Journal of Public Economics 94 (2010) 1067–1072
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Journal of Public Economics j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / j p u b e
Public goods provision, redistributive taxation, and wealth accumulation☆ Toshiki Tamai ⁎ Faculty of Economics, Kinki University, 3-4-1 Kowakae, Higashi-Osaka, 577-8502, Japan
a r t i c l e
i n f o
Article history: Received 25 February 2009 Received in revised form 10 June 2010 Accepted 29 June 2010 Available online 1 August 2010 JEL classiﬁcation: H41 Keywords: Public goods Redistributive taxation Accumulation of wealth
a b s t r a c t This paper presents an extension of a static model of public goods provision with redistributional taxation, as described by Uler [Journal of Public Economics 93 (3–4), pp. 440–453], as a dynamic model of wealth accumulation. Intertemporal consumption and saving behavior strongly affect the relation between redistributive taxation and charitable contribution. Indeed, the analyses presented herein reveal that more patience (i.e. higher saving) engenders a lower redistributional tax rate. However, the optimal redistributive tax rate is not zero because redistributive taxation improves the efﬁciency of providing public goods that are not improved by balanced growth. This paper ﬁlls the gap separating static analysis and dynamic analysis, and generalizes the results presented by static analysis. © 2010 Elsevier B.V. All rights reserved.
1. Introduction The provision of public goods and redistribution of income have long been studied by numerous public economists. One interesting result is transfer neutrality, as initially implied by Shibata (1971) and later formulated by Warr (1982, 1983). The transfer neutrality theorem states that an income transfer policy does not affect the allocation of goods under voluntary provision of public goods. This surprising result stimulated many subsequent studies (e.g., Bergstrom et al., 1986; Andreoni, 1989, 1990; Boadway et al., 1989; Buchholz and Konrad, 1995; and Ihori, 1996).1 Especially, Bergstrom et al. (1986) generalized the transfer neutrality theorem. Thereafter the emphasis of analysis has notably centered upon the condition that the neutrality of income transfer does not hold.2 ☆ I am grateful to Hikaru Ogawa, Yasuhiro Sato, Kazutoshi Miyazawa, Shinya Fujita and the seminar participants at Nagoya University for their advice and comments. I also thank the co-editor, James Andreoni, and two anonymous referees for their constructive comments and suggestions. This work was supported by a Grant-in-Aid for Young Scientists (B) (No. 20730206) of the Japan Society for the Promotion of Science. ⁎ Tel.: +81 6 6721 2332x7065. E-mail address: [email protected]
1 In related literature, numerous studies have examined the relation between public goods provision and the number of members (e.g., Pecorino, 1999; Kawachi and Ogawa, 2006). Pecorino (1999) and Kawachi and Ogawa (2006) investigate public goods provision under a repeated game. 2 Andreoni (1989, 1990) examines the transfer neutrality in the model of public goods provision with impure altruism. He shows that transfer neutrality is broken under impure altruism. Boadway et al. (1989), Buchholz and Konrad (1995), and Ihori (1996) investigate the transfer neutrality in the model with different costs of providing public goods. Boadway et al. (1989) show that the neutrality theorem holds, although Buchholz and Konrad (1995) and Ihori (1996) show that the neutrality theorem does not hold.
0047-2727/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jpubeco.2010.06.019
As presented recently, Uler (2009) examines the relation between public goods provision and redistribution of income by the introduction of redistributional taxation into the model described by Bergstrom et al. (1986).3 She not only generalizes the results of previous studies (e.g., Warr, 1982, 1983; Bergstrom et al., 1986); she shows that the supply of public goods increases with the tax rate. This result is interesting to public economists because the privatization of (some types of) public goods provision might thereby be achieved simply through tax reform. However, we cannot overlook the negative effect of taxes on welfare, especially through a negative impact on saving. Indeed, the literature related to endogenous growth shows that tax policy negatively affects growth and welfare and that the difference in tax rates engenders a difference in the rate of wealth accumulation (King and Rebelo, 1990; Rebelo, 1991). It is important for knowledge related to public economics because the negative impact on income growth also inﬂuences the future prices of private and public goods. The analysis developed in this paper provides a means for us to ﬁll the gap separating static analysis and dynamic analysis, and provides a novel view of public goods provision. Concretely, this paper presents an investigation of the interaction between public goods provision, redistributional taxation, in addition to wealth accumulation in an intertemporal model of saving. As described in this paper, we will explore the relation between public goods provision and a redistributive
3 The recent literature related to public goods provision has developed rapidly through studies of incentive mechanism design. Uler (2009) and this paper shares many similarities with reports of the literature (e.g., Falkinger, 1996; Falkinger et al., 2000).
T. Tamai / Journal of Public Economics 94 (2010) 1067–1072
tax more fully, particularly considering the impact of such a tax on future prices of goods. The most interesting implication of our analysis is that the balanced growth of wealth cannot reduce the income inequality existing between contributors and non-contributors. This ﬁnding implies that the problem of free riders will be not settled by wealth accumulation under balanced growth. It also implies that a means to settle the problem of free riders is the only wealth redistribution policy. In our analysis, an increase in the redistributional tax reduces income inequality and temporarily increases the supply of public goods. However, a redistributional tax also has a negative effect on the rate of wealth accumulation, which is the growth rate of future income. The effect of redistribution on inequality is weakened by the effect of redistribution on wealth accumulation. Consequently, in our study, there exists the positive optimal tax rate, although the standard literature on macroeconomics holds that the capital income tax should not be taxed. Furthermore, these results are consistent with previous studies. We show that a positive correlation exists between an optimal tax and impatience. If the degree of impatience is sufﬁciently large, then our model converges to previous studies such as those of Bergstrom et al. (1986) and Uler (2009). Our model provides a natural extension of the static model and generalizes its result. The remainder of this paper is organized as follows. Section 2 presents a full description of our model. We assume that the agent is inﬁnitely lived (later); our model has an agent with perfect intergenerational altruism. Section 3 characterizes the decentralized equilibrium. The tax incidence affects future prices of goods and the time preference rate serves as a mediator between Uler's analysis and ours. Finally, Section 4 presents conclusions of this paper.
Consider an economy with private goods, public goods, and n individuals. Public goods are provided through charitable giving, so that the level of public good provision is equal to total giving: G = ∑ nj = 1gj where gi signiﬁes the ith individual's contribution. The ith individual's income is yi = Aki (A N 0): the income yi depends linearly on each individual's wealth ki.4 Government taxes on income net of contributions and the collected tax revenue are distributed equally among individuals. Therefore, the budget constraint of the ith individual is
: where ki is the increment of wealth, ci stands for the ith individual's consumption, τ is the tax rate, and τ ⋅ ∑ nj = 1(yj − gj) / n denotes the income transfer.5 Lifetime utility is an improper integration of instantaneous utility. Presuming that the instantaneous utility function is deﬁned over the consumption of private goods and public goods and that it takes a logarithmic form, then the lifetime utility function is ∞
Ui = ∫0 ½α ln ci + ð1−αÞln G expð−ρtÞdt ;
where ρ stands for the subjective discount rate. 4
We assume that the wealth is not depreciated. This taxation implies perfect deduction of charitable giving from income taxes. See Uler (2009) for details of this taxation. 5
1−α ci n−1 τ ; ≤ 1− α n G
c˙ i = 1− n−1 τ A−ρ≡γ; n ci
where γ ≥ 0 is assumed.6 Furthermore, the growth rate of private consumption is characterized as ∂ γ / ∂ τ = − (n − 1)A / n b 0. Eq. (3) becomes an equality if gi N 0. On the other hand, Eq. (3) is a strict inequality if gi = 0. Eq. (4) shows that the private consumption grows at a constant rate γ; ci(t) = ci(0)exp(γt). Summing up Eq. (1) from 1 to n, we obtain K˙ = AK−G−C = Y−G−C;
where K ≡ ∑ ni = 1ki, Y ≡ ∑ ni = 1yi, and C ≡ ∑ ni = 1ci. Eq. (5) denotes the resource constraint of this economy. 3. Consequences in a decentralized equilibrium In this section, we investigate the equilibrium properties of the decentralized economy. First, we consider the case in which everyone contributes to public goods provision. The economy is always in an equilibrium in which all endogenous variables grow at the same rate γ.7 Therefore, we have ci = C / n, ci(t) = C(0)exp(γt) / n, and G(t) = G(0)exp(γt), where Cð0Þ =
½n−ðn−1ÞτðA−γÞα ðA−γÞð1−αÞ Kð0Þ and Gð0Þ = Kð0Þ:ð6Þ 1 + ðn−1Þð1−τÞα 1 + ðn−1Þð1−τÞα
Using the equations presented above and Eq. (2), we can calculate the ith individual's welfare as shown below.
2. The model
: τ n ki = ð1−τÞðyi −gi Þ−ci + ⋅ ∑ ðyj −gj Þ; n j=1
Each individual maximizes lifetime utility (2) subject to Eq. (1) and gi ≥ 0 for a given ki(0). The optimality conditions lead to
Wi = Zγ = ρ + lnðA−γÞ + α lnfn−ðn−1Þτg− ln f1 + ðn−1Þð1−τÞαg ð7Þ
−α ln n + α ln α + ð1−αÞ lnð1−αÞ + ln Kð0Þ = ρ Outcomes of our analysis are summarized as the following proposition. Proposition 1. Presuming that each individual is a contributor in the equilibrium, then the following are true. (a) The current level of public goods provision is positively affected by a rise in taxation, but the future supply of public goods will be negatively affected by it. (b) There exists an optimal degree of redistribution 0 b τ* b 1. (c) The optimal tax rate is increasing in the subjective discount rate, and decreasing in productivity. (d) Both total contributions and welfare are unaffected by the distribution of initial (pre-tax) wealth. Outcomes (a)–(d) are explained as follows. (a): At time t, the level of public goods provision is given as G(t) = G(0)exp(γt). The logarithmic derivation of G(t) yields 1 ∂GðtÞ = GðtÞ ∂τ
1 ∂γ ðn−1Þα + : A−γ ⋅ ∂τ 1 + ðn−1Þð1−τÞα
The second term of the equation presented above represents the effect of redistributional taxation. This effect is fundamentally equivalent to that of Uler (2009). Speciﬁcally addressing the relative prices of private and public goods, an increase in the tax rate 6
These conditions are derived in Appendix A. See the Appendix A for proof of this property. Furthermore, derivations of Eqs. (6) and (7) are explained in the Appendix A. 7
T. Tamai / Journal of Public Economics 94 (2010) 1067–1072
engenders a change in relative price. Then, demand for public goods increases; supply also increases. The ﬁrst term represents the negative effect on growth and positive one on consumption (i.e. negative on saving). A rise in the tax rate engenders a decrease in individual savings, which is harmful to wealth accumulation. In other words, a rise in the tax rate raises the future prices of goods. In the short run, a rise in the tax rate has a positive effect on charitable giving. However, in the long run, a rise in the tax rate lowers future consumption possibilities and raises the future prices of goods. Consequently, a rise in the tax rate negatively affects charitable giving because, in the course of time, the negative effect dominates the positive effects. (b): The effect of a change in tax rate on welfare is
∂Wi n−1 A n−1 A ðn−1Þα − + =− n ρ n A−γ n−ðn−1Þτ ∂τ
︸ ︸ ︸ ðE1Þ ðE2Þ ðE3Þ
ðn−1Þα ; 1 + ðn−1Þð1−τÞα
butional tax through the negative growth effect of a redistributional tax. Eq. (10) means that a rise in productivity reduces the level of the optimal tax rate. A rise in productivity raises the growth rate of wealth. Because a higher growth rate engenders higher future income, a high tax rate under a high level of productivity brings about heavy burden as compared with low level of productivity. Therefore, a rise in productivity has a negative effect on the optimal tax rate. (d): Public goods provision is independent of wealth (income) distribution because G(t) = G(0)exp(γt) is unaffected by wealth distribution. This result is consistent with those of previous studies (e.g. Warr, 1983; Bergstrom et al., 1986; Uler, 2009). In those studies, the consumption of private goods is also independent of wealth distribution. Consequently, welfare is independent of wealth distribution (indeed, Eq. (7) shows it). The next examination is that of a case in which non-contributors exist in equilibrium. In this case, the economy is always in balanced growth equilibrium (See Appendix A.2). Let G − i be G − gi. Similarly to the standard literature on charitable giving, the contribution of the ith individual is gi = max½Di −G−i ; 0;
where where (E1) is the negative growth effect, (E2) the negative saving effect, (E3) the negative wealth effect through redistribution, and (E4) the positive wealth effect through redistribution. Terms (E3) and (E4) are fundamentally similar to the effect of redistributional taxation on welfare in the static model of Uler (2009). A simple calculation provides (E3) + (E4) N 0. Therefore, redistribution has a positive partial welfare effect through its effect on wealth. On the other hand, terms (E1) and (E2) are speciﬁc to the dynamic model. The sign of (E1) + (E2) is negative. In other words, the redistribution has a negative partial welfare effect derived through the decrease of the growth rate. If so, will the interior optimal tax rate exist? The answer is “Yes”: We have
j ∂W ρ ∂τ j
ðn−1Þ2 ð1−αÞα N 0; ½1 + ðn−1Þαn
ðn−1Þ2 A2 b0: ½ρn + ðn−1ÞAρn
Therefore, the optimal tax rate τ is in (0, 1).8 (c): Presuming that H b 0 holds, then the optimal tax rate is unique and is deﬁnable as the function of ρ and A (see Appendix A.3): ∂τ 1 1 n−1 A N 0; =− 2− 2 n H ∂ρ ρ ðA−γÞ
ρnki + ð1−τÞnG−i + ðK−ki ÞτA : Di ≡ð1−αÞ n−ðn−1Þτ
In the expressions above, Di denotes the demand for public goods. Our second analysis establishes the following proposition: Proposition 2. Without the restriction that everybody is a contributor, (e) all individuals with wealth ki ≤ k* contribute nothing, although every individual with wealth ki N k* contributes to public goods provision if τ b ρn / A. (f) A rise in tax rate increases the current supply of public goods and decreases the future level of public goods provision. Subsequently, if the wealth distribution is k1 ≤ k2 ≤ ⋯ ≤ kn and the number of noncontributors is sufﬁciently small, then (g) the optimal degree of redistribution will be in (0, 1). Results (e)–(g) are explainable as follows (e): This result is a natural extension of Bergstrom et al. (1986) and Uler (2009), although an additional assumption τ b ρn is necessary.9 Let G*− i be the equilibrium value of G − i. The individual with Di − G*− i ≤ 0 does not contribute to providing public goods. The critical value of wealth is calculated as
∂τ 1 1 n−1 ρ b0: = 2− 2 n H ∂A ρ ðA−γÞ
Eq. (9) shows that the optimal tax rate is increasing in the subjective discount rate. In other words, more patience engenders a lower tax rate, although more impatience engenders a higher tax rate. An economy with impatient individuals is equivalent to a world with a low saving rate (or no savings in the extreme case), which is similar to that assumed in previous studies. Intertemporal decision-making of consumption makes it possible to decrease the necessity of redistri8 Then, Eq. (8) with ∂ Wi / ∂ τ = 0 might have some interior solution because the sign of the second order derivative of Wi with respect to τ is ambiguous. Indeed, we cannot 2 A 2 2 2 ðn−1Þ2 α2 − ðn−1Þ α 2 + determine the sign of ρ ∂∂τW2 i = − n−1 2 ≡H: n A−γ
½τ + ð1−τÞαnG −ð1−αÞτAK ; ð1−αÞðρn−τAÞ
where G* stands for the equilibrium value of G. Because the demand function Di is increasing in individual's wealth, individuals with ki N k* are contributors and with ki ≤ k* are non-contributors. (f): Presuming that the incident arises at time t, and differentiating Eq. (12) with respect to τ, then we obtain 1 ∂Di ðtÞ Y−i ð0Þ−G−i ð0Þ + ðn−1Þρki ð0Þ ∂γ n+ = ⋅tf0; ð14Þ Di ðtÞ ∂τ ½n−ðn−1Þτ½ρnki ð0Þ + ð1−τÞnG−i ð0Þ + τY−i ð0Þ ∂τ
︸ ︸ ⊖ ⊕
where Y − i ≡ Y − yi. A rise in the tax rate increases the demand for public goods at t = 0 because dDi(0) / dτ N 0. Therefore, contributors increase their contributions to providing public goods; some non-contributors might be
½1 + ðn−1Þð1−τÞα
However, if ρ is sufﬁciently small, then the sign of the above equation is negative. Eq. (8) is decreasing in τ. Then, ∂ Wi / ∂ τ has a unique interior solution.
9 We cannot determine a critical wealth level or obtain a reversal outcome of (e) that is not natural (because τ N ρn / A is a reversal—progressive taxation) if τ ≥ ρn.
T. Tamai / Journal of Public Economics 94 (2010) 1067–1072
switched to contributors. A rise in the tax rate increases the total supply of public goods temporarily through redistribution of wealth. In contrast, we obtain dDi(∞) / dτ b 0 if t → ∞ because a rise in the tax rate permanently reduces the growth rate of wealth. Its permanent decrease in the growth rate of wealth engenders a decrease in the future income and a decrease in demand for public goods. Then, contributors decrease their contributions to providing public goods; a rise in the tax rate negatively affects the future supply of public goods. (g): We deﬁne the Benthamian welfare function as social welfare. Consequently, social welfare W is n γ 1 W = α ∑ ln ci ð0Þ + ð1−αÞ ln Gð0Þ + : ð15Þ ρ ρ i=1 From the derivative of Eq. (15) with respect to τ, the optimal tax rate is derived as τ* = τ(θ, n, m, A, ρ).10 Note that θ ≡ ∑ ni = m + 1ki / K ∈ (0, 1]. Then, readers wonder what relation between optimal tax rate and inequality exist. Interpreting θ as the inequality index, we can examine the interaction between the optimal tax rate and inequality. The index θ is the wealth share of contributors who are rich. A rise in θ is interpreted as an expansion of inequality. Total differentiation of ∂ W / ∂ τ = 0 gives
∂τ ∂2 W ∂2 W : =− ∂θ ∂θ∂τ ∂τ2 The sign of ∂ τ* / ∂ θ is generally ambiguous. Higher θ reﬂects an expansion of inequality between contributors' wealth and noncontributors' wealth. An increase in contributors' wealth increases the supply of public goods. Its increase in supply of public goods improves welfare, although it reduces the private consumption of non-contributors (∂ G / ∂ θ N 0). That is true because the negative welfare effect on noncontributors' private consumption is dominant over the positive welfare effect on contributors' private consumption if m is sufﬁciently small. Therefore, under a sufﬁciently small number of m, the numerator of ∂ τ* / ∂ θ will be negative. Then, the optimal tax rate is decreasing in wealth inequality if m is sufﬁciently small.
As stated above, the model of the present paper is applicable to the investigation of the intertemporal scheme of providing international public goods or national public goods by interpreting “individual” as “jurisdiction” or “country”. Indeed, our results can present an answer to a problem which ﬁscal decentralization or centralization is desirable. The central (world) government can increase the supply of public goods to adopt a redistributional system that considers local (country) governments' contribution of providing public goods. Lastly, we point out some directions of future research. As described in this paper, we stress the negative impacts of income tax on growth (King and Rebelo, 1990; Rebelo, 1991). However, Easterly and Rebelo (1993) establish a positive relation between income tax and the growth rate. The difference is whether public input ﬁnanced by income tax is incorporated or not. It will provide another insight on redistributional tax if we adopt the Barro (1990) model and examine private provision of public input. Then, an endogenous determination of tax rate, as a politico-economic model, is also important to clarify the effect of redistribution (e.g., Alesina and Rodrik, 1994; Saint-Paul and Verdier, 1996). These topics present important avenues for future investigations. Appendix A. Appendicies on public goods provision, redistributive taxation, and wealth accumulation A.1. Derivation of Eqs. (3) and (4) The Hamiltonian for the individual's optimization problem is "
# τ n H = α ln ci + ð1−αÞ ln G + pi ð1−τÞðyi −gi Þ−ci + ⋅ ∑ yj −gj : n j=1
The optimality conditions are −1
= pi ; −1
n−1 1−α n−1 τ pi ; gi ≥0; − 1− τ pi gi = 0; ≤ 1− n G n ðA:1:2Þ
4. Concluding remarks This paper developed an extension of the static model of public goods provision by Uler (2009). Under the simple intertemporal model of saving, the paper presented examination of the relation among charitable giving, redistributional taxation, and wealth accumulation. Speciﬁcally addressing novel implications, we prove that there exists an inverted-U relation between welfare and redistributional tax rate. A rise in tax rate increases supply of public goods and also welfare at initial time. On the other hand, a redistributional tax is more harmful to welfare through decreased wealth accumulation as saving increases. Therefore, the optimal tax rate is uniquely determined as higher than zero and less than unity. Results also show that the tax rate of optimal redistribution is decreasing in the degree of patience. Furthermore, our dynamic analysis implies that balanced growth of wealth cannot improve inefﬁciency of providing public goods because income expansion under balanced growth has no impact on the relative distribution of wealth. The redistribution plays a certain role of improving the efﬁciency of public goods provision, although stronger patience reduces the optimal tax rate. Interpreting individuals as local (or country) governments, the central government (or world) should be expected as the executor of interregional (or international) wealth redistribution.
n−1 p˙ τ A−ρ = i ; 1− n pi
and limt → ∞pi(t)ki(t) exp(−ρt) = 0. Furthermore, pi is the costate variable of ki. Eqs. (A.1.1) and (A.1.2) engender Eq. (3). Eqs. (A.1.1) and (A.1.3) yield Eq. (4). A.2. Balanced growth equilibrium and derivation of Eqs. (6) and (7) Without non-contributors, we have ˙ ci = ϕG⇔ c˙ i = G ; ci G
c˙ i = C˙ = γ; ci C
K˙ = A− G − C ; K K K
See Appendix A for its derivation.
α n1 1− τ : 1−α n
T. Tamai / Journal of Public Economics 94 (2010) 1067–1072
A.3. Derivation of Eqs. (9) and (10)
Using Eqs. (A.2.1) and (A.2.3), we obtain K˙ = A− 1 + 1 C : nϕ K K
From Eq. (8), the optimal tax rate satisﬁes
n−1 A n−1 A ðn−1Þα ðn−1Þα − + − + = 0: n ρ n A−γ n−ðn−1Þτ 1 + ðn−1Þð1−τÞα
Eqs. (A.2.2) minus (A.2.4) is C˙ − K˙ = C K
1 C n−1 −ρ− τA: 1+ nϕ K n
Total differentiation of the above equation is
Deﬁning X as C / K, Eq. (A.2.5) can be rewritten as X˙ =
1 n−1 1+ X−ρ− τA X: nϕ n
A unique stationary point is unstable; X is a control variable. Therefore, the economy attains the stationary equilibrium immediately. In the stationary equilibrium, all endogenous variables increase at the same rate γ (so-called, balanced growth rate). Consequently, we can solve, respectively, ċi / ci = γ and Ġ / G = γ as ci(t) = ci(0)exp (γt) and G(t) = G(0)exp(γt). With non-contributors, we have Eqs. (A.2.2), (A.2.3), and
n−1 Adρ n−1 Adρ n−1 dA n−1 ρdA + − − + Hdτ = 0: n n n ρ n ð A−γÞ2 ð A−γÞ2 ρ2
Therefore, we obtain " # ∂τ 1 1 n−1 A N 0; = − 2 − n H ∂ρ ρ A−γ2 ∂τ 1 1 n−1 ρ b0: = 2− n H ∂A ρ ð A−γÞ2
Note that A-γ N ρ holds for τ N 0. A.4. Detail of result (g)
˙ ci ≤ϕG⇔ c˙ i ≤ G : ci G
The growth rate of the consumption of private goods is common among all individuals. Therefore, the growth rate of the total supply of public goods is the same as γ. Consequently, we obtain C(t) = C(0)exp (γt) and G(t) = G(0)exp(γt). We now deﬁne γk as K̇/K. Then, Eq. (A.2.3) is rewritten as h t i Gð0Þ + C ð0Þ exp ∫0 ðγ−γk ðsÞÞds : γk ðt Þ = A− K ð0Þ
Assuming that k1 ≤ k2⋯ ≤ kn holds, and letting km as k⁎, then for 1 ≤ i ≤ m (non-contributors), the value of private consumption is ci =
γk K = AK = AK−G−C⇔C + G = 0 This equation contradicts C, G ≥ 0. These contradictions arise from the assumption that γ ≠ γk. Therefore, it establishes γ = γk. In the balanced growth equilibrium, Eq. (5) leads to 1 + ðn−1Þð1−τÞα C ð0Þ: γK ð0Þ = AK ð0Þ−Gð0Þ−C ð0Þ = AK ð0Þ− ½n−ðn−1Þτα ðA:2:8Þ Rearranging Eq. (A.2.8), we obtain C(0) in Eq. (6). Using Eq. (3) (equality case) and C(0) in Eq. (6), we obtain G(0) in Eq. (6). In the stationary equilibrium, the instantaneous utility is α ln ci + ð1−αÞ ln G = γt + α ln ci ð0Þ + ð1−αÞGð0Þ = γt + α lnC ð0Þ−α ln n + ð1−αÞ ln Gð0Þ: Using Eq. (6), the equation presented above is rewritten as ui ðτ; K ð0Þ; t Þ = γt + lnð A−γÞ + α ln ½n−ðn−1Þτ − ln½1 + ðn−1Þð1−τÞα−α ln n + α ln α + ð1−αÞ lnð1−αÞ + ln K ð0Þ: ∞
Then, the welfare of ith individual is Wi = ∫ ui(τ,K(0),t)exp(−ρt) 0 dt. Therefore, it is possible to calculate Wi as Eq. (7).
For m b i ≤ n (contributors), private consumption is ci =
Presuming that γ ≠ γk holds, then the growth rate of wealth γk is negative if γ N γk for all t. When t → ∞, wealth converges to zero and the consumption of private and public goods is inﬁnite. These results are mutually contradictory. The growth rate of wealth converges to A if γ b γk for all t. When t → ∞, the resource constraint (5) and γk = A engenders.
ρn−τA ðY−GÞτ ðY −GÞτ ki + = ρki + −i : n n n
α n−1 1− τ G: 1−α n
Summing up (11) from m + 1 to n and manipulating it, we obtain G=
ð1−αÞ½ðρn−τAÞθ + ðn−mÞτAK ; n−τm + ð1−τÞðn−m−1Þαn
where θ ≡ ∑in= m + 1ki / K a (0,1]. In fact, θ is constant over time because the economy is always in balanced growth equilibrium. When m = 0 (i.e. θ = 1), Eq. (A.4.3) is equal to the supply of public goods in the case in which all individuals are contributors. The partial derivative of Eq. (A.4.3) with respect to also provides the result (f). Using Eqs. (15), (A.4.1), and (A.4.2), we obtain ρ
m ∂W 1 ∂ci ð0Þ ðn−1Þðn−mÞα 1 + ðn−m−1Þα ∂Gð0Þ + =α∑ − ðA:4:4Þ n−ðn−1Þτ Gð0Þ ∂τ ∂τ i = 1 ci ð0Þ ∂τ
1 ∂γ ; ρ ∂τ
where ∂ci ð0Þ Y ð0Þ−Gð0Þ τ ∂Gð0Þ = −i − f 0 for 1≤ i≤m; n n ∂τ ∂τ 1 ∂Gð0Þ ðn−m−θÞA m + ðn−m−1Þαn = + N 0: Gð0Þ ∂τ ðρn−τAÞθ + ðn−mÞτA n−τm + ð1−τÞðn−m−1Þαn
When m = 0 (θ = 1), Eq. (A.4.4) is the same as Eq. (8). If m is sufﬁciently small, then the property of Eq. (A.4.4) is close to that of Eq. (8). References Alesina, A., Rodrik, D., 1994. Distributive politics and economic growth. Quarterly Journal of Economics 109 (2), 465–490. Andreoni, J., 1989. Giving with impure altruism: applications to charity and Ricardian equivalence. Journal of Political Economy 97 (6), 1447–1458.
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