On Industrialization, Human Resources Training, and Policy Coordination

Abstract

This paper proposes a dynamic model using a general equilibrium approach and shows that the coordination of public policies with suitable human resources training is a key factor for an economy to industrialize. We analyze three public policy domains: innovation policies; policies regarding human resources training, including wages and employment; and push policies. The omission or implementation of public policies, as well as their coordination, defines whether an economy stagnates in a poverty trap, or on the contrary, whether the economy is activated through industrialization processes. Such outcomes depend on what the initial state of the economy is like: whether it is on the left or the right of the industrialization frontier. The viability of crossing the industrialization frontier will depend on how these types of policies are coordinated.

Appendices

Appendix 1: Stability Analysis

To study the stability of the dynamic system (36)-(37), we will analyze the characteristic polynomial $f(\epsilon )=\det (A-\epsilon I)$, where 𝜖 = (𝜖₁, 𝜖₂), I is the 2 × 2-identity matrix and A is the matrix

$$ \left[\begin{array}{ccc} {\partial G\over \partial \lambda_{\mu}} & {\partial G \over \partial \gamma_{h}} \\ {\partial H\over \partial \lambda_{\mu}} & {\partial H \over \partial \gamma_{h}} \end{array}\right] $$

(48)

where

$$ \begin{array}{@{}rcl@{}} G(\lambda_{\mu},\gamma_{h})&:=&\lambda_{\mu} (1-\lambda_{\mu})\pi_{\mu} (\lambda_{\mu},\gamma_{h}),\\ H(\lambda_{\mu},\gamma_{h})&:=&\gamma_{h} (1-\gamma_{h})[\kappa\lambda_{\mu}\omega-(1-\kappa)(\lambda_{\mu} w_{\mu}+(1-\lambda_{\mu})w_{\tau}],\\ {\partial G\over \partial \lambda_{\mu}}&=&\lambda_{\mu} (1-\lambda_{\mu}) {\partial \pi_{\mu}\over \partial \lambda_{\mu}} +(1-2\lambda_{\mu})\pi_{\mu},\\ {\partial G \over \partial \gamma_{h}}&=& \lambda_{\mu} (1-\lambda_{\mu}) \left[{\partial\pi_{\mu} \over \partial\gamma}{d \gamma \over d\gamma_{h}}+{\partial\pi_{\mu} \over \partial\gamma_{h}}\right],\\ {\partial H\over \partial \lambda_{\mu}}&=& \gamma_{h} (1-\gamma_{h}) [\kappa\omega+(1-\kappa)(w_{\tau}-w_{\mu})],\\ {\partial H \over \partial \gamma_{h}}&=& (1-2\gamma_{h}) [\kappa\lambda_{\mu}\omega-(1-\kappa)(\lambda_{\mu} w_{\mu}+(1-\lambda_{\mu})w_{\tau})]. \end{array} $$

1.1 1.1 Steady State (0, 0)

In this case

$$ \left.\left[\begin{array}{ccc} {\partial G\over \partial \lambda_{\mu}} & {\partial G \over \partial \gamma_{h}} \\ {\partial H\over \partial \lambda_{\mu}} & {\partial H \over \partial \gamma_{h}} \end{array}\right] \right|_{(0, 0)}= \left[\begin{array}{ccc} \pi_{\mu}(0, 0) & 0 \\ 0 &-(1-\kappa)w_{\tau} \end{array} \right], $$

where

$$ \pi_{\mu}(0, 0)= [(w_{\tau}-w_{\mu} c) (L/(NF)) -w_{\mu}] F. $$

By (42), π_μ(0, 0) < 0. In this case

$$ f(\epsilon)=(\pi_{\mu}(0, 0)-\epsilon_{1})(-(1-\kappa)w_{\tau}-\epsilon_{2}), $$

which implies that the steady state (0, 0) is an attractor.

1.2 1.2 Steady State (0,1)

For this steady state

$$ \left.\left[\begin{array}{ccc} {\partial G\over \partial \lambda_{\mu}} & {\partial G \over \partial \gamma_{h}} \\ {\partial H\over \partial \lambda_{\mu}} & {\partial H \over \partial \gamma_{h}} \end{array}\right] \right|_{(0,1)}= \left[\begin{array}{ccc} \pi_{\mu}(0,1) & 0 \\ 0 & (1-\kappa)w_{\tau} \end{array} \right], $$

where

$$ \pi_{\mu}(0,1)=F\big[(w_{\tau}-(w_{\mu}+\omega)c\sigma) (\kappa L/(NF)) -(w_{\mu} +\omega) \big]. $$

We have that

$$ f(\epsilon)= (\pi_{\mu}(0,1)-\epsilon_{1})((1-k)w_{\tau}-\epsilon_{2}) $$

which implies that the steady state (1, 0) is a repeller if (43) is satisfied, and it is a saddle if (43) is not satisfied.

1.3 1.3 Steady State (1, 0)

In this case

$$ \left.\left[\begin{array}{ccc} {\partial G\over \partial \lambda_{\mu}} & {\partial G \over \partial \gamma_{h}} \\ {\partial H\over \partial \lambda_{\mu}} & {\partial H \over \partial \gamma_{h}} \end{array}\right] \right|_{(1, 0)}= \left[\begin{array}{ccc} -\pi_{\mu}(1, 0) & 0 \\ 0 & \kappa(w_{\mu}+\omega)-w_{\mu} \end{array} \right], $$

where

$$ \pi_{\mu}(1, 0)= {F\over c} [(w_{\tau}-w_{\mu} c) (L/(NF)) -w_{\tau}]. $$

Since (42) is satisfied, then π_μ(1, 0) > 0. Moreover

$$ f(\epsilon)=(-\pi_{\mu}(1, 0)-\epsilon_{1})(\kappa(w_{\mu}+\omega)-w_{\mu}-\epsilon_{2}). $$

Given π_μ(1, 0) > 0, if (35) is not satisfied we have that the steady state (0,1) is an attractor. If (35) is satisfied, then it is a saddle.

1.4 1.4 Steady state (1,1)

For this steady state

$$\left.\left[\begin{array}{ccc} {\partial G\over \partial \lambda_{\mu}} & {\partial G \over \partial \gamma_{h}} \\ {\partial H\over \partial \lambda_{\mu}} & {\partial H \over \partial \gamma_{h}} \end{array}\right] \right|_{(1,1)}= \left[\begin{array}{ccc} -\pi_{\mu}(1,1) & 0 \\ 0 & -(\kappa(w_{\mu}+\omega)-w_{\mu}) \end{array} \right], $$

where

$$\pi_{\mu}(1,1)={F\over c\sigma} \left[(w_{\tau}-(w_{\mu}+\omega)c\sigma) (\kappa L/NF)-w_{\tau}\right] $$

By (16) and (42),

$$(w_{\tau}-(w_{\mu}+\omega)c\sigma) (\kappa L/(NF))-w_{\tau}~>~(w_{\tau}-w_{\mu} c) (\kappa L/(NF))-w_{\tau}~>~0,$$

then π_μ(1, 1) > 0. We have that

$$f(\epsilon)=(-\pi_{\mu}(1,1)-\epsilon_{1})(-(\kappa(w_{\mu}+\omega)-w_{\mu})-\epsilon_{2}),$$

where π_μ(1, 1) > 0. Therefore, the steady state (0,1) is a saddle if (35) is not satisfied, and it is an attractor if (35) is satisfied.

1.5 1.5 Steady State $(\lambda _{\mu _{0}^{*}}, 0)$

Consider $\lambda _{\mu _{0}^{*}}$ as in (44). In this case

$$\left.\left[\begin{array}{ccc} {\partial G\over \partial \lambda_{\mu}} & {\partial G \over \partial \gamma_{h}} \\ {\partial H\over \partial \lambda_{\mu}} & {\partial H \over \partial \gamma_{h}} \end{array}\right] \right|_{(\lambda_{\mu_{0}}^{*}, 0)}= \left[\begin{array}{ccc} \lambda_{\mu} (1-\lambda_{\mu}) \left.{\partial \pi_{\mu}\over \partial \lambda_{\mu}}\right|_{(\lambda_{\mu_{0}}^{*}, 0)} & \left.{\partial G \over \partial \gamma_{h}}\right|_{(\lambda_{\mu_{0}}^{*}, 0)} \\ 0 & \kappa \lambda_{\mu_{0}}^{*}\omega-(1-\kappa)(\lambda_{\mu_{0}}^{*} w_{\mu}+(1-\lambda_{\mu_{0}}^{*})w_{\tau}) \end{array} \right] $$

where

$$\left.{\partial \pi_{\mu}\over \partial \lambda_{\mu}}\right|_{(\lambda_{\mu_{0}}^{*}, 0)}={{(w_{\mu}-w_{\tau}) F }\over{\lambda_{\mu_{0}}^{*} c + (1-\lambda_{\mu_{0}}^{*}) }}>0. $$

The roots of the characteristic polynomial f(𝜖) are

$$ \begin{array}{@{}rcl@{}} \epsilon_{1}&=&\lambda_{\mu} (1-\lambda_{\mu}) \left.{\partial \pi_{\mu}\over \partial \lambda_{\mu}}\right|_{(\lambda_{\mu_{0}}^{*}, 0)},\\ \epsilon_{2}&=& \kappa \lambda_{\mu_{0}}^{*}\omega-(1-\kappa)(\lambda_{\mu_{0}}^{*} w_{\mu}+(1-\lambda_{\mu_{0}}^{*})w_{\tau}), \end{array} $$

where 𝜖₁ > 0. If (35) is not satisfied, then 𝜖₂ < 0. In the other case, i.e., if (35) is satisfied, then it is not possible to determine the sign of 𝜖₂. Therefore if (35) is not satisfied, then $(\lambda _{\mu _{0}}^{*}, 0)$ is saddle; in the other case, we only know that it is unstable.

1.6 1.6 Steady State $(\lambda _{\mu _{1}^{*}},1)$

Consider $\lambda _{\mu _{1}^{*}}$ as in (45), this steady state does not exist if $\lambda _{\mu _{1}^{*}}$ is not positive. If $\lambda _{\mu _{1}^{*}}>0$, then

$$\left.\left[\begin{array}{ccc} {\partial G\over \partial \lambda_{\mu}} & {\partial G \over \partial \gamma_{h}} \\ {\partial H\over \partial \lambda_{\mu}} & {\partial H \over \partial \gamma_{h}} \end{array}\right] \right|_{(\lambda_{\mu_{1}}^{*},1)}= \left[\begin{array}{ccc} \lambda_{\mu} (1-\lambda_{\mu}) \left.{\partial \pi_{\mu}\over \partial \lambda_{\mu}}\right|_{(\lambda_{\mu_{1}}^{*},1)} & \left.{\partial G \over \partial \gamma_{h}}\right|_{(\lambda_{\mu_{1}}^{*},1)} \\ 0 & -\kappa \lambda_{\mu_{1}}^{*}\omega+(1-\kappa)(\lambda_{\mu_{1}}^{*} w_{\mu}+(1-\lambda_{\mu_{1}}^{*})w_{\tau}) \end{array} \right] $$

where

$$\left.{\partial \pi_{\mu}\over \partial \lambda_{\mu}}\right|_{(\lambda_{\mu_{1}}^{*},1)}={{(w_{\mu}+\omega-w_{\tau}) F }\over{\lambda_{\mu_{1}}^{*} c\sigma + (1-\lambda_{\mu_{1}}^{*}) }}>0. $$

The roots of the characteristic polynomial f(𝜖) are

$$ \begin{array}{@{}rcl@{}} \epsilon_{1}&=&\lambda_{\mu} (1-\lambda_{\mu}) \left.{\partial \pi_{\mu}\over \partial \lambda_{\mu}}\right|_{(\lambda_{\mu_{1}}^{*},1)},\\ \epsilon_{2}&=& -\kappa \lambda_{\mu_{1}}^{*}\omega+(1-\kappa)(\lambda_{\mu_{1}}^{*} w_{\mu}+(1-\lambda_{\mu_{1}}^{*})w_{\tau}), \end{array} $$

where 𝜖₁ > 0. If (35) is not satisfied, then 𝜖₂ > 0. If (35) is satisfied, then it is not possible to determine the sign of 𝜖₂. Therefore if (35) is not satisfied, then $(\lambda _{\mu _{1}}^{*},1)$ is repeller; in the other case, we only know that it is unstable.

1.7 1.7 Steady State $(\lambda _{\mu }^{*},\gamma _{h}^{*})$

Consider $\lambda _{\mu }^{*}$ and $\gamma _{h}^{*}$ as (46) and (47), respectively. In this case

$$\left.\left[\begin{array}{ccc} {\partial G\over \partial \lambda_{\mu}} & {\partial G \over \partial \gamma_{h}} \\ {\partial H\over \partial \lambda_{\mu}} & {\partial H \over \partial \gamma_{h}} \end{array}\right] \right|_{(\lambda_{\mu}^{*},\gamma_{h}^{*})}= \left[\begin{array}{ccc} \lambda_{\mu} (1-\lambda_{\mu}) \left.{\partial \pi_{\mu}\over \partial \lambda_{\mu}}\right|_{(\lambda_{\mu}^{*},\gamma_{h}^{*})} & \left.{\partial G \over \partial \gamma_{h}}\right|_{(\lambda_{\mu}^{*},\gamma_{h}^{*})} \\ \gamma_{h}^{*}(1-\gamma_{h}^{*})[\kappa\gamma+(1-\kappa)(w_{\tau}-w_{\mu})] & 0 \end{array} \right] $$

where

$$ \begin{array}{@{}rcl@{}} \left.{\partial \pi_{\mu}\over \partial \lambda_{\mu}}\right|_{(\lambda_{\mu}^{*},\gamma_{h}^{*})}&=& {{(w_{\mu}+\omega\gamma^{*}-w_{\tau}) (w_{\mu}+\omega\gamma^{*})F }\over{\Big[\lambda_{\mu}^{*}[(1-\gamma^{*})w_{\mu} c+\gamma^{*}(w_{\mu}+\omega)c\sigma ]+(w_{\mu}+\omega\gamma^{*})(1-\lambda_{\mu}^{*}) \Big]} } >0,\\ \gamma^{*}&=&\gamma\big|_{\gamma_{h}^{*}} \text{~~with~} \gamma \text{~as~in~} (11). \end{array} $$

The roots of the characteristic polynomial f(𝜖) of A as in (48) are

$$ \epsilon_{1,2}={\text{tr}(A)\pm\big((\text{tr}(A))^{2}-4\det(A) \big)^{1/2}\over 2} $$

with

$$ \text{tr}(A)=\lambda_{\mu} (1-\lambda_{\mu}) \left.{\partial \pi_{\mu}\over \partial \lambda_{\mu}}\right|_{(\lambda_{\mu}^{*},\gamma_{h}^{*})}>0. $$

Since tr(A) > 0, then $(\lambda _{m}u^{*},\gamma _{h}^{*})$ is not a stable steady state. Therefore $(\lambda _{\mu }^{*},\gamma _{h}^{*})$ is unstable, but it is not possible to determine their classification.

Appendix 2: Comparative Statics of λ _μ in the Steady States

1.1 2.1 Parameter in Table 2

Consider the values of $\lambda _{\mu _{0}}^{*},~ \lambda _{\mu _{1}}^{*},$ and $\lambda _{\mu }^{*}$, as in (44), (45), and (46), respectively.

• (i) Partial derivatives of λ_μ with respect to σ:

  $$ \begin{array}{@{}rcl@{}} {\partial\lambda_{\mu_{0}}^{*}\over \partial\sigma}&=&0; \\ {\lambda_{\mu_{1}}^{*}\over \partial\sigma}&=&{(w_{\mu}+\omega)c \over (w_{\mu}+\omega)- w_{\tau}}{\kappa L\over NF}>0; \\ {\lambda_{\mu}^{*}\over \partial\sigma}&=& 0. \end{array} $$

• (ii) Partial derivatives of λ_μ with respect to c:

  $$ \begin{array}{@{}rcl@{}} {\partial\lambda_{\mu_{0}}^{*}\over \partial c}&=&{w_{\mu} \over (w_{\mu}-w_{\tau})} {L\over NF}>0; \\ {\lambda_{\mu_{1}}^{*}\over \partial c}&=&{(w_{\mu}+\omega)\sigma\over (w_{\mu}+\omega)- w_{\tau}}{\kappa L\over NF}>0; \\ {\lambda_{\mu}^{*}\over \partial c}&=&0. \end{array} $$

• (ii) Partial derivatives of λ_μ with respect to F:

  $$ \begin{array}{@{}rcl@{}} {\partial\lambda_{\mu_{0}}^{*}\over \partial F}&=&{(w_{\tau}-w_{\mu} c) \over (w_{\mu}-w_{\tau})} {L\over NF^{2}}>0; \\ {\lambda_{\mu_{1}}^{*}\over \partial F} &=&{w_{\tau}-(w_{\mu}+\omega)c\sigma \over (w_{\mu}+\omega)- w_{\tau}}{\kappa L\over NF^{2}}>0; \\ {\lambda_{\mu}^{*}\over \partial F}&=&0. \end{array} $$

• (ii) Partial derivatives of λ_μ with respect to N:

  $$ \begin{array}{@{}rcl@{}} {\partial\lambda_{\mu_{0}}^{*}\over \partial N}&=&{(w_{\tau}-w_{\mu} c) \over (w_{\mu}-w_{\tau})} {L\over N^{2}F}>0; \\ {\lambda_{\mu_{1}}^{*}\over \partial N} &=&{w_{\tau}-(w_{\mu}+\omega)c\sigma \over (w_{\mu}+\omega)- w_{\tau}}{\kappa L\over N^{2}F}>0; \\ {\lambda_{\mu}^{*}\over \partial N}&=&0. \end{array} $$

1.2 2.2 Parameter in Table 3

Consider the values of $\lambda _{\mu _{0}}^{*},~ \lambda _{\mu _{1}}^{*},$ and $\lambda _{\mu }^{*}$, as in (44), (45), and (46), respectively.

• (i) Partial derivatives of λ_μ with respect to L:

  $$ \begin{array}{@{}rcl@{}} {\partial\lambda_{\mu_{0}}^{*}\over \partial L}&=&-{(w_{\tau}-w_{\mu} c) \over (w_{\mu}-w_{\tau})} {1\over NF} <0; \\ {\lambda_{\mu_{1}}^{*}\over \partial L}&=&-{w_{\tau}-(w_{\mu}+\omega)c\sigma \over (w_{\mu}+\omega)- w_{\tau}}{\kappa \over NF}<0; \\ {\lambda_{\mu}^{*}\over \partial L} &=&0. \end{array} $$

• (ii) Partial derivatives of λ_μ with respect to κ:

  $$ \begin{array}{@{}rcl@{}} {\partial\lambda_{\mu_{0}}^{*}\over \partial \kappa}&=&0; \\ {\lambda_{\mu_{1}}^{*}\over \partial \kappa}&=&-{w_{\tau}-(w_{\mu}+\omega)c\sigma \over (w_{\mu}+\omega)- w_{\tau}}{L\over NF}<0; \\ {\lambda_{\mu}^{*}\over \partial \kappa} &=&{-w_{\tau} \over \kappa\omega -(1-\kappa)(w_{\mu}-w_{\tau})}-{(1-\kappa)w_{\tau} (w_{\mu}+\omega -w_{\tau} )\over \big(\kappa\omega -(1-\kappa)(w_{\mu}-w_{\tau})\big)^{2}}<0. ~~(*) \end{array} $$

  (*) If $\lambda _{\mu _{0}}^{*}>0$ we need that κω − (1 − κ)(w_μ − w_τ) > 0, which implies that ${\lambda _{\mu }^{*}\over \partial \kappa }<0$.

• (iii) Partial derivatives of λ_μ with respect to w_μ:

  $$ \begin{array}{@{}rcl@{}} {\partial\lambda_{\mu_{0}}^{*}\over \partial w_{\mu}}&=&{{-w_{\tau}\left( 1-(1-c){L\over NF}\right) }\over (w_{\mu}-w_{\tau})^{2}}>0~~(*); \\ {\lambda_{\mu_{1}}^{*}\over \partial w_{\mu}}&=&{{-w_{\tau}\left( 1-(1-c\sigma){\kappa L\over NF}\right) }\over (w_{\mu}-w_{\tau})^{2}}>0~~(**); \\ {\lambda_{\mu}^{*}\over \partial w_{\mu}} &=&{(1-\kappa)^{2} w_{\tau} \over\big(\kappa\omega -(1-\kappa)(w_{\mu}-w_{\tau})\big)^{2}}>0. \end{array} $$

  (*) If (43) is satisfied then

  $$(1-c){L\over NF}>\left( 1-{w_{\mu} c \over w_{\tau}} \right) {L\over NF}>1,$$

  which implies that ${\partial \lambda _{\mu _{0}}^{*}\over \partial w_{\mu }}>0$.

  (**) If (43) is satisfied then

  $$(1-c\sigma){\kappa L\over NF}>\left( 1-{w_{\mu} c \over w_{\tau}} \right) {L\over NF}>1,$$

  which implies that ${\partial \lambda _{\mu _{1}}^{*}\over \partial w_{\mu }}>0$.

• (iii) Partial derivatives of λ_μ with respect to ω:

  $$ \begin{array}{@{}rcl@{}} {\partial\lambda_{\mu_{0}}^{*}\over \partial \omega}&=&0; \\ {\lambda_{\mu_{1}}^{*}\over \partial \omega}&=&{{-w_{\tau}\left( 1-(1-c\sigma){\kappa L\over NF}\right) }\over (w_{\mu}-w_{\tau})^{2}}>0; \\ {\lambda_{\mu}^{*}\over \partial \omega} &=&{-(1-\kappa)\kappa w_{\tau} \over \big(\kappa\omega -(1-\kappa)(w_{\mu}-w_{\tau})\big)^{2}}<0; \\ \end{array} $$

1.3 2.3 Parameter of Figures

Table 4 Values of parameters in Figs. 1 and 2

Table 5 Values of parameters in Fig. 4

Table 6 Values of parameters in Fig. 5

Table 7 Values of parameters in Fig. 6