\documentclass[dvipdfmx]{jlreq}
\usepackage{amsmath, amssymb}
\usepackage[top=15truemm,bottom=10truemm,left=15truemm,right=10truemm]{geometry}
\title{Reconsideration of Equilibrium}
\author{Kazuhito Kitamura}
\begin{document}
\maketitle

3.2.1
\[
a_t=k_t+b_t=(1-\theta)a_t+\theta\ a_t
\]

3.2.2
\[
max\int_{0}^{\infty}{e^{-\rho t}U\left(c_t,b_t\right)dt}\qquad =\qquad max\int_{0}^{\infty}{e^{-\rho t}U\left(c,\theta a_t\right)dt}
\]

3.2.3
\[
\dot{k_t}=f\left(k_t\right)-c_t-\delta k_t-nk_t
\]

3.2.4
\[
\dot{b_t}=rb_t-nb_t-D\left(a_t,a_{ext,t}\right)
\]

3.2.5
\[
\dot{a_t}=f\left(k_t\right)+rb_t-c_t-\delta\ k_t-nk_t-nb_t-D\left(a_t,a_{ext,t}\right)
\]
\[
=f\left(\left(1-\theta\right)a_t\right)+r\theta\ a_t-c_t-\delta\left(1-\theta\right)a_t-na_t-D\left(a_t,a_{ext,t}\right)
\]
\[
=f\left(\left(1-\theta\right)a_t\right)-c_t+\left(r\theta-\left(1-\theta\right)\delta-n\right)a_t-D\left(a_t,a_{ext,t}\right)
\]

3.3.1
\[
H\left(a_t,\ c_t,\ \lambda_t\right)=U\left(c_t,\ \theta a_t\right)+\lambda_t\left.\ \left[f\left(\left(1-\theta\right)a_t\right)-c_t+\left(r\theta-\left(1-\theta\right)\delta-n\right)a_t-D\left(a_t,a_{ext,t}\right)\right.\right]
\]

3.3.2
\[
\frac{\partial H}{\partial c}=U_c-\lambda_t=0\ \qquad\Rightarrow\qquad \lambda_t=U_c
\]

3.3.3
\[
\dot{\lambda_t}=\rho\lambda_t-\frac{\partial}{\partial a_t}\left(U\left(c_t,\ \theta a_t\right)+\lambda_t\left.\ \left[f\left(\left(1-\theta\right)a_t\right)-c_t+\left(r\theta-\left(1-\theta\right)\delta-n\right)a_t-D\left(a_t,a_{ext,t}\right)\right.\right]\right)
\]
\[
=\rho\lambda_t-[U_{\left(\theta a\right)}\theta+\lambda_t\left[f^\prime\left(\left(1-\theta\right)a_t\right)\left(1-\theta\right)+r\theta-\left(1-\theta\right)\delta-n-D_a\right]]
\]
\[
=\rho\lambda_t-[U_{\left(\theta\ a\right)}\theta+\lambda_t[\left(1-\theta\right)(f^\prime\left(\left(1-\theta\right)a_t\right)-\delta)+\theta\ r-n-D_a]]
\]

3.3.4
\[
R_t=\left(1-\theta\right)\left(f^\prime\left(\left(1-\theta\right)a_t\right)-\delta\right)+\theta\ r
\]

3.3.5
\[
\frac{\dot{\lambda_t}}{\lambda_t}=\rho-R_t-\frac{U_{\left(\theta a\right)}\theta}{U_c}+n+D_a
\]

3.3.6
\[
\lim_{0\rightarrow\infty}{e^{-\rho t}\lambda_ta_t}=0
\]

3.4.1
\[
\dot{\lambda_t}=\frac{dU_c}{dt}=U_{cc}\dot{c_t}+U_{c\left(\theta a\right)}\theta\dot{a_t}
\]

3.4.2
\[
\frac{\dot{\lambda_t}}{\lambda_t}=\frac{U_{cc}}{U_c}\dot{c_t}+\frac{U_{c\left(\theta a\right)}}{U_c}\theta\dot{a_t}
\]

3.4.3
\[
\frac{\dot{\lambda_t}}{\lambda_t}=-\frac{1}{\sigma}\frac{\dot{c_t}}{c_t}+\frac{U_{c\left(\theta a\right)}}{U_c}\theta\dot{a_t}
\]

3.4.4
\[
-\frac{1}{\sigma}\frac{\dot{c_t}}{c_t}+\frac{U_{c\left(\theta a\right)}}{U_c}\theta\dot{a_t}=\rho-R_t-\frac{U_{\left(\theta a\right)}\theta}{U_c}+n+D_a
\]

3.4.5
\[
\frac{\dot{c_t}}{c_t}=\sigma\left[\left(R_t-\rho\right)+\left(\frac{U_{\left(\theta a\right)}\theta}{U_c}-D_a-n\right)\right]
\]

3.4.5'
\[
\frac{\dot{c_t}}{c_t}=\frac{1}{\gamma}\left[\left(R_t-\rho\right)+\left(\frac{U_{\left(\theta a\right)}\theta}{U_c}{-D}_a-n\right)\right]
\]

3.4.6
\[
R_t-\rho=n+D_a-\frac{U_{\left(\theta a\right)}\theta}{U_c}
\]

3.4.7
\[
U_c=\frac{U_{\left(\theta a\right)}\theta}{n+D_a-\left(R_t-\rho\right)}
\]

3.4.8
\[
c_t=f\left(\left(1-\theta\right)a_t\right)+\left(r\theta-\left(1-\theta\right)\delta-n\right)a_t-D\left(a_t,a_{ext,t}\right)
\]

3.4.8'
\[
c_t=\left[f\left(\left(1-\theta\right)a_t\right)+r\theta a_t\right]-\left[\left(\left(1-\theta\right)\delta+n\right)a_t+D\left(a_t,a_{ext,t}\right)\right]
\]

3.5.1
\[
e^{-\rho T}\lambda_Ta_T-\lambda_0a_0=\int_{0}^{T}{\frac{d}{dt}\left(e^{-\rho t}\lambda_ta_t\right)dt}
\]

3.5.2
\[
-\lambda_0a_0=\int_{0}^{\infty}{\frac{d}{dt}\left(e^{-\rho t}\lambda_ta_t\right)dt}
\]

3.5.3
\[
\int_{0}^{\infty}{\frac{d}{dt}\left(e^{-\rho t}\lambda_ta_t\right)dt}=\int_{0}^{\infty}{e^{-\rho t}\left(\dot{\lambda_t}a_t+\lambda_t\dot{a_t}-\rho\lambda_ta_t\right)dt}
\]
\[
=\int_{0}^{\infty}{e^{-\rho t}\left(\lambda_ta_t\left(\rho-R_t-\frac{U_{\left(\theta a\right)}\theta}{U_c}+n+D_a\right)+\lambda_t\dot{a_t}-\rho\lambda_ta_t\right)dt}
\]
\[
=\int_{0}^{\infty}{e^{-\rho t}\lambda_ta_t\left(\left(n+D_a\right)-\left(R_t+\frac{U_{\left(\theta a\right)}\theta}{U_c}\right)\right)dt}
\]

3.5.4
\[
-\lambda_0a_0=\int_{0}^{\infty}{e^{-\rho t}\lambda_ta_t\left(\left(n+D_a\right)-\left(R_t+\frac{U_{\left(\theta a\right)}\theta}{U_c}\right)\right)dt}
\]

3.5.5
\[
\frac{d}{dt}\left(e^{-\rho t}\lambda_ta_t\right)=-\rho e^{-\rho t}\lambda_ta_t
\]

3.5.6
\[
-\lambda_0a_0=\int_{0}^{\infty}{-\rho e^{-\rho t}\lambda_t}a_tdt
\]

3.5.7
\[
\frac{d}{dt}\left(e^{-\rho t}\lambda_ta_t\right)=e^{-\rho t}\lambda_ta_t\left(\left(n+D_a\right)-\left(R_t+\frac{U_{\left(\theta a\right)}\theta}{U_c}\right)\right)
\]

3.5.8
\[
\left(n+D_a\right)-\left(R_t+\frac{U_{\left(\theta a\right)}\theta}{U_c}\right)\lt0
\]

4.1.1
\[
f\left(k_t\right)=Ak_t^\alpha\qquad\ \left(A>0,\ \alpha\in\left[0,1\right]\right)
\]

4.1.2
\[
f\left(k_t\right)=f\left(\left(1-\theta\right)a_t\right)=A\left(\left(1-\theta\right)a_t\right)^\alpha=A\left(1-\theta\right)^\alpha a_t^{\alpha}
\]

4.1.3
\[
f^\prime\left(k_t\right)=A\alpha\ k^{\alpha-1}=A\alpha\left(\left(1-\theta\right)a_t\right)^{\alpha-1}=A\alpha\left(1-\theta\right)^{\alpha-1}a_t^{\alpha-1}
\]

4.1.4
\[
U\left(c_t,b_t\right)=\frac{c_t^{1-\gamma}}{1-\gamma}+\beta\frac{b_t^{1-\psi}}{1-\psi}\qquad\left(\gamma>0\right)
\]

4.1.5
\[
U_c=\frac{\partial U\left(c_t,b_t\right)}{\partial c_t}=c_t^{-\gamma}
\]

4.1.6
\[
U_{\left(\theta a\right)}=U_b=\frac{\partial U\left(c_t,b_t\right)}{\partial b_t}=\beta b_t^{-\psi}=\beta\left(\theta a_t\right)^{-\psi}=\beta\theta^{-\psi}a_t^{-\psi}
\]

4.1.7
\[
D\left(a_t,a_{ext,t}\right)=\phi\left(a_t-a_{ext,t}\right)\qquad\left(\phi>0\right)
\]

4.1.8
\[
D_a=\frac{\partial D\left(a_t,a_{ext,t}\right)}{\partial a_t}=\phi \quad
\]

4.2.1
\[
{\dot{a}}_t=A\left(1-\theta\right)^\alpha a_t^\alpha-c_t+\left(r\theta-\left(1-\theta\right)\delta-n\right)a_t-\phi\left(a_t{-a}_{ext,t}\right)
\]
\[
\Rightarrow\qquad\ c_t=\left(A\left(1-\theta\right)^\alpha a_t^\alpha+r\theta a_t\right)-(\left(1-\theta\right)\delta+n+\phi)a_t+\phi\ a_{ext,t}
\]

4.2.2
\[
\left(1-\theta\right)\left(A\alpha\left(1-\theta\right)^{\alpha-1}a_t^{\alpha-1}-\delta\right)+\theta\ r-\rho=n+\phi-\frac{\left(\theta a_t\right)^{-\psi}}{c_t^{-\gamma}}\beta\theta \quad
\]

4.2.3
\[
c_t=\left(\frac{a_t^\psi}{\beta\theta^{1-\psi}}\left(\rho-\left(\left(1-\theta\right)\left(A\alpha\left(1-\theta\right)^{\alpha-1}a_t^{\alpha-1}-\delta\right)+\theta r\right)+n+\phi\right)\right)^\frac{1}{\gamma}\qquad\ (\beta\neq0)
\]

4.2.4
\[
\beta\theta^{1-\psi}a_t^{-\psi}c_t^\gamma=\left(\rho-\left(\left(1-\theta\right)\left(A\alpha\left(1-\theta\right)^{\alpha-1}a_t^{\alpha-1}-\delta\right)+\theta r\right)\right)+n+\phi \quad
\]

4.2.5
\[
\dot{a_t}=G_1\left(a_t,c_t\right)=A\left(1-\theta\right)^\alpha a_t^\alpha-c_t+\left(r\theta-\left(1-\theta\right)\delta-n\right)a_t-\phi\left(a_t-a_{ext,t}\right)
\]

4.2.6
\[
{\dot{c}}_t=G_2\left(a_t,c_t\right)=\frac{c_t}{\gamma}\left[\left(\left(1-\theta\right)\left(A\alpha\left(1-\theta\right)^{\alpha-1}{a_t}^{\alpha-1}-\delta\right)+\theta r-\rho\right)+\left(\frac{\beta\theta^{1-\psi}{a_t}^{-\psi}}{c_t^{-\gamma}}-\phi-n\right)\right]
\]

4.2.7
\[
J_{11}=\frac{\partial G_1}{\partial a_t}=(\left(1-\theta\right)\left(A\alpha\left(1-\theta\right)^{\alpha-1}{a_t}^{\alpha-1}-\delta\right)+\theta\ r)-\left(n+\phi\right)
\]

4.2.8
\[
J_{12}=\frac{\partial G_1}{\partial c_t}=-1
\]

4.2.9
\[
J_{21}=\frac{\partial G_2}{\partial a_t}=\frac{c^\ast}{\gamma}\left[A\alpha\left(\alpha-1\right)\left(1-\theta\right)^\alpha{a_t}^{\alpha-2}-\frac{\beta\psi\theta^{1-\psi}{a_\ast}^{-\psi-1}}{{c_\ast}^{-\gamma}}\right]
\]

4.2.10
\[
J_{22}=\frac{\partial G_2}{\partial c_t}=\frac{c^\ast}{\gamma}\frac{\partial}{\partial c_t}\left[\ldots\right]=\frac{c^\ast}{\gamma}\frac{\partial}{\partial c_t}\left(\frac{\beta\theta^{1-\psi}{a_\ast}^{-\psi}}{c_\ast^{-\gamma}}\right)=\frac{c^\ast}{\gamma}\beta\theta^{1-\psi}{a_\ast}^{-\psi}\gamma{c_\ast}^{\gamma-1}=\frac{\beta\theta^{1-\psi}{a_\ast}^{-\psi}}{c_\ast^{-\gamma}}
\]

4.2.11
\[
J=\left[\begin{matrix}J_{11}&J_{12}\\J_{21}&J_{22}\\\end{matrix}\right]=\left[\begin{matrix}\frac{\partial G_1}{\partial a_t}&\frac{\partial G_1}{\partial c_t}\\\frac{\partial G_2}{\partial a_t}&\frac{\partial G_2}{\partial c_t}\\\end{matrix}\right]=\left[\begin{matrix}(\left(1-\theta\right)\left(A\alpha\left(1-\theta\right)^{\alpha-1}{a_t}^{\alpha-1}-\delta\right)+\theta r)-\left(n+\phi\right)&-1\\\frac{c^\ast}{\gamma}\left[A\alpha\left(\alpha-1\right)\left(1-\theta\right)^\alpha{a_t}^{\alpha-2}-\frac{\beta\psi\theta^{1-\psi}{a_\ast}^{-\psi-1}}{{c_\ast}^{-\gamma}}\right]&\frac{\beta\theta^{1-\psi}{a_\ast}^{-\psi}}{c_\ast^{-\gamma}}\\\end{matrix}\right]
\]

5.1.1
\[
U\left(c_i,b_i\right)=\ln{c_i+\beta_i\ln{b_i}}
\]

5.1.2
\[
U_c=\frac{\partial U\left(c_i,b_i\right)}{\partial c_i}=c_i^{-1}=\frac{1}{c_i}
\]

5.1.3
\[
U_{\left(\theta a\right)}=U_b=\frac{\partial U\left(c_i,b_i\right)}{\partial b_i}=\beta_i\ b_i^{-1}=\beta_i\frac{1}{b_i}=\frac{\beta_i}{\theta a_i}
\]

5.1.4
\[
{\dot{a}}_i=A\left(1-\theta\right)^\alpha a_i^\alpha-c_i+\left(\theta r-\left(1-\theta\right)\delta-n\right)a_i-\phi_i\left(a_i-a_j\right)
\]

5.1.5
\[
\phi_i\left(a_i-a_j\right)=-\phi_j\left(a_j-a_i\right)+Res.
\]

5.1.6
\[
c_i=A\left(1-\theta\right)^\alpha a_i^\alpha+\left(\theta r-\left(1-\theta\right)\delta-n\right)a_i-\phi_i\left(a_i-a_j\right)
\]
\[
=\left(A\left(1-\theta\right)^\alpha a_i^\alpha+\theta r\ a_i\right)-\left(\left(1-\theta\right)\delta+n+\phi_i\right)a_i+\phi_ia_j
\]

5.1.7
\[
{\dot{c}}_i=c_i\left[\left(\left(1-\theta\right)\left(A\alpha\left(1-\theta\right)^{\alpha-1}{a_i}^{\alpha-1}-\delta\right)+\theta r-\rho_i\right)+\left(\frac{\beta_i{a_i}^{-1}}{{c_i}^{-1}}-n-\phi\right)\right]
\]

5.1.8
\[
c_i=\frac{a_i}{\beta_i}\left(\rho_i-\left(\left(1-\theta\right)\left(A\alpha\left(1-\theta\right)^{\alpha-1}a_i^{\alpha-1}-\delta\right)+\theta r\right)+n+\phi_i\right)
\]

5.1.9
\[
\left(\left(1-\theta\right)\left(A\alpha\left(1-\theta\right)^{\alpha-1}{a_i}^{\alpha-1}-\delta\right)+\theta r\right)-\rho_i=n+\phi_i-\frac{\beta_i{a_i}^{-1}}{{c_i}^{-1}}
\]

5.2.1
\[
{\dot{a}}_i=A\left(1-\theta\right)^\alpha a_i^\alpha-\left(\frac{a_i}{\beta_i}\left(\rho_i-\left(\left(1-\theta\right)\left(A\alpha\left(1-\theta\right)^{\alpha-1}{a_i}^{\alpha-1}-\delta\right)+\theta r\right)+n+\phi_i\right)\right)+\left(\theta r-\left(1-\theta\right)\delta-n\right)a_i-\phi_i\left(a_i-a_j\right)
\]
\[
=A\left(1-\theta\right)^\alpha a_i^\alpha-\frac{a_i}{\beta_i}\left(\rho_i-\left(\left(1-\theta\right)\left(A\alpha\left(1-\theta\right)^{\alpha-1}{a_i}^{\alpha-1}-\delta\right)+\theta r\right)+n+\phi_i\right)+\theta\ ra_i-\left(1-\theta\right)\delta\ a_i-na_i-\phi_i\ a_i+\phi_i\ a_j
\]
\[
=A\left(1-\theta\right)^\alpha a_i^\alpha-\frac{a_i}{\beta_i}\rho_i+\frac{a_i}{\beta_i}\left(1-\theta\right)A\alpha\left(1-\theta\right)^{\alpha-1}{a_i}^{\alpha-1}-\frac{a_i}{\beta_i}\left(1-\theta\right)\delta+\frac{a_i}{\beta_i}\theta\ r-\frac{a_i}{\beta_i}n-\frac{a_i}{\beta_i}\phi_i+\theta\ ra_i-\left(1-\theta\right)\delta\ a_i-na_i-\phi_i\ a_i+\phi_i\ a_j
\]
\[
{
=\left(1-\theta\right){a_iA\left(1-\theta\right)}^{\alpha-1}a_i^{\alpha-1}+\frac{\alpha}{\beta_i}\left(1-\theta\right)a_iA\left(1-\theta\right)^{\alpha-1}{a_i}^{\alpha-1}-\left(1-\theta\right)a_i\delta-\frac{a_i}{\beta_i}\left(1-\theta\right)\delta+\theta\ ra_i+\frac{a_i}{\beta_i}\theta\ r-\frac{a_i}{\beta_i}\rho_i-na_i-\frac{a_i}{\beta_i}n-\phi_ia_i-\frac{a_i}{\beta_i}\phi_i+\phi_i\ a_j}
\]
\[
{
=\left(1+\frac{\alpha}{\beta_i}\right)\left(1-\theta\right){a_iA\left(1-\theta\right)}^{\alpha-1}a_i^{\alpha-1}-\left(1+\frac{1}{\beta_i}\right)\left(1-\theta\right)a_i\delta+\left(1+\frac{1}{\beta_i}\right)\theta\ ra_i-\frac{a_i}{\beta_i}\rho_i-\left(1+\frac{1}{\beta_i}\right)na_i-\left(1+\frac{1}{\beta_i}\right)\phi_i\ a_i+\phi_i\ a_j}
\]
\[
=\left(\left(1-\theta\right)a_i\left(\left(1+\frac{\alpha}{\beta_i}\right)A\left(1-\theta\right)^{\alpha-1}a_i^{\alpha-1}-\left(1+\frac{1}{\beta_i}\right)\delta\right)+\left(1+\frac{1}{\beta_i}\right)\theta a_ir\right)-\frac{1}{\beta_i}\rho_ia_i-\left(1+\frac{1}{\beta_i}\right)\left(na_i+\phi_i\ a_i\right)+\phi_i\ a_j
\]

5.2.2
\[
\left(\left(1-\theta\right)\left(\left(1+\frac{\alpha}{\beta_i}\right)A\left(1-\theta\right)^{\alpha-1}a_i^{\alpha-1}-\left(1+\frac{1}{\beta_i}\right)\delta\right)+\left(1+\frac{1}{\beta_i}\right)\theta r\right)-\frac{1}{\beta_i}\rho_i-\left(1+\frac{1}{\beta_i}\right)\left(n+\phi_i\right)+\phi_i\frac{a_j}{a_i}=0
\]

5.2.3
\[
\Rightarrow\qquad\ \left(\left(1-\theta\right)\left(\left(1+\frac{\alpha}{\beta_i}\right)A\left(1-\theta\right)^{\alpha-1}a_i^{\alpha-1}-\left(1+\frac{1}{\beta_i}\right)\delta\right)+\left(1+\frac{1}{\beta_i}\right)\theta r\right)-\frac{1}{\beta_i}\rho_i=\left(1+\frac{1}{\beta_i}\right)\left(n+\phi_i\right)-\phi_i\frac{a_j}{a_i}
\]

5.2.4H
\[
\left(\left(1-\theta\right)\left(\left(1+\frac{\alpha}{\beta_H}\right)A\left(1-\theta\right)^{\alpha-1\ }a_H^{\alpha-1}-\left(1+\frac{1}{\beta_H}\right)\delta\right)+\left(1+\frac{1}{\beta_H}\right)\theta r\right)-\frac{1}{\beta_H}\rho_H-\left(1+\frac{1}{\beta_H}\right)\left(n+\phi_H\right)+\phi_H\frac{a_L}{a_H}=0
\]

5.2.4L
\[
\left(\left(1-\theta\right)\left(\left(1+\frac{\alpha}{\beta_L}\right)A\left(1-\theta\right)^{\alpha-1\ }a_L^{\alpha-1}-\left(1+\frac{1}{\beta_L}\right)\delta\right)+\left(1+\frac{1}{\beta_L}\right)\theta r\right)-\frac{1}{\beta_L}\rho_L-\left(1+\frac{1}{\beta_L}\right)\left(n+\phi_L\right)+\phi_L\frac{a_H}{a_L}=0
\]

5.2.5H
\[
{
{\dot{a}}_H=\left(\left(1-\theta\right)a_H\left(\left(1+\frac{\alpha}{\beta_H}\right)A\left(1-\theta\right)^{\alpha-1}a_H^{\alpha-1}-\left(1+\frac{1}{\beta_H}\right)\delta\right)+\left(1+\frac{1}{\beta_H}\right)\theta a_Hr\right)-\frac{1}{\beta_H}\rho_Ha_H-\left(1+\frac{1}{\beta_H}\right)\left(na_H+\phi_H\ a_H\right)+\phi_H\ a_L}
\]

5.2.5L
\[
{
{\dot{a}}_L=\left(\left(1-\theta\right)a_L\left(\left(1+\frac{\alpha}{\beta_L}\right)A\left(1-\theta\right)^{\alpha-1}a_L^{\alpha-1}-\left(1+\frac{1}{\beta_L}\right)\delta\right)+\left(1+\frac{1}{\beta_L}\right)\theta a_Lr\right)-\frac{1}{\beta_L}\rho_La_L-\left(1+\frac{1}{\beta_L}\right)\left(na_L+\phi_L\ a_L\right)+\phi_L\ a_H}
\]

5.2.6H
\[
{
J_{11}=\frac{\partial{\dot{a}}_H}{\partial a_H}=\frac{\partial}{\partial a_H}\left(\left(\left(1-\theta\right)a_H\left(\left(1+\frac{\alpha}{\beta_H}\right)A\left(1-\theta\right)^{\alpha-1}a_H^{\alpha-1}-\left(1+\frac{1}{\beta_H}\right)\delta\right)+\left(1+\frac{1}{\beta_H}\right)\theta a_Hr\right)-\frac{1}{\beta_H}\rho_Ha_H-\left(1+\frac{1}{\beta_H}\right)\left(na_H+\phi_H\ a_H\right)+\phi_H\ a_L\right)}
\]
\[
{
=\frac{\partial}{\partial a_H}\left(\left(1+\frac{\alpha}{\beta_H}\right)A\left(1-\theta\right)^\alpha a_H^\alpha-\left(1+\frac{1}{\beta_H}\right)\left(1-\theta\right)\delta a_H+\left(1+\frac{1}{\beta_H}\right)\theta a_Hr-\frac{1}{\beta_H}\rho_Ha_H-\left(1+\frac{1}{\beta_H}\right)\left(na_H+\phi_H\ a_H\right)+\phi_H\ a_L\right)}
\]
\[
=\left(1+\frac{\alpha}{\beta_H}\right)A{\alpha\left(1-\theta\right)}^\alpha a_H^{\alpha-1}-\left(1+\frac{1}{\beta_H}\right)\left(1-\theta\right)\delta+\left(1+\frac{1}{\beta_H}\right)\theta\ r-\frac{1}{\beta_H}\rho_H-\left(1+\frac{1}{\beta_H}\right)\left(n+\phi_H\right)
\]
\[
=\left(1+\frac{\alpha}{\beta_H}\right)A{\alpha\left(1-\theta\right)}^\alpha a_H^{\alpha-1}+\left(1+\frac{1}{\beta_H}\right)(\theta\ r-\left(1-\theta\right)\delta-n-\phi_H)-\frac{1}{\beta_H}\rho_H
\]

5.2.6L
\[
J_{22}=\frac{\partial{\dot{a}}_L}{\partial a_L}=\left(1+\frac{\alpha}{\beta_L}\right)A{\alpha\left(1-\theta\right)}^\alpha a_L^{\alpha-1}+\left(1+\frac{1}{\beta_L}\right)\left(\theta r-\left(1-\theta\right)\delta-n-\phi_L\right)-\frac{1}{\beta_L}\rho_L
\]

5.2.7H
\[
J_{12}=\frac{\partial\dot{a_H}}{\partial a_L}=\frac{\partial}{\partial a_L}\left(\left(1-\theta\right)a_H\ldots\ldots\ldots+\phi_H\ a_L\right)=\phi_H
\]

5.2.7L
\[
J_{21}=\frac{\partial\dot{a_L}}{\partial a_H}=\frac{\partial}{\partial a_H}\left(\left(1-\theta\right)a_L\ldots\ldots\ldots+{\phi_La}_H\right)=\phi_L
\]

5.2.8H
\[
\left(\left(1-\theta\right)\left(\left(1+\frac{\alpha}{\beta_H}\right)A\left(1-\theta\right)^{\alpha-1\ }a_H^{\alpha-1}-\left(1+\frac{1}{\beta_H}\right)\delta\right)+\left(1+\frac{1}{\beta_H}\right)\theta r\right)-\frac{1}{\beta_H}\rho_H-\left(1+\frac{1}{\beta_H}\right)\left(n+\phi_H\right)+\phi_H\frac{a_L}{a_H}=0
\]
\[
\left(1+\frac{\alpha}{\beta_H}\right)A\left(1-\theta\right)^{\alpha\ }a_H^{\alpha-1}-\left(1+\frac{1}{\beta_H}\right)\left(1-\theta\right)\delta+\left(1+\frac{1}{\beta_H}\right)\theta r-\frac{1}{\beta_H}\rho_H-\left(1+\frac{1}{\beta_H}\right)\left(n+\phi_H\right)=-\phi_H\frac{a_L}{a_H}
\]
\[
\left(1+\frac{\alpha}{\beta_H}\right)A\left(1-\theta\right)^{\alpha\ }a_H^{\alpha-1}+\left(1+\frac{1}{\beta_H}\right)\left(\theta r-\left(\left(1-\theta\right)\delta+n+\phi_H\right)\right)-\frac{1}{\beta_H}\rho_H=-\phi_H\frac{a_L}{a_H}
\]
\[
\phi_H\frac{a_L}{a_H}=\frac{1}{\beta_H}\rho_H-\left(1+\frac{\alpha}{\beta_H}\right)A\left(1-\theta\right)^{\alpha\ }a_H^{\alpha-1}-\left(1+\frac{1}{\beta_H}\right)\left(\theta r-\left(\left(1-\theta\right)\delta+n+\phi_H\right)\right)
\]
\[
a_L=\frac{a_H}{\phi_H}\left(\frac{1}{\beta_H}\rho_H-\left(1+\frac{\alpha}{\beta_H}\right)A\left(1-\theta\right)^{\alpha\ }a_H^{\alpha-1}-\left(1+\frac{1}{\beta_H}\right)\left(\theta r-\left(\left(1-\theta\right)\delta+n+\phi_H\right)\right)\right)
\]

5.2.8L
\[
a_H=\frac{a_L}{\phi_L}\left(\frac{1}{\beta_L}\rho_L-\left(1+\frac{\alpha}{\beta_L}\right)A\left(1-\theta\right)^{\alpha\ }a_L^{\alpha-1}-\left(1+\frac{1}{\beta_L}\right)\left(\theta r-\left(\left(1-\theta\right)\delta+n+\phi_L\right)\right)\right)
\]
\end{document}