Endogenous sector-biased technical change and perpetual and transient structural change

Abstract

This paper provides a new definition of structural change and then categorizes such changes into two types. Transient structural change is related to the short-term adjustment of sectoral labor employment, while perpetual structural change concerns the long-run trends of labor reallocation among sectors. If we focus on supply-side reasons, we find that the one and only fundamental driving force of perpetual structural change is sector-biased technical change. We also investigate the structural parameters of the model that determine the direction of sector-biased technical change.

Appendices

Appendix

In the appendix, we will analyze the ABGP and the transitional dynamics of the present model, in which the following lemma is useful.

Lemma A.1

When $\mathop {\lim }\limits _{t\rightarrow \infty } n_{-i} >0$ and $\mathop {\lim }\limits _{t\rightarrow \infty } u_{-i} >0$, the optimal trajectory of the present model can be described by an autonomous nonlinear dynamical system consisting of the dynamic trajectories of C, K, $u_i$, and L in Eqs. (A.1), (A.2), and (A.3) as well as $L(t)=\exp (\nu t)$.

$$\begin{aligned}&\frac{\dot{C}}{C}=-\frac{(\rho -\nu \sigma )}{\sigma }+\frac{\left( \alpha _i u_{-i} +\alpha _{-i} u_i\right) ^{\frac{1}{\varepsilon -1}}}{\sigma }\left\{ \gamma _i^{\frac{\varepsilon }{\varepsilon -1}} \alpha _i \alpha _{-i}^{\frac{1}{1-\varepsilon }} \left[ u_i^{\frac{1}{1-\varepsilon }} ({Y_i }/K)\right] \right. \nonumber \\&\quad \left. +\gamma _{-i}^{\frac{\varepsilon }{\varepsilon -1}} \alpha _{-i} \alpha _i^{\frac{1}{1-\varepsilon }} \left[ u_{-i}^{\frac{1}{1-\varepsilon }} ({Y_{-i} }/K)\right] \right\} \end{aligned}$$

(A.1)

$$\begin{aligned}&\frac{\dot{K}}{K}=\xi _i^{-1} \left[ {\frac{\varepsilon }{(\varepsilon -1)u_{-i} }-{\mathbb {F}}_i (u_i)} \right] \frac{\dot{u}_i }{u_i }-(\alpha _{-i} -\alpha _i)\xi _i^{-1} \nu \end{aligned}$$

(A.2)

$$\begin{aligned}&\frac{C}{K}+\left\{ {\mathbb {S}}(u_i )\xi _i^{-1} \left[ {\frac{\varepsilon }{(\varepsilon -1)u_{-i} }-{\mathbb {F}}_i (u_i )} \right] +\left[ \alpha _i (1-\varphi _i)\right] ^{-1}u_i\right. \nonumber \\&\quad \left. -\left[ \alpha _{-i} (1-\varphi _{-i} )\right] ^{-1}u_i \right\} \frac{\dot{u}_i}{u_i}\nonumber \\&\quad =\left[ {\gamma _i ({Y_i}/K)^{\frac{\varepsilon -1}{\varepsilon }}+\gamma _{-i} ({Y_{-i}}/K)^{\frac{\varepsilon -1}{\varepsilon }}} \right] ^{\frac{\varepsilon }{\varepsilon -1}}+{\mathbb {S}}(u_i )(\alpha _{-i}-\alpha _i)\xi _i^{-1} \nu \end{aligned}$$

(A.3)

where

$$\begin{aligned}&{Y_i}/K=A_i u_i \left( {\frac{(1-\alpha _i )\alpha _{-i} }{(1-\alpha _{-i} )\alpha _i +(\alpha _{-i} -\alpha _i )u_i }} \right) ^{1-\alpha _i }(L/K)^{1-\alpha _i }\\&\quad {\mathbb {F}}_i (u_i )\equiv \frac{1}{1-\varphi _i }+\alpha _i +\frac{(1-\alpha _i )(1-\alpha _{-i} )\alpha _i }{(1-\alpha _{-i} )\alpha _i +(\alpha _{-i} -\alpha _i )u_i } \\&\quad +\left[ {\frac{1}{1-\varphi _{-i} }+\alpha _{-i} +\frac{(1-\alpha _{-i} )(1-\alpha _i )\alpha _{-i} }{(1-\alpha _{-i} )\alpha _i +(\alpha _{-i} -\alpha _i )u_i }} \right] \frac{u_i }{u_{-i}} \\&\quad {\mathbb {S}}(u_i )\equiv 1+\frac{u_i }{\alpha _i (1-\varphi _i )}+\frac{u_{-i} }{\alpha _{-i} (1-\varphi _{-i} )} \end{aligned}$$

Proof of Lemma A.1

The derivative of the Hamiltonian with respect to total consumption, C, implies that

$$\begin{aligned} {\dot{C}}/C=-\sigma ^{-1}\left[ {\dot{\lambda }}/\lambda +(\rho -\nu \sigma )\right] \end{aligned}$$

(A.4)

Substituting ${\dot{\lambda }}/\lambda$ in Eq. (15) into (A.4) yields Eq. (A.1). Substituting ${\dot{n}_i}/{n_i}$ in Eq. (7) and the state of the art technology in Eq. (17) into (8), we can determine the growth rate of real output in the labor-outflowing sector i by

$$\begin{aligned} \frac{\dot{Y}_i }{Y_i }=\left[ {\frac{1}{1-\varphi _i }+\alpha _i } \right] \frac{\dot{K}}{K}+\left[ {\frac{1}{1-\varphi _i }+\alpha _i +\frac{(1-\alpha _i )(1-\alpha _{-i} )\alpha _i }{\alpha _i (1-\alpha _{-i})+(\alpha _{-i} -\alpha _i )u_i }} \right] \frac{\dot{u}_i }{u_i }+(1-\alpha _i )\nu \nonumber \\ \end{aligned}$$

(A.5)

Substituting ${\dot{Y}_i}/{Y_i}$ in Eq. (A.5) into (6) yields (A.2).

Finally, combining the R&D expenditures in Eq. (19) and the capital accumulation rate in Eq. (A.2) with the budget constraint $Y=C+\dot{K}+X_i +X_{-i}$ gives Eq. (A.3).

Note that when sector $-i$ instead of i is the labor-outflowing sector, $\mathop {\lim }\limits _{t\rightarrow \infty } u_{-i} =0$, and $u_{-i}$ cannot appear alone in the denominator in ${\mathbb {F}}_i (u_i )$ and Eq. (A.3) with the exception of ${\dot{u}_{-i} }/{u_{-i} }$. Thus, when $\mathop {\lim }\limits _{t\rightarrow \infty } u_{-i} =0$, the optimal trajectory of the present model can be described by an autonomous nonlinear dynamical system consisting of the dynamic trajectories of C, K, $u_{-i}$ instead of $u_i$, and L, with i and $-i$ being switched in Lemma A.1. $\square$

Proof of Proposition 3

When technical change is sector unbiased, Proposition 1 implies that the fractions of labor allocated in both sectors should lie on the unit interval, i.e., $\mathop {\lim }\limits _{t\rightarrow \infty } n_i \in (0,1)$ and $\mathop {\lim }\limits _{t\rightarrow \infty } n_{-i} \in (0,1)$. Substituting $\mathop {\lim }\limits _{t\rightarrow \infty } {\dot{u}_i }/{u_i }=0$ and $\mathop {\lim }\limits _{t\rightarrow \infty } {\dot{n}_i }/{n_i }=0$ into Eq. (A.2), we can yield the capital accumulation rate in the ABGP in the case of sector-unbiased technical change by $\mathop {\lim }\limits _{t\rightarrow \infty } {\dot{K}}/K=(\alpha _i -\alpha _{-i} )\xi _i^{-1} \nu$. Sustainable economic growth in the case of sector-unbiased technical change requires $(\alpha _i -\alpha _{-i} )\xi _i^{-1} >1$.

Along the ABGP, $\mathop {\lim }\limits _{t\rightarrow \infty } {\dot{K}}/K=\mathop {\lim }\limits _{t\rightarrow \infty } {\dot{Y}}/Y$ requires that

$$\begin{aligned} \mathop {\lim }\limits _{t\rightarrow \infty } {\dot{K}}/K=\mathop {\lim }\limits _{t\rightarrow \infty } {\dot{Y}}/Y=\mathop {\lim }\limits _{t\rightarrow \infty } {\dot{Y}_i }/{Y_i }=\mathop {\lim }\limits _{t\rightarrow \infty } {\dot{Y}_{-i} }/{Y_{-i} }=\mathop {\lim }\limits _{t\rightarrow \infty } {\dot{C}}/C=(\alpha _i -\alpha _{-i} )\xi _i^{-1} \nu \end{aligned}$$

Substituting $\mathop {\lim }\limits _{t\rightarrow \infty } {\dot{K}}/K=(\alpha _i -\alpha _{-i} )\xi _i^{-1} \nu$, $\mathop {\lim }\limits _{t\rightarrow \infty } {\dot{u}_i }/{u_i }=0$ and $\mathop {\lim }\limits _{t\rightarrow \infty } {\dot{u}_{-i} }/{u_{-i} }=0$ into Eq. (17), we can determine the purely labor-augmenting technical change rate in the case of sector-unbiased technical change by

$$\begin{aligned}&(1-\alpha _i )^{-1}\mathop {\lim }\limits _{t\rightarrow \infty } {\dot{A}_i }/{A_i }=(1-\alpha _{-i} )^{-1}\mathop {\lim }\limits _{t\rightarrow \infty } {\dot{A}_{-i} }/{A_{-i} }\\&\quad =(1-\alpha _i )^{-1}(1-\varphi _i )^{-1}(\alpha _i -\alpha _{-i} )\xi _i^{-1} \nu \end{aligned}$$

Substituting $\mathop {\lim }\limits _{t\rightarrow \infty } {\dot{C}}/C=(\alpha _i -\alpha _{-i} )\xi _i^{-1} \nu$ into Eq. (A.4), we can then express the growth rate for the co-state variable on physical capital in the long run in the case of sector-unbiased technical change as

$$\begin{aligned} \mathop {\lim }\limits _{t\rightarrow \infty } {\dot{\lambda }}/\lambda =-\sigma (\alpha _i -\alpha _{-i} )\xi _i^{-1} \nu -(\rho -\nu \sigma ) \end{aligned}$$

In addition, the growth rate for the co-state variable on the state of the art technology in sector j in the long run can be calculated from Eq. (14). It is easy to observe that the STVC in the case of sector-unbiased technical change can be satisfied provided that $(1-\sigma )(\alpha _i -\alpha _{-i} )\xi _i^{-1} \nu <(\rho -\nu \sigma )$. This process completes the proof of Proposition 3 in the main text.

Now, we will turn to the transitional dynamics of the present model in the case of sector-unbiased technical change. Sustainable economic growth implies that C, K, $Y_i$ and $Y_{-i}$ will grow without bound; thus, we need to transform these four variables into constants along the ABGP and then investigate the evolution of the transformed variables. The foregoing analysis implies that C / K, $u_i^{\frac{1}{1-\varepsilon }} {Y_i}/K$, $u_{-i}^{\frac{1}{1-\varepsilon }}{Y_{-i}}/K$, and $u_i$ ($u_{-i}$) are all positive constants along the ABGP when technical change is sector unbiased.

Let us define $\Theta \equiv C/K$, $\Phi _i\equiv u_i^{\frac{1}{1-\varepsilon }}{Y_i}/K$, and $\Phi _{-i}\equiv u_{-i}^{\frac{1}{1-\varepsilon }}{Y_{-i}}/K$. Regarding the transitional dynamics in the case of sector-unbiased technical change, we can apply Lemma A.1 to obtain the following proposition. $\square$

Proposition A.1

Suppose that the assumptions in Proposition 3 are satisfied, given $L_0 =1$ and any initial conditions $K(0)>0$; then, in the case of sector-unbiased technical change, the present economy can be described as either (a) a four-dimensional autonomous dynamical system consisting of the evolution of $\Theta$ in Eq. (A.6), the evolution of $\Phi _i$ in Eq. (A.7), the evolution of $\Phi _{-i}$ in Eq. (A.8), and the evolution of $u_i$ in Eq. (A.9) with $u_{-i}$ being replaced by $u_{-i} =1-u_i$; or (b) a four-dimensional autonomous dynamical system consisting of the evolution of $\Theta$ in Eq. (A.6), the evolution of $\Phi _i$ in Eq. (A.7), the evolution of $\Phi _{-i}$ in Eq. (A.8), and the evolution of $u_{-i}$ in Eq. (A.9) with $u_i$ being replaced by $u_i =1-u_{-i}$.

$$\begin{aligned}&\frac{\dot{\Theta }}{\Theta }=-\frac{(\rho -\nu \sigma )}{\sigma }+\left( {\alpha _i u_{-i} +\alpha _{-i} u_i } \right) ^{\frac{1}{\varepsilon -1}}\sigma ^{-1}\left[ \gamma _i^{\frac{\varepsilon }{\varepsilon -1}} \alpha _i \alpha _{-i}^{\frac{1}{1-\varepsilon }} \Phi _i +\gamma _{-i}^{\frac{\varepsilon }{\varepsilon -1}} \alpha _{-i} \alpha _i^{\frac{1}{1-\varepsilon }} \Phi _{-i} \right] -\frac{\dot{K}}{K} \nonumber \\\end{aligned}$$

(A.6)

$$\begin{aligned}&\frac{\dot{\Phi }_i }{\Phi _i }=\left[ {\frac{1}{1-\varphi _i }+\alpha _i -1} \right] \frac{\dot{K}}{K}+(1-\alpha _i )\nu \nonumber \\&\quad +\left[ {\frac{1}{1-\varepsilon }+\frac{1}{1-\varphi _i }+\alpha _i +\frac{(1-\alpha _i )(1-\alpha _{-i} )\alpha _i }{\alpha _i (1-\alpha _{-i} )+(\alpha _{-i} -\alpha _i )u_i }} \right] \frac{\dot{u}_i }{u_i} \end{aligned}$$

(A.7)

$$\begin{aligned}&\frac{\dot{\Phi }_{-i} }{\Phi _{-i} }=\left[ {\frac{1}{1-\varphi _{-i} }+\alpha _{-i} -1} \right] \frac{\dot{K}}{K}+(1-\alpha _{-i} )\nu \nonumber \\&\quad +\left[ {\frac{1}{1-\varepsilon }+\frac{1}{1-\varphi _{-i} }+\alpha _{-i} +\frac{(1-\alpha _{-i} )(1-\alpha _i )\alpha _{-i} }{\alpha _{-i} (1-\alpha _i )+(\alpha _i -\alpha _{-i} )u_{-i}}} \right] \frac{\dot{u}_{-i} }{u_{-i}} \end{aligned}$$

(A.8)

$$\begin{aligned}&\Theta +\left\{ {{\mathbb {S}}(u_i )\xi _i^{-1} \left[ {\frac{\varepsilon }{(\varepsilon -1)u_{-i} }-{\mathbb {F}}_i (u_i )} \right] +\left[ \alpha _i (1-\varphi _i )\right] ^{-1}u_i -\left[ \alpha _{-i} (1-\varphi _{-i} )\right] ^{-1}u_i } \right\} \frac{\dot{u}_i }{u_i } \nonumber \\&\quad =\left[ {\gamma _i ({Y_i }/K)^{\frac{\varepsilon -1}{\varepsilon }}+\gamma _{-i} ({Y_{-i} }/K)^{\frac{\varepsilon -1}{\varepsilon }}} \right] ^{\frac{\varepsilon }{\varepsilon -1}}+{\mathbb {S}}(u_i)(\alpha _{-i} -\alpha _i )\xi _i^{-1} \nu \end{aligned}$$

(A.9)

where ${\dot{K}}/K$ is given by Eq. (A.2).

Proof of Proposition 4

When technical change is sector biased, in which both factors will flow away from the labor-outflowing sector i to the labor-inflowing sector $-i$ without cessation, sustainable economic growth in Eq. (20) requires $(1-\alpha _{-i} )(1-\varphi _{-i} )-1>0$. Substituting the R&D expenditures in Eq. (19) and the capital accumulation rate in Eq. (20) into the budget constraint $Y=C+\dot{K}+X_i +X_{-i}$, we can obtain the aggregate consumption growth rate in the long run as

$$\begin{aligned} \mathop {\lim }\limits _{t\rightarrow \infty } {\dot{C}}/C=\left[ (1-\alpha _{-i} )(1-\varphi _{-i} )-1\right] ^{-1}(1-\alpha _{-i} )(1-\varphi _{-i} )\nu \end{aligned}$$

(A.10)

Equations (A.4) and (A.10) imply that the growth rate for the co-state variable on physical capital along the ABGP can be expressed as

$$\begin{aligned} \mathop {\lim }\limits _{t\rightarrow \infty } {\dot{\lambda }}/\lambda =-\sigma (1-\alpha _{-i} )(1-\varphi _{-i})\left[ (1-\alpha _{-i} )(1-\varphi _{-i} )-1\right] ^{-1}\nu -(\rho -\nu \sigma )\nonumber \\ \end{aligned}$$

(A.11)

In addition, the growth rate for the co-state variable on the state of the art technology in sector j along the ABGP can be calculated by substituting Eqs. (21), (22) and (A.11) into (14). It is easy to observe that the STVC in the case of sector-biased technical change, in which both factors will flow away from the labor-outflowing sector i to the labor-inflowing sector $-i$ without cessation, can be satisfied provided that $(1-\sigma )(1-\alpha _{-i} )(1-\varphi _{-i} )[(1-\alpha _{-i} )(1-\varphi _{-i} )-1]^{-1}\nu <(\rho -\nu \sigma )$.

Finally, the arithmetic manipulation in Eq. (24) implies that (a) $(1-\alpha _i )^{-1}\mathop {\lim }\limits _{t\rightarrow \infty } {\dot{A}_i }/{A_i }>(1-\alpha _{-i} )^{-1}\mathop {\lim }\limits _{t\rightarrow \infty } {\dot{A}_{-i} }/{A_{-i} }$ if $0<\varepsilon <1$ and $(1-\alpha _i )(1-\varphi _i )<(1-\alpha _{-i} )(1-\varphi _{-i} )$; (b) $(1-\alpha _i )^{-1}\mathop {\lim }\limits _{t\rightarrow \infty } {\dot{A}_i }/{A_i }<(1-\alpha _{-i} )^{-1}\mathop {\lim }\limits _{t\rightarrow \infty } {\dot{A}_{-i} }/{A_{-i} }$ if $1<\varepsilon <2-\varphi _i$ and $(1-\alpha _{-i} )(1-\varphi _{-i} )<(1-\alpha _i )(1-\varphi _i )$; and (c) $(1-\alpha _i )^{-1}\mathop {\lim }\limits _{t\rightarrow \infty } {\dot{A}_i }/{A_i }<(1-\alpha _{-i} )^{-1}\mathop {\lim }\limits _{t\rightarrow \infty } {\dot{A}_{-i} }/{A_{-i} }$ if $\varepsilon >2-\varphi _i$ and $(1-\alpha _i )(1-\varphi _i )<(1-\alpha _{-i} )(1-\varphi _{-i} )$. This completes the proof of Proposition 4 in the main text. $\square$

With regard to the transitional dynamics in the case of sector-biased technical change in which both factors will flow away from the labor-outflowing sector i to the labor-inflowing sector $-i$ without cessation, we can apply Lemma A.1 to obtain the following proposition.

Proposition A.2

Suppose that the assumptions in Proposition 4 are satisfied, given $L(0)=1$ and any initial conditions $K(0)>0$; then, the present economy in the case of sector-biased technical change, in which both factors will flow away from the labor-outflowing sector i to the labor-inflowing sector $-i$ without cessation, can be described as a four-dimensional autonomous dynamical system consisting of the evolution of $\Theta$ in Eq. (A.12), the evolution of $\Phi _{-i}$ in Eq. (A.13), the evolution of $\Phi _i$ in Eq. (A.14), and the evolution of $u_i$ in Eq. (A.15).

$$\begin{aligned}&\frac{\dot{\Theta }}{\Theta }=-\frac{(\rho -\nu \sigma )}{\sigma }+\left( \alpha _i u_{-i} +\alpha _{-i} u_i\right) ^{\frac{1}{\varepsilon -1}}\sigma ^{-1}\left[ \gamma _i^{\frac{\varepsilon }{\varepsilon -1}} \alpha _i \alpha _{-i}^{\frac{1}{1-\varepsilon }} \Phi _i +\gamma _{-i}^{\frac{\varepsilon }{\varepsilon -1}} \alpha _{-i} \alpha _i^{\frac{1}{1-\varepsilon }} \Phi _{-i}\right] -\frac{\dot{K}}{K} \nonumber \\\end{aligned}$$

(A.12)

$$\begin{aligned}&\frac{\dot{\Phi }_{-i} }{\Phi _{-i} }=\left[ {\frac{1}{1-\varphi _{-i} }+\alpha _{-i} -1} \right] \frac{\dot{K}}{K}+(1-\alpha _{-i} )\nu \nonumber \\&\quad - \frac{u_i }{u_{-i} }\left[ {\frac{1}{1-\varepsilon }+\frac{1}{1-\varphi _{-i} }+\alpha _{-i} +\frac{(1-\alpha _{-i} )(1-\alpha _i )\alpha _{-i} }{\alpha _{-i} (1-\alpha _i )+(\alpha _i -\alpha _{-i} )(1-u_i )}} \right] \frac{\dot{u}_i }{u_i } \end{aligned}$$

(A.13)

$$\begin{aligned}&\frac{\dot{\Phi }_i }{\Phi _i }=\left[ {\frac{1}{1-\varphi _i }+\alpha _i -1} \right] \frac{\dot{K}}{K}+(1-\alpha _i )\nu \nonumber \\&\quad + \left[ {\frac{1}{1-\varepsilon }+\frac{1}{1-\varphi _i }+\alpha _i +\frac{(1-\alpha _i )(1-\alpha _{-i} )\alpha _i }{\alpha _i (1-\alpha _{-i} )+(\alpha _{-i} -\alpha _i )u_i }} \right] \frac{\dot{u}_i }{u_i } \end{aligned}$$

(A.14)

$$\begin{aligned}&\Theta +\left\{ {{\mathbb {S}}(u_i )\xi _i^{-1} \left[ {\frac{\varepsilon }{(\varepsilon -1)u_{-i} }-{\mathbb {F}}_i (u_i )} \right] +\left[ \alpha _i (1-\varphi _i )\right] ^{-1}u_i -\left[ \alpha _{-i} (1-\varphi _{-i} )\right] ^{-1}u_i } \right\} \frac{\dot{u}_i }{u_i } \nonumber \\&\quad =\left[ {\gamma _i u_i^{\frac{1}{\varepsilon }} (\Phi _i )^{\frac{\varepsilon -1}{\varepsilon }}+\gamma _{-i} u_{-i}^{\frac{1}{\varepsilon }} (\Phi _{-i} )^{\frac{\varepsilon -1}{\varepsilon }}} \right] ^{\frac{\varepsilon }{\varepsilon -1}}+{\mathbb {S}}(u_i)(\alpha _{-i} -\alpha _i )\xi _i^{-1} \nu \end{aligned}$$

(A.15)

where ${\dot{K}}/K$ is given by Eq. (A.2).

The Jacobian of the dynamic system in the case of sector-biased technical change

When technical change is sector biased, in which both factors will flow away from the labor-outflowing sector i to the labor-inflowing sector $-i$ without cessation, both $\mathop {\lim }\limits _{t\rightarrow \infty } C/K$ and $\mathop {\lim }\limits _{t\rightarrow \infty } u_{-i}^{\frac{1}{1-\varepsilon }} {Y_{-i} }/K$ are positive constants, and $\mathop {\lim }\limits _{t\rightarrow \infty } u_i^{\frac{1}{1-\varepsilon }} {Y_i }/K=0$ according to $\mathop {\lim }\limits _{t\rightarrow \infty } {\dot{Y}_i }/{Y_i }=\mathop {\lim }\limits _{t\rightarrow \infty } {\dot{K}}/K+\varepsilon /{(\varepsilon -1)}\mathop {\lim }\limits _{t\rightarrow \infty } {\dot{u}_i }/{u_i }$. Let us use ${*}$ as a superscript to indicate the constant value of a variable in the long run; we observe that $\Theta ^{{*}}=\mathop {\lim }\limits _{t\rightarrow \infty } C/K>0$, $\Phi _{-i}^{*} =\mathop {\lim }\limits _{t\rightarrow \infty } u_{-i}^{\frac{1}{1-\varepsilon }} {Y_{-i} }/K>0$, $\Phi _i^{*} =\mathop {\lim }\limits _{t\rightarrow \infty } u_i^{\frac{1}{1-\varepsilon }} {Y_i }/K=0$, and $u_i^{*} =0$. Substituting $\mathop {\lim }\limits _{t\rightarrow \infty } {\dot{\Theta }}/\Theta =0$, $u_i^{*} =0$, and $\Phi _i^{*} =0$ into Eqs. (A.12) and (A.15), we can yield the steady-state value of $\Theta ^{{*}}$ and $\Phi _{-i}^{*}$ with the following linear system of equations with two variables.

$$\begin{aligned} 0= & {} -(\rho -\nu \sigma )\sigma ^{-1}-\mathop {\lim }\limits _{t\rightarrow \infty } {\dot{K}}/K+\sigma ^{-1}(\gamma _{-i} )^{\frac{\varepsilon }{\varepsilon -1}}\alpha _{-i} \Phi _{-i}^{*}\\&\Theta ^{{*}}+\mathop {\lim }\limits _{u_i \rightarrow 0} {\mathbb {S}}(u_i )\xi _i^{-1} \left[ {\frac{\varepsilon }{(\varepsilon -1)u_{-i} }-{\mathbb {F}}_i (u_i )} \right] \frac{\dot{u}_i }{u_i }=(\gamma _{-i} )^{\frac{\varepsilon }{\varepsilon -1}}\Phi _{-i}^{*}\\&+\mathop {\lim }\limits _{u_i \rightarrow 0} {\mathbb {S}}(u_i )(\alpha _{-i} -\alpha _i )\xi _i^{-1} \nu \end{aligned}$$

Substituting $\mathop {\lim }\limits _{u_i \rightarrow 0} {\mathbb {F}}_i (u_i )=(1-\varphi _i )^{-1}+1$, $\mathop {\lim }\limits _{u_i \rightarrow 0} {\mathbb {S}}(u_i )\equiv 1+[\alpha _{-i} (1-\varphi _{-i} )]^{-1}$, $\mathop {\lim }\limits _{t\rightarrow \infty } {\dot{K}}/K$ in Eq. (20) and $\xi _i =(1-\varphi _i )^{-1}-(1-\varphi _{-i} )^{-1}+\alpha _i -\alpha _{-i}$ and $\mathop {\lim }\limits _{t\rightarrow \infty } {\dot{u}_i }/{u_i }=\mathop {\lim }\limits _{t\rightarrow \infty } {\dot{n}_i }/{n_i }$ in Eq. (23) into the above linear system of equations, we can obtain the explicit solutions of $\Phi _{-i}^{*}$ and $\Theta ^{{*}}$ as

$$\begin{aligned} \Phi _{-i}^{*}= & {} \left[ (\gamma _{-i} )^{\frac{\varepsilon }{\varepsilon -1}}\alpha _{-i}\right] ^{-1}\left\{ (\rho -\nu \sigma )+(1-\alpha _{-i} )(1-\varphi _{-i} )\sigma \right. \\&\left. \left[ (1-\alpha _{-i} )(1-\varphi _{-i})-1\right] ^{-1}\nu \right\} \\ \Theta ^{{*}}= & {} (\alpha _{-i} )^{-1}(1-\alpha _{-i} )\left\{ {(1-\varphi _{-i} )\sigma -\left[ {\alpha _{-i} (1-\varphi _{-i})+1} \right] } \right\} \\&\left[ (1-\alpha _{-i})(1-\varphi _{-i} )-1\right] ^{-1}\nu +(\rho -\nu \sigma ) \end{aligned}$$

The Jacobian of the dynamical system in the case of sector-biased technical change can be written as

$$\begin{aligned} \left[ {{\begin{array}{llll} {J_{11}^0}&{} {J_{12}^0}&{} {J_{13}^0}&{} {J_{14}^0} \\ {J_{21}^0}&{} {J_{22}^0}&{} {J_{23}^0}&{} {J_{24}^0} \\ {J_{31}^0}&{} {J_{32}^0}&{} {J_{33}^0}&{} {J_{34}^0} \\ {J_{41}^0}&{} {J_{42}^0}&{} {J_{43}^0}&{} {J_{44}^0} \\ \end{array}}}\right] =\mathop {\lim }\limits _{t\rightarrow \infty } \left[ {{\begin{array}{llll} {{\partial \dot{\Theta }}/{\partial \Theta }}&{}{{\partial \dot{\Theta }}/{\partial u_i}}&{} {{\partial \dot{\Theta }}/{\partial \Phi _{i}}}&{} {{\partial \dot{\Theta }}/{\partial \Phi _{-i}}}\\ {{\partial \dot{u}_i}/{\partial \Theta }}&{} {{\partial \dot{u}_i}/{\partial u_i }}&{} {{\partial \dot{u}_i }/{\partial \Phi _i }}&{} {{\partial \dot{u}_i }/{\partial \Phi _{-i}}}\\ {{\partial \dot{\Phi }_i}/{\partial \Theta }}&{}{{\partial \dot{\Phi }_i}/{\partial u_i }}&{} {{\partial \dot{\Phi }_i }/{\partial \Phi _i }}&{} {{\partial \dot{\Phi }_i}/{\partial \Phi _{-i}}} \\ {{\partial \dot{\Phi }_{-i}}/{\partial \Theta }}&{}{{\partial \dot{\Phi }_{-i} }/{\partial u_i }}&{} {{\partial \dot{\Phi }_{-i} }/{\partial \Phi _i }}&{} {{\partial \dot{\Phi }_{-i}}/{\partial \Phi _{-i}}}\\ \end{array}}}\right] \end{aligned}$$

It is obvious $J_{21}^0 =0$, $J_{23}^0 =0$, $J_{24}^0 =0$, $J_{31}^0 =0$, $J_{32}^0 =0$, and $J_{34}^0 =0$ owing to $u_i^{*} =0$ and $\Phi _i^{*} =0$. Moreover, $J_{22}^0 =J_{33}^0 <0$ is the direct result of $\mathop {\lim }\limits _{t\rightarrow \infty }{\dot{\Phi }_i }/{\Phi _i}=\mathop {\lim }\limits _{t\rightarrow \infty } {\dot{u}_i}/{u_i}<0$.³³

Equation (A.12) implies that $J_{11}^0$ and $J_{14}^0$, respectively, are given by

$$\begin{aligned} J_{11}^0 =-\Theta ^{{*}}\mathop {\lim }\limits _{t\rightarrow \infty } \frac{\partial ({\dot{K}}/{K)}}{\partial \Theta }\hbox { and }J_{14}^0 =\Theta ^{{*}}\left[ {-\mathop {\lim }\limits _{t\rightarrow \infty } \frac{\partial ({\dot{K}}/{K)}}{\partial \Phi _{-i} }+\sigma ^{-1}(\gamma _{-i} )^{\frac{\varepsilon }{\varepsilon -1}}\alpha _{-i} } \right] \end{aligned}$$

where $\mathop {\lim }\limits _{t\rightarrow \infty } \frac{\partial ({\dot{K}}/{K)}}{\partial \Theta }=-\frac{\alpha _{-i} (1-\varphi _{-i} )}{1+\alpha _{-i} (1-\varphi _{-i} )}$ and $\mathop {\lim }\limits _{t\rightarrow \infty } \frac{\partial ({\dot{K}}/{K)}}{\partial \Phi _{-i} }=\frac{\alpha _{-i} (1-\varphi _{-i} )(\gamma _{-i} )^{\frac{\varepsilon }{\varepsilon -1}}}{\alpha _{-i} (1-\varphi _{-i} )+1}$ from Eqs. (A.15) and (A.16).

Equation (A.13) implies that $J_{41}^0$ and $J_{44}^0$, respectively, are given by

$$\begin{aligned} J_{41}^0= & {} \Phi _{-i}^{*} \left[ {\frac{1}{1-\varphi _{-i} }+\alpha _{-i} -1} \right] \mathop {\lim }\limits _{t\rightarrow \infty } \frac{\partial ({\dot{K}}/{K})}{\partial \Theta }\hbox { and }J_{44}^0\\= & {} \Phi _{-i}^{*} \left[ {\frac{1}{1-\varphi _{-i} }+\alpha _{-i} -1} \right] \mathop {\lim }\limits _{t\rightarrow \infty } \frac{\partial ({\dot{K}}/{K})}{\partial \Phi _{-i}} \end{aligned}$$

It is obvious that $J_{11}^0 +J_{44}^0 =(\sigma -1)(1-\alpha _{-i} )(1-\varphi _{-i} )[(1-\alpha _{-i} )(1-\varphi _{-i} )-1]^{-1}\nu +(\rho -\nu \sigma )>0$ from the assumption of the STVC in Proposition 4, and $J_{11}^0 J_{44}^0 -J_{41}^0 J_{14}^0$ is given by

$$\begin{aligned}&J_{11}^0 J_{44}^0 -J_{41}^0 J_{14}^0 =-\left[ (1-\alpha _{-i} )(1-\varphi _{-i} )-1\right] \sigma ^{-1}\\&\left[ 1+\alpha _{-i} (1-\varphi _{-i} )\right] ^{-1}(\alpha _{-i} )^{2}\Theta ^{{*}}\Phi _{-i}^{*} (\gamma _{-i} )^{\frac{\varepsilon }{\varepsilon -1}} \end{aligned}$$

The assumption that guarantees sustainable economic growth in Proposition 4, $(1-\alpha _{-i} )(1-\varphi _{-i} )>1$, implies $J_{11}^0 J_{44}^0 -J_{41}^0 J_{14}^0 <0$. Therefore, the Jacobian of the dynamical system for sector-biased technical change has three negative real roots and a positive real root; thus, the dynamic system will be locally indeterminate.