 34104. Bazarra, MS. and C.M. Shetty, 1979, Nonlinear programming: Theory and algorithm (Wiley, New York). D’Aspremont, C. and F. Tulkens, 1980, Commodity exchanges as gradient processes, Econometrica 48, 387~400. Diamond, P. and J. Mirrlees, 1971, Optimal taxation and public production, American Economic Review 61, 8 -27, 261-278. Diewert, W.E., 1979, Optimal tax perturbations, Journal of Public Economics 10, 139-- 177. Dixit, A., 1975, Welfare effects of tax and price changes, Journal of Public Economics 4, 1033 123. Dixit, A., 1979, Price changes and optimum taxation in a many-consumer economy, Journal of Public Economics 11, 143-157. Fogelman, F., R. Guesnerie and M. Quinzii. 1978, Dynamic processes for tax reform theory, Journal of Economic Theory 17, 20&225. Guesnerie, R., 1977, On the direction of tax reform. Journal of Public Economics 7, 179~ 222. Guesnerie, R., 1980, Modeles de 1’Economie publique, Monographies du Seminaire d’Economt_trie. Hadley, G., 1964, Nonlinear and dynamic programming (Addison-Wesley, Reading, Mass.). Heal, G., 1973, The theory of economic planning (North-Holland, Amsterdam). Hirsch, M.W. and S. Smale, 1974, Differential equations dynamical systems and linear algebra (Academic Press, New York). Hurewicz, W., 1958, Lectures on ordinary differential equations (M.I.T., Cambridge, Mass.). La Salle and S. Lefschetz, 1961, Stability by Liapunov’s direct method (Academic Press, New York). Mirrlees, J., 1976. Optimal tax theory, a synthesis, Journal of Public Economics 6, 3299358. Rosen, J.B., 1961, The gradient projection method for nonlinear programming Part II: Nonlinear constraint, SIAM Journal. Tirole, J., 1978, Essais sur le calcul economique public et le taux d’actualisation, These pour le doctorat de 3eme cycle. Universite de Paris IX-Dauphine. Weymark. J., 1979, A reconciliatron of recent results in taxation theory, Journal of Public Economics 12. 171-190.
of Public Economics
TAX REFORM
15 (1981) 2755293.
FROM
North-Holland
Publishing
THE GRADIENT VIEWPOINT
Company
PROJECTION
J. TIROLE* Ecole Nationale des Pants et Chausstes and M.I.T, Cambridge, MA 02139, USA
R. GUESNERIE CEPREMAP, Received
75013 Paris, France
April 1979, revised version
received July 1980
This paper examines the problems of gradual tax reform. Desirable small moves of the tax system are determined using the principles of the well-known gradient projection algorithm. Three classical models of public finance are considered. For each of them a continuous time dynamic process is exhibited along which the social welfare function increases until convergence to local second best extrema. Although the selection of trajectories according to the gradient projection method has some arbitrariness, it leads to intuitively and economically appealing rules such as a tax reform analog of the classical optimal taxation result on production efficiency.
1. Introduction The recent literature on theoretical public finance has shown an increased concern for the problems of gradual tax reform, putting a particular emphasis on the local analysis of the feasible and desirable moves of a system from non-optimal points. Desirability can be evaluated according to different criteria. For example, in the context of the well-known Diamond-Mirrlees model (1971), which will be considered here, Guesnerie (1977) looks for directions of tax changes making every consumer better off, whereas Diewert (1979) and Dixit (1979) rely on a social welfare function to construct what they call respectively ‘optimal tax perturbations’ and ‘best small improvemements’.’ These procedures will remind the reader of the classical algorithm of search of the direction of steepest constrained ascent in applied mathematics. Such a direction is obtained by projecting the gradient of the objective *We are grateful to A. Dixit, H. Tulkens, and especially to the referee, whose helpful comments considerably improved the exposition. Part of the material presented here can be found in I irole (1978) who gratefully acknowledges CEPREMAP’s hospitality. ‘See also Weymark (1979) for a comparison of papers by Guesnerie and Diewert.
004772727/81/0000~000/$02.50
0 North-Holland
Publishing
Company
216
J. Tirole und R. Guesnerie, Tax reform
function on the linear local approximation to the feasible set; this is the gradient projection method or the gradient projection algorithm.2S3 This paper applies the gradient projection method to three classical models in public finance. For each of them a continuous time dynamic process will be exhibited which leads to a steady increase in a given social welfare function until convergence to the set of points satisfying the (second best) optimality conditions is achieved. We should make at this point an important warning. The gradient projection algorithm that we single out here is certainly not the only one which could be used for our purpose (although it is certainly the simplest one). Furthermore, for a given algorithm, there exists different processes according to the choice of control variables; in other words, the trajectory given by the gradient projection method depends on the space in which the projection is made. This is the counterpart, here, of the problems faced by Dixit and Diewert in choosing the vectors, the norm of which is constrained. Hence, we do not claim to provide a unique solution to the problems of tax reform. Among the many monotone dynamic processes which can be expected to converge to local welfare optima, we single out one. This is in itself a desirable goal since tax reform theory might well content itself with an algorithm prescribing changes in taxes and public production, however arbitrary they may be. However, we do more here: the tax and public good production revision formulae (which rely on particularly natural principles) are both simple and economically appealing, as shown by the following prescriptions. The consumer prices revision can be decomposed into two moves, the first (socially desirable one) proportional to the marginal social surplus, and the second being the cheapest (in terms of private production) correction in order to ensure feasibility (section 4). Public production is guided at each instant by the private production prices; this rule can be considered as the reform analog of the Diamond-Mirrlees production efficiency rule at a second best optimum (section 5). Lastly, in a one consumer economy with lump sum transfer, consumer prices are revised according to their absolute index of discouragement (section 6). We also discuss the possibility of non-desirability of private production efficiency, and its effect on the above prescriptions. The rest of the paper is organized as follows: section 2 gives some background material on the gradient projection method, and section 3 presents the classical Diamond-Mirrlees model. The application of the gradient projection method to the model and the results mentioned above ’ ‘Notice that the use of gradient process in economics goes back at least to Arrow-Hurwicz. A systematic analysis of the usual planning processes of the first best theory in light of the gradient projection method has been recently attempted by d’Aspremont-Tulkens (1980). 3Gradient projection algorithms are usually defined for processes operating in discontinuous time [see Rosen (1961)]. We consider here their continuous time version.
J. Tit-ok and R. Guesnerie,
Tax reform
are derived in sections 4-6. Section 7 provides analysis and proves existence and convergence.
2. The gradient projection method -
a more careful
217
mathematical
A reminder
Let p,(x)
be an objective function, where x belongs to the Euclidean space that the feasible set is defined by M equality constraints {pi(x) = 0, i=l ,. . ., M}. All these functions are assumed to be continuously differentiable, and let VP,(X). V’pl (x), . . ., VpM(x) be their gradients (column vectors), S(x) the space spanned by Vpl(x),.. ., V’pM(x), and S(x) its orthogonal complement. S(x) is also the set of locally feasible moves at x. Let Q(x) be a matrix, derived from the matrix A(x) formed by Vpl(x), . . ., VpM(x), and composed of linearly independent vectors spanning S(x). The projection operator on S(x) is then RN. Assume
P(x)=I-Q(x)(Q’(x)Q(x))-‘Q’(x), where t denotes the transposition. The gradient projection of the objective
function
is
An alternative, but equivalent, development of this problem is as follows: Since RN= S(x)@s(x), any vector in RN can be decomposed unique/y as the sum of vectors in S(x) and S(x). This VpO(-l-) is the sum of P/)P(x) and some vector in S(x); that is, there exists a column vector a(x) such that
~PO(X)=~~p(x)+Q(x)&).
w
When the number of equality constraints is low, it is computationally easier to posit form (/l), and determine the components of a(x) by imposing the feasibility constraints (Vpi(x). Vpp(x) =O, Vi), than to use the general formula (u). The gradient projection is shown in fig. 1 for the case of one equality constraint, where Q(x)= (Vpl (x)). In the following, we shall work with two constraints (at most) and therefore shall systematically use this second method. Let us now discuss the link between the gradient projection problem and the problem of finding the ‘best small improvement associated with the Euclidean norm’. At a point x, the best small improvement associated with
278
the Euclidean
J. Tirole and R. Guesnerie,
norm,
where ll~xll= Jm
Tax reform
6x*, is given by
is the Euclidean
norm.
Fig. 1
It is easy to check that 6x* is proportional to the gradient projection V~,,(X).~ Thus the best small welfare improvement associated with the Euclidean norm (with equality constraints A’(x)Gx =0) or ‘optimal tax perturbation problem’ and the gradient projection problem are equivalent. The gradient projection method is just the algorithm corresponding to the one-time ‘best small improvement’. Note that if, in the best small improvement problem, one does not impose equality constraints, but only A’(x) 6x 5 0, the two methods may give different results. We shall discuss at length the implications of this observation for the models considered in sections 4. 5 and 6. ‘See for example Bazarra-Shetty (1979). Graphically, the best small improvement tries to minimize on s(x) the angle with VP,(X); clearly this leads to the projection of BP,(X) on s(x). The whole issue of the relationship between the gradient projection and the best small improvement was clarified thanks to conversations with A. Dixit.
219
J. Tirole and R. Guesnerie, Tax reform 3.
The economy There are 1 commodities
(h = 1,. . ., 1); the different
(all vectors in the paper will be column
price vectors belong
to
vectors).
(a) Consumers are indexed by i= 1,. . ., m; consumer i has ;I utility function Ui, and for consumer prices q and income Ii, solves the program: max Ui(xi) s.t. q .xisZi. The solution of this program, supposed unique, is denoted xi(q, Zi) (or xi(q) in sections 4 and 5, where Ii =O). We shall assume that xi is a C2 function on the interior of its domain. v(q,Zi)U,(x,(q, Zi)) is the indirect utility function and is assumed to be C2, and x=&xi is the total demand function. dX denotes the (1 x I) Jacobian matrix associated with the partial derivatives of x with respect to the consumer prices. For the economy with one consumer we shall call x1 the vector of derivatives of x with respect to income. (b) There is one private production sector with production set Y and production plan y. Given the production price system p, the private sector supplies y(p) which, we assume, is single valued and C2 on .P. 8Y denotes Jacobian matrix whose kernel contains, as is well known, the ray spanned p. Furthermore
we suppose that vectors 6u such that p .6u =0 and words, one can accommodate any demand) 6u such that p. 6u=O = dY _ ’ 6u (when small changes in production price production price to be taxed away.
its by
8Y is a one-to-one mapping between the the vectors 6p such that p ‘6~ = 0; in other desired small change in production (or net by a small production price change 6p
6Y - ’ is restricted to the appropriate &AS), and conversely net demand which have a zero value measured with the system are the only ones to be implementable through a change [see Guesnerie (1977)].5 Lastly, profits are assumed
(c) Finally, there is a public production sector controlled government, the production plan of which is denoted z. z satisfies where g is a C2 function. Vg denotes the gradient of g. For comments on the policy assumptions in this framework the reader is to Diamond-Mirrlees (197 1) or Guesnerie (1977).
‘Formally the assumption if the production function Guesnerie (1980)].
by the g(z)iO, general referred
requires that iY is of rank I- 1. This property holds, for example, 1s differentiable and has a non-zero Gaussian curvature 1 [see
280
J. Tirole and R. Guesnerie,
Tax reform
4. The gradient projection method in a model without a public sector In this section and the next we assume the system is in tight equilibrium:
Ii E 0, Vi. Suppose
that
at time t
(1) and let
(2) The following remark helps in interpreting C. Suppose that some policy change induces a change in aggregate consumption 6x. The production cost of the change is p .6x. Hence C, = xi pi(8xi/o?q,) is the production cost associated with a unit increase in the consumer price of commodity h, and C may be called the vector of production costs of consumer price changes. Following Guesnerie (1977) and section 3, the only directions of consumer price changes dq/dt which can be accommodated by a producer price change dp/dt are such that:
=04(p,q).dt=o.
dq
(3)
Thus the set of feasible consumer price changes can be represented by only one constraint, (3); the accommodating change in production prices will be given in a second stage by (4):
dp -
pY-
I,@ cxdc.
(4)
A change (dq/dt,dp/dt) satisfying (3) and (4) is said to be tight equilibrium preserving. For simplicity, the social welfare function will be assumed to be linear:
Let pi E ai(2K/ZIi)
be the marginal
social utility
of i’s income,
then
J. Tide
and R. Guesnerie,
281
Tux reform
Thus ( --&Pixih) is the marginal social surplus associated with a unit increase in the consumer price of h. We define a particular gradient projection algorithm in the consumer prices space by projecting ( -ci /&xi) on the hyperplane tangent to the set of feasible moves : {dq/dt 1C. (dq/dt)=O} (first stage), and then by accommodating dq/dt by the change in producer prices given by (4) (second stage). Proposition 1. Assume that C(p, q)#O. 7he gradient defined above leads to the following process:
4
dt=
projection
algorithm
(5)
(-FBlh)xi(Y))!-PIP,‘I)C(I’,q~,
where c(P2q)
’
-C (
Pi(qlxi(q) i
>
(6)
AP,4)= Ip3P~q)llz
’
(7)
Proof: From the analysis in section 2, the projection of the gradient of the objective function on the local approximation of the feasible set must have the form given by (5). To find p, we write the feasibility condition as C (dq/dt) =O or C (-xi/&xi) -~1) Cl)* =O. Eq. (7) is the accommodation formula, and recall that 11II is the Euclidean norm (see $2). n It is clear that the simplicity of the computations is due to our two stage procedure which allows us to project the gradient of the objective function on a single hyperplane in the consumer prices space. Another (one stage) procedure is to project the gradient of the objective function (-xi bixi, 0) in the (dq/dt,dp/dt) space on the intersection of the I hyperplanes defined by the market clearing conditions: ZX (dq/dt)-aY (dp/dt)=0.6 This problem is far more difficult, and would give results which are not easy to interpret; this remark vindicates our choice of the control variables (dq/dt) in the projection stage. But it is clear that it makes a difference into which space the projection is being made. In the local welfare optimization associated with our algorithm, the ‘optimal tax perturbation’ depends on whether only changes in consumer
282
J. Tirole and R. Guesnerie,
Tax reform
prices are constrained (according to the norm ((dq/dt\\, with corresponding accommodation by a move of production prices) or if both changes in consumer and production prices are constrained (according to the norm is important sensitivity problem is unavoidable in tax [[(dqldt), (dpldr)ll). Th’ reform theory. Of course we shall face it again when dealing with the other two models. Since we shall not systematically discuss this point again, the reader should keep in mind that the simplicity of our results is due to the particular choice of control variables in the projection stage. Let us now develop the relationship between the gradient projection technique, and Dixit’s (1979) ‘local welfare optimization’ and Diewert’s (1979) ‘optimal tax perturbation problem’. As explained in section 2, if only equality constraints are used in the definition of the feasible set, the three methods are equivalent. Dixit imposes only an inequality constraint: C(p,q) .(dq/dt) 5 0 (3’). Thus if in his problem the constraint is not binding, the ‘best change in consumer prices’ is proportional to the marginal social surplus vector and differs from our tight equilibrium preserving result. Actually Dixit shows that if ~(p, q) > 0, private production efficiency is desirable (while it is imposed in our model). Hence, in this case his result (eg. (29)) coincides with the one given by eqs. (5) and (6). It is only when p(p, q) < 0, that production efficiency is no longer desirable and that our tax revision differs from Dixit’s best small improvement. Let us comment now on proposition 1 and stress the economic meaning of our tax revision formulae.7 Let us suppose first that p(p,q)>O. The direction of change dq,/dt is the combination of two opposite changes: (1) In the first one ( -~ijlixi,), the consumption price of commodity h is ‘reduced’ in proportion to the sum of the quantities consumed by each consumer and weighted by the social marginal utilities of incomes. Hence, the price of a consumption good h should be reduced more if this good is consumed more or if it is demanded by people who have a high social marginal utility of income. (2) In a second step, corrections should be made in order to satisfy feasibility. These corrections consist in increasing (p>O) the price of commodity h in proportion of the production surplus generated by a unit increase in qh. Hence the price of a commodity whose price reduction is costly in terms of equivalent production should be increased more. The appropriate weights of these two opposite changes ~ the first one is the most socially desirable and the second one gives the cheapest correction ~ have to be inversely correlated with their production costs C ( --xi pixi) and C. (-C). Hence, the directions are defined by eqs. (5) and (6). ‘It is worth noticing here that the process given by proposition 1 is (as desirable) independent of the choice of units in which quantities are measured. Furthermore the direction defined by eqs. (5) and (6) - which is the only relevant information ~ is not modified if all pi are doubled, i.e. it does not depend upon the cardinal representation of W
J. Tirole and R. Guesnerie, Tax reform
283
p is negative if the first step price reduction induces a positive production surplus: C . (-xi pixi) ~0, i.e. if it releases, instead of consumes, resources. In this case, the second step, which is both costly and socially harmful, would not be desirable.’
5. The gradient projection method and the best investment firm
policy of a public
We now add a public firm to the preceding model. We impose public production efficiency and tightness of the equilibrium; that is, at time t the state of the economy is entirely defined by {p(t), q(t), z(t)} satisfying:
gMt))=O,
(8)
Cxi(q(t))=Y(p(t))+z(t).
(9)
L
Repeating the argument given in the preceding section we see that a tight equilibrium preserving direction of price and public production changes (dqldt, dpldt, dzldt) meets the following constraints:
C(p,q)
J&p+o,
(10)
Pg(z).$=O;
(11)
. We again use a two stage procedure. First in the (dq/dt,dz/dt) compute the projection of the gradient of the objective function
(12) space, we
(r > 1 /&Y; 0
on the intersection of the two hyperplanes defined by (10) and (11); we then accommodate the change in net demand for the private goods according to (12). ‘It can be argued that the desirability of production inefficiency (p(
284
J. Tirole and R. Guesnerie, Tax reform
Proposition the revisions
2. Assuring that either C # 0 or: p and Vg are not proportional, associated with the gradient projection are given by:
dq
(13)
z= dz -= dt
(14)
V(P, 4, z) proj,(,,p,
(15)
,
and (17)
is the whereprojHcz,p
production
H(z)={UI Proof:
following
projection
of p on the hyperplane
tangent
to the public
set:
According form:
Vg(z).U=O}.
to section
1 and p are determined
2 (eq. (/?)) the gradient
by the feasibility
constraints
projection
takes
the
(10) and (11):
(10’)
-IL11 vgll2+vp.
vg=o.
(11’)
J. Tirole and R. Guesnerie,
Thus 3.= v[(p . Vg)/ll Vg11’], and substituting dz
(14).
>, into (10’) gives (16). Finally,
(P. Vg) Vg IIBgl12
dt=vp-v
gives
28.5
Tax wforform
(Note
that
by
the
Schwarz
(/pII’
inequality,
W
2 (P. ~gY/ll ~gl12).
The most remarkable characteristic 2 is given by (14): supposing v>O, direction production
of
public
price
production
system
of the process exhibited one sees that ut euch
chunge
on the tangent
is
obtained
hyperplane
by
in proposition time
projecting
to the public
the
‘best’ p,
production
the set,
and this conclusion holds irrespective of the specific social weights pi. Hence, the production price system which is an adequate reference shadow price system in any second best optimum of this economy, is also an appropriate all Ilong a reference, in the sense just mentioned, outside the optimum process directed to the steepest ascent (see fig. 2).
Fig. 2
286
J. Tirole and R. Guesnerie, Tax reform
However, this latter condition would not hold if v(p, 4, z)
Let us now recapitulate eqs. (13t(17) when v>O. moves :
and give an intuitive economic understanding of The best move is obtained as the sum of three
(1) First, a move of public production j in the direction magnitude CI.This move generates a social surplus
=P . projH,,,p
= a
9We owe this remark
(
(P . VgJ2
-p+llPl12) II w
to the referee
(positive
if cc>O)
proj,,,,p,
with
J. Tide
(2) Second, -C(P,q),
reform
287
system social
in the direction of surplus generated is
consumption 6. The cost
price system in the associated is
and R. Guesnerie,
a move of the consumption with the same magnitude
Tax
price CL The
+l12. (3) Third, and finally, direction -cpixi,
a move of the with magnitude
Observing that the first and second step surpluses have to be integrally consumed in the third step, we conclude that 2 and 6 have to be inversely proportional respectively to C.(-CiBiXi) and (IIP112-[(P’~g)2/11~gl121 + IICJI”), which explains eg. (16). As in section 4, a ‘reduction’ of prices in the direction of the gradient of the social welfare function is combined with a correction in the direction of the vector of social costs of price changes. However, the weights of the two basic moves are modilied, as compared to the previous section: the weight of the correction is smaller thanks to the additional surplus generated by the public investment.
6. The gradient distortions
projection
method
in a one consumer
economy
with initial
We consider in this section a one consumer economy in which a price distortion exists and where income can be transferred from the government to the consumer in a lump sum way.” For such a model the desirability of a partial reduction of price distortion from the distorted initial situation is an issue which has often been considered in the literature. The problem actually has a reasonably simple structure and strong and appealing results have been obtained (see in particular Dixit (1975) for the derivation of the most important ones, and a more comprehensive set of references). It seems worth exhibiting the best infinitesimal change defined in this model by the gradient projection method. Dropping index i, we define as before a tight equilibrium preserving change (dqldt, dpldt, dlldt) as satisfying
dq
C(p,q,I).dt+r(PIq,l)dt=O.
where r is the production
dl
cost of a change
“We do not consider the public not alter the nature of the result.
firm any longer.
(18) in income: However,
r-p
introducing
.xr. a public
firm would
J. Tirole and R. Guesnerie, Tax reform
288
For a given change changes are now
(dqldt, dlldt),
the accommodating
production
dp d+y-’
The changes
(19)
in real income
dl’ dl -=--X’dt dt
price
I’ is given by
dq
(20)
dt ’
and is, by definition, proportional to the change in social welfare. We shall use real income as a control variable, and therefore we transform (18): dq dl’ (C+rx)~-++-=O~CU~-+r--~O, dt dt
C” has actually
dq
dl’
dt
dt
(21)
a simple expression:
CU=dX’p+p
‘X,X’
(axt+xx;)p
= 2x”p,
(22)
where 2X is the (symmetric) matrix of compensated derivatives. C” is thus the vector of production costs of compensated price changes. Another interpretation is given below. In the (dq/dt,dI’/dt) space, we project the gradient of the objective function (y) on the hyperplane defined by (21), and we then accommodate the change in production by (19). Proposition 3. The gradient prqjection income changes proportional to
dq
-= dt
method
-r(q,p,I)C”(q,p,I)=r(q,p,I)(~X”(q,I)T),
dl’ ~=llC”(q>PJl~“,
selects
prices
and
real
(23)
(24)
289
J. Tirole and R. Guesnerie, Tax rgform
dp z=aY
-; (p) -u ax
(25)
)
where T-q-p
(26)
is the tax vector. Proof:
In the (dq/dt, dZ’/dt) space, the gradient
projection
is given by
(27)
(26). Substituting Imposing constraint (21) gives -~~~C”~~2+r-~r2=0 into (27) and multiplying by 11cull2 + r2, we obtain (23) and (24). Note that q belongs to the kernel of 8X”‘, which implies dX”p=c?X”(q-T)= Lastly, (25) is the accommodating move in production prices. The main result is eq. (23). By analogy
the index of discouragement
of commodity
with Mirrlees
-2X”T.
n
(1976), who calls
h, we shall call
the absolute index of discouragement of h. C’, which is the vector of production costs of compensated consumer price changes, can thus be interpreted as (the opposite of) the vector of absolute indices of discouragement. Assuming 2~0 (that is, an increase in income induces a change of consumption which has a positive value at production prices) the consumption price of a good h should be reduced proportionally to its absolute index of discouragement. It is only in the case where these indices are
perfectly correlated with the distortions that the direction of steepest ascent actually coincides with a proportional reduction of distortions. Note also that, following a line of reasoning inspired by sections 4 and 5, one can argue
Tax reform
J. Tirole and R. Guesnerie,
290
that the case where r
7. Existence
of trajectories and convergence for gradient algorithms
In this section we will consider existence system of differential equations (13), (14) model with one public firm. Very similar specific problem of section 4, and similar process of section 6. First we have to check that a solution properties expected from its design. This is Proposition
4.
A
solution
p(t),
q(t),
and convergence problems for the and (15) of section 5, i.e. for the results would apply to the more methods would also apply to the of the process actually easily done in proposition
z(t)
of
the
system
of
has the 4.
differential
equations (13), (14) and (15) of section 5 defined on [0, T] and starting from P(O), q(O), z(0) which is u tight equilibrium, has, z$ it exists, the following Properties: (1) at each time t, {P(t), q(t), z(t)) is u tight equilibrium u_nd function oft. IlP(0ll= llP(O)ll> jlq(t)ll= 11q(O)ll; (2) W zs a non-decreusing Proof:
The proof of the first part of (1) is straightforward. The fact that the Euclidean norm of prices is preserved results from the integration of p. (dp/dt)=O, which [as in Guesnerie (1977)] is a consequence of the definition of 8Y - ’ and q (dq/dt) = 0, which is easily shown from (13). To prove the second part of proposition 4, let us call
~(~)=J~C~~2+~~P~~2--
(P . vgj2
IIvgl12 .
One then has
and thus d Wldt is strictly
positive
unless:
“Let us note, however, that the desirability of production inefficiency is much less likely than previously. When all commodities are not inferior r>O [see Dixit (1975, p. 107) for very closely related remarks].
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or (b) In the following
the corresponding
points
will be said to be of type (a) or
n
type (b).
The next question to be raised concerns the existence of solutions. Local existence is assured by the differentiability assumptions as soon as D > 0, but we need the existence of trajectories, i.e. of solutions defined (0, + a). For that we shall introduce the following assumptions. (HI). There does not exist p, q, z, and i, >O s.t. xi xi(q)= y(p)+ z, C(p, q) =o, p=i.Vg(z), g(z)=O.‘2 (Hl) ensures that D is always strictly positive. One can compare this assumption with assumption (H7) in Folgelman-Guesnerie-Quinzii (1978) and argue, as they do, that (Hl) is likely to be generically true. z, xicXi, (H2). Let A={x,,y,~I~~x~=y+ ‘physically feasible states’. A is compact. This is a standard
as’+.umption. It ensures
KEY,
g(z)=01
be the
that any trajectory
set of
will remain
in
the compact set K x projzA, where K = ip, q 1[(PJI= Ilp(O)ll and llq[\ = ~~q(O)~~>. To avoid problems at the boundary of 9, we add two other assumptions to analogous to assumptions (A) and (H3) in FogelmanGuesnerie-Quinzii, which the reader is referred for justifications. (H3). Let q” be a sequence of consumption prices converging to 4, such that for some (but not all) h:q,=O. Then either 11x(q”)1I++x or W(...,xi(q”),...)
to p, such that
Proposition 5. For any initial tight equilibrium p(O), q(O), z(O), the system defined by (13), (14) and (15) has a solution defined on (0, + cc). Proof: (H2), (H3) and (H4) imply that any trajectory stays subset of feasible states N contained in {K n 9 x P} x proj,A. “The reader will check that there is a close connection D >O and the constraints qualifications of the optimisation
between both literature.
in a compact
(Hl) and the fact that
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292
Because of (Hl), there exists an open set 0 containing N and such that the second member of eqs. (13), (14) and (15) is continuously differentiable. Hirsch-Smale’s (1974) theorem gives the conclusion (p. 196). n Finally,
we have the following
Proposition 6. Every points of type (b). ProoJ: function
proposition.
trajectory
We use an extension
converges
either to points of type (a) or to
of Liapounov
theorem,
with
the Liapounov
I
V= max W(p,q,2) - WMt), 4(t), z(t)). i N V is non-increasing limit
Ps i
on trajectory
and being
bounded
above
from zero has a
max W - W(p(O), q(O), z(O)). N 1
Let I’+ be the limit set of the process, accumulation points of p(t,), q(tn), z(t,) One can see that on r+, V= P and dV/dt (1958), any trajectory (which is bounded) the set of points (a) and (b) which contains
i.e. the set of points which are for infinite sequences t,, t,+ +r;c. =O. Then, according to Hurewicz converges to I-+ and therefore to r+. n
Points of type (a) are local social welfare extrema in the sense that the necessary conditions of optimality are satisfied: they may be either global optima or local optima or saddle points. Points of type (b) are points at which an increase of social welfare is impossible if one is constrained, as supposed here, to have an efficient public production. Hence at such points temporary production inefficiencies in the public sector are desirable.
8. Conclusions It is clear there are generally many ways by which an objective function can be increased at a non-optimal point. This point makes tax reform somehow arbitrary, and has already been recognized by authors considering small moves of a system. The same point, though somewhat disguised, arises when choosing an algorithm and its control variables. Therefore the issue might well be the following. Does there exist a process which has given
J. Tirole und R. Guesnerie,
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properties ~ monotonicity and convergence to the set of points satisfying the second best optimality conditions ~ and is easy to interpret and to implement? We show in this paper that the question has a positive answer. The use of the simplest algorithm ~ the gradient projection algorithm ~~~~ and a judicious choice of control variables leads to appealing rules in spite of their arbitrariness. In particular public production is guided by the private production prices. (See section 5. This is the tax reform equivalent of the Diamond-Mirrlees efficiency result). In a one consumer economy with lump sum transfers the change in consumer prices is proportional to the vector of absolute indices of discouragement (section 6). We feel that the economic insights and implementability of these rules vindicate the use of gradient projection methods in tax reform theory. References Arrow, K. and L. Hurwicz, 1960, Decentralization and computation in resource allocation, in: R.W. Pfouts, ed., Essays in economics and econometrics (University of North Carolina Press, Chapel Hill) 34104. Bazarra, MS. and C.M. Shetty, 1979, Nonlinear programming: Theory and algorithm (Wiley, New York). D’Aspremont, C. and F. Tulkens, 1980, Commodity exchanges as gradient processes, Econometrica 48, 387~400. Diamond, P. and J. Mirrlees, 1971, Optimal taxation and public production, American Economic Review 61, 8 -27, 261-278. Diewert, W.E., 1979, Optimal tax perturbations, Journal of Public Economics 10, 139-- 177. Dixit, A., 1975, Welfare effects of tax and price changes, Journal of Public Economics 4, 1033 123. Dixit, A., 1979, Price changes and optimum taxation in a many-consumer economy, Journal of Public Economics 11, 143-157. Fogelman, F., R. Guesnerie and M. Quinzii. 1978, Dynamic processes for tax reform theory, Journal of Economic Theory 17, 20&225. Guesnerie, R., 1977, On the direction of tax reform. Journal of Public Economics 7, 179~ 222. Guesnerie, R., 1980, Modeles de 1’Economie publique, Monographies du Seminaire d’Economt_trie. Hadley, G., 1964, Nonlinear and dynamic programming (Addison-Wesley, Reading, Mass.). Heal, G., 1973, The theory of economic planning (North-Holland, Amsterdam). Hirsch, M.W. and S. Smale, 1974, Differential equations dynamical systems and linear algebra (Academic Press, New York). Hurewicz, W., 1958, Lectures on ordinary differential equations (M.I.T., Cambridge, Mass.). La Salle and S. Lefschetz, 1961, Stability by Liapunov’s direct method (Academic Press, New York). Mirrlees, J., 1976. Optimal tax theory, a synthesis, Journal of Public Economics 6, 3299358. Rosen, J.B., 1961, The gradient projection method for nonlinear programming Part II: Nonlinear constraint, SIAM Journal. Tirole, J., 1978, Essais sur le calcul economique public et le taux d’actualisation, These pour le doctorat de 3eme cycle. Universite de Paris IX-Dauphine. Weymark. J., 1979, A reconciliatron of recent results in taxation theory, Journal of Public Economics 12. 171-190.