Optimal fiscal policies in an economy with externalities from public spending

Abstract

This paper examines the optimal fiscal policies in an economy with externalities from government expenditure. We extend Lucas (Oxf Econ Pap 42:293–316, 1990) two-sector endogenous growth model and consider the spillover effects from the public spending on infrastructure and education. We compare the optimal fiscal policies derived from the Ramsey allocation problem with those in the centrally-planned economy. The results of this paper are as follows. First, the optimal share of public spending on infrastructure is smaller than its relative contribution in the production function. Next, the optimal share of public spending on education is smaller than its relative contribution in the accumulation of human capital, and does not affect the tax rate of capital income. Finally, the optimal tax rate of capital income is positive if the externality from public productive spending exists.

Appendix

1.1 The calculation of the centrally-planned problem

The central planner’s objective is to maximize the household’s discounted lifetime utility, (1b), subject to the aggregate goods market clearing constraint, (10), and the accumulation equation of human capital, (11). Let $\lambda ^1_t$ and $\lambda ^2_t$ denote the Lagrange multipliers associated with (10) and (11), respectively. The first-order conditions with respect to $g_{xt}$, $g_{yt}$, $c_t$, $n_t$, $v_t$, $k_{t+1}$ and $h_{t+1}$ are as follows:

$$\begin{aligned}&{\textstyle \lambda ^1_t y_t =\lambda ^2_t (1-\beta ) \frac{x_t}{g_{xt}} }, \end{aligned}$$

(19a)

$$\begin{aligned}&{\textstyle \lambda ^1_t y_t=\lambda ^1_t \frac{1-\alpha -\eta }{\alpha +\eta }(1-g_{xt}-g_{yt})\frac{y_t}{g_{yt}} +\lambda ^2_t \frac{(1-\beta )(1-\alpha -\eta )}{\alpha +\eta } \frac{x_t}{g_{yt}} }, \end{aligned}$$

(19b)

$$\begin{aligned}&u_{1t}=\lambda ^1_t, \end{aligned}$$

(19c)

$$\begin{aligned}&{\textstyle u_{2t}=\lambda ^1_t (1-g_{xt}-g_{yt}) \frac{\eta }{\alpha +\eta } \frac{y_t}{n_t} +\lambda ^2_t\frac{\eta (1-\beta )}{\alpha +\eta } \frac{x_t}{n_t} }, \end{aligned}$$

(19d)

$$\begin{aligned}&{\textstyle u_{2t}=\lambda ^2_t \gamma \frac{x_t}{v_t} }, \end{aligned}$$

(19e)

$$\begin{aligned}&{\textstyle \lambda ^1_t= \frac{1}{1+\rho } \left\{ \lambda ^1_{t+1} \left[ (1-g_{xt+1}-g_{yt+1})\frac{y_{t+1}}{k_{t+1}} \frac{\alpha }{\alpha +\eta }+1-\delta \right] + \lambda ^2_{t+1} \frac{\alpha (1-\beta )}{\alpha +\eta }\frac{x_{t+1}}{k_{t+1}} \right\} }, \nonumber \\\end{aligned}$$

(19f)

$$\begin{aligned}&{\textstyle \lambda ^2_t=\frac{1}{1+\rho } \left\{ \lambda ^2_{t+1}\left[ 1+(-\frac{\alpha (1-\beta )}{\alpha +\eta }+1) \frac{x_{t+1}}{h_{t+1}}\right] +\lambda ^1_{t+1} (1-g_{xt+1}-g_{yt+1})\frac{\eta }{\alpha +\eta }\frac{y_{t+1}}{h_{t+1}} \right\} }.\nonumber \\ \end{aligned}$$

(19g)

Now we can derive the equilibrium conditions in the centrally-planned problem. First, combining (19a) and (19b) yields the value of the public spending on infrastructure, (12a). Next, by substituting (12a), (19a) and (19c) into (19d) we obtain the consumption-leisure tradeoff condition, (12c). Moreover, (12a), (19a), (19d) and (19e) jointly produce the value of public spending on education, (12b). In addition, using (12a), (19a), (19c) and (19f) results in the consumption Euler equation, (12d). Finally, (19a), (19d), (19e) and (19g) together yield the intertemporal condition for the household’s time allocation, (12e).

1.2 The calculation of the Ramsey allocation problem

The planner chooses the allocation including $g_{yt}$ and $g_{xt}$ in order to maximize the representative household’s welfare in (1b), subject to the implementability constraint (13), the intertemporal condition for the household’s time allocation (5c), the goods market clearing condition (10) and the accumulation equation of human capital (11). The first-order conditions with respect to $g_{xt}$, $g_{yt}$, $c_t$, $n_t$, $v_t$, $k_{t+1}$ and $h_{t+1}$ are as follows:

$$\begin{aligned} {\textstyle -\xi ^2_t \frac{(1-\beta ) \Lambda _t}{g_{xt} (1+\rho )} +\xi ^2_{t-1} \frac{\Phi _{t-1}}{\Phi _t} \frac{h_t}{h_{t+1}} \frac{1-\beta }{g_{xt}}-\xi ^3_t y_t + \xi ^4_t (1-\beta ) \frac{x_t}{g_{xt}}=0 }, \end{aligned}$$

(20a)

$$\begin{aligned}&{\textstyle -\xi ^1 u_{1t}\frac{(1-\alpha -\eta )^2}{\alpha +\eta } \frac{y_t}{g_{yt}} -\xi ^2_t \frac{\Lambda _t}{(1+\rho )} \frac{(1-\beta )(1-\alpha -\eta )}{\alpha +\eta } \frac{1}{g_{yt}} +\xi ^2_{t-1} \frac{\Phi _{t-1}}{\Phi _t} \frac{h_t}{h_{t+1}} \frac{1}{g_{yt}} \frac{(1-\beta )(1-\alpha -\eta )}{\alpha +\eta } } \nonumber \\&{\textstyle +\xi ^3_t \left[ -y_t+ \frac{(1-g_{xt}-g_{yt})(1-\alpha -\eta )}{\alpha +\eta } \frac{y_t}{g_{yt}}\right] + \xi ^4_t \frac{(1-\beta )(1-\alpha -\eta )}{\alpha +\eta } \frac{x_t}{g_{yt}}=0 }, \end{aligned}$$

(20b)

$$\begin{aligned} {\textstyle u_{1t}+\xi ^1 \{ u_{11t}[c_t-(1-\alpha -\eta )y_t]+u_{1t} \} -\xi ^3_t=0 }, \end{aligned}$$

(20c)

$$\begin{aligned}&{\textstyle -u_{2t}+\xi ^1 \left[ -u_{1t} \frac{(1-\alpha -\eta )\eta }{\alpha +\eta } \frac{y_t}{n_t}+ u_{22t}n_t-u_{2t}\right] +\xi ^2_t \left[ \frac{-u_{22t}}{u_{2t+1}}-\frac{1}{1+\rho } \frac{\eta (1-\beta )}{\alpha +\eta } \frac{\Lambda _t}{n_t}\right] } \nonumber \\&{\textstyle + (1+\rho )\xi ^2_{t-1} \{ \frac{u_{2t-1} u_{22t}}{(u_{2t})^2}-\frac{\Phi _{t-1}}{(1+\rho )} [1-\frac{\eta (1-\beta )}{\alpha +\eta }\frac{1}{\Phi _t} \frac{h_t}{h_{t+1}} \frac{1}{n_t}] \} } \nonumber \\&{\textstyle +\xi ^3_t (1-g_{xt}-g_{yt}) \frac{y_t}{n_t} \frac{\eta }{\alpha +\eta } + \xi ^4_t \frac{\eta (1-\beta )}{\alpha +\eta } \frac{x_t}{n_t} =0 }, \end{aligned}$$

(20d)

$$\begin{aligned}&{\textstyle -u_{2t}+ \xi ^1 u_{22t} n_t +\xi ^2_t \left[ \frac{-u_{22t}}{u_{2t+1}}-\frac{\gamma -1}{1+\rho } \frac{\Lambda _t}{v_t}\right] } \nonumber \\&{\textstyle +(1+\rho )\xi ^2_{t-1}\left[ \frac{u_{2t-1} u_{22t}}{(u_{2t})^2}-\frac{\Phi _{t-1}}{(1+\rho )}( \frac{1}{\Phi _t} \frac{h_t}{h_{t+1}} \frac{1-\gamma }{v_t} +\frac{\beta }{\gamma })\right] +\xi ^4_t \gamma \frac{x_t}{v_t} =0 }, \end{aligned}$$

(20e)

$$\begin{aligned}&{\textstyle -\xi ^1 \frac{u_{1t+1}}{1+\rho } \frac{(1-\alpha -\eta )\alpha }{\alpha +\eta } \frac{y_{t+1}}{k_{t+1}} +\frac{\xi ^2_t}{1+\rho } \frac{\Phi _t}{\Phi _{t+1}} \frac{h_{t+1}}{h_{t+2}} \frac{\alpha (1-\beta )}{\alpha +\eta } \frac{1}{k_{t+1}}- \frac{\xi ^2_{t+1}}{1+\rho } \frac{\Lambda _{t+1}}{1+\rho } \frac{\alpha (1-\beta )}{\alpha +\eta } \frac{1}{k_{t+1}} } \nonumber \\&{\textstyle - \xi ^3_t +\frac{\xi ^3_{t+1}}{1+\rho } \left[ \frac{(1-g_{xt+1}-g_{yt+1})\alpha }{\alpha +\eta } \frac{y_{t+1}}{k_{t+1}}+1-\delta \right] +\frac{\xi ^4_{t+1}}{1+\rho } \frac{\alpha (1-\beta )}{\alpha +\eta } \frac{x_{t+1}}{k_{t+1}}=0 }, \end{aligned}$$

(20f)

$$\begin{aligned}&{\textstyle \!-\! \frac{\xi ^1 u_{1t+1}}{1\!+\!\rho } \frac{(1-\alpha -\eta )\eta }{\alpha +\eta } \frac{y_{t+1}}{h_{t+1}} \!+\!\frac{\xi ^2_t}{1+\rho } \left[ \frac{\Lambda _t}{h_{t+1}}\!-\! \frac{\Phi _t}{\Phi _{t+1}} \frac{1}{h_{t+2}} \frac{\alpha (1-\beta )}{\alpha +\eta }\right] \!-\!\frac{\xi ^2_{t+1}}{(1+\rho )^2} \left[ -\frac{\alpha (1-\beta )}{\alpha +\eta }+1\right] \frac{\Lambda _{t+1}}{h_{t+1}} } \nonumber \\&{\textstyle + \frac{\xi ^3_{t+1}}{1+\rho } \frac{(1-g_{xt+1}-g_{yt+1})\eta }{\alpha +\eta } \frac{y_{t+1}}{h_{t+1}} -\xi ^4_t +\frac{\xi ^4_{t+1}}{1+\rho } \left[ 1+(\frac{-\alpha (1-\beta )}{\alpha +\eta }+1)\frac{x_{t+1}}{h_{t+1}}\right] =0 }, \end{aligned}$$

(20g)

where $\Lambda _t \equiv {\textstyle \frac{\gamma x_t}{v_t} \left[ \frac{ n_{t+1}}{h_{t+1}} + \frac{ v_{t+1}}{\gamma x_{t+1}} (1+\beta \frac{x_{t+1}}{h_{t+1}})\right] }$ and $\Phi _t \equiv \frac{\gamma x_t}{v_t h_{t+1}}$.

Now we need to prove that consumption $c_t$, physical capital $k_t$, and human capital $h_t$ all grow at the same rate, denoted by $\theta$, and $\xi ^3_t k_t$ and $\xi ^4_t k_t$ are constant in the long run. First, we define $\frac{h_{t+1}-h_t}{h_t}\equiv \theta _t$ which is constant in the long run. In addition, $\Lambda =1+\rho$ and $\Phi =\frac{\gamma }{v} \frac{\theta }{1+\theta }$ in the long run. According to (3) and (11), we obtain that $\frac{x_t}{h_t}=\theta _t$, and thus $\frac{h_t}{k_t}$ is constant in the long run. Since $\frac{y_t}{k_t}=A^{\frac{1}{\alpha +\eta }} g_{yt}^{\frac{1-\alpha -\eta }{\alpha +\eta }} n_t^{\frac{\eta }{\alpha +\eta }} \left( \frac{h_t}{k_t}\right) ^{\frac{\eta }{\alpha +\eta }}$, thus $\frac{y_t}{k_t}$ is constant in the long run. Besides, (10) suggests that $\theta _t+\delta +\frac{c_t}{k_t}=(1-g_{yt}-g_{xt}) \frac{y_t}{k_t}$, so that $\frac{c_t}{k_t}$ is also constant in the long run.

Moreover, combining (20b) and (20c), along with (20a), we obtain that $1+\xi ^3_t k_t \left[ (1-\frac{g_{yt}}{1-\alpha -\eta })\frac{y_t}{k_t}-\frac{c_t}{k_t}\right] =0$. Thus $\xi ^3_t k_t$ is constant in the long run. As (20b) implies that $\xi ^1=\xi ^3_t k_t \frac{c_t}{k_t} \left[ \frac{1-\alpha -\eta -g_{yt}}{(1-\alpha -\eta )^2}\right]$, by using the conclusion that $\xi ^3_t k_t$ is a constant, we obtain that $\xi ^1$ is constant in the long run.

Furthermore, combining (20a) and (20d) provides that $-u_{2t}+\xi ^1 \left[ u_{22t}n_t-u_{2t}-\frac{(1-\alpha -\eta )\eta }{\alpha +\eta } \frac{1}{n_t} \frac{k_t}{c_t} \frac{y_t}{k_t}\right] -\xi ^2_t \frac{u_{22t}}{u_{2t+1}} + (1+\rho )\xi ^2_{t-1} \left[ \frac{u_{2t-1} u_{22t}}{(u_{2t})^2}-\frac{\Phi _{t-1}}{(1+\rho )}\right] +\xi ^3_t k_t (1-g_{yt}) \frac{1}{n_t} \frac{y_t}{k_t} \frac{\eta }{\alpha +\eta }=0$, in which we can conclude that $\xi ^2_t=\xi ^2_{t-1}$ is constant in the long run. Finally, (20a) suggests that $-\xi ^2_t + \frac{\xi ^2_{t-1}}{1+\theta }- \xi ^3_t k_t \frac{y_t}{k_t} \frac{g_{xt}}{1-\beta }+\xi ^4_t k_t \theta \frac{h_t}{k_t} =0$. Thus $\xi ^4_t k_t$ is also a constant in the long run. In order to analyze the BGP, we define $\xi ^3_t k_t \equiv \xi _3$ and $\xi ^4_t k_t \equiv \xi _4$, and denote $\xi _2$ as the value of $\xi ^2_t$ along the BGP.

In the BGP, (5c) becomes

$$\begin{aligned} {\textstyle 1+\beta \theta -(1+\rho )(1+\theta )=-\frac{\theta \gamma n}{v}}. \end{aligned}$$

(21)

The first-order conditions (20a)–(20g) in the BGP can be rearranged as follows:

$$\begin{aligned}&{\textstyle -\xi _2 \frac{\theta }{1+\theta }-\xi _3 \frac{y}{k} \frac{g_x}{1-\beta } +\xi _4 \frac{h}{k} \theta =0 }, \end{aligned}$$

(22a)

$$\begin{aligned}&{\textstyle \xi _3\left[ 1-\frac{g_y}{(1-\alpha -\eta )}\right] =\xi ^1 \frac{k}{c}(1-\alpha -\eta ) }, \end{aligned}$$

(22b)

$$\begin{aligned}&{\textstyle 1+\xi ^1 (1-\alpha -\eta )\frac{y}{k}\frac{k}{c}=\xi _3\frac{c}{k} }, \end{aligned}$$

(22c)

$$\begin{aligned}&{\textstyle -u^2+\xi _2 \left[ \frac{\rho u_{22}}{u_2}- \frac{\gamma }{v} \frac{\theta }{1+\theta }\right] } {\textstyle +\xi _3 \frac{g_y \eta }{1-\alpha -\eta }\frac{y}{k n} +\xi ^1\left[ u_{22}n-u_2\right] =0 }, \end{aligned}$$

(22d)

$$\begin{aligned}&{\textstyle -u^2+\xi _2\left[ \frac{\rho u_{22}}{u_2}+ \frac{\theta }{1+\theta } \frac{1-\gamma -\beta }{v}\right] +\xi _4 \frac{\gamma }{v} \frac{h}{k} \theta +\xi ^1 u_{22}n=0 }, \end{aligned}$$

(22e)

$$\begin{aligned}&{\textstyle -(1+\rho )(1+\theta )+\frac{\alpha g_y y}{(1-\alpha -\eta )k}+1-\delta =0 }, \end{aligned}$$

(22f)

$$\begin{aligned}&{\textstyle \xi _2\left[ 1+\rho -\beta -\frac{1-\beta }{1+\theta }\right] } {\textstyle + \xi _3 \frac{g_y \eta }{1-\alpha -\eta } \frac{y}{k} +\xi _4 \frac{h}{k} \left[ 1+\beta \theta -(1+\rho )(1+\theta )\right] =0 }.\nonumber \\ \end{aligned}$$

(22g)

It is worth noting that (22b) is the optimality condition with respect to $g_y$ in the long run, i.e., (16), and (22f) is the optimality condition with respect to $k_{t+1}$ in the long run, i.e., (18a). Due to the binding implementability constraint and the goods market clearance condition, i.e., $\xi ^1>0$ and $\xi _3>0$, (22b) gives rise to $g_y<1-\alpha -\eta$.

Moreover, by combining (22d) with (22e), along with (22a), we obtain the following equation:

$$\begin{aligned} {\textstyle \xi _2 \frac{\theta }{1+\theta } \frac{\beta -1}{v} +\xi _3 \frac{y}{k n} \frac{g_y \eta }{1-\alpha -\eta } =\xi _4 \frac{h}{k} \theta \frac{\gamma }{v} +\xi ^1 u_2 }. \end{aligned}$$

(23a)

We then use (21), (22g) and (23a) to derive the value of $\xi _2$ as follows:

$$\begin{aligned} {\textstyle -\xi _2 \left[ \frac{\theta }{1+\theta }\frac{1-\beta }{v} n+\rho + \frac{(1-\beta )\theta }{1+\theta }\right] =\xi ^1 u_2 n }. \end{aligned}$$

(23b)

According to (23b), we obtain that $\xi _2<0$.

Finally, putting the combination of (23a) and (23b) into (22a), along with (21), yields the condition with respect to $g_x$ in the long run, i.e., (17).