Fiscal policy and social infrastructure provision under alternative growth and distribution regimes

Abstract

We built a Kaleckian dynamic model that can comprehensively analyse the growth, distribution, and employment rate of the government's social infrastructure and debt accumulation. The model allows for not only wage-led growth and profit-led growth regimes but also labour-market-led and goods-market-led distribution regimes. Particular attention is paid to the demand effects of fiscal policy and the productivity growth effect of social infrastructure investment. Our model derives the following results. A combination of alternative growth and distribution regimes is important for stability. When government debt also changes in the long-run, the Domar condition is required for stability. Despite the principally Kaleckian nature, the long-run economic growth rate depends not on demand or fiscal parameters but on supply-side parameters. We conclude that the government can still play an important role in stabilising the economy, improving the quality of social infrastructure, and achieving a resilient economy.

Appendices

Appendix 1. Conditions for the existence of long-run steady states

The long-run steady state must simultaneously satisfy Eqs. (36), (37), and (38). As shown in Eq. (38), an economy may have two potential growth regimes: the shape of the economic growth rate is a convex quadrant in the domain of

$$\begin{array}{*{20}c} {m \in \left( {\frac{\tau }{{s\left( {1 - t} \right)}}\left( {\left( {1 + \gamma } \right)\left( {\theta_{C} + \theta_{I} } \right) - 1} \right),1} \right).} \\ \end{array}$$

(44)

Therefore, a unique profit share, $\tilde{m}$, switches the growth regime from WLG to PLG in this domain, and the actual growth rate (i.e. the LHS of Eq. (38)) takes the minimum value. The minimum growth rate is represented by:

$$\begin{array}{*{20}c} {g_{{\tilde{m}}} \equiv \frac{{\left( {\alpha + \beta \left( {1 - \tau } \right)\tilde{m}} \right)\left( {s\tilde{m}\left( {1 - \tau } \right) - \tau \left( {\theta_{C} + \theta_{I} - 1} \right)} \right)}}{{s\tilde{m}\left( {1 - \tau } \right) - \tau \left( {\left( {1 + \gamma } \right)\left( {\theta_{C} + \theta_{I} } \right) - 1} \right)}}.} \\ \end{array}$$

(45)

Hence, if

$$\begin{array}{*{20}c} {g_{{\tilde{m}}} < n + \frac{{\varepsilon_{0} }}{{1 - \varepsilon_{2} }}} \\ \end{array}$$

(46)

is satisfied in the above domain, the economy has two steady-state values for profit share and the associated growth regimes (i.e. WLG for smaller profit share and PLG for larger profit share). The value of a larger profit share must be less than unity to be economically meaningful. The capital composition is then determined by Eq. (36) according to the steady-state profit share. Independently, the employment rate is principally given by the parameters in Phillips curves.

Appendix 2. Proof of propositions 1 to 4

The dynamic system consists of Eqs. (19), (25), and (32), for which the Jacobian matrix, $J^{*}$, evaluated at the long-run steady state is given as follows:

$$j_{11} = \frac{{\partial \dot{\chi }}}{\partial \chi } = - \frac{{\left( {\alpha + \beta \left( {1 - \tau } \right)m^{*} } \right)\left( {sm^{*} \left( {1 - \tau } \right) - \tau \left( {\theta_{C} + \theta_{I} - 1} \right)} \right)}}{{sm^{*} \left( {1 - \tau } \right) - \tau \left( {\left( {1 + \gamma } \right)\left( {\theta_{C} + \theta_{I} } \right) - 1} \right)}}$$

$$j_{12} = \frac{{\partial \dot{\chi }}}{\partial m} = - \frac{{s\left( {1 - \tau } \right)\tau \left( {\alpha + \beta \left( {1 - \tau } \right)m^{*} } \right)\theta_{I} }}{{\left( {sm^{*} \left( {1 - \tau } \right) - \tau \left( {\theta_{C} + \theta_{I} - 1} \right)} \right)\left( {sm^{*} \left( {1 - \tau } \right) - \tau \left( {\left( {1 + \gamma } \right)\left( {\theta_{C} + \theta_{I} } \right) - 1} \right)} \right)}}$$

$$j_{13} = \frac{{\partial \dot{\chi }}}{\partial e} = 0$$

$$j_{21} = \frac{{\partial \dot{m}}}{\partial \chi } = - \frac{{\left( {1 - m^{*} } \right)\left( {\alpha + \beta \left( {1 - \tau } \right)m^{*} } \right)\left( {\varepsilon_{1} + \varepsilon_{2} } \right)\left( {sm^{*} \left( {1 - \tau } \right) - \tau \left( {\theta_{C} + \theta_{I} - 1} \right)} \right)^{2} \rho_{q} }}{{\tau \left( {1 - \varepsilon_{2} } \right)\theta_{I} \left( {sm^{*} \left( {1 - \tau } \right) - \tau \left( {\left( {1 + \gamma } \right)\left( {\theta_{C} + \theta_{I} } \right) - 1} \right)} \right)}}$$

$$j_{22} = \frac{{\partial \dot{m}}}{\partial m} = - \frac{{\left( {1 - m^{*} } \right)s\left( {1 - \tau } \right)\left( {\alpha + \beta \left( {1 - \tau } \right)m^{*} } \right)\left( {\varepsilon_{1} + \varepsilon_{2} } \right)\rho_{q} }}{{\left( {1 - \varepsilon_{2} } \right)\left( {sm^{*} \left( {1 - \tau } \right) - \tau \left( {\left( {1 + \gamma } \right)\left( {\theta_{C} + \theta_{I} } \right) - 1} \right)} \right)}}$$

$$j_{23} = \frac{{\partial \dot{m}}}{\partial e} = \left( {1 - m^{*} } \right)\left( {\rho_{P} - \rho_{W} } \right)$$

$$j_{31} = \frac{{\partial \dot{e}}}{\partial \chi } = \frac{{e^{*} \left( {\alpha + \beta \left( {1 - \tau } \right)m^{*} } \right)\left( {\varepsilon_{1} + \varepsilon_{2} } \right)\left( {sm^{*} \left( {1 - \tau } \right) - \tau \left( {\theta_{C} + \theta_{I} - 1} \right)} \right)^{2} }}{{\tau \left( {1 - \varepsilon_{2} } \right)\theta_{I} \left( {sm^{*} \left( {1 - \tau } \right) - \tau \left( {\left( {1 + \gamma } \right)\left( {\theta_{C} + \theta_{I} } \right) - 1} \right)} \right)}}$$

$$j_{32} = \frac{{\partial \dot{e}}}{\partial m} = e^{*} \left( {\left( {\frac{{1 + \varepsilon_{1} }}{{1 - \varepsilon_{2} }}} \right)\frac{\partial g}{{\partial m}} - \left( {\frac{{\varepsilon_{1} + \varepsilon_{2} }}{{1 - \varepsilon_{2} }}} \right)\frac{{\partial g_{s} }}{\partial m} } \right)$$

$$j_{33} = \frac{{\partial \dot{e}}}{\partial e} = 0$$

Note that, for simplicity, the steady-state value of $\chi^{*}$ is substituted into the Jacobian matrix elements to obtain these values.

The characteristic equation associated with the Jacobian matrix can be defined by

$$\begin{array}{*{20}c} {\lambda^{3} + a_{1} \lambda^{2} + a_{2} \lambda + a_{3} = 0,} \\ \end{array}$$

(47)

where $\lambda$ is the characteristic root. Coefficients $a_{1}$, $a_{2}$, and $a_{3}$ are given as:

$$\begin{array}{*{20}c} {a_{1} = g^{*} \left( {1 + \frac{{s\left( {1 - m^{*} } \right)\left( {1 - \tau } \right)\left( {\varepsilon_{1} + \varepsilon_{2} } \right)\rho_{q} }}{{\left( {1 - \varepsilon_{2} } \right)\left( {sm^{*} \left( {1 - \tau } \right) - \tau \left( {\theta_{C} + \theta_{I} - 1} \right)} \right)}}} \right)} \\ \end{array}$$

(48)

$$\begin{array}{*{20}c} {a_{2} = - \frac{{e^{*} \left( {1 - m^{*} } \right)\left( {1 - \tau } \right)\left( {\rho_{P} - \rho_{W} } \right)}}{{\left( {1 - \varepsilon_{2} } \right)\left( {sm^{*} \left( {1 - \tau } \right) - \tau \left( {\left( {1 + \gamma } \right)\left( {\theta_{C} + \theta_{I} } \right) - 1} \right)} \right)^{2} }}\left( {F\left( {m^{*} } \right) + \omega_{1} \varepsilon_{1} + \omega_{2} \varepsilon_{2} } \right),} \\ \end{array}$$

(49)

and

$$\begin{array}{*{20}c} {a_{3} = - \frac{{e^{*} \left( {1 - m^{*} } \right)g^{*} \left( {1 - \tau } \right)}}{{\left( {sm^{*} \left( {1 - \tau } \right) - \tau \left( {\left( {1 + \gamma } \right)\left( {\theta_{C} + \theta_{I} } \right) - 1} \right)} \right)^{2} }}\left( {\rho_{P} - \rho_{W} } \right)F\left( {m^{*} } \right),} \\ \end{array}$$

(50)

where

$$F\left( {m^{*} } \right) \equiv am^{*2} + bm^{*} + c\frac{ > }{ < }0$$

$$\omega_{1} \equiv s\left( {\alpha + \beta \left( {1 - \tau } \right)m^{*} } \right)\left( {sm^{*} \left( {1 - \tau } \right) - \tau \left( {\left( {1 + \gamma } \right)\left( {\theta_{C} + \theta_{I} } \right) - 1} \right)} \right) > 0$$

$$\omega_{2} \equiv \left( {sm^{*} \left( {1 - \tau } \right) - \tau \left( {\theta_{C} + \theta_{I} - 1} \right)} \right)\left( {s\alpha + \tau \beta \left( {\left( {1 + \gamma } \right)(\theta_{C} + \theta_{I} } \right) - 1} \right)) > 0$$

According to the Routh–Hurwitz criterion, the necessary and sufficient condition for the local asymptotic stability of the long-run steady state is:

$$\begin{array}{*{20}c} {a_{1} > 0, a_{2} > 0, a_{3} > 0,\; {\text{and}}\; a_{1} a_{2} - a_{3} > 0,} \\ \end{array}$$

(51)

where $a_{1} > 0$ is obvious under our assumptions, whereas the rest of the conditions are not a priori clear. Based on these preliminaries, we proceed to the proof of the propositions in order.

Proof of proposition 1. An economy with a WLG regime has $F\left({m}^{*}\right)<0$, whereas that with a PLG regime has $F\left({m}^{*}\right)>0$. In addition, ${\rho }_{P}-{\rho }_{W}<0$ in the LML regime, whereas ${\rho }_{P}-{\rho }_{W}>0$ in the GML regime. Based on the combination of these regimes, an economy with the WLG/LML and, or PLG/GML regimes does not satisfy ${a}_{3}>0$.

Q.E.D.

Proof of proposition 2. In an economy with PLG/LML regimes, we have $F\left( {m^{*} } \right) > 0$ and $\rho_{P} - \rho_{W} < 0$. Then, as the signs of both $\omega_{1}$ and $\omega_{2}$ are positive, $a_{2} > 0$ and $a_{3} > 0$ are ensured. Regarding $a_{1} a_{2} - a_{3}$, we have

$$\begin{array}{*{20}c} {a_{1} a_{2} - a_{3} = \frac{{e^{*} g^{*} \left( {1 - m^{*} } \right)\left( {1 - \tau } \right)\left( {\rho_{P} - \rho_{W} } \right)}}{{\left( {1 - \varepsilon_{2} } \right)\left( {sm^{*} \left( {1 - \tau } \right) - \tau \left( {\left( {1 + \gamma } \right)\left( {\theta_{C} + \theta_{I} } \right) - 1} \right)} \right)^{2} }} \cdot \Theta ,} \\ \end{array}$$

(52)

where

$\Theta \equiv F\left( {m^{*} } \right)\left( {1 - \varepsilon_{2} } \right) - \left( {1 + \frac{{\left( {1 - m^{*} } \right)s\left( {1 - \tau } \right)\left( {\varepsilon_{1} + \varepsilon_{2} } \right)\rho_{q} }}{{\left( {1 - \varepsilon_{2} } \right)\left( {sm^{*} \left( {1 - \tau } \right) - \tau \left( {\theta_{C} + \theta_{I} - 1} \right)} \right)}}} \right)\left( {F\left( {m^{*} } \right) + \omega_{1} \varepsilon_{1} + \omega_{2} \varepsilon_{2} } \right)$.

As $\rho_{P} - \rho_{W} < 0$ in the LML regime, the sign for $\Theta$ must be negative. Suppose $\Theta$ is negative; then, we have

$$\begin{array}{*{20}c} {F\left( {m^{*} } \right)\left( {1 - \varepsilon_{2} } \right) < \left( {1 + \frac{{\left( {1 - m^{*} } \right)s\left( {1 - \tau } \right)\left( {\varepsilon_{1} + \varepsilon_{2} } \right)\rho_{q} }}{{\left( {1 - \varepsilon_{2} } \right)\left( {sm^{*} \left( {1 - \tau } \right) - \tau \left( {\theta_{C} + \theta_{I} - 1} \right)} \right)}}} \right)\left( {F\left( {m^{*} } \right) + \omega_{1} \varepsilon_{1} + \omega_{2} \varepsilon_{2} } \right).} \\ \end{array}$$

(53)

It follows from this inequality that the following condition must be satisfied

$$\begin{array}{*{20}c} {\frac{{(1 - m^{*} )s\left( {1 - \tau } \right)\left( {\varepsilon_{1} + \varepsilon_{2} } \right)\rho_{q} }}{{\left( {1 - \varepsilon_{2} } \right)\left( {sm^{*} \left( {1 - \tau } \right) - \tau \left( {\theta_{C} + \theta_{I} - 1} \right)} \right)}} > - \left( {\frac{{F\left( {m^{*} } \right)\varepsilon_{2} + \omega_{1} \varepsilon_{1} + \omega_{2} \varepsilon_{2} }}{{F\left( {m^{*} } \right) + \omega_{1} \varepsilon_{1} + \omega_{2} \varepsilon_{2} }}} \right).} \\ \end{array}$$

(54)

Because $\omega_{1} > 0$, $\omega_{2} > 0$, and $F\left( {m^{*} } \right) > 0$ in the PLG regime, the sign of the RHS is always negative. In contrast, the value of the LHS is always positive, satisfying the above inequality. Hence, $a_{1} a_{2} - a_{3} > 0$ was also ensured under the PLG/LML regimes.

Q.E.D.

Proof of proposition 3. The steady-state values were independent of $\varepsilon_{1}$. In an economy with WLG/GML regimes, we have $F\left( {m^{*} } \right) < 0$ and $\rho_{P} - \rho_{W} > 0$. It follows from Eq. (49) that we need

$$\begin{array}{*{20}c} {F\left( {m^{*} } \right) + \omega_{1} \varepsilon_{1} + \omega_{2} \varepsilon_{2} < 0,} \\ \end{array}$$

(55)

to ensure $a_{2} > 0$. Because $\omega_{1} \varepsilon_{1} + \omega_{2} \varepsilon_{2} > 0$, the absolute value of $F\left( {m^{*} } \right)$, which we call the degree of being wage-led, must be sufficiently large. In other words, if the degree of wage-led is sufficiently weak and $\omega_{1} \varepsilon_{1} + \omega_{2} \varepsilon_{2} > \left| {F\left( {m^{*} } \right)} \right|$, we have $a_{2} < 0,$ and one of the stability conditions is violated.

Conversely, if the degree of being wage-led is sufficiently strong and $\omega_{1} \varepsilon_{1} + \omega_{2} \varepsilon_{2} < \left| {F\left( {m^{*} } \right)} \right|$, we have $a_{2} > 0$ in the following conditions:

$$\begin{array}{*{20}c} {0 < \varepsilon_{1} < \varepsilon_{1D} \equiv - \left( {\frac{{F\left( {m^{*} } \right) + \omega_{2} \varepsilon_{2} }}{{\omega_{1} }}} \right),} \\ \end{array}$$

(56)

where the sign of $\varepsilon_{1D}$ is positive by a sufficiently strong degree of being wage-led.

Meanwhile, because $\rho_{P} - \rho_{W} > 0$ in an economy with a GML regime, we need $\Theta > 0$ to satisfy $a_{1} a_{2} - a_{3} > 0$. Thus, the following conditions must be guaranteed:

$$\begin{array}{*{20}c} {\frac{{(1 - m^{*} )s\left( {1 - \tau } \right)\left( {\varepsilon_{1} + \varepsilon_{2} } \right)\rho_{q} }}{{\left( {1 - \varepsilon_{2} } \right)\left( {sm^{*} \left( {1 - \tau } \right) - \tau \left( {\theta_{C} + \theta_{I} - 1} \right)} \right)}} > - \left( {\frac{{F\left( {m^{*} } \right)\varepsilon_{2} + \omega_{1} \varepsilon_{1} + \omega_{2} \varepsilon_{2} }}{{F\left( {m^{*} } \right) + \omega_{1} \varepsilon_{1} + \omega_{2} \varepsilon_{2} }}} \right),} \\ \end{array}$$

(57)

where the denominator of the RHS is negative for $a_{2} > 0$.

Let us consider both sides of inequality (57) as a function of $\varepsilon_{1}$, which does not affect the steady-state values of our system, and illustrate them on a plane coordinate in Fig. 3 below. Taking the value of $\varepsilon_{1}$ on the x-axis, the plot of the LHS has a positive slope and intercept. For the RHS of inequality (57), by differentiating it with respect to $\varepsilon_{1}$ in a row, we obtain

$$\frac{{\partial {\text{RHS}}}}{{\partial \varepsilon_{1} }} = - \frac{{F\left( {m^{*} } \right)\left( {1 - \varepsilon_{2} } \right)\omega_{1} }}{{\left( {F\left( {m^{*} } \right) + \omega_{1} \varepsilon_{1} + \omega_{2} \varepsilon_{2} } \right)^{2} }} > 0,$$

and

$$\frac{{\partial^{2} {\text{RHS}}}}{{\partial \varepsilon_{1}^{2} }} = \frac{{ - 2F\left( {m^{*} } \right)\left( {1 - \varepsilon_{2} } \right)\omega_{1}^{2} }}{{\left( {F\left( {m^{*} } \right) + \omega_{1} \varepsilon_{1} + \omega_{2} \varepsilon_{2} } \right)^{3} }} > 0.$$

Therefore, the plot of the RHS of Inequality (57) shows an increasing curve, asymptotically approaching the value of $\varepsilon_{1D}$.

Meanwhile, the sign of the numerator in (57) is not obvious, and we may have:

$$\begin{array}{*{20}c} {F\left( {m^{*} } \right)\varepsilon_{2} + \omega_{1} \varepsilon_{1} + \omega_{2} \varepsilon_{2} \frac{ > }{ < }0,} \\ \end{array}$$

(58)

that is,

$$\begin{array}{*{20}c} {\varepsilon_{1} \frac{ > }{ < }\varepsilon_{1N} \equiv - \left( {\frac{{F\left( {m^{*} } \right) + \omega_{2} }}{{\omega_{1} }}} \right)\varepsilon_{2} = - \varepsilon_{2} ,} \\ \end{array}$$

(59)

where it is $\frac{{F\left( {m^{*} } \right) + \omega_{2} }}{{\omega_{1} }} = 1$. The following relationship is, therefore, confirmed:

$$\begin{array}{*{20}c} {\varepsilon_{1D} - \varepsilon_{1N} = - \frac{{F\left( {m^{*} } \right)}}{{\omega_{1} }}\left( {1 - \varepsilon_{2} } \right) > 0,} \\ \end{array}$$

(60)

and we always have $\varepsilon_{1D} > \varepsilon_{1N}$ in the WLG regime. The values of both sides in Eq. (57) are zero when $\varepsilon_{1} = \varepsilon_{1N} = - \varepsilon_{2}$ holds. Observing them together, we can find a unique value of $\varepsilon_{1}^{*} > 0$ between $0$, and $\varepsilon_{1D}$ guarantees that the value of LHS is equal to that of RHS, realising $a_{1} a_{2} - a_{3} = 0.$ These arguments are illustrated graphically in Fig. 3.

Fig. 3

Parameter configuration for stability condition $a_{1} a_{2} - a_{3}$

If $0 < \varepsilon_{1} < \varepsilon_{1}^{*}$, then both $a_{2} > 0$ and $a_{1} a_{2} - a_{3} > 0$ are guaranteed. If $\varepsilon_{1}^{*} < \varepsilon_{1}$, then both $a_{1} a_{2} - a_{3} > 0$ are violated. Moreover, if $\varepsilon_{1D} < \varepsilon_{1}$, then $a_{2} > 0$ is violated.

Thus, the following properties emerge according to the value of $\varepsilon_{1}$: for $0 < \varepsilon_{1} < \varepsilon_{1}^{*}$, the value of LHS is larger than that of RHS in Inequality (A14), and we have $a_{1} a_{2} - a_{3} > 0$. However, if $\varepsilon_{1}^{*} < \varepsilon_{1} < \varepsilon_{1D}$, we have $a_{1} a_{2} - a_{3} < 0$. Accordingly, there also exists a unique value of $0 < \varepsilon_{1}^{*} < \varepsilon_{1D}$, on which $a_{1} a_{2} - a_{3} = 0$ is established. Thus, a limit cycle occurs by Hopf bifurcation for $\varepsilon_{1}$ sufficiently close to $\varepsilon_{1}^{*}$ for the combination of WLG/GML regimes.

Q.E.D.

Proof of proposition 4

The dynamics of the debt ratio is

$$\begin{array}{*{20}c} \dot{\delta } = \left( {\theta_{C} + \theta_{I} - 1} \right)\tau u^{*} + i\delta - \left( {\hat{p}^{*} + g^{*} } \right)\delta , \end{array}$$

(61)

where the rates of capacity utilisation rate, inflation, and output growth as the fast variables will all follow the long-run steady-state values given by Eqs. (36), (37), and (38), respectively. By substituting them into Eq. (35), we obtain

$$\begin{array}{*{20}c} {\dot{\delta } = \frac{{\left( {\theta_{C} + \theta_{I} - 1} \right)\tau \left( {\alpha + \beta \left( {1 - \tau } \right)m^{*} } \right)}}{{sm^{*} \left( {1 - \tau } \right) - \tau \left( {\left( {1 + \gamma } \right)\left( {\theta_{C} + \theta_{I} } \right) - 1} \right)}} + i\delta - \left( {n + \frac{{\mu_{1} \left( {1 - \nu_{2} } \right)}}{{\mu_{1} \left( {1 - \nu_{2} } \right) - \mu_{2} \left( {1 - \nu_{1} } \right)}}\left( {\frac{{\varepsilon_{0} }}{{1 - \varepsilon_{2} }}} \right)} \right)\delta .} \\ \end{array}$$

(62)

It has the following unique steady-state value, $\delta^{*}$, shown by Eq. (40). The steady state of the government's debt ratio is locally and asymptotically stable if

$$\begin{array}{*{20}c} {\frac{{{\text{d}}\dot{\delta }}}{{{\text{d}}\delta }} < 0,} \\ \end{array}$$

(63)

is ensured at a steady state. Hence, we have

$$\begin{array}{*{20}c} {i < n + \frac{{\mu_{1} \left( {1 - \nu_{2} } \right)}}{{\mu_{1} \left( {1 - \nu_{2} } \right) - \mu_{2} \left( {1 - \nu_{1} } \right)}}\left( {\frac{{\varepsilon_{0} }}{{1 - \varepsilon_{2} }}} \right),} \\ \end{array}$$

(64)

which is equivalent to

$$\begin{array}{*{20}c} {i < \hat{p}^{*} + g^{*} .} \\ \end{array}$$

(65)

This shows that the nominal economic growth rate is higher than the nominal interest rate or the economic growth rate is higher than the real interest rate.

Q.E.D.