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Fiscal Federalism Revisited
Journal of Economic Theory 92, 300317 (2000) doi:10.1006jeth.2000.2643, available online at http:www.idealibrary.com on
NOTES, COMMENTS, AND LETTERS TO THE EDITOR
Fiscal Federalism Revisited 1 Charles Blackorby Department of Economics, University of British Columbia, 997-1873 East Mall, Vancouver, BC, Canada V6T 1Z1; and GREQAM, Centre de la Vieille Charite, 2 rue de la Charite, 13002 Marseille, France orbyecon.ubc.ca, orbyehess.cnrs-mrs.fr
and Craig Brett Department of Economics, University of Essex, Wivenhoe Park, Colchester, Essex CO4 3SQ, United Kingdom craigessex.ac.uk Received September 1, 1998; revised January 3, 2000
We analyze how constitutional restrictions on tax bases within a federation affect the nature of Pareto-improving directions of tax reform and the design of optimal federal taxes. We show that constraints on federal taxation entail production inefficiency at the optimum, except under very restrictive circumstances. In passing, we show that using consumer prices as control variablesa standard procedure in tax-reform analysisrather than the taxes themselves, leads to incorrect conclusions when not all taxes or prices can be controlled. Journal of Economic Literature Classification Numbers: D61, H21, H70. 2000 Academic Press Key Words: fiscal federalism; tax reform; production efficiency.
1. INTRODUCTION Most discussions of fiscal federalism center on the devolution of power in light of both efficiency and equity considerations. When the GST (Goods and Services Tax) replaced the manufacturers' sales tax in Canada, the argument was primarily based on efficiency: it was to be a revenue-neutral change that eliminated many distortions generated by the manufacturer's sales tax. However, adding the GST to the existing provincial sales taxes 1 We thank Michael Keen, an associate editor, and a seminar audience at the University of Essex for comments on an earlier version of this paper. Financial assistance from the Social Sciences and Humanities Research Council of Canada is gratefully acknowledged.
300 0022-053100 35.00 Copyright 2000 by Academic Press All rights of reproduction in any form reserved.
FISCAL FEDERALISM REVISITED
301
raised again the issue of tax harmonization. This dialogue has been accentuated recently by the tax-harmonization agreement between the federal government and the Atlantic provinces; the other Canadian provinces have refused to enter into such an agreement. We address one aspect of the tax-harmonization issue in this paper. In the context of a simple model we show that the failure to harmonize taxes entails production inefficiency; that is, without tax harmonization, the second-best optimum of the economy is necessarily inside its production possibility frontier. It is therefore no surprise that a move towards tax harmonization yields a strict Pareto-improvement. Further this strict Pareto-improvement can be achieved without resorting to individual lumpsum transfersonly a demogrant 2 and tax harmonization are required as instruments. There are, of course, many potential benefits to tax harmomization beyond the simple efficiency considerations discussed here, in particular, the reduction of administrative costs. Even these simple efficiency considerations seem however quite compelling. In our model, the provincial government provides a public good that it finances by means of commodity taxes. We assume that the federal government takes the taxes set by the provinces as given; that is, the provinces are assumed to have already made their decisions and must stick with them. However, the federal government recognizes that its taxation decisions have effects on the provincial budget constraint. Indeed, overall feasibility requires that federal tax changes leave the provincial budget in balance. The only way to fully avoid this issue would be to assume complete revenue sharing on the part of the two governments. That is, one could consider the case in which the federal government actually compensates the province (or receives compensation from the province) for changes in provincial revenue due to changes in federal policy. Although attempts at compensation are sometimes observed, we view them as exceptions. 3 Hence, we choose to model a world without complete revenue sharing. We begin with the case where taxes are harmonized, and characterize the second-best optimum. The fiscal-federal problem is introduced by only permitting the federal government to control the demogrant and the taxes on those commodities not taxed by the provincial authorities. Until the last section we act as if there were but one province but two levels of tax authority. Although artificial, this is sufficient to capture some of the problems that arise in a fiscal-federal system. In particular, it allows us to focus on the issue of constitutional restrictions on the power of some levels of government to tax. To fix ideas, we consider the case in which 2
A demogrant is a lump-sum tax that is the same for all individuals, often called a poll tax. The Canadian federal government paid compensation to provincial governments as part of its move toward tax harmonization. 3
302
BLACKORBY AND BRETT
there are some commodities the federal government cannot tax. There is, in general, a three part partition of the the commodity vector: commodities that are zero-rated by the province and taxed by the federal government, commodities that are taxed both both levels of government, and those that are taxed by the province and zero-rated by the federal tax authority. Given that the federal government takes the taxes of the province as given, the existence of the first category generates no new possibilities for the federal tax authority. Hence we deal only with the latter two. Consumer prices are given by producer prices plus the provincial tax vector and the federal tax vector. The model is analyzed in two supposedly equivalent ways: letting the federal government control taxes indirectly by means of final consumer prices or directly by means of the federal tax vector. When the federal government can control all prices or all taxes these two procedures are equivalent. It is common to let the planner in these models control consumer prices directly and taxes only indirectly for technical convenience; 4 however, when the federal government can control only a subset of consumer prices or taxes these two modeling strategies are no longer equivalent. When the federal government controls a subset of taxes indirectly through consumer prices the social shadow prices are equal to producer prices. However, when the planner controls only a subset of the taxes directly, social shadow prices are no longer equal to producer prices and the economy is inside its production possibility frontier at the second-best optima. In addition, when a subset of consumer prices is used as the set of control variables, the provincial budget constraint, although satisfied, is not strictly binding; that is, the multiplier on the provincial budget constraint is always zero. When a subset of the tax vector is controlled directly this constraint is generally strictly binding. This is of some interest in itself as actual fiscal-federal systems control taxes directly and not indirectly; the choice of modeling strategies is no longer innocuous. In this context we show that there are strict Pareto-improvements possible by means of tax harmonization. In addition we show that these improvements come by changing relative prices in the sector under provincial control and not simply by adjusting their level. In the last section we extend these results to two provinces. (Except for notation this comprises the general case.)
2. DIRECT CONTROL OF TAXES We begin with the case where the federal government can tax all commodities and levy a demogrant, m. The provincial government provides a 4
See for example Guesnerie [6, 7] and Myles [10].
FISCAL FEDERALISM REVISITED
303
public good and finances it by means of commodity taxation. Throughout, the level of the public good is fixed. We then move to the case where the federal government cannot adjust the taxes on some commodities. Consumer prices are given by q= p+t+{, where t is the vector of provincial specific taxes and { is the federal specific tax vector. The individual indirect utility functions are given by u h =V h(q, m, g),
(2.1)
where g is the fixed amount of the public good provided by the provincial government. V h is differentially strongly quasi-convex 5 in prices and increasing in income and the level of the public good. The competitive sector is represented by a profit function, ?( p), 6 which is assumed to be differentially strongly convex; 7 the competitive sector supply is given by y*c ={ p ?( p).
(2.2)
We assume that the technology for producing the public good is given by g F( y g )
where
F(0)=0;
(2.3)
F is continuously differentiable, increasing, and concave. At the second-best optimum the authorities could simply direct the producer of the public good to use a specific set of inputs. Alternatively, the government can give the managers of this firm a set of shadow prices at which to minimize costs, say v. Then the managers minimize the cost of producing a given level of the public good by solving min[v Ty g | g F( y g )].
(2.4)
If it is optimal for the government to set these shadow prices proportional to producer prices then there is production efficiency; the marginal rates of transformation in the public sector enterprise and the competitive private sector are equal. If however, the optimal shadow prices are not proportional to the producer prices then there is production inefficiency at the optimum; that is, the economy is inside its production possibility frontier. We do not assume in advance that there is production efficiency but rather demonstrate the circumstances under which it holds. Net supply is given by y*=y*c & y g. 5
(2.5)
See Blackorby and Diewert [1] for details. There is no loss of generality in assuming that the competitive sector has a single firm; see Bliss [2], page 68 for example. 7 See Diewert, Avriel, and Zang [5] for details. This implies that the Hessian of the profit function has rank N&1. 6
304
BLACKORBY AND BRETT
Equilibrium is given by *+y* 0 &x
where
x*=: x*h.
(2.6)
h
In addition, the provincial government must balance its budget so that * p Ty g. tT x
(2.7)
When (2.6) and (2.7) hold with equality we say that the equilibrium is tight. Implicit in this formulation is that the federal government levies a hundred percent profit tax. Walras's Law guarantees that the federal budget is balanced. Utility increases are given by du h >0 W { Tq V h(q, m, g) dp+{ Tq V h(q, m, g) d{+{ m V h(q, m, g) dm>0 * hTd{+dm>0, * hTdp&x W &x
(2.8)
where the last step uses Roy's theorem. Let the matrix of demands be given by * X T=
x* 1T b . * HT x
_ &
(2.9)
Starting from a tight equilibrium, and assuming that the federal government has as instruments the commodity tax vector and the demogrant, there exist strict Pareto-improving equilibrium-preserving directions of change if and only if A has a solution where
*T [ &X
*T &X
1H
A=
_
dp d{ 0 H_N ] > >0 dm dy g
_&
*+{ y* c &{ x * &{ x * &{ q x p q m *& y gT t T { x* t T { x * tT { q x q m 0 TN
0 TN
0
&I N_N & pT { Tyg F
&_
dp d{ 0. dm dy g
(2.10)
&
By Motzkin's theorem, 8 A has a solution if and only if a complementary system, B, does not. B is given by 8
See Mangasarian [9], page 28.
FISCAL FEDERALISM REVISITED T T gT * hT =&v T { x * *c * : sh x , q +v { p y +zt { q x &zy
305 (2.11)
h T * hT =&v T { x * * : sh x q +zt { q x,
(2.12)
h
*&zt T { x*, : s h =v T { m x m
(2.13)
h
and &v T &zp T +r { Tyg F=0,
(2.14)
where 0{s 0,
v 0,
r 0,
and z 0.
The system of relations B characterizes the local second-best optima for the economy. The vector v is the vector of social shadow prices of goods, while s is the vector of social marginal utilities of income. These optima are characterised by Theorem 1. Assume that the federal government can levy taxes on all commodities as well as a universal demogrant. At all Pareto-optima, shadow prices are proportional to producer prices, i.e., there is production efficiency, and the provincial budget constraint is not strictly binding. Proof.
Subtract (2.12) from (2.11) to obtain v T { p y* c &zy gT =0.
(2.16)
Postmultiplying (2.16) by p, and using the homogeneity of supply yields zy gTp=0.
(2.17)
z>0 is a contradiction given that y g and p are positive. Therefore, z=0; that is, the provincial budget constraint is not strictly binding. It follows that social shadow prices, v, are proportional to producer prices. To see this note that (2.16) can now be written as v T { p y*=0.
(2.18)
Because the profit function is strongly convex, its Hessian has rank N&1 and p T { p y*=0.
(2.19)
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BLACKORBY AND BRETT
This implies that v is proportional to p. From (2.14) this implies that p=% { y g F( y g, g)
(2.20)
for some %>0. That is, the firm producing the public good should minimize costs using producer prices. K Altogether, this looks exactly like revenue sharing. The provincial budget constraint has no shadow value. This is precisely because of the fact that, when all taxes are under federal control, the model admits the separate normalization of producer and consumer prices. 9 In such circumstances, the exact numerical values of taxes have little significance. They can be chosen in such a way to ensure budget balance for the province. We are interested in the case where a subset of commodity taxes is under exclusive control of the provincial government, say sector two. We continue to assume that the federal government can effect local changes in tax rates in sector one. Thus, we construct a restricted problem, A, by adding the constraint d{ 2 =0, that is,
[0 n2_N
0 n2_n1 I n2_n2
0 n2
dp d{ 0 n2_N ] =0, dm dy g
_&
(2.21)
to problem A. A has a solution if and only if B does not where B is given by gT *+v T { y*+zt T { x * , : s h x hT =&v T { q x p q &zy
(2.22)
h T T * hT =&v T { x * * : s h x T ), q +zt { q x +(0 n1 , w
(2.23)
h
*&zt T { x*, : s h =v T { m x m
(2.24)
h
and &v T &zp T +r { y g F=0,
(2.25)
where 0{s , 9
v 0,
r 0,
and z 0.
Guesnerie [8], pp. 7781 provides a detailed discussion of this issue.
(2.26)
307
FISCAL FEDERALISM REVISITED
If the lack of tax harmonization is an effective constraint, then A has a solution while A does not. Equivalently, B does not have a solution and B does with w nonzero. Theorem 2. Suppose that the federal government can control only a subset of commodity taxes and a universal demogrant. Then, at all Pareto-optima, shadow prices are not proportional to producer prices, i.e., there is production inefficiency. Proof.
First, suppose that z {0. Subtracting (2.23) from (2.22) yields v T { p y* c &zy gT =(0 Tn1 , w T ).
(2.27)
From (2.27) it is obvious that v{%p, that is, shadow prices are not equal to producer prices; there is production inefficiency. Now, suppose that z =0. Evaluating (2.27) when z =0 yields v T { p y=(0, w T ).
(2.28)
Given strong convexity of the profit function, (2.28) implies that v=+p only if w =0. That is, production efficiency holds only if the fiscal federalism constraints are not strictly binding. It remains to show the conditions under which the fiscal federal restrictions, d{ 2 =0, are strictly binding even when the provincial budget constraint is not strictly binding. The fiscal federalism constraints are not binding if and only if A has a solution whenever A does, where A is given by
* [ &X T
* &X T1 1 H 0 H_N ]
A =
_
dp d{ 1 > >0 dm dy g
_&
*+{ y* c &{ x * &{ x * &{ q x p q1 m *& y gT t T { x* t T { x * tT { q x q1 m 0 TN
0 TN
0
&I N_N & pT { Tyg F
&_
dp d{ 1 0. dm dy g
(2.29)
&
Clearly if A has a solution then A does. Suppose that whenever A has a solution then A does as well. This is the same as saying that if A does not have a solution then A does not have a solution which in turn says that when B (the dual of A ) has a solution then B does as well.
308
BLACKORBY AND BRETT
By Motzkin's theorem, if A does not have a solution, then B does; it is given by * hT =&v~ T { q x *+v~ T { p y* c +z~t T { q x*&z~y gT, : s~ h x
(2.30)
h T *hT =&v~ T { x * * : s~ h x q1 +z~t { q1 x, 1
(2.31)
h
: s~ h =v~ T { m x*&z~t T { m x*,
(2.32)
h
and &v~ T &z~p T +r~ { Tyg F=0,
(2.33)
where v~ 0,
0{s~ 0,
r~ 0,
and z~ 0.
(2.34)
Remembering that we are in the case where the provincial budget constraint is not strictly binding, subtract (2.31) from (2.30) to obtain 0 Tn1 =v~ T { p1 y* c, *hT =&v~ T[{ q x *+{ p y* c ] : s~ h x 2 2 2
(2.35) (2.36)
h
and, from (2.31), T *. : s~ h x*hT { q1 x 1 =&v~
(2.37)
h
Given our regularity conditions, B does not imply B. That is, the fiscalfederal constraints on the taxation power of the federal government are strictly binding. This in turn implies that v{%p; there is production inefficiency whether or not the provincial budget constraint is strictly binding. K Two other characteristics of these Pareto-optima are also of interest. Multiplying (2.24) by m and subtracting it from (2.22) after having multiplied the latter by q yields 0=[v T { p y*&zy gT ] q.
(2.38)
FISCAL FEDERALISM REVISITED
309
Substituting (2.27) into (2.38) yields 0=w Tq 2 .
(2.39)
That is the multiplier on the fiscal federalism constraints is orthogonal to the consumer price vector in the provincial sector. Post-multiplying (2.27) by p shows that w Tp 0.
(2.40)
v T { p yv=w Tv 2 >0;
(2.41)
Furthermore, from (2.28),
that is, the shadow prices in sector two do not lie in the subspace orthogonal to w which mean that sector two shadow prices cannot be written as a convex combination of producer prices and consumer prices in that sector. Condition (2.39) provides insight into the nature of the adjustments the federal government would like to make in sector two prices, if it could. Formally, we have that the vector of shadow values on the fiscal federalism constraints is orthogonal to sector two consumer prices. Thus, changes in the relative prices in that sector would be required to bring about a Pareto improvement. A mere rescaling of the prices (relative to those prevailing in sector one) would not suffice. This is reminiscent of the work by Newbery [11], in which it was shown that if the government cannot impose a consumption tax on a good, it may wish to impose an input tax on that same good, resulting in production inefficiency. The production inefficiency we expose is far more general. Usually, a tax on all inputs would be needed to decentralize the second best optima.
3. INDIRECT CONTROL AND THE ROLE OF PRODUCER PRICES We now model a situation in which, as before, the federal government sets the taxes, {, but does so indirectly by the control of final consumer prices. 10 Although the federal government can control consumer prices and the demogrant, producer prices adjust so as to preserve equilibrium. We begin again with the case where the federal government controls all consumer prices that the province does and then add the constraints that prevent it from doing so. Remembering that q= p+t+{, where t is the provincial tax vector and { is the federal tax vector, there exist strict 10
Wildasin [12] uses this assumption in his analysis of a question very similar to our own.
310
BLACKORBY AND BRETT
Pareto-improving changes, starting from a tight equilibrium, if and only if system A below has a solution
* [ &X T A=
_
* &{ q x * tT { x q
0 TN
dq dm 1 H 0 H_N 0 H_N ] > >0 dp dy g dq * { p y*c &I N_N &{ m x dm t T { m x* & y gT & pT 0. dp 0 0 TN { Tyg F dy g
_&
&_
(3.1)
&
A has a solution if and only if B does not have a solution; B is given by &v T &zp T +r{ y g F=0,
(3.2)
v { p y* &zy =0,
(3.3)
*&zt T { x*, : s h =v T { m x m
(3.4)
T
c
gT
h
and *+zt T { q x *, : s h x*T =&v T { q x
(3.5)
h
where 0{s 0,
v 0,
r 0,
and z 0.
(3.6)
Just as in the preceding analysis, one can show that the provincial budget is not strictly binding at the second-best optimum and that production efficiency prevails. We now turn our attention to the case where a subset of commodity taxes is not under the control of the federal authorities, here, { 2 . However, we continue to assume that the federal government can implement its control over sector one taxes by means of consumer prices in sector one. Thus we consider a problem more restrictive than A, named A, by setting dq 2 =0; it is given by adjoining
[0 n2_n1
I n2_n2 0 n2 0 n2_N 0 n2_N ]
dq dm =0 dp dy g
_&
(3.7)
311
FISCAL FEDERALISM REVISITED
to A. By Motzkin's theorem, A has a solution if and only if B does not have a solution; B is given by &v T &zp T +r { y g F=0, v T { y*=zy gT,
(3.8)
*&zt T { x*, : s h =v T { m x m
(3.10)
(3.9)
p
h
and T *+zt T { x * : s h x*T =&v T { q x T ). q +(0 n1 , w
(3.11)
h
where 0{s 0,
v 0,
r 0,
and z 0.
Postmultiplying (3.9) by p shows that z is zero and hence, that the provincial budget constraint is not strictly binding. This, in turn, implies that v is proportional to p-production efficiency. We summarize this as Theorem 3. Suppose that the federal government can control only a subset of consumer prices as well as a universal demogrant. At all Pareto-optima the provincial budget constraint fails to be strictly binding and shadow prices are proportional to producer prices, i.e., there is production efficiency. What is surprising is that the federal government now appears to have enough instruments to essentially circumvent the provincial budget constraint. Even though the province is guaranteed a level of public good that is financed by provincial taxes, by controlling sector one prices, the demogrant, and letting all producer prices adjust to the new equilibrium, any production inefficiency that might have been generated by this fiscalfederal arrangement can be avoided. Nevertheless, there are strict Paretoimprovements possible by moving to tax harmonization. So, what is the fundamental difference between this result and the situation described in the previous section? In order to assess this issue, it is helpful to ask when complete local control of { 1 , p, m and y g generates the same feasible directions of reform as complete local control of q 1 , p, m and y g. The former set of controls generates feasible directions given by
_
*+{ y*c &{ x * &{ x * &{ q x p q1 m T gT T T * * * t { q x& y t { q1 x t { m x T 0 TN 0 0N
The latter generates feasible directions.
&I N_N & pT { Tyg F
&_
dp d{ 1 0. dm dy g
&
(3.13)
312
BLACKORBY AND BRETT
_
* &{ x * { y*c &{ q1 x m p * t T { m x* & y gT t T { q1 x 0
T N
0
0
T N
&I N_N & pT T yg
{ F
&_
dq 1 dm 0. dp dy g
&
(3.14)
If we substitute the identity dq 1 =dp 1 +d{ 1 into (3.14) and expand, we find that (3.13) and (3.14) are equivalent exactly when * dp={ q x * dp 1 , {q x 1
(3.15)
* dp =0. { q2 x 2
(3.16)
which is equivalent to
Thus, the two approaches are equivalent only when changes in demand caused by changes in sector two producer prices (through their effect on consumer prices) have no effect on the value of aggregate demand. The form of condition (3.16) is reminiscent of Proposition 1 of Guesneries [6], p. 187. The intuition behind that result can help us to interpret the present result. The focus on changes in consumer prices in standard tax reform theory is without loss of generality when the aggregate production set is sufficiently smooth, as adjustments in producer prices can be found to decentralise any small change in demand along the production possibilities frontier. In general, such adjustments require a change in all producer prices. When the federal government changes the taxes under its control, it induces changes in demands for all goods. The concomitant adjustments in producer prices required to maintain equilibrium induce changes in consumer prices in sector two precisely because the federal government does not have the power to initiate offsetting taxes. Consumers then respond to these changes in sector two prices. Only when (3.16) is satisfied do these responses have no effect on overall feasibility. In addition, note that (3.16) generates n 2 linear restrictions in the system. Thus, using consumer prices as control variables amounts to imposing n 2 normalizations on prices. For example, if we set all producer prices in sector two to unity, all changes must satisfy dp 2 =0, and, hence (3.16) would hold. It is interesting to compare this source of inefficiency with the existing literature on vertical fiscal externalities. 11 In that literature, different levels of government share a common tax base. An increase in the rate of taxation by one jurisdiction reduces the tax base, leading to a revenue loss for other governments reliant on that base. Failure to take the revenue of others into account leads to nonoptimal taxation, and provides a justification for cooperation (possibly harmonization) among levels of government. 11
Boadway and Keen [3], and Dahlby [4] are among the contributions to this literature.
313
FISCAL FEDERALISM REVISITED
In our model, the federal government is aware of the injury (or service) it is doing to provincial coffers. Indeed, the federal government can modify the taxes over which it has control to take account of changes in the pattern of demand in sector one. It does not, however, have the ability to counteract the general equilibrium effects of its tax changes on sector two.
4. TWO PROVINCES In this section we sketch briefly the extension of the above to the case of two provinces and show that all of the above results are robust to this extension. The provinces, : and ;, each provide a public good for their respective citizens which is financed by provincial taxes. In general, therefore, the consumers in the two provinces do not face the same prices. We consider only the case of direct control of taxes by the federal government, and begin with the case of full harmonization, the federal government can levy commodity taxes in the two provinces independently, as well as a universal demogrant. We show that, in this case, neither of the provincial budget constraints is strictly binding, that there is production efficiency. We then show that Pareto-optimality does not imply consumption efficiency, that is, consumer prices are not equalized across provinces. Consumer prices are given by q : =p+t : +{ : where t : is the vector of provincial specific taxes in province : and { : is the federal specific tax vector for that province. For province ;, the prices are q ; =p+t ; +{ ;. The individual indirect utility functions are given by u h =V h(q :, m, g : )
for all h # A
(4.1)
u h =V h(q ;, m, g ; )
for all h # B,
(4.2)
and
where A (respectively B) is the set of people living in province : (respectively ;). Let H : and H ; be the number people living in : and ; respectively. The competitive sector is still represented by a profit function, ?( p) so that supply is given by y*c ={ p ?( p).
(4.3)
We assume that the technologies for producing the public goods are given by g i F i ( y gi )
where F i (0)=0,
for
i=:, ;.
(4.4)
314
BLACKORBY AND BRETT
Again, we do not assume in advance that there is production efficiency but rather demonstrate the circumstances under which it holds. Net supply is given by y*=y* c & y g: & y g;.
(4.5)
*+y* 0, &x
(4.6)
Equilibrium is given by
where * ; = : x* h + : x * h. *=x * : +x x h#A
(4.7)
h#B
In addition, the provincial governments must balance their respective budgets so that * i p Ty gi tiT x
for
i=:, ;.
(4.8)
Implicit in this formulation is that there is free trade between the provinces and that the federal government levies a hundred percent profit tax. Walras's Law guarantees that the federal budget is balanced. Utility increases are given by du h >0 W { Tq V h(q i, m, g i ) dp+{ Tq V h(q i, m, g i ) d{ i +{ m V h(q i, m, g i ) dm>0 W &x* hTdp+ &x* hTd{ i +dm>0, for all h, and, for
i=:, ;.
(4.9)
Let the matrices of demands be given by * X :T
and
* X ;T.
(4.10)
Starting from a tight equilibrium, and assuming that the federal government has as instruments the commodity tax vectors and the demogrant, there exist strict Pareto-improving directions of change if and only if A has a solution where
315
FISCAL FEDERALISM REVISITED
_
* &X :T * &X ;T
* &X :T 0 H :_N * 0 H ;_N &X ;T
1H : 1H;
0 H :_N 0 H :_N 0 H ;_N 0 H ;_N
&
A=
_
&{ q x*+{ p y* c &{ q x* : t :T { q x* : &y g:T t :T { q x* : t ;T { q x* ; &y g;T 0 TN T 0N 0 TN 0 TN 0 TN
dp d{ : d{ ; dm dy :g dy ; g
_&
> >0
*; * &I N_N &{ q x &{ m x 0 TN t :T { m x* : &p T t ;T { q x* ; t ;T { m x* ; 0 TN T 0N 0 { Tyg F : 0 TN 0 0 TN
&I N_N 0 TN & pT 0 TN { Tyg F ;
dp d{ : d{ ; dm dy :g dy ;g
&_ &
0.
(4.11) By Motzkin's theorem, A has a solution if and only if B does not have a solution; B is given by &v T &z i p T +r i { y g F i =0,
for
i=:, ;,
H
*&z t :T { x * : &z t ;T { x* ;, : s h =v T { m x : m ; m h=1
and * ;T =&v T { x * ; +z ;T { x * ;, : sh x q t q
(4.14)
h#B
* :T =&v T { x * : +z t :T { x * :, : sh x q : q
(4.15)
h#A H
* hT =&v T { q x *+v T { p y*+z : t :T { q x * : &z : y g:T : sh x h=1
+z ; t ;T { q x* ; &z ; y g;T,
(4.16)
where 0{s 0,
v 0,
r 0,
and z 0.
(4.17)
First, subtract (4.14) and (4.15) from (4.16) to obtain v T { p y*=z : y g:T +z ; y g;T.
(4.18)
316
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If either z : or z ; are positive, postmultiplying (4.18) by p yields a contradiction. Hence both multipliers are zero and neither provincial budget constraint is strictly binding. This in turn implies that v=+p
(4.19)
so that, by (4.12), we again have production efficiency. From this it is clear that the results of the two previous sections can be reinterpreted in this context. If the federal government can control only a subset of taxes in either province as well as the universal demogrant, then there is production inefficiency. If however, the control is exercised by means of consumer prices then production efficiency rules again. Proceeding as in the previous sections, it is clear that tax harmonization in only one province is Pareto-improving. But, as long as there is any interaction or trade between the two provinces, only complete tax harmonization is sufficient to obtain production efficiency. These results may seem surprising, given the universal nature of the demogrant. But, with unrestricted taxation, consumer prices in the two provinces can be normalized separately. Thus, in real terms, the common demogrant is essentially two demogrants.
5. CONCLUDING REMARKS We have shown that, in a simple model of a fiscal-federal system, there exist strict Pareto-improving changes from tax harmonization. Although the model is simple there is nothing to suggest making the model more complex would make the case for tax harmonization less compelling. Perhaps the most restrictive feature of our model is the passive nature of our provincial government. In particular, we have made no presumption that provincial governments set their taxes optimally. However, as long as there exists some sort of imperfection in the taxation powers of provincial governments, our analysis is well-grounded. After all, provincial governments, too, have constraints placed on them. In the process we have noted a modeling problem. Using specific taxes the consumer price vector is given by producer prices plus the provincial and federal tax vectors. If the federal government can control all taxes then it makes no difference if the federal government controls its taxes indirectly via the consumer price vector or directly via the tax vector itself. In the case where there exist some taxes that the federal government cannot control, for example are zero-rated, then this equivalence disappears. If the planner could control a demogrant and a subsector of final goods prices (controlling taxes indirectly) the social shadow prices are always the
FISCAL FEDERALISM REVISITED
317
producer prices and hence there is at least production efficiency even in face of this fiscal-federal problem. However, no one has ever suggested that this is the way government behave; governments set their taxes directly and consumer prices indirectly, via market adjustments. When modeled this way, the social shadow prices are no longer proportional to producer prices and the lack of tax harmonization leads to production inefficiency. Harmonization alleviates this problem by giving tax setters sufficient power to take account of the feedback effects of changes in tax rates.
REFERENCES 1. C. Blackorby and W. Diewert, Expenditure functions, local duality, and second order approximations, Econometrica 47 (1979), 579601. 2. C. Bliss, ``Capital Theory and the Distribution of Income,'' NorthHolland, Amsterdam, 1975. 3. R. Boadway and M. Keen, Efficiency and the Optimal Direction of FederalState Transfers, Internat. Tax Public Finance 3 (1996), 137155. 4. B. Dahlby, Fiscal externalities and the design of intergovernmental grants, Internat. Tax Public Finance 3 (1996), 397412. 5. W. Diewert, M. Avriel, and I. Zang, Nine Kinds of Quasi-Concavity and Concavity, J. Econ. Theory 25 (1981), 397420. 6. R. Guesnerie, On the direction of tax reform, J. Public Econ. 7 (1977), 179202. 7. R. Guesnerie, General statements on second best pareto optimality, J. Math. Econ. 6 (1979), 169194. 8. R. Guesnerie, ``A Contribution to the Pure Theory of Taxation,'' Cambridge University Press, Cambridge, 1995. 9. O. Mangasarian, ``Nonlinear Programming,'' McGrawHill, New York, 1969. 10. G. Myles, ``Public Economics,'' Cambridge University Press, Cambridge, 1995. 11. D. Newbery, On the desirability of input taxes, Econ. Lett. 20 (1986), 267270. 12. D. Wildasin, Production efficiency in tax-distorted economies with multiple revenue constraints, mimeo, University of Illinois, 1978.
NOTES, COMMENTS, AND LETTERS TO THE EDITOR
Fiscal Federalism Revisited 1 Charles Blackorby Department of Economics, University of British Columbia, 997-1873 East Mall, Vancouver, BC, Canada V6T 1Z1; and GREQAM, Centre de la Vieille Charite, 2 rue de la Charite, 13002 Marseille, France orbyecon.ubc.ca, orbyehess.cnrs-mrs.fr
and Craig Brett Department of Economics, University of Essex, Wivenhoe Park, Colchester, Essex CO4 3SQ, United Kingdom craigessex.ac.uk Received September 1, 1998; revised January 3, 2000
We analyze how constitutional restrictions on tax bases within a federation affect the nature of Pareto-improving directions of tax reform and the design of optimal federal taxes. We show that constraints on federal taxation entail production inefficiency at the optimum, except under very restrictive circumstances. In passing, we show that using consumer prices as control variablesa standard procedure in tax-reform analysisrather than the taxes themselves, leads to incorrect conclusions when not all taxes or prices can be controlled. Journal of Economic Literature Classification Numbers: D61, H21, H70. 2000 Academic Press Key Words: fiscal federalism; tax reform; production efficiency.
1. INTRODUCTION Most discussions of fiscal federalism center on the devolution of power in light of both efficiency and equity considerations. When the GST (Goods and Services Tax) replaced the manufacturers' sales tax in Canada, the argument was primarily based on efficiency: it was to be a revenue-neutral change that eliminated many distortions generated by the manufacturer's sales tax. However, adding the GST to the existing provincial sales taxes 1 We thank Michael Keen, an associate editor, and a seminar audience at the University of Essex for comments on an earlier version of this paper. Financial assistance from the Social Sciences and Humanities Research Council of Canada is gratefully acknowledged.
300 0022-053100 35.00 Copyright 2000 by Academic Press All rights of reproduction in any form reserved.
FISCAL FEDERALISM REVISITED
301
raised again the issue of tax harmonization. This dialogue has been accentuated recently by the tax-harmonization agreement between the federal government and the Atlantic provinces; the other Canadian provinces have refused to enter into such an agreement. We address one aspect of the tax-harmonization issue in this paper. In the context of a simple model we show that the failure to harmonize taxes entails production inefficiency; that is, without tax harmonization, the second-best optimum of the economy is necessarily inside its production possibility frontier. It is therefore no surprise that a move towards tax harmonization yields a strict Pareto-improvement. Further this strict Pareto-improvement can be achieved without resorting to individual lumpsum transfersonly a demogrant 2 and tax harmonization are required as instruments. There are, of course, many potential benefits to tax harmomization beyond the simple efficiency considerations discussed here, in particular, the reduction of administrative costs. Even these simple efficiency considerations seem however quite compelling. In our model, the provincial government provides a public good that it finances by means of commodity taxes. We assume that the federal government takes the taxes set by the provinces as given; that is, the provinces are assumed to have already made their decisions and must stick with them. However, the federal government recognizes that its taxation decisions have effects on the provincial budget constraint. Indeed, overall feasibility requires that federal tax changes leave the provincial budget in balance. The only way to fully avoid this issue would be to assume complete revenue sharing on the part of the two governments. That is, one could consider the case in which the federal government actually compensates the province (or receives compensation from the province) for changes in provincial revenue due to changes in federal policy. Although attempts at compensation are sometimes observed, we view them as exceptions. 3 Hence, we choose to model a world without complete revenue sharing. We begin with the case where taxes are harmonized, and characterize the second-best optimum. The fiscal-federal problem is introduced by only permitting the federal government to control the demogrant and the taxes on those commodities not taxed by the provincial authorities. Until the last section we act as if there were but one province but two levels of tax authority. Although artificial, this is sufficient to capture some of the problems that arise in a fiscal-federal system. In particular, it allows us to focus on the issue of constitutional restrictions on the power of some levels of government to tax. To fix ideas, we consider the case in which 2
A demogrant is a lump-sum tax that is the same for all individuals, often called a poll tax. The Canadian federal government paid compensation to provincial governments as part of its move toward tax harmonization. 3
302
BLACKORBY AND BRETT
there are some commodities the federal government cannot tax. There is, in general, a three part partition of the the commodity vector: commodities that are zero-rated by the province and taxed by the federal government, commodities that are taxed both both levels of government, and those that are taxed by the province and zero-rated by the federal tax authority. Given that the federal government takes the taxes of the province as given, the existence of the first category generates no new possibilities for the federal tax authority. Hence we deal only with the latter two. Consumer prices are given by producer prices plus the provincial tax vector and the federal tax vector. The model is analyzed in two supposedly equivalent ways: letting the federal government control taxes indirectly by means of final consumer prices or directly by means of the federal tax vector. When the federal government can control all prices or all taxes these two procedures are equivalent. It is common to let the planner in these models control consumer prices directly and taxes only indirectly for technical convenience; 4 however, when the federal government can control only a subset of consumer prices or taxes these two modeling strategies are no longer equivalent. When the federal government controls a subset of taxes indirectly through consumer prices the social shadow prices are equal to producer prices. However, when the planner controls only a subset of the taxes directly, social shadow prices are no longer equal to producer prices and the economy is inside its production possibility frontier at the second-best optima. In addition, when a subset of consumer prices is used as the set of control variables, the provincial budget constraint, although satisfied, is not strictly binding; that is, the multiplier on the provincial budget constraint is always zero. When a subset of the tax vector is controlled directly this constraint is generally strictly binding. This is of some interest in itself as actual fiscal-federal systems control taxes directly and not indirectly; the choice of modeling strategies is no longer innocuous. In this context we show that there are strict Pareto-improvements possible by means of tax harmonization. In addition we show that these improvements come by changing relative prices in the sector under provincial control and not simply by adjusting their level. In the last section we extend these results to two provinces. (Except for notation this comprises the general case.)
2. DIRECT CONTROL OF TAXES We begin with the case where the federal government can tax all commodities and levy a demogrant, m. The provincial government provides a 4
See for example Guesnerie [6, 7] and Myles [10].
FISCAL FEDERALISM REVISITED
303
public good and finances it by means of commodity taxation. Throughout, the level of the public good is fixed. We then move to the case where the federal government cannot adjust the taxes on some commodities. Consumer prices are given by q= p+t+{, where t is the vector of provincial specific taxes and { is the federal specific tax vector. The individual indirect utility functions are given by u h =V h(q, m, g),
(2.1)
where g is the fixed amount of the public good provided by the provincial government. V h is differentially strongly quasi-convex 5 in prices and increasing in income and the level of the public good. The competitive sector is represented by a profit function, ?( p), 6 which is assumed to be differentially strongly convex; 7 the competitive sector supply is given by y*c ={ p ?( p).
(2.2)
We assume that the technology for producing the public good is given by g F( y g )
where
F(0)=0;
(2.3)
F is continuously differentiable, increasing, and concave. At the second-best optimum the authorities could simply direct the producer of the public good to use a specific set of inputs. Alternatively, the government can give the managers of this firm a set of shadow prices at which to minimize costs, say v. Then the managers minimize the cost of producing a given level of the public good by solving min[v Ty g | g F( y g )].
(2.4)
If it is optimal for the government to set these shadow prices proportional to producer prices then there is production efficiency; the marginal rates of transformation in the public sector enterprise and the competitive private sector are equal. If however, the optimal shadow prices are not proportional to the producer prices then there is production inefficiency at the optimum; that is, the economy is inside its production possibility frontier. We do not assume in advance that there is production efficiency but rather demonstrate the circumstances under which it holds. Net supply is given by y*=y*c & y g. 5
(2.5)
See Blackorby and Diewert [1] for details. There is no loss of generality in assuming that the competitive sector has a single firm; see Bliss [2], page 68 for example. 7 See Diewert, Avriel, and Zang [5] for details. This implies that the Hessian of the profit function has rank N&1. 6
304
BLACKORBY AND BRETT
Equilibrium is given by *+y* 0 &x
where
x*=: x*h.
(2.6)
h
In addition, the provincial government must balance its budget so that * p Ty g. tT x
(2.7)
When (2.6) and (2.7) hold with equality we say that the equilibrium is tight. Implicit in this formulation is that the federal government levies a hundred percent profit tax. Walras's Law guarantees that the federal budget is balanced. Utility increases are given by du h >0 W { Tq V h(q, m, g) dp+{ Tq V h(q, m, g) d{+{ m V h(q, m, g) dm>0 * hTd{+dm>0, * hTdp&x W &x
(2.8)
where the last step uses Roy's theorem. Let the matrix of demands be given by * X T=
x* 1T b . * HT x
_ &
(2.9)
Starting from a tight equilibrium, and assuming that the federal government has as instruments the commodity tax vector and the demogrant, there exist strict Pareto-improving equilibrium-preserving directions of change if and only if A has a solution where
*T [ &X
*T &X
1H
A=
_
dp d{ 0 H_N ] > >0 dm dy g
_&
*+{ y* c &{ x * &{ x * &{ q x p q m *& y gT t T { x* t T { x * tT { q x q m 0 TN
0 TN
0
&I N_N & pT { Tyg F
&_
dp d{ 0. dm dy g
(2.10)
&
By Motzkin's theorem, 8 A has a solution if and only if a complementary system, B, does not. B is given by 8
See Mangasarian [9], page 28.
FISCAL FEDERALISM REVISITED T T gT * hT =&v T { x * *c * : sh x , q +v { p y +zt { q x &zy
305 (2.11)
h T * hT =&v T { x * * : sh x q +zt { q x,
(2.12)
h
*&zt T { x*, : s h =v T { m x m
(2.13)
h
and &v T &zp T +r { Tyg F=0,
(2.14)
where 0{s 0,
v 0,
r 0,
and z 0.
The system of relations B characterizes the local second-best optima for the economy. The vector v is the vector of social shadow prices of goods, while s is the vector of social marginal utilities of income. These optima are characterised by Theorem 1. Assume that the federal government can levy taxes on all commodities as well as a universal demogrant. At all Pareto-optima, shadow prices are proportional to producer prices, i.e., there is production efficiency, and the provincial budget constraint is not strictly binding. Proof.
Subtract (2.12) from (2.11) to obtain v T { p y* c &zy gT =0.
(2.16)
Postmultiplying (2.16) by p, and using the homogeneity of supply yields zy gTp=0.
(2.17)
z>0 is a contradiction given that y g and p are positive. Therefore, z=0; that is, the provincial budget constraint is not strictly binding. It follows that social shadow prices, v, are proportional to producer prices. To see this note that (2.16) can now be written as v T { p y*=0.
(2.18)
Because the profit function is strongly convex, its Hessian has rank N&1 and p T { p y*=0.
(2.19)
306
BLACKORBY AND BRETT
This implies that v is proportional to p. From (2.14) this implies that p=% { y g F( y g, g)
(2.20)
for some %>0. That is, the firm producing the public good should minimize costs using producer prices. K Altogether, this looks exactly like revenue sharing. The provincial budget constraint has no shadow value. This is precisely because of the fact that, when all taxes are under federal control, the model admits the separate normalization of producer and consumer prices. 9 In such circumstances, the exact numerical values of taxes have little significance. They can be chosen in such a way to ensure budget balance for the province. We are interested in the case where a subset of commodity taxes is under exclusive control of the provincial government, say sector two. We continue to assume that the federal government can effect local changes in tax rates in sector one. Thus, we construct a restricted problem, A, by adding the constraint d{ 2 =0, that is,
[0 n2_N
0 n2_n1 I n2_n2
0 n2
dp d{ 0 n2_N ] =0, dm dy g
_&
(2.21)
to problem A. A has a solution if and only if B does not where B is given by gT *+v T { y*+zt T { x * , : s h x hT =&v T { q x p q &zy
(2.22)
h T T * hT =&v T { x * * : s h x T ), q +zt { q x +(0 n1 , w
(2.23)
h
*&zt T { x*, : s h =v T { m x m
(2.24)
h
and &v T &zp T +r { y g F=0,
(2.25)
where 0{s , 9
v 0,
r 0,
and z 0.
Guesnerie [8], pp. 7781 provides a detailed discussion of this issue.
(2.26)
307
FISCAL FEDERALISM REVISITED
If the lack of tax harmonization is an effective constraint, then A has a solution while A does not. Equivalently, B does not have a solution and B does with w nonzero. Theorem 2. Suppose that the federal government can control only a subset of commodity taxes and a universal demogrant. Then, at all Pareto-optima, shadow prices are not proportional to producer prices, i.e., there is production inefficiency. Proof.
First, suppose that z {0. Subtracting (2.23) from (2.22) yields v T { p y* c &zy gT =(0 Tn1 , w T ).
(2.27)
From (2.27) it is obvious that v{%p, that is, shadow prices are not equal to producer prices; there is production inefficiency. Now, suppose that z =0. Evaluating (2.27) when z =0 yields v T { p y=(0, w T ).
(2.28)
Given strong convexity of the profit function, (2.28) implies that v=+p only if w =0. That is, production efficiency holds only if the fiscal federalism constraints are not strictly binding. It remains to show the conditions under which the fiscal federal restrictions, d{ 2 =0, are strictly binding even when the provincial budget constraint is not strictly binding. The fiscal federalism constraints are not binding if and only if A has a solution whenever A does, where A is given by
* [ &X T
* &X T1 1 H 0 H_N ]
A =
_
dp d{ 1 > >0 dm dy g
_&
*+{ y* c &{ x * &{ x * &{ q x p q1 m *& y gT t T { x* t T { x * tT { q x q1 m 0 TN
0 TN
0
&I N_N & pT { Tyg F
&_
dp d{ 1 0. dm dy g
(2.29)
&
Clearly if A has a solution then A does. Suppose that whenever A has a solution then A does as well. This is the same as saying that if A does not have a solution then A does not have a solution which in turn says that when B (the dual of A ) has a solution then B does as well.
308
BLACKORBY AND BRETT
By Motzkin's theorem, if A does not have a solution, then B does; it is given by * hT =&v~ T { q x *+v~ T { p y* c +z~t T { q x*&z~y gT, : s~ h x
(2.30)
h T *hT =&v~ T { x * * : s~ h x q1 +z~t { q1 x, 1
(2.31)
h
: s~ h =v~ T { m x*&z~t T { m x*,
(2.32)
h
and &v~ T &z~p T +r~ { Tyg F=0,
(2.33)
where v~ 0,
0{s~ 0,
r~ 0,
and z~ 0.
(2.34)
Remembering that we are in the case where the provincial budget constraint is not strictly binding, subtract (2.31) from (2.30) to obtain 0 Tn1 =v~ T { p1 y* c, *hT =&v~ T[{ q x *+{ p y* c ] : s~ h x 2 2 2
(2.35) (2.36)
h
and, from (2.31), T *. : s~ h x*hT { q1 x 1 =&v~
(2.37)
h
Given our regularity conditions, B does not imply B. That is, the fiscalfederal constraints on the taxation power of the federal government are strictly binding. This in turn implies that v{%p; there is production inefficiency whether or not the provincial budget constraint is strictly binding. K Two other characteristics of these Pareto-optima are also of interest. Multiplying (2.24) by m and subtracting it from (2.22) after having multiplied the latter by q yields 0=[v T { p y*&zy gT ] q.
(2.38)
FISCAL FEDERALISM REVISITED
309
Substituting (2.27) into (2.38) yields 0=w Tq 2 .
(2.39)
That is the multiplier on the fiscal federalism constraints is orthogonal to the consumer price vector in the provincial sector. Post-multiplying (2.27) by p shows that w Tp 0.
(2.40)
v T { p yv=w Tv 2 >0;
(2.41)
Furthermore, from (2.28),
that is, the shadow prices in sector two do not lie in the subspace orthogonal to w which mean that sector two shadow prices cannot be written as a convex combination of producer prices and consumer prices in that sector. Condition (2.39) provides insight into the nature of the adjustments the federal government would like to make in sector two prices, if it could. Formally, we have that the vector of shadow values on the fiscal federalism constraints is orthogonal to sector two consumer prices. Thus, changes in the relative prices in that sector would be required to bring about a Pareto improvement. A mere rescaling of the prices (relative to those prevailing in sector one) would not suffice. This is reminiscent of the work by Newbery [11], in which it was shown that if the government cannot impose a consumption tax on a good, it may wish to impose an input tax on that same good, resulting in production inefficiency. The production inefficiency we expose is far more general. Usually, a tax on all inputs would be needed to decentralize the second best optima.
3. INDIRECT CONTROL AND THE ROLE OF PRODUCER PRICES We now model a situation in which, as before, the federal government sets the taxes, {, but does so indirectly by the control of final consumer prices. 10 Although the federal government can control consumer prices and the demogrant, producer prices adjust so as to preserve equilibrium. We begin again with the case where the federal government controls all consumer prices that the province does and then add the constraints that prevent it from doing so. Remembering that q= p+t+{, where t is the provincial tax vector and { is the federal tax vector, there exist strict 10
Wildasin [12] uses this assumption in his analysis of a question very similar to our own.
310
BLACKORBY AND BRETT
Pareto-improving changes, starting from a tight equilibrium, if and only if system A below has a solution
* [ &X T A=
_
* &{ q x * tT { x q
0 TN
dq dm 1 H 0 H_N 0 H_N ] > >0 dp dy g dq * { p y*c &I N_N &{ m x dm t T { m x* & y gT & pT 0. dp 0 0 TN { Tyg F dy g
_&
&_
(3.1)
&
A has a solution if and only if B does not have a solution; B is given by &v T &zp T +r{ y g F=0,
(3.2)
v { p y* &zy =0,
(3.3)
*&zt T { x*, : s h =v T { m x m
(3.4)
T
c
gT
h
and *+zt T { q x *, : s h x*T =&v T { q x
(3.5)
h
where 0{s 0,
v 0,
r 0,
and z 0.
(3.6)
Just as in the preceding analysis, one can show that the provincial budget is not strictly binding at the second-best optimum and that production efficiency prevails. We now turn our attention to the case where a subset of commodity taxes is not under the control of the federal authorities, here, { 2 . However, we continue to assume that the federal government can implement its control over sector one taxes by means of consumer prices in sector one. Thus we consider a problem more restrictive than A, named A, by setting dq 2 =0; it is given by adjoining
[0 n2_n1
I n2_n2 0 n2 0 n2_N 0 n2_N ]
dq dm =0 dp dy g
_&
(3.7)
311
FISCAL FEDERALISM REVISITED
to A. By Motzkin's theorem, A has a solution if and only if B does not have a solution; B is given by &v T &zp T +r { y g F=0, v T { y*=zy gT,
(3.8)
*&zt T { x*, : s h =v T { m x m
(3.10)
(3.9)
p
h
and T *+zt T { x * : s h x*T =&v T { q x T ). q +(0 n1 , w
(3.11)
h
where 0{s 0,
v 0,
r 0,
and z 0.
Postmultiplying (3.9) by p shows that z is zero and hence, that the provincial budget constraint is not strictly binding. This, in turn, implies that v is proportional to p-production efficiency. We summarize this as Theorem 3. Suppose that the federal government can control only a subset of consumer prices as well as a universal demogrant. At all Pareto-optima the provincial budget constraint fails to be strictly binding and shadow prices are proportional to producer prices, i.e., there is production efficiency. What is surprising is that the federal government now appears to have enough instruments to essentially circumvent the provincial budget constraint. Even though the province is guaranteed a level of public good that is financed by provincial taxes, by controlling sector one prices, the demogrant, and letting all producer prices adjust to the new equilibrium, any production inefficiency that might have been generated by this fiscalfederal arrangement can be avoided. Nevertheless, there are strict Paretoimprovements possible by moving to tax harmonization. So, what is the fundamental difference between this result and the situation described in the previous section? In order to assess this issue, it is helpful to ask when complete local control of { 1 , p, m and y g generates the same feasible directions of reform as complete local control of q 1 , p, m and y g. The former set of controls generates feasible directions given by
_
*+{ y*c &{ x * &{ x * &{ q x p q1 m T gT T T * * * t { q x& y t { q1 x t { m x T 0 TN 0 0N
The latter generates feasible directions.
&I N_N & pT { Tyg F
&_
dp d{ 1 0. dm dy g
&
(3.13)
312
BLACKORBY AND BRETT
_
* &{ x * { y*c &{ q1 x m p * t T { m x* & y gT t T { q1 x 0
T N
0
0
T N
&I N_N & pT T yg
{ F
&_
dq 1 dm 0. dp dy g
&
(3.14)
If we substitute the identity dq 1 =dp 1 +d{ 1 into (3.14) and expand, we find that (3.13) and (3.14) are equivalent exactly when * dp={ q x * dp 1 , {q x 1
(3.15)
* dp =0. { q2 x 2
(3.16)
which is equivalent to
Thus, the two approaches are equivalent only when changes in demand caused by changes in sector two producer prices (through their effect on consumer prices) have no effect on the value of aggregate demand. The form of condition (3.16) is reminiscent of Proposition 1 of Guesneries [6], p. 187. The intuition behind that result can help us to interpret the present result. The focus on changes in consumer prices in standard tax reform theory is without loss of generality when the aggregate production set is sufficiently smooth, as adjustments in producer prices can be found to decentralise any small change in demand along the production possibilities frontier. In general, such adjustments require a change in all producer prices. When the federal government changes the taxes under its control, it induces changes in demands for all goods. The concomitant adjustments in producer prices required to maintain equilibrium induce changes in consumer prices in sector two precisely because the federal government does not have the power to initiate offsetting taxes. Consumers then respond to these changes in sector two prices. Only when (3.16) is satisfied do these responses have no effect on overall feasibility. In addition, note that (3.16) generates n 2 linear restrictions in the system. Thus, using consumer prices as control variables amounts to imposing n 2 normalizations on prices. For example, if we set all producer prices in sector two to unity, all changes must satisfy dp 2 =0, and, hence (3.16) would hold. It is interesting to compare this source of inefficiency with the existing literature on vertical fiscal externalities. 11 In that literature, different levels of government share a common tax base. An increase in the rate of taxation by one jurisdiction reduces the tax base, leading to a revenue loss for other governments reliant on that base. Failure to take the revenue of others into account leads to nonoptimal taxation, and provides a justification for cooperation (possibly harmonization) among levels of government. 11
Boadway and Keen [3], and Dahlby [4] are among the contributions to this literature.
313
FISCAL FEDERALISM REVISITED
In our model, the federal government is aware of the injury (or service) it is doing to provincial coffers. Indeed, the federal government can modify the taxes over which it has control to take account of changes in the pattern of demand in sector one. It does not, however, have the ability to counteract the general equilibrium effects of its tax changes on sector two.
4. TWO PROVINCES In this section we sketch briefly the extension of the above to the case of two provinces and show that all of the above results are robust to this extension. The provinces, : and ;, each provide a public good for their respective citizens which is financed by provincial taxes. In general, therefore, the consumers in the two provinces do not face the same prices. We consider only the case of direct control of taxes by the federal government, and begin with the case of full harmonization, the federal government can levy commodity taxes in the two provinces independently, as well as a universal demogrant. We show that, in this case, neither of the provincial budget constraints is strictly binding, that there is production efficiency. We then show that Pareto-optimality does not imply consumption efficiency, that is, consumer prices are not equalized across provinces. Consumer prices are given by q : =p+t : +{ : where t : is the vector of provincial specific taxes in province : and { : is the federal specific tax vector for that province. For province ;, the prices are q ; =p+t ; +{ ;. The individual indirect utility functions are given by u h =V h(q :, m, g : )
for all h # A
(4.1)
u h =V h(q ;, m, g ; )
for all h # B,
(4.2)
and
where A (respectively B) is the set of people living in province : (respectively ;). Let H : and H ; be the number people living in : and ; respectively. The competitive sector is still represented by a profit function, ?( p) so that supply is given by y*c ={ p ?( p).
(4.3)
We assume that the technologies for producing the public goods are given by g i F i ( y gi )
where F i (0)=0,
for
i=:, ;.
(4.4)
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BLACKORBY AND BRETT
Again, we do not assume in advance that there is production efficiency but rather demonstrate the circumstances under which it holds. Net supply is given by y*=y* c & y g: & y g;.
(4.5)
*+y* 0, &x
(4.6)
Equilibrium is given by
where * ; = : x* h + : x * h. *=x * : +x x h#A
(4.7)
h#B
In addition, the provincial governments must balance their respective budgets so that * i p Ty gi tiT x
for
i=:, ;.
(4.8)
Implicit in this formulation is that there is free trade between the provinces and that the federal government levies a hundred percent profit tax. Walras's Law guarantees that the federal budget is balanced. Utility increases are given by du h >0 W { Tq V h(q i, m, g i ) dp+{ Tq V h(q i, m, g i ) d{ i +{ m V h(q i, m, g i ) dm>0 W &x* hTdp+ &x* hTd{ i +dm>0, for all h, and, for
i=:, ;.
(4.9)
Let the matrices of demands be given by * X :T
and
* X ;T.
(4.10)
Starting from a tight equilibrium, and assuming that the federal government has as instruments the commodity tax vectors and the demogrant, there exist strict Pareto-improving directions of change if and only if A has a solution where
315
FISCAL FEDERALISM REVISITED
_
* &X :T * &X ;T
* &X :T 0 H :_N * 0 H ;_N &X ;T
1H : 1H;
0 H :_N 0 H :_N 0 H ;_N 0 H ;_N
&
A=
_
&{ q x*+{ p y* c &{ q x* : t :T { q x* : &y g:T t :T { q x* : t ;T { q x* ; &y g;T 0 TN T 0N 0 TN 0 TN 0 TN
dp d{ : d{ ; dm dy :g dy ; g
_&
> >0
*; * &I N_N &{ q x &{ m x 0 TN t :T { m x* : &p T t ;T { q x* ; t ;T { m x* ; 0 TN T 0N 0 { Tyg F : 0 TN 0 0 TN
&I N_N 0 TN & pT 0 TN { Tyg F ;
dp d{ : d{ ; dm dy :g dy ;g
&_ &
0.
(4.11) By Motzkin's theorem, A has a solution if and only if B does not have a solution; B is given by &v T &z i p T +r i { y g F i =0,
for
i=:, ;,
H
*&z t :T { x * : &z t ;T { x* ;, : s h =v T { m x : m ; m h=1
and * ;T =&v T { x * ; +z ;T { x * ;, : sh x q t q
(4.14)
h#B
* :T =&v T { x * : +z t :T { x * :, : sh x q : q
(4.15)
h#A H
* hT =&v T { q x *+v T { p y*+z : t :T { q x * : &z : y g:T : sh x h=1
+z ; t ;T { q x* ; &z ; y g;T,
(4.16)
where 0{s 0,
v 0,
r 0,
and z 0.
(4.17)
First, subtract (4.14) and (4.15) from (4.16) to obtain v T { p y*=z : y g:T +z ; y g;T.
(4.18)
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BLACKORBY AND BRETT
If either z : or z ; are positive, postmultiplying (4.18) by p yields a contradiction. Hence both multipliers are zero and neither provincial budget constraint is strictly binding. This in turn implies that v=+p
(4.19)
so that, by (4.12), we again have production efficiency. From this it is clear that the results of the two previous sections can be reinterpreted in this context. If the federal government can control only a subset of taxes in either province as well as the universal demogrant, then there is production inefficiency. If however, the control is exercised by means of consumer prices then production efficiency rules again. Proceeding as in the previous sections, it is clear that tax harmonization in only one province is Pareto-improving. But, as long as there is any interaction or trade between the two provinces, only complete tax harmonization is sufficient to obtain production efficiency. These results may seem surprising, given the universal nature of the demogrant. But, with unrestricted taxation, consumer prices in the two provinces can be normalized separately. Thus, in real terms, the common demogrant is essentially two demogrants.
5. CONCLUDING REMARKS We have shown that, in a simple model of a fiscal-federal system, there exist strict Pareto-improving changes from tax harmonization. Although the model is simple there is nothing to suggest making the model more complex would make the case for tax harmonization less compelling. Perhaps the most restrictive feature of our model is the passive nature of our provincial government. In particular, we have made no presumption that provincial governments set their taxes optimally. However, as long as there exists some sort of imperfection in the taxation powers of provincial governments, our analysis is well-grounded. After all, provincial governments, too, have constraints placed on them. In the process we have noted a modeling problem. Using specific taxes the consumer price vector is given by producer prices plus the provincial and federal tax vectors. If the federal government can control all taxes then it makes no difference if the federal government controls its taxes indirectly via the consumer price vector or directly via the tax vector itself. In the case where there exist some taxes that the federal government cannot control, for example are zero-rated, then this equivalence disappears. If the planner could control a demogrant and a subsector of final goods prices (controlling taxes indirectly) the social shadow prices are always the
FISCAL FEDERALISM REVISITED
317
producer prices and hence there is at least production efficiency even in face of this fiscal-federal problem. However, no one has ever suggested that this is the way government behave; governments set their taxes directly and consumer prices indirectly, via market adjustments. When modeled this way, the social shadow prices are no longer proportional to producer prices and the lack of tax harmonization leads to production inefficiency. Harmonization alleviates this problem by giving tax setters sufficient power to take account of the feedback effects of changes in tax rates.
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