Locational efficiency in a federal system
without transfers or land rent taxation
Robert Philipowski
∗
Abstract
While it is generally believed that efficiency in a federal state with mobile population
requires location-based taxes and an undistortive land tax, we show that if there are only
two types of regions, it suffices to tax labor and capital tax income.
Keywords: Federalism, Mobile population, Locational efficiency
JEL classification: H21, H71
1
Introduction
There seems to be a general agreement that achieving an efficient allocation in a federal state with
mobile population requires interregional transfers or “a tax system consisting of (a) locationbased taxes on mobile individuals (. . .) and (b) an undistortive land tax” (Wellisch, 2000, Proposition 2.3).1 Namely, the location-based taxes are needed to internalize congestion costs, and
the land tax is needed to finance efficient public good provision.
While this reasoning seems quite convincing, we show that if a federation consists of two
types of regions (e.g. large and small, or urban and rural regions) such that the regions of each
type are identical to each other, it is in most cases possible to achieve the efficient allocation by
taxing only labor and capital income.
In short, our result can be explained as follows: In a federation with n regions, there are
3n−2 first-order conditions for an efficient allocation (n Samuelson conditions for efficient public
good provision, n − 1 conditions for efficient labor allocation, and n − 1 conditions for efficient
capital allocation), while the local governments have 2n tax rates at their disposal (n wage tax
rates and n capital tax rates). In the case of two regions we have 3n − 2 = 2n, so that typically
there is a unique solution, and this solution extends to the case of two types of regions.
2
The model
Our model is a simplified version of the one presented in Wellisch (2000, Section 2.1.1). We consider a federation consisting of n regions. If region i is inhabited by Li people and if the amount
of capital invested there equals Ki , then total production in that region equals Fi (Li , Ki ).2 The
utility of a person living in region i is given by U (Xi , Zi ), where Xi and Zi are the consumption
levels of private and public good in region i, respectively. By assumption the inhabitants of the
federation are all identical, so that they all have the same utility function U . The price of the
private good is normalized to 1. If region i is inhabited by Li people, the cost of providing the
public good level
Pn Zi there is given by Ci (Zi , Li ). We assume that
Pn the total population of the federation, L := i=1 Li , and the total supply of capital, K := i=1 Ki , are fixed, but that labor
and capital are perfectly mobile within the federation. Homogeneity and perfect mobility of the
population imply that the utility levels in all regions coincide, i.e. U (X1 , Z1 ) = . . . = U (Xn , Zn ).
∗
Unit´e de Recherche en Math´ematiques, Universit´e du Luxembourg, 6, rue Richard Coudenhove-Kalergi, 1359
Luxembourg, Grand Duchy of Luxembourg. E-mail address: [email protected]
1
This insight goes back to Wildasin (1980).
2
In particular, this assumption implies that labor is supplied inelastically.
1
3
First-order conditions for efficiency
While the total population L, the total amount of capital K, the production functions Fi , the
cost functions Ci and the utility function U are exogeneously given, the quantities Li , Ki , Xi
and Zi should be chosen optimally, i.e. in such a way as to
maximize U (X1 , Z1 )
subject to the constraints
U (X1 , Z1 ) = . . . = U (Xn , Zn ),
n
X
Li = L,
(1)
(2)
i=1
n
X
Ki = K,
(3)
i=1
n
X
n
X
Fi (Li , Ki ) =
(Li Xi + Ci (Zi , Li )).
i=1
(4)
i=1
Using the method of Lagrange multipliers one finds the following 3n − 2 necessary first-order
conditions for an interior3 maximizer (cf. Wellisch, 2000, Section 2.1.2):
∂U/∂Zi
∂U/∂Xi
∂F1
∂K1
∂F1
∂C1
− X1 −
∂L1
∂L1
Li
=
∂Ci
,
∂Zi
i = 1, . . . , n,
∂Fn
,
∂Kn
∂Fn
∂Cn
= ... =
− Xn −
.
∂Ln
∂Ln
= ... =
(5)
(6)
(7)
Condition (5) is the usual Samuelson condition for efficient public good provision, (6) means
that the marginal product of capital should be the same in all regions, and (7) means that
the net social benefit of a marginal individual (marginal product of labor minus private good
consumption and marginal congestion cost) should be the same in all regions.4
4
Market behavior
We now make the following assumptions (cf. Wellisch, 2000, Section 2.2.1):
1. In all regions labor and capital are paid according to their marginal products. Hence in
region i the wage rate is given by ∂Fi /∂Li and the gross return to capital by ∂Fi /∂Ki .
Consequently, the total land rent is given by
Ri (Li , Ki ) = Fi (Li , Ki ) − Li
∂Fi
∂Fi
− Ki
.
∂Li
∂Ki
(8)
2. Capital and land in the federation are owned exclusively by its inhabitants, and each of
them owns an equal share of the capital and of the land of each region.
3. Labor and capital income are taxed at source. The wage tax rate in region i is denoted
by τwi , and the capital income tax rate by τki . Land rents, however, are not taxed.
3
In principle, an efficient allocation could involve complete depopulation of one or several regions. Since this
does not occur in practice, we exclude this possibility.
4
If there were no public good (so that ∂U/∂Zi ≡ 0 and Ci ≡ 0 for all i), (1) would simplify to X1 = . . . = Xn ,
and consequently (7) would reduce to the standard efficiency condition ∂F1 /∂L1 = . . . = ∂Fn /∂Ln . The reason
why this condition has to be replaced with (7) is that workers are not only a production factor, but also consumers
of the public good.
2
These assumptions imply that each inhabitant of region i has a total net income (wage income
+ capital income + rent income) of
Xi = (1 −
∂Fi
τwi )
∂Li
n
n
j=1
j=1
∂Fj
1X
1X
+
+
(1 − τkj )Kj
Rj (Lj , Kj )
L
∂Kj
L
or, in view of (8),
Xi = (1 −
∂Fi
τwi )
∂Li
n ∂Fj
∂Fj
1X
j
+
Fj (Lj , Kj ) − Lj
− τk Kj
.
L
∂Lj
∂Kj
(9)
j=1
Perfect mobility of labor and capital implies that the utility level and the net return to capital
are the same in all regions, i.e.
U (X1 , Z1 ) = . . . = Un (Xn , Zn )
and
(1 − τk1 )
∂F1
∂Fn
= . . . = (1 − τkn )
.
∂K1
∂Kn
(10)
(11)
Eqs. (9)–(11), together with the budget constraints of local governments,
Ci (Zi , Li ) = τwi Li
∂Fi
∂Fi
+ τki Ki
,
∂Li
∂Ki
(12)
determine the quantities Li , Ki , Xi and Zi as implicit functions of the tax rates τwj and τkj .
In order to achieve efficiency, Li , Ki , Xi and Zi have to satisfy the 3n − 2 conditions (5)–(7).
Since the local governments have only 2n tax rates at their disposal, this is impossible in general;
however, it is possible (except in a certain degenerate case, see Section 5) if there are only two
regions (because then 3n − 2 = 2n) or, more generally, two types of regions such that the regions
of each type are identical to each other.
In the case of more than two types of regions we are left with the following second-best
problem: Maximize U (X1 , Z1 ) subject to the constraints (2)–(3) and (9)–(12). In the appendix
we will show that in the second-best optimum at least the Samuelson conditions (5) are satisfied.
5
Explicit computation of the efficient tax rates in the case of
two (types of ) regions
In order to compute the optimal tax rates in the case n = 2, let us start from an optimal interior
allocation (L1 , L2 , K1 , K2 , X1 , X2 , Z1 , Z2 ) which, in particular, satisfies (1)–(7), and determine
the tax rates τw1 , τw2 , τk1 , τk2 in such a way that this allocation is compatible with market behavior,
i.e. satisfies
∂F1
∂F2
(1 − τk1 )
= (1 − τk2 )
(13)
∂K1
∂K2
and the budget constraints
X1 = (1 −
X2 = (1 −
∂F1
τw1 )
∂L1
2 ∂Fj
∂Fj
1X
j
+
Fj (Lj , Kj ) − Lj
− τk Kj
,
L
∂Lj
∂Kj
(14)
∂F2
τw2 )
∂L2
2 ∂Fj
∂Fj
1X
j
+
Fj (Lj , Kj ) − Lj
− τk Kj
,
L
∂Lj
∂Kj
(15)
j=1
j=1
∂F1
∂F1
+ τk1 K1
,
∂L1
∂K1
∂F2
∂F2
C1 (Z2 , L2 ) = τw2 L2
+ τk2 K2
.
∂L2
∂K2
C1 (Z1 , L1 ) = τw1 L1
3
(16)
(17)
Note that these four budget constraints are not independent; if three of them are satisfied, the
remaining one is satisfied as well.5 In view of (6), (13) forces us to choose
τk1 = τk2 =: τk ,
so that it remains to determine τw1 , τw2 and τk as the solution of the system
∂F1 1
∂F1
τw + K1
τk = C1 (Z1 , L1 ),
∂L1
∂K1
∂F2
∂F2 2
τw + K2
τk = C2 (Z2 , L2 ),
L2
∂L2
∂K2
∂F1 1 ∂F2 2
∂F1
∂F2
τw −
τw = X2 − X1 +
−
,
∂L1
∂L2
∂L1 ∂L2
L1
(18)
(19)
(20)
where (20) is obtained by subtracting (14) from (15). Combining (20) with (7) we obtain
∂F2 2 ∂C2
∂F1 1 ∂C1
τ −
=
τ −
,
∂L1 w ∂L1
∂L2 w ∂L2
cf. Wellisch (2000, Eq. (2.24)). That is, “the difference between location-based taxes and
marginal congestion costs must be identical across regions in order to avoid fiscal externalities” (Wellisch, 2000, p. 37).
Writing
∂F1 1 ∂C1
∂F2 2 ∂C2
∂F1
tw :=
τw −
=
τw −
and
tk :=
τk ,
(21)
∂L1
∂L1
∂L2
∂L2
∂K1
(18)–(19) simplify to
∂C1
,
∂L1
∂C2
= C2 (Z2 , L2 ) − L2
.
∂L2
L1 tw + K1 tk = C1 (Z1 , L1 ) − L1
(22)
L2 tw + K2 tk
(23)
The economic idea behind (21) is the following: The wage tax τwi ∂Fi /∂Li in region i is split into
1. a congestion fee equal to the marginal congestion cost ∂Ci /∂Li , and
2. a remainder term tw which has to be the same in all regions (in order to assure efficient
labor allocation) and serves to finance the efficient level of public good provision.
If the optimal capital-labor ratios of both regions differ, i.e. if K1 /L1 6= K2 /L2 , the system
(22)–(23) has a unique solution given by
∂C2
1
K2 C1 (Z1 , L1 ) − L1 ∂C
∂L1 − K1 C2 (Z2 , L2 ) − L2 ∂L2
tw =
,
K2 L1 − K1 L2
∂C2
1
L2 C1 (Z1 , L1 ) − L1 ∂C
−
L
C
(Z
,
L
)
−
L
1
2
2
2
2
∂L1
∂L2
tk =
,
K1 L2 − K2 L1
which, via (21), determines the tax rates τwi and τk . In the degenerate case K1 /L1 = K2 /L2
however, there will typically be no solution.
As one can easily see, the solution computed in this section extends to the case of two types
of regions if the regions of each type are identical to each other in the sense that they share the
same production function and the same cost function for public good provision.
5
Economically, this should be clear. Mathematically, it can be seen by multiplying (14) with L1 and (15)
with L2 and then summing up (14)–(17), which yields (4).
4
6
Conclusion
We have shown that the standard view that efficiency requires location-based taxes and an
undistortive land tax is not completely correct: If there are only two types of regions, it suffices
to tax labor and capital tax income. This result is of considerable practical importance since, as
remarked by e.g. H¨
ulshorst and Wellisch (1996, pp. 387–388), massive land rent taxation seems
institutionally infeasible in many countries.
Appendix: The case of more than two types of regions
In this appendix we assume that there are more than two types of regions so that, as remarked
at the end of Section 4, it is in general impossible to achieve global efficiency. Hence we are left
with the second-best problem:
Maximize U (X1 , Z1 )
subject to the constraints (2)–(3) and (9)–(12).6 This problem leads to the Lagrangian
L(L1 , . . . , Ln , K1 , . . . Kn , X1 , . . . , Xn , Z1 , . . . , Zn , τw1 , . . . , τwn , τk1 , . . . , τkn ,
λ2U , . . . , λnU , λ2K , . . . , λnK , λL , λK , λ1X , . . . , λnX , λ1Z , . . . , λnZ )
n
n
X
X
∂Fi
∂F1
= U (X1 , Z1 ) +
λiU (U (Xi , Zi ) − U (X1 , Z1 )) +
λiK (1 − τki )
− (1 − τk1 )
∂Ki
∂K1
i=2
i=2
!
!
n
n
X
X
+ λL
Li − L + λ K
Ki − K
i=1
i=1

n X
∂Fj 
∂Fj
1
∂Fi
−
Fj (Lj , Kj ) − Lj
− τkj Kj
+
λiX Xi − (1 − τwi )
∂Li L
∂Lj
∂Kj
j=1
i=1
n
X
∂Fi
∂Fi
i
i
i
+
− τk Ki
.
λZ Ci (Zi , Li ) − τw Li
∂Li
∂Ki
n
X

i=1
P
∂F
∂F
Denoting the non-labor income term L1 nj=1 Fj (Lj , Kj ) − Lj ∂Ljj − τkj Kj ∂Kjj by I and writing
P
P
λ1U := 1 − ni=2 λiU and λ1K := − ni=2 λiK the Lagrangian simplifies to
!
!
n
n
n
n
X
X
X
X
∂F
i
L =
λiU U (Xi , Zi ) +
λiK (1 − τki )
+ λL
Li − L + λK
Ki − K
∂Ki
i=1
i=1
i=1
i=1
X
n
n
X
∂Fi
∂Fi
i
i ∂Fi
i
i
i
+
λX Xi − (1 − τw )
−I +
λZ Ci (Zi , Li ) − τw Li
− τk Ki
.
∂Li
∂Li
∂Ki
i=1
i=1
Taking the derivative w.r.t. τwi yields
∂L
∂Fi
∂Fi
= λiX
− λiZ Li
=0
∂τwi
∂Li
∂Li
and therefore
λiX = λiZ Li .
6
Note that this problem is loosely related to the one studied by H¨
ulshorst and Wellisch (1996), Wellisch and
H¨
ulshorst (2000) and Wellisch (2000, Chapter 3). These authors consider the case of small regions, where each
region takes the federation-wide utility level and the federation-wide net return to capital as exogeneously given.
In this case, however, utility maximization is not a sensible objective anymore. Instead, H¨
ulshorst and Wellisch
assume that local governments maximize land rent.
5
Consequently, taking derivatives w.r.t. Xi and Zi yields
∂U
∂U
∂L
= λiU
+ λiX = λiU
+ λiZ Li = 0
∂Xi
∂Xi
∂Xi
and
∂L
∂U
∂Ci
= λiU
+ λiZ
= 0,
∂Zi
∂Zi
∂Zi
and combining these two equations yields the Samuelson condition (5).7
Taking the derivative of L w.r.t. τki yields
∂L
∂τki
n
∂Fi
1X j
∂Fi
∂Fi
+
− λiZ Ki
λX Ki
∂Ki L
∂Ki
∂Ki
j=1



n
∂Fi  i
1X j

−λK + Ki
λZ Lj − λiZ  = 0
∂Ki
L
= −λiK
=
j=1
and hence

n
X
1
λjZ Lj − λiZ .
= Ki 
L

λiK
j=1
Consequently, taking derivatives w.r.t. Li and Ki yields
2
∂ 2 Fi
∂I
∂L
i
i
i
i ∂ Fi
+
= λK (1 − τk )
+ λL − λX (1 − τw )
∂Li
∂Li ∂Ki
∂Li
∂L2i
2
2
∂ Fi
∂ Fi
∂Ci
i ∂Fi
i
i
i
− τw
− τw Li
− τk Ki
+ λZ
∂Li
∂Li
∂Li ∂Ki
∂L2i
n
2
1 ∂ Fi X j
= λL + (1 − τki )Ki
λZ Lj
L ∂Li ∂Ki
j=1
∂ 2 Fi
∂Ci
∂ 2 Fi
∂I
i
i ∂Fi
− Ki
+ λZ
− τw
− Li
− Li
=0
∂Li
∂Li
∂Li ∂Ki
∂Li
∂L2i
(24)
and
∂L
∂Ki
=
=
2F
∂ 2 Fi
∂I
i
i
i
−
+
λ
−
λ
(1
−
τ
)
+
K
X
w
2
∂Li ∂Ki ∂Ki
∂Ki
2
∂ Fi
∂ 2 Fi
i
i
i ∂Fi
i
+ τk
+ τk Ki
− λZ τw Li
∂Li ∂Ki
∂Ki
∂Ki2
n
1 ∂ 2 Fi X j
λZ Lj
λK + (1 − τki )Ki
L ∂Ki2
j=1
∂ 2 Fi
∂ 2 Fi
∂I
i
i ∂Fi
− λZ Li
+ τk
+ Ki
+ Li
= 0.
∂Li ∂Ki
∂Ki
∂Ki
∂Ki2
λiK (1
∂
τki )
(25)
Since we have not succeeded in interpreting these conditions in general, we now assume for
simplicity that I depends only negligibly on Li and Ki and that the second derivatives of Fi
vanish at the second-best optimum. In this case (24) and (25) simplify to
∂Ci
∂Fi
i ∂Fi
i
λL + λZ
− τw
=0
and
λK − λiZ τki
= 0,
∂Li
∂Li
∂Ki
7
Note that this result is similar to H¨
ulshorst and Wellisch (1996, Proposition 4) and Wellisch and H¨
ulshorst
(2000, Proposition 4).
6
so that
τwi ∂Fi /∂Li − ∂Ci /∂Li
τki ∂Fi /∂Ki
does not depend on i. Hence in the second-best optimum the quotient of the net fiscal benefit
of an additional inhabitant and the fiscal benefit of an additional unit of capital is the same in
all regions. This result should however be taken with caution since the simplifying assumptions
used in its derivation are very strong.
References
H¨
ulshorst, J. and Wellisch, D. (1996), Optimal local environmental and fiscal policies in secondbest situations. Finanzarchiv 53, 387–410.
Wellisch, D. (2000), Theory of Public Finance in a Federal State. Cambridge University Press.
Wellisch, D. and H¨
ulshorst, J. (2000), A second-best theory of local government policy. International Tax and Public Finance 7, 5–22.
Wildasin, D. E. (1980), Locational efficiency in a federal system. Regional Science and Urban
Economics 10, 453–471.
7