A Multi-sector Model of Public Expenditure and Growth

Abstract

A large and rich body of empirical and theoretical research examines the role of public expenditure in fostering economic growth. This paper develops a model of endogenous growth with a particular focus on the role of public expenditure in structural changes. We consider the dynamics of a multi-sector economy that allows for differences among sectors in the output elasticity of public expenditure, and demonstrate that there exists a unique balanced growth path that is locally saddle-path stable. Along the balanced growth path, the endogenous growth of public expenditure and its disproportionate effect on different sectors result in changes in the relative prices of goods, which, in turn, cause structural changes represented by a reallocation of resources across sectors. Quantitatively, the faster the economic growth triggered by the policy, the greater the changes in the reallocation of labor.

Appendices

Appendix A

1.1 Derivation of Eq. (19)

From (15) and (18), we have

$$\begin{aligned} x_i =\frac{P_i \,(P\frac{a_i }{P_i })^\sigma }{\sum \nolimits _{i=1}^N {\left[ {P_i (P\frac{a_i }{P_i })^\sigma } \right] } } \end{aligned}$$

(31)

Substituting (16) into (31), we can write an expression for the output value share of intermediate sector $i$ as

$$\begin{aligned} x_i =\frac{( {G_s^\alpha G_p^{1-\alpha } })^{(\sigma -1)\beta _i }a_i ^\sigma }{\sum \nolimits _{i=1}^N {\left[ {( {G_s^\alpha G_p^{1-\alpha } })^{(\sigma -1)\beta _i }a_i ^\sigma } \right] } } \end{aligned}$$

(32)

Combining (7) with (15) yields

$$\begin{aligned} \frac{( {G_s^\alpha G_p^{1-\alpha } })^{\beta _i }K_i ^{1-\mu }L_i ^\mu }{( {G_s^\alpha G_p^{1-\alpha } })^{\beta _j }K_j ^{1-\mu }L_j ^\mu }=\left( \frac{P_j }{P_i }\frac{a_i }{a_j }\right) ^\sigma \end{aligned}$$

(33)

Using (12), (33) reduces to

$$\begin{aligned} \frac{( {G_s^\alpha G_p^{1-\alpha } })^{\beta _i }L_i }{( {G_s^\alpha G_p^{1-\alpha } })^{\beta _j }L_j }=\left( \frac{P_j }{P_i }\frac{a_i }{a_j }\right) ^\sigma \end{aligned}$$

(34)

Substituting Eq. (16) into (34), we obtain an equation for the relative employment shares between intermediate sectors,

$$\begin{aligned} \frac{\lambda _i }{\lambda _j }=\frac{L_i }{L_j }=( {G_s^\alpha G_p^{1-\alpha } })^{(\beta _i -\beta _j )(\sigma -1)}\big (\frac{a_i }{a_j }\big )^\sigma \end{aligned}$$

(35)

Thus, the employment share is given by

$$\begin{aligned} \lambda _i =\frac{L_i }{\sum \nolimits _{i=1}^N {L_i } }=\frac{( {G_s^\alpha G_p^{1-\alpha } })^{\beta _i (\sigma -1)}a_i ^\sigma }{\sum \nolimits _{i=1}^N {\left[ {( {G_s^\alpha G_p^{1-\alpha } })^{\beta _i (\sigma -1)}a_i ^\sigma } \right] } } \end{aligned}$$

(36)

Combining (32) with (36), we get Eq. (19).

Appendix B

1.1 Derivation of Eq. (20)

To analyze the aggregate growth, we assume that there exists an intermediate sector $i'$ in our economy and that the public expenditure elasticity for the intermediate sector $i'$ is $\mu (\beta _1 \le \mu \le \,\beta _N )$. For simplicity, we normalize the price of the intermediate good in sector $i'$ to one at each instance, $P_{i'} =1$. Therefore, (8) and (9) can be rewritten as

$$\begin{aligned} r=(1-\mu )( {G_s^\alpha G_p^{1-\alpha } })^\mu \,(K/L)^{-\mu } \end{aligned}$$

(37)

and

$$\begin{aligned} w=\mu ( {G_s^\alpha G_p^{1-\alpha } })^\mu \,(K/L)^{1-\mu } \end{aligned}$$

(38)

Using (4), (10), (17), (37) and (38), the level of public services can be written as

$$\begin{aligned} G_s =( {\tau \,\nu _s })^{\frac{1}{1-\alpha \mu }}G_p ^{\frac{(1-\alpha )\mu }{1-\alpha \mu }}K^{\frac{1-\mu }{1-\alpha \mu }} \end{aligned}$$

(39)

This result can be substituted into (37) to obtain an expression for the interest rate of

$$\begin{aligned} r=(1-\mu )( {\tau \,\nu _s })^{\frac{\alpha \mu }{1-\alpha \mu }}( {G_p /K})^{\frac{(1-\alpha )\mu }{1-\alpha \mu }} \end{aligned}$$

(40)

The problem for the representative household is to maximize the discounted stream of utility, defined in (1), over an infinite time horizon subject to its budget constraint in (2), taking factor prices as a given. The variables are $c$ and $k$, so that the first-order conditions for optimality are Eq. (2) and the condition

$$\begin{aligned} \dot{C}=C\left[ {(1-\tau )r-\rho } \right] \end{aligned}$$

(41)

The necessary conditions (2) and (41) are completed with the addition of the usual transversality condition,

$$\begin{aligned} \mathop {\lim }\limits _{t\rightarrow \infty } \left[ {\xi _t K_t e^{-\rho t}} \right] =0 \end{aligned}$$

(42)

where $\xi$ is the co-state variable for the shadow price of private capital.

Put Eq. (40) into (41) to yield (20).

Appendix C

1.1 Proof of Proposition 1

If a balanced growth path (BGP) exists for the dynamic systems (23) and (24), the relationship of $\mathop {g_p }\limits ^\bullet /g_p =\mathop c\limits ^\bullet /c=0$ will be satisfied.

Let the values for $g_p$ and $c$ on the BGP denote $\bar{g}_p$ and $\bar{c}$, respectively. Set $\mathop c\limits ^\bullet =0$ in (23) to obtain $\bar{c}=\rho +( {1-\tau })\mu ( {\tau \nu _s })^{\frac{\alpha \mu }{1-\alpha \mu }}( {g_p })^{\frac{(1-\alpha )\mu }{1-\alpha \mu }}$and then substitute $\bar{c}$ in (24) to yield the implicit form

$$\begin{aligned} f(g_p )\!=\!(1-\nu _s )\,\tau ^{\frac{1}{1-\alpha \mu }}( {\nu _s })^{\frac{\alpha \mu }{1-\alpha \mu }}( {g_p })^{\frac{\!-\!(1\!-\!\mu )}{1-\alpha \mu }}\!-\!( {1-\tau })(1-\mu )( {\nu _s \tau })^{\frac{\alpha \mu }{1-\alpha \mu }}( {g_p })^{\frac{(1-\alpha )\mu }{1-\alpha \mu }}+\rho . \end{aligned}$$

Simple calculation leads to $\frac{\partial f(g_p )}{\partial g_p }<0\mathop {\lim }\limits _{g_p \rightarrow 0} f(g_p )=+\infty$, and $\mathop {\lim }\limits _{g_p \rightarrow +\infty } f(g_p )=-\infty$. Given that $f(g_p )$ is a continuous, monotonically decreasing function of $g_p$, there is a unique positive value of $\bar{g}_p$ that satisfies $f(\bar{g}_p )=0$. This, in turn, implies that there exists a unique BGP where both $\bar{c}>0$ and $\bar{g}_p >0$. The aggregate economy’s growth rate along the BGP is

$$\begin{aligned} \bar{\gamma }=(1-\nu _s )\,\tau ^{\frac{1}{1-\alpha \mu }}( {\nu _s })^{\frac{\alpha \mu }{1-\alpha \mu }}( {\bar{g}_p })^{\frac{-(1-\mu )}{1-\alpha \mu }} \end{aligned}$$

(43)

which is determined by

$$\begin{aligned} ( {1-\tau })(1-\mu )( {\nu _s \tau })^{\frac{\alpha \mu }{1-\alpha \mu }}\bar{g}_p ^{\frac{(1-\alpha )\mu }{1-\alpha \mu }}-\rho -(1-\nu _s )\,\,\tau ^{\frac{1}{1-\alpha \mu }}( {\nu _s })^{\frac{\alpha \mu }{1-\alpha \mu }}\bar{g}_p ^{\frac{-(1-\mu )}{1-\alpha \mu }}=0\quad \end{aligned}$$

(44)

Appendix D

1.1 The saddle-path stability of the equilibrium

In this appendix, we prove that the balanced growth path (BGP) is locally saddle-path stable$.$

Because the dynamic system in (23) and (24) is nonlinear, to examine the local stability of the transitional path, we linearize the system around its BGP

$$\begin{aligned} \left( {\begin{array}{l} {\mathop c\limits ^\bullet } \\ {\mathop {g_p }\limits ^\bullet } \\ \end{array}}\right) =\left( {{\begin{array}{l@{\quad }l} {a_{11} } &{} {a_{12} } \\ {a_{21} } &{} {a_{22} } \\ \end{array} }}\right) \left( {\begin{array}{l} c-\bar{c} \\ g_p -\bar{g}_p \\ \end{array}}\right) , \end{aligned}$$

where the values of $a_{i,j}$ are given by

$$\begin{aligned} a_{11} =\bar{c}>0,\,a_{21} =\bar{g}_p >0, \end{aligned}$$

$$\begin{aligned} a_{12} =(\,1-\tau \,)\,\mu \,( {\tau \nu _s })^{\frac{\alpha \mu }{1-\alpha \mu }}\frac{\mu (1-\alpha )}{1-\alpha \mu }\bar{c}( {\bar{g}_p })^{\frac{\mu -1}{1-\alpha \mu }}\,>\,0, \end{aligned}$$

$a_{22} \!=\!-\!\frac{(1-\mu )}{1-\alpha \mu }\tau ^{\frac{1}{1-\alpha \mu }}(1-\nu _s )( {\nu _s })^{\frac{\alpha \mu }{1-\alpha \mu }}( {\bar{g}_p })^{\frac{\mu -1}{1-\alpha \mu }\!-\!1}\!-\!(1-\tau )( {\tau \nu _s })^{\frac{\alpha \mu }{1-\alpha \mu }}\frac{(1-\alpha )\mu }{1\!-\!\alpha \mu }( {\bar{g}_p })^{\frac{\mu (1\!-\!\alpha )}{1\!-\!\alpha \mu }\!-\!1}<0$ and

$$\begin{aligned} \text{ J }=\left| {\begin{array}{l} a_{11} \quad a_{12} \\ a_{21} \quad a_{22} \\ \end{array}} \right| <0 \end{aligned}$$

Hence, at least one eigenvalue is negative (or has a negative real part). Because Det J is always negative, we find that the equilibrium is locally unique, so that the dynamic system is saddle-path stable in the neighborhood of the BGP.