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A reconciliation of recent results in optimal taxation theory
Journal
of Public
Economics
12 (1979) 171~ 189. 0 North-Holland
A RECONCILIATION
OF RECENT RESULTS TAXATION THEORY John
Duke
Received
Publishmg
IN OPTIMAL
A. WEYMARK*
UniuersitJ,
February
Company
Durhurn,
NC 17706, USA
1978, revised version received July 1979
The recent papers by Guesnerie and Diewert on tax reforms are interpreted as contributions to the characterization of second-best optima. This paper demonstrates that when it is possible to achieve any feasible direction of change in supplies by a differential change in producer prices, there are unique producer support prices. Under these circumstances, the apparent differences between Guesnerie and Diewert are reconciled. Optimality conditions with nonunique support prices are also considered.
1. Introduction One of the major concerns of optimal taxation theory is the presentation of conditions which characterize the Pareto optimal states of an economy which must operate with a limited set of tax instruments. Recently Diewert (1978) and Guesnerie (1977) have addressed themselves to the slightly different problem of analyzing tax reform proposals. Weymark (1978b) provides further results on tax reform. Diewert and Guesnerie consider whether it is possible to propose a set of marginal changes in the tax instruments so as to achieve a Pareto improvement. Necessary conditions for the nonexistence of Pareto-improving tax changes are necessary conditions for the initial allocation to be Pareto optimal. Consequently, it is possible to consider this aspect of Diewert’s and Guesnerie’s contributions in terms of the more traditional approach of characterizing optima. Unfortunately, it is difficult to see the relationship between the conditions developed by Diewert and Guesnerie. To correspond with Guesnerie’s model, Diewert’s problem must be reformulated by replacing his ad valorem commodity taxes by specific taxes, by fixing profit tax rates at 100x, and by replacing compensated demand functions with ordinary demand functions. *I have benefited from conversations with W.E. Diewert and R.A. Jones as well as written comments from A.K. Dixit, R. Guesnerie, and an editor of this Journal. I am particularly mdebted to A.K. Dixit for suggestions which have simplified the presentation. Much of the research was carried out while visiting at the University of British Columbia.
However, applying Diewert’s techniques to this respecified model yields conditions which appear to depend upon the derivatives of the aggregate supply function. This seems to contradict Guesnerie’s conclusions, since his conditions do not contain terms of this sort. The main purpose of this paper is to reconcile this apparent difference. In section 2 necessary conditions for second-best optimality are developed for an economy with specific commodity taxes and lOO”,, profit taxation. These optimality conditions do not employ supply derivatives. Furthermore. the criteria derived in this section reduce to Guesnerie’s results when it is assumed that the production vector has a unique support price. When this is the case, any feasible direction of change in supplies can be achieved by a differential change in producer prices, there is local controllability of the production sector. If the support prices are not unique, finite changes in producer prices are required to induce supply changes in some feasible directions. In section 3, following Diewert (1978), feasible changes in supply are modelled by a linear approximation to the supply function at the initial prices. To use this technique only differential changes in the control variables can be considered. with the resulting criteria containing explicit reference to supply derivatives. However, when there are unique support prices this technique yields conditions identical to those found in section 2. Section 4 briefly considers variable profit taxation. Dixit (1979) investigates this topic in some detail. The results of this section support the general conclusion that optimality conditions can be expressed without reference to supply derivatives if there is local controllability of the production sector, even if the optimality conditions are derived for differential changes in the control variables. Section 5 presents conclusions. An appendix summarizes results on convex cones which are used in the text.
2. Second-best
optimality
2.1. Statement
of‘ the problem
Given the preferences of consumers, the technologies of firms, and initial resource endowments, one of the central concerns of welfare economics is the establishment of conditions which characterize the Pareto optimal states of an economy. This characterization will, of course, depend upon the set of feasible allocations. In the first-best model an allocation is feasible if each consumption bundle belongs to the appropriate consumption set, each production plan is technically feasible, and the allocation satisfies a materials balance constraint. There is no requirement that consumer or firm decisions must be decentralized.
J.A. Wrymark, Optimal taxation theory
173
In the second-best literature some further constraints are imposed on the set of feasible allocations. In optimal taxation problems these constraints take the form of a requirement that private decision-making be decentralized and by a specification of the limited tax instruments available to the government. The consume1 sector consist of H households, indexed by h = 1,. . ., H, each with continuous strictly quasiconcave monotone utility functions uh = u”(x~), where xh is an n-dimensional net consumption vector. The (net) consumption sets Xh are convex and bounded from below. Let x =Chxh and Z? =(x,,...,xH). There are F private firms. indexed by f= 1,. ., F, plus a government production sector indexed by f = F + 1. Individual production possibilities sets Y/ contain typical elements J,I. It is assumed that these sets are closed, convex, contain the origin, and satisfy free disposability. Let Yp=CT= i Yr and Y = Y, + Y,, i. Y is assumed to be irreversible, i.e. Y n ( - Y) = (0). The first-best problem may now be stated. Partially order by the Pareto criterion the set of all ?~ci where ci = (.?(xE Y
and
xh~Xh
for all
h]
(1)
No behavioural constraints appear in cl. In the second-best problem considered in this section the additional constraints correspond to Guesnerie’s (1977) model which in turn is based upon Diamond and Mirrlees (1971). All private decisions must be decentralized with all private firms behaving competitively. The government may use per unit (specific) commodity taxes and is required to use 100% profit taxation. Accordingly, consumers’ incomes do not depend upon producers’ prices. There is no lump-sum income. Consumer prices are q= (qi,. . ., q,,)T, producer prices are p= (pi,. . .,P”)~, and commodity taxes are t= (t 1,. ., t,)’ with p + t = q. To simplify the exposition, boundary problems are avoided by assuming that q + 0 and ~9 0.’ Throughout, p and q will be used as control variables with t defined implicitly. Consumer net demand functions are xh(q) which are assumed to be continuously differentiable. The indirect utility functions are d(q) = uh(xh(q)). One can easily demonstrate that each uh(q) is continuous, locally nonsatiated, quasiconvex, and homogeneous of degree zero in q. Differentiability of each uh(q) is assumed. Firm supply correspondences are denoted /(p). Using the Pareto criterion, the solution to the second-best problem is a partial ordering of the set of all ,~Ec’ where
‘All vectors are column vectors unless superscripted with a transpose operator, T. Notation: b % 0 means bi > 0 for all i, b z-0 means bi 2 0 for all i with b # 0, and b 2 0 means bi 2 0 for all i.
c2 = (.
F,_V’+‘EYF+,,
for some and
p
and, for that
.<= J;(q)
for some
p,
for all Y).
(2)
By construction all households face the same prices so there will be equality of marginal rates of substitution. Similarly. all priwtr firms face common prices so production is on the frontier of Y, (equality of marginal rates of transformation). However, in general t+O so the marginal rates of substitution do not equal the marginal rates of transformation. 2.2. Production
@cienc,j~
One of the major questions considered by the optimal taxation literature is whether it is desirable (i.e. whether it is a property of maximal elements of the Pareto partial ordering) for the government sector to determine its production vector so as to equate its marginal rates of transformation with those of the private sector. In other words, is aggregate production efficiency desirable? Given the assumptions on Y,, , this is equivalent to asking if public production managers should act as profit maximizers with shadow prices equal to p. To answer this question the concept of Pareto-improving (consumer) price changes is introduced. Definition. For 2(q), y is a strictly Pareto-improving tial) price changes if and only if Vvh(q j'y>O for all h.
direction
of (differen-
The use of strict Pareto improvements follofls Diewert (1978) and Guesnerie (1977).’ Employing the gradient vectors of the indirect utility functions in the evaluation of price changes involves the use of first-order approximations; when Vch(q)#O this is of negligible significance.” When F’ch(q)=O the use of gradient vectors is inappropriate since any small finite change in prices makes the consumer better off while for differential changes no improvement is possible. Except for theorem 2, where it is unavoidable, zero gradients are excluded from consideration. Now consider .C(q*), a Pareto optimal allocation, with corresponding aggregate net demand vector x(q*). Diamond and Mirrlees (1971) provided a methodology for determining whether x(q*) is on the frontier of Y. Suppose the contrary, x(q*)Eint Y, the interior of Y. If at i(q*) there exists a strictly Pareto-improving direction of price changes, the induced change in demands ‘Fimte weak Pareto-improvmg 3With vh(y) quasiconvex, if sufficiently small magnitude the finite changes in the direction y (1978) considers the relationship
changes
are considered in Weymark (1978a). then for all finite changes m the direction y of consumer is made better off while if Vuh(y)Tj,=O then for small the consumer is no worse off (and could be better off). Matthews between differential and small finite changes in some detail. Vd’(q)Ty>O
J.A. Wevmark,
Optimal
taxation
175
rheor)
will be feasible (since x(q*)~int Y) which contradicts the assumption that ,$(q*) is Pareto optimal.4 Thus, x(q*) must be on the frontier of Y. Hence, if for all a(q) in c2 it is possible to find a strictly Pareto-improving direction of price changes, then aggregate production efficiency is desirable in this second-best problem. Let T(q) be the set of all vectors orthogonal to q. Ignoring non-negativity constraints, for all consumers T(q) is the net demands budget hyperplane. Since the demand functions are homogeneous of degree zero in prices, all price changes could be restricted to be orthogonal to q. Thus, T(q) can also be viewed as the set of all possible directions of price changes. In fig. 1 the
Fig. 1. The normalization
procedure
for price changes.
change in prices from q to q+ Aq’ results in the same demand behaviour as the change from q to q+ Aq’, where Aq’ E T(q). Since only differential changes in prices are considered and q$O, there is no need explicitly to consider non-negativity constraints. Henceforth all directions of change in consumer prices will be elements of T(q). Define
4By a suitable choice on the frontier of Y,.
of p it is possible
to induce
private
firms to produce
any supply
vector
176
J.A. Weymark, Optimal tuxation theory
A(q)
is the cone generated by the net demand Using Roy’s Identity, Vuh(q) is proportional cone generated by the gradient vectors. Let T(q).
P(q)= bli!ET(q)
and
A=~20
vectors and is contained in to -x”(q) so --A(q) is the
for all
AEA(q)}.
(4)
P(q) is the negative polar cone to n(q) (the positive polar cone to -A(q)). Figure 2 illustrates these sets for the two-consumer three-good case. 7’(q) is a plane which has been rotated to be coincident with the page. n(q) is formed by drawing the rays from the origin through x’(q) and x’(q) and then taking all convex combinations of points on these rays. P(q) is the intersection of the halfspace normal to x’(q) (which does not contain x’(q)) with the corresponding halfspace for x’(q).
Fig. 2. Geometry
of the two-person,
three-good
case.
Theorem 1. For _-2(q) with x”(q)#O f or all h, the set of directions of strictly Pareto-improving dt@rential price changes is intP(q)’ which is nonempty iff A(q) is pointed. Proof The theorem is a trivial application to theorem A.2 found in the appendix. Since P(q) is the positive ‘int P(q) is the interior
of theorem
A.1 and the corollary
polar cone to the cone generated
of P(q) relative
to
T(q).
by the gradient
J.A. Wepmark,
Optimal taxation
177
theory
vectors, it is intuitively clear that int P(q) is the set of Pareto-improving directions of price changes. An alternative interpretation of this result is provided by the fact that P(q) is the negative polar cone to A(q); int P(q) is the set of directions of price changes which reduce the value of all households’ net demands. Intuitively, if A(q) is pointed there is some Hicksian composite good in net demand (or net supply) by all households. Lowering (raising) its ‘price’ makes all households better off. This generalizes the condition found in Diamond and Mirrlees (1971) that there be some good in net demand (or net supply) by all households. Guesnerie (1977) assumes that the Diamond and Mirrlees condition holds for all goods. Returning to the efficiency question, the previous remarks may be summarized in the following theorem. Theorem 2. -If desirable.6
2.3. Motzkin’s
A(q)
is pointed
for
all q, then
production
efficiency
is
Theorem
As in Diewert (1978) optimality conditions are developed using Motzkin’s Theorem of the Alternative. A theorem of the alternative states that either a system of homogeneous linear equations has a solution or, if not, a related system of inequalities does have a solution. Motzkin’s Theorem.” For two matrices E and F containing of columns, exactly one of the following holds: 36
s.t.
EbzO
3w
and
3+
and s.t.
FbgO, wTE+GTF=OT,
the same number
with
FfO.
(5)
with
wzO,+>O.
(6)
The use of this theorem can be illustrated by reconsidering the existence Pareto-improving price changes. The (net) expenditure function for household h is defined as mh(Uh,q) = min {qxhluh(xh) 2 uh, xh E X,}. x*
of
(7)
@fo rigorously establish theorem 2 it must be demonstrated that when A(q) is pointed it is possible to make local Pareto improvements even if some x*(q)=O. The conclusions of theorem 1 remain valid for i(q) with xh(q)=O for some h if attention is restricted to consumers with nonzero demands. Noting that VtP(q)=O when x”(q)=O, the earlier discussion suggested that any change in prices can be considered to be an improvement for this consumer. ‘This is a special case of Motzkin’s theorem. A closely related proposition is Tucker’s theorem which has Fb>O in (5) and JpO in (6). Tucker’s theorem would be used to study weak Paretoimproving changes. Both theorems are discussed in Mangasarian (1969).
J..4. Wermurh.
17x
Opt~md tauution
theor\
It is assumed that mh(nh,q) is twice continuously made of the fact that the compensated net demand
Household
net expenditure
mh(uh,4) 5 0, If initially
differentiable. functions are
Use will be
must not exceed zero. h = 1.. ., H.
(8) holds with equality,
(8)
for differential
changes
(9)
where M xhT du is ha;e been A strict
is a diagonal matrix with diagonal entries Zmh/?uh >O, X has rows a vector with components dub, etc. For later use, producer prices treated explicitly. Pareto improvement occurs if du
0
(r,O,Ol dq
$0,
(10)
dp
where I is an identity matrix. If initially (8) holds with equality, strictly Pareto-improving price changes exist if and only if (9) and (10) can be solved simultaneously. Using Motzkin’s theorem, this occurs if and only if $(M’~O
and
G>O)
s.t.
+vT(M,-%,O)=GT(I,O,O).
(11)
Since i;mh/8uh >O, if (11) has a solution with G >O, then u’#O. Consequently, Pareto-improving price changes exist if and only if jw>O
s.t. CwhXh=O. h
(12)
If xh#O for all h, by theorem A.3 in the appendix, (12) is equivalent being pointed. This is the result found in theorem 1. 2.4. Optimality Continuing
to A(q)
conditions
on the assumption
that production
efficiency
is desirable,
it is
J.A. Weymark, Optimal tax&ion theory
179
possible to exhibit further properties of Pareto optimal allocations. The individual profit maximizing decisions of firms and the shadow profit maximizing decision of the government can be replaced by profit maximization on the aggregate production possibilities set Y resulting in the aggregate supply correspondence y(p). The constraint set for the second-best problem may then be rewritten as c3 where c3={$~y(p)
forsome
p,.C=x*(q)
forsome
q,
and
x(q)syJ.
(13) If all production would be c4:
decisions
c4 = {2]2 = i(q)
were under
for some
q
central
and
control
x(q)sy
the constraint
forsome
set
KEY}. (14)
With this paper’s production assumptions, c3 and c4 describe the same set of consumption allocations. Given any point on the frontier of Y, the closedness, convexity, and free disposability of Y ensures that there is some price vector that will support the choice of this supply vector. Thus, the requirement that production decisions be decentralized places no additional restriction on the problem if the materials balance constraint is written as a weak inequality. Consider the net supply vector YE y(p). Let p(y) be the cone of support prices for y. If Y is ‘ridged’ or ‘kinked’ at y there will be at least two linearly independent support prices. Suppose x(q) is a production efficient allocation, i.e. x(q)= y for YE y(p). Let V.x(q) be the matrix of aggregate demand partial derivatives. With PEP(J), pTVx(q)y represents the cost in terms of these support prices of the demand changes induced by a differential change in consumer prices y.’ Let ~(4)=‘(~1~=~T~-~(q),6EPt4’),!‘=x(q),yEy(p) To a first-order
forsome
P}. (15)
approximation
is the set of all-directions of consumer price changes which lead to a feasible change in demands. In other words, the value of the demands decreases or remains constant regardless of which support prices are used in the evaluation. Assuming A(q) is pointed, from theorem 1 it is known which directions of consumer price changes are strictly Pareto-improving. Now (16) provides the ‘It is easy
to show that
wTVx(q)e
T(q) for any weights
w.
J.A. Weymark, Optimal taxation theory
IRO
directions of consumer price changes which lead to feasible changes in demands. A necessary condition for i(q) (with xh(q)#O for all h) to be Pareto optimal is that these sets have an empty intersection. Theorem 3. Suppose A(q) is pointed, x”(q)10 for all h, y =x(q) where ye y(p) for some p, and p(y) is a polyhedral cone. A necessary condition for f(q)
to be Pareto
optimul
is: fi’Vx(q)E
-A(q)\{O}
for some o~p(y).
Proof. Suppose Q(q) n int P(q) =$3. Since p(y) is a polyhedral a finite number of generators pg, g = 1,. ., G. In this case, Q(q)={ylpgTVx(q)YsO,
By theorem
for all
cone, p(y) has
pg).
(17)
1, intP(q)=jyl-xh(q)‘y>O.
By supposition, no Motzkin’s theorem, 3~20 _
y satisfies
and
3\i,>O
for all (17)
s.t.
and
(18)
h).
(18)
simultaneously.
Hence,
-~~,xh(q)T=Cwgp9T~x(q).
by
(19)
9 By
(12)
dTVx(q)E
neither -A(q)\{O}
side of (19) is zero. for some $~p(y).~
Letting
bT = CgwgpgT,
this
implies
Theorem 3 is illustrated in fig. 1. The construction of -A(q) and P(q) were considered in the description of fig. 2. The derivation of Q(q) from $(q) is analogous to the derivation of P(q) from A(q). If 2(q) is Pareto optimal, then Q(q) n int P(q) =$J, as shown in fig. 3. 2.5.
Optimality
Now p.‘O
Let evaluated
with unique
prices
the local assumption that y(p) is a differentiable function at be the symmetric matrix of supply partial derivatives at p. From the first-order conditions for profit maximization,
make
By(p)
pVy(p)
From
support
= 0.
(20) the rank
(20)
of Vy(p)
is at most
n - 1. Guesnerie
(1977) assumes
‘This method of proof follows a suggestion made by A. Dixit. I conjecture that the assumption that p(y) is polyhedral could be relaxed. “Using y(p) to denote a function Involves an abuse of the notation introduced earlier. If Y is not smooth at y(p). a sufficient condition for this differentiability assumption would be to suppose p is in the relative interior of p(y(p)).
J.A. Weymark, Optimal taxation theory
Fig. 3. Necessary
conditions
for second-best
that Pq’(p) is precisely of rank n- 1. implies that the government has local sense that it is possible to induce a direction along the frontier of Y by a prices. Another important implication of unique (to a positive scaling factor) established as a corollary to a theorem Theorem
4.
If p’Ep(y(p)),
then
181
optimality.
He demonstrates that this assumption control of the production sector in the differential change in supplies in any suitable differential change in producer this rank assumption is that p is the support price to y(p). This will be which generalizes (20).
p”Vy(p)=O.
Proof.
Since p and p” support y(p), so does p(p)=p+p(p”-p) Differentiating y(p(p)) with respect to p and evaluating at p=O, =O. From (20), pTVy(p)=O so dTVy(p)=O as well.
Corollary 1. If the rank of positive scaling factor) support Proof. “A.
Suppose Dixit suggested
Vy(p) price.’
p and p’ are linearly
is n-
1, then
y(p)
has
for 05_~51. (p”-~)~Vy(p)
a unique
(to
a
’ independent
and both support
this result. I have used his proof to establish
the more general
y(p).
By
theorem
4.
J.A. Wepark,
182
theorem 4 both assumption. Corollary Proof.
2.
P~VL.(P)=O
and
If the runk of Vy(p)
The result is a trivial
Optimal taxation
pTVy(p)=O
is n-
implication
theory
contradicts
which
optimal
rank
if und only if b = Ap.
1, pTVy(p)=O
of (20) and the rank assumption.
Combining these results with theorem 3 establishes the necessary for optimality corresponding to Guesnerie’s (1977) assumptions. Theorem 5. p, and Vy(p)
the
conditions
Suppose A(q) is pointed, xh(q) # 0 for all h, x(q) = y(p) for some is of runk n- 1. A necessary condition for z?(q) to be Pareto
is pTVx(q)E
3. Optimality
-fl(q)\jO).
conditions using supply functions
3.1. Optimulity
conditions
‘If a change in demands is evaluated using the change is nonpositive for all support decision, then the change is feasible. In this the supply functions at the initial producer feasibility of a change. Market balance requires
producer prices and the value of prices of the initial production section linear approximations of prices are used to determine the
(21)
Eqs. (9) and (21) are the equilibrium conditions for the model. Supposing that (9) and (21) hold with equality initially, assuming y(p) is differentiable at p, for differential changes
-M,
i
0,
-X.
0
- Vx(q),
(22)
VY(P)
A necessary condition for second-best optimality solution to (22) which also satisfies (10). By Motzkin’s if and only if
3
20
and
3E>O
s.t.
is that there be no theorem, this obtains
X,
0
V-x(q),
- V)‘(P)
=*r(I,o,o).
(23)
183
J.A. Weymark, Optimal taxation theory
Theorem
6.
necessary
condition
Suppose
xh(q)#O
for
i(q)
for
all
to be Pareto
h
and
optimal
x(q)=p(p)
for
is: 3w’>O
and 3~~20
some
p.
A
such
that w’Tx
= - w2TVx(q)
(24)
and w2Vy(p)=O.
Proof.
(25)
With ?mh/C7uh> 0, (24) and (25) follow immediately
Corollary
1.
Suppose
A(q)
is pointed,
some p. A necessary condition that (25) holds and w2TBx(q)E Proof.
Since
w2TVx(q)#0
Corollary
2.
for
a(q)
xh(q)#O
from (23).
for all h, and x(q)=;(p)
to be Pareto
optimal
is: 3wz>0
-A(q)\!(O).
A(q) is pointed, (12) implies wlTX #O for ~1~>O and w2 #O. The result follows from the definition of n(q). Suppose
A(q)
p, and
V’y(p) is of rank
optimal
is pTVx(q)E
Proof.
By corollary
n-
for such
so
pointed, xh(q)#Ofor all h, x(q)=y(p) for some 1. A necessary condition for g(q) to be Pareto
is
-A(q)\${0],.12
1, IV’ #O. By corollary
2 to theorem
4 and (25), w2 =Lp
for 3.>0.
3.2. A reconciliation Theorem 6 together with its corollaries are the major results from the use of supply functions. But corollary 2 is identical to theorem 5. In other words, by adding to the assumptions of theorem 6 the requirement that strictly Pareto-improving price changes exist and by assuming Vy(p) is precisely of rank n- 1 reconciles the results of Guesnerie (1977) and Diewert (1978). The key to this reconciliation was noting that in these circumstances the only solutions to (25) must be proportional to p.
12Diewert (1978) requires (9) and (21) to hold with equality: this necessitates the use of a slightly different form of Motzkin’s theorem. In (8) he has Eh=O which means that w in (6) obeys no sign restrictions. With the assumptions of corollary 2, it is only possible to conclude that IV’ = Lp for i #O rather than for i >O. In effect, by working only with the frontier of Y these techniques cannot determine the sign of the support prices. Dixit (1979) avoids this problem by working wrth weak inequalities. Diewert also develops his conditions using compensated demand functions rather than ordinary demand functions. His theorems can be reduced to those of this paper using the Slutsky equation.
184
J.A. Weymark, Optimal taxatidn theory
Further insight can be gained by comparing theorem 3’ with these new results. The relevant comparison is to corollary 1 as strictly Paretoimproving price changes were assumed to exist in theorem 3. Knowledge of the semipositive solutions to (25) is needed for corollary 1, i.e. one must solve for the kernel to (25) and consider semipositive vectors in this kernel. From theorem 4 is it known that all support prices in p(y(p)) are solutions. However, only when the dimension of the kernel to (25) is one will the support prices coincide with the set of all semipositive solutions to (25). In other words, the converse to theorem 4 is false when the rank of Vy(p) is less than n- 1. So, while theorem 5 and corollary 2 are identical, theorem 3 and corollary 1 differ. In section 2 two features of-the assumption that Vy(p) is of rank n- 1 were noted: (a) there are unique (to a scaling factor) support prices and (b) it is possible to induce a differential change in supplies in any direction along the frontier of Y by a suitable differential change in producer prices. When the rank of By(p) is less than n- 1 there are some directions of supply changes which cannot be achieved by differenetial changes in producer prices, although all directions can be attained by suitable finite changes. When considering the feasibility of a change in demands, in section 2 the change was not required to be achieved by a differential change in producer prices. However, to use the techniques developed in this section only differential changes in producer prices could be considered. In tax reform problems only differential changes are allowed, so it is possible that reforms could terminate although a small change in demands would lead to a feasible Pareto improvement. An example is instructive. Suppose n =2 and the initial supply vector is at a kink on the frontier of Y with p in the interior of p(y(p)). This is illustrated in fig. 4(a). With differential price changes it is not possible to achieve any change in supplies. The matrix Vy(p)=O, so the semipositive solutions to (25) are all of R:\(O). This set is all the support prices to the technology depicted in fig. 4(b). For differential changes these technologies are indistinguishable. The only changes in demands considered in this paper were changes achieved by differential changes in consumer prices. If Vxh(q) has rank n- 1 for all h, where Vxh(q) is defined analogously to Vx(q), each consumer will have q as the unique prices supporting 2(q). In this situation theorems 3 and 5 and corollary 2 to theorem 6 provide necessary and sufficient conditions for a(q) to be a local optimum, where an allocation is considered to be Pareto optimal if and only if it is impossible to make aI1 consumers better Off.
4. Variable profit taxation In Diewert
(1978) and Dixit (1979) profits
are returns
to factors
which are
J.A. Weymnrk, Optimal taxation
theory
185
(a)
P(Y(P)) P ///N////--/////i/
Y(P)5
5 Y
// /
-
Fig. 4. Technologies
YI
with Vy(p)= 0.
in inelastic supply. Each household is endowed with a vector of these fixed factors uh for which they receive an after-profit-taxation return of s. Given 1; =C,v”, the aggregate technology for variable inputs and outputs is Y, as before. As v is supplied inelastically, there is no need explicitly to consider the prices firms pay for these factors.13 13Diewert (1978) and Dixit (1979) provide further details of the model. In the presence of lixed factors the assumption that Y is irreversible should be relaxed to the requirement that Y -, R” hs h
186
J.A. Weymark,
The two equilibrium
conditions
mh(u,q) 5 SUh,
Optimal taxation
theory
are now
h=l,...,H
(26)
and x(q,vfi)5L’(p),
(27)
where li = (ml,. ., m”). For differential hold with equality initially,
-M,
-X,
0,
0,
V4x(q>h),
VY(P),
changes
and supposing
(26) and (27)
V
(28)
-N
where V has rows c,hl‘, N has entries &(Zx~/?mh)v~, and V,x(q,ti) is the matrix of derivatives of x(q,rG) with respect to q. Ignoring feasibility, it is a simple matter to apply Motzkin’s theorem to establish that strict Pareto improvements are possible if and only if jM’>O
s.t.
w’.Y =\\” b’=O.
(29 I
If rh > 0 for all h, strict Pareto improvements can be achieved by increasing s proportionally. Now taking account of feasibility, attention is restricted to the case where strict Pareto improvements exist and where Vq’(p) is of rank n - 1. It is clear from (28) that (25) must hold for Pareto optimality. Theorem
6.
Supposr
xh(y)fOfbr
Vy(p) is of’ runk n - 1. A newswry is: 3~’ > 0 and 32 > 0 such that /lpTV,x(q.Gl)=
-w’TX+o
~11 h, (29), .u(q,&)=y(p),fbr condition
jbr
.sonw p, and
.G(q, n^l) to he Pareto
optimul
(30)
and ip=N =w’=V#O.
(31)
The proof of this theorem is essentially the same as theorem 5 together with its corollary 2, so it is omitted. Eq. (30) simply requires pTVqx(qr h) E -A(q,ti)\,(OJ. Dropping the requirement that a strict Pareto improvement exists, (29), A could be zero. With this modification (30) and (31) are eqs. (21) and (22) in Dixit (1979).
Diewert (1978) suggests that the presence of supply derivatives in his conditions and their absence in Guesnerie’s (1977) conditions is related to the fact that there is 100% profit taxation in Guesnerie’s model. From (30) and (31) it is clear that optimality conditions can be written without reference to supply derivatives even in the case of variable profit taxation. With only differential changes in instruments allowed, this would not be possible if the rank of V’y(p) is less than n- 1.
5. Conclusions The key feature of the model thus appears to be the local controllability of the production sector. When it is possible to induce an arbitrary feasible direction of supply changes by a differential change in producer prices, the results of sections 2 and 3 coincide. Since both Guesnerie (1977) and Diewert (1978) explicitly restricted themselves to differential changes, the relative simplicity of Guesnerie’s propositions can now be understood to depend largely on his local controllability assumption. In tax reform problems, where the optimality conditions are derived for small changes in the instruments, the optimality conditions can always be written with explicit reference to the supply derivatives. It is only in the case where local controllability is assumed is it possible to express the optimality conditions with no explicit reference to the supply derivatives for, in this case, there are unique support prices. However, in characterizing second-best optima, the use of supply derivatives is inappropriate except when local controllability is present, and even then their use complicates the statement of the results. This resolves the apparent conflict between Guesnerie and Diewert. The possibility of variable profit taxation (or the existence of fixed factors) is not at issue; rather, it is the extent to which the private production sector may be controlled in a decentralized fashion.
Appendix This appendix summarizes various results on cones which are used in the text. Theorems stated here without proofs may be found in either Berman (1973) or Gale (1960). The universal set is assumed to be a real vector space. To avoid ambiguity, 1 will be a vector while p will be a scalar. A set K is a cone iff k E K implies pk E K for all p 10. A convex cone K is pointed iff K n (-K) = (0). A convex cone K is solid iff intK #g. The negative polar cone of a convex cone K is K-={k’jkTk’SO
for all
kgK}.
IX8
J.A.
Weymitrk,
Optimul
tcaution
theor\,
The positive polur cone of a convex cone K is K f = - (K ). A cone K is finite or poiyhedrul iff there exists a matrix M such that K={klk=Mj.
for some
AZOJ.
In this definition the columns of M may be chosen to be nonzero, in which case the columns of M are called generutors of K. As a notational convention denote a column of M by M’ and the cone generated by M by K(M). K-
It is straightforward is closed.
to establish
that a polyhedral
Theorem
A.1.
If K is u polyhedrul
Theorem
A.2.
If K is u pointed polyhedral
int(K-)=(k’Ik’EKCorollary.
For
kEint(K(M)-)
a matrix
cone, then K is pointed
and
A.3.
For a matrix
iff K - is solid.
kEK\{O)).
if
columns,
ifK(M)
is pointed,
then
all i.
Proof. (a) Suppose MiTk
and that
cone, then
k’k’
M with nonzero
iff MiTk
cone is convex
i, then and
M with nonzero
(CiliMi)Tk K(M)-
to
columns, K(M)
for all i >O. theorem A.2,
is pointed
#Ji
>0 such that C,I,M’=O. Proof.
(a) Suppose K(M) is not pointed. Then, 3a,b~K(M) with a= That is, a=&&! M’ for i,’ >O and b=Ci#Mi for A2 >O with C,i! M’ =-- C,,I?M’. Letting Ai =A,! +$, this implies C,&M’=O with /1>O. (b) Suppose 31 >O such that C,&M’=O. Consider any 2’ >O and A2 >O such that 1.’ +A2=A. Then, a= -b with a,bEK(M) where a=C,L!M’ and b =&A! M’. So K(M) is not pointed. -b#O.
References Berman, A., 1973,Cones, matrices and mathematical programming (Springer-Verlag. Berlin). Diamond, P.A. and J.A. Mirrlees, 1971, Optimal taxation and public production IJII, American Economic Review 61, 8-27, 261-278. Diewert, W.E., 1978, Optimal tax perturbations, Journal of Public Economics 10, 139-177. Dixit, A.K., 1979, Price changes and optimum taxation in a many-person economy, Journal of Public Economics 11, 1433157.
J.A. Weymark,
Optimal taxation
theory
189
Gale, D., 1960, The theory of linear economic models (McGraw-Hill, New York). Guesnerie, R., 1977, On the direction of tax reform, Journal of Public Economics 7, 179-202. Mangasarian, O.L., 1969, Nonlinear programming (McGraw-Hill, New York). Matthews, S., 1978, Directional cores in simple dynamic games, California Institute of Technology, July. Weymark, J.A., 1978a, On Pareto-improving price changes, Journal of Economic Theory 19, 338-346. Weymark, J.A., 1978b, Undominated directions of tax reform, University of British Columbia Discussion Paper 78826, August,
of Public
Economics
12 (1979) 171~ 189. 0 North-Holland
A RECONCILIATION
OF RECENT RESULTS TAXATION THEORY John
Duke
Received
Publishmg
IN OPTIMAL
A. WEYMARK*
UniuersitJ,
February
Company
Durhurn,
NC 17706, USA
1978, revised version received July 1979
The recent papers by Guesnerie and Diewert on tax reforms are interpreted as contributions to the characterization of second-best optima. This paper demonstrates that when it is possible to achieve any feasible direction of change in supplies by a differential change in producer prices, there are unique producer support prices. Under these circumstances, the apparent differences between Guesnerie and Diewert are reconciled. Optimality conditions with nonunique support prices are also considered.
1. Introduction One of the major concerns of optimal taxation theory is the presentation of conditions which characterize the Pareto optimal states of an economy which must operate with a limited set of tax instruments. Recently Diewert (1978) and Guesnerie (1977) have addressed themselves to the slightly different problem of analyzing tax reform proposals. Weymark (1978b) provides further results on tax reform. Diewert and Guesnerie consider whether it is possible to propose a set of marginal changes in the tax instruments so as to achieve a Pareto improvement. Necessary conditions for the nonexistence of Pareto-improving tax changes are necessary conditions for the initial allocation to be Pareto optimal. Consequently, it is possible to consider this aspect of Diewert’s and Guesnerie’s contributions in terms of the more traditional approach of characterizing optima. Unfortunately, it is difficult to see the relationship between the conditions developed by Diewert and Guesnerie. To correspond with Guesnerie’s model, Diewert’s problem must be reformulated by replacing his ad valorem commodity taxes by specific taxes, by fixing profit tax rates at 100x, and by replacing compensated demand functions with ordinary demand functions. *I have benefited from conversations with W.E. Diewert and R.A. Jones as well as written comments from A.K. Dixit, R. Guesnerie, and an editor of this Journal. I am particularly mdebted to A.K. Dixit for suggestions which have simplified the presentation. Much of the research was carried out while visiting at the University of British Columbia.
However, applying Diewert’s techniques to this respecified model yields conditions which appear to depend upon the derivatives of the aggregate supply function. This seems to contradict Guesnerie’s conclusions, since his conditions do not contain terms of this sort. The main purpose of this paper is to reconcile this apparent difference. In section 2 necessary conditions for second-best optimality are developed for an economy with specific commodity taxes and lOO”,, profit taxation. These optimality conditions do not employ supply derivatives. Furthermore. the criteria derived in this section reduce to Guesnerie’s results when it is assumed that the production vector has a unique support price. When this is the case, any feasible direction of change in supplies can be achieved by a differential change in producer prices, there is local controllability of the production sector. If the support prices are not unique, finite changes in producer prices are required to induce supply changes in some feasible directions. In section 3, following Diewert (1978), feasible changes in supply are modelled by a linear approximation to the supply function at the initial prices. To use this technique only differential changes in the control variables can be considered. with the resulting criteria containing explicit reference to supply derivatives. However, when there are unique support prices this technique yields conditions identical to those found in section 2. Section 4 briefly considers variable profit taxation. Dixit (1979) investigates this topic in some detail. The results of this section support the general conclusion that optimality conditions can be expressed without reference to supply derivatives if there is local controllability of the production sector, even if the optimality conditions are derived for differential changes in the control variables. Section 5 presents conclusions. An appendix summarizes results on convex cones which are used in the text.
2. Second-best
optimality
2.1. Statement
of‘ the problem
Given the preferences of consumers, the technologies of firms, and initial resource endowments, one of the central concerns of welfare economics is the establishment of conditions which characterize the Pareto optimal states of an economy. This characterization will, of course, depend upon the set of feasible allocations. In the first-best model an allocation is feasible if each consumption bundle belongs to the appropriate consumption set, each production plan is technically feasible, and the allocation satisfies a materials balance constraint. There is no requirement that consumer or firm decisions must be decentralized.
J.A. Wrymark, Optimal taxation theory
173
In the second-best literature some further constraints are imposed on the set of feasible allocations. In optimal taxation problems these constraints take the form of a requirement that private decision-making be decentralized and by a specification of the limited tax instruments available to the government. The consume1 sector consist of H households, indexed by h = 1,. . ., H, each with continuous strictly quasiconcave monotone utility functions uh = u”(x~), where xh is an n-dimensional net consumption vector. The (net) consumption sets Xh are convex and bounded from below. Let x =Chxh and Z? =(x,,...,xH). There are F private firms. indexed by f= 1,. ., F, plus a government production sector indexed by f = F + 1. Individual production possibilities sets Y/ contain typical elements J,I. It is assumed that these sets are closed, convex, contain the origin, and satisfy free disposability. Let Yp=CT= i Yr and Y = Y, + Y,, i. Y is assumed to be irreversible, i.e. Y n ( - Y) = (0). The first-best problem may now be stated. Partially order by the Pareto criterion the set of all ?~ci where ci = (.?(xE Y
and
xh~Xh
for all
h]
(1)
No behavioural constraints appear in cl. In the second-best problem considered in this section the additional constraints correspond to Guesnerie’s (1977) model which in turn is based upon Diamond and Mirrlees (1971). All private decisions must be decentralized with all private firms behaving competitively. The government may use per unit (specific) commodity taxes and is required to use 100% profit taxation. Accordingly, consumers’ incomes do not depend upon producers’ prices. There is no lump-sum income. Consumer prices are q= (qi,. . ., q,,)T, producer prices are p= (pi,. . .,P”)~, and commodity taxes are t= (t 1,. ., t,)’ with p + t = q. To simplify the exposition, boundary problems are avoided by assuming that q + 0 and ~9 0.’ Throughout, p and q will be used as control variables with t defined implicitly. Consumer net demand functions are xh(q) which are assumed to be continuously differentiable. The indirect utility functions are d(q) = uh(xh(q)). One can easily demonstrate that each uh(q) is continuous, locally nonsatiated, quasiconvex, and homogeneous of degree zero in q. Differentiability of each uh(q) is assumed. Firm supply correspondences are denoted /(p). Using the Pareto criterion, the solution to the second-best problem is a partial ordering of the set of all ,~Ec’ where
‘All vectors are column vectors unless superscripted with a transpose operator, T. Notation: b % 0 means bi > 0 for all i, b z-0 means bi 2 0 for all i with b # 0, and b 2 0 means bi 2 0 for all i.
c2 = (.
F,_V’+‘EYF+,,
for some and
p
and, for that
.<= J;(q)
for some
p,
for all Y).
(2)
By construction all households face the same prices so there will be equality of marginal rates of substitution. Similarly. all priwtr firms face common prices so production is on the frontier of Y, (equality of marginal rates of transformation). However, in general t+O so the marginal rates of substitution do not equal the marginal rates of transformation. 2.2. Production
@cienc,j~
One of the major questions considered by the optimal taxation literature is whether it is desirable (i.e. whether it is a property of maximal elements of the Pareto partial ordering) for the government sector to determine its production vector so as to equate its marginal rates of transformation with those of the private sector. In other words, is aggregate production efficiency desirable? Given the assumptions on Y,, , this is equivalent to asking if public production managers should act as profit maximizers with shadow prices equal to p. To answer this question the concept of Pareto-improving (consumer) price changes is introduced. Definition. For 2(q), y is a strictly Pareto-improving tial) price changes if and only if Vvh(q j'y>O for all h.
direction
of (differen-
The use of strict Pareto improvements follofls Diewert (1978) and Guesnerie (1977).’ Employing the gradient vectors of the indirect utility functions in the evaluation of price changes involves the use of first-order approximations; when Vch(q)#O this is of negligible significance.” When F’ch(q)=O the use of gradient vectors is inappropriate since any small finite change in prices makes the consumer better off while for differential changes no improvement is possible. Except for theorem 2, where it is unavoidable, zero gradients are excluded from consideration. Now consider .C(q*), a Pareto optimal allocation, with corresponding aggregate net demand vector x(q*). Diamond and Mirrlees (1971) provided a methodology for determining whether x(q*) is on the frontier of Y. Suppose the contrary, x(q*)Eint Y, the interior of Y. If at i(q*) there exists a strictly Pareto-improving direction of price changes, the induced change in demands ‘Fimte weak Pareto-improvmg 3With vh(y) quasiconvex, if sufficiently small magnitude the finite changes in the direction y (1978) considers the relationship
changes
are considered in Weymark (1978a). then for all finite changes m the direction y of consumer is made better off while if Vuh(y)Tj,=O then for small the consumer is no worse off (and could be better off). Matthews between differential and small finite changes in some detail. Vd’(q)Ty>O
J.A. Wevmark,
Optimal
taxation
175
rheor)
will be feasible (since x(q*)~int Y) which contradicts the assumption that ,$(q*) is Pareto optimal.4 Thus, x(q*) must be on the frontier of Y. Hence, if for all a(q) in c2 it is possible to find a strictly Pareto-improving direction of price changes, then aggregate production efficiency is desirable in this second-best problem. Let T(q) be the set of all vectors orthogonal to q. Ignoring non-negativity constraints, for all consumers T(q) is the net demands budget hyperplane. Since the demand functions are homogeneous of degree zero in prices, all price changes could be restricted to be orthogonal to q. Thus, T(q) can also be viewed as the set of all possible directions of price changes. In fig. 1 the
Fig. 1. The normalization
procedure
for price changes.
change in prices from q to q+ Aq’ results in the same demand behaviour as the change from q to q+ Aq’, where Aq’ E T(q). Since only differential changes in prices are considered and q$O, there is no need explicitly to consider non-negativity constraints. Henceforth all directions of change in consumer prices will be elements of T(q). Define
4By a suitable choice on the frontier of Y,.
of p it is possible
to induce
private
firms to produce
any supply
vector
176
J.A. Weymark, Optimal tuxation theory
A(q)
is the cone generated by the net demand Using Roy’s Identity, Vuh(q) is proportional cone generated by the gradient vectors. Let T(q).
P(q)= bli!ET(q)
and
A=~20
vectors and is contained in to -x”(q) so --A(q) is the
for all
AEA(q)}.
(4)
P(q) is the negative polar cone to n(q) (the positive polar cone to -A(q)). Figure 2 illustrates these sets for the two-consumer three-good case. 7’(q) is a plane which has been rotated to be coincident with the page. n(q) is formed by drawing the rays from the origin through x’(q) and x’(q) and then taking all convex combinations of points on these rays. P(q) is the intersection of the halfspace normal to x’(q) (which does not contain x’(q)) with the corresponding halfspace for x’(q).
Fig. 2. Geometry
of the two-person,
three-good
case.
Theorem 1. For _-2(q) with x”(q)#O f or all h, the set of directions of strictly Pareto-improving dt@rential price changes is intP(q)’ which is nonempty iff A(q) is pointed. Proof The theorem is a trivial application to theorem A.2 found in the appendix. Since P(q) is the positive ‘int P(q) is the interior
of theorem
A.1 and the corollary
polar cone to the cone generated
of P(q) relative
to
T(q).
by the gradient
J.A. Wepmark,
Optimal taxation
177
theory
vectors, it is intuitively clear that int P(q) is the set of Pareto-improving directions of price changes. An alternative interpretation of this result is provided by the fact that P(q) is the negative polar cone to A(q); int P(q) is the set of directions of price changes which reduce the value of all households’ net demands. Intuitively, if A(q) is pointed there is some Hicksian composite good in net demand (or net supply) by all households. Lowering (raising) its ‘price’ makes all households better off. This generalizes the condition found in Diamond and Mirrlees (1971) that there be some good in net demand (or net supply) by all households. Guesnerie (1977) assumes that the Diamond and Mirrlees condition holds for all goods. Returning to the efficiency question, the previous remarks may be summarized in the following theorem. Theorem 2. -If desirable.6
2.3. Motzkin’s
A(q)
is pointed
for
all q, then
production
efficiency
is
Theorem
As in Diewert (1978) optimality conditions are developed using Motzkin’s Theorem of the Alternative. A theorem of the alternative states that either a system of homogeneous linear equations has a solution or, if not, a related system of inequalities does have a solution. Motzkin’s Theorem.” For two matrices E and F containing of columns, exactly one of the following holds: 36
s.t.
EbzO
3w
and
3+
and s.t.
FbgO, wTE+GTF=OT,
the same number
with
FfO.
(5)
with
wzO,+>O.
(6)
The use of this theorem can be illustrated by reconsidering the existence Pareto-improving price changes. The (net) expenditure function for household h is defined as mh(Uh,q) = min {qxhluh(xh) 2 uh, xh E X,}. x*
of
(7)
@fo rigorously establish theorem 2 it must be demonstrated that when A(q) is pointed it is possible to make local Pareto improvements even if some x*(q)=O. The conclusions of theorem 1 remain valid for i(q) with xh(q)=O for some h if attention is restricted to consumers with nonzero demands. Noting that VtP(q)=O when x”(q)=O, the earlier discussion suggested that any change in prices can be considered to be an improvement for this consumer. ‘This is a special case of Motzkin’s theorem. A closely related proposition is Tucker’s theorem which has Fb>O in (5) and JpO in (6). Tucker’s theorem would be used to study weak Paretoimproving changes. Both theorems are discussed in Mangasarian (1969).
J..4. Wermurh.
17x
Opt~md tauution
theor\
It is assumed that mh(nh,q) is twice continuously made of the fact that the compensated net demand
Household
net expenditure
mh(uh,4) 5 0, If initially
differentiable. functions are
Use will be
must not exceed zero. h = 1.. ., H.
(8) holds with equality,
(8)
for differential
changes
(9)
where M xhT du is ha;e been A strict
is a diagonal matrix with diagonal entries Zmh/?uh >O, X has rows a vector with components dub, etc. For later use, producer prices treated explicitly. Pareto improvement occurs if du
0
(r,O,Ol dq
$0,
(10)
dp
where I is an identity matrix. If initially (8) holds with equality, strictly Pareto-improving price changes exist if and only if (9) and (10) can be solved simultaneously. Using Motzkin’s theorem, this occurs if and only if $(M’~O
and
G>O)
s.t.
+vT(M,-%,O)=GT(I,O,O).
(11)
Since i;mh/8uh >O, if (11) has a solution with G >O, then u’#O. Consequently, Pareto-improving price changes exist if and only if jw>O
s.t. CwhXh=O. h
(12)
If xh#O for all h, by theorem A.3 in the appendix, (12) is equivalent being pointed. This is the result found in theorem 1. 2.4. Optimality Continuing
to A(q)
conditions
on the assumption
that production
efficiency
is desirable,
it is
J.A. Weymark, Optimal tax&ion theory
179
possible to exhibit further properties of Pareto optimal allocations. The individual profit maximizing decisions of firms and the shadow profit maximizing decision of the government can be replaced by profit maximization on the aggregate production possibilities set Y resulting in the aggregate supply correspondence y(p). The constraint set for the second-best problem may then be rewritten as c3 where c3={$~y(p)
forsome
p,.C=x*(q)
forsome
q,
and
x(q)syJ.
(13) If all production would be c4:
decisions
c4 = {2]2 = i(q)
were under
for some
q
central
and
control
x(q)sy
the constraint
forsome
set
KEY}. (14)
With this paper’s production assumptions, c3 and c4 describe the same set of consumption allocations. Given any point on the frontier of Y, the closedness, convexity, and free disposability of Y ensures that there is some price vector that will support the choice of this supply vector. Thus, the requirement that production decisions be decentralized places no additional restriction on the problem if the materials balance constraint is written as a weak inequality. Consider the net supply vector YE y(p). Let p(y) be the cone of support prices for y. If Y is ‘ridged’ or ‘kinked’ at y there will be at least two linearly independent support prices. Suppose x(q) is a production efficient allocation, i.e. x(q)= y for YE y(p). Let V.x(q) be the matrix of aggregate demand partial derivatives. With PEP(J), pTVx(q)y represents the cost in terms of these support prices of the demand changes induced by a differential change in consumer prices y.’ Let ~(4)=‘(~1~=~T~-~(q),6EPt4’),!‘=x(q),yEy(p) To a first-order
forsome
P}. (15)
approximation
is the set of all-directions of consumer price changes which lead to a feasible change in demands. In other words, the value of the demands decreases or remains constant regardless of which support prices are used in the evaluation. Assuming A(q) is pointed, from theorem 1 it is known which directions of consumer price changes are strictly Pareto-improving. Now (16) provides the ‘It is easy
to show that
wTVx(q)e
T(q) for any weights
w.
J.A. Weymark, Optimal taxation theory
IRO
directions of consumer price changes which lead to feasible changes in demands. A necessary condition for i(q) (with xh(q)#O for all h) to be Pareto optimal is that these sets have an empty intersection. Theorem 3. Suppose A(q) is pointed, x”(q)10 for all h, y =x(q) where ye y(p) for some p, and p(y) is a polyhedral cone. A necessary condition for f(q)
to be Pareto
optimul
is: fi’Vx(q)E
-A(q)\{O}
for some o~p(y).
Proof. Suppose Q(q) n int P(q) =$3. Since p(y) is a polyhedral a finite number of generators pg, g = 1,. ., G. In this case, Q(q)={ylpgTVx(q)YsO,
By theorem
for all
cone, p(y) has
pg).
(17)
1, intP(q)=jyl-xh(q)‘y>O.
By supposition, no Motzkin’s theorem, 3~20 _
y satisfies
and
3\i,>O
for all (17)
s.t.
and
(18)
h).
(18)
simultaneously.
Hence,
-~~,xh(q)T=Cwgp9T~x(q).
by
(19)
9 By
(12)
dTVx(q)E
neither -A(q)\{O}
side of (19) is zero. for some $~p(y).~
Letting
bT = CgwgpgT,
this
implies
Theorem 3 is illustrated in fig. 1. The construction of -A(q) and P(q) were considered in the description of fig. 2. The derivation of Q(q) from $(q) is analogous to the derivation of P(q) from A(q). If 2(q) is Pareto optimal, then Q(q) n int P(q) =$J, as shown in fig. 3. 2.5.
Optimality
Now p.‘O
Let evaluated
with unique
prices
the local assumption that y(p) is a differentiable function at be the symmetric matrix of supply partial derivatives at p. From the first-order conditions for profit maximization,
make
By(p)
pVy(p)
From
support
= 0.
(20) the rank
(20)
of Vy(p)
is at most
n - 1. Guesnerie
(1977) assumes
‘This method of proof follows a suggestion made by A. Dixit. I conjecture that the assumption that p(y) is polyhedral could be relaxed. “Using y(p) to denote a function Involves an abuse of the notation introduced earlier. If Y is not smooth at y(p). a sufficient condition for this differentiability assumption would be to suppose p is in the relative interior of p(y(p)).
J.A. Weymark, Optimal taxation theory
Fig. 3. Necessary
conditions
for second-best
that Pq’(p) is precisely of rank n- 1. implies that the government has local sense that it is possible to induce a direction along the frontier of Y by a prices. Another important implication of unique (to a positive scaling factor) established as a corollary to a theorem Theorem
4.
If p’Ep(y(p)),
then
181
optimality.
He demonstrates that this assumption control of the production sector in the differential change in supplies in any suitable differential change in producer this rank assumption is that p is the support price to y(p). This will be which generalizes (20).
p”Vy(p)=O.
Proof.
Since p and p” support y(p), so does p(p)=p+p(p”-p) Differentiating y(p(p)) with respect to p and evaluating at p=O, =O. From (20), pTVy(p)=O so dTVy(p)=O as well.
Corollary 1. If the rank of positive scaling factor) support Proof. “A.
Suppose Dixit suggested
Vy(p) price.’
p and p’ are linearly
is n-
1, then
y(p)
has
for 05_~51. (p”-~)~Vy(p)
a unique
(to
a
’ independent
and both support
this result. I have used his proof to establish
the more general
y(p).
By
theorem
4.
J.A. Wepark,
182
theorem 4 both assumption. Corollary Proof.
2.
P~VL.(P)=O
and
If the runk of Vy(p)
The result is a trivial
Optimal taxation
pTVy(p)=O
is n-
implication
theory
contradicts
which
optimal
rank
if und only if b = Ap.
1, pTVy(p)=O
of (20) and the rank assumption.
Combining these results with theorem 3 establishes the necessary for optimality corresponding to Guesnerie’s (1977) assumptions. Theorem 5. p, and Vy(p)
the
conditions
Suppose A(q) is pointed, xh(q) # 0 for all h, x(q) = y(p) for some is of runk n- 1. A necessary condition for z?(q) to be Pareto
is pTVx(q)E
3. Optimality
-fl(q)\jO).
conditions using supply functions
3.1. Optimulity
conditions
‘If a change in demands is evaluated using the change is nonpositive for all support decision, then the change is feasible. In this the supply functions at the initial producer feasibility of a change. Market balance requires
producer prices and the value of prices of the initial production section linear approximations of prices are used to determine the
(21)
Eqs. (9) and (21) are the equilibrium conditions for the model. Supposing that (9) and (21) hold with equality initially, assuming y(p) is differentiable at p, for differential changes
-M,
i
0,
-X.
0
- Vx(q),
(22)
VY(P)
A necessary condition for second-best optimality solution to (22) which also satisfies (10). By Motzkin’s if and only if
3
20
and
3E>O
s.t.
is that there be no theorem, this obtains
X,
0
V-x(q),
- V)‘(P)
=*r(I,o,o).
(23)
183
J.A. Weymark, Optimal taxation theory
Theorem
6.
necessary
condition
Suppose
xh(q)#O
for
i(q)
for
all
to be Pareto
h
and
optimal
x(q)=p(p)
for
is: 3w’>O
and 3~~20
some
p.
A
such
that w’Tx
= - w2TVx(q)
(24)
and w2Vy(p)=O.
Proof.
(25)
With ?mh/C7uh> 0, (24) and (25) follow immediately
Corollary
1.
Suppose
A(q)
is pointed,
some p. A necessary condition that (25) holds and w2TBx(q)E Proof.
Since
w2TVx(q)#0
Corollary
2.
for
a(q)
xh(q)#O
from (23).
for all h, and x(q)=;(p)
to be Pareto
optimal
is: 3wz>0
-A(q)\!(O).
A(q) is pointed, (12) implies wlTX #O for ~1~>O and w2 #O. The result follows from the definition of n(q). Suppose
A(q)
p, and
V’y(p) is of rank
optimal
is pTVx(q)E
Proof.
By corollary
n-
for such
so
pointed, xh(q)#Ofor all h, x(q)=y(p) for some 1. A necessary condition for g(q) to be Pareto
is
-A(q)\${0],.12
1, IV’ #O. By corollary
2 to theorem
4 and (25), w2 =Lp
for 3.>0.
3.2. A reconciliation Theorem 6 together with its corollaries are the major results from the use of supply functions. But corollary 2 is identical to theorem 5. In other words, by adding to the assumptions of theorem 6 the requirement that strictly Pareto-improving price changes exist and by assuming Vy(p) is precisely of rank n- 1 reconciles the results of Guesnerie (1977) and Diewert (1978). The key to this reconciliation was noting that in these circumstances the only solutions to (25) must be proportional to p.
12Diewert (1978) requires (9) and (21) to hold with equality: this necessitates the use of a slightly different form of Motzkin’s theorem. In (8) he has Eh=O which means that w in (6) obeys no sign restrictions. With the assumptions of corollary 2, it is only possible to conclude that IV’ = Lp for i #O rather than for i >O. In effect, by working only with the frontier of Y these techniques cannot determine the sign of the support prices. Dixit (1979) avoids this problem by working wrth weak inequalities. Diewert also develops his conditions using compensated demand functions rather than ordinary demand functions. His theorems can be reduced to those of this paper using the Slutsky equation.
184
J.A. Weymark, Optimal taxatidn theory
Further insight can be gained by comparing theorem 3’ with these new results. The relevant comparison is to corollary 1 as strictly Paretoimproving price changes were assumed to exist in theorem 3. Knowledge of the semipositive solutions to (25) is needed for corollary 1, i.e. one must solve for the kernel to (25) and consider semipositive vectors in this kernel. From theorem 4 is it known that all support prices in p(y(p)) are solutions. However, only when the dimension of the kernel to (25) is one will the support prices coincide with the set of all semipositive solutions to (25). In other words, the converse to theorem 4 is false when the rank of Vy(p) is less than n- 1. So, while theorem 5 and corollary 2 are identical, theorem 3 and corollary 1 differ. In section 2 two features of-the assumption that Vy(p) is of rank n- 1 were noted: (a) there are unique (to a scaling factor) support prices and (b) it is possible to induce a differential change in supplies in any direction along the frontier of Y by a suitable differential change in producer prices. When the rank of By(p) is less than n- 1 there are some directions of supply changes which cannot be achieved by differenetial changes in producer prices, although all directions can be attained by suitable finite changes. When considering the feasibility of a change in demands, in section 2 the change was not required to be achieved by a differential change in producer prices. However, to use the techniques developed in this section only differential changes in producer prices could be considered. In tax reform problems only differential changes are allowed, so it is possible that reforms could terminate although a small change in demands would lead to a feasible Pareto improvement. An example is instructive. Suppose n =2 and the initial supply vector is at a kink on the frontier of Y with p in the interior of p(y(p)). This is illustrated in fig. 4(a). With differential price changes it is not possible to achieve any change in supplies. The matrix Vy(p)=O, so the semipositive solutions to (25) are all of R:\(O). This set is all the support prices to the technology depicted in fig. 4(b). For differential changes these technologies are indistinguishable. The only changes in demands considered in this paper were changes achieved by differential changes in consumer prices. If Vxh(q) has rank n- 1 for all h, where Vxh(q) is defined analogously to Vx(q), each consumer will have q as the unique prices supporting 2(q). In this situation theorems 3 and 5 and corollary 2 to theorem 6 provide necessary and sufficient conditions for a(q) to be a local optimum, where an allocation is considered to be Pareto optimal if and only if it is impossible to make aI1 consumers better Off.
4. Variable profit taxation In Diewert
(1978) and Dixit (1979) profits
are returns
to factors
which are
J.A. Weymnrk, Optimal taxation
theory
185
(a)
P(Y(P)) P ///N////--/////i/
Y(P)5
5 Y
// /
-
Fig. 4. Technologies
YI
with Vy(p)= 0.
in inelastic supply. Each household is endowed with a vector of these fixed factors uh for which they receive an after-profit-taxation return of s. Given 1; =C,v”, the aggregate technology for variable inputs and outputs is Y, as before. As v is supplied inelastically, there is no need explicitly to consider the prices firms pay for these factors.13 13Diewert (1978) and Dixit (1979) provide further details of the model. In the presence of lixed factors the assumption that Y is irreversible should be relaxed to the requirement that Y -, R” hs h
186
J.A. Weymark,
The two equilibrium
conditions
mh(u,q) 5 SUh,
Optimal taxation
theory
are now
h=l,...,H
(26)
and x(q,vfi)5L’(p),
(27)
where li = (ml,. ., m”). For differential hold with equality initially,
-M,
-X,
0,
0,
V4x(q>h),
VY(P),
changes
and supposing
(26) and (27)
V
(28)
-N
where V has rows c,hl‘, N has entries &(Zx~/?mh)v~, and V,x(q,ti) is the matrix of derivatives of x(q,rG) with respect to q. Ignoring feasibility, it is a simple matter to apply Motzkin’s theorem to establish that strict Pareto improvements are possible if and only if jM’>O
s.t.
w’.Y =\\” b’=O.
(29 I
If rh > 0 for all h, strict Pareto improvements can be achieved by increasing s proportionally. Now taking account of feasibility, attention is restricted to the case where strict Pareto improvements exist and where Vq’(p) is of rank n - 1. It is clear from (28) that (25) must hold for Pareto optimality. Theorem
6.
Supposr
xh(y)fOfbr
Vy(p) is of’ runk n - 1. A newswry is: 3~’ > 0 and 32 > 0 such that /lpTV,x(q.Gl)=
-w’TX+o
~11 h, (29), .u(q,&)=y(p),fbr condition
jbr
.sonw p, and
.G(q, n^l) to he Pareto
optimul
(30)
and ip=N =w’=V#O.
(31)
The proof of this theorem is essentially the same as theorem 5 together with its corollary 2, so it is omitted. Eq. (30) simply requires pTVqx(qr h) E -A(q,ti)\,(OJ. Dropping the requirement that a strict Pareto improvement exists, (29), A could be zero. With this modification (30) and (31) are eqs. (21) and (22) in Dixit (1979).
Diewert (1978) suggests that the presence of supply derivatives in his conditions and their absence in Guesnerie’s (1977) conditions is related to the fact that there is 100% profit taxation in Guesnerie’s model. From (30) and (31) it is clear that optimality conditions can be written without reference to supply derivatives even in the case of variable profit taxation. With only differential changes in instruments allowed, this would not be possible if the rank of V’y(p) is less than n- 1.
5. Conclusions The key feature of the model thus appears to be the local controllability of the production sector. When it is possible to induce an arbitrary feasible direction of supply changes by a differential change in producer prices, the results of sections 2 and 3 coincide. Since both Guesnerie (1977) and Diewert (1978) explicitly restricted themselves to differential changes, the relative simplicity of Guesnerie’s propositions can now be understood to depend largely on his local controllability assumption. In tax reform problems, where the optimality conditions are derived for small changes in the instruments, the optimality conditions can always be written with explicit reference to the supply derivatives. It is only in the case where local controllability is assumed is it possible to express the optimality conditions with no explicit reference to the supply derivatives for, in this case, there are unique support prices. However, in characterizing second-best optima, the use of supply derivatives is inappropriate except when local controllability is present, and even then their use complicates the statement of the results. This resolves the apparent conflict between Guesnerie and Diewert. The possibility of variable profit taxation (or the existence of fixed factors) is not at issue; rather, it is the extent to which the private production sector may be controlled in a decentralized fashion.
Appendix This appendix summarizes various results on cones which are used in the text. Theorems stated here without proofs may be found in either Berman (1973) or Gale (1960). The universal set is assumed to be a real vector space. To avoid ambiguity, 1 will be a vector while p will be a scalar. A set K is a cone iff k E K implies pk E K for all p 10. A convex cone K is pointed iff K n (-K) = (0). A convex cone K is solid iff intK #g. The negative polar cone of a convex cone K is K-={k’jkTk’SO
for all
kgK}.
IX8
J.A.
Weymitrk,
Optimul
tcaution
theor\,
The positive polur cone of a convex cone K is K f = - (K ). A cone K is finite or poiyhedrul iff there exists a matrix M such that K={klk=Mj.
for some
AZOJ.
In this definition the columns of M may be chosen to be nonzero, in which case the columns of M are called generutors of K. As a notational convention denote a column of M by M’ and the cone generated by M by K(M). K-
It is straightforward is closed.
to establish
that a polyhedral
Theorem
A.1.
If K is u polyhedrul
Theorem
A.2.
If K is u pointed polyhedral
int(K-)=(k’Ik’EKCorollary.
For
kEint(K(M)-)
a matrix
cone, then K is pointed
and
A.3.
For a matrix
iff K - is solid.
kEK\{O)).
if
columns,
ifK(M)
is pointed,
then
all i.
Proof. (a) Suppose MiTk
and that
cone, then
k’k’
M with nonzero
iff MiTk
cone is convex
i, then and
M with nonzero
(CiliMi)Tk K(M)-
to
columns, K(M)
for all i >O. theorem A.2,
is pointed
#Ji
>0 such that C,I,M’=O. Proof.
(a) Suppose K(M) is not pointed. Then, 3a,b~K(M) with a= That is, a=&&! M’ for i,’ >O and b=Ci#Mi for A2 >O with C,i! M’ =-- C,,I?M’. Letting Ai =A,! +$, this implies C,&M’=O with /1>O. (b) Suppose 31 >O such that C,&M’=O. Consider any 2’ >O and A2 >O such that 1.’ +A2=A. Then, a= -b with a,bEK(M) where a=C,L!M’ and b =&A! M’. So K(M) is not pointed. -b#O.
References Berman, A., 1973,Cones, matrices and mathematical programming (Springer-Verlag. Berlin). Diamond, P.A. and J.A. Mirrlees, 1971, Optimal taxation and public production IJII, American Economic Review 61, 8-27, 261-278. Diewert, W.E., 1978, Optimal tax perturbations, Journal of Public Economics 10, 139-177. Dixit, A.K., 1979, Price changes and optimum taxation in a many-person economy, Journal of Public Economics 11, 1433157.
J.A. Weymark,
Optimal taxation
theory
189
Gale, D., 1960, The theory of linear economic models (McGraw-Hill, New York). Guesnerie, R., 1977, On the direction of tax reform, Journal of Public Economics 7, 179-202. Mangasarian, O.L., 1969, Nonlinear programming (McGraw-Hill, New York). Matthews, S., 1978, Directional cores in simple dynamic games, California Institute of Technology, July. Weymark, J.A., 1978a, On Pareto-improving price changes, Journal of Economic Theory 19, 338-346. Weymark, J.A., 1978b, Undominated directions of tax reform, University of British Columbia Discussion Paper 78826, August,