The Marginal Cost of Public Funds is One
at the Optimal Tax System∗
Bas Jacobs∗∗
August 22, 2016
Abstract
This paper develops a Mirrlees (1971) framework with skill and preference heterogeneity to
analyze optimal linear and non-linear redistributive taxes, optimal provision of public goods
and the marginal cost of public funds (MCF). It is shown that MCF equals one at the optimal tax system, for both lump-sum and distortionary taxes, for linear and non-linear taxes,
and for both income and consumption taxes. By allowing for redistributional concerns, the
marginal excess burden of distortionary taxes is shown to be equal to the marginal distributional gain at the optimal tax system. Consequently, the modified Samuelson rule should
not be corrected for the marginal cost of public funds. Outside the optimum, the marginal
cost of public funds for both non-distortionary and distortionary taxes can be either smaller
or larger than one. The findings of this paper have potentially important implications for
applied tax policy and social-cost benefit analysis.
Keywords: marginal cost of funds, marginal excess burden, optimal taxation, optimal redistribution, optimal provision of public goods, Samuelson rule
JEL-codes: H20, H40, H50.
∗
A previous version of this paper circulated under a slightly different title (Jacobs, 2010). I thank the Dutch
Ministry of Economic Affairs and the Netherlands Organisation for Scientific Research (Vidi-grant No. 452-07013) for financial support. I am very grateful to Ruud de Mooij, whose contributions to this paper have been
invaluable. Parts of this paper have been written when I was a guest researcher at the Center for Fiscal Studies, and I thank Uppsala University for its hospitality. Numerous discussions over the years with many people
have contributed to the ideas that are developed in this paper. I would like to thank Geir Bjertnaes, Sören
Blomquist, Robin Boadway, Lans Bovenberg, Vidar Christiansen, Josse Delfgaauw, Casper van Ewijk, Aart Gerritsen, James Heckman, Martin Hellwig, Laurence Jacquet, Carl Koopmans, Etienne Lehmann, Joana Pereira,
Rick van der Ploeg, Agnar Sandmo, Erik Schokkaert, Dirk Schindler, Paul Tang and Floris Zoutman in particular. Furthermore, I thank seminar participants at the Dutch Ministry of Economic Affairs, CPB Netherlands
Bureau for Economic Policy Analysis, NHH-Bergen, Uppsala University, University of St. Gallen, Tinbergen
Institute, Erasmus University Rotterdam, Max Planck Institute Bonn, and Statistics Norway for their comments
and suggestions. In addition, valuable comments were made by participants at the following congresses: ‘Taxation, Public Goods Provision and the Future of the Nordic Welfare Model’, Helsinki, June 15, 2009; Congress of
the European Economic Association, Barcelona, August 24, 2009; the 6th CESifo Norwegian-German Seminar
on Public Economics, Munich, November 13-14, 2009; and the 67th Congress of the International Institute for
Public Finance, Ann Arbor, August 7–11, 2011. Naturally, any remaining errors are my own.
∗∗
Bas Jacobs, Erasmus University Rotterdam, Tinbergen Institute, and CESifo. Address: Erasmus
School of Economics, Erasmus University Rotterdam, PO Box 1738, 3000 DR Rotterdam, The Netherlands. Phone: +31–10–4081452/1441. Fax: +31–10–4089161. E-mail: [email protected]. Homepage:
http://people.few.eur.nl/bjacobs.
1
Pigou (1947, p.34): “Where there is indirect damage, it ought to be added to
the direct loss of satisfaction involved in the withdrawal of the marginal unit of
resources by taxation, before this is balanced against the satisfaction yielded by
the marginal expenditure. It follows that, in general, expenditure ought not to be
carried out so far as to make the real yield of the marginal unit of resources expended
by the government equal to the real yields of the last unit left in the hands of the
representative citizen.”
1
Introduction
The marginal cost of public funds is the ratio of the marginal value of a unit of resources raised
by the government and the marginal value of that unit of resources in the private sector. The
marginal cost of public funds is therefore a measure indicating the scarcity of public resources.
Ever since Pigou (1947) many scholars and policymakers are convinced that the marginal cost
of public funds must be larger than one, since the government relies on distortionary taxes
to finance its outlays. If the marginal cost of public funds is indeed larger than one, this has
important normative consequences for the determination of optimal public policy in many fields.
In early theoretical contributions, Stiglitz and Dasgupta (1971) and Atkinson and Stern
(1974) demonstrated that the Samuelson (1954) rule for the optimum provision of public goods
needs to be modified to account for tax distortions.1 The optimal level of public goods provision
should be lower, and the optimal size of the government should thus be smaller, if the marginal
cost of public funds is higher. Many applied cost-benefit analyses multiply the cost of public
projects with a measure for the marginal cost of public funds that is larger than one. As a result,
public projects are less likely to pass a cost-benefit test. For example, Heckman et al. (2010)
evaluate the Perry Preschool Program and add 50 cents per dollar spent to account for the
deadweight costs of taxation. Sandmo (1975) and Bovenberg and de Mooij (1994) demonstrate
that the optimal corrective tax is generally set below the Pigouvian tax, since a high marginal
cost of public funds raises the cost of internalizing of externalities. Hence, governments should
pursue less ambitious environmental policies if taxation is more costly. Many other examples
can be given, but the message is clear: the marginal cost of public funds has a tremendous
impact on how governments should evaluate the desirability of public policies.2
This paper challenges the conventional wisdom that the marginal cost of public funds is
necessarily larger than one. It does so by explicitly introducing distributional concerns to
motivate tax distortions. Most of the literature, on which the analysis of the marginal cost
of public funds is based, has focused on Ramsey (1927) frameworks with homogeneous agents
where redistributional concerns are absent by assumption see, e.g., Browning (1976), Browning
(1987), Wildasin (1984), and Ballard and Fullerton (1992).3 However, in this context Sandmo
1
Ballard and Fullerton (1992), Dahlby (2008), and Jacobs (2009) provide extensive reviews of the large literature that subsequently emerged.
2
Other examples include Laffont and Tirole (1993), who put the marginal cost of public funds at the center
of their theories on optimal procurement and regulation. A larger cost of public funds renders rent extraction
more valuable compared to provision of cost-reducing incentives. . Similarly, Barro (1979) and Lucas and Stokey
(1983) derive that the optimal path of public debt ensures that tax rates are smoothed over time. Consequently,
debt policy is aimed at equalizing the marginal cost of public funds over time.
3
Some authors have allowed for heterogeneous agents, see, for example, the contributions by Browning and
2
(1998, pp. 378-379) notes the following:
“The observation that many discussions of the marginal cost of public funds are
based on theoretical models where distributional issues have been assumed away is
somewhat paradoxical, since the distortionary effects of taxation, on which thinking
around the MCF is based, can only be justified from a welfare economics point of
view by their positive effects on the distribution of income.”
A whole literature has thus analyzed the marginal cost of public funds for distortionary taxes,
without paying attention to the ultimate reasons why there are distortionary taxes in the first
place.
This paper contributes to the literature on the marginal cost of public funds in a number of
ways. First, it follows the literature on optimal income taxation – pioneered by Mirrlees (1971)
– so as to provide a proper microeconomic foundation for analyzing tax distortions. Individuals
are heterogeneous with respect to their earning ability (or skill level). The government cannot achieve any desired distribution of welfare due to informational constraints, since earning
abilities are private information. Therefore, individualized lump-sum taxes are not incentivecompatible, and the government needs to rely on taxing observable income (or consumption),
which results in labor supply distortions. Distributional concerns in combination with informational constraints thus provide the fundamental reason why distortionary taxation is optimal
in second-best settings. Second, non-distortionary, non-individualized lump-sum taxes are not
ruled out to obtain a non-trivial second-best policy problem. Non-individualized lump-sum
taxes (or transfers) are incentive compatible, and, therefore, part of the instrument set of the
government. It is shown that non-distortionary transfers (‘demogrant’) are always part of the
optimal tax schedule. Moreover, a non-distortionary general tax exemption or general tax credit
is part of virtually all tax systems in the real world.
The main finding of this paper is that, at the optimal tax system, the marginal cost of
public funds is equal to one for all tax instruments. The marginal cost of public funds for
the non-distortionary non-individualized lump-sum tax equals one at the optimal tax system.
This comes at no surprise, since non-individualized lump-sum taxes neither feature distortions
nor distributional benefits. Moreover, the marginal cost of public funds for distortionary taxes
should be one as well, since it should be equal to the marginal cost of public funds for nondistortionary taxes. Intuitively, the excess burden of distortionary taxation should be exactly
compensated by the social marginal redistributional benefits of taxation.
To demonstrate how allowing for heterogeneous agents and non-individualized lump-sum
taxes drive our main findings, this paper also analyzes the special case in which the government
cannot optimize lump-sum transfers. Even then it is not correct to conclude that the marginal
costs of public funds is necessarily larger than one. The marginal cost of public funds for distortionary taxes can either be larger or smaller than one depending on whether the distributional
Johnson (1984) and Allgood and Snow (1998), but these studies focus mainly on the efficiency costs of taxation.
Others have explicitly introduced distributional aspects of public goods and taxes, see, e.g., Christiansen (1981),
Wilson (1991), Boadway and Keen (1993), Kaplow (1996), Sandmo (1998), Slemrod and Yitzhaki (2001), and
Dahlby (2008).
3
gains are smaller or larger than the excess burden of the distortionary tax. The representativeagent models are nested as a special case of the model without non-individualized lump-sum
taxes, where all distributional benefits of taxation are assumed to be zero. Only in that special
case one can unambiguously conclude that the marginal cost of public funds is larger than one,
since all available tax instruments only cause distortions, but yield no redistributional gains.
This case, however, is of no practical relevance, since individuals are not identical in the real
world.
Ramsey (1927) frameworks with representative agents thus produce misleading results on
the marginal cost of public funds by excluding the reasons for distortionary taxation from the
analysis and eliminating the non-individualized lump-sum tax from the government instrument
set. In particular, the government is forced to use distortionary taxes in a setting whereas all
individuals would prefer the government to levy non-distortionary (non-individualized) lumpsum taxes instead. If tax system is optimized, marginal distributional benefits of distortionary
taxes are equal to marginal deadweight losses of distortionary taxes. The marginal cost of
public funds is therefore not a measure for the social cost of taxation, since it also includes the
distributional gains of taxation. Indeed, the social cost of distortionary taxation is properly
measured by standard measures for the marginal excess burden of taxation.
The findings of this paper have important consequences for applied public policy. In particular, tax distortions as such should not lower public good provision. The marginal excess
burden of distortionary taxation is the price of redistribution, and not the marginal cost of
public funds, i.e., the price of government activities that exclude redistributional tasks. In
particular, if the government would not be interested in redistribution at all, then all distortionary taxes would be zero, and all public goods would be financed with non-distortionary
non-individualized lump-sum taxes. Moreover, there should be no corrections for distortionary
taxes in social cost-benefit analysis of public projects if taxes are optimized. Alternatively,
there should be corrections for both the marginal excess burden and the marginal distributional
benefit of distortionary taxation if tax systems are not optimized. This policy prescription goes
against practices in social cost-benefit analysis, where deadweight costs of taxation are included
at the cost side, but the distributional benefits of taxation are ignored at the benefit side. These
practices are bound to yield policy errors, because many public projects can now fail the social
cost-benefit test, whereas they are in fact socially desirable. Jacobs and de Mooij (2015) apply the concepts from this paper to demonstrate that by allowing for distributional concerns,
the optimal corrective tax to internalize consumption externalities should no longer set below
the Pigouvian tax. The distributional benefits of distortionary taxes cancel out against their
deadweight losses. Hence, it is not desirable to pursue less aggressive environmental policies
even in the presence of distortionary taxation, as Sandmo (1975) and Bovenberg and de Mooij
(1994) suggested. Finally, Werning (2007) finds that the optimal path of public debt becomes
indeterminate and Ricardian equivalence is restored under distortionary taxation. Intuitively,
tax distortions are compensated by distributional gains. As a result, the optimal path of public
debt can no longer be determined by smoothing out tax distortions over time.
The literature on the marginal cost of public funds has generated substantial confusion,
see also the reviews in Ballard and Fullerton (1992), Dahlby (2008), and Jacobs (2009). In
4
particular, earlier literature has not settled down on a commonly agreed definition for the
marginal cost of public funds.4 Moreover, for the most regularly used definition, e.g. in Atkinson
and Stern (1974), Ballard and Fullerton (1992), and Sandmo (1998), the marginal cost of
public funds for lump-sum taxes is generally not equal to one, whereas there is no theoretical
presumption that it should differ from one, given that lump-sum taxes are non-distortionary
and produce no distributional benefits.5 It remains unclear how measures for the marginal cost
of public funds relate to the marginal excess burden of taxation, even though this relationship
is often suggested by, e.g., Pigou (1947), Harberger (1964), and Browning (1976). Indeed, in
the absence of distributional concerns, one would expect a direct correspondence between the
marginal cost of public funds and the deadweight loss of taxation. Finally, standard measures
for the marginal cost of public funds are highly sensitive to the choice of the untaxed numéraire
good – even if allocations are identical under each normalization. Many authors have tried to
remedy these issues, but as of today applied policy analysts are still confronted with a host of
unsettled definitional and conceptual problems in the literature on the marginal costs of public
funds, which seriously limits its practical applicability.
Clearing up the morass of conceptual problems in the literature is a prerequisite before a
meaningful analysis of the marginal cost of public funds can be made. To that end, this paper
defines the marginal cost of public funds (M CF ) as the ratio of the social marginal value of
public income and (the average of) the social marginal value of private income, i.e., M CF ≡
∆ social welfare 1 unit public resources
∆ social welfare 1 unit private resources , where Diamond (1975)’s definition of the
of private income is used.6 This paper demonstrates that most conceptual
social marginal value
problems in the liter-
ature originate from employing the private marginal value of private income as a measure for the
social marginal value of private income, i.e., M CF ≡
∆ social welfare 1 unit public resources 7
∆ private welfare 1 unit private resources .
How-
ever, the private marginal social value of private income does not measure the social marginal
value of private income due to policy-induced income effects on taxed bases, which the Diamondbased measure takes into account. The Diamond-based measure for the marginal cost of public
funds resolves most problems in the literature. First, the marginal cost of public funds is equal
to one if lump-sum taxes are optimized. Moreover, a direct correspondence is obtained between
4
On the one hand, the so-called Pigou-Harberger-Browning approach (also called ‘differential analysis’) relates
the marginal cost of public funds to the marginal excess burden of taxation, which is determined by the compensated tax elasticity of earnings, see also Pigou (1947), Harberger (1964), Browning (1976), and Browning (1987).
On the other hand, the Atkinson-Stern-Ballard-Fullerton approach (also called ‘balanced-budget approach’) bases
the marginal cost of public funds on the ratio of the marginal value of income of the public and the private sector
(i.e., the ratio of Lagrange multipliers on the government budget constraint and private budget constraint). In
that case, the uncompensated tax elasticity of earnings supply determines the marginal cost of funds, see also
Atkinson and Stern (1974) and Ballard and Fullerton (1992). By using the latter approach to the marginal cost
of funds Sandmo (1998) and Slemrod and Yitzhaki (2001) include distributional aspects of distortionary linear
taxation, while Gahvari (2006) extends it to non-linear taxation. Using models with representative agents, Triest
(1990), Håkonsen (1998) and Dahlby (1998) develop yet another MCF concept relying on correcting the standard
MCF measures for the shadow value of public resources before and after the introduction of distortionary taxes.
Up to this date, it remains unclear which MCF measure should be used in applied policy analysis.
5
Lump-sum taxes have no distributional benefits in optimal tax models, because the covariance between social
welfare weights and lump-sum taxes is zero.
6
This paper is related to an unpublished paper by Lundholm (2005) that also analyzes optimal second-best
public goods provision under optimal linear income taxation using the Diamond approach to the social marginal
value of income.
7
The conceptual problems remain if the private marginal value of private income is multiplied by a social
welfare weight to capture a social preference for redistribution.
5
the marginal cost of public funds and the marginal excess burden of taxation if distributional
concerns are absent. In particular, the marginal cost of public funds of a distortionary tax then
equals the inverse of one minus the excess burden of the tax. Finally, the marginal cost of public
funds measures have become insensitive to the normalization of the tax code.
This paper also contributes to the literature on optimal non-linear taxation and public
goods provision, as in Christiansen (1981), Kaplow (1996) and Gahvari (2006), that assumed
identical utility functions for all agents. This paper allows for fully general preference structures,
which can be dependent on skill levels.8 The optimal non-linear tax structure and modified
Samuelson rule for public goods provision are derived under non-linear income taxation in
terms of sufficient statistics as in Saez (2001): earnings, willingness to pay for public goods,
behavioral elasticities, and social welfare weights. The modified Samuelson rules under optimal
linear and non-linear income taxation are shown to be nearly identical, and do not contain
corrections for the marginal cost of public funds. Moreover, when preference heterogeneity is
allowed for, public goods provision under optimal non-linear income taxation is also determined
by distributional concerns, which is not the case with identical preferences. In particular, if the
marginal willingness to pay for the public good declines with earning ability, the government
will provide more public goods, compared to the first-best Samuelson rule for public goods
provision. The reverse reasoning holds if the willingness to pay for public goods increases with
earning ability.9
The contributions of this paper are, furthermore, related to Kaplow (1996), Kaplow (2004),
Laroque (2005) and Gauthier and Laroque (2009). These authors show that, if preferences are
identical across individuals and weakly separable between labor supply and other commodities,
the government can design benefit-absorbing changes in the non-linear tax schedule that perfectly offset the distributional impact of public good provision without changing labor supply
incentives. Consequently, the benefit-absorbing tax change perfectly imitates a pure benefit
tax. Standard first-best policy rules can be applied in second-best, and no corrections for the
marginal cost of public funds need to be made. In contrast, this paper shows that the marginal
cost of public funds should never play a role in second-best policy rules, as long as the income
tax is optimized.10 This result holds true without changing the entire tax schedule to neutralize
the distributional impact of the public good, without imposing weakly separable or identical
utility functions, and without assuming that the tax system is non-linear.
The remainder of this paper is structured as follows. Section 2 introduces the model. Section
3 is devoted to optimal taxation, optimal provision of public goods, and the marginal cost of
8
This paper is related to Boadway and Keen (1993) who developed a 2-type Stiglitz (1982) model with
preference heterogeneity. Most of their applications, however, assume identical preferences. This paper fully
traces the implications of preference heterogeneity on optimal non-linear taxes and public goods provision in the
continuous-type Mirrlees (1971) model. Moreover, we express all tax rules in terms of sufficient statistics. Similarly, Kreiner and Verdelin (2012) explore tax reforms and public good provision with heterogeneous preferences
under sub-optimal non-linear income taxation, whereas the current paper analyzes optimal taxation and optimal
public-good provision.
9
These findings extend the earlier results of Mirrlees (1976) and Saez (2002) on optimal commodity taxation
with preference heterogeneity to optimal public goods provision with preference heterogeneity.
10
Naturally, second-best interactions of public good provision with labor supply can enter the Samuelson rule,
see also Atkinson and Stern (1974). Moreover, the fact that the policy rule is not affected by the marginal cost
of public funds does not imply that the policy is invariant to taxation. As the second-best allocation will be
different from the first-best allocation, the various elements of the policy rule will be different as well.
6
public funds under linear tax instruments. Section 4 discusses optimal taxation and public
goods provision under non-linear instruments. Section 5 discusses the results and Section 6
concludes. An Appendix contains a long proof of a Proposition.
2
Model
The model consists of heterogeneous individuals optimally supplying labor on the intensive
margin and a benevolent government optimally setting taxes and public goods.11 Without loss
of generality, a partial-equilibrium setting is assumed in which prices are fixed, so that firms
can be ignored.12 Attention is initially focused on the optimal setting of linear tax instruments.
Later, optimal non-linear taxation is explored. Real-world tax systems are neither completely
flat, nor completely non-linear. Hence, it is useful to explore both cases.
2.1
Individuals
There is a mass N of individuals that differ by a single-dimensional parameter n ∈ N = [n, n],
where the upper bound n could be infinite. n is the individual’s earning ability (‘skill level’),
which equals the productivity per hour worked. The density of individuals of type n is denoted
R
by f (n) and the cumulative distribution function by F (n). Note that N N f (n)dn = N .
Each individual n derives utility u(n) from consumption c(n) and pure public goods G.
Furthermore, it derives disutility from supplying labor l(n). Each individual has an endowment
of time, which is allocated between leisure and working. Consumption and leisure are both
assumed to be normal goods. This paper generalizes the analysis of optimal taxation and
public goods provision by allowing for preference heterogeneity: the utility function depends
on the skill level n. The utility function is strictly (quasi-)concave, and is twice continuously
differentiable:
u(n) ≡ u(c(n), l(n), G, n),
uc , −ul , uG > 0,
∀n.
(1)
Subscripts denote partial derivatives. The Spence-Mirrlees conditions are imposed on the utility
function so as to guarantee the implementability of the non-linear policies derived in the next
section (see also Lemma 2 below).
The government employs a tax schedule consisting of a linear tax rate t on gross labor
earnings z(n) ≡ nl(n), a linear tax rate τ on consumption goods c(n), and a non-individualized
lump-sum transfer g. The informational requirements for employing linear taxes are that the
government should be able to verify aggregate labor income or consumption. Note that in
Mirrlees (1971)-analyses non-individualized lump-sum transfers are always part of the instrument set of the government, since the government can always provide each individual with an
11
It is based on earlier contributions by Ramsey (1927), Mirrlees (1971), Diamond and Mirrlees (1971), Sheshinski (1972), Diamond (1975), Christiansen (1981), Boadway and Keen (1993), Kaplow (1996) and Sandmo (1998).
12
Jacobs (2010) analyzes the model in general-equilibrium settings allowing for a representative firm operating
a constant-returns-to-scale technology and none of the results change. Almost all of the papers in the literature fix
the marginal rates of transformation between all commodities at one. Hence, all prices are constant, and allowing
for general equilibrium provides no additional insights, see previous footnote. Moreover, our partial-equilibrium
results fully generalize to general-equilibrium settings with non-constant prices, since optimal second-best tax
rules in general equilibrium are identical to the ones in partial equilibrium as long as there are constant returns
to scale in production and all labor types are perfect substitutes, see also Diamond and Mirrlees (1971).
7
equal amount of resources. The individual budget constraint states that net expenditures on
consumption are equal to net labor earnings:
(1 + τ )c(n) = (1 − t)z(n) + g,
∀n.
(2)
One tax instrument is redundant, since the consumption tax is equivalent to the income tax.
Thus, without loss of generality, one tax instrument can always be normalized to zero.
The individual maximizes utility (1) subject to its budget constraint (2). This yields the
standard first-order condition for labor supply:13
−ul (c(n), l(n), G, n)
(1 − t)n
=
,
uc (c(n), l(n), G, n)
1+τ
∀n.
(3)
Taxation is distortionary as it drives a wedge between the marginal social benefits (n) and the
marginal private benefits ((1 − t)n/(1 + τ )) of an increase in labor supply. Higher levels of
taxation (income or consumption taxes) reduce the price of leisure in terms of consumption and
individuals substitute leisure for consumption.
Indirect utility of individual n can be written as: v(n) ≡ v(t, τ, g, G, n) ≡ u(ĉ(n), ˆl(n), G, n),
where hats denote the optimized values for consumption and labor supply. Straightforward
application of Roy’s identity yields the following properties of v(·):
−λ(n)c(n),
∂v(n)
∂g
= λ(n) and
∂v(n)
∂G
=
λ(n) uuGc ,
∂v(n)
∂t
= −λ(n)z(n),
∂v(n)
∂τ
=
where λ(n) is the private marginal utility of
income of individual n. Given that preferences depend on the skill level n, the private marginal
utility of income is not necessarily non-increasing in n. To ensure that a well-defined redistribution problem is obtained, it is assumed that
∂λ(n)
∂n
≤ 0. This inequality always holds if
preferences are identical for all individuals due to the assumptions on the derivatives of the
utility function.
Some additional notation is introduced to express the optimal policy rules in terms of
well-known elasticity concepts. In particular, the compensated, uncompensated and income
elasticities of labor supply and consumption demand with respect to the tax rates and the
∂lu (n) 1−t
∂lc (n) 1−t
∂l(n) G
c
u
c
∂t l(n) , εlt ≡
∂t l(n) < 0, εlG ≡ ∂G l(n) , εlG ≡
u
∂cc (n) 1+τ
∂cu (n) G
c
u
0, εucτ ≡ ∂c∂τ(n) 1+τ
∂τ c(n) , εcG ≡
∂G c(n) ,
c(n) , εcτ ≡
∂c(n)
τ ) ∂g > 0, where the superscript u (c) denotes an un-
public good are denoted by: εult ≡
∂lc (n) G
∂G l(n) , εlg ≡
c
G
εccG ≡ ∂c∂G(n) c(n)
,
(1 − t)n ∂l(n)
<
∂g
and εcg ≡ (1 +
compensated (compensated) change. In addition, the Slutsky equations will be employed to
c
c
∂lu (n)
∂cu (n)
= ∂l∂t(n) − z(n) ∂l(n)
= ∂c∂τ(n) − c(n) ∂c(n)
∂t
∂g ,
∂τ
∂g ,
∂lu (n)
∂lc (n)
∂cc (n)
uG ∂l(n) ∂cu (n)
uG ∂c(n)
∂G = ∂G + λ(n) ∂g ,
∂G = ∂G + λ(n) ∂g . Rewriting the Slutsky equations yields
u
c
u
c
εlt = εlt − εlg , εcτ = εcτ − εcg , εclG = εulG − ωωGl εlg and εclG = εulG − ωωGc εlg , where ωG ≡ uGuG ,
ωl ≡ −ulul(n) and ωc ≡ uc c(n)
denote the shares of public goods, labor and consumption in utilu
simplify the optimal tax expressions:
ity. In the remainder of the paper, a bar is used to indicate an income-weighted elasticity, e.g.,
R
R
−1
ε̄clt ≡ N εclt z(n)dF (N ) N z(n)dF (n) .
13
Under linear instruments, first-order conditions are both necessary and sufficient given the restrictions on
the derivatives of the utility function. Under non-linear policies, however, budget constraints are non-linear and
additional assumptions, i.e., monotonicity and Spence-Mirrlees conditions, are required to ensure that first-order
conditions are both necessary and sufficient, see also section 4.3.
8
2.2
Government
The social objectives is a Samuelson-Bergson social welfare function, which is a sum of concave
individual utilities:
Z
Ψ(u(n))dF (n),
N
Ψ0 (u(n)) > 0,
Ψ00 (u(n)) ≤ 0.
(4)
N
Maximizing social welfare under a Utilitarian criterion (Ψ0 = 1) implies a social preference for
redistribution if the private marginal utility of income λ(n) declines with skill n at the individual
level. However, due to the concavity of Ψ(·) the government may exhibit a stronger preference
for redistribution than individuals do.14
The government budget constraint states that total tax revenues equal spending on transfers
and public goods:
Z
(tz(n) + τ c(n))dF (n) = N g + pG.
N
(5)
N
where p denotes the constant marginal rate of transformation (i.e., the price) of public goods
in terms of private goods.
3
Optimal linear taxation and public goods provision
The government maximizes social welfare subject to its budget constraint by choosing the nonindividualized lump-sum transfer g, the tax rate on income t or consumption τ , and the level
of public goods provision G. Optimal policies are derived under both tax normalizations (i.e.,
either consumption or income remains untaxed). The social marginal value of one unit of public
resources is denoted by η. The Lagrangian for maximizing social welfare is given by:
Z
L≡N
Z
Ψ(v(n))dF (n) + η N
(tnl(n) + τ c(n))dF (n) − N g − pG .
N
The first-order conditions for a maximum are:
Z ∂L
∂l(n)
∂c(n)
0
=N
Ψ (v(n))λ(n) − η + ηtn
+ ητ
dF (n) = 0,
∂g
∂g
∂g
N
Z ∂l(n)
∂c(n)
−Ψ (v(n))λ(n)nl(n) + ηnl(n) + ηtn
+ ητ
dF (n) = 0,
∂t
∂t
N
Z ∂L
∂l(n)
∂c(n)
0
=N
−Ψ (v(n))λ(n)c(n) + ηc(n) + ηtn
+ ητ
dF (n) = 0,
∂τ
∂τ
∂τ
N
Z ∂L
∂l(n)
∂c(n)
0
=N
Ψ (v(n))uG + ηtn
+ ητ
dF (n) − pη = 0,
∂G
∂G
∂G
N
∂L
=N
∂t
(6)
N
0
(7)
(8)
(9)
(10)
where the derivatives of indirect utility have been used in each first-order condition.15
14
Alternatively, exogenously given Pareto weights δ(n) can be used, i.e., Ψ(u(n)) ≡ δ(n)u(n), where Ψ0 = δ(n)
and Ψ00 = 0. None of the results below would be affected by using this social welfare function instead.
15
The first-order conditions are necessary, but may not be sufficient to characterize the optimum (Diamond
and Mirrlees, 1971). In the remainder it’s assumed that the second-order conditions for maximization of social
welfare are always respected.
9
3.1
Marginal cost of public funds – The standard approach
This section first derives optimal policies using the standard measure for the marginal cost
of public funds, see, e.g., Wilson (1991), Ballard and Fullerton (1992), Sandmo (1998), and
Gahvari (2006). To that end, the following definitions are introduced: marginal social value
of private income, the marginal cost of public funds, and the Feldstein (1972) distributional
characteristics of the tax bases and public goods.
Definition 1 The social marginal value of one unit of private income accruing to individual n
equals
αs (n) ≡ Ψ0 (u(n))λ(n),
∀n.
(11)
According to the traditional definition, the social marginal value of private income is captured
by the mechanical increase in social welfare of transferring one unit of resources to individual
n. Utility rises with λ(n), and this increase in utility is valued by the government by Ψ0 (u(n)).
In the remainder of this section the superscript s is used to denote variables are derived under
the standard approach.
Definition 2 The marginal cost of public funds based on the standard measure of the social
marginal value of private income is given by
η
M CF s ≡ R
N
αs (n)dF (n)
.
(12)
The traditional marginal cost of public funds M CF s is the ratio of the social marginal value of
public income η and the average of the social marginal value of private income αs (n).
Definition 3 The distributional characteristics ξys of tax bases y(n) = {z(n), c(n)} are given
by
ξys ≡ − R
N
cov[αs (n), y(n)]
R
> 0.
N y(n)dF (n)
αs (n)dF (n)
(13)
ξys is a normalized covariance between a tax base and the welfare weights. It represents the
gain in social welfare (expressed in monetary equivalents and then divided by the taxed base) of
redistributing a marginal unit of resources through base y(n) = {z(n), c(n)}. The distributional
characteristics of the tax bases are positive, since the covariance between tax base y(n) and
the social welfare weights αs (n) is negative. Individuals with higher incomes or consumption
levels feature lower welfare weights because the social marginal utility of income is diminishing
in income or consumption due to diminishing private marginal utility of income or concavity
of the social welfare function. The positive distributional characteristic ξys therefore implies
that redistribution through taxing income or consumption yields distributional benefits. A
stronger social desire for redistribution or greater inequality in the skill distribution raise the
distributional characteristic. Since ξys is a positive normalized covariance, it ranges between
zero and one. ξys = 0 is obtained either if the government is not interested in redistribution
because it attaches the same social welfare weights αs (n) to all individuals or if the base y(n)
is the same for all n so that there is no inequality.
10
Definition 4 The distributional characteristic of the public good is
i
h
(·)
cov αs (n), uuGc (·)
s
.
ξG
≡ −R
R uG (·)
s (n)dF (n)
α
dF
(n)
N
N uc (·)
(14)
s is the negative normalized covariance
The distributional characteristic for the public good ξG
between the social marginal valuation of income αs (n) and the marginal willingness to pay for
the public good
uG
uc .
s > 0 if the public good mainly benefits the rich, who feature the lowest
ξG
s = 0 if the government is not interested in
social welfare weights αs (n), and vice versa. ξG
redistribution and attaches the same welfare weights αs (n) to all individuals or if all individuals
benefit equally from the public good, i.e.,
uG
uc
is equal for all n.
The next Lemma derives the marginal excess burden of the income or consumption tax. The
excess burden measures the reduction in social welfare, measured in monetary units, expressed
as a fraction of the tax base, of raising the distortionary income or consumption tax.
Lemma 1 The marginal excess burdens for the income tax and the consumption tax are given
by
M EBt = −
t c
ε̄ ,
1 − t lt
M EBτ = −
τ
ε̄c .
1 + τ cτ
(15)
Proof. The welfare effect of a rise in the tax rate is evaluated, while public goods provision
remains constant. A rise in the tax rate is considered, while each individual n receives a
individual-specific lump-sum transfer s(n) so as to keep its utility constant.16 The excess
burden is equal to the resulting loss in tax revenue, which is summed over all individuals.
To determine the excess burden of the income tax, assume that τ = 0. The change in taxes
dt and lump-sum income ds(n) for each individual n that leaves utility unaffected: dv(n) =
λ(n)ds(n) − λ(n)nl(n)dt = 0. Hence, ds(n) = nl(n)dt for all n. Public revenue changes
c
c
according to dRn ≡ −ds(n) + nl(n)dt + tn ∂l∂t(n) dt = tn ∂l∂t(n) dt for all n. Since utility remains
constant, changes in labor supply are compensated changes. Rewriting yields a revenue loss
of dRn =
∂lc (n) 1−t
t
1−t nl(n) ∂t l(n) dt Rfor
dividing by taxable income N
of taxed income:
N
each individual n. Summing
dRn
dt
over all individuals, and
nl(n)dF (n), yields the marginal excess burden as a fraction
R
M EBt ≡ − R N
N
dRn
dt dF (n)
nl(n)dF (n)
=−
t c
ε̄ .
1 − t lt
(16)
Recall that the bar indicates an income-weighted elasticity. Using similar steps and setting t = 0 gives the marginal excess burden of the consumption tax as a fraction of taxed
consumption:
R
M EBτ ≡ − R N
N
dRn
dτ dF (n)
c(n)dF (n)
=−
τ
ε̄c .
1 + τ cτ
(17)
Proposition 1 replicates Sandmo (1998) by characterizing optimal tax policies and public
goods provision under the standard measure for the marginal cost of public funds, and assuming
that the consumption tax is normalized to zero. Each tax instrument has its own marginal cost
16
Of course this instrument does not exist, since it boils down to an individualized lump-sum tax. However,
this thought-exercise allows us to calculate the excess burden of the tax.
11
of public funds. Therefore, M CFgs and M CFts are used to denote the marginal cost of public
funds of the lump-sum tax and the tax rate, respectively, see also Sandmo (1998).
Proposition 1 With the standard MCF definition, and with the consumption tax normalized
to zero, the optimal rules for public goods provision and the linear income tax are given by
s
(1 − ξG
)N
Z
N
uG (·)
dF (n) = (1 − γt ε̄ulG ) · M CF s · p,
uc (·)
Z
1
M CFgs =
<
1,
ε̃
≡
εlg dF (n) < 0,
lg
t
1 − 1−t
ε̃lg
N
1 − ξzs
1 − ξzs
M CFts =
,
t u =
t
1 + 1−t ε̄lt
1 − M EBt − 1−t
ε̄lg
M CF s = M CFgs = M CFts < 1.
(18)
(19)
(20)
(21)
Proof. Equation (7) can be simplified by setting τ = 0 and substituting equation (12) to find
equation (19). Equation (8) is simplified by using equation (13) and setting τ = 0 to find the
first part of equation (20). The second part of equation (20) follows upon substitution of the
Slutsky equation ε̄ult = ε̄clt − ε̄lg and using equation (15). Equation (10) is simplified by using
R
equation (14), setting τ = 0, and using γt ≡ N t N nl(n)dF (n)/pG to find equation (18).
Equation (18) is the modified Samuelson rule for the optimal provision of public goods.
The modified Samuelson rule equates the sum of the marginal social benefits with the marginal
social costs of providing the public good. The benefits – the sum of the marginal rates of
R
substitution N N uuGc dF (n) – are deflated by the distributional characteristic of the public
s . If poor individuals value the public good more (less) than rich individuals do, then
good ξG
s < 0 (ξ s > 0), and the level of public goods provision increases (decreases) – ceteris paribus.
ξG
G
The right-hand side gives the marginal cost of providing the public good. p equals the marginal
rate of transformation between the public good and the reference private good. The cost
side of the Samuelson rule is corrected for tax distortions via the marginal cost of public funds
M CF s . A higher cost of public funds lowers the level of public goods provision – ceteris paribus.
Providing the public good may exacerbate (reduce) pre-existing labor-tax distortions if the
public good reduces (boosts) labor supply. This correction is governed by the uncompensated
cross-elasticity of labor with the public good (if ε̄ulG 6= 0), see also Atkinson and Stern (1974).
R
γt ≡ N t N z(n)dF (n)/pG denotes the share of public goods that is financed with distortionary
taxes. If γt = 1, then public goods are financed exclusively with distortionary taxes. If the
public good and labor supply are complementary (ε̄ulG > 0), the provision of the public good
boosts labor supply and this raises tax revenue in the presence of a positive labor tax (γt > 0).
These positive revenue effects decrease the cost of public goods provision. If the public good
reduces labor supply (ε̄ulG < 0), the cost of the public good rises, since lower labor supply erodes
the labor tax base.
Equation (19) gives the marginal cost of public funds for the lump-sum tax. If there is a
positive income tax (t > 0), the marginal cost of public funds for the lump-sum tax is always
smaller than one (M CFgs < 1) as long as leisure is a normal good (ε̃lg < 0 ). Intuitively,
transferring an extra unit of funds from individuals to the government via a larger lump-sum
tax generates an income effect in labor supply, and this raises revenues from the income tax if
12
it is positive (t > 0). Consequently, one can raise the lump-sum tax by less than one unit to
raise one unit of public funds. However, one would expect the lump-sum tax to have a marginal
cost of public funds equal to one, since the lump-sum tax neither causes distortions nor features
distributional effects (i.e., the normalized covariance between the social welfare weights αs (n)
and the lump-sum tax g is zero). The result that the lump-sum tax does not have a marginal
cost of public funds of unity is the first problem that is identified in the literature.
Equation (20) gives the marginal cost of public funds for the tax rate (M CFts ). M CFts
depends on the income-weighted uncompensated tax elasticity of labor supply ε̄ult . The sign
of this elasticity is ambiguous due to offsetting income and substitution effects.17 Clearly, by
allowing for distributional concerns (ξzs > 0), distortionary taxes have not only costs, but also
distributional benefits. These distributional gains lower the marginal cost of public funds for
the distortionary income tax.
In Ramsey (1927) frameworks the non-individualized lump-sum tax is ruled out and distributional concerns are absent. Hence, M CFgs is not applicable and ξzs = 0 in M CFts , see also
Atkinson and Stern (1974). M CFts > 1 is then obtained if the labor supply curve is upwardsloping (ε̄ult < 0) and M CFts < 1 if there is a backward-bending labor supply curve (ε̄ult > 0).
The results that (in the absence of redistributional concerns) the M CFts can be smaller than
one and that it cannot be directly related to the marginal excess burden of the tax led to a
large literature trying to resolve these counterintuitive findings, see for example Triest (1990),
Ballard and Fullerton (1992) and Dahlby (2008).
In the absence of distributional concerns (i.e., ξzs = 0), equation (20) demonstrates that
it is not possible to directly relate the marginal cost of public funds of the distortionary income tax to the marginal excess burden of the income tax, as suggested by the analyses in
Pigou (1947), Harberger (1964) and Browning (1976). The often-used approximation that the
marginal cost of public funds equals one plus the marginal excess burden of the tax will only
apply as long as income effects are absent in labor supply (ε̄lg = 0), since then ε̄clt = ε̄ult and
M CFts = [1 − M EBt ]−1 ≈ 1 + M EBt at low levels of t. However, assuming away income
effects is theoretically not very appealing and is not empirically warranted, see also Blundell
and MaCurdy (1999) and Meghir and Phillips (2010). Thus, in the absence of distributional
benefits of taxes, the marginal cost of public funds of the distortionary tax is not directly related
to the excess burden of the tax instrument. This is the second problem in the literature on the
marginal cost of public funds.
Equation (21) states that, in a second-best optimum, the marginal cost of public funds for all
tax instruments should be equalized (i.e., M CF s = M CFgs = M CFts ). Consequently, Sandmo
(1998) concluded that the marginal cost of public funds at the optimal tax system is smaller
than one in a second-best setting with redistributional concerns. However, note that this result
holds as long as there is a positive labor tax, and irrespective of whether there are distributional
concerns or not, see also Wilson (1991).
The next Proposition demonstrates that the standard definition for the marginal cost of
public funds is highly sensitive to the normalization of the tax system. In particular, it gives
17
Most empirical studies conducted in recent decades suggest a positive value for the uncompensated wage
elasticity of labor supply, implying that M CFts will exceed one, see, for example, Blundell and MaCurdy (1999)
and Meghir and Phillips (2010).
13
the optimality conditions for the tax system and public goods provision when the income tax is
normalized to zero and consumption is taxed instead. In this case, M CFgs and M CFτs denote
the marginal cost of public funds of the lump-sum tax and the consumption tax.
Proposition 2 With the standard MCF definition, and with the income tax normalized to zero,
the optimal rules for public goods provision and the linear consumption tax are given by
(1 −
s
ξG
)N
Z
N
uG (·)
dF (n) = (1 − γτ ε̄ucG ) · M CF s · p,
uc (·)
Z
1
s
M CFg =
> 1, ε̃cg ≡
εcg dF (n) > 0,
τ
1 − 1+τ
ε̃cg
N
1 − ξcs
1 − ξcs
M CFτs =
=
,
τ
τ
1 + 1+τ ε̄ucτ
1 − M EBτ − 1+τ
ε̄cg
M CF s = M CFgs = M CFτs > 1.
(22)
(23)
(24)
(25)
Proof. Equation (7) can be simplified by setting t = 0 and substituting equation (12) to find
equation (23). Equation (8) is simplified by using equation (13), and setting t = 0 to find the
first part of equation (24). The second part of equation (24) follows upon substitution of the
Slutsky equation ε̄ucτ = ε̄ccτ − ε̄cg and using equation (15). Equation (10) is simplified by using
R
equation (14), setting t = 0, and using γτ ≡ N τ N c(n)dF (n)/pG to find equation (22).
The interpretations of equations (22) and (24) are identical to those of (18) and (20) that
appear below Proposition 1. Hence, these will not be repeated.
Equation (23) reveals that the marginal cost of public funds for the lump-sum tax is always
higher than (or equal to) one, rather than always smaller than (or equal to) one when income is
taxed. The reason is that the income elasticity of consumption is positive (if consumption is a
normal good): ε̃cg > 0. If the lump-sum tax is increased (transfer is reduced), there is a negative
income effect in consumption demand, which reduces the revenue from the consumption tax.
In order to raise one unit of public revenue by raising the lump-sum tax, the lump-sum tax
must therefore be increased by more than one unit to compensate for the revenue loss from the
consumption tax. Hence, as the income effect on the taxed base switches in sign, M CFgs switches
from a number below one under income taxation to a number above one under consumption
taxation.
In Ramsey (1927) frameworks with a representative agent (ξcs = 0) and no lump-sum taxes,
equation (24) gives the marginal cost of public funds of the consumption tax. It reveals that
M CFτs > 1, since substitution effects and income effects in consumption demand are reinforcing
rather than offsetting. Consequently, the marginal cost of public funds for the consumption
tax is always larger than one, whereas it could be smaller than one for the income tax. In
the presence of distributional concerns (ξcs > 0), equation (25) shows that at the optimal tax
system the marginal cost of public funds for all tax instruments are equalized. As a result,
M CF s is larger than one if consumption is taxed, and M CF s is smaller than one if income
is taxed. A different normalization of the tax system therefore produces completely different
marginal cost of public funds measures for both lump-sum and distortionary taxes. However,
the normalization of the tax system does not influence the optimal second-best allocation. This
14
is the third problem with the standard marginal cost of public funds measures.18
The normalization explains the findings of Wilson (1991) and Sandmo (1998). They both
suggest that distributional concerns are the reason why the marginal cost of public funds is
smaller than one. However, their findings stem from the normalization of the tax system,
and would be reversed if consumption would be taken as the tax base instead of income, as
Proposition 2 demonstrates.
The standard M CF measure thus suffers from three shortcomings. First, the marginal cost
of public funds for lump-sum taxes are not equal to one – irrespective of the normalization of
the tax system. Second, in the absence of distributional concerns, the marginal cost of public
funds for a distortionary tax instrument cannot be directly related to the excess burden of
the tax instrument – irrespective of the normalization of the tax system. Hence, the M CF
measure does not properly capture the welfare costs of taxation in the absence of distributional
concerns. Third, the M CF measures for both distortionary and lump-sum taxes are shown to
be critically sensitive to the normalization of the tax system. All these problems are suggestive
of a conceptually problematic M CF measure. From a practical point of view, these problems
render the applicability of standard M CF measures in policy analysis doubtful.
3.2
Marginal cost of public funds – The Diamond approach
This section argues that the three problems with the standard M CF measures can be resolved
by appropriately specifying the social marginal value of private income, which is used in the
definition of the marginal cost of public funds. In particular, it is argued that the social marginal
value of private income should also include the income effects on the taxed bases, as in Diamond
(1975). If the government transfers one unit of funds to the private sector, it should not only
take into account the direct utility gains to individuals – as the standard measure does –, but
also include the indirect effect of a higher transfer on revenues from other taxes as social losses
(under income taxation) or gains (under consumption taxation).
According to the traditional approach, the income effects on the taxed bases are not included
in the social marginal value of private income, see also Definition 1. Consequently, to calculate
the M CF , the social value of a unit of funds to the public sector is compared with the private
value of private funds – up to a transformation of the latter through the social welfare function
if non-utilitarian social preferences are adopted.19 However, the marginal cost of public funds
should measure the ratio between social marginal value of a unit of public funds and the social
marginal value of a unit of private funds, and not the private marginal value of private funds. By
defining the marginal cost of public funds based on Diamond (1975), it will be demonstrated
that the three problems identified above are nullified. The social marginal value of private
income is thus defined as follows.
18
The sensitivity of M CF s to the normalization of the tax system was already pointed out by Atkinson and
Stern (1974) without using the M CF terminology, however. In particular, M CF s can be shown to be bigger
(smaller) than one if consumption goods (factor supplies) are taxed, and factor supplies (consumption goods) are
not taxed.
19
Naturally, this does not mean that one cannot define the marginal cost of public funds to be as such, since
a definition cannot be wrong in and of itself. However, from an economic point of view it would be incorrect
to interpret the traditional measure as the relative social marginal value of public funds in terms of the social
marginal value of private funds.
15
Definition 5 The Diamond definition for the social marginal value of one unit of private income accruing to individual n equals
α(n) ≡ Ψ0 (u(n))λ(n) + ηtn
∂l(n)
∂c(n)
+ ητ
,
∂g
∂g
∀n.
(26)
The direct utility effect is still measured by Ψ0 (u(n))λ(n). In addition, transferring one unit of
∂c(n)
funds to the individual reduces labor supply ( ∂l(n)
∂g < 0) and raises consumption demand ( ∂g >
0), since leisure and consumption are assumed to be normal goods. Hence, the government loses
∂c(n)
−tn ∂l(n)
∂g in tax revenues from the income tax (if t > 0) or gains τ ∂g in revenues from the
consumption tax (if τ > 0) if the individual receives an additional unit of funds. The welfare
effects are obtained by multiplication of the revenue changes with η, the social marginal value
of one unit of public resources. Adopting α(n) as the social marginal value of private income
makes it possible to define the marginal cost of public funds.
Definition 6 The marginal cost of public funds based on the Diamond measure of the social
marginal value of private income is given by
M CF ≡ R
N
η
.
α(n)dF (n)
(27)
Analogously to the traditional measure for the marginal cost of public funds, the Diamondbased measure for the marginal cost of public funds M CF is defined as the ratio of the social
marginal value of public resources η and the average social marginal value of private income
α(n). Similarly, the definitions for the distributional characteristics can be adjusted by using
α(n) instead of αs (n) as the social welfare weights.
Definition 7 The distributional characteristics ξy of tax bases y(n) = {z(n), c(n)} based on
the Diamond measure of the social marginal value of private income are given by
ξy ≡ − R
N
cov[α(n), y(n)]
R
> 0.
α(n)dF (n) N y(n)dF (n)
(28)
Definition 8 The distributional characteristic of the public good based on the Diamond measure
of the social marginal value of private income is
h
i
(·)
cov α(n), uuGc (·)
ξG ≡ − R
.
R uG (·)
α(n)dF
(n)
dF
(n)
N
N uc (·)
(29)
Using the new definitions, the optimal policies under linear income taxation are expressed
as follows.
Proposition 3 Under the Diamond based MCF definition, and the consumption tax normalized
16
to zero, the optimal rules for public goods provision and the linear income tax are given by
Z
(1 − ξG )N
N
uG (·)
dF (n) = (1 − γt ε̄clG ) · p,
uc (·)
M CFg = 1,
M CFt =
1 − ξz
1 − ξz
t c = 1 − M EB ,
1 + 1−t ε̄lt
t
M CF = M CFg = M CFt = 1.
(30)
(31)
(32)
(33)
Proof. Equation (7) can be simplified by setting τ = 0 and substituting equation (27) to find
equation (31). Equation (8) is simplified by using equation (28), the Slutsky equation
∂lc (n)
∂t
−
nl(n) ∂l(n)
∂g ,
∂lu (n)
∂t
=
and setting τ = 0 to find the first part of equation (32). The second part
follows from substituting equation (15). Equation (10) is simplified by using equation (29), the
R
u
c (n)
uG ∂l(n)
Slutsky equation ∂l∂G(n) = ∂l∂G
+ λ(n)
∂g , setting τ = 0, and using γt ≡ N t N nl(n)dF (n)/pG
to find equation (30).
Equation (30) reveals that the modified Samuelson rule is almost the same as before, see
equation (18), except that the compensated cross-elasticity of labor supply with respect to public
goods ε̄clG enters the expression (rather than the uncompensated cross-elasticity).20 However,
the most important difference with Proposition 1 is that there should be no correction for the
marginal cost of public funds on the cost side of the modified Samuelson rule. Consequently, tax
distortions do not affect the decision rule for the optimal supply of public goods. Although the
decision rule does not contain a correction for the marginal cost of public funds in second-best,
the level of public goods provision is generally different, since the allocations are not identical
in first- and second-best. Therefore, one cannot conclude that tax distortions do not affect the
second-best level of public goods provision.
Equation (31) demonstrates that, at the optimal tax system, the marginal cost of public
funds for the lump-sum tax (M CFg ) is always equal to one. The reason is that the lump-sum
tax does neither cause deadweight losses nor have distributional gains (losses). Indeed, there is
a zero covariance between the lump-sum tax g and the social welfare weights α(n), so that the
distributional characteristic ξ for the lump-sum tax is zero. Hence, the result that the marginal
cost of public funds is equal to one is, therefore, merely a statement that the tax system is
optimal. Intuitively, at the optimal tax system, one unit of resources should be equally valuable
in the private as in the public sector. Thus, the government should be indifferent between
transferring funds from the public to the private sector. The first problem that the lump-sum
tax does not have a marginal cost of public funds equal to one is thereby resolved.21
Equation (32) shows that in Ramsey (1927) analyses without lump-sum taxes and redistributional concerns (ξz = 0), the marginal cost of public funds for the distortionary tax would be
20
The compensated cross-elasticity is a different measure than the uncompensated cross-elasticity. Whereas
the uncompensated elasticity is zero with utility functions exhibiting (weak) separability between public goods
and labor (leisure), the compensated cross-elasticity is generally different from zero and positive for a wide class
of utility functions including the separable ones, see Jacobs (2009).
21
Gahvari (2014) suggested that M CF = 1 is a definition rather than a result. This is not correct, since
M CF = 1 is an optimality condition, not a definition. Outside the tax optimum, the marginal cost of public
funds forRthe lump-sum tax does not equal one (i.e., M CFT 6= 1) if the average social marginal value of private
income ( N α(n)dF (n)) is unequal to the social marginal value of public income (η).
17
directly related to the marginal excess burden of the tax. Indeed, without distributional concerns it is exactly equal to the inverse of 1 − M EBt . This finding resolves the second problem
that there is no link between the marginal cost of public funds and the excess burden of taxation.
The standard definition of the marginal cost of public funds does not include the income effects
on taxed bases in the social marginal value of private income. Hence, these income effects pop
up in the expression for the marginal cost of public funds, see equation (20).
In Mirrlees (1971) analyses, however, the government only introduces distortionary taxes
if doing so contributes to equality (i.e., if taxing labor income yields distributional benefits).
With distributional concerns, ξz > 0, and M CFt is lowered, as equation (32) reveals. At the
optimal tax system, the marginal cost of public funds for the income tax should be equal to the
marginal cost of public funds for the lump-sum tax, as equation (33) demonstrates. Therefore,
the marginal deadweight losses of income taxes should be exactly equal to the marginal distributional gains of income taxes: M EBt =
t c
1−t ε̄lt
= ξz . At the optimum, the marginal excess
burden of a distorting tax rate (expressed in monetary units, as a fraction of taxed income)
exactly equals the marginal benefits of redistribution (expressed in monetary units, as a fraction of taxed income). Tax distortions are therefore only introduced insofar as these distortions
contribute to a more equal distribution of utility. The more society cares about distribution,
the larger is ξz , and the higher is the optimal income tax. The more elastically labor supply
responds to taxes, the larger is −ε̄clt , and the lower is the optimal income tax. This is the
standard trade-off between equity and efficiency.
Naturally, there should be no correction of the modified Samuelson rule in equation (30) if the
public good is financed at the margin with the lump-sum tax, since there is no deadweight loss
involved (and no distributional gains either). However, neither should it contain a correction if
the marginal source of finance for the public good is the distortionary tax. This is an application
of the envelope theorem: the deadweight loss of a marginally higher tax exactly cancels against
the distributional gain of the tax if the tax is optimally set.
Equation (30) is mathematically identical to the modified Samuelson rule in equation (18),
only it is expressed in terms of compensated elasticities by using the Slutsky equations, cf.
Wildasin (1984). Although both equations provide equally valid characterizations of the modified Samuelson rule for the optimal provision of public goods, the policy conclusions derived
from both equations may seem to be different (but are not). Indeed, only by correcting the
definition for the marginal cost of public funds, it becomes clear that it should not play a role
in the modified Samuelson rule.
Tax distortions are introduced only for redistributional reasons, not for public goods provision. If the government would not be interested in income redistribution (i.e., ξz = ξG = 0),
distortionary income taxes would be optimally zero (t = 0), see Proposition 3. Thus, in the absence of a preference for redistribution, all public goods would be financed with non-distortionary
non-individualized taxes. The marginal excess burden of the income tax is therefore the price
of equality and not the price of public goods provision.
The next Proposition demonstrates that all results remain valid also using the different tax
normalization, where consumption rather than income is taxed.
Proposition 4 Under the Diamond based MCF definition, and with the income tax normalized
18
to zero, the optimal rules for public goods provision and the linear consumption tax are given by
Z
(1 − ξG )N
N
uG (·)
dF (n) = (1 − γτ ε̄ccG ) · p,
uc (·)
M CFg = 1,
M CFτ =
1 − ξc
1 − ξc
,
=
τ
1 + 1+τ
1 − M EBτ
ε̄ccτ
M CF = M CFg = M CFτ = 1.
(34)
(35)
(36)
(37)
Proof. Equation (7) can be simplified by setting t = 0 and substituting equation (27) to find
equation (35). Equation (8) is simplified by using equation (28), the Slutsky equation
∂cc (n)
∂τ
−
c(n) ∂c(n)
∂g ,
∂cu (n)
∂τ
=
and setting t = 0 to find the first part of equation (36). The second part
follows from substituting equation (15). Equation (10) is simplified by using equation (29), the
R
c
u
uG ∂c(n)
Slutsky equation ∂c∂G(n) = ∂c∂G(n) + λ(n)
∂g , setting t = 0, and using γτ ≡ N τ N c(n)dF (n)/pG
to find equation (34).
Proposition 4 demonstrates that the marginal cost of public funds measures have become
independent from the particular normalization of the tax system. At the optimal tax system,
the marginal cost of public funds remains equal to one for all tax instruments, as shown in
equation (35) and (37). Equation (36) derives that the excess burden of the consumption tax
τ
equals its distributional gain: M EBτ = − 1+τ
ε̄ccτ = ξc . Moreover, the characterization of the
optimal policy rule for public good provision in equation (34) has become independent from the
particular tax normalization.
The marginal excess burdens of consumption and income taxes – as defined in Lemma 1 – are
not quantitatively identical under different tax normalizations in the absence of distributional
concerns (ξz = ξc = 0), even though allocations are, see also Håkonsen (1998). The contributions
by Triest (1990) and Dahlby (2008) are related. If there are no distributional concerns, and
lump-sum taxes are ruled out, then M CFt = (1 − M EBt )−1 6= M CFτ = (1 − M EBτ )−1 ,
even if the Diamond concept of the social marginal value of private income is adopted. To
resolve this apparent inconsistency, Triest (1990), Håkonsen (1998) and Dahlby (2008) develop
alternative M CF concepts relying on correcting the M CF measures for the shadow values of
public resources (η), both before and after the introduction of distortionary taxes. However,
the explanation for the difference in the M CF measures is that both tax instruments have
equal marginal excess burdens in absolute terms, but these excess burdens are expressed as
fractions of different tax bases. Hence, the M EB measures differ quantitatively. However, once
distributional concerns are included, this issue has become moot, since marginal distortionary
costs and marginal distributional gains are equal, and the normalization of the costs or benefits
of the tax instrument with a particular tax base has become immaterial.
The next Proposition derives a special case in which the first-best Samuelson rule for public
goods provision is obtained in second-best settings with distortionary taxation.22
Proposition 5 If utility is given by u(n) ≡ c(n) − v(l(n)) + Γ(G), v 0 , Γ0 > 0, v 00 > 0, Γ00 ≤ 0,
then the optimal provision of public goods follows the first-best Samuelson rule in second-best
22
Note that in this particular case it does not matter how one defines the marginal cost of public funds as the
traditional and the Diamond measures of M CF are identical in the absence of income effects.
19
settings with optimal distortionary taxation:
Z
N
N
uG (·)
dF (n) = p.
uc (·)
(38)
Proof. The first-order condition for labor supply is given by v 0 (l(n)) =
εclG
εclG
= 0. Furthermore,
uG
uc
=
Γ0 (G)
(1−t)n
(1+τ ) , ∀n.
Therefore,
is independent from skill n, hence, ξG = 0. Substitution of
= 0 and ξG = 0 in equation (30) or equation (34) yields the result.
This simple, special case demonstrates why it is misleading to ignore distributional concerns
in the analysis of the marginal cost of public funds. When distributional concerns are ignored,
the cost side of the Samuelson rule would be multiplied with M CF , whereas the distributional
benefits of the distortionary tax would be left out of the picture. The explanation why the
first-best Samuelson rule is found in a second-best setting is twofold. First, the public good
does not increase or decrease compensated labor supply. Therefore, the public good cannot be
used to alleviate the tax distortions on labor supply. Second, the public good does not affect
the welfare distribution, since every individual benefits to an equal extent from the public good.
Indeed, the public good is a perfect substitute for the lump-sum cash transfer g. Since the
provision of the public good neither affects efficiency nor equity, provision of the public good
should follow the first-best policy rule.
3.3
Sub-optimal taxation
To conclude the discussion on linear taxation, this subsection explores to what extent the results
are driven by allowing for heterogeneous agents and non-individualized lump-sum transfers.
To that end, suppose that the government cannot optimize the lump-sum tax. Then, the
government has to resort to distortionary taxation as the marginal source of finance for public
goods. For brevity, this section only discusses income taxation. The following Proposition
derives optimal policy for any level of lump-sum taxation.23
Proposition 6 Under the Diamond based MCF definition, the lump-sum tax exogenously given,
and the consumption tax normalized to zero, the optimal rules for public goods provision and
the marginal cost of public funds are given by
Z
(1 − ξG )N
N
uG (·)
dF (n) = (1 − γt ε̄clG ) · M CFt · p,
uc (·)
1 − ξz
1 − ξz
M CFt =
t c = 1 − M EB R 1.
1 + 1−t ε̄lt
t
(39)
(40)
Proof. This follows from Proposition 3.
The modified Samuelson rule in equation (39) now features a correction (M CFt ) for the
marginal cost of public funds of the income tax. Since M CFt =
1−ξz
1−M EBt ,
a correction needs to
be made for both the excess burden of the income tax M EBt and the distributional benefits ξz
23
If the marginal source of public finance is the lump-sum tax, and the government jointly optimizes the lumpsum tax and public goods provision, for a given level of the income tax, then the marginal cost of public funds is
one. In that case, there should be no correction of the the modified Samuelson rule. This follows trivially from
Proposition 3.
20
of the income tax. These distributional benefits are omitted in models adopting a representative
agent. Clearly, when the lump-sum tax cannot be optimized, the marginal cost of public funds
ceases to be equal to one. However, even when lump-sum taxes are unavailable, one can not
conclude that the marginal cost of public funds for distortionary taxation is necessarily larger
than one. This depends on the distributional benefits of the distortionary tax. If the income tax
is sub-optimally low (high) from a distributional perspective (i.e., ξz > (<)M EBt ), then the
marginal cost of public funds is smaller (bigger) than one, i.e., M CFt < 1 (> 1), and optimal
public goods provision is larger (smaller), everything else equal. Intuitively, if M CFt < 1,
the government over-provides public goods – relative to the second-best rule with optimized
transfers – to compensate for the sub-optimal income redistribution by the income tax (and
vice versa if M CFt > 1).
The representative-agent models are nested as a special case of the model where lumpsum taxes are excluded and the distributional effects of taxation or public goods are absent
(ξz = ξG = 0). In that particular case, the marginal cost of public funds are unambiguously
larger than one, since M CFt =
1
1−M EBt
> 1 if ξz = 0. Indeed, the modified Samuelson rule
from Proposition 6 then collapses to the expression derived by Atkinson and Stern (1974). This
case, however, is of no practical relevance, since individuals in the real world are heterogeneous.
Hence, it would be incorrect to ignore the distributional effects of taxes and public goods
provision, and only count for the distortions of taxation and the interaction of the public good
with labor supply in social cost-benefit analysis.
If tax systems are not optimized because lump-sum taxes are not available, the standard
Samuelson rule in equation (39) should thus be adjusted for four factors in second-best settings
with distortionary taxation: i) interactions of the public good with distorted labor supply (ε̄clG ),
ii) distributional benefits of the public good (ξG ), iii) distortions of the financing of the public
good, i.e., the marginal excess burden of distortionary taxation (M EBt ), and iv) distributional
benefits of the financing of the public good (ξz ). If tax systems are optimized, the distributional
gains equal the deadweight losses of financing the public good, so that M EBt = ξz , and effects
iii) and iv) cancel from the modified Samuelson rule.
4
Optimal non-linear taxation and public goods provision
The analysis has so far been confined to linear policy instruments. In the real world, however,
most tax systems are non-linear. This section extends the model to allow for non-linear income
taxation, as in Mirrlees (1971). It is demonstrated that all major findings derived under linear
policies carry over to non-linear policies. Moreover, the formulae for optimal non-linear taxation
and optimal provision of public goods are expressed in terms of sufficient statistics so as to obtain
the closest correspondence with the corresponding expressions derived under linear instruments.
Finally, this section extends previous literature by analyzing optimal taxation and public goods
provision with preference heterogeneity.
21
4.1
Individuals
Let the non-linear tax schedule be denoted by T (z(n)), where T 0 (z(n)) ≡ dT (z(n))/dz(n)
denotes the marginal income tax rate at income z(n). The tax function is assumed to be
continuous and differentiable. The government can only verify total income of an individual
(z(n) = nl(n)), not its labor supply l(n) or ability n, which rules out individualized lump-sum
taxes. The individual budget constraint is modified:
c(n) = z(n) − T (z(n)),
∀n.
(41)
The first-order condition for labor supply of each individual reads as
−ul (c(n), l(n), G, n)
= (1 − T 0 (z(n)))n,
uc (c(n), l(n), G, n)
∀n.
(42)
This first-order condition is the same as before, see equation (3), except that the non-linear
marginal tax rate replaces the linear one.
4.2
Government
The social welfare function remains the same as in equation (4). In the Mirrlees setup it is more
convenient to work with the economy’s resource constraint rather than the government budget
constraint.24 The economy’s resource constraint is:
Z
(nl(n) − c(n))dF (n) = pG.
N
(43)
N
4.3
Incentive compatibility
To determine the non-linear tax schedule T (·) and optimal public goods provision G, a standard mechanism-design approach is employed. Any second-best allocation must satisfy the
incentive-compatibility constraints. Since n is not observable by the government, every bundle {c(n), z(n)} for individual n must be such that each individual self-selects into this bundle
and does not prefer another bundle {c(m), z(m)} intended for individual m 6= n. By adopting
the first-order approach, as in Mirrlees (1971), the incentive-compatibility constraints can be
summarized by a differential equation on utility:
du(c(n), l(n), G, n)
l(n)
= un (c(n), l(n), G, n) −
ul (c(n), l(n), G, n),
dn
n
∀n.
(44)
The first-order approach yields an implementable allocation only if the second-order conditions for utility maximization are fulfilled at the optimum allocation. Lemma 2 states the
conditions under which the first-order approach is both necessary and sufficient to describe the
optimum.25
24
If the government maximizes social welfare subject to the resource constraint, and all households respect
their budget constraints, then the government budget constraint is automatically satisfied by Walras’ law.
25
Lemma 2 implies that the second-order conditions are locally satisfied, which is a standard result in the
Mirrlees model without public goods provision, see also Hellwig (2010).
22
Lemma 2 The optimal allocation derived under the first-order approach is implementable with
the non-linear income tax function if i) utility U (c(n), z(n), G, n) ≡ u(c(n), z(n)/n, G, n) satisfies the Spence-Mirrlees condition, i.e., d(−Uz /Uc )/dn < 0, and ii) gross incomes are nondecreasing with skill n, i.e., dz(n)/dn ≥ 0.
Proof. See Jacobs (2010).
Since the public good is identical for all agents, incentive compatibility constraints are not
affected by introducing public goods, even if different individuals exhibit a different valuation
for the public good, see also Kreiner and Verdelin (2012). Intuitively, one individual cannot
mimic another individual’s ability to benefit from the public good, since his utility function
remains the same. In the remainder of this section it’s assumed that Lemma 2 holds.
4.4
Optimal taxation and public goods provision
To characterize the optimal second-best policy, Proposition 7 provides the conditions for the
optimal non-linear income tax, the marginal cost of public funds, and the optimal provision of
public goods. The optimal non-linear tax and modified Samuelson rule are expressed in terms
of sufficient statistics, i.e., the earnings distribution, the willingness to pay for public goods,
behavioral elasticities, and social welfare weights. The earnings distribution equals the skill
distribution, i.e., F̃ (z(n)) ≡ F (n), in view of the one-to-one mapping between earnings and
skills (Saez, 2001). The corresponding density of earnings equals f˜(z(n)).
Proposition 7 The optimal non-linear income tax, the marginal cost of public funds, and
optimal public goods provision are given by
T 0 (z(n))
1 − T 0 (z(n))
=
1
−εclT 0
R z(n) z(n)
1−
(1 − ξG )N
N
f˜(z(m))dz(m) 1 − F̃ (z(n))
1 − F̃ (z(n))
z(n)f˜(z(n))
,
η
= 1,
˜(z(m))dz(m)
α(m)
f
z(n)
Z
uG (·)
N T 0 (nl(n))nl(n)
c
dF (n) = p 1 −
γn εlG dF (n) , γn ≡
.
uc (·)
pG
N
M CF ≡ R z(n)
Z
α(m)
η
∀n,
(45)
(46)
(47)
Proof. See Appendix.
The expression for the optimal non-linear income tax in equation (45) is identical to Saez
(2001). This expression is well known in the literature and the reader is referred to Mirrlees
(1971), Seade (1977), Tuomala (1984), Diamond (1998) and Saez (2001) for more elaborate
discussions of the optimal non-linear tax.
At the optimum, the marginal cost of public funds in equation (46) is once again equal to
one. The formulation of optimal tax policy in terms of sufficient statistics formally demonstrates
that the marginal cost of public funds equals one also in the Mirrlees (1971) model. Intuitively,
the government has acces to a non-distortionary marginal source of public finance: the intercept
of the tax function T (0). The intercept −T (0) is equivalent to the non-individualized lump-sum
transfer g under linear taxation. The government lowers the intercept of the tax function T (0)
until the marginal utility gains plus income effects (α(n)) are on average equal to its marginal
23
cost in terms of revenue (η). When the tax system is optimized, the government is indifferent
to a marginal redistribution of income from the public to the private sector.
The marginal cost of public funds for non-distortionary taxes should be equal to the marginal
cost of public funds for all distortionary taxes. Thus, tax distortions should be equal to distributional gains for marginal tax rates at each point in the income distribution. This can be seen
by rewriting the expression for the optimal non-linear income tax in equation (45):
M EB(n) · z(n)f˜(z(n)) =
z̄
α(m) ˜
1−
f (z(m))dz(m),
η
z(n)
Z
M EB(n) ≡ −
(48)
T 0 (z(n)) c
ε 0.
1 − T 0 (z(n)) lT
Clearly, the total marginal excess burden of a higher marginal tax rate at earnings level z(n)
equals the marginal excess burden per unit of tax base M EB(n) multiplied by the tax base
f˜(z(n))z(n). The distributional gain of a higher marginal tax rate at income z(n) equals the
social value of extracting a marginal unit of revenue from all individuals above z(n). The latter
equals 1 – the unit of revenue – minus the Diamond (1975) social marginal value of income
α(n)/η – the utility cost in money equivalents of paying a unit more tax.
If marginal tax rates are not optimized, equation (48) does not hold with equality. In that
case, the marginal excess burden of a marginal tax rate is not equal to the marginal distributional
benefit of the marginal tax rate. Moreover, if equation (48) does not hold with equality at all
points in the income distribution, the marginal cost of public funds is no longer equal to one,
hR
i−1
z(n)
i.e., M CF ≡ η z(n) α(m)f˜(z(m))dz(m)
6= 1.
The expression for the optimal provision of public goods in equation (47) is identical to
expression (30) for optimal public goods provision under linear taxation. The only (minor)
difference is that the share parameters γn take into account the non-linearity of the tax schedule.
The interpretation of the modified Samuelson rule otherwise remains completely the same, and
the reader is referred to the previous discussion. Clearly, there does not need to be a correction
in the modified Samuelson rule for the marginal cost of public funds under optimal non-linear
income taxation, since M CF = 1 at the optimal tax system. The non-linear case therefore
confirms the linear case.
To gain more intuition for the reasons why the modified Samuelson rule differs from the
standard Samuelson rule, it can be expressed as well in terms of the model’s primitives (see
Appendix):
Z
N
(1 + ∆(n))
N
where
∆(n) ≡
T 0 (z(n)) εclT 0
1 − T 0 (z(n)) εuzn
uG (·)
dF (n) = p,
uc (·)
∂ ln(uG /uc ) ∂ ln(uG /uc )
−
∂ ln l(n)
∂ ln n
(49)
,
∀n.
(50)
The modified Samuelson rules in equations (47) and (49) are the same, except that equation
(47) is expressed in terms of sufficient statistics rather than the model’s primitives, i.e. the skill
distribution and the utility function (recall that 1 − T 0 (z(n)) = −ul (·)/uc (·)). ∆(n) indicates
the extent to which public goods are overprovided relative to the first-best Samuelson rule, cf.
equation (38). ∆(n) can be interpreted as an implicit subsidy on public goods provision at skill
24
level n. ∆(n) is different from zero if the marginal willingness to pay for the public good
uG (·)
uc (·)
varies with labor supply or with ability.
Whether public goods are under- or overprovided compared to the first-best rule is determined by the presence of the distortionary income tax (T 0 (z(n)) > 0). The more distortionary
is income taxation, as indicated by a larger tax rate T 0 (z(n)) or a larger compensated elasticity
εclT 0 , the more public goods provision deviates from the first-best policy rule. In the absence
of redistributional concerns, marginal tax rates are zero (T 0 (z(n)) = 0), and so is the implicit
subsidy on public goods provision (∆(n) = 0). In that case, public goods provision satisfies the
first-best Samuelson rule (38).
The public good is more (less) complementary to work than private consumption, if the
marginal willingness to pay for the public good rises (falls) with labor effort (i.e.,
∂ ln(uG /uc )
∂ ln l
>
0 (< 0)).26 The provision of public goods then increases. The intuition is the same as in
Atkinson and Stiglitz (1976). The government should provide more public goods if they are
more complementary to work than private goods are, and provide fewer public goods if they are
more complementary to leisure than private goods are. In doing so, the government alleviates
the distortions of the labor income tax on work effort. To put it differently, if
∂ ln(uG /uc )
∂ ln l
>0
(< 0), a higher (lower) level of public goods provision relaxes the incentive constraints associated
with the redistribution of income, as individuals with a high ability are less tempted to mimic
individuals with a lower ability. The Atkinson and Stern (1974) term capturing the interaction
of the public goods with labor supply is therefore also present under non-linear taxation. See
also Christiansen (1981) and Boadway and Keen (1993). The term
is therefore associated with the compensated cross-elasticity
εclG
∂ ln(uG /uc )
∂ ln l
in equation (49)
of labor supply with respect to
the public good in the modified Samuelson rule under linear taxation in equation (47).
If the utility function differs across individuals, public goods provision should also be employed for redistribution. Public goods provision can extract additional information on the skill
level n. Intuitively, the willingness to pay for public goods varies with ability even if all individuals would have the same labor earnings. Therefore, the willingness to pay for public goods
reveals information about earnings ability that is independent from labor income. This makes
over- or underprovision of public goods attractive for redistribution. Over- or under-provision
of public goods – relative to the first-best Samuelson rule – results in more redistribution than
the government can achieve with the non-linear income tax alone. Naturally, under- or overprovision causes inefficiencies, and these need to be traded off against the distributional gains. If
the marginal willingness to pay for the public good rises (falls) with ability (i.e.,
∂ ln(uG /uc )
∂ ln n
>0
(< 0)), then the public good benefits the individuals with higher skill levels relatively more
(less).27 Consequently, the optimal level of public goods provision falls (increases) for redistriG /uc )
l
The term ∂ ln(u
can be rewritten as ωl (ρlG − ρlc ), where ωl ≡ −lu
> 0 is the utility share of labor,
∂ ln l
u
ulG u
ρlG ≡ (−ul )uG is the Hicksian partial elasticity of complementarity between public goods and labor, and ρlc ≡
ulc u
is the Hicksian partial elasticity of complementarity between private consumption and labor. Thus, the
(−ul )uc
willingness to pay for the public good increases with labor supply if ρlG > ρlc . That is, if the public good is a
stronger Hicksian complement to labor than private consumption is.
27
G /uc )
can be rewritten as ωn (ρnG − ρnc ), where ωn ≡ nuun > 0 is the utility share of
The term ∂ ln(u
∂ ln n
u
ability, ρnG ≡ uunnG
is
the
Hicksian partial elasticity of complementarity between public goods and ability, and
uG
ρnc ≡ uunncuuc is the Hicksian partial elasticity of complementarity between private consumption and ability. The
willingness to pay for the public good increases with ability if ρnG > ρnc . That is, if public goods are stronger
Hicksian complements to ability than private goods are.
26
25
butional reasons. The term
∂ ln(uG /uc )
∂ ln n
in equation (49) is thus associated with the distributional
characteristic ξG of the public good in the modified Samuelson rule under linear taxation in
equation (47).
Alternatively, one can understand this result by looking at the incentive constraints. If
∂ ln(uG /uc )
∂ ln n
> 0 (< 0), then high-skill types are less (more) tempted to mimic low-skill types
if public goods provision expands, since they have a stronger (weaker) preference for public
goods. Consequently, incentive-compatibility constraints are relaxed (tightened), and public
goods provision reduces (increases) the distortions of redistributing income.
If the utility function is identical for all individuals n, the distributional term drops out (i.e.,
∂ ln(uG /uc )
∂ ln n
= 0). Intuitively, if differences in earning ability are the only source of heterogeneity,
and everyone has the same utility function, the willingness to pay for public good is the same if
labor earnings are the same. Therefore, differences in the willingness to pay for public goods only
originate from differences in labor earnings. In that case, under- or or overprovision of the public
good – relative to the first-best Samuelson rule – does not result in more income redistribution
than can be achieved with the non-linear income tax alone. However, under- or overprovision
of public goods – relative to the first-best rule – causes inefficiencies. These distortions can
be avoided by organizing all redistribution via the non-linear income tax and providing public
goods according to the Samuelson rule. See also Christiansen (1981) and Boadway and Keen
(1993) for the case of homogeneous preferences. Hence, when the utility function is the same
for all individuals, public goods are not provided for redistributional reasons.
The finding that public goods are not used for redistribution if utilities are identical contrasts
with linear case. Under linear income taxation, optimal provision of the public good is always
found to be dependent on the distributional impact of the public good, except for the (trivial)
cases in which the marginal willingness to pay is equal or is linear in income for all individuals.
With linear taxation, the government uses an informationally inferior instrument to redistribute
income by ignoring the information on individual earnings. Hence, the government optimally
complements the income tax by using indirect instruments for redistribution, such as public
goods provision.
Christiansen (1981), Boadway and Keen (1993) and Kaplow (1996) discuss the case in
which the utility function is identical for all individuals and is (weakly) separable between private/public goods and leisure. They find that the optimal provision of public goods then follows
the first-best Samuelson rule. The next Proposition demonstrates this formally. Naturally, the
allocations will differ between first- and second-best.
Proposition 8 If utility is given by u(n) ≡ u(h(c(n), G), l(n)), ∀n, then the optimal provision of public goods follows the first-best Samuelson rule in second-best settings with optimal
distortionary taxation:
Z
N
N
uG (·)
dF (n) = p.
uc (·)
Proof. If u(n) ≡ u(h(c, G), l), it is immediately established that
(51)
∂ ln(uG /uc )
∂ ln l
=
∂ ln(uG /uc )
∂ ln n
= 0.
Substitution in (49) yields (51).
Proposition 8 is the non-linear counterpart of Proposition 5. It demonstrates that, once
second-best interactions of the public good with labor supply are absent, or when redistribution
26
via public goods has no value added over direct income redistribution, the optimal rule for public
good provision follows the first-best Samuelson rule. Hence, there should be no correction for
the marginal cost of public funds as distortions of taxation cancel against the distributional
gains of taxation.
5
Discussion
5.1
Instrument set
This paper has demonstrated that ruling out non-individualized lump-sum taxes – as a shortcut for an unspecified distributional problem – has large and policy-relevant consequences for
the analysis of the marginal cost of public funds. However, some may question whether in the
real world governments have in fact access to non-distortionary lump-sum taxes as a marginal
source of public finance. One may argue that ruling out non-individualized lump-sum taxes is
in fact a realistic feature due to practical or even legal problems in implementing such taxes.
However, from a theoretical point of view, the lump-sum part of the tax system generally
consists of subsidies (i.e., transfers) to individuals, since the intercept of the tax function is
optimally negative. Therefore, even if one likes to rule out non-individualized lump-sum taxes
as a marginal source of public finance, it is certainly feasible to marginally reduce lump-sum
transfers to the private sector to finance public goods. Moreover, virtually all real-world tax
systems entail a general tax credit or a general tax exemption, which works as a lump-sum
subsidy and can be (and often is) changed by policy makers without running into all kinds
of practical or legal obstacles. Hence, this paper’s assumption that non-individualized lumpsum transfers are a marginal source of public finance can be defended on both theoretical and
empirical grounds.
5.2
Kaplow and sub-optimal taxation
Kaplow (1996), Kaplow (2004) and Kaplow (2008) argue that even if the tax system is not
optimal, neither distributional concerns nor incentive effects of the financing of public goods
with distortionary taxes should be included in the discussion of second-best policy analysis.
Laroque (2005) and Gauthier and Laroque (2009) contain similar ideas and provide rigorous
proofs of this claim. In this argument, the government simultaneously does two things: i) it
changes the provision of the public good, and ii) it applies a benefit-absorbing change in the
non-linear tax schedule that fully extracts each individual’s willingness to pay for the public
good. Given that the utility function is identical across agents and weakly separable, this tax
change does not affect labor supply incentives. Hence, the tax adjustment perfectly imitates a
pure benefit tax. Consequently, the traditional Samuelson rule can be used to judge whether
public goods provision should increase or not, without making corrections for the marginal cost
of public funds.
Kaplow’s approach has a number of disadvantages. First, preferences must be weakly separable and identical, otherwise the benefit-absorbing change in the non-linear tax schedule is
generally not incentive compatible, and cannot be implemented as a result, see also Laroque
27
(2005) and Gahvari (2006). Consequently, Kaplow’s approach cannot be generalized to nonseparable or heterogenous preferences. Second, while the tax system is not assumed to be
optimal, the non-linear income tax schedule is flexible enough to perfectly off-set any distributional effect of public good provision at each income level. However, if there are binding
constraints that make non-linear tax schedule sub-optimal, the very same constraints may prevent the implementation of the benefit-absorbing tax changes. Hence, it may not be possible
to implement the benefit-absorbing tax change in the non-linear tax schedule. Third, Kaplow’s
analysis cannot be readily generalized to linear tax schedules, because linear taxes generally do
not neutralize all distributional effects of public goods – except in the knife-edge case where
the benefits of public goods are proportional in income. Fourth, in real-world policy making,
changes in the provision of public goods generally occur without neutralizing the distributional
effects with benefit-absorbing changes in the tax system (Gahvari, 2006).
This paper avoided these disadvantages by analyzing public good provision and income
taxation with non-separable and heterogeneous preferences, allowing for both linear and nonlinear tax schedules, and without adjusting the (linear or non-linear) tax schedules to fully
neutralize the distributional impact of public goods. It demonstrated that the marginal cost of
public funds should not be present in the modified Samuelson rule for public goods provision.
Intuitively, if the tax system is optimized, a marginal change in distortionary taxes to finance
public goods produces exactly offsetting distortions and distributional gains, which cancel out in
the modified Samuelson rule. This is an application of the envelope theorem for optimal taxes,
which also explains why this result generalizes to non-seperable and heterogeneous preferences,
linear and non-linear tax systems, and without the need to implement benefit-absorbing tax
changes. Consequently, the marginal cost of public funds should not play a role in the Samuelson
rule for public goods provision if taxes are optimized.
5.3
Implications for other public policies
The results of this paper have larger relevance to other public policies than public goods provision. First, Jacobs and Boadway (2014) analyze optimal linear or non-linear commodity taxes
jointly with optimal non-linear income taxes in the frameworks of Atkinson and Stiglitz (1976)
and Mirrlees (1976). They show that the marginal cost of public funds – based on Diamond’s
social value of income – remains equal to one in the full tax optimum. Hence, distortions from
commodity taxation do not drive up the marginal cost of public funds above one when income
and commodity taxes are optimized.
Second, Kleven and Kreiner (2006) extend the literature on the marginal cost of public
funds – based on the traditional definition – to account for extensive-margin distortions in
labor supply, besides intensive-margin distortions. They argue that participation distortions
tend to raise the marginal cost of public funds. Jacquet, Lehmann, and Van der Linden (2013)
merge the Mirrlees model of optimal income taxation with labor supply on the intensive margin
with Diamond (1980)’s model of optimal income taxation with labor supply on the extensive
margin. Zoutman, Jacobs, and Jongen (2016) follow their analysis and demonstrate that in
this model the marginal cost of public funds is once more equal to one in the tax optimum if
the Diamond-based measure for the marginal costs of public funds is employed. Participation
28
distortions are therefore not a reason why the marginal cost of public funds would rise above
one in the tax optimum.
Third, Sandmo (1975), Bovenberg and de Mooij (1994) and Bovenberg and van der Ploeg
(1994) show that the optimal corrective tax is driven below the Pigouvian level if the marginal
cost of public funds is larger than one. Intuitively, providing environmental quality directly
competes with provision of ordinary public goods. Jacobs and de Mooij (2015) analyze optimal
corrective taxes alongside optimal linear and non-linear income taxes in the models of Atkinson
and Stiglitz (1976) and Mirrlees (1976), which are extended with externalities. They demonstrate that the marginal cost of public funds is once more equal to one in the tax optimum with
optimal corrective taxation. They derive that it is generally incorrect to set optimal corrective
taxes below Pigouvian levels, because the marginal cost of public funds is one. Consequently,
governments should not pursue less ambitious environmental policies if tax rates are high.
Fourth, Barro (1979) and Lucas and Stokey (1983) show that under distortionary taxation
Ricardian equivalence breaks down, and debt-financing and tax-financing of public spending
cease to be equivalent. Since tax distortions are convex in tax rates, it is better to smooth
tax rates over time, rather than having time-varying tax rates. Therefore, public debt should
optimally be used to smooth tax rates over time. Werning (2007), however, demonstrates that
ignoring distributional concerns to motivate tax distortions has fundamental consequences for
the theory of optimal debt management. The reason is that, at the margin, the government
always has access to non-distortionary sources of finance (i.e., non-individualized lump-sum
transfers). Werning (2007)’s analysis implies that the marginal cost of public funds is equal
to one at all times if governments optimize tax systems, see also Jacobs (2009). As a result,
debt and (lump-sum) tax financing are equivalent, Ricardian equivalence is restored, and the
optimal path of public debt becomes indeterminate.
Fifth, if the marginal cost of public funds is equal to one, revenue-raising instruments (such
as taxes) are not superior to revenue-neutral instruments (such as regulation), or to revenuereducing instruments (such as subsidies). The reason is that the fundamental informational
constraint in second-best analysis – ability is private information – cannot be relaxed by simply adopting a different policy implementation. Many indirect mechanisms may exist that can
sustain the second-best optimal allocation derived from an incentive-compatible direct mechanism. Quantity controls, regulation or auctions may achieve the second-best optimal allocation
equally as well as price instruments. Hence, one needs to include additional, instrument-specific
constraints into the analysis in order to assess the desirability of revenue-raising over revenueneutral or revenue-reducing policy instruments.
5.4
Social cost-benefit analysis
In social cost-benefit analysis it is probably best not to make corrections for the marginal cost
of public funds. Tax distortions cancel out against distributional gains if taxes are optimally
set, which renders the marginal cost of public funds equal to one.28 Alternatively, if one does
not want to assume that taxes are optimally set, then not only the deadweight costs of taxation,
28
This does not mean that tax distortions do not affect the allocations, and thereby affect both the costs and
the benefits of a project, however.
29
but also the distributional benefits of taxation should be included in social cost-benefit analysis.
A common practice to add, say, 50 cents for every dollar spent on a public project to account
for tax distortions, but completely ignoring the distributional benefits of tax instruments at the
same time, is incorrect. Indeed, too few public projects would then pass the social cost-benefit
test.
To justify setting the marginal cost of public funds equal to one in social cost-benefit analysis, policy makers may invoke Becker (1983)’s efficient redistribution hypothesis, which argues
that the political system should achieve an outcome in which all opportunities for political gains
to redistribute incomes are exhausted. Alternatively, one may invoke the principle of insufficient reason, since the calculation of deadweight costs and distributional benefits of taxation is
fraught with difficulties. In particular, if tax systems are not considered to be optimal, policy
analysts should estimate both the marginal distortions and the marginal distributional benefits of taxation. It is in principle possible to estimate the deadweight losses of taxation, even
though uncertainties in deadweight loss estimates can be substantial. The estimation of the
distributional benefits of taxation, however, is a hazardous task for any policy analyst, since
this inevitably involves political judgements about the desirability of income redistribution from
which policy analysts should preferably abstain. Indeed, any policy analyst claiming that the
marginal cost of public funds is necessarily larger than one is merely revealing his/her political
preferences for income redistribution: apparently the deadweight losses of taxation are always
considered to be larger than his/her perceived benefits of redistribution.
6
Conclusions
This paper analyzed the simultaneous setting of optimal taxes and the provision of public goods
in standard optimal tax models with heterogeneous agents having heterogeneous preferences.
Both optimal linear and non-linear tax schedules are analyzed. It has been demonstrated that
tax distortions are the ultimate result of redistributional concerns. Using Diamond’s definition
for the social marginal value of private income, which includes policy-induced income effects on
tax bases, the marginal cost of public funds at the optimal tax system is shown to be one, for
optimal linear and non-linear taxes, and for income and consumption taxes. At the optimal tax
system, the marginal cost of public funds for all tax instruments should be equalized. Hence, the
marginal cost of public funds for distortionary taxation equals the marginal cost of public funds
for non-distortionary taxation, which equals one. The distributional benefits of distortionary
tax rates are therefore equal to the marginal excess burden of tax rates. The modified Samuelson
rule is derived using general preference structures. It is demonstrated that the marginal cost
of public funds does not determine optimal public goods provision under optimal taxation.
Modified Samuelson rules for public good provision under linear and non-linear taxation are
identical.
Applied policy economists should use the marginal excess burden as the social cost of income
redistribution, and the marginal cost of public funds as the social cost of financing public activities that are unrelated to income redistribution. It should be remembered that the marginal
cost of public funds of a distortionary tax does not only capture the deadweight loss of a dis-
30
tortionary tax, but also its distributional benefits. When tax systems are not optimal, the
marginal cost of public funds could either be smaller or larger than one, depending on whether
the distributional benefits are smaller than the marginal excess burden of distortionary taxes.
This paper followed common practice in the second-best literature by assuming that the
government is a benevolent social planner and markets are Pareto-efficient in the absence of
government intervention. Government failure could explain why taxes and public goods are not
set at second-best optimal levels. One may therefore be tempted to conclude that government
failure raises the marginal cost of public funds above one. Moreover, not only governments, but
also markets can fail. The main lesson of this paper is that without explicitly incorporating
the fundamental reasons why governments or markets fail into the analysis, it is premature to
conclude that the marginal cost of public funds lies above or below one. In future research, the
marginal cost of public funds should be derived in settings where market or government failure
is explicitly modeled from first principles and not derived from ad hoc constraints.
One could also extend the model to allow for migration of individuals, resulting in participation constraints as in, for example, Hellwig (2004) or Lehmann, Simula, and Trannoy (2014).
This is a large deviation from the assumptions used in the standard literature on the marginal
cost of public funds. Nevertheless, even lump-sum taxes/transfers are probably no longer neutral from an efficiency point of view. In particular, a marginal reduction in the transfer induces
some individuals to migrate. It can be conjectured that the marginal cost of public funds is no
longer one at the optimal tax system if no non-distortionary government instrument exists at
all. Future research could explore this further.
A
Proof Proposition 7
A.1
Elasticities
First, the behavioral elasticities of model are derived. These elasticities will be used later in
deriving the optimal tax expressions. As in Jacquet, Lehmann, and Van der Linden (2013),
define the following shift functionm – omitting the indices n:
L(l, G, n, τ, ρ) ≡ n(1 − T 0 (nl) − τ )uc (nl − T (nl) − τ (nl − nl(n)) + ρ, l, G, n)
+ ul (nl − T (nl) − τ (nl − nl(n)) + ρ, l, G, n).
(52)
L(l, n, τ, ρ, G) measures a shift in the first-order condition for labor supply when one of the
variables l, G, n, τ , or ρ changes. τ captures an exogenous increase in the marginal tax rate
(i.e., for any level of earnings). ρ is introduced to retrieve the income effect when the individual
receives an exogenous amount of income ρ. The first-order condition for labor supply of the
individual n is thus equivalent to L(l, G, n, 0, 0) = 0.
The following partial derivatives of the shift function L are found, using the first-order
31
condition −ul = n(1 − T 0 )uc :
2
ul
T 00
Ll (l, G, n, 0, 0) = ull +
ucc − 2
ucl + nul
,
(53)
uc
(1 − T 0 )
2
!
T 00
ul
ul
l
uln ucn
−ul
+ nul
+
ucc −
ulc
+
ul , (54)
Ln (l, G, n, 0, 0) =
−
l
(1 − T 0 )
uc
uc
n
ul
uc
ul
uc
Lτ (l, G, n, 0, 0) = −nuc ,
ulc uc − ul ucc
Lρ (l, G, n, 0, 0) =
.
uc
(55)
(56)
The envelope theorem gives the following formula for the partial derivatives:
∂l
∂q
L
= − Lql , q =
n, τ, ρ.
The uncompensated wage elasticity of labor supply εuln is equal to:
εuln
∂lu n
Ln n
≡
=−
=−
∂n l
Ll l
−ul
l
2
ul
ul
nuln
T 00
+
u
−
+ nul (1−T
0)
cc
uc
uc ulc +
ul −
2
T 00
ull + uucl ucc − 2 uucl ucl + nul (1−T
0)
nucn
uc
ul
l
.
(57)
The income elasticity of labor supply εlρ is defined as:
ul ucc
u
−
lc
uc
Lρ
∂l
εlρ ≡ (1 − T 0 )n
= −(1 − T 0 )n
=
.
2
∂ρ
Ll
T 00
ull + uucl ucc − 2 uucl ucl + nul (1−T
0)
ul
uc
(58)
The compensated wage elasticity of labor supply εcln is defined residually by the Slutsky
equation (εcln ≡ εuln − εlρ ):
εcln
nuln
nucn ul
T 00
−
nu
−
−
l (1−T 0 )
ul
uc
l
n
≡
= εuln − εlρ =
.
2
∂n l
T 00
ull + uucl ucc − 2 uucl ucl + nul (1−T
0)
ul
l
∂lc
(59)
The compensated tax elasticity of labor supply εclT 0 is:
εclT 0 ≡ −
∂lc (1 − T 0 )
Lτ (1 − T 0 )
=
=
∂τ
l
Ll
l
ull +
ul
uc
2
ul /l
.
T 00
ucc − 2 uucl ucl + nul (1−T
0)
(60)
Note that the compensated tax elasticity of earnings supply εczT 0 equals the compensated tax
elasticity of labor supply (since the wage rate n is not affected by an increase in marginal taxes,
only labor supply).
The uncompensated wage elasticity of earnings supply εuzn is equal to:
εuzn ≡
∂z n
= 1 + εuln
∂n z
ul
1 + luulll − luuccl − nuulln − nuuccn
l
=
.
2
T 00
ull + uucl ucc − 2 uucl ucl + nul (1−T
0)
32
(61)
A.2
Optimal taxation and public goods provision
Consumption is defined as a function of the allocation: c(n) ≡ c(l(n), u(n), G, n), which is
obtained by inverting the utility function u(c(n), l(n), G, n). The derivatives of c(·) are found
by applying the implicit function theorem: cu = 1/uc , cl = −ul /uc , cG = −uG /uc , cn = −un /uc .
For notational compactness the incentive compatibility constraint can be rewritten as:
du(n)
l(n)
= un (c(l(n), u(n), G, n), l(n), G, n) −
ul (c(l(n), u(n), G, n), l(n), G, n)
dn
n
≡ ϕ(l(n), u(n), G, n),
(62)
where the function ϕ(l(n), u(n), G, n) has partial derivatives – omitting the n-indices:
ul
lull lulc
nunl nunc
ϕl = −
1+
−
,
−
−
n
ul
uc
ul
uc
1 nunc lulc
ϕu =
−
,
n
uc
uc
uG
l
uG
ϕG = unG − unc
−
ulG − ulc
.
uc
n
uc
(63)
(64)
(65)
The Hamiltonian for maximizing social welfare is:
H = Ψ(u(n))f (n) + η [nl(n) − c(l(n), u(n), G, n) − pG/N ] f (n) − θ(n)ϕ(l(n), u(n), G, n), (66)
where θ(n) is the co-state variable associated with state variable u(n), which has to satisfy
incentive compatibility constraint (62). Furthermore, η denotes the Lagrange multiplier on the
economy’s resource constraint in equation (43). θ(n) is multiplied with a minus sign to obtain
a positive multiplier θ(n).
The first-order conditions for an optimal allocation are – omitting the n-indices:
∂H
ul
lull lulc
θul
nunl nunc
=η n+
f+
1+
−
−
−
= 0,
∂l
uc
n
ul
uc
ul
uc
Z
N
∀n,
(67)
∂H
η
θ nunc lulc
dθ
= Ψ0 −
, ∀n,
(68)
f−
−
=
∂u
uc
n
uc
uc
dn
Z ∂H
uG
p
θuG
nunG nunc
lulG lulc
dn =
−
−
−
η
f−
−
dn = 0, (69)
∂G
uc
N
n
uG
uc
uG
uc
N
where the derivatives of the c(·) function are used in each line. The transversality conditions
for this optimal control problem are given by limn→n θ(n) = 0, and limn→n θ(n) = 0.
A.2.1
Optimal non-linear taxes
The first-order conditions for l in equation (67) can be rewritten using
−ul
uc
= (1 − T 0 (z))n:
T0
uc θ/η
lull lulc
nunl nunc
=
1+
−
−
−
.
1 − T0
nf (n)
ul
uc
ul
uc
33
(70)
Next, employ the definitions for the elasticities for εclT 0 in equation (60) and εuzn in equation
(61) to find:
1+
lull lulc
−
ul
uc
−
nunl nunc
−
ul
uc
=
εuzn
.
εclT 0
(71)
Hence, the first-order condition for optimal taxes can be written as:
T 0 (z(n))
uc (·)θ/η εuzn
.
=
1 − T 0 (z(n))
nf (n) εclT 0
(72)
The first-order condition for u in equation (68) can be rewritten as:
Ψ0 uc
θ
θ lulc
uc dθ
− 1 f (n) − ucn +
=
.
η
η
η n
η dn
(73)
Introduce the composite multiplier Θ(n):
Θ(n) ≡
θ(n)uc (c(n), l(n), G, n)
θ(n)uc (c(l(n), u(n), G, n), l(n), G, n)
=
,
η
η
(74)
which has total derivative (note that G does not vary with n):
dθ uc θucc
dΘ
=
+
dn
dn η
η
∂c dl
∂c du
∂c
+
+
∂l dn ∂u dn ∂n
θ
dl
θ
+ ucl
+ ucn .
η dn η
(75)
Simplify the last expression using the derivatives of the c-function and substituting equation
(62) for
du
dn :
dθ uc
θ
dΘ
=
−
dn
dn η
η
l ul ucc
− ucn
n uc
θ
−
η
ul ucc
− ucl
uc
dl
.
dn
(76)
Use the elasticity εlρ in equation (58) and the elasticity εclT 0 in equation (60) to find an expression
for
ucc ul
uc
− ucl :
εlρ uc
ul ucc
− ulc = − c
.
uc
εlT 0 l
(77)
Substitute equation (77) into equation (76), and use εuln from equation (57) to derive:
dΘ
dθ uc
θ l ucc ul
θ
θ εlρ uc u
=
−
+ ucn +
ε .
dn
dn η
η n uc
η
η εclT 0 n ln
(78)
Substitute the first-order condition for u in equation (73) into equation (78) to find:
dΘ θ εlρ uc u
−
ε =
dn
η εclT 0 n ln
Ψ0 uc
θ ucc ul
l
− 1 f (n) −
− ulc
.
η
η
uc
n
(79)
Substitute equation (77) in equation (79) and note that 1 + εuln = εuzn from equations (57) and
(61) to find:
dΘ uc θ εuzn ul
−
εlρ =
dn
ul η εclT 0 n
Ψ0 uc
− 1 f (n).
η
(80)
Rewrite first-order condition for zn in equation (72):
T 0 nf (n) = −
34
θ ul εuzn
.
η n εclT 0
(81)
Substitute equation (81) in (80) to find:
dΘ
=
dn
Ψ0 uc
T0
+
ε
−
1
f (n).
lρ
η
1 − T0
(82)
Equation (82) can be integrated – using a transversality condition – to obtain:
θ(n)uc (c(n), l(n), G, n)
Θ(n) =
=
η
Ψ0 (u(m))uc (m)
T 0 (z(m))
1−
−
εlρ f (m)dm.
η
1 − T 0 (z(m))
(83)
n
Z
n
The ABC-formula for the optimal non-linear tax results upon substituting equation (83) in
equation (72):
T 0 (z(n))
1 − T 0 (z(n))
=
εuzn
εclT 0
Rn
n
1−
Ψ0 (u(m))uc (m)
η
−
T 0 (z(m))
1−T 0 (z(m)) εlρ
1 − F (n)
f (m)dm 1 − F (n)
nf (n)
.
(84)
Finally, the optimal tax formula can be written in terms of the earnings density as in Saez
(2001) and Jacquet, Lehmann, and Van der Linden (2013). Define the earnings distribution as
F̃ (z(n)) ≡ F (n), with corresponding density function f˜(z(n)). Then, it can be derived that
εuzn f˜(z(n))z(n) = f (n)n. By noting that there is a perfect mapping between earnings z(n) and
ability n the expression for the optimal income tax is written as:
T 0 (z(n))
1 − T 0 (z(n))
=
1
R z(n) z(n)
1 − g(m) −
εclT 0
T 0 (z(m))
1−T 0 (z(m)) εlρ
f˜(z(m))dz(m) 1 − F̃ (z(n))
1 − F̃ (z(n))
z(n)f˜(z(n))
.
(85)
0
T (z(n))
∂l(n)
0
0
Using the definition of α(n) ≡ Ψ0 (u(n))λ(n)+η 1−T
0 (z(n)) εlρ = Ψ (u(n))λ(n)+ηnT (z(n)) ∂(−T (0)) ,
where the second step follows from εlρ = εl(−T (0)) , this can be rewritten as the expression in
the main text.
A.2.2
Marginal cost of public funds
0
T (z(n))
Use α(n) ≡ Ψ0 (u(n))λ(n) + η 1−T
0 (z(n)) εlρ and the transversality condition θ(n) = 0 in equation
(83) to derive:
Z
n
(1 − α(m)) f (m)dm.
0=
(86)
n
Hence, switching the bound of the integral, the marginal cost of public funds equals unity in
the policy optimum:
M CF ≡ R z(n)
z(n)
A.2.3
η
= 1.
α(m)f˜(z(m))dm
(87)
Optimal public goods provision
In order to express the optimal formula for public goods provision in terms of sufficient statistics, the optimal level of public goods is evaluated along an optimized non-linear tax system
T ∗ (nl(n)). The Lagrangian for maximizing social welfare with respect to the public good can
35
then be formulated as:
Z
L≡N
Z
∗
T (nl(n))dF (n) − pG ,
Ψ(v(n))dF (n) + η N
N
(88)
N
where v(n) is indirect utility of the individual. The first-order condition for G is given by:
∂L
=N
∂G
Z N
∂l(n)
Ψ (v(n))uG + ηnT (nl(n))
∂G
0
0
dF (n) − pη = 0,
(89)
where the derivatives of indirect utility have been substituted. Substituting the Slutsky equation
∂lu (n)
∂G
∂lc (n)
∂G
=
+
uG ∂l(n)
λ(n) ∂ρ
Z
N
N
yields:
α(n) uG
dF (n) + N
η uc
Z N
∂lc (n)
nT (nl(n))
∂G
0
dF (n) = p,
(90)
Use the Feldstein characteristic for the public good ξG to derive:
Z
uG (·)
dF (n) =
uc (·)
(1 − ξG )
N
Z
N
α(n) uG (·)
dF (n).
η uc (·)
(91)
Hence, optimal provision of the public good follows from:
Z
uG (·)
dF (n) = p 1 −
γn εclG dF (n) ,
uc (·)
N
Z
(1 − ξG )N
N
where γn ≡
(92)
N T 0 (nl(n))nl(n)
.
pG
Alternatively, one can derive the optimal rule for public goods provision in terms of the
model’s primitives, as in Jacobs (2010). Derive that
∂ ln(uG /uc )
.
∂ ln l
Z
N
N
nunG
uG
− nuucnc =
∂ ln(uG /uc )
∂ ln n
lulG
uG
− luuclc =
dF (n).
(93)
and
Substitute this in equation (69), and rearrange to obtain:
uG (·)
dF (n) = p + N
uc (·)
Z
N
θ(n)uG (·)/η
nf (n)
∂ ln(uG /uc ) ∂ ln(uG /uc )
−
∂ ln n
∂ ln l(n)
Substitute equation (72) for the optimal income tax in equation (93) to find:
Z
N
N
where
uG (·)
(1 + ∆(n))dF (n) = p,
uc (·)
T 0 (z(n)) εclT 0
∆(n) ≡
1 − T 0 (z(n)) εuzn
∂ ln(uG /uc ) ∂ ln(uG /uc )
−
∂ ln l(n)
∂ ln n
(94)
.
(95)
References
Allgood, Sam and Arthur Snow. 1998. “The Marginal Cost of Raising Tax Revenue and Redistributing Income.” Journal of Political Economy 106 (6):1264–1273.
Atkinson, Anthony B. and Nicholas H. Stern. 1974. “Pigou, Taxation and Public Goods.”
Review of Economic Studies 41 (1):119–128.
36
Atkinson, Anthony B. and Joseph E. Stiglitz. 1976. “The Design of Tax Structure: Direct
versus Indirect Taxation.” Journal of Public Economics 6 (1-2):55–75.
Ballard, Charles L. and Don Fullerton. 1992. “Distortionary Taxes and the Provision of Public
Goods.” Journal of Economic Perspectives 6 (3):117–131.
Barro, Robert J. 1979. “On the Determination of the Public Debt.” Journal of Political
Economy 87 (5):940–971.
Becker, Gary S. 1983. “A Theory of Competition Among Pressure Groups for Political Influence.” Quarterly Journal of Economics 98 (3):371–400.
Blundell, Richard and Thomas MaCurdy. 1999. “Labor Supply: A Review of Alternative
Approaches.” In Handbook of Labor Economics, vol. 3A, edited by Orley Ashenfelter and
David Card, chap. 27. Elsevier, 1559–1695.
Boadway, Robin and Michael Keen. 1993. “Public Goods, Self-Selection and Optimal Income
Taxation.” International Economic Review 34 (3):463–478.
Bovenberg, A. Lans and Ruud A. de Mooij. 1994. “Environmental Levies and Distortionary
Taxation.” American Economic Review 84 (4):1085–1085.
Bovenberg, A. Lans and Frederick van der Ploeg. 1994. “Environmental Policy, Public Finance
and the Labour Market in a Second-Best World.” Journal of Public Economics 55 (3):349–
390.
Browning, Edgar K. 1976. “The Marginal Cost of Public Funds.” Journal of Political Economy
84 (2):283–298.
———. 1987. “On the Marginal Welfare Cost of Taxation.” American Economic Review
77 (1):11–23.
Browning, Edgar K. and William R. Johnson. 1984. “The Trade-Off Between Equity and
Efficiency.” Journal of Political Economy 92 (2):175–203.
Christiansen, Vidar. 1981. “Evaluation of Public Projects under Optimal Taxation.” Review of
Economic Studies 48 (3):447–457.
Dahlby, Bev. 1998. “Progressive Taxation and the Social Marginal Cost of Public Funds.”
Journal of Public Economics 67 (1):105–122.
———. 2008. The Marginal Cost of Public Funds: Theory and Applications. Cambridge-MA:
MIT Press.
Diamond, Peter A. 1975. “A Many-Person Ramsey Tax Rule.” Journal of Public Economics
4 (4):335–342.
———. 1980. “Income Taxation with Fixed Hours of Work.” Journal of Public Economics
13 (1):101–110.
37
———. 1998. “Optimal Income Taxation: An Example with a U-Shaped Pattern of Optimal
Marginal Tax Rates.” American Economic Review 88 (1):83–95.
Diamond, Peter A. and James A. Mirrlees. 1971. “Optimal Taxation and Public Production I:
Production Efficiency.” American Economic Review 61 (1):8–27.
Feldstein, Martin. 1972. “Distributional Equity and the Optimal Structure of Public Prices.”
American Economic Review 62 (1):32–36.
Gahvari, Firouz. 2006. “On the Marginal Cost of Public Funds and the Optimal Provision of
Public Goods.” Journal of Public Economics 90 (6-7):1251–1262.
———. 2014. “Second-Best Pigouvian Taxation: A Clarification.” Environmental and Resource
Economics 59 (4):525535.
Gauthier, Stephane and Guy R. Laroque. 2009. “Separability and Public Finance.” Journal of
Public Economics 93 (11-12):1168–1174.
Håkonsen, Lars. 1998. “An Investigation into Alternative Representations of the Marginal Cost
of Public Funds.” International Tax and Public Finance 5 (3):329–343.
Harberger, Arnold C. 1964. “Taxation, Resource Allocation, and Welfare.” In The Role of Direct
and Indirect Taxes in the Federal Revenue System. Princeton University Press for NBER and
the Brookings Institution, 25–70.
Heckman, James J., Seong Hyeok Moon, Rodrigo Pinto, Peter A. Savelyev, and Adam Yavitz.
2010. “The Rate of Return to the High Scope Perry Preschool Program.” Journal of Public
Economics 94 (1-2):114–128.
Hellwig, Martin. 2004. “Optimal Income Taxation, Public Goods Provision, and Public Sector
Pricing. A Contribution to the Foundations of Public Economics.” MPI Discussion Paper
2004/14, Bonn: Max Planck Institute for Research on Collective Goods.
———. 2010. “Incentive Problems with Unidimensional Hidden Characteristics: A Unified
Approach.” Econometrica 78 (4):1201–1237.
Jacobs, Bas. 2009. “The Marginal Cost of Public Funds and Optimal Second-Best Policy Rules.”
Erasmus University Rotterdam.
———. 2010. “The Marginal Cost of Public Funds is One.” CESifo Working Paper No. 3250,
Munich: CESifo.
Jacobs, Bas and Robin Boadway. 2014. “Optimal Linear Commodity Taxation under Optimal
Non-Linear Income Taxation.” Journal of Public Economics 117 (1):201–210.
Jacobs, Bas and Ruud A. de Mooij. 2015. “Pigou Meets Mirrlees: On the Irrelevance of Tax
Distortions for the Optimal Corrective Tax.” Journal of Environmental Economics and Management 71:90–108.
38
Jacquet, Laurence, Etienne Lehmann, and Bruno Van der Linden. 2013. “The Optimal Marginal
Tax Rates with both Intensive and Extensive Responses.” Journal of Economic Theory
148 (5):1770–1805.
Kaplow, Louis. 1996. “The Optimal Supply of Public Goods and the Distortionary Cost of
Taxation.” National Tax Journal 49 (4):513–533.
———. 2004. “On the (Ir)relevance of Distribution and Labor Supply Distortion to Public
Goods Provision and Regulation.” Journal of Economic Perspectives 18 (4):159–175.
———. 2008. The Theory of Taxation and Public Economics. Princeton and Oxford: Princeton
University Press.
Kleven, Henrik Jacobsen and Claus Thustrup Kreiner. 2006. “The Marginal Cost of Public
Funds: Hours of Work versus Labor Force Participation.” Journal of Public Economics
90 (10-11):1955–1973.
Kreiner, Klaus Thustrup and Nicolai Verdelin. 2012. “Optimal Provision of Public Goods: A
Synthesis.” Scandinavian Journal of Economics 114 (2):384–408.
Laffont, Jean-Jacques and Jean Tirole. 1993. A Theory of Incentives in Procurement and
Regulation. Cambridge-MA: MIT-Press.
Laroque, Guy R. 2005. “Indirect Taxation is Superfluous under Separability and Taste Homogeneity: A Simple Proof.” Economics Letters 87 (1):141–144.
Lehmann, Etienne, Laurent Simula, and Alain Trannoy. 2014. “Tax Me If You Can! Optimal
Nonlinear Income Tax Between Competing Governments.” Quarterly Journal of Economics
129 (4):1995–2030.
Lucas, Robert E. jr. and Nancy Stokey. 1983. “Optimal Fiscal and Monetary Policy in an
Economy without Capital.” Journal of Monetary Economics 12 (1):55–93.
Lundholm, Michael. 2005. “Cost-Benefit Analysis and Marginal Cost of Public Funds.” Mimeo
Stockholm University.
Meghir, Costas and David Phillips. 2010. “Labour Supply and Taxes.” In The Mirrlees Review
– Dimensions of Tax Design, edited by James A. Mirrlees, Stuart Adam, Timothy J. Besley,
Richard Blundell, Steven Bond, Robert Chote, Malcolm Gammie, Paul Johnson, Gareth D.
Myles, and James M. Poterba, chap. 3. Oxford: Oxford University Press, 202–274.
Mirrlees, James A. 1971. “An Exploration in the Theory of Optimum Income Taxation.” Review
of Economic Studies 38 (2):175–208.
———. 1976. “Optimal Tax Theory: A Synthesis.” Journal of Public Economics 6 (4):327–358.
Pigou, Arthur C. 1947. A Study in Public Finance - 3rd Edition. London: Macmillan.
Ramsey, Frank P. 1927. “A Contribution to the Theory of Taxation.” Economic Journal
37 (145):47–61.
39
Saez, Emmanuel. 2001. “Using Elasticities to Derive Optimal Income Tax Rates.” Review of
Economic Studies 68 (1):205–229.
———. 2002. “The Desirability of Commodity Taxation Under Non-Linear Income Taxation
and Heterogeneous Tastes.” Journal of Public Economics 83 (2):217–230.
Samuelson, Paul A. 1954. “The Pure Theory of Public Expenditure.” Review of Economics
and Statistics 36 (4):387–389.
Sandmo, Agnar. 1975. “Optimal Taxation in the Presence of Externalities.” Scandinavian
Journal of Economics 77 (1):86–98.
———. 1998. “Redistribution and the Marginal Cost of Public Funds.” Journal of Public
Economics 70:365–382.
Seade, Jesus K. 1977. “On the Shape of Optimal Tax Schedules.” Journal of Public Economics
7 (2):203–235.
Sheshinski, Eytan. 1972. “The Optimal Linear Income Tax.” Review of Economic Studies
39 (3):297–302.
Slemrod, Joel B. and Schlomo Yitzhaki. 2001. “Integrating Expenditure and Tax Decisions:
The Marginal Cost of Funds and the Marginal Benefit of Projects.” National Tax Journal
52 (2):189–201.
Stiglitz, Joseph E. 1982. “Self-Selection and Pareto Efficient Taxation.” Journal of Public
Economics 17 (2):213–240.
Stiglitz, Joseph E. and Partha Dasgupta. 1971. “Differential Taxation, Public Goods and
Economic Efficiency.” Review of Economic Studies 38 (2):151–174.
Triest, Robert K. 1990. “The Relationship between the Marginal Cost of Public Funds and
Marginal Excess Burden.” American Economic Review 80 (3):557–566.
Tuomala, Matti. 1984. “On the Optimal Income Taxation: Some Further Numerical Results.”
Journal of Public Economics 23 (3):351–366.
Werning, Ivan. 2007. “Optimal Fiscal Policy with Redistribution.” Quarterly Journal of Economics 122 (3):925–967.
Wildasin, David E. 1984. “On Public Good Provision with Distortionary Taxation.” Economic
Inquiry 22 (2):227–243.
Wilson, John D. 1991. “Optimal Public Good Provision with Limited Lump-sum Taxation.”
American Economic Review 81 (1):153–166.
Zoutman, Floris T., Bas Jacobs, and Egbert L.W. Jongen. 2016. “Redistributive Politics and
the Tyranny of the Middle Class.” CESifo Working Paper No. 5881, Munich: CESifo.
40