$a_t=k_t+b_t=(1-\theta)a_t+\theta a_t \qquad<1.4.1>$
$max\int_0^\infty e^{-\rho t}U(c_t,b_t)dt \quad\Rightarrow\quad max\int_0^\infty e^{-\rho t} U(c_t,\theta a_t)dt\qquad<1.4.2>$
$\dot{k}_t=f(k_t)-c_t-\delta k_t-nk_t\qquad<1.5.1>$
$\dot{b}_t=rb_t-nb_t-D(a_t,a_{ext,t})\qquad<1.5.2>$
$\dot{a}_t=f(k_t)+rb_t-c_t-\delta k_t-nk_t-nb_t-D(a_t,a_{ext,t})=f((1-\theta)a_t)+r\theta a_t-c_t-\delta(1-\theta)a_t-na_t-D(a_t,a_{ext,t})=f((1-\theta)a_t)-c_t+(r\theta-(1-\theta)\delta-n)a_t-D(a_t,a_{ext,t})\qquad<1.6.1>$
$H(a_t,c_t,λ_t )=U(c_t,\theta a_t)+λ_t[f((1-\theta)a_t)-c_t+(r\theta-(1-\theta)\delta-n)a_t-D(a_t,a_{ext,t})]\quad<1.7.1>$
$\frac{\partial H}{\partial c}=U_c-\lambda_t=0\quad\Rightarrow\quad\lambda_t=U_c\qquad<1.7.2>$
$\dot{\lambda}_t=\rho\lambda_t-\frac{\partial}{\partial a_t}(U(c_t,\theta a_t)+\lambda_t[f((1-\theta)a_t)-c_t+(r\theta-(1-\theta)\delta-n) a_t-D(a_t,a_{ext,t})])=\rho\lambda_t-[U_{(\theta a)}\theta+\lambda_t[f^\prime((1-\theta)a_t)(1-\theta)+r\theta-(1-\theta)\delta-n-D_a]]=\rho\lambda_t-[U_((\theta a))\theta+\lambda_t[(1-\theta)f^\prime((1-\theta)a_t)-\delta)+\theta r-n-D_a]]\qquad<1.7.3>$
$R_t=(1-\theta)f^\prime((1-\theta)a_t)-\delta)+\theta r\qquad<1.7.4>$
$\frac{\dot{\lambda}_t}{\lambda_t}=\rho-R_t-\frac{U_{(\theta a)}\theta}{U_c}+n+D_a\qquad<1.7.5>$
$\dot{\lambda}_t=\frac{dU_c}{dt}=U_{cc}\dot{c}_t+U_{c(\theta a)}\theta\dot{a}_t\qquad<1.8.1>$
$\frac{\dot{\lambda}_t}{\lambda_t}=\frac{U_{cc}}{U_c}\dot{c}_t+\frac{U_{c(\theta a)}}{U_c}\theta\dot{a}_t\qquad<1.8.2>$
$\frac{\dot{\lambda}_t}{\lambda_t}=-\frac{1}{\sigma}\frac{\dot{c}_t}{c_t} +\frac{U_{c(\theta a)}}{U_c}\theta\dot{a}_t\qquad<1.8.3>$
$-\frac{1}{\sigma}\frac{\dot{c}_t}{c_t}+\frac{U_{c(\theta a)}}{U_c}\theta\dot{a}_t=\rho-R_t-\frac{U_{(\theta a)}\theta}{U_c}+n+D_a\qquad<1.8.4>$
$\frac{\dot{c}_t}{c_t}=\frac{1}{\gamma}[(R_t-\rho)+(\frac{U_{(\theta a)}\theta}{U_c}-D_a-n)]\qquad<1.8.5'>$
$R_t-\rho=n+D_a-\frac{U_{(\theta a)}\theta}{U_c}\qquad<1.9.1>$
$U_c=\frac{U_{(\theta a)}\theta}{n+D_a-(R_t-\rho)}\qquad<1.9.2>$
$c_t=f((1-\theta)a_t)+(r\theta-(1-\theta)\delta-n)a_t-D(a_t,a_{ext,t})\qquad<1.9.3>$
$c_t=(f((1-\theta)a_t)+r\theta a_t)-(((1-\theta)\delta+n)a_t+D(a_t,a_{ext,t}))\qquad<1.9.3'>$
$f(k_t)=Ak_t^\alpha\quad(A\gt0,\alpha \in[0,1])\qquad<2.1.1>$
$f(k_t)=f((1-\theta)a_t)=A((1-\theta)a_t)^\alpha=A(1-\theta)^\alpha a_t^\alpha\qquad<2.1.2>$
$f^\prime(k_t)=A\alpha k^{\alpha-1}=A\alpha((1-\theta)a_t)^{\alpha-1}=A\alpha(1-\theta)^{\alpha-1}a_t^{\alpha-1}\qquad<2.1.3>$
$U(c_t,b_t)=\frac{c_t^{1-\gamma}}{1-\gamma}+\beta \frac{b_t^{1-\psi}}{1-\psi}\quad(\gamma,\psi\gt0)\qquad<2.1.4>$
$U_c=\frac{\partial U(c_t,b_t)}{\partial c_t}=c_t^{-\gamma}\qquad<2.1.5>$
$U_{(\theta a)}=U_b=\frac{\partial U(c_t,b_t)}{\partial b_t}=\beta b_t^{-\psi}=\beta (\theta a_t)^{-\psi}=\beta\theta^{-\psi}a_t^{-\psi}\qquad<2.1.6>$
$D(a_t,a_{ext,t})=\phi(a_t-a_{ext,t})\quad(\phi\gt0)\qquad<2.1.7>$
$D_a=\frac{\partial D(a_t,a_{ext,t}}{\partial a_t}=\phi\qquad<2.1.8>$
$\dot{a}_t=A(1-\theta)^\alpha a_t^\alpha-c_t+(r\theta-(1-\theta)\delta-n) a_t-\phi(a_t-a_{ext,t})\Rightarrow\qquad c_t=(A(1-\theta)^\alpha a_t^\alpha+r\theta a_t)-((1-\theta)\delta+n+\phi)a_t+\phi a_{ext,t}\qquad<2.2.1>$
$(1-\theta)(A\alpha(1-\theta)^{\alpha-1} a_t^{\alpha-1}-\delta)+\theta r-\rho=n+\phi-\frac{\beta(\theta a_t)^{-\psi} \theta}{c_t^{-\gamma}}$
$(1-\theta)(A\alpha(1-\theta)^{\alpha-1} a_t^{\alpha-1}-\delta)+\theta r-\rho=n+\phi-\frac{\beta\theta^{1-\psi}}{a_t^\psi} c_t^\gamma\qquad<2.2.2>$
$c_t=(\frac{a_t^\psi}{\beta\theta^{1-\psi}}(\rho-((1-\theta)(A\alpha(1-\theta)^{\alpha-1}a_t^{\alpha-1}-\delta)+\theta r)+n+\phi))^{\frac{1}{\gamma}}\quad(\beta\neq0)\qquad<2.2.3>$
$\beta\theta^{1-\psi} a_t^{-\psi} c_t^\gamma=(\rho-((1-\theta)(A\alpha(1-\theta)^{\alpha-1}a_t^{\alpha-1}-\delta)+\theta r))+n+\phi\qquad<2.3.1>$
$\dot{a}_t=G(a_t,c_t)=A(1-\theta)^\alpha a_t^\alpha-c_t+(r\theta-(1-\theta)\delta-n)a_t-\phi(a_t-a_{ext,t})\qquad<2.3.2>$
$\dot{c}_t=H(a_t,c_t)=\frac{c_t}{\gamma}[((1-\theta)(A\alpha(1-\theta)^{\alpha-1}a_t^(\alpha-1)-\delta)+\theta r-\rho)+((\frac{\beta\theta^{1-\psi}a_t^{-\psi}}{c_t^{-\gamma}}-\phi-n)]\qquad<2.3.3>$
$J_{11}=\frac{\partial G}{\partial a_t}=((1-\theta)(A\alpha(1-\theta)^{\alpha-1} a_t^{\alpha-1}-\delta)+\theta r)-(n+\phi)\qquad<2.3.4>$
$J_{12}=\frac{\partial G}{\partial c_t}=-1\qquad<2.3.5>$
$J_{21}=\frac{\partial H}{\partial a_t}=\frac{c^*}{\gamma}[A\alpha(\alpha-1)(1-\theta)^\alpha a_t^{\alpha-2}-\frac{\beta\psi\theta^{1-\psi}a_*^{-\psi-1}}{c_*^{-\gamma}}]\qquad<2.3.6>$
$J_{22}=\frac{\partial H}{\partial c_t}=\frac{c^*}{\gamma}\frac{\partial}{\partial c_t} […]=\frac{c^*}{\gamma}\frac{\partial}{\partial c_t}(\frac{\beta\theta^{1-\psi}a_*^{-\psi}}{c_*^{-\gamma}})=\frac{c^*}{\gamma}\beta\theta^{1-\psi}a_*^{-\psi}\gamma c_*^{\gamma-1}=\frac{\beta\theta^{1-\psi}a_*^{-\psi}}{c_*^{-\gamma}}\qquad<2.3.7>$
$J=\begin{bmatrix}J_{11}&J_{12}\\J_{21}&J_{22}\end{bmatrix}=\begin{bmatrix}\frac{\partial G}{\partial a_t}&\frac{\partial G}{\partial c_t}\\ \frac{\partial H}{\partial a_t}&\frac{\partial H}{\partial c_t}\end{bmatrix}=\begin{bmatrix}((1-\theta)(A\alpha(1-\theta)^{\alpha-1} a_t^{\alpha-1}-\delta)+\theta r)-(n+\phi)&-1\\\frac{c^*}{\gamma}[A\alpha(\alpha-1)(1-\theta)^\alpha a_t^{\alpha-2}-\frac{\beta\psi\theta^{1-\psi}a_*^{-\psi-1}}{c_*^{-\gamma}}]&\frac{\beta\theta^{1-\psi}a_*^{-\psi}}{c_*^{-\gamma}} \end{bmatrix}\qquad<2.3.8>$
$U(c_i,b_i)=ln c_i+\beta_i ln b_i\qquad<3.1.1>$
$U_c=\frac{\partial U(c_i,b_i)}{\partial c_i}=c_i^{-1}=\frac{1}{c_i}\qquad<3.1.2>$
$U_{(\theta a)}=U_b=\frac{\partial U(c_i,b_i)}{\partial b_i}=\beta_i b_i^{-1}=\beta_i\frac{1}{b_i}=\frac{\beta_i}{\theta a_i}\qquad<3.1.3>$
$\dot{a}_i=A(1-\theta)^\alpha a_i^\alpha-c_i+(\theta r-(1-\theta)\delta-n)a_i-\phi_i(a_i-a_j)\qquad<3.1.4>$
$\phi_i(a_i-a_j)=-\phi_j(a_j-a_i)+Res.\qquad<3.1.5>$
$c_i=A(1-\theta)^\alpha a_i^\alpha+(\theta r-(1-\theta)\delta-n)a_i-\phi_i(a_i-a_j)=(A(1-\theta)^\alpha a_i^\alpha+\theta r a_i)-((1-\theta)\delta+n+\phi_i)a_i+\phi_i a_j\qquad<3.2.1>$
$\dot{c}_i=c_i[((1-\theta)(A\alpha(1-\theta)^{\alpha-1}a_i^{\alpha-1}-\delta)+\theta r-\rho_i)+\frac{\beta_i a_i^{-1}}{c_i^{-1}}-n-\phi_i)]\qquad<3.2.2>$
$c_i=\frac{a_i}{\beta_i}(\rho_i-((1-\theta)(A\alpha(1-\theta)^{\alpha-1}a_i^{\alpha-1}-\delta)+\theta r)+n+\phi_i)\qquad<3.2.3>$
$((1-\theta)(A\alpha(1-\theta)^{\alpha-1}a_i^{\alpha-1}-\delta)+\theta r)-\rho_i=n+\phi_i-\frac{\beta_i a_i^{-1}}{c_i^{-1}}\qquad<3.2.4>$
$\dot{a}_i=A(1-\theta)^\alpha a_i^\alpha-(\frac{a_i}{\beta_i}(\rho_i-((1-\theta)(A\alpha(1-\theta)^{\alpha-1}a_i^{\alpha-1}-\delta)+\theta r)+n+\phi_i))+(\theta r-(1-\theta)\delta-n)a_i-\phi_i(a_i-a_j)=A(1-\theta)^\alpha a_i^\alpha-\frac{a_i}{\beta_i}(\rho_i-((1-\theta)(A\alpha(1-\theta)^{\alpha-1}a_i^{\alpha-1}-\delta)+\theta r)+n+\phi_i)+\theta r a_i-(1-\theta)\delta a_i-n a_i-\phi_i a_i+\phi_i a_j=A(1-\theta)^\alpha a_i^\alpha-\frac{a_i}{\beta_i}\rho_i+\frac{a_i}{\beta_i}(1-\theta)A\alpha(1-\theta)^{\alpha-1}a_i^{\alpha-1}-\frac{a_i}{\beta_i}(1-\theta)\delta+\frac{a_i}{\beta_i}\theta r-\frac{a_i}{\beta_i}n-\frac{a_i}{\beta_i}\phi_i+\theta r a_i-(1-\theta)\delta a_i-n a_i-\phi_i a_i+\phi_i a_j=(1-\theta)a_iA(1-\theta)^{\alpha-1}a_i^{\alpha-1}+\frac{\alpha}{\beta_i}(1-\theta)a_i A(1-\theta)^{\alpha-1}a_i^{\alpha-1}-(1-\theta)a_i\delta-\frac{a_i}{\beta_i}(1-\theta)\delta+\theta r a_i+\frac{a_i}{\beta_i}\theta r-\frac{a_i}{\beta_i}\rho_i-na_i-\frac{a_i}{\beta_i}n-\phi_i a_i-\frac{a_i}{\beta_i}\phi_i+\phi_i a_j=(1+\frac{\alpha}{\beta_i})(1-\theta)a_i A(1-\theta)^{\alpha-1}a_i^{\alpha-1}-(1+\frac{1}{\beta_i})(1-\theta)a_i\delta+(1+\frac{1}{\beta_i})\theta r a_i-\frac{a_i}{\beta_i}\rho_i-(1+\frac{1}{\beta_i})na_i-(1+\frac{1}{\beta_i})\phi_i a_i+\phi_i a_j=((1-\theta)a_i((1+\frac{\alpha}{\beta_i})A(1-\theta)^{\alpha-1}a_i^{\alpha-1}-(1+\frac{1}{\beta_i})\delta)+(1+\frac{1}{\beta_i})\theta a_i r)-\frac{1}{\beta_i}\rho_i a_i-(1+\frac{1}{\beta_i})(na_i+\phi_i a_i)+\phi_i a_j)\qquad<3.3.1>$
$((1-\theta)((1+\frac{\alpha}{\beta_i})A(1-\theta)^{\alpha-1}a_i^{\alpha-1}-(1+\frac{1}{\beta_i})\delta)+(1+\frac{1}{\beta_i})\theta r)-\frac{1}{\beta_i}\rho_i-(1+\frac{1}{\beta_i})(n+\phi_i)+\phi_i\frac{a_j}{a_i}=0\qquad<3.3.2>$
$\Rightarrow\qquad((1-\theta)((1+\frac{\alpha}{\beta_i})A(1-\theta)^{\alpha-1}a_i^{\alpha-1}-(1+\frac{1}{\beta_i})\delta)+(1+\frac{1}{\beta_i})\theta r)-\frac{1}{\beta_i}\rho_i=(1+\frac{1}{\beta_i})(n+\phi_i)-\phi_i\frac{a_j}{a_i}\qquad<3.3.3>$
$\left((1-\theta)\left(\left(1+\frac{\alpha}{\beta_H}\right)A(1-\theta)^{\alpha-1}a_H^{\alpha-1}-\left(1+\frac{1}{\beta_H}\right)\delta\right)+\left(1+\frac{1}{\beta_H}\right)\theta r\right) - \frac{1}{\beta_H}\rho_H - \left(1+\frac{1}{\beta_H}\right)(n+\phi_H) + \phi_H\frac{a_L}{a_H}=0 \qquad<3.3.4H>$
$\left((1-\theta)\left(\left(1+\frac{\alpha}{\beta_L}\right)A(1-\theta)^{\alpha-1}a_L^{\alpha-1}-\left(1+\frac{1}{\beta_L}\right)\delta\right)+\left(1+\frac{1}{\beta_L}\right)\theta r\right) - \frac{1}{\beta_L}\rho_L - \left(1+\frac{1}{\beta_L}\right)(n+\phi_L) + \phi_L\frac{a_H}{a_L}=0 \qquad<3.3.4L>$
$\dot{a}_H=((1-\theta)a_H((1+\frac{\alpha}{\beta_H})A(1-\theta)^{\alpha-1}a_H^{\alpha-1}-(1+\frac{1}{\beta_H})\delta)+(1+\frac{1}{\beta_H})\theta a_H r)-\frac{1}{\beta_H}\rho_H a_H-(1+\frac{1}{\beta_H})(na_H+\phi_H a_H)+\phi_H a_L)\qquad<3.3.5H>$
$\dot{a}_L=((1-\theta)a_L((1+\frac{\alpha}{\beta_L})A(1-\theta)^{\alpha-1}a_L^{\alpha-1}-(1+\frac{1}{\beta_L})\delta)+(1+\frac{1}{\beta_L})\theta a_L r)-\frac{1}{\beta_L}\rho_L a_L-(1+\frac{1}{\beta_L})(na_L+\phi_L a_L)+\phi_L a_H)\qquad<3.3.5L>$
$J_{11}=\frac{\partial \dot{a}_H}{\partial a_H}=\frac{\partial}{\partial a_H}(((1-\theta)a_H((1+\frac{\alpha}{\beta_H})A(1-\theta)^{\alpha-1}a_H^{\alpha-1}-(1+\frac{1}{\beta_H})\delta)+(1+\frac{1}{\beta_H})\theta a_H r)-\frac{1}{\beta_H}\rho_H a_H-(1+\frac{1}{\beta_H})(na_H+\phi_H a_H)+\phi_H a_L))=\frac{\partial}{\partial a_H}((1+\frac{\alpha}{\beta_H})A(1-\theta)^\alpha a_H^\alpha-(1+\frac{1}{\beta_H})(1-\theta)\delta a_H+(1+\frac{1}{\beta_H})\theta a_H r-\frac{1}{\beta_H}\rho_H a_H-(1+\frac{1}{\beta_H})(na_H+\phi_H a_H)+\phi_H a_L)=(1+\frac{\alpha}{\beta_H})A\alpha(1-\theta)^\alpha a_H^{\alpha-1}-(1+\frac{1}{\beta_H})(1-\theta)\delta+(1+\frac{1}{\beta_H})\theta r-\frac{1}{\beta_H}\rho_H-(1+\frac{1}{\beta_H})(n+\phi_H)=(1+\frac{\alpha}{\beta_H})A\alpha(1-\theta)^\alpha a_H^{\alpha-1}+(1+\frac{1}{\beta_H})(\theta r-(1-\theta)\delta-n-\phi_H)-\frac{1}{\beta_H}\rho_H\qquad<3.3.6H>$
$J_{22}=\frac{\partial \dot{a}_L}{\partial a_L}=(1+\frac{\alpha}{\beta_L})A\alpha(1-\theta)^\alpha a_L^{\alpha-1}+(1+\frac{1}{\beta_L})(\theta r-(1-\theta)\delta-n-\phi_L)-\frac{1}{\beta_L}\rho_L\qquad<3.3.6L>$
$J_{12}=\frac{\partial \dot{a}_H}{\partial a_L}=\frac{\partial}{\partial a_L}(((1-\theta)a_H………+\phi_H a_L)=\phi_H\qquad<3.3.7H>$
$J_{21}=\frac{\partial \dot{a}_L}{\partial a_H}=\frac{\partial}{\partial a_H}(((1-\theta)a_L………+\phi_L a_H)=\phi_L\qquad<3.3.7L>$
$\left((1-\theta)\left(\left(1+\frac{\alpha}{\beta_H}\right)A(1-\theta)^{\alpha-1}a_H^{\alpha-1}-\left(1+\frac{1}{\beta_H}\right)\delta\right)+\left(1+\frac{1}{\beta_H}\right)\theta r\right) - \frac{1}{\beta_H}\rho_H - \left(1+\frac{1}{\beta_H}\right)(n+\phi_H) + \phi_H\frac{a_L}{a_H}=0$
$(1+\frac{\alpha}{\beta_H})A(1-\theta)^\alpha a_H^{\alpha-1}-(1+\frac{1}{\beta_H})(1-\theta)\delta+(1+\frac{1}{\beta_H})\theta r-\frac{1}{\beta_H}\rho_H-(1+\frac{1}{\beta_H})(n+\phi_H)=-\phi_H\frac{a_L}{a_H}$
$(1+\frac{\alpha}{\beta_H})A(1-\theta)^\alpha a_H^{\alpha-1}+(1+\frac{1}{\beta_H})(\theta r-((1-\theta)\delta+n+\phi_H))-\frac{1}{\beta_H}\rho_H=-\phi_H\frac{a_L}{a_H}$
$\phi_H\frac{a_L}{a_H}=\frac{1}{\beta_H}\rho_H-(1+\frac{\alpha}{\beta_H})A(1-\theta)^\alpha a_H^{\alpha-1}-(1+\frac{1}{\beta_H})(\theta r-((1-\theta)\delta+n+\phi_H))$
$a_L=\frac{a_H}{\phi_H}(\frac{1}{\beta_H}\rho_H-(1+\frac{\alpha}{\beta_H})A(1-\theta)^\alpha a_H^{\alpha-1}-(1+\frac{1}{\beta_H})(\theta r-((1-\theta)\delta+n+\phi_H)))\qquad<3.3.8H>$
$a_H=\frac{a_L}{\phi_L}(\frac{1}{\beta_L}\rho_L-(1+\frac{\alpha}{\beta_L})A(1-\theta)^\alpha a_L^{\alpha-1}-(1+\frac{1}{\beta_L})(\theta r-((1-\theta)\delta+n+\phi_L)))\qquad<3.3.8L>$