Public spending on maintenance and imperfect competition.

Lee, Jin-wen

I. Introduction

Since the seminal works of Barro (1990) and Futagami, et al. (1993), much attention has been paid to studying the role of productive government expenditure in the endogenous growth context. This framework implicitly assumes that total expenditure related to the public capital accumulation process of the economy is oriented toward the formation of 'new' public capital. However, the government faces an option between building 'new' infrastructure and increasing maintenance expenditure. The maintenance work done in the present is not only a contribution to current final output, but to the final output of future years. Since increasing maintenance spending affects the durability of public capital, it therefore raises the productivity of public capital. Rioja (2002) pointed out that the benefit from a doubling of maintenance expenditures financed by a cut in expenditures for new projects was substantial in Latin America.

However, little work has been done so far in exploring the effects of maintenance spending on public capital formation. This issue is illustrated by Rioja (2003a, 2003b) and Kalaitzidakis and Kalyvitis (2004). Rioja (2003a) assumed that the depreciation rate of public capital is determined by public expenditure on maintenance. Rioja (2003b) demonstrated the optimal maintenance level, but did not analyze the trade-off between maintenance and new investment. By contrast, Kalaitzidakis and Kalyvitis (2004) developed an endogenous growth model where the durability of public capital depends on its usage and the level of maintenance expenditure. They also accounted for the trade-offs between infrastructural spending and other components of government expenditure. However, they did not analyze the impact of taxation on the consumption decision.

We develop an infrastructure-led two-sector endogenous growth model in which public and private capital stocks are entered directly into the production function, as in Turnovsky (1997). As in Agenor (2005), we assume that maintenance spending by the public sector not only increases the durability of public capital, but also raises the efficiency of the infrastructure. We also discuss the impact of maintenance spending on the private capital stock. It now becomes plausible to suggest that the maintenance spending by the public sector enhances the productivity of the existing public capital as well as the private capital stock. Maintaining the quality of infrastructure (e.g., public roads, the system of electric wires or pipes carrying gas) enhances the durability of private equipment (e.g., trucks and electric equipment).

Dixon (1987) noted that imperfect competition is a pervasive part of modern industrial economies. Since the importance of imperfect competition has long been recognized in the literature, this paper extends the Agenor (2005) model to an imperfect endogenous growth model by introducing monopolistic competition in the intermediate-goods sector. Based on this framework, we study how the degree of imperfect competition can affect the growth-maximizing tax rate, the growth-maximizing share of maintenance spending, and the steady-state growth rate.

The remainder of the paper is organized as follows. In Section 2, we present the analytical framework. In Section 3, we discuss the dynamic properties of the system. Finally, in Section 4 we conclude the paper.

II. The Model

The model is comprised of three types of agents: firms, households, and the government. The production side consists of two sectors: the monopolistically competitive intermediate-goods sector and the perfectly competitive final-goods sector. The final goods are produced from the set of intermediate goods. The household derives utility from consumption, but incurs disutility from work. The government invests in infrastructure and spends on maintenance. It balances its budget each period by levying a flat tax rate on output.

1. Firms

We consider a monopolistically competitive intermediate-goods sector as in Dixit and Stiglitz (1977). There is a continuum of intermediate goods [y.sub.i], i [member of] [0, 1], which are used by a single representative firm to produce a final good y. The final good production technology is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1)

Let m = 1/[sigma] [member of] [0, 1) indicate the degree of monopoly power with [sigma] representing the elasticity of substitution between any two intermediate goods. When 1 < [sigma] < [infinity] (0 < m < 1), the intermediate-goods producers own a degree of monopoly power.

Accordingly, the profit-maximization problem for the final-goods firm is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

where [p.sub.i] denotes the relative prices of the i-th intermediate good and the final good. The first-order condition for profit maximization leads to the following inverse demand function for the i-th intermediate good:

[p.sub.i] = [(y/[y.sub.i].sup.1/[sigma]]. (3)

It is easy to learn that the price elasticity of demand for the i-th intermediate good is [sigma]. The intermediate-goods firms face a downward-sloping demand curve as long as [sigma] > 1 (m > 0). The producers in the perfectly competitive final-goods sector earn zero profit, and the zero-profit condition implies that:

1 = [([[integral].sup.1.sub.0] [p.sup.1 - [sigma].sub.i] di).sup.1/1 - [sigma]]. (4)

The production technology of the i-th intermediate good takes the Cobb-Douglas form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [n.sub.i] are the private capital and labor inputs of the i-th firm, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the physical stock of public capital, and [e.sub.i] is its efficiency. The effective stock of public infrastructure is expressed as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

As noted in the Introduction, if maintenance spending increases the quality of infrastructure, then the equipment used by the private sector may wear down less. As in Agenor (2005), we assume that the depreciation rate of private capital [[delta].sub.P] exhibits a linear relationship with the ratio of government spending on maintenance to the private capital stock:

[[delta].sub.P] = [[bar.[delta]].supb.P] - [[theta].sub.P](M/[k.sub.P]), [[theta].sub.P] [member of] [0, 1), (6)

where [[theta].sub.P] indicates the marginal effect of maintenance spending on the private depreciation rate. Equation (6) implies that maintenance spending on public capital may also enhance the durability of private capital. When [[theta].sub.P] = 0, then maintenance spending does not affect the depreciation rate of private capital.

Following Agenor (2005), efficiency is a concave function of the ratio of public maintenance spending to the stock of public capital:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (7)

where [chi] indicates the efficiency effect.

The optimization problem of the i-th intermediate-good firm is to choose [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and [n.sub.i] so as to maximize profit [[pi].sub.i], that is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (8)

where w and r are the wage rate and the interest rate in terms of the final goods, respectively. The first-order conditions for this optimization problem yield:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (9)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (10)

Equations (9) and (10) indicate that the demand for inputs is decreasing in the monopoly power index m.

Our analysis is confined to a symmetric equilibrium under which [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. As a result, the intermediate-goods production function can be restated as:

y = [k.sup.1 - [alpha].sub.p][n.sup.[alpha]][([ek.sub.g]).sup.[alpha]]. (11)

Given the symmetric equilibrium, Equations (9) and 10) can be written as:

r = (1 - [alpha])(1 - m)[k.sup.-[alpha].sub.p][n.sup.[alpha]][([ek.sup.g]).sup.[alpha]]. (12)

w = [alpha](1 - m)[k.sup.1 - [alpha].sub.p][n.sup.[alpha] - 1][([ek.sub.g]).sup.[alpha]]. (13)

In view of the existence of monopoly power, the interest rate is below the marginal productivity of private capital. Consequently, the profit of intermediate-goods producers is given by:

[pi] = y - wn - [rk.sub.p] = my. (14)

We can easily learn that m not only measures the degree of monopoly, but also represents the equilibrium profit share of output. When m > 0, intermediate firms earn an economic profit.

2. Households

The economy is populated by an infinitely-lived representative household. The household will choose consumption c and hours worked n, so as to maximize the discounted stream of future utility. We consider the intertemporal labor supply substitution and the substitution between consumption and labor supply. Thus, we specify that the representative household lifetime utility is of the form:

U = [[integral].sup.[infinity].sub.0] [ln c - [eta] n.sup.1 + [epsilon]]/1 + [epsilon]] exp(- [rho]dt, (15)

where [epsilon] ([less than or equal to] 0) denotes the inverse of the intertemporal labor supply substitution elasticity, [eta] denotes the substitution elasticity of consumption and labor supply, [rho] represents the constant rate of time preference, and t is the time index. The household's budget constraint is described by:

[[??].sub.p] = (1 - [tau])(wn + [rk.sub.t] + [pi]) - c - [[delta].sub.p][k.sub.p], (16)

where [tau] [member of] (0, 1) is the tax rate on output. As the owners of all firms, the households receive profits in the form of dividends.

The household treats w, r, [pi], [tau], and e as given and maximizes utility subject to the budget constraint. The Hamiltonian function is written as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (17)

The necessary conditions for the consumer's optimum involve:

1/c = [lambda], (18a)

[eta]/[n.sup.[epsilon]] = [lambda](1 - [tau])w, (18b)

[??] = [lambda][[rho] - r(1 - [tau]) + [[delta].sub.p]], (18c)

[[??].sub.p] = (1 - [tau])(wn + [rk.sub.p], + [tau]) - c - [[delta].sub.p][k.sub.p], (18d)

together with the household's budget constraint equation (16) and the transversality condition

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (19)

3. Government

The government is assumed to collect a proportional income tax revenue to finance the public spending. Let g denotes the public spending, and [v.sub.g] and [v.sub.M] denote the fraction of government spending devoted to investment in infrastructure [I.sub.g] and spending on maintenance M, respectively. Therefore, we have:

g = [[tau].sub.y], (20)

[I.sub.g] = [v.sub.g][tau]y, (21)

M = [v.sub.M][tau]y, (22)

where [v.sub.g] + [v.sub.M] = 1, and [V.sub.g], [v.sub.M] [member of] (0, 1). Equation (20) states the government budget constraint. Using equations (21) and (22), the change in [k.sub.g] is given by:

[[??].sub.g] = [I.sub.g] - [[delta].sub.g][k.sub.g], (23)

where [[delta].sub.g] is the depreciation rate of public capital in infrastructure. Following Agenor (2005), we assume that [[delta].sub.g] depends negatively on and is linearly related to the ratio of maintenance spending to the stock of public capital:

[[delta].sub.g] = 1 - [[delta].sub.g](M/[k.sub.g]), [[theta].sub.g] (0, 1). (24)

Equation (24) implies that maintenance spending enhances the durability of public capital.

4. The Steady-Growth Equilibrium

By imposing the conditions of symmetric equilibrium, the equilibrium of this economy may be described by the following equations:

1/c = [lambda], (25a)

[eta][n.sup.[epsilon]] = [lambda](l - [tau])w, (25b)

[??] = [lambda] [[rho] + [[delta].sub.s] - [sn.sup.[alpha]][([ek.sub.g]/[k.sub.p]).sup.[alpha]], (25c)

[??] = (1 - [tau])[([enk.sub.g]/[k.sub.p]).sup.[alpha]] [k.sub.p] = [[delta].sub.p][k.sub.p] - c, (25d)

[I.sub.g] + M = [[tau].sub.y], (25e)

[[??].sub.g] = [I.sub.g] - [[delta].sub.g][k.sub.g], (25f)

where s = (1 - m)(1 - [alpha])(1 - [tau]). Equation (25c) gives the equilibrium path of the shadow price of wealth, while equation (25d) is the market clearing condition in the product market. Equation (25e) is the government budget constraint.

Following Barro and Sala-i-Martin (1995), we define h = c/[k.sub.p] and z = [k.sub.g]/[k.sub.p]. By differentiating equation (25a) with respect to time and substituting equation (25c) into the resulting equation, the Keynes-Ramsey rule is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (26a)

Using equations (25d) and (25f), respectively, we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (26b)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (26c)

where v = [v.sub.g] + [[theta].sub.g][v.sub.M]. The dynamic system in terms of the transformed variables h and z can be summarized as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (27a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (27b)

III. Transitional Dynamics and Policy Change

We now linearize the dynamic system equations (27a) and (27b) around the steady state to obtain:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let [??] and [??] denote the stationary values of h and z, respectively. From equation (28), the general solution for h and z can be expressed as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (29a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (29b)

Obviously, the two characteristic roots of the dynamic system have opposite signs. We let [[zeta].sub.1] be the negative root and [[zeta].sub.2] the positive root. As addressed in the literature on dynamic rational-expectations models, if the number of unstable roots equals the number of jump variables, then there exists a unique perfect-foresight equilibrium solution. Since the dynamic system has only one jump variable h, the unique steady-state equilibrium depicted in Figure 1 is thus locally determined. From equations (27a) and (27b), the h = 0 locus is steeper than the [??] = 0 locus. (1) The stable path SS curve is steeper than the unstable path UU curve. (2)

At the balanced growth equilibrium, the economy is characterized by [??] = [??] = 0. Thus, consumption, private capital, public capital, and national product grow at the same rate [??], that is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The steady-state growth rate [??] is given by: (3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (30a)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (30b)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (30c)

1. The Effect of Income Taxation

We first analyze the impact of income taxation on the growth rate. By differentiating equation (26c) with respect to [tau], we can obtain the following result:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4) (31)

Equation (31) indicates that an increase in the tax rate has an ambiguous impact on the growth rate. It is clear from the Keynes-Ramsey rule in equation (26a) that a rise in the tax rate can affect the growth rate through two channels. On the one hand, a rise in the tax rate raises the ratio of maintenance spending to the stock of public capital, and the more effective the stock of the public capital, the more it will tend to raise the marginal productivity of private capital. On the other hand, a rise in the tax rate tends to reduce the post-tax marginal productivity of private capital. The net effect of a rise in the tax rate on the growth rate depends on the relative strength of these channels.

Setting [partial derivative][??]/[partial derivative][tau] = 0 in equation (31) yields the growth-maximizing tax rate. Therefore, we can establish the following proposition:

Proposition 1: With maintenance spending also affecting the private depreciation rate, the growth-maximizing tax rate is given by:

[[tau].sup.*] [alpha](1 - [alpha])(1 - m)/(1 - [alpha])(1 - m) - [[theta].sub.p][v.sub.M] > [alpha]. (32)

This solution indicates that the growth-maximizing tax rate is positive in relation to the marginal effect of maintenance spending on the private depreciation rate [[theta].sub.p]. This is in contrast to the Barro rule ([[tau].sup.*] = [alpha]), since it is clear that a higher initial share of maintenance spending will produce a higher growth-maximizing tax rate. These results are in line with Agenor (2005). In addition, the growth-maximizing tax rate is positively related to the degree of imperfect competition, since an increase in the degree of imperfect competition will lead to a reduction in capital accumulation and labor employment, which in turn will lower the growth rate. However, the tax revenue has a positive external effect on private production. Therefore, the greater the monopoly power, the higher the growth-maximizing tax rate will be.

However, the growth-maximizing tax rate is equal to the elasticity of effective public capital in production a when private depreciation is exogenous. This result is consistent with Agenor's basic model (2005) and exactly follows the Barro rule. The growth-maximizing tax rate is irrelevant to the degree of monopoly power index as well as to the maintenance spending. (5)

2. The Growth-maximizing Share of Spending on Maintenance

We next turn to the question of how the degree of imperfect competition can affect the growth-maximizing share of maintenance spending. Setting [partial derivative][??]/[partial derivative][v.sub.M] = 0 in equation (30a) with [dV.sub.g] = -[dv.sub.M] yields the following optimal condition:

[[alpha]s + [[theta].sub.p][tau][v.sup.*.sub.M]][v.sup.*.sub.M] = [(1 - [[theta].sub.g]).sup.-1] ([PHI][[theta].sub.p][tau][v.sup.*.sub.M] + [alpha]S[chi]). (33)

Equation (33) can be written as two distinct components:

[G.sub.l] ([v.sup.*.sub.M]) = [[alpha]s + [[theta].sub.p][tau][v.sup.*.sub.M][v.sup.*.sub.M], (34a)

[G.sub.2]([v.sup.*.sub.M]) = [(1 - [[theta].sub.g]).sup.-1] ([PHI][[theta].sub.p][tau][v.sup.*.sub.M] + [alpha]s[chi]), (34b)

where [PHI] = [1 - [alpha](1 - [chi])]. The growth-maximizing requires that [G.sub.1]([v.sup.*.sub.M]) = [G.sub.2]([v.sup.*.sub.M]) which is obtained at point E in Figure 2. (6)

A reduction in the degree of imperfect competition m rotates curve [G.sub.1] upward, whereas curve [G.sub.2] shifts upward. If the sum of the net benefit of the maintenance spending ([v.sup.*.sub.M][[theta].sub.g] - [v.sup.*.sub.M]) and the efficiency effect [chi] is positive, then the growth-maximizing share of maintenance spending will rise as the degree of imperfect competition decreases. (7) This is illustrated in Figure 3. Intuitively, a reduction in the degree of imperfect competition tends to raise the capital accumulation and labor employment, which in turn will enhance the output and the growth rate. On the other hand, the larger the marginal effect of maintenance spending on the depreciation rate of public capital, and/or the efficiency effect, the more likely it is that the maintenance spending will increase the growth rate. Therefore, the share of maintenance spending should increase to achieve the higher growth rate as the monopoly power declines. That is, sufficiently productive maintenance spending on public capital leads to a higher growth-maximizing share of maintenance spending as the degree of imperfect competition declines. This implies the following proposition:

Proposition 2: If ([v.sup.*.sub.M][[theta].sub.g] - [v.sup.*.sub.M]) + [chi] > 0, the smaller the degree of imperfect competition, the higher will be the growth-maximizing share of maintenance spending.

Then, let us examine the relationship between the marginal effect of maintenance spending on the private depreciation rate and the growth-maximizing share of maintenance spending. If the sum of the net benefit of the maintenance spending ([v.sup.*.sub.M][[theta].sub.g], - [v.sup.*.sub.M]), the elasticity of the private capital stock in production (1 - [alpha]), and the product of the elasticity of effective public capital in production [alpha] and the efficiency effect [chi] is positive, then curve [G.sub.1] rotates upward relatively less than curve [G.sub.2] as [[theta].sub.P] rises. This is illustrated in Figure 4 which corresponds to the case where the growth-maximizing share of maintenance spending will rise as [[theta].sub.p] increases. (8) Intuitively, an increase in the marginal effect of maintenance spending on the private depreciation rate tends to raise the private capital accumulation, which in turn will enhance the output and the growth rate. Besides, the larger the elasticity of the private capital stock in production, and/or the elasticity of the effective public capital in production, and/or the efficiency effect, the more likely it is that the maintenance spending will increase the growth rate. Therefore, the share of maintenance spending should increase to achieve the higher growth rate as the marginal effect of maintenance spending on the private depreciation rate increases. Thus, we can establish the following proposition:

Proposition 3: If ([v.sup.*.sub.M][[theta].sub.g] - [v.sup.*.sub.M]) + (1 - [alpha]) + [alpha][chi] > 0, then the higher the marginal effect of maintenance spending on the private depreciation rate [[theta].sub.p], the higher will be the growth-maximizing share of maintenance spending [v.sup.*.sub.M].

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

However, when private depreciation is exogenous, the result is consistent with Agenor's basic model (2005). The growth-maximizing share of maintenance spending is positively related to the efficiency effect and the marginal effect of maintenance spending on the depreciation rate of public capital. The growth-maximizing share of maintenance spending does not depend on the elasticity of effective public capital in production. Moreover, the growth-maximizing share of maintenance spending is irrelevant to the degree of monopoly power. (9)

3. Monopoly Power and Economic Growth

Finally, we deal with the relationship between monopoly power and economic growth. Differentiating equation (2.26a) with respect to m, we can obtain:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (10) (35)

where [PSI] = (1 - [alpha])(1 - [tau])(1 - [tau])(1 - [alpha][chi]). Equation (35) indicates that an increase in monopoly power in the intermediate-goods sector reduces the employment and economic growth in the long run. Therefore, we can establish the following proposition.

Proposition 4: The regulation of monopoly power promotes economic growth in the long run.

This result is consistent with the common notion in the existing literature. Since the interest rate is lower than the marginal productivity of capital and the wage is lower than the marginal productivity of labor, an increase in the degree of imperfect competition leads to a reduction in capital accumulation and labor employment. As such, the aggregate output decreases as well. Therefore, the regulation of monopoly power will promote the steady-state economic growth.

IV. Conclusion

The main contribution of this paper is that it addresses the issue of maintenance spending by the public sector when there is imperfect competition. Two major conclusions emerge from our analysis. First, when we discuss the impact of maintenance spending by the public sector on the private capital stock, the growth-maximizing tax rate will be higher than the elasticity of effective public capital in production. A larger degree of monopoly power will produce a higher growth-maximizing tax rate. Moreover, if the maintenance spending has a sufficiently strong positive effect on efficiency and the depreciation rate of public capital, then a reduction in the degree of imperfect competition and an increase in the marginal effect of maintenance spending on the private depreciation rate will both raise the growth-maximizing share of maintenance spending. Secondly, when maintenance spending does not affect the private depreciation rate, the growth-maximizing tax rate and the share of maintenance spending are consistent with Agenor (2005). The growth-maximizing tax rate only relates to the elasticity of effective public capital in production. The growth-maximizing share of maintenance spending depends only on the efficiency effect and the marginal effect of maintenance on the private depreciation rate. Moreover, the growth-maximizing share of maintenance spending is irrelevant to the degree of monopoly power.

Mathematical Appendix

From equation (28), the trace and the determinant of [??], respectively, are: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Notes

(1.) Note that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], since [[zeta].sub.1] [[zeta].sub.2] = [a.sub.11][a.sub.22] - [a.sub.21][a.sub.12] < 0.

(2.) Note that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(3.) When labor is omitted in the model, then the steady-state growth rate [??] is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(4.) Setting [??] = 0 in equation (27b) and differentiating it with respect to [tau], we obtain the following:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(5.) Setting [[theta].sub.p] = 0 in equation (32), the growth-maximizing tax rate is equal to the elasticity of effective public capital in production [alpha].

(6.) Note that curve [G.sub.1] is a convex function of [v.sup.*.sub.M] from the origin. However, curve [G.sub.2] is a linear function of [v.sup.*.sub.M] with the vertical intercept [alpha][(1 - [[theta].sub.g]).sup.-1]s[chi]. The slopes of curves [G.sub.1] and [G.sub.2] are 2[[theta].sub.p][tau][v.sup.*.sub.M] + [alpha]s and [(1 - [[theta].sub.g]).sup.- 1][PHI][[theta].sub.p][tau], respectively.

(7.) By differentiating equations (34a) and (34b) with respect to m, respectively, we obtain:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(8.) By differentiating equations (34a) and (34b) with respect to [[theta].sub.p], respectively, we obtain:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(9.) The growth-maximizing share of spending on maintenance is equal to [chi]/1 - [[theta].sub.g] (see Agenor (2005)).

(10.) By setting z = 0 in equation (27b), with respect to m, we obtain the following:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

References

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Kalaitzidakis, P. and Kalyvitis, S. (2004). "On the Macroeconomic Implications of Maintenance in Public Capital," Journal of Public Economics 88, 695-712.

Rioja, F. K. (2003a). "The Penalties of Inefficient Infrastructure," Review of Development Economics 7, 127-37.

Rioja, F. K. (2003b). "Filling Potholes: Macroeconomic Effects of Maintenance versus New Investment in Public Infrastructure," Journal of Public Economics 87, 2281-304.

Turnovsky, S. J. (1996). "Optimal Tax, Debt, and Expenditure Policies in a Growing Economy," Journal of Public Economics 60, 21-44.

Jin-wen Lee, Department of Finance, National Taichung Institute of Technology and Institute of Industrial Economics, National Central University, Taiwan.

I am grateful to an anonymous referee of this journal for excellent guidance in revising the paper. Needless to say, any remaining deficiencies are the author's responsibility.