Discussion Papers In Economics
And Business

Optimal Capital Income Taxation in the Case ofPrivate Donations to Public Goods
Shigeo Morita and Takuya Obara
Discussion Paper 16-21

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANOptimal Capital Income Taxation in the Case ofPrivate Donations to Public Goods
Shigeo Morita and Takuya Obara
Discussion Paper 16-21

August 2016

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANOptimal Capital Income Taxation in the Case ofPrivate Donations to Public Goods∗
Shigeo Morita†

Takuya Obara‡

Abstract
In this study, we investigate optimal nonlinear labor and capital income taxation and subsidies for contribution goods in a dynamic setting. We show thatwhen individuals can contribute to a public good—even if additive and separable preference between consumption and labor supply is assumed and individuals
diﬀer only in earning ability—marginal capital income tax rate for low-income
earners is not zero, indicating that the Atkinson–Stiglitz theorem does not hold.
In particular, heterogeneous tastes for private consumptions endogenously occur.
In addition, we derive a formula for optimal tax treatment of a public good, whichis expressed in terms of the Pigouvian eﬀect and the eﬀect on an incentive compatibility constraint.
JEL Classification: H21, H23
Keywords: Capital income tax, Private donations, Tax treatment∗

We thank our supervisors Motohiro Sato and Yukihiro Nishimura for their invaluable comments,suggestions, and support. We also thank Yoshihiro Takamatsu for agreeing to be a discussant for thepresentation of our paper and the participants at the Japanese Economic Association, June 19, 2016.
Financial support from Hitotsubashi University is gratefully acknowledged.
†
Graduate School of Economics, Osaka University, Machikaneyama 1-7, Toyonaka, Osaka 560-0043,
Japan. Email: ngp024ms@student.econ.osaka-u.ac.jp.
‡
Graduate School of Economics, Hitotsubashi University, Naka 2-1, Kunitachi, Tokyo 186-8601,
Japan. Email: ed142003@g.hit-u.ac.jp

1

Introduction

Optimal tax theory plays an important role in designing income redistribution policies
and implementing public projects. For several years, researchers have explored aspectsof the field: in particular, economists have been concerned with the question, “Should
capital income be taxed? This question arises from the fact that the government canreinforce redistributive policies by levying taxes on savings; however, this is a form of
double taxation. Although the literature has discussed whether taxation on capitalincome is justified, it remains an ongoing research issue.
This study aims to investigate the desirability of capital income tax from the viewpoint of economic behaviors, specifically when individuals can contribute to public
goods. The motivation stems from policy discussions and empirical evidence. An important policy discussion being held in the United States is whether the government
should levy higher taxes on those who earn more. In fact, the topic has even been placedon the agenda in a 2016 presidential election: the candidate plans on increasing taxes
for the very rich, in particular, their capital gains and charitable giving. This is because
a high tax rate on capital gains generally leads to taxpayers choosing charitable giving
as strategies to avoid recognizing taxable gains. Hood, Martin, and Osberg (1977) findthat Canada’s 1971 Tax Reform, which introduced a 50% capital gain tax, brought
about a decrease in individual charitable donations. More recently, Auten, Seig, andClotfelter (2002) estimate the price elasticity of donation, where price is a weighted average of the price of giving cash and appreciated properties. Their estimations present
that the value of the price elasticity of donation is negative. These findings imply that
individuals account for capital income tax when donating toward public goods. Policy
debates and empirical evidence suggest that individuals’ charitable giving cannot be
discussed independently of capital income taxation. Therefore, our study takes a step
toward theoretically clarifying how capital income tax schemes should be designed when
individuals can contribute to a public good.
Our analysis comprises a dynamic setting in which individuals live for two periods.
We assume that in the first period, individuals can spend a part of their savings on
donations to a charity. For simplicity, there are two types of individuals: high- andlow-skilled individuals. The government designs three types of tax schedules: nonlinear
taxes on labor and capital income and nonlinear subsidies for contribution to public
goods. We demonstrate that although a utility function is represented by the preference
that private goods are additively separable from leisure, marginal tax on capital is zerofor the high skilled but not low skilled when private contributions are made to public
goods. The amount of donation to a public good diﬀers between high- and low-skilledindividuals, which aﬀect the marginal rate of substitution between consumption in thefirst and second period. If a high-skilled individual’s valuation of future consumption
is higher than a low-skilled individual’s one, the distortion on savings behavior for thelatter relaxes the self-selection constraint for the former. Thus, the complementarity

1between private consumption and donation to a public good plays an important role incharacterizing the optimal tax rates for marginal capital income.

1.1

Related literature

An important contribution to the literature on capital income taxation is Ordover andPhelps (1979). They examine optimal nonlinear taxation on income and savings in an
overlapping generation economy in the case of unobservable earnings ability. Their main
conclusion states that if preferences are weakly separable between private goods andleisure, taxes on savings are redundant. This is consistent with the Atkinson and Stiglitz(1976) theorem. On the other hand, Saez (2002) investigates conditions necessary toobtain the Atkinson–Stiglitz theorem and show that if individuals have heterogeneous
tastes in private consumptions, the results are violated, even if the utility function is
weakly separable between private goods and leisure.
The present study is closely related to the model explaining the desirability of capital income taxes on the basis of heterogeneous tastes for goods between high- andlow-income earners, which stems from Saez (2002). However, the extant literature
treats diﬀerentiation in taste based on initial endowments and discount rates as an
assumption (Boadway, Marchand, and Pestieau (2000), Cremer, Pestieau, and Rochet(2003), Diamond and Spinnewijn (2011)). However, taste diﬀerentiation can also result from individuals’ behavior without explicitly assuming additional characteristics.
Thus, we present the desirability of capital income taxes by establishing the theoreticalfoundation that taste diﬀerentiation occurs. To the best of our knowledge, this study
provides new evidence justifying capital taxation since private donations have not been
considered in the context of individual behaviors in capital income taxation theory.
In our model, the government oﬀers subsidies for contributions to public goods.
Prior studies have investigated optimal tax policy assuming the presence of charitablegiving (Andreoni (1988), Saez (2004)). The study most closely related in terms oftax treatment of private donations is Diamond (2006), who shows that the welfareimproving eﬀect is achieved by introducing a subsidy on private donations toward a
public good under nonlinear income taxes on labor. However, Diamond (2006) adopts
a static model that does not allow the government to impose income taxes on capital anddoes not attempt to derive an optimal tax treatment formula. We extend Diamond’s
model as a two-period model to investigate the desirability of capital income taxes andrigorously characterize a tax treatment formula at the optimum level, which depends
on the Pigouvian eﬀect and the eﬀects on an incentive compatibility constraint undera more general utility function.
The optimal tax rate formulas in terms of capital income taxes and subsidies for private donations are conceptually related to the findings of Cremer, Gahvari, and Ladoux(1998), who examine both optimal linear and nonlinear taxation on commodities andincome in the presence of goods that cause externalities. They show that if tastes
2

for private consumption are not identical, goods without externalities are taxed. The
crucial diﬀerence between models proposed by Diamond (2006) and Cremer, Gahvari,and Ladoux (1998) is that the former considers a finite number of individuals, while
the latter assumes an infinite number. In the present setting, changes in contributions
to a public good due to a mimicker aﬀects the aggregate level of a public good. 1 If
consumption is not weakly separable for a public good, the mimicker’s behavior causesa variation in intertemporal substitution between the mimicker and mimicked, whichallows for taxation on capital income without exogenously assuming taste diﬀerentiation.
The remainder of this paper is organized as follows. Section 2 describes the framework of the basic model. Section 3 characterizes optimal tax formulas. Section 4
concludes the paper.

2

Model

2.1

Environment

We consider an economy in which individuals live for two periods and work only in thefirst. There are two types of individuals: low-skilled individuals whose earning wage
rate is w1 and high-skilled ones with earning wage rate w2 , where w2 > w1 . Their
before-tax income is yi ≡ wi li , where li denotes the labor supply of type i individuals.
The number of type i individuals is defined by π i , which is a natural number greaterthan two and for now, is invariant. 2 The utility function of type i individuals is
U i (ci , xi , G, li ) = u(ci , xi , G) − v(li )(1)where ci denotes consumption of a private good in the first period, xi is consumption
in the second period, and G is the amount of public good. Individuals can contributeto a public good; then, the aggregate amount of public goods is∑G=
πigi + gG(2)iwhere g G denotes the amount of public good provision by the government. 3 The
sub-utility function, u(·), is strictly increasing, concave, and twice diﬀerentiable and1

In the case of infinite population, as in Cremer, Gahvari, and Ladoux (1998), the behavioral change
of a mimicker does not aﬀect the aggregate level of a public good.
2
Using a finite number of individuals, Piketty (1993) and Hamilton and Slutsky (2007) show thatthe first-best allocation can be achieved if an individual’s tax schedule depends on the behavior of otherindividuals. Following traditional optimal taxation literature, the present study restricts an individual’s
tax schedule to a function of the value of his/her labor income, capital income, and private donation
to a public good.
3
We consider public goods financed by not only individuals but also the government such as health,education, and social services. According to Charitable Giving Statistics by National Philanthropic3

satisfies the Inada condition and v(·) is strictly increasing, convex, and twice diﬀerentiable. Hereafter, we consider the case in which public and consumption goods are
2u
2ucomplements, that is, ∂c∂i ∂G
and ∂x∂i ∂G
are positive. The complementarity between private consumption and donation to a public good is supported by experimental studies.
For example, Strahilevitz and Myers (1998) show that charity incentives are more effective in the case of frivolous products than practical ones, that is, charitable donation
has a complementary relationship with private consumption.
Let si denotes savings of type i individuals and r is the interest rate. The budget
constraints type i individuals face can be written as follows:ci + si + g i − τ (g i ) = y i − T (y i )(3)

si (1 + r) − Φ(si r) = xi

(4)where τ (g i ) is a subsidy for private donations by type i individuals to a public good,T (y i ) is an income tax payment, and Φ(rsi ) is the capital income tax payment, which
respectively, are nonlinear functions of g i , y i , and rsi . Individuals choose ci , xi , si , g i ,and li to maximize the utility function (equation (1)) subject to their budget constraints(equations (3) and (4)). Combining this with the first-order conditions yields−uc (ci , xi , G) + {(1 + r) − rΦ′ (rsi )}ux (ci , xi , G) = 0(5)∂uwhere uc (ci , xi , G) ≡ ∂ci denotes the marginal utility of consumption in the first period,∂uiiux (c , x , G) ≡ ∂xi is the marginal utility of consumption in the second period, anddΦΦ′ (rsi ) ≡ drs
i is the marginal capital income tax rate function corresponding to returnsof savings rsi . The first-order condition for donation g i yields 4−(1 − τ ′ (g i ))uic (ci , xi , G) + uiG (ci , xi , G) = 0(6)dτ
where τ ′ (g i ) ≡ dg
i is the marginal subsidy rate function of a private donation to a public
good.
For simplicity, the production sector utilizes labor and capital. Production technology exhibits a constant return to scale. This means that each unit of eﬀective labor

Trust, in the United States, individuals’ charitable giving accounts for 71% of total giving and amajority of donation are made to religious, educational, and healthcare organizations. A donation toreligious organizations is a suitable example for the outcomes where only type-2 individuals contributeto a public good because the US government cannot contribute to them. These outcomes are discussed
below.
4
To derive the optimal condition for private contributions to public goods, we introduce notationG∼i , which is the total amount of public good contributed by the government and other individuals,
including others of the same type. The sub-utility function u(·) can be seen as u(ci , xi , G∼i + gi ). As∂uequals that of a private
the marginal utility of the total amount of public good uG (ci , xi , G) ≡ ∂Gidonation to a public good, the first-order condition with respect to g can be written as equation (6).

4

wi li is required to produce one unit of private good and each unit of private good savedin the first period produces (1 + r) units of a private good in the second period.5

2.2

Planning problem

The objective of the government is represented by the following utilitarian social welfare
function:∑W =π i U i (ci , xi , G, li )(7)i

The budget constraint for the government is 6∑∑
π i T (y i ) −π i τ (g i ) − sG − g G ≥ 0
i

(8)

i

∑

π i Φ(rsi ) + (1 + r)sG ≥ 0(9)iwhere sG denotes government saving. The saving technology available to the government is the same as that available to individuals. Using the budget constraints that
individuals face, these can be equivalently written as∑∑(10)yiπi −(ci + si + g i )π i − sG − g G ≥ 0
i

i

(∑
) ∑
i i
G
(1 + r)sπ +s−xi π i ≥ 0
i

(11)

i

The informational assumptions are conventional: the government can observe individuals’ donation, labor income, and capital income, while their ability is never observable.
We focus on the case in which the government attempts to redistribute from type-2
to type-1 individuals. This means the following incentive compatibility constraint isbinding at the social optimum:
U 2 (c2 , x2 , G,

y2
y1
2 1
1
)
≥
U
(c
,
x
,
Ĝ,
)
w2
w2

5

(12)

Pirttila and Tuomala (2001) show that capital income taxation is justified when wages are endogenously determined and the relative wage rate is aﬀected by the amount of savings. By contrast,
we assume no general-equilibrium eﬀects of wage rates. Thus, our model can be seen as the twoperiod, partial equilibrium version of Pirttila and Tuomala’s model. At the optimum, where only thegovernment contributes to a public good, our model’s outcome is consistent with that of their model.
6
Following Diamond (2006), we assume that there is no response of government budget constraints
to a deviation from individuals’ anticipated revealing strategies.

5

where Ĝ ≡ G − g 2 + g 1 denotes the aggregate level of a public good achieved when
type-2 individuals mimic.
The social planning problem is to maximize the social welfare function (equation(7)), subject to the equations for public goods (equation (2)), resource constraints(equations (10) and (11)), and incentive compatibility constraints (equation (12)). The
Lagrangean corresponding to this planning problem can be formulated as follows:
[∑
]
[∑
]
∑
i i
G
i i
i
i
i
i
G
G
L=W +µg π + g − G + γ1yπ −(c + s + g )π − s − g (13)
i
i
(∑ i
) ∑
]
[
]
y2
y1
i i
G
i i
2 2
2
2 1
1
+γ2 (1 + r)sπ +s−x π + λ U (c , x , G, 2 ) − U (c , x , Ĝ, 2 )
w
w
i
i

[

where µ, γ1 , γ2 , and λ are the Lagrange multipliers.

3

Characterizing optimal taxation

Here, we present the key features of our model’s outcomes. The results imply that the
government should design taxes on capital income such that it supplements the tax
treatment of private donations to a public good.

3.1

Optimal capital income taxation

Leti
M RScx≡uc (ci , xi , G)ux (ci , xi , G)

and M ˆRS cx ≡uc (c1 , x1 , Ĝ)ux (c1 , x1 , Ĝ)

denote the marginal rate of substitution between private consumption in the first andsecond period faced by type i individuals and the corresponding marginal rate of substitution that the mimicker faces. Combining the optimality condition regarding ci andxi yields the optimal capital income tax rate for type i individuals:)(
λux (c1 , x1 , Ĝ)1′
1
M RScx − M ˆRS cx(14)Φ (rs ) =rπ 1 γ2Φ′ (rs2 ) = 0(15)

The derivation is presented in Appendix A. Equation (14) implies that the deviationof the optimal tax rate on capital income from the Atkinson–Stiglitz theorem depends
on the term in the brackets on the right-hand side. These equations give the following
proposition:
6

Proposition 1. When a public good has a more complementary relationship with theconsumption good in the first period than in the second, even if individual preferences
can be separated between labor and consumption, the marginal capital income tax rate
is positive for type-1 individuals and zero for the type-2 individuals.
The result of Proposition 1 is crucially related to the diﬀerence between G and Ĝ.
At the optimum, the level of a public good is higher when a type-2 individual chooses
a truth-telling strategy than a mimicking-one, that is, G > Ĝ. As shown in the Appendix B, it is optimal that only type-2 individuals contribute to a public good, g 1 = 0,
g 2 > 0, and g G = 0. Namely, in our context, public provision and private provision havediﬀerent impact on the incentive compatibility constraint and social welfare. This isconsistent with Diamond (2006), which shows that inducing type-2 individuals to contribute improves their level of social welfare from the allocation, where no one makes
a private donation to public goods. Our interest is the implication of the property ofprivate provision on the capital income tax. Assuming that a public good has a complementary relationship with the private good in the first period than in the second,
the intertemporal marginal rate of substitution for the mimicker is larger than the cor1
responding marginal rate of substitution for the mimicked, that is, M RScx
> M ˆRS cx .
This implies that distorting the capital income of type-1 individuals downward relaxes
the incentive compatibility constraint and then creates an informational advantage forthe government. Therefore, the marginal capital income tax rate should be positive.
On the other hand, equation (15) shows that the government should not distort type-2
individuals’ saving behavior, making zero marginal capital income tax rate desirable.

3.2

Optimal subsidy for a public good

A new issue emerges owing to the welfare gain from private donations by type-2 individuals, that is, how should optimal subsidies for donations be characterized? LetuG (ci , xi , G)i
M RSGc ≡uc (ci , xi , G)11uG (c , x , Ĝ)and M ˆRS Gc ≡uc (c1 , x1 , Ĝ)

denote the marginal rate of substitution between public good and private consumption
in the first period for type i individuals and the corresponding marginal rate of substitution for the mimicker. The optimality conditions with respect to g 2 and c2 gives
us
[
]
[
]
λuc (c1 , x1 , Ĝ)
1
2
2
′ 2
1
1
2
2
ˆ
M RSGc π − M RS Gc (π − 1) (16)
τ (g ) = M RSG,c π + M RSGc (π − 1) +γ1 π 2
{z
} |
{z
}
|
Pigouvian eﬀect

The eﬀect of type 2’s donation on IC constraint

The derivations are included in Appendix C.
7

Proposition 2. At the optimum, where only type-2 individuals contribute to a public
good, the optimal subsidy for their private donation diﬀers from the standard Pigouvian
subsidy.
The right-hand side of these equations comprises two terms. The first and second
terms can be seen as an externality from a public good correcting eﬀect. The first term
has an externality eﬀect on the other type, while the second term has an externalityeﬀect on other individuals of the same type. Therefore, the terms for optimal tax
conditions act as Pigouvian tax and these signs are positive. The third term reflects themarginal eﬀect of private donation on the incentive compatibility constraint. 7 Because
G > Ĝ, type-1 individuals’ marginal utility from a public good is less than that of a
mimicker. When type-1 individuals’ consumption and public good are complements,
1
M RSGc< M ˆRS Gc . Then, distorting a private donation by type-2 individuals upward
makes type-1 individuals worse oﬀ, but leaves mimickers well oﬀ. By contrast, using
the definition of G and Ĝ, it is easy to show that∂G
∂ Ĝ= π2 > 2 = π2 − 12∂g∂gThis implies that the marginal eﬀect of type-2 individuals’ donation on G is larger thanthe eﬀect on Ĝ, thus inducing type-2 individuals to contribute appears as a welfare gain.
Therefore, ”the eﬀect of type 2’s donation on the IC constraint” cannot be signed.

4

Concluding Remarks

This study is largely relevant to debates on the desirability of capital income taxes.
Since the seminal work of Ordover and Phelps (1979), a large body of literature has
accumulated on whether capital income taxes are required from the viewpoint of heterogeneous tastes in private consumptions, even though the utility function is weakly
separable between private goods and leisure. For instance, Boadway, Marchand, andPestieau (2000), Cremer, Pestieau, and Rochet (2003), and Diamond and Spinnewijn(2011) consider a multidimensional heterogeneity setting in which individuals diﬀer in
not only earning abilities but also other characteristics such as initial endowments (bequest or inheritance) and discount rates, which are assumptions. By contrast, this study
provides additional economic rationale for capital income taxes from the viewpoint of
economic behavior that is, in reality, individuals deduct charitable contributions. Under the standard optimal tax approach, we show that the government should design
taxes on capital income to supplement its redistribution policy when individuals can
7

The Samuelson rule derived in our model is modified and the social marginal benefit should beequal to the marginal cost of a public good provision and ”the eﬀect of type 2’s donation on the IC
constraint.” This is consistent with the corresponding rule derived in Diamond (2006). See equation(22) in Diamond (2006).

8

contribute to a public good. This persists even if the additive and separable preference
between consumption and labor supply is satisfied and individuals diﬀer in only earning
abilities.
The theoretical contribution of this paper is as follows. Although we show that
Atkinson–Stiglitz theorem breaks down as a result of heterogeneous preferences, as in
the case of Saez (2002), we justify capital income taxes by clarifying the source of heterogeneity on the basis of individual behavior and not assumptions. Furthermore, ourfindings have important implications that the crucial condition to design capital income
tax policies is the relationship between contribution to a public good and intertemporalconsumption choice.

References[1] Andreoni J. (1988). “Privately provided public goods in a large economy: The
limits of altruism” in Journal of Public Economics. 35(1), 57-73.
[2] Auten G. E and H. Seig and C. T Clotfelter. (2002). “Charitable giving, income,and taxes: an analysis of panel date” in The American Economic Review. 92(1),371-382.
[3] Atkinson A. B and J. Stiglitz. (1976). “The design of tax structure: direct versus
indirect taxation” in Journal of Public Economics. 6(1-2), 55-75.
[4] Boadway R. and M. Marchand and P. Pestieau. (2000). “Redistribution with unobservable bequests: a case for taxing capital income” in Scandinavian Journal of
Economics. 102(2), 253-267.
[5] Cremer H. and F. Gahvari and N. Ladoux. (1998). ‘Externalities and optimal
taxation” in Journal of Public Economics. 70(3), 343-364.
[6] Cremer H. and P. Pestieau and J-C. Rochet. (2001). “Direct versus indirect taxation: the design of the tax structure revisited” in International Economic Review.
42(3), 781-800.
[7] Diamond P. (2006). “Optimal tax treatment of private contributions for public
goods with and without warm glow preferences” in Journal of Public Economics.
90(4-5), 897-919.
[8] Diamond P. and J. Spinnewijn. (2011). “Capital income taxes with heterogeneous
discount rates” in American Economic Journal: Economic Policy. 3(4), 52-76.
[9] Hamilton J. and S. Slutsky. (2007). “Optimal nonlinear income taxation with afinite population” in Journal of Economic Theory. 132(3), 548-556.
9[10] Hood R. D. and S. A. Martin and L. S. Osberg. (1977). “Economic determinantsof individuals charitable donations in Canada” in The Canadian Journal of Economics. 10(4), 653-669.
[11] Ordover J. A. and S. Phelps. (1979). “The concept of optimal taxation in theoverlapping-generations model of capital and wealth” in Journal of Public Economics. 12(1), 1-26.
[12] Piketty T. (1993). “Implementation of first-best allocations via generalized tax
schedules” in Journal of Economic Theory. 61(1), 23-41.
[13] Pirttila J. and M. Tuomala (2001). “On optimal non-linear taxation and public
good provision in an overlapping generations economy” in Journal of Public Economics. 79, 485-501.
[14] Saez E. (2002). “The desirability of commodity taxation under non-linear income
taxation and heterogeneous tastes” in Journal of Public Economics. 83(2), 217-230.
[15] Saez E. (2004). “The optimal treatment of tax expenditures” in Journal of Public
Economics. 88(12), 2657-2684.
[16] Strahilevitz M. and J. G. Myers. (1998). “Donations to charity as purchase incentives: How well they work may depend on what you are trying to sell” in Journalof Consumer Research. 24(4), 434-446.

Appendix A
The first order conditions associated with c1 , x1 , c2 , x2 , s1 , and s2 are∂L= π 1 uc (c1 , x1 , G) − γ1 π 1 − λuc (c1 , x1 , Ĝ) = 0
∂c1

(A.1)

∂L
= π 1 ux (c1 , x1 , G) − γ2 π 1 − λux (c1 , x1 , Ĝ) = 0
∂x1

(A.2)

∂L
= π 2 uc (c2 , x2 , G) − γ1 π 2 + λuc (c2 , x2 , G) = 0
2
∂c

(A.3)

∂L
= π 2 ux (c2 , x2 , G) − γ2 π 2 + λux (c2 , x2 , G) = 0
∂x2

(A.4)

∂L
= −γ1 + γ2 (1 + r) = 0 i = 1, 2, G
∂si

(A.5)

10

Substituting equation (A.1) and (A.2) into equation (A.5) yields:
π 1 {uc (c1 , x1 , G) − (1 + r)ux (c1 , x1 , G)} = λ{uc (c1 , x1 , Ĝ) − (1 + r)ux (c1 , x1 , Ĝ)} (A.6)
Combining equation (5) with equation (A.6) yields:(π 1 ux (c1 , x1 , G) − λux (c1 , x1 , Ĝ))rΦ′ (rs1 ) = λux (c1 , x1 , Ĝ)(uc (c1 , x1 , Ĝ)uc (c1 , x1 , G)
−
(A.7)
)
ux (c1 , x1 , G) ux (c1 , x1 , Ĝ)

Substituting equation (A.2) into the term in the brackets of the left hand side, we
obtain equation (14). Similarly, substituting equation (A.3) and (A.4) into equation(A.5) yields:−π 2 {uc (c2 , x2 , G) − (1 + r)ux (c2 , x2 , G)} = λ{uc (c2 , x2 , G) − (1 + r)ux (c2 , x2 , G)}(A.8)
This can be rewritten as follows:(π 2 + λ)ux (c2 , x2 , G)rΦ′ (rs2 ) = 0(A.9)

Equation (A.3) implies that π 2 +λ is positive. Then, equation (A.9) implies that Φ′ (rs2 )is zero.

Appendix B
Diﬀerentiating L with respect to g G , g 1 , and g 2 implies∂L= −γ1 + µ∂g G∂L
= −γ1 π 1 − λûG + µπ 1
1
∂g

(B.1)
(B.2)

∂L
= −γ1 π 2 + λûG + µπ 2
(B.3)
∂g 2
If equation (B.1) is equal to zero, equation (B.3) is as follows.
∂L= λûG > 0
(B.4)
∂g 2
In this case, the optimal solution does not exist because of diverging. Therefore, at the∂LGoptimum, we must have ∂g= 0 to satisfy Kuhn-Tucker conditions.
G < 0 and gGiven this condition, from equation (B.2), no contribution to a public good of type
1 individuals is optimal, that is, g 1 = 0. On the other hand, the private donation to a
public good of type 2 individuals is not zero because the second term in equation (B.3)
is suﬃciently larger than the sum of the first and third term by the Inada condition
when g 2 is close to zero given g 1 = g G = 0. Therefore, g 2 is positive. In addition, g 2∂L22
which isis an interior solution. As g 2 is close to infinity, ∂g
2 converges to −γ1 π + µπ
2negative. This implies that g must not be corner solution at the optimum.
11

Appendix C
The first order condition associated with G is∂L
= π 1 uG (c1 , x1 , G) + π 2 uG (c2 , x2 , G) + λuG (c2 , x2 , G) − λuG (c1 , x1 , Ĝ) − µ = 0(C.1)∂G
Taking the product of equation (C.1) and π 2 yields:
{
}
∂L 2
1
1
1
2
2
2
π = π uG (c , x , G) + π uG (c , x , G) π 2 − µπ 2
∂G
{
}
2
2
2
1
1
2
+λ uG (c , x , G)π − uG (c , x , Ĝ)π = 0(C.2)

Substituting the first order condition associated with g 2 and c2 into (C.2) yields:
{
}
1
1
1
2
2
2
π uG (c , x , G) + π uG (c , x , G) π 2 − π 2 uc (c2 , x2 , G) − λuc (c2 , x2 , G)
(C.3)
{
}
+λ uG (c2 , x2 , G)π 2 − uG (c1 , x1 , Ĝ)(π 2 − 1) = 0
Dividing equation (C.3) by γ1 π 2 yields:}{
11
λuc (c2 , x2 , G)
uc (c2 , x2 , G) 2 uG (c2 , x2 , G)
1 uG (c , x , G)
−1 +π
π
−
γ1
uc (c2 , x2 , G)γ1
π 2 γ1
{
}
λ
2
2
2
1
1
2
+ 2
uG (c , x , G)π − uG (c , x , Ĝ)(π − 1) = 0
π γ1
Rearranging equation (C.4) yields:
{
}{
}
2
2
1
1
uc (c2 , x2 , G) λuc (c2 , x2 , G)
2 uG (c , x , G)
1 uG (c , x , G)
−
1
+
+
π
π
uc (c2 , x2 , G)γ1
π 2 γ1
γ1
−

(C.4)

(C.5)

λuc (c1 , x1 , Ĝ) uG (c1 , x1 , Ĝ) 2(π − 1) = 0
π 2 γ1uc (c1 , x1 , Ĝ)

Substituting equation (6) and (A.2) into the first term of equation (C.5) yields:{}11uG (c2 , x2 , G) 2
uG (c2 , x2 , G)
′ 2
1 uG (c , x , G)
π
−
τ
(g
)
−
+
π
(C.6)
uc (c2 , x2 , G)
uc (c2 , x2 , G)
γ1−

λuc (c1 , x1 , Ĝ) uG (c1 , x1 , Ĝ) 2(π − 1) = 0
π 2 γ1uc (c1 , x1 , Ĝ)

12

Substituting equation (A.1) into the fourth term of equation (C.6) yields:uG (c1 , x1 , G)uG (c2 , x2 , G)2
τ ′ (g 2 ) = π 1
+
(π
−
1)
uc (c1 , x1 , G)
uc (c2 , x2 , G)}{
λuc (c1 , x1 , Ĝ) u1G (c1 , x1 , G) 2 uG (c1 , x1 , Ĝ) 2+
π −(π − 1)γ1 π 2uc (c1 , x1 , G)
uc (c1 , x1 , Ĝ)iUsing the notation M RSGc
, we can rewrite equation (C.7) for (16).

13(C.7)