Bridging the Tax-Expenditure Gap: Green Taxes and the Marginal Cost of Funds

Bridging the Tax-Expenditure Gap: Green Taxes and the Marginal
Cost of Funds.
Agnar Sandmo
Norwegian School of Economics and Business Administration
Abstract
The marginal cost of public funds is usually seen as a number greater than one,
reflecting the efficiency cost of distortionary taxes. But economic intuition suggests
that since green taxes are efficiency-enhancing the MCF with such taxes will be less
than one. The paper demonstrates that this intuition is not necessarily true, even when
a green tax is the sole source of funds. The analysis also considers the MCF with a
proportional income tax, given the presence of green taxes. It compares the
optimization approach to the MCF with that of a balanced budget reform and shows
that they lead to equivalent results.
JEL Classification: D62, H21, H41.
This paper was prepared for the conference “Public Finances and Public Policy in the
New Millenium” on the occasion of Richard Musgrave’s 90th and CES’s 10th
birthdays, Munich 12-13 January 2001. I am grateful to the discussant, Jeremy
Edwards, for his comments on the original version of the paper.
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1. Introduction.
On several occasions Richard Musgrave has lamented the tendency in the theory of
public finance to analyze questions of taxation and of the supply of public goods1 in
separate compartments. Although this practice can often be justified in terms of
analytical tractability, it is true that a joint perspective on taxes and public expenditure
is sometimes very important. In the recent literature this point has been emphasized in
numerous studies of the concept of the marginal cost of (public) funds, or the MCF
for short. The basic idea in this literature is that when public goods are financed by
distortionary taxes, the efficiency costs that this entails should, in a cost-benefit
analysis of public projects, be reflected in an adjustment of the marginal resource cost
of increased supply. If public goods supply could have been financed by lump sum
taxes, an increased supply involving a cost of 1 million euros and benefits of 1.2
million euros should definitely be carried out. But if each euro of tax revenue involves
0.3 euros of tax efficiency cost, the social cost should be computed as 1.3 times the
direct resource cost, 1.3 being the MCF. With a social cost of 1.3 million euros the
proposed increase in public goods supply no longer passes the cost-benefit test. Thus,
the concept of the marginal cost of public funds is the modern theory’s response to
Musgrave’s critique. Its origin lies in the tax side of the public budget, and its
application is to the determination of the expenditure side.
Like a number of other fundamental ideas in public finance, this one can be traced
back to Pigou (1928). It reentered the literature through the theory of optimal taxation,
notably in a famous article by Atkinson and Stern (1974), although they did not use
the MCF terminology, which was apparently introduced by Browning (1976). More
recent contributions include Wildasin (1984), Mayshar (1991), Ballard and Fullerton
(1992) and Håkonsen (1998). While most analyses of the MCF interpret it as a pure
measure of inefficiency, some authors, like Wilson (1991), Dahlby (1998) and
Sandmo (1998), have argued that the MCF should also incorporate a measure of the
possible distributional gains from distortionary taxes. The basic argument for this is
that taxes are distortionary precisely because one wants to achieve some distributional

1 Or, more generally, publicly provided goods. These might – and indeed do – also comprise private
goods in areas like health and education.
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objective; hence the MCF should reflect the redistributional gain as well as the
efficiency loss.
Underlying most of this literature is the crucial assumption that when lump sum taxes
are not available, taxes used to finance the supply of public goods must be
distortionary. But this is not necessarily the case. In the case of commodities or
factors of production generating negative external effects, we know that the
imposition of a tax reflecting the difference between marginal social and private cost
(or between marginal private and social benefit) does not create any inefficiency; on
the contrary, it leads to an efficiency gain. This insight has recently given rise to a
large number of analyses of the so-called double dividend from a green tax reform, in
which one studies the substitution of green or Pigouvian taxes for standard
distortionary taxes, assuming that government revenue is to be held constant. That the
existence of a double dividend turns out not to be so obvious as might be suggested by
partial equilibrium analysis comes essentially from the cross-price effects between
markets, an aspect not captured in the partial equilibrium approach2.
The definition of the double dividend with constant tax revenue as the point of
reference is, however, not the only one possible. If one believes that a distortionary
tax system keeps the supply of public goods at an inefficiently low level, one way in
which to reap the benefits of a less distortionary system would be to expand public
expenditure, seeing that the MCF is now lower than it used to be. This idea has also a
considerable appeal to economic intuition. In fact, partial equilibrium analysis would
suggest that if increased public expenditure could be financed by Pigouvian or green
taxes, the MCF should be less than one, since there is now an efficiency gain from tax
finance which should be subtracted from the direct resource cost. But experience from
following the double dividend debate should warn us that there may be complications
ahead and that a more general analysis is called for.
Among the contributions that already address this or related questions from a
theoretical angle, van der Ploeg and Bovenberg (1994) and Kaplow (1996), are
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particularly noteworthy. van der Ploeg and Bovenberg study the effects of varying
environmental preferences on the optimal supply of public goods, but they do not
discuss the role of environmental taxes in determining the MCF. Kaplow’s main
concern is to study the role of optimal non-linear income taxation; under special
assumptions about preferences he shows that we should think of the MCF in first-best
terms3. The articles by Ballard and Medema (1993) and Brendemoen and Vennemo
(1996) use computable general equilibrium models to study alternative sources of
finance for public projects and find that the MCF for environmental taxes are much
lower than for traditional taxes, sometimes indeed considerably below unity.
2. Individual behaviour and the first best allocation.
A desire for redistribution is essential for understanding why existing tax systems are
distortionary. The efficiency loss from distortionary taxes therefore has to be balanced
against redistributional gains, and to focus solely on the loss side, as one does in most
of the literature on the marginal cost of funds, may therefore be misleading. However,
in the interests of analytical simplicity, this is nevertheless what we shall do in the
following, keeping in mind that distributional concerns can relatively easily be added
on to the model, e.g. in the way in which it has been done in Sandmo (1998). Hence it
is assumed that all consumers are alike, and that the representative consumer’s utility
function can be written as
U = U(y, x, l, z, e),
(1)
where y and x are the quantities of two consumer goods, l is leisure, z is the supply of
a public good and e is environmental pollution. U is increasing in the first four
arguments and decreasing in the fifth. Environmental pollution is generated by the
aggregate consumption of the x-good, so that e = nx. Labour supply is denoted by h,
with h + l = T, which is the time endowment.

2 For a more detailed analysis see Sandmo (2000, ch. 6) and the review of the literature by Bovenberg
(1999).
3 This is closely related to an earlier result in an important paper by Christiansen (1981).
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Each consumer maximizes his utility, taking the supply of public goods and the
amount of environmental pollution as given. His budget constraint is
y + Px = w(1-t)h + a.
(2)
The y-good is the numéraire, while the price of the x-good is P = p + ?, with p being
the producer price and ? the tax rate. Labour income is subject to tax at the rate t. a is
any exogenous income that the consumer might have; if a<0, it is a lump sum tax.
Utility maximization leads to the first order conditions
Ul/ Uy = w(1-t),
(3)
Ux/ Uy = P.
(4)
This gives rise to a supply function for labour
h = h(w(1-t), P, a, z, e),
(5)
and demand functions for the two consumer goods. In particular, the demand function
for the x-good or “dirty good” is
x = x(w(1-t), P, a, z, e).
(6)
We assume that the dirty good is normal (?x/?a>0), implying that demand is a
decreasing function of price (?x/?P<0).
Note the dependence of these functions on the state of the environment, e. While this
is an exogeneous variable from the point of view of each single individual4, changes

4 This may require a comment in view of the assumption that all individuals are identical. The essential
part of the assumption is that each consumer’s use of the dirty good is small relative to aggregate
consumption and pollution. Under that assumption, even when individuals are not identical, each one
of them may know that others respond to prices and income in the same way as he does himself, but it
is still not rational for him to take this into account in his own consumption decisions. This is simply
the assumption of perfectly competitive behaviour.
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in prices, taxes and public goods supply will in the aggregate affect individual
behaviour through their effects on e and the feedback effects on labour supply and
commodity demands. Many writers have chosen to neglect these feedback effects; the
case in which there is a rigorous justification for it is of course where the utility
function is weakly separable between the state of the environment and other goods, so
that
U = U(?(y, x, l, z), e),
(1’)
Separability is hardly a realistic assumption, and for a number of environmental
problems, such as traffic congestion, non-separability and feedback effects are
obviously very important. Nevertheless, it will be adopted in what follows, basically
because it simplifies the analysis without distorting the conclusions that can be drawn
from it.
Optimizing behaviour also implies the indirect utility function
V = V(w(1-t), P, a, z, e),
(7)
with the Roy conditions
Vt = -?wh; VP = -?x; Va = ?.
(8)
We now turn from individual behaviour to social welfare maximization. With all
individuals being alike the natural choice for a social welfare function is the utilitarian
sum of utilities, which is simply W = nU. The production possibility schedule is
assumed to be of the linear Ricardian form, so that it can be written as
-wnh + ny + pnx +qz = 0.
(9)
Here w, p and q are the technical production coefficients. The symbols have been
chosen to reflect the fact that under competitive conditions the coefficients will be
equal to equilibrium producer prices, again with the y-good as the numéraire.
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Social welfare maximization is now characterized by the first order optimality
conditions
Ul/ Uy = w,
(10)
Ux/ Uy + n Ue/ Uy = p,
(11)
n Uz/ Uy = q.
(12)
Comparing (10) and (11) with the conditions for individual utility maximization (3)
and (4), we can characterize the first best optimal tax structure. This is simply t = 0
and ? = - nUe/ Uy. There should be no distortionary tax in the labour market. The tax
on the dirty good should reflect the marginal social damage, and this is the sum of the
marginal damages imposed on all individuals. Finally, the public good should be
supplied according to the Samuelson (1954) optimality rule; the sum of the marginal
willingness to pay across all individuals should equal the marginal cost or the
marginal rate of transformation, and the MCF is unity. If this combination of taxes
and public goods supply leads to a deficit or surplus in the government’s budget
constraint, the gap should be filled by a lump sum transfer from or to the consumers,
i.e. by an adjustment of the lump sum income term, a.
3. Public goods supply with distortionary taxes.
We now abandon the assumption that lump sum taxes are feasible. Of course, in a
model economy of identical individuals, there is no real justification why it should be
impossible to collect the same amount in taxes from all individuals. This must be seen
simply as an ad hoc device to concentrate on the efficiency properties of a second best
optimum situation. The government has to finance the cost of supplying the public
good partly by means of the distortionary income tax, partly through the Pigouvian
tax on the dirty good. As a natural point of reference, we begin by deriving the
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conditions for a second best optimum. What is the optimal supply of the public good,
and what is the best combination of the labour income tax and the Pigouvian tax?
The government’s budget constraint says that taxes collected must equal expenditure,
so that
ntwh + n?x = qz,
(13)
while the social welfare function can be written on dual form as
W = n V(w(1-t), P, a, z, e),
(14)
where a must now be understood as being equal to zero.
We are now in a position to study how the cost of public goods supply depends on the
costs of tax finance. But there are in principle two ways in which this can be done.
We could, as Atkinson and Stern (1974) did, adopt the framework of optimal taxation
and public goods, or we could, as is more or less implicit in cost-benefit analysis,
consider a balanced budget change in public expenditure and taxes without assuming
anything about optimality. The first approach gives the most straightforward
interpretation of the MCF as a shadow price emerging from the optimality conditions.
The second, however, is much less restrictive and more relevant for the view of the
MCF as a practical tool for the evaluation of public projects. In the following we shall
pursue both approaches and see how they are related.
Starting within the optimality framework, the problem is to maximize (14) with
respect to the tax rates t and ?, subject to the budget constraint (13). The Lagrangian
can be written as
? = n V(w(1-t), P, z, e) + µ[ntwh + n?x - qz].
(15)
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The first-order conditions for this optimization problem5 are
??/?t = - n?wh + nVen(?x/?t) + µ[nwh + ntw(?h/?t) +n?(?x/?t)] = 0,
(16)
??/?? = - n?x + nVen(?x/?P) + µ[nx + ntw(?h/?P) +n?(?x/?P)] = 0,
(17)
??/?z = nVz + nVen(?x/?z) + µ[ntw(?h/?z) +n?(?x/?z) - q] = 0.
(18)
Although the three conditions provide a joint characterization of the optimal tax-
expenditure policy, it is natural to see (18) as the optimality condition for public
goods supply. Dividing through this equation by ? and rearranging terms, we obtain
n(Vz/?) +( nVe/?)n(?x/?z) = ?[q - ntw(?h/?z) - n?(?x/?z)],
(19)
where ? = µ/?. The interpretation of this condition is straightforward. The first term
on the left is the Samuelson sum of the marginal rates of substitution, i.e. the direct
benefit of the increase in public goods supply. The second term is the indirect benefit
that arises because the public good may cause a change in the amount of
environmental damage. This benefit is positive if the dirty good and the public good
are substitutes (?x/?z<0) and negative if they are complements (?x/?z>0). On the
right-hand side, q is as before the direct resource cost of the public good. The direct
resource cost is modified by the remaining two terms in square brackets. These terms
represent the change in tax revenue that is generated by an increased public goods
supply; to the extent that the public good increases the tax bases, it counteracts the
adverse distortionary effects of the taxes, so that real resource costs are lowered.
Finally, the parameter ? represents the ratio of the marginal utilities of income in the
private and public sector and is a measure of the inefficiency of the tax system. It is
this parameter that will be identified with the marginal cost of public funds.
However, a question may be raised as to whether ? alone is not too restrictive as a
measure of the MCF. In particular, one might argue that the tax revenue effects should

5 The form of the optimality conditions reflects the assumption of separability. In the general case the
partial derivatives ?h/?t etc. would have to be replaced by derivatives dh/dt etc., which would take
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also somehow be included, since they too characterize the second best optimality
condition in contrast to the first best Samuelson rule. There may be something to be
said for this, but the issue depends on how one sees the practical role of the concept of
the MCF. The point of view taken here is that the potential usefulness of the MCF lies
in cost-benefit analyses of public goods projects funded by general tax finance, and
that it should be the same for all projects. But the bracketed expression in (19) is
project specific, since the only realistic assumption is that each public good is
characterized by a different degree of substitutability or complementarity with private
taxed goods. ?, on the other hand, is a characteristic of the system of tax finance and
does not vary with the nature of the project. Thus, the modification of the direct
resource cost via the effect of the public good on the tax base should be seen as a
separate operation, which is to be performed before the MCF is applied to the net
resource cost of the project.
4. An optimal tax structure.
When both tax rates have been chosen in accordance with the second best optimal tax
criterion6, it follows that the MCF at the optimum must be the same, whatever the
source of tax finance. This follows by noting that when (16) and (17) both hold, we
must have that

[wh –n(Ve/?)(?x/?t)]/[wh+tw(?h/?t)+?(?x/?t)]
?={
(20)

[x-n(Ve/?)(?x/?P)]/[x+ tw(?h/?P) +?(?x/?P)]

account of the environmental feedback on demands and supplies. See Sandmo (2000, ch. 6 for details).
6 The reader may check that conditions (16) and (17) together imply the property of additivity, as it was
called in Sandmo (1975), or the principle of targeting. Solving the two equations for t and ?, it can be
shown that the characterization formula for the income tax rate is a generalized version of the Ramsey
inverse elasticity and is independent of the marginal social damage, while the formula for the green tax
is the weighted sum of a Ramsey term and one reflecting the marginal social damage. Of the available
taxes, it is only the tax on the dirty good which, in the optimal design of the tax system, is targeted on
improving the environment.
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