Multi-regional economic growth with public good and regional fiscal policies in a small-open economy

Abstract

This paper is concerned with multi-regional economic growth with environment, capital accumulation and regional public goods. The economy has a fixed number of regions, and there are a production sector and a public sector in a region. The production sector provides goods in perfectly competitive markets. The public sector, which is financed by the regional government’s tax incomes, supplies regional public goods. The public goods affect both firms and households. We show the existence of a unique equilibrium in the dynamic system. We simulate the equilibrium of 3-region economy and examine effects of changes in some parameters on the spatial economy. The comparative statics analysis provides some important insights. For instance, as the technologically least advanced region (TLAR) improves its productivity or amenity, the national output and wealth are reduced, and more people are attracted to the region from the more productive regions. The labor forces in the TLAR’s two sectors are increased, and the labor forces in the other two regions are reduced. The change pattern for the capital distribution is similar to the change in the population distribution. The output levels of the two sectors in the TLAR are increased and in the other two regions are reduced. The TLAR’s total and per capita expenditures on public goods are increased; the other two regions’ total and per capita expenditures are reduced. The per-worker output level, wealth and consumption level per capita, wage rate in the TLAR are increased, and those variables in the other two regions are reduced. The lot size falls and the land rent rises in the TLAR, and the trends are opposite in the other two regions.

Appendix proving Lemma 1

By (9), we have $s_j = \bar{{k}}_j$ at steady state. From this equation and $s_j = \lambda \hat{{y}}_j$, we have $\hat{{y}}_j = \bar{{k}}_j /\lambda$. From (2) and (14), we solve

$$\begin{aligned} \tilde{k}_j \equiv \frac{K_{pj} }{K_{ij} } = \frac{\alpha N_{pj} }{N_{ij} }, \end{aligned}$$

where $\alpha \equiv \alpha _{p}\beta _{i}/ \alpha _{i}\beta _{p}$. From this equation, $N_{j}=N_{ij}+N_{pj}$ and $K_{j}=K_{ij}+K_{pj}$, we solve

$$\begin{aligned} \begin{aligned}&N_{ij} = \frac{\alpha N_j }{\alpha + \tilde{k}_j }, N_{pj} = \frac{\tilde{k}_j N_j }{\alpha + \tilde{k}_j },\\&K_{ij} = \frac{K_j }{1 + \tilde{k}_j }, K_{pj} = \frac{\tilde{k}_j K_j }{1 + \tilde{k}_j }. \end{aligned} \end{aligned}$$

(23)

From the definition of $\Omega _{j}(t)$, (11) and (23), we can express the production functions as follows

$$\begin{aligned} F_{ij} = \frac{\bar{{A}}_j \tilde{k}_j^{\bar{{\theta }}} K_j^{\bar{{\alpha }}} N_j^{\bar{{\beta }}} }{\left( {1 + \tilde{k}_j } \right) ^{\bar{{\alpha }}} \left( {\alpha + \tilde{k}_j } \right) ^{\bar{{\beta }}}}, \end{aligned}$$

(24)

where

$$\begin{aligned} \bar{{\theta }} \equiv \theta _p \alpha _{0p} + \theta _c + \theta _p \beta _{0p}, \bar{{\alpha }} \equiv \alpha _i + \theta _e + \theta _p \alpha _{0p}, \bar{{\beta }} \equiv \beta _i + \theta _p \beta _{0p}, \bar{{A}}_j \equiv A_{ij} A_{pj}^{\theta _p } \alpha ^{\beta _i }. \end{aligned}$$

From (1), (23) and (24), we have

$$\begin{aligned} r^{*} + \delta _k = \frac{\alpha _i \bar{{\tau }}_{ij} \bar{{A}}_j \tilde{k}_j^{\bar{{\theta }}} K_j^{\bar{{\alpha }}-1} N_j^{\bar{{\beta }}} }{\left( {1 + \tilde{k}_j } \right) ^{\bar{{\alpha }}-1} \left( {\alpha + \tilde{k}_j } \right) ^{\bar{{\beta }}}}, w_j = \frac{\left( {r^{*} + \delta _k } \right) \beta _i K_j \left( {\alpha + \tilde{k}_j } \right) }{\alpha \alpha _{i} N_{j}\left( {1 + \tilde{k}_{j}}\right) }. \end{aligned}$$

(25)

From the definition of $\hat{{y}}_j$ and $\hat{{y}}_j = \bar{{k}}_j /\lambda$, we solve $w_j = r_j^{*} \bar{{k}}_j$,where

$$\begin{aligned} r_j^*\equiv \frac{1/\lambda - 1 - \bar{{\tau }}_{rj} r^{*}}{\bar{{\tau }}_{wj} }. \end{aligned}$$

From $w_j = r_j^*\bar{{k}}_j$ and (25)

$$\begin{aligned} \bar{{k}}_j = \frac{\left( {r^{*} + \delta _k } \right) \beta _i K_j \left( {\alpha + \tilde{k}_j } \right) }{\alpha \alpha _i r_j^{*} N_j \left( {1 + \tilde{k}_j } \right) }. \end{aligned}$$

(26)

From the marginal conditions for capital in (25), we have

$$\begin{aligned} K_j = \Lambda _{K} K_{1}, j = 1,\ldots , J, \end{aligned}$$

where

$$\begin{aligned} \Lambda _{Kj} \equiv \left[ {\frac{\bar{{\tau }}_{i1} \bar{{A}}_1 \tilde{k}_1^{\bar{{\theta }}} N_1^{\bar{{\beta }}} \left( {\alpha + \tilde{k}_j } \right) ^{\bar{{\beta }}}}{\bar{{\tau }}_{ij} \bar{{A}}_j \tilde{k}_j^{\bar{{\theta }}} N_j^{\bar{{\beta }}} \left( {\alpha + \tilde{k}_1 } \right) ^{\bar{{\beta }}}}} \right] ^{1/\left( {\bar{{\alpha }}-1} \right) }\frac{1 + \tilde{k}_j }{1 + \tilde{k}_1 }. \end{aligned}$$

From $K_{2}=\Lambda _{K}K_{1}$ and $\sum _{j}K_{j}=K$, we have

$$\begin{aligned} K_j = \frac{\Lambda _{Kj} K}{\sum _1^J {\Lambda _{Kj} } }, j = 1, \ldots , J, \end{aligned}$$

(27)

where $\Lambda _{K1} = 1$. Insert $l_{j}=L_{j}/ N_{j}$, $c_j = \xi \hat{{y}}_j$, and $s_j = \lambda \hat{{y}}_j$ in the utility function

$$\begin{aligned} U_j = \bar{{\theta }}_j \xi ^{\xi _h } \lambda ^{-\xi _h }A_{pj}^{v_h } L_j^{\eta _h } \frac{K_j^{v_h \alpha _{0p} } \tilde{k}_j^{v_h \beta _{0p} +v_h \alpha _{0p} } N_j^{b-\eta _h +v_h \beta _{0p} } \bar{{k}}_j^{\xi _h +\lambda _h } }{\left( {\alpha + \tilde{k}_j } \right) ^{v_h \beta _{0p} } \left( {1 + \tilde{k}_j } \right) ^{v_h \alpha _{0p} }}, \end{aligned}$$

(28)

where we use (7), (11) and $\hat{{y}}_j = \bar{{k}}_j /\lambda$. From (28), (26) and $U_{j}=U_{1}$, we have

$$\begin{aligned} N_j = \Lambda _{Nj} N_1, j = 2,...\ldots , J, \end{aligned}$$

where

$$\begin{aligned} \Lambda _{Nj} \left( {\tilde{k}_1, \tilde{k}_j } \right) \!\equiv \! \theta _{uj} \left( {\frac{\tilde{k}_1}{\tilde{k}_j}} \right) ^{\left[ {v_h \beta _{0p} \!+\!v_h \alpha _{0p} \!-\!\bar{{\theta }}\xi _u /\left( {\bar{{\alpha }}\!-\!1} \right) } \right] \lambda _u }\left( {\frac{\alpha + \tilde{k}_j }{\alpha \!+\! \tilde{k}_1 }} \right) ^{\left[ {v_h \beta _{0p} \!-\!\xi _h \!-\!\lambda _h \!-\!\bar{{\beta }}\xi _u /\left( {\bar{{\alpha }}-1} \right) } \right] \lambda _u }, \end{aligned}$$

where

$$\begin{aligned} \xi _u&\equiv v_h \alpha _{0p} +\xi _h +\lambda _h, \lambda _u \equiv \left( {b - \eta _h + v_h \beta _{0p} - \xi _h - \lambda _h - \frac{\xi _u \bar{{\beta }}}{\bar{{\alpha }}-1}} \right) ^{-1}, \\ \theta _{uj}&\equiv \left[ {\frac{r_j^{*\left( {\xi _h +\lambda _h } \right) } \bar{{\theta }}_1 A_{p1}^{v_h } L_1^{\eta _h } }{r_1^{*\left( {\xi _h +\lambda _h } \right) } \bar{{\theta }}_j A_{pj}^{v_h } L_j^{\eta _h } }} \right] ^{\lambda _u } \left( {\frac{\bar{{\tau }}_{ij} \bar{{A}}_j }{\bar{{\tau }}_{i1} \bar{{A}}_1 }} \right) ^{\xi _u \lambda _u /\left( {\bar{{\alpha }}-1} \right) } \end{aligned}$$

From $N_{j}=\Lambda _{N}N_{1}$ and the full labor employment condition, we have

$$\begin{aligned} N_j = \frac{\Lambda _{Nj} N}{\sum _{j=1}^J {\Lambda _{Nj} } }, j = 1 , \ldots , J, \end{aligned}$$

(29)

where $\Lambda _{N1} = 1$. From the definition of $\Lambda _{Kj}$ and (28)

$$\begin{aligned} \Lambda _{Kj} \left( {\tilde{k}_1, \tilde{k}_j} \right) = \left[ {\frac{\bar{{\tau }}_{i1} \bar{{A}}_1 \tilde{k}_1^{\bar{{\theta }}} \left( {\alpha + \tilde{k}_j } \right) ^{\bar{{\beta }}}}{\bar{{\tau }}_{ij} \bar{{A}}_j \tilde{k}_j^{\bar{{\theta }}} \Lambda _{Nj}^{\bar{{\beta }}} \left( {\alpha + \tilde{k}_1 } \right) ^{\bar{{\beta }}}}} \right] ^{1/\left( {\bar{{\alpha }}-1} \right) }\frac{1 + \tilde{k}_j }{1 + \tilde{k}_1 }, j = 2, \ldots , J. \end{aligned}$$

(30)

From (14) and (23), we have

$$\begin{aligned} K_j = \frac{\alpha _p \left( {1 + \tilde{k}_j } \right) }{\left( {r^{*} + \delta _k } \right) \tilde{k}_j }Y_{pj}. \end{aligned}$$

(31)

From the definition of $Y_{pj}$, we have

$$\begin{aligned} Y_{pj} = \left( {\frac{\alpha \tau _{ij} r_j^*}{\beta _i \bar{{\tau }}_{ij} \left( {\alpha + \tilde{k}_j } \right) } + \tau _{rj} r^{*} + \tau _{wj} r_j^*} \right) \bar{{k}}_j N_j, \end{aligned}$$

(32)

where we use $F_{ij} = w_j N_{ij} / \beta _i \bar{{\tau }}_{ij}, \quad w_j = r_j^*\bar{{k}}_j$, and the equation for $N_{ij}$ in (23). Substituting (32) into (31) yields

$$\begin{aligned} \tilde{k}_j = \left( {\frac{\alpha \tau _{ij} r_j^*}{\beta _i \bar{{\tau }}_{ij} } + \alpha \left( {\tau _{rj} r^{*} + \tau _{wj} r_j^*} \right) } \right) \left( {\frac{r_j^*}{\beta _p } - \tau _{rj} r^{*} - \tau _{wj} r_j^*} \right) ^{-1}. \end{aligned}$$

(33)

The above equations determine the equilibrium values of $\tilde{k}_j$. From (29), we determine $N_{j}$ as functions of $\tilde{k}_j$. By (23), $N_{ij}$ and $N_{pj}$ are determined. The equilibrium values of $K_{j}$ are found by the marginal conditions for capital in (25). We have $K=\sum _{j}K_{j}$ and $w_{j}$ by (25). We determine $K_{ij}$ and $K_{pj}$ by (23) and $Y_{pj}$ by (32). It is straightforward to get $F_{ij}$ by (24) and $F_{pj}$ by (11). The equilibrium values of $\bar{{k}}_j$ are solved by (26) and $\hat{{y}}_j$ by (5). From (8), we have $l_{j}=L_{j}/ N_{j}$ and $R_j = \eta \hat{{y}}_j /l_j$. We determine $c_{j}$ and $s_{j}$ by (8). We have $C_{j}=c_{j}N_{j}$ and $S_{j}=s_{j}N_{j}$. We thus proved Lemma 1.