Abstract

Introduction
1
In nearly every developed country, the government finances the cost of various types of public good provision by issuing public debt. Public debt issuance affects household savings and thus, has crucial effects on long-term economic growth and welfare. Several studies show that public debt crowds out physical capital accumulation and so slows down economic growth (e.g., Saint-Paul, 1992; Josten, 2000; Bräuninger, 2005).  This model prediction fits recent empirical evidence (e.g., Checherita-Westphal and Rother, 2012; Reinhart, Reinhart, and Rogoff, 2012; Kumar and Woo, 2015; Chudik, Mohaddes, Pesaran, and Raissi, 2017). 1 Public debt issuance implies an inter-temporal transfer of income, because debt repayment costs are passed onto the future. This suggests a conflict over fiscal pol- icy among different generations, giving fertile ground for politico-economic analysis of public debt.  Given this political background, several studies analyze the politics of pub- lic debt in overlapping-generations frameworks (e.g., Song, Storesletten, and Zilibotti, 2012; Müller, Storesletten and Zilibotti, 2016; Röhrs, 2016).  However, these studies abstract from physical capital accumulation and thus, show nothing about how public debt affects capital accumulation and economic growth via the political process. Two notable exceptions are Cukierman and Meltzer (1989) and Barseghyan and Battaglini (2016). Cukierman and Meltzer (1989) consider majority voting on debt- financed social security in an overlapping-generations model with a neoclassical pro- duction technology. The authors assume that within a generation, there are two types of agents, bequest-constrained and unconstrained agents, and focus on an intra- generational conflict over fiscal policy. An intergenerational conflict is inherent in their model, but little attention is given to that conflict and its impact on growth and welfare across generations. Barseghyan and Battaglini (2016) present an infinitely lived agent model demon- strating economic growth via technology accumulation. Within this framework, they consider fiscal policy determined through legislative bargaining, and investigate its im- pact on economic growth. In particular, they use the model to evaluate the welfare implications of an austerity program that reduces debt below a given debt-ceiling level. However, their analysis is silent on the issue of intergenerational conflict owing to the model assumption of the infinitely lived agent.2 To resolve the above-mentioned issues, this study presents a standard two-period overlapping-generations model with physical capital accumulation. Each  individual lives two periods, youth and old age. We assume a technology represented as a Romer (1986)-type production function to demonstrate endogenous growth. The government 1 provides a public good financed by labor income taxation and/or public debt issuance. The policies are determined in a probabilistic voting modeled by Lindbeck and Weibull (1987), in which in each period, a weighted sum of utility of the young and old is maximized in a competition between political candidates.  Specifically, we focus on Markov perfect equilibrium in which policy proposal today depends on the current payoff-relevant state variables, namely, physical capital and public debt. Based on the above setting, we first demonstrate a case in which the government is allowed to issue public debt in the absence of any legal rules or constraints. We show that the ratio of public debt to GDP decreases as the political power of the old increases. Greater power of the old incentivizes the government to increase public good expenditure. To finance increased expenditure, the government issues more debt and raises the tax rate. A rise in the tax rate in turn works to control public debt issuance. Thus, there are two opposing effects on debt issuance and in the present framework, the negative effect is shown to outweigh the positive one. We also show that the ratio of capital to GDP decreases as the political power of the old increases. The two opposing effects on public debt issuance imply that, given a crowding-out effect of public debt, there are two opposing effects on capital formation. In addition, there is a negative effect on capital accumulation via a rise in the tax rate. Taking these effects together, we show that increased political power of the old results in a decrease in the ratio of physical capital to GDP. In other words, the growth rate decreases as the political power of the old increases. In reality, several developed countries have introduced fiscal rules to control their debt issues from the viewpoint of fiscal sustainability.  For example, the Maastricht Treaty convergence criteria require EU member countries to keep public debt within 60% of GDP. In the United States, the total amount of new bonds that can be issued is limited by the Second Liberty Bond Act of 1917. However, in Japan, there is no such law associated with public debt issuance, although Japan has experienced the highest debt-to-GDP ratio among Organisation for Economic Co-operation and Development (OECD) countries for the past decade. Motivated by these contrasting examples, we undertake the analysis in the presence of a debt-ceiling rule that controls the ratio of public debt to GDP. We show that the introduction of the debt-ceiling rule mitigates the crowding-out effect, raises the growth rate, and thereby improves the welfare of future generations. However, to compensate for the loss of revenue from issuing public debt, the government raises the initial- period tax rate and thereby harms the current generation. Thus, introduction of the debt-ceiling rule creates a trade-off between current and future generations in terms of welfare. The assumption of fixed debt-ceiling rule may fit the Maastricht Treaty convergence criteria, which have not been basically modified since its establishment. However, according to the US Department of the Treasury, the US debt ceiling has been raised 78 times since 1960.3 This case suggests that we should view the debt-ceiling rule as endogenous rather than exogenous. Given this background, we take a step further by introducing voting on the debt-ceiling rule, and show that voters in each period choose no rule, because the rule constrains their choice of fiscal policy. This result could be viewed as a possible explanation for why the US has relaxed its debt-ceiling rules many times since 1960. 4 In addition to the abovementioned studies, the present study is related to the fol- lowing three strands of literature.  The first is the literature on Markov voting on public policy in overlapping-generations models (Hassler, Rodríguez Mora, Storesletten, and Zilibotti, 2003; Forni, 2005; Hassler, Krusell, Storesletten, and Zilibotti, 2005; Hassler, Storesletten, and Zilibotti, 2007; Bassetto, 2008; Gonzalez-Eiras and Niepelt, 2008, 2012; Song, 2011). However, public debt issuance is omitted from their analyses, be- cause they assume a balanced government budget. The present study contributes to the literature by exploring the politics of public policy when public expenditures are financed by taxes as well as debt issues. The second strand is the literature on dynamic political economy analysis of public debt in two-period models (Alesina and Tabellini, 1989, 1990; Persson and Svensson, 1989; Tabellini, 1990) and infinitely lived agent models (Battaglini and Coate, 2008; Caballero and Yared, 2010; Yared, 2010; Azzimonti, Battaglini, and Coate, 2016). The present study departs from these studies by assuming overlapping generations to demonstrate an intergenerational conflict over public debt issuance and its impacts on growth and welfare across generations. The third strand is the literature on time-consistent optimal fiscal policy (Klein and Rios-Rull, 2003; Klein, Krusell, and Rios-Rull, 2008; Ortigueira, Pereira, and Pichler, 2012). In this framework with infinitely lived agents, in each period, the government chooses Markov strategy, that is, current policies depend on payoff-relevant state vari- ables. The present study follows the equilibrium concept of these works but departs from theirs by assuming a short-lived government, representing only existing genera- tions. Under this alternative assumption, we consider the conflict of interest between generations and its generational consequence. The rest of this paper is organized as follows. Section 2 presents the model. Section 3 considers voting on fiscal policy in the absence of any legal constraints on debt issues, and its impact on economic growth.  Section 4 introduces the debt-ceiling rule and investigates its effects on growth and welfare across generations. Section 5 provides concluding remarks.
2
Model
We consider a closed economy model with overlapping generations. Individuals who are born in period t are called generation t (= 0, 1, 2, . . . ). They are homogeneous within each generation and live for two periods, youth and old age. There is no population growth, and the size of each generation is normalized to be unity. Individuals obtain utility from consumption of private and public goods in both periods. Their preferences are specified by
where c i and c i +1 are private consumption when young and old, respectively, and g t and g t +1 are per capita public good consumption when young and old, respectively. The parameter β is the discount factor, and the parameter θ represents the degree of individuals' preferences for public good consumption. When young individuals supply one unit of labor inelastically to firms, obtain wages, and allocate their disposable income between consumption and savings:
where s t , w t , and τ t denote savings, a wage rate, and a labor income tax rate, respec- tively. When old, individuals retire and consume the proceeds of savings:
where R t+1 is the gross interest rate. As economic agents, individuals choose consumption and savings in order to max- imize their utility, taking w t , R t+1, T t , g t , and g t+1 as given.  Solving the utility- maximization problem, we obtain
In the initial period, each old individual, called generation −1, is endowed with s−1 units of physical capital and receives R os_1 units of return from saving. The utility of individuals in generation −1 is represented as log c o + θ log go. There is a continuum of identical firms with a unit mass.  They are perfectly compet- itive profit maximizers that produce output by using a type of Romer (1986) production function, Yt = Akçl where y t is output, A(> 0) is the productivity parameter, k t is physical capital, l t is labor, k t is the aggregate physical capital that works as a technological externality, and α ∈ (0,1) is a constant parameter representing capital share in production. Physical capital is assumed to depreciate fully within each period. Each firm chooses k t and l t in order to maximize its profit, Ak e l l − α − R ι k t − w ι l t, where R t is the rental price of capital and w t is the wage rate.  Because of the assumption of competitive markets, each firm takes R t and w t as given. The first-order conditions with respect to kt and lt are
respectively. Given that k t = k t , the conditions are reformulated as
(3) (4) The government finances public good provision by levying labor income tax and issuing new debt. The budget constraint of the government in period t is
(5) where b t+1 is the one-period debt issued in period t , and G t ≡ 2g t is the aggregate public good provision. We assume that the government is not allowed to hold positive assets, so that bt+1 ≥ 0 holds for all t ≥ 0. The market-clearing condition for capital is
According to this condition, the saving by the young agents in generation t is equal to the sum of the stocks of public debt and physical capital at the beginning of period t + 1. Summarizing the results thus far, we can express the evolution of public debt and physical capital by the following two equations:
(6) and (7) show how public good provision and labor income tax affect Eqs. capital and debt accumulation.  First, an increase in public good provision leads to further debt accumulation, but slows down physical capital accumulation: ∂Z B /∂g > 0 and ∂Z K /∂g < 0. Second, an increase in the labor income tax reduces public debt (∂Z B /∂τ < 0) and promotes physical capital accumulation. However, it reduces the disposable income of young individuals, implying a negative income effect on physical capital accumulation. Thus, the labor income tax has two opposing effects on capital. The positive effect dominates the negative one in the present framework: ∂Z K /∂τ > 0.
The Politics
3
Up to now, the analysis has assumed that fiscal policy is taken as given. However, in the real world, it is determined through political competition, and this in turn affects eco- nomic growth and welfare across generations. In order to demonstrate the competition and its economic impacts, the present study employs probabilistic voting à la Lindbeck and Weibull (1987). In this voting scheme, there is electoral competition between two office-seeking candidates. Each candidate announces a set of fiscal policies subject to the government budget constraint. As demonstrated in Persson and Tabellini (2000), the two candidates' platforms converge in equilibrium to the same fiscal policy that maximizes the weighted-average welfare of voters. Formally, the political objective function in period t is given by
where V t ° and V t ' are the welfare of the old and the young, respectively, and ω ∈ [0,1] and 1 – ω are relative weights on the old and the young in period t , respectively:
V 9 (gt, Tt, gt+1, kt, bt) ~ (1 + β) log kt + (1 + β) log(1 − τι) + θ log gt + βθ log gt+1,    (9) where some irrelevant terms are omitted from the expressions. 5  Thus, the weighted sum of welfare is given by
We employ the concept of Markov-perfect equilibrium in which the fiscal policy today depends on the current payoff-relevant state variables. In the present framework, the payoff-relevant state variables are the physical capital, k t , and the public debt, b t . Thus, the public good provision, g t , the labor income tax rate τ t , and the public debt issue, b t+1 , which are determined in the period- t voting, are represented as functions of these two state variables:
In what follows, we denote the next period's variable by a prime symbol: x t+1 = x ′ , x = k, b, and g. The public debt issue crowds out physical capital accumulation. This implies that when the stock of public debt, b , is high, such that b/k > (1 − α)/α holds, the economy falls into a trivial state even if the government provides no public good and imposes 100% taxation on individuals. Thus, we restrict the domain of the state variables in the following range: Throughout the analysis, we assume (ko, bo) E S. Hereafter, we attach the subscript “ NC ” to each variable to emphasize that apart from the domain of the state variables, S , there is no constraint on public debt issues. The following gives the definition of the Markov-perfect politico–economic equilibrium. Definition 1. A Markov-perfect politico-economic equilibrium is a set of policy func- tions ( G NC, T NC, B NC ), where G NC is a public good provision rule, g = G NC (k, b), T NC is a tax rule, τ = T NC (k,b), and B NC is a public debt rule, b' = B NC (k,b), such that given k  and b, 1. the pair of functions ( GNC,TNC ) satisfies the following:
2. given ( G NC , T NC ), the function B NC satisfies the government budget constraint: B NC (k, b) = Z B [G NC (k, b), T NC (k, b), k, b].
3.1
Politico–economic Equilibrium
We focus on a situation in which the public good provision is represented as a linear function of physical capital and public debt. In particular, we conjecture the following public good provision function:
where δ 1 > 0 and δ 2 > 0 are constant variables. We use the capital market-clearing condition and the government budget constraint to reformulate this expression as
Eq. (12) indicates opposing effects of current public good provision and tax rate on the next-period public good provision.  An increase in the current public good provision, g , slows down physical capital accumulation but accelerates public debt issue, and thus, lowers public good provision in the next period, g ′ (i.e., ∂g ′ /∂g < 0). However, an increase in the current tax rate, τ , promotes physical capital accumulation but reduces public debt issue, and hence, raises public good provision in the next period, g ′ (i.e., ∂g'/∂τ > 0). The first-order conditions of the functional equation (11) with respect to g and τ are given by
and
respectively. (13) and (14) present the optimal choice of g and τ , respectively, by the Eqs. political candidates. They choose g (or τ ) to equate its marginal cost and benefit in terms of welfare. In Eq. (13), the left-hand side is the marginal benefit of the current public good provision in terms of welfare of the young and old, whereas the right-hand side is its marginal cost; an increase in g reduces the public good provision in the next period and thus, reduces the benefits that the current young will enjoy in their old age. (14), the left-hand side is the marginal benefit of the increased public good In Eq. provision in the next period, which is created by an increase in the current tax rate; and the right-hand side is the marginal cost of the increased tax rate, which implies a decrease in disposable income and thereby a decrease in lifetime income of the young. The first-order conditions in (13) and (14) lead to the following relationship between g and τ :
Substituting (1 – α) Ak τ of Eq. (15) into Eq. (12) yields
Furthermore, by substituting g ′ of Eq. (16) into Eq. (13), we obtain
8 where φ is defined as φ ≡ θ + (1 − ω)[1 + β(1 + θ)]. Thus, the initial guess is verified as long as the following holds:
We next derive the tax function, T NC . Given δ 1 and δ 2 derived above, the expression in Eq. (15) is reformulated as
The function T NC is increasing in the public debt/physical capital ratio, b/k , and sat- isfies the following:
If the parameters satisfy
then, T NC (k, b) ∈ [0, 1] holds for any (k, b) ∈ S . Finally, we derive the public debt function, BNC . Substituting g = GNC(k,b) and T = T NC (k,b) into the government budget constraint and rearranging the terms, we obtain
If the parameters satisfy
then, B NC (k,b) ≥ 0 for any (k,b) ∈ S. Furthermore, substituting g = G NC (k,b) and T = T NC (k, b) into the transition equation of physical capital yields
9 The functions B NC and Z K imply
Thus, ( b ′ , k ′ ) ∈ S as long as (A.1) holds. The analysis up to now is summarized in the following proposition. Proposition 1. Suppose that (A1) and (A2) hold: θ ∈ [θ,θ] .  There is a Markov-perfect politico-economic equilibrium distinguished by the following policy functions:
The physical capital accumulates according to
The policy functions in Eqs. (18), (19), and (20) and the physical capital formation in Eq.  (21) suggest that they depend on the debt-to-capital ratio, b/k, the preference for the public good , θ , and the relative political weight of the old, ω . We investigate how these factors affect the formation of physical capital and policy functions. In particular, we eliminate the scale effect by taking the ratios of aggregate public good expenditure, debt issues, and physical capital to GDP as
b/k, θ, and ω. Corollary 1. Consider the Markov-prefect politico-economic equilibrium demonstrated in Proposition 1. 1. A higher ratio of public debt to physical capital raises the tax rate, but lowers the ratio of aggregate public good expenditure to GDP, the ratio of debt issues to GDP, and the ratio of physical capital to GDP; that is, ∂T NC/∂(b/k) > 0, ∂Ğ NC /∂(b/k) < 0, ∂Ğ NC /∂(b/k) < 0, and ∂Ğ NC /∂(b/k) < 0. 2. A stronger preference for public good provision raises the tax rate and the ratio of aggregate public expenditure to GDP; lowers the ratio of public debt issues to GDP; and raises the ratio of physical capital to GDP if and only if α < 1 – ω; that is , ∂T NC /∂θ > 0, ∂Ğ NC /∂θ > 0, ∂Ğ NC /∂θ < 0, and ∂Ř NC /∂θ ≥ 0 if and only if α ≤ 1 − ω. 3. A higher relative political weight on the old raises the tax rate and the ratio of aggregate public good expenditure to GDP, but lowers the ratio of public debt issues to GDP and the ratio of physical capital to GDP; that is, ∂T NC /∂ω > 0, ∂ĞNC/∂ω > 0, ∂ĞNC/∂ω < 0, and ∂歐NC/∂ω < 0.
Proof. See Appendix A.

The effects on the tax rate and the ratio of aggregate public good expenditure to GDP are discussed in Appendix B. Here, we focus on the ratio of debt issues to GDP and that of physical capital to GDP, which are relevant to the following analysis. Corollary 1 shows that the ratio B NC decreases as the ratio of debt to physical capital, the preferences for public good, and the relative political weight on the old increase. To understand the mechanism behind this result, recall the government budget constraint that is rewritten in terms of B NC , G NC , and T NC as
where the term α · ( b / k ) = ( α Ab )/( Ak ) represents the ratio of repayment cost to GDP. Differentiation of B NC with respect to b/k leads to
The expression shows that there are three effects of an increased b/k on the ratio of debt issues to GDP. First, the government issues more public debt to finance increased debt repayment. Second, given the revenue, the government cuts public good expendi- ture in response to increased debt repayment. This enables the government to reduce public debt issues.  Finally, the government raises the tax rate to finance increased debt repayment. This in turn incentivizes the government to reduce debt issues. In the present framework, the last two negative effects on public debt issues outweigh the first positive effect. Therefore, the ratio of public debt issues to GDP decreases as the ratio of debt to physical capital increases. This result implies that increased debt repayment cost induces the government to strengthen fiscal discipline. Corollary 1 shows that the ratio of public debt issues to GDP is also affected by the relative political weight on the old (ω) and the preferences for public good (θ). To inspect the effects, we differentiate BNC with respect to θ and ω to obtain
expenditure.  To finance increased expenditure, the government increases public debt issues and raises the tax rate. However, an increased tax rate enables the government to control public debt issues. In other words, there are two opposing effects of θ on public debt issues, but the negative effect outweighs the positive one in the present framework. Therefore, fiscal discipline could be strengthened as the preference for the public good increases. Next, consider the effect of increased weight on the old, ω , on public debt issues. A greater weight on the old incentivizes the government to increase public good ex- penditure, because the old have no tax burden but benefit from increased expenditure. This incentive implies that the government finances increased expenditure by issuing more public debt as well as raising the tax rate. A rise in the tax rate in turn works to control public debt issues. Thus, increased ω exhibits two opposing effects on public debt issue, and the negative effect outweighs the positive one in the present framework. To consider the effects of b/k, θ, and ω on the ratio of physical capital to GDP, k'/Ak = K NC (k,b), recall the capital market-clearing condition presented in Section 2, which is rewritten as
or
Eq. (25) indicates that the ratio K NC (k,b) decreases as the ratio of debt to physical This is because the government is incentivized to issue more capital, b/k, increases. public debt to finance its increased debt repayment. This creates a crowding-out effect on physical capital formation, and thereby results in a decrease in the ratio of physical capital to GDP. The ratio K NC (k,b) also decreases as the political weight on the old increases. As already argued above, a greater weight on the old incentivizes the government to increase public good expenditure.  To finance increased expenditure, the government increases public debt issues, which produces a crowding-out effect on physical capital formation.  The government also raises the tax rate to finance part of its increased expenditure, which produces two opposing effects on the physical capital formation, as observed in the above expression, but the net effect is positive. Taking these effects together, we find that increased political weight on the old leads to a decrease in the ratio of physical capital to GDP. Finally, consider the effect of the preference for public good, θ , on the ratio of phys- ical capital to GDP. A higher preference for the public good also gives the government an incentive to increase public good expenditure and to raise the tax rate, as in the case of increased political weight on the old. However, these effects are qualitatively different from those of increased political power. Our analysis shows that the positive effect via the tax rate outweighs the negative effect via the public good expenditure if and only if α < 1 – ω holds.
Economic Growth
3.2
Based on the characterization of the politico-economic equilibrium in the previous sub- section, we consider how public debt and physical capital evolve over time. For this purpose, recall Eq.  (17), indicating that the ratio of public debt to physical capital, bt/kt, remains constant for any t ≥ 1:
The ratio converges to x N C within one period, and thereafter, the stock of public debt and physical capital grow at the same rate. In other words, the economy exhibits a balanced growth path (BGP). We use Eq.  (26) to reformulate the equation of physical capital formation in Eq. (21) as follows:
where γ NC , the growth rate of physical capital in the BGP, is defined by
The growth rate along the BGP has the following properties. Proposition 2. Consider the Markov-perfect politico-economic equilibrium presented in Proposition 1. The growth rate of physical capital in the BGP, YNC, is increasing in θ and decreasing in ω; that is, θγνς/θθ > 0 and θγνς/θω < 0. Proof.  Differentiating γ NC with respect to θ and ω , we obtain the result in Proposition 2. To understand the intuition behind the result in Proposition 2, recall the capital market-clearing condition in Eq. (25),
When the debt-to-capital ratio, b/k, is given, the effects of increased θ and ω on the growth rate k'/k are immediate from the result in Corollary 1. However, along the BGP, the ratio b/k is constant across periods and is affected by θ and ω , as follows:
where the second equality comes from Eq.  (17). This expression suggests that we must take into account the effects of increased θ and ω on the debt-to-capital ratio when we consider their effects along the BGP. Based on the argument thus far, we present the growth rate along the BGP, denoted by γ NC = k'/k, as follows:
where G* NC and T NC are the corresponding values of X NC
These expressions indicate that the ratio of public good expenditure to GDP and the tax rate are affected by increased θ and ω via the debt-to-capital ratio, x NC To observe the effects of θ and ω through the term x N C in more detail, recall the functions B NC (k,b) and K NC (k,b), which could be included in the expression of x NC as follows:
where k_ and b_ denote the previous-period values of k and b , respectively. As demon- strated in Corollary 1, B NC is decreasing in θ and ω ; K NC is decreasing in ω , and is increasing (decreasing) in θ if and only if α < (>)1 − ω . After some manipulation, we find that the net effect of θ on x N C is negative: ∂x N C/∂θ < 0; and the effects of ω on B NC and K NC are cancelled out: ∂x NC /∂ω = 0. This result, associated with the result in Corollary 1, provides the growth effects of increased θ and ω along the BGP as presented in Proposition 2.
4
Debt-ceiling  Rule
In the previous section, we undertake the analysis without imposing any constraint or limit on public debt issues except the domain S . However, in the real world, some fiscal rules are introduced in developed countries to control their debt issues from the viewpoint of fiscal sustainability. We here introduce into the model a debt rule that controls the ratio of public debt to GDP in order to investigate its impacts on growth and welfare across generations. In particular, we consider a rule, b ′ /k ′ ≤ η, implying that the ratio of new debt issue to physical capital should be equal to or below η(> 0). Given the AK technology, this is qualitatively equivalent to the upper limit constraint of the debt-to-GDP ratio. Thus, our analysis provides some insight into the effects of the debt rule, like the Maastricht Treaty convergence criteria, from the viewpoint of growth and welfare.
4.1
Characterization
To proceed with the analysis, we first consider the ratio b /k ′ in the absence of the constraint, given by Eq. (17). The constraint, b /k' ≤ η , becomes binding if the ratio in Eq. (17) is above η :
Given this condition associated with (A.1) and (A.2) imposed in the previous section, we obtain the following characterization of the political equilibrium in the presence of the debt-ceiling rule. The subscript " DC " in each variable means that the debt-ceiling constraint is binding. Proposition 3. Suppose that (A.1) and (A.2) hold: θ ∈ [θ,θ]. 1. If (A.3) fails to hold, the debt-ceiling constraint, b'/k' ≤ η, is non-binding. The allocation and the set of fiscal policies are identical to those in the absence of the debt-ceiling rule. 2.  If (A.3) holds, the debt ceiling is binding. The set of policy functions are given as follows:
Proof. See Appendix C.
Figure 1 illustrates (A.1)–(A.3) in an η-θ space. The shaded area illustrates the set of parameters that satisfy (A.1) and (A.2). They are below the downward-sloping curve, which corresponds to the set of parameters that satisfy (A.3). Therefore, the debt-ceiling rule is binding if a pair of parameters ( η,θ ) is located in the wave area; meanwhile, the rule is not binding if the pair is located in the shaded area but not the wave area. The figure shows that the debt-ceiling constraint is more likely to be binding if η and θ are lower. The intuition behind a lower η is straightforward. A lower θ implies that the young are less concerned about future public good provision. This gives the government representing the young less incentive to control current fiscal policy and thus, public debt issues.  Because of this lack of fiscal disciplining effect, the debt ceiling becomes more likely to be binding as θ decreases. In what follows, we assume that conditions (A.1)–(A.3) hold. Under these assump- tions, we demonstrate the dynamic motion of public debt and physical capital accumu- lation. Because the constraint, b ′ k ′ ≤ η , is binding by assumption, b t+1 / k t+1 = η holds for any t ≥ 0. In other words, the ratio of public debt to physical capital converges to η within one period, and thereafter, public debt and physical capital grow at the same rate. From Eq. (29), the growth rate of physical capital when the debt ceiling is binding is given by
where γ DC denotes the growth rate of physical capital in the BGP when the debt-ceiling constraint is binding.
4.2
Effects of Debt-ceiling Rule
The result up to now implies that we could obtain different policy functions of tax and public debt issues and the law of motion of capital for the two scenarios, that is, for the absence and presence of a debt ceiling, although the policy functions of public goods are identical. To investigate the differences in detail, we compare the two scenarios in terms of the growth rate of capital, k ′ / k | j , the tax rate, T j , and the ratio of aggregate public expenditure to GDP, Ğ j , where j = NC , DC . We first compare them in period 0. Proposition 4. Suppose that (A1)–(A3) hold.  The introduction of the debt ceiling in period 0 raises the growth rate and the tax rate but has no effect on the ratio of aggregate public expenditure to GDP in period 0: k1/k0| NC < k1/k0| DC , T NC (k0, b0) < TDC(k0, b0), and GNC(k0, b0) = GDC(k0, b0). Proof. See Appendix D.
●
The result in Proposition 4 indicates that in period 0, the growth rate in the presence of the debt-ceiling rule is higher than in its absence, because the rule mitigates the crowding-out effect of public debt on capital accumulation. The result also indicates that in period 0, the tax rate in the presence of the rule is higher than in its absence. This is because when the debt-ceiling constraint is binding, the government needs to compensate for its loss of revenue from issuing public debt by raising the tax rate. Thus, the introduction of the debt-ceiling rule creates a tax-hike effect in period 0. The result in Proposition 4 also suggests that the introduction of the debt-ceiling rule has no effect on the ratio of aggregate public expenditure to GDP in period 0. To understand this result, recall the political objective function in Eq.  (10).  The introduction of the rule raises the tax rate, and thus, increases the tax burden on the young, as observed by the third term in Eq. (10). However, the rule accelerates physical capital accumulation, and thus, increases the future provision of the public goods. This benefit, observed by the fifth term in Eq.  (10), is offset by the aforementioned cost.  This result implies that the benefit stemming from current public good provision, observed by the fourth term, is unaffected by the other terms. Therefore, the introduction of the rule has no effect on the ratio of aggregate public expenditure to GDP in period 0. We next consider the effects of the debt-ceiling rule from period t = 1 onward. As demonstrated in Section 3 , the ratio b/k is constant for t ≥ 1 because the economy follows the BGP . The growth rate, the ratio of aggregate public expenditure to GDP, and the tax rate are denoted by γ j ,    , and Proposition 5. Suppose that (A1)–(A3) hold. The introduction of the debt ceiling in period 0 raises the growth rate and the ratio of aggregate public expenditure to GDP for period t ≥ 1: YNC < YDC and G N C < G DC ; and it raises the tax rate for period t ≥ 1, that is, T N C < T N C if and only if the following condition holds:
Proof. See Appendix E.
The effect on economic growth from period 1 onward is qualitatively similar to that in period 0. However, the effect on public good provision differs from that in period 0. For period t ≥ 1, the available resources for the government, (1 — α) Ak — Rb, are larger in the presence of the constraint than in its absence, because the constraint stimulates capital accumulation and lowers debt accumulation.  Because of this difference, the ratio of the public good to GDP is higher in the presence of the constraint than in its absence. In addition, the effect on the tax rate in period t (≥ 1) differs from that in period 0. As demonstrated in Proposition 4, the debt-ceiling rule creates a tax-hike effect. However, the introduction of the rule lowers the debt-repayment cost from period 1 onward, and this produces an opportunity for the government to cut the tax rate. Proposition 5 shows that this tax-cut effect is outweighed by the tax-hike effect if (A.4) holds. The growth rate is higher in the presence of the debt-ceiling rule than in its absence, because the rule mitigates the crowding-out effect of public debt on physical capital accumulation. This implies a positive income effect on future generations. However, owing to the limitation of public debt issues, the government might want to raise the tax rate to compensate for the loss of revenue from public debt issues. This outcome would imply a negative income effect on current and future generations. Thus, there would be two opposing effects of the debt ceiling on welfare across generations. The following proposition summarizes the welfare effects for each generation. Proposition 6. Suppose that (A.1)–(A.3) hold. 1. The welfare of generation –1 is unaffected; generation 0 is made worse off by the introduction of the debt ceiling. 2. There is a critical period, denoted by Î(> 1), such that generation t ≤ Î is made worse off whereas generation t > Î is made better off by the introduction of the debt-ceiling rule. Proof. See Appendix G.

Voting on Rules
4.3
The analysis up to now has assumed that the debt-ceiling rule is exogenously given. This assumption fits the Maastricht Treaty convergence criteria, which have not been modified since its establishment. However, according to the US Department of the Treasury, the US debt ceiling has been raised 78 times since 1960. This case suggests that we should view the debt ceiling as an endogenously determined institution rather than an unchanged constitutional rule. Given this background, we now introduce a voting-based fiscal rule in the following way. First, in each period, office-seeking candidates propose two alternatives, that is, no constraint on public debt issues, as presented in the previous section, and the debt- ceiling rule, as presented in this section. One alternative is chosen through voting. Second, for a given rule determined in the first stage, the candidates propose a set of fiscal policies (new public debt issues, public good provision, and the tax rate). We solve this two-stage game by backward induction. We have already solved the second-stage problem, so we now solve the first-stage problem. Let Ω NC,t denote the period-t political objective functional in the absence of the constraint on public debt issues, and Ω DC.t denote the period- t political objective functional in the presence of the debt-ceiling rule. Voters in period t choose no rule (the debt-ceiling rule) if Ω NC,t ≥ (<)Ω DC,t. By comparing Ω NC,t and Ω DC,t , we obtain the following result. Proposition 7. Suppose that (A.1)–(A.3) hold. In each period, voters choose no rule: Ω NC,t > Ω DC,t holds for any (k,b) ∈ S. The intuition behind the result is straightforward. In the absence of the rule, the government proposes fiscal policy that maximizes its objective.  However, in the presence of the rule, the government's choice is constrained by the rule. In particular, the government is forced to issue less debt in order to meet the debt-ceiling rule, which mitigates the crowding-out effect on physical capital accumulation and thereby creates However, in order to compensate for the loss of revenue a positive growth effect. stemming from the debt-issue constraint, the government might choose a higher tax rate. Proposition 6 shows that this increased tax burden outweighs the benefit from the increased growth rate for the government representing currently living young and old agents. The result is in line with the finding of Azzimonti, Battaglini, and Coate (2016), who report that the balanced budget rule could be costly in the short run but may offer benefit in the long run. Alesina and Passalacqua (2016, p.2632) argue that their result leads to "interesting and immediate consequences on the political economy implications on voting upon a balanced budget rule in say, an overlapping generations model." Our result suggests a difficulty of preventing the introduction of a less severe budget rule in the overlapping-generations model. Thus, the result could be interpreted as providing a possible explanation for why the US has raised its debt-ceiling rules many times since 1960.
5
Conclusion
This study presents a two-period overlapping-generations model with physical capital accumulation. The technology is represented as a Romer-type production function to demonstrate endogenous growth.  The government representing young and old gen- erations provides public goods financed by labor income taxation and/or public debt issuance. The policies are determined in probabilistic voting, in which in each period, the weighted sum of utility of the young and old is maximized in a competition between political candidates. Within this framework, we show that increased political power of the old incentivizes the government to increase public good expenditure. To finance increased expenditure, the government issues more debt and raises the tax rate. A rise in the tax rate in turn works to control public debt issuance. Thus, there are two opposing effects on the debt issuance. Given a crowding-out effect of public debt, this result implies two opposing effects on capital formation, that is, economic growth. In addition, there is a negative effect on capital accumulation via a rise in the tax rate. Taking these effects together, we find that increased political power of the old is harmful to economic growth. To resolve the negative growth effect, we introduce a debt-ceiling rule that controls the ratio of public debt to GDP. The introduction of the debt-ceiling rule mitigates the crowding-out effect, raises the growth rate, and thus, benefits future generations. How- ever, it creates an inter-generational trade-off in terms of welfare, since the government is incentivized to raise the initial-period tax rate and thus, harms the current gener- ation. We further investigate voting on the choice of the debt-ceiling rule, and show that it is never implemented, since current generations are made worse off. This result provides a possible explanation for why some countries fail to strengthen an existing debt rule.
Appendix
A. Proof of Corollary 1
The effects of b/k, θ, and ω on T NC , Ğ NC , and Ř NC are immediate from Eqs.  (19), (22), and (23). The effects of b/k and ω on Î NC (k,b) are also immediate from Eq. (24). To observe the effect of θ on
Given that θ/ { 1 + β( 1 + θ)} is increasing in θ, we can conclude that increasing in θ if and only if α < (1 − ω).
B. Comparative statics on G NC and T NC
In this appendix, we preset the comparative statics analysis for the ratio of aggregate public expenditure to GDP, G NC , and the tax rate, T NC . First, consider the ratio of aggregate public expenditure to GDP. Given g , the transition equations of k and b are represented as
and
respectively, and the level of public good provision in the next period is given by
We use this expression and 2( δ 1 + δ 2 ) = (θ A )/φ to reformulate the first-order condition with respect to g in Eq. (13) as follows:
or,
where ğ ≡ (2g)/( Ak ). The left-hand side of Eq.  (B.1) corresponds to the marginal benefit of increased current public good provision, and the right-hand side corresponds to the marginal cost of increased current public good provision. We use Eq.  (B.1) to understand how b/k, θ, and ω affect the ratio,  GNC(k, b). We can summarize the properties of M B g and M C g as follows. ● The marginal benefit, M B g , is independent of b/k . An increase in b/k not only raises the repayment cost/output ratio, ( αb )/k , but also raises the tax rate, T NC . In the present framework, the former effect dominates the latter one, and hence, the marginal cost, MC g , is increasing in b/k . Thus, G NC is decreasing in b/k . ● An increase in θ equally raises both the marginal benefit, M B g , and the marginal cost, MC 3 , as observed in the numerators on both sides. In addition, an increase in θ raises the tax rate, T NC , and thus, lowers the marginal cost. Therefore, an increase in θ raises the marginal benefit more than the marginal cost. This result implies that G NC is increasing in θ . ● The marginal benefit, M B g , is independent of ω . An increase in ω not only lowers the marginal cost, MC g , as observed in the numerator of the term representing MC g , but also raises the tax rate, T NC , and thus, lowers the marginal cost.  Owing to these negative effects on the marginal costs, G NC is increasing in ω . Next, consider the tax rate. Given τ , the transition equations of k and b are repre- sented as
and
respectively, and the public good provision in the next period is given by
Given that δ 1 /(1+β)+δ 2 = (1/2)(θ/φ)[(1+αβ)/(1+β)] A holds, the first-order condition with respect to τ in Eq. (14) is rewritten as
The left-hand side (right-hand side) of Eq.  (B.2) represents the marginal cost (marginal benefit) of increasing the current tax rate. We use Eq. (B.2) to understand how b/k, θ, and ω affect the tax rate, T NC ( k, b ). We summarize the properties on MC τ and M B τ as follows. ● The marginal cost, MC r , is independent of b/k . An increase in b/k raises the repayment cost/output ratio, ( α b )/k , and lowers the public good/output ratio, G NC . In the present framework, the former effect dominates the latter one, and thus, the marginal benefit, M B r , is increasing in b/k . Thus, T NC , is increasing in b/k. ● The marginal cost, MC r , is independent of θ .  An increase in θ raises the marginal benefit, M B r , as observed in the numerator. An increase in θ also raises the ratio G NC , and thus, raises the marginal benefit. Thus, T NC , is increasing in θ . ● An increase in ω equally lowers both the marginal benefit, MC τ , and the marginal cost, MC r , as observed in the numerators of both terms. In addition, an increase in ω raises the ratio, an increase in ω raises the marginal benefit, M B τ , more than the marginal cost, MC 7 . This result implies that the tax rate, T NC , is increasing in ω .
C. Proof of Proposition 3
To prove the first part, suppose that the constraint b ′ / k ′ ≤ η is not binding. Then, the solutions of b ′ and k ′ are given by Eqs.  (20) and (21), respectively. Thus, the constraint is actually non-binding if and only if the following condition holds:
To prove the second part, assume that (A.3) holds. Recall the motions of public debt and physical capital presented in Eqs. (6) and (7):
These expressions hold regardless of the presence of the constraint.  Substitution of these into the constraint b' = ηκ' leads to
We also use Eq.  (C.1) to represent the disposable income of the young, (1 − τ )(1 − α ) Ak, as follows: The welfare of young individuals in Eq. (9) is now reformulated as V Bc (k, b, g, g') ~ (1 + β) log[(1 − α) Ak − αAb − 2g] + θ log g + βθ log g', and the corresponding political objective function, denoted by Ω DC , becomes Ω DC ≥ (1 − ω)(1 + β) log[(1 − α) Ak − αAb − 2g] + θ log g + (1 − ω)βθ log g ′ .   (C.4) Thus, the problem of the government is to choose g to maximize Ω DC subject to Eqs. (C.2) and (C.3). To solve the maximization problem, we conjecture the following linear public good provision function: where ε 1 and ε 2 are constant.  With Eqs.  (C.2) and (C.3), this is reformulated as follows: With Eq. (C.5), we can reformulate Eq. Ω DC as Ω DC ~ (1 − ω)[1 + β(1 + θ)] log[(1 − α) Ak − αAb − 2g] + θ log g. The first-order condition with respect to g gives The left-hand side (right-hand side) of Eq.  (C.6) represents the marginal benefit (marginal cost) of increasing the current public good provision, g. Solving Eq. (C.6) for g , we obtain
Note that GDC(k,b) ≥ 0 holds for any (k,b) ∈ S. Therefore, the function g = ε 1 k − ε 2 b constitutes a Markov perfect politico–economic equilibrium as long as the following holds:
Substituting g = GDC(k,b) into Eqs.  (C.1) and (C.2), we obtain the corresponding policy functions of τ and b':

D. Proof of Proposition 4
Recall the policy functions and the law of motion of physical capital presented in Propo- sitions 1 and 3. First, we compare GNC(k 0 , b 0 ) and GDC(k 0 , b 0 ), and obtain
Next, we compare the growth rate in period 0 and obtain
where the last expression is equivalent to (A.3). Direct comparison of the tax rate in period 0 also yields
Therefore, k 1 /k 0 | NC < k 1 /k 0 | DC and T NC (k 0 , b 0 ) < T DC (k 0 , b 0 ) hold if (A.3) holds.
E. Proof of Proposition 5
Recall that k 1 /k 0 | NC < k 1 /k 0 | DC for a given k 0 /b 0 , as demonstrated in Proposition 4. This result implies that given b t /k t , k t +1/k t /k t +1/k t |DC holds for t ≥ 1. The growth rate is decreasing in bt/kt , and the ratio bt/kt is lower in the presence of the debt-ceiling constraint than in its absence. Therefore, kt+1/kt| NC < kt+1/kt| DC holds for t ≥ 1, that is, γ NC < γ DC · Next, recall that call also that ¯ j ( k t , b t ), j = NC , DC and is decreasing in b t /k t , and that the ratio b t /k t is lower in the presence of the debt-ceiling constraint than in its absence. Therefore, we obtain Ã¶¢ < G¶¢· Finally, we compute T * C and T * C (t ≥ 1) as follows:
To find a set of parameters that satisfy Eq. (E.1), we differentiate LHS and RHS in Eq. (E.1) with respect to θ and obtain
increasing and strictly concave in θ , as illustrated in Figure 2. After some manipulation, we obtain the following equivalence relation:
where
In addition, under θ = χ(η), we obtain
We also obtain
where the proof of Eq. (E.3) is provided in Appendix F. Thus, we obtain
that is,
Figure 3 illustrates the graph of ψ(η) = 0. From the figure, we find that ψ(η) < 0 holds if and only if
where η = − (1 + β) + √β² + β/α is a solution to ψ(η) = 0.  Given that η is bounded above (1 − α)/ [α + β (1 − ω)], as illustrated in Figure 1, the latter condition is sum-
and only if α < β / (1 + 2 β ) and Eq. (E.4) hold.
F. Proof of Eq.  (E.3)
We compare LHS and RHS in Eq. (E.1) at θ = θ and obtain
LHS and RHS, respectively. We differentiate LHS with respect to η and obtain
This leads to
Suppose that ψ( η ) ≥ 0 holds. From Eq. (E.2), ∂LHS/∂θ ≤ ∂RHS/∂θ| g =x( η ) holds, as illustrated in Panel (a) of Figure 2.  Thus, we obtain LHS > RHS for any θ ∈ [θ, χ (η)). Alternatively, suppose that ψ(η) < 0 holds. From Eq. (E.2), ∂LHS/∂θ > ∂RHS/∂θ| θ = χ(η) holds, as illustrated in Panel (b) of Figure 2. In addition, when ψ(η) < 0, ∂LHS/∂η > 0 holds from Eq.  (F.1).  Thus, LHS is maximized at the upper limit of η, η = (1 − α)/ [α + β (1 − ω)]. Furthermore, some manipulation leads to
This result implies that LHS ≤ RHS, that is, LHS| o − e < RHS| o − holds for any η ∈ (0, (1 − α)/ [α + β (1 − ω)]) . Therefore, when ψ(η) < 0, LHS < RHS holds for θ ∈ [θ, χ (η)), as illustrated in Panel (b) of Figure 2.
G. Proof of Proposition 6
1) The welfare of generation −1 is unchanged by the introduction of the debt-ceiling rule, because the period-0 public good provision is unchanged, as demonstrated in Proposition 4. Under (A.3), the debt-ceiling constraint is binding. This result implies that the choice of the period-0 government is constrained by the debt ceiling and the period-0 government attains a lower value of its objective in the presence of the constraint than in its absence, Ω NC,0 > Ω DC,0 , where Ω j,0 denotes the period-0 government objective when j = NC , DC . The expression Ω NC ,0 > Ω DC ,0 is reformulated as
where V j 0 denotes the period-0 old utility and V j 0 denotes the period-0 young utility when j = NC, DC. Given that V NC,0 = V BC,0, we obtain
2) As for the welfare of generation t (≥ 1), recall its indirect utility in Eq. (9): V j,t ~ (1 + β) log k j,t + (1 + β) log(1 − τ j,t ) + θ log g j,t + βθ log g j,t +1, where j = NC , DC . Along the BGP, the tax rate is constant at T* | j , and the physical capital and the public good grows at a constant rate:  kt+1/kt+1/kt+1/g = γ j and gt+1/gt+1/gt+1/j = γ j . Thus, for t ≥ 1, we obtain
The indirect utility above is reformulated as

29
We compare V NC,t and V NC,t directly and obtain
where X is defined by
Therefore, we conclude that V N C,t ≤ V B C,t if and only if ˆ t ≤ t.
Acknowledgements
We are grateful to Akihisa Shibata, Kazuo Mino, Koichi Kawamoto, Ryoji Ohdoi, and Motohiro Sato for their helpful comments and suggestions. We also thank the par- ticipants of ETPW at Tokyo Metropolitan University, the CAES Seminar at Fukuoka University, 5th macroeconomics conference for young economists, the 2011 JEA Spring Meeting at Kumamoto Gakuen University, the 2013 Japan Public Choice Society Meet- ing at Komazawa University, and the Public Economics Workshop at Hitotsubashi University. Funding: Ono acknowledges financial support from JSPS KAKENHI Grant Num- ber JP15K03509.  Arai acknowledges financial support from JSPS KAKENHI Grant Number 23730298.