DRAFT
Public inputs as common property
May 4, 2004
James P. Feehan (corresponding author)
Department of Economics
Memorial University
St. John’s, Newfoundland
Canada A1C 5S7
and
Raymond G. Batina
Department of Economics
Washington State University
Pullman, Washington
USA
Abstract: Some forms of public intermediate goods, i.e., public inputs, are congestible.
They enter production functions for private goods such that the returns to scale in primary
factors are decreasing. If provided on a free-access basis then a public input of this type
is equivalent to a common property resource. Recent literature has failed to take account
of the well known consequent implications of free-access to such resources. This paper
identifies those implications and investigates design of factor-tax financing for public
inputs of this type. In particular, the structure of the optimal tax on internationally
mobile capital is derived and the tax itself is found to be positive.
Keywords: common property; rent dissipation; congestible public inputs; factor taxes
JEL classification: H5; H2; H4
__________________
* This paper is based upon research that was financially supported by Japan’s Economic
and Social Research Institute. We are grateful to Mutsumi Matsumoto for comments on
an earlier draft. We remain solely responsible for the contents of this paper and any errors
herein.
1. Introduction
Government support for basic education, basic R&D, public infrastructure and
other public inputs is generally recognized as a mechanism for enhancing an economy’s
production possibilities. For the most part, theoretical models in the public finance
literature have generally treated public inputs in two alternate ways. First, a public input
may enter the production functions for private goods such that those functions exhibit
constant returns to scale in the private inputs, thus giving rise to increasing returns to
scale overall, as in Haughwout (2002) and Feehan and Matsumoto (2002). In this
circumstance, the public input is often referred to as pure, or factor-augmenting, or, in the
terminology of Meade (1952), atmospheric. The second approach has the private
production function exhibiting constant returns to scale in the primary inputs, and thus
decreasing returns to scale in the primary factors. As discussed in Feehan (1989)
sometimes public inputs under this specification are known as unpaid factors. In this
paper, and for reasons that should become clear, we choose to refer to this second type as
a congestible public input.1
It is an empirical issue as to whether a particular public input is of one type or the
other. However, it seems likely that some public inputs are pure while others are not.
For instance, agricultural R&D is likely to be atmospheric. Doubling of all other factors
1
A third variety is known as firm-augmenting. Under that specification, the public input
provides services to firms. However, a firm can be divided into smaller firms, the effect of which
is to create more benefit from the same level of the public input, leading to limitless production
possibilities. This unboundedness outcome can be avoided by employing various assumptions;
e.g., a minimum size for firms, or a fixed number of firms or a fixed cost of establishing firms.
However, a lack of realistic examples of this type of public input and the idea of it being
collective among firms, which are institutional entities, have led Henderson (1974) and McMillan
(1979) among others, to conclude that it is a rather implausible specification. For more
elaboration see Feehan (1989).
1
employed in the agricultural sector, with the same level of R&D, is likely to lead to a
doubling of agricultural output. On the other hand, many forms of public services to
industry, e.g., public infrastructure, are congestible.2
The purpose of this paper is twofold. First, we argue that this type of public
input, once provided, is equivalent to a common property resource. It is well know that a
rationing mechanism is needed to avoid the inefficiency of rent dissipation. In the
absence of such a mechanism, primary factors will not be hired according to the values of
their marginal products and no government revenue can be realized from taxing rents
arising from the public input; such rents are dissipated through factor misallocation.
Secondly, we present and utilize a standard model to explore the issue of factor-tax
financing taking explicit account of the common-property attribute. The design of the
appropriate factor taxes is derived.
2. A Congestible Public Input as Common Property
Consider an economy in which an aggregate private consumption good is
produced. Industry production is given by
X = F(K, L, G)
(1)
where L and K are, respectively, the inputs of labour and private capital in the industry; X
is the output of the private good; G denotes the amount of a public input made
2
Aschauer (1989) sparked a considerable empirical literature on public infrastructure. In its
footnote 3, Cassou and Lansing (1998) points out that, based on that literature, there is empirical
support for constant-returns-to-scale in all factors, implying that many forms of public capital fall
within the category of congestible public inputs. Batina (1999), in an aggregate model with
private capital, labour and public capital infrastructure, even finds decreasing returns to scale in
all three; this is also consistent with the notion of public infrastructure being a congestible form of
public input. However, to be consistent with the theoretical literature, and also reflecting an
appeal to the replication argument, we assume that there are constant returns to scale in all inputs.
2
exogenously available by the authorities; and F is the production function. It is taken to
exhibit the general neo-classical characteristics, e.g., strict quasi-concavity in all
arguments, positive continuous first-order partial derivatives with respect to all factors,
and positive cross-partial derivatives.
For convenience of exposition, we normalize the price of the private consumption
good at unity, and take the marginal opportunity cost of the public input, expressed in
units of the consumption good, to be q, a constant. It is also assumed that the factor
markets as well as the market for the private commodity are competitive. Finally, it is
assumed that competitive firms produce the public input, so its per-unit price is q.
If the production function is characterized by constant returns to scale in the
primary inputs, K and L, then the public input is pure. In that case, competition leads to
private factors being paid the value of their respective marginal products, so factor
payments exactly exhaust output. It is well established that if government revenue can
be raised in a non-distortionary manner then efficiency is achieved once the appropriate
level of the public input is provided.3 However, matters are more complex when the
public input is not pure. Those complexities are the crux of this paper.
2.1 Factor Mis-allocation and Rent Dissipation
With a congestible public input then F, in (1) is linearly homogeneous in all three
inputs. Thus, by Euler’s theorem,
3
If factors are in fixed supply then factor income taxation is non-distortionary. In that situation,
Manning, Markusen and McMillan (1985) derive the appropriate uniform tax on factor incomes,
and Feehan and Matsumoto (2000) derive differential factor taxes based on the benefit principle.
With variable factors, factor taxation is distortionary and Feehan and Matsumoto (2002) show
that the standard optimal taxation rules for a second-best situation should then be followed when
financing a pure public input.
3
F(K,L,G) = FL(K, L, G)L + FK(K, L, G)K + FG(K, L, G)G
(2)
where Fj(K, L,G) denotes the marginal product of factor j = K, L, and G. All marginal
products are assumed to be positive and diminishing. For notational convenience let R =
FGG.
The private-good producing industry takes G as parametric. Normalizing the
price of the private good to unity, industry profit, assuming free-access to G, is given by:
F(K,L,G) – wL – rK
(3)
where w and r represent the respective factor prices and c denotes a fee for the public
input. For any exogenously given amount of G, the efficiency conditions for resource
allocation are, of course,
FL(K, L, G) = w
(4)
FK(K, L, G) = r.
(5)
and
However, substituting these conditions into the Euler equation, (2), and then into the
profit function, (3), gives FG(K,L,G)G, which is positive. This is incompatible with a
competitive equilibrium. The existence of positive profit is an incentive for economic
agents to employ resources to capture that rent. This will lead to complete rent
dissipation.4 In short, once provided this type of public input is analytically equivalent to
a common-property resource. Without efficient rationing of it, firms will not utilize the
private inputs in quantities that accord with the equality of the values of their marginal
products to their respective factor prices. As with an open access fishery, firms will
4
Henderson (1974, 324) makes this observation as well. Gramlich (1994, 1185-1186) also makes
a similar remark.
4
employ more of the private factors, i.e., fishing boats and fishers, in order to seek the
rent, which ultimately dissipates the rent.
Thus, in the case of two primary factors, and with open-access to an exogenous
amount of G, one would expect that firms in the X industry would hire labour and capital
until:
FL + SR/L = w
(6)
FK + (1-S)R/K = r
(7)
and
where S represents the share of the average rent per worker that the industry obtains from
hiring an additional worker; (1-S) has a corresponding interpretation for capital.5 With
(6) and (7) holding, factor payments now exactly exhaust firm revenues, which reflects
the outcome of free-access. Clearly, production efficiency, as given by (4) and (5) is not
achieved. Rent-seeking behaviour arising from the free-access to the public input leads
the economy to operate inside its production possibilities frontier.
The value of S in (6) and (7) varies with the production technology.6 Through the
homogeneity property, capital-labour ratio will be determined solely by their respective
factor-price ratio. Thus, given factor prices, hiring additional amounts of those factors in
5
Note that had labour been the only primary factor, then S =1. In that case, it is well known
that labour would be hired up to where its average product equaled the wage; see for example
Varian (2003, 618). Hence, reflecting the homogeneity property, labour is hired up to where to
the point where all industry rent is fully dissipated: FL + R/L = w. Despite this classic result,
Dahlby and Wilson (2003) assume that firms hire labour according to FL = w and that the rent
arising from G accrues to firms in the form of persistent equilibrium profit. Keen and Marchand
(1997) make a similar assumption in a two-factor model.
6
Negishi (1973) assumed that the rent is captured through the use of more capital only, i.e., S = 0
and (1-S) = 1. However, this is difficult to rationalize as a general proposition. It requires some
form of Leontif-type fixed-proportions technology.
5
an attempt to capture the rent would entail hiring more of both capital and labour in
proportion. Therefore:
S = S(K/L)
(8)
∂S/∂(K/L) ≥ 0.
(9)
and
Note that in the limiting case of Cobb-Douglas production technology, since income
shares are constant, (9) would hold with strict equality.
2.2 Implications
The policy implications of free-access to public capital are substantial. The
argument that rent-seeking firms will increase their use of primary factors to capture rents
from a public input may explain why there is increasing congestion on roads, highways
and other public facilities over time. An increase in a public input will induce economic
profit. Firms react by increasing their use of primary inputs and thus increase output in
an effort to capture those rents. This will increase utilization of the public input, possibly
increasing congestion. Thus, building new infrastructure, for example, will not
necessarily solve the problem of congestion in the long run. Even if it does “solve” the
congestion problem, it is an inefficient solution since the level of public input would be
excessive.
A first-best approach for achieving efficiency is to charge each firm for its
utilization of the public input according to the value of its marginal product to the firm, as
in Sandmo (1972). That is the standard solution to the problem of the commons. No rent
would be then freely available and firms would revert to hiring the primary factors
according to (4) and (5). This option, however, may be difficult to implement in many
6
cases, e.g., a multi-access highway or city streets, since exclusion may be either
impossible or highly costly. Other options must be considered.
If a congestible public input is provided on a free-access basis then, even if it is
financed by lump-sum taxes and there are no other distortions, efficiency cannot be
expected to prevail. Free-entry would severely limit the chances of the firms themselves
solving the dissipation problem along the lines of the Coase Theorem. The authorities
would not only have to provide the public input but also deal with consequent commons
problem. For instance, they could ration fixed amounts of it to a fixed number of firms;
this is analogous to a quota system in a fishery. Profits then would not be dissipated and
the rents would be realized by firms. Such rents would also provide an attractive tax
source for raising funds to finance the public input. Without such rationing, rent-seeking
factor allocation occurs, and the economy fails to achieve the first-best equilibrium even
if lump-sum taxation is possible. Other rationing mechanisms and revenue-raising
instruments, such as factor taxes or output taxes, would be needed to halt the inefficiency
arising from rent dissipation.
3. Financing
This section considers the financing of a common-property public input that is
provided on a free-access basis. To explore the implications for public finance, this
section incorporates that common-property feature into a simple model in which both
factors are variable.7
7
This model is entirely consistent with that used by Keen and Marchand (1997) which, as here,
assumes that the production function is homogeneous of degree less than one in K and L. Indeed,
our model is simply an adaptation of their model but one in which we explicitly incorporate the commonproperty aspects of the public input. Also, we assume a small economy in order to avoid the issue of fiscal
7
3.1 The Model
Consider a one-good, two-factor, public-input economy in which there is a
representative household. That household consumes the private good subject to the
following budget constraint:
C = (w-t)E + iK*
(10)
where C denotes consumption of the private good, t is an employment tax, E is labour
supply, K* is the quantity of domestically owned capital which is assumed fixed but
internationally mobile, and i is the after-tax return on capital that can be realized on world
markets. The economy is assumed to be small so i is exogenous. Next, r = i + T is the
gross return on capital in this economy, where T is the domestic source-based tax on
capital. Note that the price of the private good has been normalized to unity.
The household maximizes utility over a well defined utility function U(C, E)
subject to its budget constraint (10). The associated demand for the private good and
supply of labour are given by C = C(w-t, iK*) and E = E(w-t, iK*), respectively. The
indirect utility function is then
V = V(w-t, iK*).
(11)
The instruments available to finance the public input are the capital tax and the
employment tax. Recalling that the marginal cost of G is q, a constant, the public sector
budget constraint is:
tE + TK = qG
(12)
competition, the subject matter of their paper. This assumption simply permits a sharper focus on the issue
of rent dissipation.
8
where K is the entire amount of capital located within the small economy, whether
domestically owned or not. The supply of capital is perfectly elastic at the world net
return.
For future reference, note that based on the Euler decomposition of the production
function, as given in (2), the partial derivatives of F(L,K,G) with respect to L, K, and G,
can expressed as follows.
Fj = FLjL + FKjK + FGjG + Fj for j = L, K and G.
(13)
Consequently,
FLLL+FKLK+FGLG = FLKL+FKKK+FGKG = FLGL+FKGK+FGGG = 0,
(14)
where Fij denotes the second order partial derivates of the production function; i,j = K, L,
and G.
Next, turn to the hiring decisions of firms. As discussed earlier, if the authorities
do not charge for G according to its marginal value then rents are open to dissipation and
there will be excessive use of the primary factors. Following from (6) and (7), firms hire
labour and capital such that8
FL + SR/L = w
(15)
FK + (1-S)R/K = r = i + T.
(16)
and
It follows from these conditions that the demands for labour and capital are functions of
w, r and G; so K = K(w,r,G) and L = L(w,r,G). G is a policy variable, and r is
8
If the authorities charged some price “p” for access to each unit of the public input then the rent
available for dissipation would be R = (FG-p)G. If p is set equal to the marginal product of G,
rent dissipation would not arise. The hiring conditions would be the standard marginal ones, and
the first-best outcome could be achieved by financing G to the level at which p = q. If G were
rationed according to a quota system then rents would accrue to firms but that would enable the
9
determined by the world market and the domestic tax on capital. Therefore, a wagedetermination mechanism is all that is necessary to complete the model. It is given by the
labour-market clearing condition:
E(w-t, iK*) = L(w,r,G).
(17)
3.2 Factor Demands
As a preliminary to the policy analysis, it is necessary to elaborate on the nature
of factor demands, K(w,r,G) and L(w,r,G). More specifically, the partial derivatives of
each are needed, namely, Lw, Lr, LG, Kw, Kr, and KG. To obtain these, conditions (15)
and (16) must be totally differentiated with respect to w, r and G. That derivation is
completed in the Appendix A, which shows that, using the following notation:
A = -(1-S)FLLL/K + SFKL – SLFGG/K > 0;
B = (1-S)FLK - SFKKK/L– SLFGG/K > 0
J = FGKK + (1-S)FGGG -SFG.
and
D = (FGG/LK)[SB + (1-S)A + S(1-S)FGG/LK] > 0,
the partial derivatives can be expressed as:
Lw = [(-L/K)B – (1-S)FGG/K2]/D < 0
(18)
Lr = -B/D < 0
(19)
Kw = -A/D < 0
(20)
Kr = [(-K/L)A - SFGG/L2]/D < 0
(21)
LG = (FG/LK2)[BLK – (1-S)GJ]/D
(22)
and
authorities to apply a profits tax that actually yielded positive revenues, in which case a first-best
outcome could again be realized.
10
KG = (FG/KL2)[ALK +SGJ + SFGG]/D.
(23)
Two observations are in order. First, Lw, Lr, Kw and Kr are all negative. Given the
problem of the commons this makes sense. There is excessive factor allocation to start
with; an increase in a factor price reduces the overall incentive to hire factors for the
purpose of rent seeking. Secondly, note that if J, which is -∂(FL + SFGG/L)/∂G, is
positive (negative) then KG (LG) is unambiguously positive (positive) but the sign of LG
(KG) is ambiguous. Any increase in G initially creates more rent and this will attract
more factors to the industry, so the demand for at least one factor increases.
3.3 Policy Analysis
The policy problem faced by the authorities may now be represented by the
following Lagrangean:
M = V(w-t, iK*)+ a[tL(w,r,G)+ TK(w,r,G)–qG] + b[L(w,r,G)- E(w-t, iK*)]
(24)
where a is the Lagrangean multiplier on the budget constraint and b is the one on the
labour-market clearing condition. That condition is incorporated into the optimization
problem because the authorities recognize that labour-supply decisions are a constraint on
the optimization exercise. Following Keen and Marchand (1997) this constraint is
modeled through the inclusion of the market-clearing condition and treating w as if is a
choice variable.
The first-order conditions corresponding to G, t, T and w are, respectively:
(at + b)LG + a(TKG –q) = 0
(25)
-Vw + aL + bEw = 0
(26)
(at + b)Lr+ a(TKr + K) = 0
(27)
and
11
Vw + (at + b)Lw + aTKw – bEw = 0.
(28)
Consider, first, the implications for the taxation of capital. To do this, substitute
(26) into (28) and then ratio the resulting expression to the expression in (27) to obtain:
(TKr + K)(Lw/Lr) = (TKw + L).
(29)
Using the results of the previous subsection, this expression simplifies, as shown in
Appendix B, to yield the following rule for the taxation of capital:
T = (1-S)FGG/K.
(30)
The optimal per-unit tax on capital is thus proportional to the public input’s contribution
to output per unit of capital.
The next step in this policy analysis is to establish the optimal spending rule. To
do so, use (27) to obtain an expression for (at + b) and substitute it into (25). Then using
the expressions for LG, KG, Lr and Kr, and with some simplification, as in Appendix B,
the result is:
FG = q.
(31)
The level of the public input should be increased up to where the value of its marginal
product coincides with its marginal cost.9
In light of (31), (30) and the budget constraint, it immediately follows that the tax
on labour has the following form:
t = SFGG/L.
(32)
The optimal per-unit tax on labour is proportional to the public input’s contribution to
output per unit of labour.
9
This efficiency condition is sometimes referred to as the Kaizuka condition in light of Kaizuka
(1965), which appears to be the first formal derivation of general equilibrium efficiency
conditions for a competitive economy in the presence of a public input.
12
These three key results – the labour tax, the capital tax, and the spending
condition - indicate that the first-best optimum is achievable if factor taxes are available.
The factor taxes serve dual purposes: raising revenue to finance the public spending, and
correcting the distortion that otherwise occurs as a result of making the public services
available to industry without charge. In short, adopting factor taxes in the correct
proportions is a perfect substitute for charging directly for the use of the public input.10
Three remarks are now in order. First, the conditions given by (30), (31) and (32)
not only are efficient but also are consistent with the equity criterion of the benefit
principle. Secondly, taxation of capital, despite complete mobility of capital, is required.
This is in contrast to standard optimal tax theory, which suggests that no taxation of a
revenue base that is perfectly mobile. However, that zero-capital-tax prescription fails to
recognize the “commons” problem: free access to public services leads to excessive use
of capital, as well as labour, as firms expand to capture the rent attributable to an increase
in the public input so a capital tax is needed to stem an inefficiently high inflow of
capital. Rather than reliance on capital and labour taxation leading to a second-best
outcome such taxes are needed for a first-best outcome in the presence of a public input,
even if lump-sum taxation is available. Thirdly, analyses that rely on the taxation of
equilibrium positive rents generated by a congestible public input are not as general as
they appear. Specifying the public input as congestible leads to the commons’ problem
of inefficient resource allocation and associated rent dissipation. To avoid those
consequent complications, special assumptions are needed. For instance, either the Coase
Theorem must apply or the authorities must be allocating quotas for utilization of the
10
The factor taxes are of course benefit taxes; each factor being taxed according to the share of
the rent attributable to it.
13
public input among a fixed number of firms. Neither of these options seems likely to
occur in practice.
4. Conclusion
Many forms of public inputs are congestible. This paper has pointed out that
there is an equivalence between a common property resource and a congestible public
input. It is well known that allowing free-access to a common property resource can
induce firms to hire more than the efficient amounts of factors in order to capture the rent
generated by the resource. The result is rent dissipation. Thus, analyses of public
inputs, e.g., Keen and Marchand (1997) and Dahlby and Wilson, that assume that factors
are hired according to values of their marginal products and that rents accrue to firms in
equilibrium are limited in their application because they do incorporate the commonproperty characteristics of congestible public inputs.
The key contribution of this paper is to explicitly incorporate those commonproperty features in deriving the design of factor taxes that can be employed to finance a
congestible public input. The tax on each factor is proportionate to the public input’s
contribution to total output per unit of the respective factor. Moreover, these taxes are of
a Pigovian nature, and thus support a first-best efficiency outcome. Second-best
considerations, as with optimal taxation theory, do not apply. Hence, even if capital is
completely mobile a tax on capital is still called for. Indeed, in the absence of these
factor taxes, or their equivalent in the form a direct user charge à la Sandmo (1972), it
would be difficult to assess whether congested public infrastructure is a signal of
inadequate supply or is simply the result of excessive use due to free access.
14
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16
APPENDIX A
Start with
and
FL + SR/L = w
(A1)
FK + (1-S)R/K = r.
(A2)
Totally differentiate (A1) and (A2) with to w, r and G. The results are:
[FLL+SFGL/L+SLFGG/L-SFGG/L2]dL +[FLK+SFGKG/L+SKFGG/L]dK
= dw – [FGL+SFGGG/L + SFG/L]dG
(A3)
[FKL+(1-S)FGLG/K-SLFGG/K]dL +[FKK+(1-S)FGKG/K-SKFGG/K-(1-S)FGG/K2]dK
= dr – [FGK+(1-S)FGGG/K +(1-S)FG/K]dG
(A4)
Next, note that ∂S/∂K = SK = -SL(L/K). Also, recall from (14) that
and
FLLL+FKLK+FGLG = 0,
(A5)
FLKL+FKKK+FGKG = 0
(A6)
FLGL+FKGK+FGGG = 0.
(A7)
Substitute for SK in (A3) and (A4). Use (A5) to substitute for FGLG on both sides
of (A3) and use (A6) to substitute for FGKG on the left-hand-side of (A4). Those
substitutions and some simplification allow (A3) and (A4) to be expressed as
[(-K/L)(A + SFGLG/LK]dL + [B]dK = dw – [J/L]dG
(A8)
[A]dL + [(-L/K)(B+(1-S)FGG/LK]dK = dr – [J/K +FG/K]dG
(A9)
where A = -(1-S)FLLL/K + SFKL – SLFGG/K > 0
B = (1-S)FLK - SFKKK/L– SLFGG/K > 0
J = FGKK + (1-S)FGGG -SFG.
Next, the determinant of the matrix formed by the left-hand-sides of (A8) and (A9) is
D = (FGG/LK)[SB + (1-S)A + S(1-S)FGG/LK] > 0.
(A10)
The expressions for Lw, Lr, LG, Kw, Kr, and LG as given in the text follow
accordingly from Cramer’s rule applied to the two-equation system given by (A8) and
(A9).
17
APPENDIX B
The Optimal Capital Tax:
Following from (29), one can write:
[-K(Lw/Lr) + L] + T[Kw - Kr(Lw/Lr)] = 0
(B1)
Next, substituting for (Lw/Lr) gives
or
[-(1-S)FGG/BK] + T[Kw - Kr(Lw/Lr)] = 0
(B2)
[-(1-S)/K] + T(B/FGG)[Kw - Kr(Lw/Lr)] = 0.
(B3)
However, it can be shown that B[Kw - Kr(Lw/Lr)] = 1. Hence, the optimal tax on capital
is simply:
T = (1-S)FGG/K.
(B4)
The Spending Condition
Using the first-order condition (27) to obtain an expression for (at + b) and
substituting it into (25) gives
(K + TKr)(LG /Lr) - TKG = - q.
(B5)
Substituting for LG /Lr yields
(K + TKr){-1 + [(1-S)GJ/KL]/B}(FG/K) - TKG = - q,
(B6)
where, as in the text, B=(1-S)FLK-SFKKK/L-SLFGG/K and J = FGKK+(1-S)FGGG -SFG.
Re-write (B6) as
FG{-1+[(1-S)GJ/KL]/B} + TFG(Kr/K){-1+[(1-S)GJ/KL]/B}- TKG = -q.
(B7)
Next, it can be shown that
TFG(Kr/K){-1+[(1-S)GJ/KL]/B}- TKG = -TJ/BL.
(B8)
Substituting (B8) into (B7) then gives
FG{-1+[(1-S)GJ/KL]/B} - TJ/BL = -q.
(B9)
Finally, using the expression in (B4) for T in (B9) yields the spending rule:
FG = q.
18
(B10)