Land, Housing, Growth and Inequality

Abstract

This paper incorporates productive assets, residential land and residential structures in a growth model with two social classes: capitalists, who invest in productive assets and housing but do not work, and workers, who invest only in housing and decide on their labor effort. It is shown that the relative price of land grows in the long run at the same rate as the economy’s GDP, while the quantity of housing services and their price grow slower than it. Moreover, numerical examples show that (i) shifting taxation away from income and towards the property of land enhances long-term GDP growth and leads in the long-run to more equalitarian (more favorable to workers) income and wealth distributions, (ii) a marginal increase in the fraction of investment expenditures in residential structures that is tax deductible reduces income and wealth inequality, (iii) a change in preferences giving more weight in the utility function to residential services leads in the long run to a distribution of income and wealth more favorable to capitalists, (iv) changes in taxation or in preferences increasing the fraction of total investment devoted to the accumulation of residential wealth rather than to the accumulation of productive assets brings about a balanced growth path characterized by a higher wealth-income ratio. Finally, endogenous fluctuations may be generated along the equilibrium trajectory converging to the balanced growth path, in a model where housing wealth and residential land are distinguished from productive capital and only fundamentals (initial endowments, preferences and technologies) drive the economy.

Appendices

Appendices

1.1 A Derivation of the Difference Equations Governing an Equilibrium Path

• A.1 From firms’ profit maximization and labor market equilibrium we obtain:

$$ {\mathrm{W}}_{\mathrm{t}} = \eta \left( {{\mathrm{nE}}_{\mathrm{t}} } \right)^{\eta - 1} {\mathrm{K}}_{\mathrm{t}} ,\quad \mathrm{t} = 0,1 \ldots , $$

(A.1)

$$ {\mathrm{R}}_{\mathrm{t}} = (1 - \eta )\left( {{\mathrm{nE}}_{\mathrm{t}} } \right)^{\eta } ,\quad {\mathrm{t}} = 0,1 \ldots . $$

(A.2)

• A.2 From the representative worker’s optimization problem we obtain:

$$ \frac{ ( 1- \beta - \gamma )}{{{\mathrm{C}}_{\mathrm{wt}} }} = \lambda_{\mathrm{wt}} ,\quad {\mathrm{t}} = 0,1 \ldots , $$

(A.3)

$$ \frac{\gamma }{{ 1- {\mathrm{E}}_{\mathrm{t}} }} = ( 1- \tau ) {\mathrm{W}}_{\mathrm{t}} \lambda_{\mathrm{wt}} ,\quad {\mathrm{t}} = 0,1 \ldots , $$

(A.4)

$$ \frac{\beta }{{{\mathrm{L}}_{\mathrm{wt}}^{\phi } {\mathrm{H}}_{\mathrm{wt}}^{1 - \phi } + {\mathrm{h}}_{\mathrm{wt}} }} = \left\{ {\begin{array}{*{20}l} {{\mathrm{P}}_{\mathrm{t}} \lambda_{\mathrm{wt}} \quad {\mathrm{if}}\,{\mathrm{h}}_{\mathrm{wt}} \ge 0} \hfill \\ { ( 1- \tau ) {\mathrm{P}}_{\mathrm{t}} \lambda_{\mathrm{wt}} \;{\mathrm{otherwise,}}\quad {\mathrm{t}} = 0 , 1\ldots ,} \hfill \\ \end{array} } \right. $$

(A.5)

$$ \frac{{\rho_{\mathrm{w}} \beta \phi \left( {\frac{{{\mathrm{H}}_{{{\mathrm{wt}} + 1}} }}{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }}} \right)^{1 - \phi } }}{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}}^{\phi } {\mathrm{H}}_{{{\mathrm{wt}} + 1}}^{1 - \phi } + {\mathrm{h}}_{{{\mathrm{wt}} + 1}} }} + \rho_{\mathrm{w}} {\mathrm{Q}}_{{{\mathrm{t}} + 1}} (1 - \zeta )\lambda_{{{\mathrm{wt}} + 1}} = {\mathrm{Q}}_{\mathrm{t}} \lambda_{\mathrm{wt}} ,\quad {\mathrm{t}} = 0,1 \ldots , $$

(A.6)

$$ \frac{{\rho_{\mathrm{w}} \beta (1 - \phi )\left( {\frac{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }}{{{\mathrm{H}}_{{{\mathrm{wt}} + 1}} }}} \right)^{\phi } }}{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}}^{\phi } {\mathrm{H}}_{{{\mathrm{wt}} + 1}}^{1 - \phi } + {\mathrm{h}}_{{{\mathrm{wt}} + 1}} }} + \rho_{\mathrm{w}} ( 1- \tau {\mathrm{b}}_{\mathrm{H}} )(1 - \delta_{\mathrm{H}} )\lambda_{{{\mathrm{wt}} + 1}} = ( 1- \tau {\mathrm{b}}_{\mathrm{H}} )\lambda_{\mathrm{wt}} ,\quad {\mathrm{t}} = 0,1 \ldots , $$

(A.7)

where $\lambda_{\mathrm{wt}}$ is the (current value) multiplier of the Hamiltonian associated with the representative worker’s problem.¹¹

• A.3 From the representative capitalist’s optimization problem and the equilibrium condition in the market for housing services we obtain:

$$ \frac{ ( 1- \alpha )}{{{\mathrm{C}}_{\mathrm{ct}} }} - \lambda_{\mathrm{ct}} = 0,\quad {\mathrm{t}} = 0,1, \ldots , $$

(A.8)

$$ \frac{\alpha }{{ ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{wt}} )^{\phi } {\mathrm{H}}_{\mathrm{ct}}^{1 - \phi } - {\mathrm{nh}}_{\mathrm{wt}} }} = \left\{ {\begin{array}{*{20}l} { ( 1- \tau ) {\mathrm{P}}_{\mathrm{t}} \lambda_{\mathrm{ct}} \quad {\mathrm{if}}\;{\mathrm{h}}_{\mathrm{wt}} \ge 0} \hfill \\ {{\mathrm{P}}_{\mathrm{t}} \lambda_{\mathrm{ct}} \;{\mathrm{otherwise}},\quad {\mathrm{t}} = 0,1 \ldots ,} \hfill \\ \end{array} } \right. $$

(A.9)

$$ \frac{{\rho_{\mathrm{c}} \alpha \phi \left( {\frac{{{\mathrm{H}}_{{{\mathrm{ct}} + 1}} }}{{{\mathrm{L}} - {\mathrm{nL}}_{{{\mathrm{wt}} + 1}} }}} \right)^{1 - \phi } }}{{ ( {\mathrm{L}} - {\mathrm{nL}}_{{{\mathrm{wt}} + 1}} )^{\phi } {\mathrm{H}}_{{{\mathrm{ct}} + 1}}^{1 - \phi } - {\mathrm{nh}}_{{{\mathrm{wt}} + 1}} }} + \rho_{\mathrm{c}} {\mathrm{Q}}_{{{\mathrm{t}} + 1}} (1 - \zeta )\lambda_{{{\mathrm{ct}} + 1}} = {\mathrm{Q}}_{\mathrm{t}} \lambda_{\mathrm{ct}} ,\quad {\mathrm{t}} = 0,1 \ldots , $$

(A.10)

$$\begin{aligned}& \frac{{\rho_{\mathrm{c}} \alpha (1 - \phi )\left( {\frac{{{\mathrm{L}} - {\mathrm{nL}}_{{{\mathrm{wt}} + 1}} }}{{{\mathrm{H}}_{{{\mathrm{ct}} + 1}} }}} \right)^{\phi } }}{{ ( {\mathrm{L}} - {\mathrm{nL}}_{{{\mathrm{wt}} + 1}} )^{\phi } {\mathrm{H}}_{{{\mathrm{ct}} + 1}}^{1 - \phi } - {\mathrm{nh}}_{{{\mathrm{wt}} + 1}} }} + \rho_{\mathrm{c}} ( 1- \tau {\mathrm{b}}_{\mathrm{H}} )(1 - \delta_{\mathrm{H}} )\lambda_{{{\mathrm{ct}} + 1}}\\ & \quad= ( 1- \tau {\mathrm{b}}_{\mathrm{H}} )\lambda_{\mathrm{ct}} ,\quad {\mathrm{t}} = 0,1 \ldots , \end{aligned} $$

(A.11)

$$ \rho_{\mathrm{c}} [ {\mathrm{R}}_{{{\mathrm{t}} + 1}} ( 1- \tau ) + ( 1- \tau {\mathrm{b}}_{\mathrm{K}} )(1 - \delta_{\mathrm{K}} )]\lambda_{{{\mathrm{ct}} + 1}} = ( 1- \tau {\mathrm{b}}_{\mathrm{K}} )\lambda_{\mathrm{ct}} ,\quad {\mathrm{t}} = 0,1 \ldots , $$

(A.12)

where $\lambda_{\mathrm{ct}}$ is the (current value) multiplier of the Hamiltonian associated with the representative capitalist’s problem.¹²

• A.4 Let suppose from now on that ${\mathrm{h}}_{\mathrm{wt}} \ge 0$. Hence, from (A.3) and (A.5) we get

$$ {\mathrm{C}}_{\mathrm{wt}} = \frac{{ ( {\mathrm{L}}_{\mathrm{wt}}^{\phi } {\mathrm{H}}_{\mathrm{wt}}^{1 - \phi } + {\mathrm{h}}_{\mathrm{wt}} ) ( 1- \beta - \gamma ) {\mathrm{P}}_{\mathrm{t}} }}{\beta },\quad {\mathrm{t}} = 0,1 \ldots , $$

(A.13)

while from (A.8) and (A.9) we get

$$ {\mathrm{P}}_{\mathrm{t}} = \frac{{\alpha {\mathrm{C}}_{\mathrm{ct}} }}{{ [ ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{wt}} )^{\phi } {\mathrm{H}}_{\mathrm{ct}}^{1 - \phi } - {\mathrm{nh}}_{\mathrm{wt}} ] ( 1- \alpha ) ( 1- \tau )}},\quad {\mathrm{t}} = 0,1 \ldots . $$

(A.14)

Finally, by substituting (A.14) for ${\mathrm{P}}_{\mathrm{t}}$ in (A.13), we get

$$ {\mathrm{C}}_{\mathrm{wt}} = \frac{{ ( {\mathrm{L}}_{\mathrm{wt}}^{\phi } {\mathrm{H}}_{\mathrm{wt}}^{1 - \phi } + {\mathrm{h}}_{\mathrm{wt}} ) ( 1- \beta - \gamma )\alpha {\mathrm{C}}_{\mathrm{ct}} }}{{\beta [ ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{wt}} )^{\phi } {\mathrm{H}}_{\mathrm{ct}}^{1 - \phi } - {\mathrm{nh}}_{\mathrm{wt}} ] ( 1- \alpha ) ( 1- \tau )}},\quad {\mathrm{t}} = 0,1 \ldots . $$

(A.15)

• A.5 By using (A.2), (A.8) and (A.12), one can obtain

$$ \frac{{{\mathrm{C}}_{{{\mathrm{ct}} + 1}} }}{{{\mathrm{C}}_{\mathrm{ct}} }} = g ( {\mathrm{E}}_{{{\mathrm{t}} + 1}} ),\quad {\mathrm{t}} = 0,1 \ldots , $$

(A.16)

where $g\left( {{\mathrm{E}}_{{{\mathrm{t}} + 1}} } \right) = \rho_{\mathrm{c}} \left[ {\frac{{(1 - \eta )(1 - \tau )\left( {{\mathrm{nE}}_{{{\mathrm{t}} + 1}} } \right)^{\eta } }}{{ ( 1- \tau {\mathrm{b}}_{\mathrm{K}} )}} + 1 - \delta_{\mathrm{K}} } \right]$

• A.6 By using (A.8), (A.11) and (A.16), one can obtain

$$ {\mathrm{C}}_{\mathrm{ct}} = {\mathrm{H}}_{\mathrm{ct}} d ( {\mathrm{F}}_{\mathrm{t}} , {\mathrm{L}}_{\mathrm{wt}} , {\mathrm{E}}_{\mathrm{t}} ),\;{\mathrm{F}}_{\mathrm{t}} \equiv \frac{{{\mathrm{h}}_{\mathrm{wt}} }}{{{\mathrm{H}}_{\mathrm{ct}}^{1 - \phi } }},\quad {\mathrm{t}} = 0,1 \ldots , $$

(A.17)

where $d ( {\mathrm{F}}_{\mathrm{t}} , {\mathrm{L}}_{\mathrm{wt}} , {\mathrm{E}}_{\mathrm{t}} ) = \frac{{ [ ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{wt}} )^{\phi } - {\mathrm{nF}}_{\mathrm{t}} ] ( 1- \tau {\mathrm{b}}_{\mathrm{H}} ) ( 1- \alpha ) [ ( 1- \eta ) ( 1- \tau ) ( {\mathrm{nE}}_{\mathrm{t}} )^{\eta } - ( 1- \tau {\mathrm{b}}_{\mathrm{K}} ) (\delta_{\mathrm{K}} - \delta_{\mathrm{H}} )]}}{{\alpha ( 1- \tau {\mathrm{b}}_{\mathrm{K}} ) ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{wt}} )^{\phi } ( 1- \phi )}}$

• A.7 By using (A.1), (A.3), (A.4) and (A.15), one can obtain

$$ \frac{{{\mathrm{C}}_{\mathrm{ct}} }}{{{\mathrm{K}}_{\mathrm{t}} }} = m ( {\mathrm{V}}_{\mathrm{t}} , {\mathrm{F}}_{\mathrm{t}} , {\mathrm{L}}_{\mathrm{wt}} , {\mathrm{E}}_{\mathrm{t}} ),{\mathrm{ V}}_{\mathrm{t}} \equiv \frac{{{\mathrm{H}}_{\mathrm{ct}} }}{{{\mathrm{H}}_{\mathrm{wt}} }},\quad {\mathrm{t}} = 0,1 \ldots , $$

(A.18)

where $m ( {\mathrm{V}}_{\mathrm{t}} ,{\mathrm{F}}_{\mathrm{t}} ,{\mathrm{L}}_{\mathrm{wt}} ,{\mathrm{E}}_{\mathrm{t}} ) = \frac{{[({\mathrm{L}} - {\mathrm{nL}}_{\mathrm{wt}} )^{\phi } - {\mathrm{nF}}_{\mathrm{t}} ]\beta \eta (1 - \alpha )(1 - \tau )^{ 2} ( 1- {\mathrm{E}}_{\mathrm{t}} )}}{{\alpha \gamma ( {\mathrm{nE}}_{\mathrm{t}} )^{1 - \eta } ( {\mathrm{L}}_{\mathrm{wt}}^{\phi } {\mathrm{V}}_{\mathrm{t}}^{{\phi { - 1}}} + {\mathrm{F}}_{\mathrm{t}} )}}.$

Notice that it derives from (A.17) and (A.18) that

$$ \begin{aligned} \frac{{{\mathrm{H}}_{\mathrm{ct}} }}{{{\mathrm{K}}_{\mathrm{t}} }} & = \frac{{m ( {\mathrm{V}}_{\mathrm{t}} , {\mathrm{F}}_{\mathrm{t}} , {\mathrm{L}}_{\mathrm{wt}} , {\mathrm{E}}_{\mathrm{t}} )}}{{d ( {\mathrm{F}}_{\mathrm{t}} , {\mathrm{L}}_{\mathrm{wt}} , {\mathrm{E}}_{\mathrm{t}} )}} \\ & = \frac{{ ( 1- {\mathrm{E}}_{\mathrm{t}} ) ( 1- \tau )^{ 2} \beta \eta ( 1- \tau {\mathrm{b}}_{\mathrm{K}} ) ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{wt}} )^{\phi } (1 - \phi ) ( {\mathrm{L}}_{\mathrm{wt}}^{\phi } {\mathrm{V}}_{\mathrm{t}}^{\phi - 1} + {\mathrm{F}}_{\mathrm{t}} )^{ - 1} }}{{\gamma [ ( 1- \eta ) ( 1- \tau ) ( {\mathrm{nE}}_{\mathrm{t}} )^{\eta } - ( 1- \tau {\mathrm{b}}_{\mathrm{K}} ) (\delta_{\mathrm{K}} - \delta_{\mathrm{H}} ) ] ( 1- \tau {\mathrm{b}}_{\mathrm{H}} ) ( {\mathrm{nE}}_{\mathrm{t}} )^{ 1- \eta } }},\\ & \quad\quad {\mathrm{t}} = 0,1 \ldots .\end{aligned} $$

(A.19)

• A.8 By using (A.3), (A.6) and (A.7), one can obtain

$$ \begin{aligned}& \frac{{\beta \phi \left( {\frac{{{\mathrm{H}}_{{{\mathrm{wt}} + 1}} }}{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }}} \right)^{1 - \phi } }}{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}}^{\phi } {\mathrm{H}}_{{{\mathrm{wt}} + 1}}^{1 - \phi } + {\mathrm{h}}_{{{\mathrm{wt}} + 1}} }} + \frac{{{\mathrm{Q}}_{{{\mathrm{t}} + 1}} ( 1- \zeta ) ( 1- \beta - \gamma )}}{{{\mathrm{C}}_{{{\mathrm{wt}} + 1}} }} \\ &\quad = {\mathrm{Q}}_{\mathrm{t}} \left[ {\frac{{\beta (1 - \phi )\left( {\frac{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }}{{{\mathrm{H}}_{{{\mathrm{wt}} + 1}} }}} \right)^{\phi } }}{{ ( 1- \tau {\mathrm{b}}_{\mathrm{H}} ) [ {\mathrm{L}}_{{{\mathrm{wt}} + 1}}^{\phi } {\mathrm{H}}_{{{\mathrm{wt}} + 1}}^{1 - \phi } + {\mathrm{h}}_{{{\mathrm{wt}} + 1}} ]}} + \frac{{ ( 1- \beta - \gamma ) ( 1- \delta_{\mathrm{H}} )}}{{{\mathrm{C}}_{{{\mathrm{wt}} + 1}} }}} \right], \\ & \quad {\mathrm{t}} = 0,1 \ldots , \end{aligned} $$

(A.20)

• A.9 By using (A.8), (A.10) and (A.16), one can obtain

$$ \begin{aligned} {\mathrm{Q}}_{\mathrm{t}} & = \frac{{\rho_{\mathrm{c}} \alpha \phi \left( {\frac{{{\mathrm{H}}_{{{\mathrm{ct}} + 1}} }}{{{\mathrm{L}} - {\mathrm{nL}}_{{{\mathrm{wt}} + 1}} }}} \right)^{1 - \phi } {\mathrm{C}}_{\mathrm{ct}} }}{{ [ ( {\mathrm{L}} - {\mathrm{nL}}_{{{\mathrm{wt}} + 1}} )^{\phi } {\mathrm{H}}_{{{\mathrm{ct}} + 1}}^{1 - \phi } - {\mathrm{nh}}_{{{\mathrm{wt}} + 1}} ](1 - \alpha )}} \\ & \quad + \frac{{ ( 1- \zeta ) ( 1- \tau {\mathrm{b}}_{\mathrm{K}} ){\mathrm{Q}}_{{{\mathrm{t}} + 1}} }}{{(1 - \eta )(1 - \tau )\left( {{\mathrm{nE}}_{{{\mathrm{t}} + 1}} } \right)^{\eta } + ( 1- \tau {\mathrm{b}}_{\mathrm{K}} )(1 - \delta_{\mathrm{K}} )}},\;{\mathrm{t}} = 0 , 1\ldots \, . \\ \end{aligned} $$

(A.21)

• A.10 By using (A.16) and (A.21) for substituting Q_t, (A.20) can be rewritten as

$$ \begin{aligned} & \frac{{ ( 1- \zeta ) {\mathrm{Q}}_{{{\mathrm{t}} + 1}} ( 1- \beta - \gamma )}}{{{\mathrm{C}}_{{{\mathrm{wt}} + 1}} }}\left[ {1 - \frac{{ ( 1- \tau {\mathrm{b}}_{\mathrm{K}} ) ( 1- \delta_{\mathrm{H}} )}}{{(1 - \eta )(1 - \tau )\left( {{\mathrm{nE}}_{{{\mathrm{t}} + 1}} } \right)^{\eta } + ( 1- \tau {\mathrm{b}}_{\mathrm{K}} )(1 - \delta_{\mathrm{K}} )}}} \right] \\ & \quad - \frac{{ ( 1- \zeta ) {\mathrm{Q}}_{{{\mathrm{t}} + 1}} \beta (1 - \phi ) ( 1- \tau {\mathrm{b}}_{\mathrm{K}} ){\mathrm{L}}_{{{\mathrm{wt}} + 1}}^{\phi } {\mathrm{V}}_{{{\mathrm{t}} + 1}} [ {\mathrm{L}}_{{{\mathrm{wt}} + 1}}^{\phi } + {\mathrm{V}}_{{{\mathrm{t}} + 1}}^{1 - \phi } {\mathrm{F}}_{{{\mathrm{t}} + 1}} ]^{ - 1} }}{{ ( 1- \tau {\mathrm{b}}_{\mathrm{H}} )[(1 - \eta )(1 - \tau )\left( {{\mathrm{nE}}_{{{\mathrm{t}} + 1}} } \right)^{\eta } + ( 1- \tau {\mathrm{b}}_{\mathrm{K}} )(1 - \delta_{\mathrm{K}} )]{\mathrm{H}}_{{{\mathrm{ct}} + 1}} }}\\ & \quad = - \frac{{\beta \phi {\mathrm{L}}_{{{\mathrm{wt}} + 1}}^{\phi - 1} }}{{ ( {\mathrm{L}}_{{{\mathrm{wt}} + 1}}^{\phi } + {\mathrm{V}}_{{{\mathrm{t}} + 1}}^{1 - \phi } {\mathrm{F}}_{{{\mathrm{t}} + 1}} )}} \\ & \quad \quad + \frac{{\alpha \phi [(1 - \eta )(1 - \tau )\left( {{\mathrm{nE}}_{{{\mathrm{t}} + 1}} } \right)^{\eta } + ( 1- \tau {\mathrm{b}}_{\mathrm{K}} )(1 - \delta_{\mathrm{K}} )]^{ - 1} ( 1- \tau {\mathrm{b}}_{\mathrm{K}} ){\mathrm{C}}_{{{\mathrm{ct}} + 1}} }}{{ ( {\mathrm{L}} - {\mathrm{nL}}_{{{\mathrm{wt}} + 1}} )^{1 - \phi } [ ( {\mathrm{L}} - {\mathrm{nL}}_{{{\mathrm{wt}} + 1}} )^{\phi } - {\mathrm{nF}}_{{{\mathrm{t}} + 1}} ](1 - \alpha )}} \\ & \quad \quad \times \left[ {\frac{{\beta (1 - \phi ){\mathrm{L}}_{{{\mathrm{wt}} + 1}}^{\phi } {\mathrm{V}}_{{{\mathrm{t}} + 1}} }}{{ ( 1- \tau {\mathrm{b}}_{\mathrm{H}} ) ( {\mathrm{L}}_{{{\mathrm{wt}} + 1}}^{\phi } + {\mathrm{V}}_{{{\mathrm{t}} + 1}}^{1 - \phi } {\mathrm{F}}_{{{\mathrm{t}} + 1}} ) {\mathrm{H}}_{{{\mathrm{ct}} + 1}} }} + \frac{{ ( 1- \beta - \gamma ) ( 1- \delta_{\mathrm{H}} )}}{{{\mathrm{C}}_{{{\mathrm{wt}} + 1}} }}} \right],\quad {\mathrm{t}} = 0,1 \ldots . \\ \end{aligned} $$

(A.22)

• A.11 By using (A.15), (A.17) and (A.22), one can obtain

$$ {\mathrm{Q}}_{\mathrm{t}} = \frac{{{\mathrm{H}}_{\mathrm{ct}} q ( {\mathrm{V}}_{\mathrm{t}} , {\mathrm{L}}_{\mathrm{wt}} , {\mathrm{E}}_{\mathrm{t}} )}}{ ( 1- \zeta )},\quad {\mathrm{t}} = 1,2 \ldots , $$

(A.23)

where

$$ \begin{aligned} & q ( {\mathrm{V}}_{\mathrm{t}} , {\mathrm{L}}_{\mathrm{wt}} , {\mathrm{E}}_{\mathrm{t}} )\\ & \quad = \frac{{\phi ( 1- \tau {\mathrm{b}}_{\mathrm{H}} ){\mathrm{V}}_{\mathrm{t}} {\mathrm{L}}_{\mathrm{wt}}^{\phi } [(1 - \eta )(1 - \tau )\left( {{\mathrm{nE}}_{\mathrm{t}} } \right)^{\eta } - ( 1- \tau {\mathrm{b}}_{\mathrm{K}} ) (\delta_{\mathrm{K}} - \delta_{\mathrm{H}} )]}}{{(1 - \phi ) ( 1- \tau {\mathrm{b}}_{\mathrm{K}} ) ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{wt}} )[ ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{wt}} )^{\phi } {\mathrm{V}}_{\mathrm{t}}^{1 - \phi } (1 - \tau ) - {\mathrm{V}}_{\mathrm{t}} {\mathrm{L}}_{\mathrm{wt}}^{\phi } ]}} \\ & \quad \quad + \frac{{\phi ( 1- \tau {\mathrm{b}}_{\mathrm{H}} )\left\{ {\frac{{{\mathrm{V}}_{\mathrm{t}}^{1 - \phi } (1 - \tau ) ( 1- \tau {\mathrm{b}}_{\mathrm{K}} ) ( 1- \delta_{\mathrm{H}} )}}{{ ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{wt}} )^{1 - \phi } }} - \frac{{[(1 - \eta )(1 - \tau )\left( {{\mathrm{nE}}_{\mathrm{t}} } \right)^{\eta } + ( 1- \tau {\mathrm{b}}_{\mathrm{K}} )(1 - \delta_{\mathrm{K}} )]}}{{{\mathrm{L}}_{\mathrm{wt}}^{1 - \phi } }}} \right\}}}{{(1 - \phi ) ( 1- \tau {\mathrm{b}}_{\mathrm{K}} )[ ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{wt}} )^{\phi } {\mathrm{V}}_{\mathrm{t}}^{1 - \phi } (1 - \tau ) - {\mathrm{V}}_{\mathrm{t}} {\mathrm{L}}_{\mathrm{wt}}^{\phi } ]}}. \\ \end{aligned} $$

• A.12 By dividing the period budget constraint of the representative worker by C_ct, one can obtain:

$$ {{\Psi} } ( {\mathrm{E}}_{{{\mathrm{t}} + 1}} , {\mathrm{V}}_{{{\mathrm{t}} + 1}} , {\mathrm{F}}_{{{\mathrm{t}} + 1}} , {\mathrm{L}}_{{{\mathrm{wt}} + 1}} , {\mathrm{E}}_{\mathrm{t}} , {\mathrm{V}}_{\mathrm{t}} , {\mathrm{F}}_{\mathrm{t}} , {\mathrm{L}}_{\mathrm{wt}} )= 0,\quad {\mathrm{t}} = 1,2 \ldots , $$

(A.24)

where

$$ \begin{aligned} {{\Psi} } (. )& = \frac{\xi \tau }{\mathrm{n}} \left \{ \frac{{[\left( {{\mathrm{nE}}_{\mathrm{t}} } \right)^{\eta } + {\mathrm{b}}_{\mathrm{K}} (1 - \delta_{\mathrm{k}} )]}}{{m ( {\mathrm{V}}_{\mathrm{t}} ,{\mathrm{F}}_{\mathrm{t}} ,{\mathrm{L}}_{\mathrm{wt}} ,{\mathrm{E}}_{\mathrm{t}} )}} - \frac{{{\mathrm{b}}_{\mathrm{K}} g({\mathrm{E}}_{{{\mathrm{t}} + 1}} )}}{{m ( {\mathrm{V}}_{{{\mathrm{t}} + 1}} ,{\mathrm{F}}_{{{\mathrm{t}} + 1}} ,{\mathrm{L}}_{{{\mathrm{wt}} + 1}} ,{\mathrm{E}}_{{{\mathrm{t}} + 1}} )}} \right. \\ & \quad + \frac{{\alpha {\mathrm{nF}}_{\mathrm{t}} }} {{(1 - \alpha )(1 - \tau )[ ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{wt}} )^{\phi } - {\mathrm{nF}}_{\mathrm{t}} ]}} \\& \quad + \left. {\frac{{{\mathrm{b}}_{\mathrm{H}} (1 - \delta_{\mathrm{H}} )}}{{d ( {\mathrm{F}}_{\mathrm{t}} ,{\mathrm{L}}_{\mathrm{wt}} ,{\mathrm{E}}_{\mathrm{t}} )}}\left( { 1+ \frac{\mathrm{n}}{{{\mathrm{V}}_{\mathrm{t}} }}} \right) - \frac{{{\mathrm{b}}_{\mathrm{H}} g({\mathrm{E}}_{{{\mathrm{t}} + 1}} )}}{{d ( {\mathrm{F}}_{{{\mathrm{t}} + 1}} ,{\mathrm{L}}_{{{\mathrm{wt}} + 1}} ,{\mathrm{E}}_{{{\mathrm{t}} + 1}} )}}\left( { 1+ \frac{\mathrm{n}}{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }}} \right)} \right\} \\& \quad + \frac{{\xi \zeta q ( {\mathrm{V}}_{\mathrm{t}} ,{\mathrm{L}}_{\mathrm{wt}} ,{\mathrm{E}}_{\mathrm{t}} ) {\mathrm{L}}}}{{{\mathrm{n}}(1 - \zeta )d ( {\mathrm{F}}_{\mathrm{t}} ,{\mathrm{L}}_{\mathrm{wt}} ,{\mathrm{E}}_{\mathrm{t}} )}} + \frac{{\left( {{\mathrm{nE}}_{\mathrm{t}} } \right)^{\eta } \eta (1 - \tau )}}{{{\mathrm{n}}m ( {\mathrm{V}}_{\mathrm{t}} ,{\mathrm{F}}_{\mathrm{t}} ,{\mathrm{L}}_{\mathrm{wt}} ,{\mathrm{E}}_{\mathrm{t}} )}} \\& \quad + \frac{{ ( 1- \tau {\mathrm{b}}_{\mathrm{H}} )(1 - \delta_{\mathrm{H}} )}}{{{\mathrm{V}}_{\mathrm{t}} d ( {\mathrm{F}}_{\mathrm{t}} ,{\mathrm{L}}_{\mathrm{wt}} ,{\mathrm{E}}_{\mathrm{t}} )}} - \frac{{ ( 1- \tau {\mathrm{b}}_{\mathrm{H}} )g({\mathrm{E}}_{{{\mathrm{t}} + 1}} )}}{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} d ( {\mathrm{F}}_{{{\mathrm{t}} + 1}} ,{\mathrm{L}}_{{{\mathrm{wt}} + 1}} ,{\mathrm{E}}_{{{\mathrm{t}} + 1}} )}} \\& \quad - \frac{{\alpha [(1 - \beta - \gamma ){\mathrm{L}}_{\mathrm{wt}}^{\phi } {\mathrm{V}}_{\mathrm{t}}^{\phi - 1} + (1 - \gamma ){\mathrm{F}}_{\mathrm{t}} ]}}{{(1 - \alpha )(1 - \tau )\beta [ ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{wt}} )^{\phi } - {\mathrm{nF}}_{\mathrm{t}} ]}}\\& \quad - \frac{{q ( {\mathrm{V}}_{\mathrm{t}} ,{\mathrm{L}}_{\mathrm{wt}} ,{\mathrm{E}}_{\mathrm{t}} ) [ {\mathrm{L}}_{{{\mathrm{wt}} + 1}} - {\mathrm{L}}_{\mathrm{wt}} ( 1- \zeta ) ]}}{{(1 - \zeta )d ( {\mathrm{F}}_{\mathrm{t}} ,{\mathrm{L}}_{\mathrm{wt}} ,{\mathrm{E}}_{\mathrm{t}} )}}. \\\end{aligned}$$

• A.13 By dividing the period budget constraint of the representative capitalist by C_ct, one can obtain:

$$ {{\Omega} } ( {\mathrm{E}}_{{{\mathrm{t}} + 1}} , {\mathrm{V}}_{{{\mathrm{t}} + 1}} , {\mathrm{F}}_{{{\mathrm{t}} + 1}} , {\mathrm{L}}_{{{\mathrm{wt}} + 1}} , {\mathrm{E}}_{\mathrm{t}} , {\mathrm{V}}_{\mathrm{t}} , {\mathrm{F}}_{\mathrm{t}} , {\mathrm{L}}_{\mathrm{wt}} )= 0,\quad {\mathrm{t}} = 1,2 \ldots , $$

(A.25)

where

$$ \begin{aligned} {{\Omega} } (. )& = ( 1- \xi )\tau \left\{ \frac{{[\left( {{\mathrm{nE}}_{\mathrm{t}} } \right)^{\eta } + {\mathrm{b}}_{\mathrm{K}} (1 - \delta_{\mathrm{k}} )]}}{{m ( {\mathrm{V}}_{\mathrm{t}} ,{\mathrm{F}}_{\mathrm{t}} ,{\mathrm{L}}_{\mathrm{wt}} ,{\mathrm{E}}_{\mathrm{t}} )}} - \frac{{{\mathrm{b}}_{\mathrm{K}} g({\mathrm{E}}_{{{\mathrm{t}} + 1}} )}}{{m ( {\mathrm{V}}_{{{\mathrm{t}} + 1}} ,{\mathrm{F}}_{{{\mathrm{t}} + 1}} ,{\mathrm{L}}_{{{\mathrm{wt}} + 1}} ,{\mathrm{E}}_{{{\mathrm{t}} + 1}} )}} \right. \\ & \quad + \frac{{\alpha {\mathrm{nF}}_{\mathrm{t}} }}{{(1 - \alpha )(1 - \tau )[ ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{wt}} )^{\phi } - {\mathrm{nF}}_{\mathrm{t}} ]}} \\ & \quad + \left. {\frac{{{\mathrm{b}}_{\mathrm{H}} (1 - \delta_{\mathrm{H}} )}}{{d ( {\mathrm{F}}_{\mathrm{t}} ,{\mathrm{L}}_{\mathrm{wt}} ,{\mathrm{E}}_{\mathrm{t}} )}}\left( { 1+ \frac{\mathrm{n}}{{{\mathrm{V}}_{\mathrm{t}} }}} \right) - \frac{{{\mathrm{b}}_{\mathrm{H}} g({\mathrm{E}}_{{{\mathrm{t}} + 1}} )}}{{d ( {\mathrm{F}}_{{{\mathrm{t}} + 1}} ,{\mathrm{L}}_{{{\mathrm{wt}} + 1}} ,{\mathrm{E}}_{{{\mathrm{t}} + 1}} )}}\left( { 1+ \frac{\mathrm{n}}{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }}} \right)} \right\} \\ & \quad - \frac{{\xi \zeta q ( {\mathrm{V}}_{\mathrm{t}} ,{\mathrm{L}}_{\mathrm{wt}} ,{\mathrm{E}}_{\mathrm{t}} ) {\mathrm{L}}}}{{(1 - \zeta )d ( {\mathrm{F}}_{\mathrm{t}} ,{\mathrm{L}}_{\mathrm{wt}} ,{\mathrm{E}}_{\mathrm{t}} )}} + \frac{{\left( {{\mathrm{nE}}_{\mathrm{t}} } \right)^{\eta } ( 1- \eta )(1 - \tau )}}{{m ( {\mathrm{V}}_{\mathrm{t}} ,{\mathrm{F}}_{\mathrm{t}} ,{\mathrm{L}}_{\mathrm{wt}} , {\mathrm{E}}_{\mathrm{t}} )}} \\ & \quad + \frac{{ ( 1- \tau {\mathrm{b}}_{\mathrm{H}} )(1 - \delta_{\mathrm{H}} )}}{{d ( {\mathrm{F}}_{\mathrm{t}} ,{\mathrm{L}}_{\mathrm{wt}} ,{\mathrm{E}}_{\mathrm{t}} )}} - \frac{{ ( 1- \tau {\mathrm{b}}_{\mathrm{H}} )g({\mathrm{E}}_{{{\mathrm{t}} + 1}} )}}{{d ( {\mathrm{F}}_{{{\mathrm{t}} + 1}} ,{\mathrm{L}}_{{{\mathrm{wt}} + 1}} ,{\mathrm{E}}_{{{\mathrm{t}} + 1}} )}} + \frac{{\alpha {\mathrm{nF}}_{\mathrm{t}} }}{{(1 - \alpha )[ ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{wt}} )^{\phi } - {\mathrm{nF}}_{\mathrm{t}} ]}} \\ & \quad + \frac{{q ( {\mathrm{V}}_{\mathrm{t}} , {\mathrm{L}}_{\mathrm{wt}} , {\mathrm{E}}_{\mathrm{t}} ) {\mathrm{n[L}}_{{{\mathrm{wt}} + 1}} - {\mathrm{L}}_{\mathrm{wt}} ( 1- \zeta ) ]}}{{(1 - \zeta )d ( {\mathrm{F}}_{\mathrm{t}} ,{\mathrm{L}}_{\mathrm{wt}} ,{\mathrm{E}}_{\mathrm{t}} )}} \\ & \quad - 1- \frac{{ ( 1- \tau {\mathrm{b}}_{\mathrm{K}} )g({\mathrm{E}}_{{{\mathrm{t}} + 1}} )}}{{m ( {\mathrm{V}}_{{{\mathrm{t}} + 1}} ,{\mathrm{F}}_{{{\mathrm{t}} + 1}} ,{\mathrm{L}}_{{{\mathrm{wt}} + 1}} ,{\mathrm{E}}_{{{\mathrm{t}} + 1}} )}} + \frac{{ ( 1- \tau {\mathrm{b}}_{\mathrm{K}} )(1 - \delta_{\mathrm{K}} )}}{{m ( {\mathrm{V}}_{\mathrm{t}} ,{\mathrm{F}}_{\mathrm{t}} ,{\mathrm{L}}_{\mathrm{wt}} ,{\mathrm{E}}_{\mathrm{t}} )}}. \\ \end{aligned} $$

• A.14 From (A.6), one can obtain:

$$ \Gamma ( {\mathrm{E}}_{{{\mathrm{t}} + 1}} , {\mathrm{V}}_{{{\mathrm{t}} + 1}} , {\mathrm{F}}_{{{\mathrm{t}} + 1}} , {\mathrm{L}}_{{{\mathrm{wt}} + 1}} , {\mathrm{E}}_{\mathrm{t}} , {\mathrm{V}}_{\mathrm{t}} , {\mathrm{F}}_{\mathrm{t}} , {\mathrm{L}}_{\mathrm{wt}} )= 0,\quad {\mathrm{t}} = 1,2 \ldots , $$

(A.26)

where

$$ \begin{aligned}\Gamma (.) & = \frac{{\rho_{\mathrm{w}} \beta \phi {\mathrm{L}}_{{{\mathrm{wt}} + 1}}^{\phi - 1} {\mathrm{V}}_{{{\mathrm{t}} + 1}}^{\phi - 1} }}{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}}^{\phi } {\mathrm{V}}_{{{\mathrm{t}} + 1}}^{\phi - 1} + {\mathrm{F}}_{{{\mathrm{t}} + 1}} }} \\ & + \frac{{\rho_{\mathrm{w}} ( 1- \alpha )\beta (1 - \tau )q ( {\mathrm{V}}_{{{\mathrm{t}} + 1}} , {\mathrm{L}}_{{{\mathrm{wt}} + 1}} , {\mathrm{E}}_{{{\mathrm{t}} + 1}} )[ ( {\mathrm{L}} - {\mathrm{nL}}_{{{\mathrm{wt}} + 1}} )^{\phi } - {\mathrm{nF}}_{{{\mathrm{t}} + 1}} ]}}{{\alpha ({\mathrm{L}}_{{{\mathrm{wt}} + 1}}^{\phi } {\mathrm{V}}_{{{\mathrm{t}} + 1}}^{\phi - 1} + {\mathrm{F}}_{{{\mathrm{t}} + 1}} )d ( {\mathrm{F}}_{{{\mathrm{t}} + 1}} , {\mathrm{L}}_{{{\mathrm{wt}} + 1}} , {\mathrm{E}}_{{{\mathrm{t}} + 1}} )}} \\ & \quad - \frac{{( 1- \alpha )\beta (1 - \tau )q ( {\mathrm{V}}_{\mathrm{t}} , {\mathrm{L}}_{\mathrm{wt}} , {\mathrm{E}}_{\mathrm{t}} )[ ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{wt}} )^{\phi } - {\mathrm{nF}}_{\mathrm{t}} ]}}{{\alpha ({\mathrm{L}}_{\mathrm{wt}}^{\phi } {\mathrm{V}}_{\mathrm{t}}^{\phi - 1} + {\mathrm{F}}_{\mathrm{t}} )d ( {\mathrm{F}}_{\mathrm{t}} , {\mathrm{L}}_{\mathrm{wt}} , {\mathrm{E}}_{\mathrm{t}} )(1 - \zeta )}}. \\ \end{aligned} $$

• A.15 From (A.7), one can obtain:

$$ \Lambda ( {\mathrm{E}}_{{{\mathrm{t}} + 1}} , {\mathrm{V}}_{{{\mathrm{t}} + 1}} , {\mathrm{F}}_{{{\mathrm{t}} + 1}} , {\mathrm{L}}_{{{\mathrm{wt}} + 1}} , {\mathrm{V}}_{\mathrm{t}} , {\mathrm{F}}_{\mathrm{t}} , {\mathrm{L}}_{\mathrm{wt}} )= 0,\quad {\mathrm{t}} = 1,2 \ldots , $$

(A.27)

where

$$ \begin{aligned}\Lambda (.) & = \frac{{\rho_{\mathrm{w}} \alpha (1 - \phi ){\mathrm{L}}_{{{\mathrm{wt}} + 1}}^{\phi } {\mathrm{V}}_{{{\mathrm{t}} + 1}}^{\phi } d ( {\mathrm{F}}_{{{\mathrm{t}} + 1}} , {\mathrm{L}}_{{{\mathrm{wt}} + 1}} , {\mathrm{E}}_{{{\mathrm{t}} + 1}} )}}{{( 1- \alpha )(1 - \tau )[ ( {\mathrm{L}} - {\mathrm{nL}}_{{{\mathrm{wt}} + 1}} )^{\phi } - {\mathrm{nF}}_{{{\mathrm{t}} + 1}} ]}} + \rho_{\mathrm{w}} ( 1- \tau {\mathrm{b}}_{\mathrm{H}} )(1 - \delta_{\mathrm{H}} ) \\ & \quad - \frac{{(1 - \tau {\mathrm{b}}_{\mathrm{H}} )({\mathrm{L}}_{{{\mathrm{wt}} + 1}}^{\phi } {\mathrm{V}}_{{{\mathrm{t}} + 1}}^{\phi - 1} + {\mathrm{F}}_{{{\mathrm{t}} + 1}} )[ ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{wt}} )^{\phi } - {\mathrm{nF}}_{\mathrm{t}} ]g({\mathrm{E}}_{{{\mathrm{t}} + 1}} )}}{{({\mathrm{L}}_{\mathrm{wt}}^{\phi } {\mathrm{V}}_{\mathrm{t}}^{\phi - 1} + {\mathrm{F}}_{\mathrm{t}} )[ ( {\mathrm{L}} - {\mathrm{nL}}_{{{\mathrm{wt}} + 1}} )^{\phi } - {\mathrm{nF}}_{{{\mathrm{t}} + 1}} ]}}. \\ \end{aligned} $$

Notice that (A.24)–(A.27) is a system of difference equations in ${\mathrm{E}}_{\mathrm{t}} , {\mathrm{V}}_{\mathrm{t}} , {\mathrm{F}}_{\mathrm{t}}$ and ${\mathrm{L}}_{\mathrm{wt}}$ that governs the equilibrium path of the economy.

1.2 B Balanced Growth Path (BGP)

• B.1 By setting ${\mathrm{E}}_{{{\mathrm{t}} + 1}} = {\mathrm{E}}_{\mathrm{t}} = {\mathrm{E}}$, ${\mathrm{V}}_{{{\mathrm{t}} + 1}} = {\mathrm{V}}_{\mathrm{t}} = {\mathrm{V}}$, ${\mathrm{F}}_{{{\mathrm{t}} + 1}} = {\mathrm{F}}_{\mathrm{t}} = {\mathrm{F}}$ and ${\mathrm{L}}_{{{\mathrm{wt}} + 1}} = {\mathrm{L}}_{\mathrm{wt}} = {\mathrm{L}}_{\mathrm{w}}$, Eq. (A.27) becomes

$$ \frac{{\rho_{\mathrm{w}} {\mathrm{L}}_{\mathrm{w}}^{\phi } {\mathrm{V}}^{\phi } [ ( 1- \eta ) ( 1- \tau ) ( {\mathrm{nE)}}^{\eta } - ( 1- \tau {\mathrm{b}}_{\mathrm{K}} ) (\delta_{\mathrm{K}} - \delta_{\mathrm{H}} )]}}{{ ( 1- \tau )(1 - \tau {\mathrm{b}}_{\mathrm{K}} ) ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}} )^{\phi } }} + \rho_{\mathrm{w}} (1 - \delta_{\mathrm{H}} ) - g({\mathrm{E)}} = 0, $$

from which one can obtain

$$ {\mathrm{V}} = [\chi ( {\mathrm{E)]}}^{{\frac{ 1}{\phi }}} \frac{{ ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}} )}}{{{\mathrm{L}}_{\mathrm{w}} }}, $$

(B.1)

where $\chi ( {\mathrm{E)}} = \frac{{ [g({\mathrm{E)}} - \rho_{\mathrm{w}} (1 - \delta_{\mathrm{H}} )](1 - \tau {\mathrm{b}}_{\mathrm{K}} )(1 - \tau )}}{{\rho_{\mathrm{w}} [ ( 1- \eta ) ( 1- \tau ) ( {\mathrm{nE)}}^{\eta } - ( 1- \tau {\mathrm{b}}_{\mathrm{K}} ) (\delta_{\mathrm{K}} - \delta_{\mathrm{H}} )]}}.$

• B.2 By setting ${\mathrm{E}}_{{{\mathrm{t}} + 1}} = {\mathrm{E}}_{\mathrm{t}} = {\mathrm{E}}$, ${\mathrm{F}}_{{{\mathrm{t}} + 1}} = {\mathrm{F}}_{\mathrm{t}} = {\mathrm{F}}$, ${\mathrm{V}}_{{{\mathrm{t}} + 1}} = {\mathrm{V}}_{\mathrm{t}} = {\mathrm{V}}$ and ${\mathrm{L}}_{{{\mathrm{wt}} + 1}} = {\mathrm{L}}_{\mathrm{wt}} = {\mathrm{L}}_{\mathrm{w}}$, Eq. (A.26) becomes

$$ \begin{aligned} & \rho_{\mathrm{w}} {\mathrm{L}}_{\mathrm{w}}^{\phi - 1} {\mathrm{V}}^{\phi - 1} \\ & \quad - \frac{{\left[ {\frac{1}{(1 - \zeta )} - \rho_{\mathrm{w}} } \right](1 - \tau ) ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}} )^{\phi } }}{{[(1 - \eta )(1 - \tau )({\mathrm{nE)}}^{\eta } - ( 1- \tau {\mathrm{b}}_{\mathrm{K}} ) (\delta_{\mathrm{K}} - \delta_{\mathrm{H}} )] [ ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}} )^{\phi } {\mathrm{V}}^{ - \phi } (1 - \tau ) - {\mathrm{L}}_{\mathrm{w}}^{\phi } ]}}\\ & \quad \left\{ \frac{{( 1- {\mathrm{b}}_{\mathrm{k}} \tau )(1 - \tau ) ( 1- \delta_{\mathrm{H}} )}}{{ ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{wt}} )^{1 - \phi } {\mathrm{V}}^{\phi } }} \right. \\ & \quad + \frac{{[(1 - \eta )(1 - \tau )({\mathrm{nE)}}^{\eta } - ( 1- \tau {\mathrm{b}}_{\mathrm{K}} ) (\delta_{\mathrm{K}} - \delta_{\mathrm{H}} )]{\mathrm{L}}_{\mathrm{w}}^{\phi } }}{{ ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}} )}} \\ & \quad \left. - \frac{{( 1- {\mathrm{b}}_{\mathrm{k}} \tau )g ( {\mathrm{E)}}}}{{\rho_{\mathrm{c}} {\mathrm{VL}}_{\mathrm{w}}^{1 - \phi } }} \right\} = 0, \\ \end{aligned} $$

which—by using (B.1) to substitute for V—can be rewritten as

$$ e ( {\mathrm{E)}} = 0, $$

(B.2)

where

$$ \begin{aligned} e({\mathrm{E)}} & = \rho_{\mathrm{w}} [(1 - \eta )({\mathrm{nE)}}^{\eta } (1 - \tau ) - ( 1- {\mathrm{b}}_{\mathrm{k}} \tau ) (\delta_{\mathrm{K}} - \delta_{\mathrm{H}} ) ] \\ & \quad - [g({\mathrm{E)}} - \rho_{\mathrm{w}} ( 1- \delta_{\mathrm{H}} ) ]( 1- {\mathrm{b}}_{\mathrm{k}} \tau ) \\ & \quad + \left[ {\frac{1}{(1 - \zeta )} - \rho_{\mathrm{w}} } \right]\left\{ \frac{{( 1- {\mathrm{b}}_{\mathrm{k}} \tau )}}{{\rho_{\mathrm{c}} }} \right. \\ & \left. \quad \quad - \frac{{ [(1 - \eta )({\mathrm{nE)}}^{\eta } (1 - \tau ) - ( 1- {\mathrm{b}}_{\mathrm{k}} \tau ) (\delta_{\mathrm{K}} - \delta_{\mathrm{H}} ) ][\chi ({\mathrm{E}})]^{{\frac{1}{\phi }}} }}{{[g ( {\mathrm{E)}} - \rho_{\mathrm{w}} ( 1- \delta_{\mathrm{H}} ) ]}} \right\}g ( {\mathrm{E)}} \\ \end{aligned} $$

Notice that any value of E satisfying (B.2) and such that $g({\mathrm{E)}} > \rho_{\mathrm{w}} ( 1- \delta_{\mathrm{H}} )$, say E*, is a BGP value of E_t.

• B.3 By setting ${\mathrm{E}}_{{{\mathrm{t}} + 1}} = {\mathrm{E}}_{\mathrm{t}} = {\mathrm{E}}$, ${\mathrm{F}}_{{{\mathrm{t}} + 1}} = {\mathrm{F}}_{\mathrm{t}} = {\mathrm{F}}$, ${\mathrm{V}}_{{{\mathrm{t}} + 1}} = {\mathrm{V}}_{\mathrm{t}} = {\mathrm{V}}$ and ${\mathrm{L}}_{{{\mathrm{wt}} + 1}} = {\mathrm{L}}_{\mathrm{wt}} = {\mathrm{L}}_{\mathrm{w}}$, Eq. (A.24) becomes

$$ \begin{aligned} & \frac{{\xi \tau \alpha \gamma ({\mathrm{L}}_{\mathrm{w}}^{\phi } {\mathrm{V}}^{\phi - 1} + {\mathrm{F}})[\left( {\mathrm{nE}} \right)^{\eta } - {\mathrm{b}}_{\mathrm{K}} g({\mathrm{E}}) + {\mathrm{b}}_{\mathrm{K}} (1 - \delta_{\mathrm{K}} )]}}{{{\mathrm{n}}(1 - \alpha )(1 - \tau )^{2} \beta \eta \left( {\mathrm{nE}} \right)^{\eta - 1} (1 - {\mathrm{E}})}} \\ & \quad + \frac{{\xi \tau \alpha {\mathrm{F}}}}{(1 - \alpha )(1 - \tau )} - \left( {\frac{\mathrm{n}}{\mathrm{V}} + 1} \right)[g({\mathrm{E}}) - 1+ \delta_{\mathrm{H}} ] \\ & \quad \times \frac{{\xi \tau \alpha ({\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}} )^{\phi } (1 - \phi )(1 - \tau {\mathrm{b}}_{\mathrm{K}} ){\mathrm{b}}_{\mathrm{H}} (1 - \tau {\mathrm{b}}_{\mathrm{H}} )^{ - 1} }}{{{\mathrm{n}}(1 - \alpha )[(1 - \tau )(1 - \eta )\left( {\mathrm{nE}} \right)^{\eta } - (1 - \tau {\mathrm{b}}_{\mathrm{K}} )(\delta_{\mathrm{K}} - \delta_{\mathrm{H}} )]}} \\ & \quad - \frac{{\alpha [(1 - \beta )(1 - {\mathrm{E)}} - \gamma ]({\mathrm{L}}_{\mathrm{w}}^{\phi } {\mathrm{V}}^{\phi - 1} + {\mathrm{F}})}}{{(1 - \alpha )(1 - \tau )\beta (1 - {\mathrm{E}})}} - \frac{{\alpha {\mathrm{F}}}}{(1 - \alpha )(1 - \tau )} \\ &\quad - \frac{{\alpha ({\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}} )^{\phi } (1 - \phi )(1 - \tau {\mathrm{b}}_{\mathrm{K}} )(1 - \alpha )^{ - 1} }}{{[(1 - \tau )(1 - \eta )\left( {\mathrm{nE}} \right)^{\eta } - (1 - \tau {\mathrm{b}}_{\mathrm{K}} )(\delta_{\mathrm{K}} - \delta_{\mathrm{H}} )]}} \\ & \quad \left\{ {\frac{{[g ( {\mathrm{E)}} - 1+ \delta_{\mathrm{H}} ]}}{\mathrm{V}} - \frac{{\zeta q ( {\mathrm{V,L}}_{\mathrm{w}} ,{\mathrm{E)}}}}{{(1 - \zeta )(1 - \tau {\mathrm{b}}_{\mathrm{H}} )}}\left( {\frac{{\xi {\mathrm{L}}}}{\mathrm{n}} - {\mathrm{L}}_{\mathrm{w}} } \right)} \right\} = 0, \\ \end{aligned} $$

from which (by using (B.1) to substitute for V and by setting E = E*) one can obtain

$$ {\mathrm{F}} = z({\mathrm{L}}_{\mathrm{w}} ,{\mathrm{E*}}), $$

(B.3)

where

$$ \begin{aligned} z(.) & = \left\{ {\frac{{\alpha {\mathrm{L}}_{\mathrm{w}} [\chi ({\mathrm{E*}})]^{{\frac{\phi - 1}{\phi }}} [ ( 1- \beta )(1 - {\mathrm{E*}}) - \gamma ]}}{{\beta (1 - {\mathrm{E*}})(1 - \tau )({\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}} )^{1 - \phi } }} - \frac{{\xi \tau \alpha \gamma {\mathrm{L}}_{\mathrm{w}} [\chi ({\mathrm{E*}})]^{{\frac{\phi - 1}{\phi }}} [\left( {\mathrm{nE*}} \right)^{\eta } - {\mathrm{b}}_{\mathrm{K}} g({\mathrm{E*}}) + {\mathrm{b}}_{\mathrm{K}} (1 - \delta_{\mathrm{K}} )]}}{{{\mathrm{n}}(1 - \tau )^{2} \beta \eta \left( {\mathrm{nE*}} \right)^{\eta - 1} (1 - {\mathrm{E*}})({\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}} )^{1 - \phi } }}} \right. \\ & \quad + \left[ {\frac{{\xi \tau {\mathrm{b}}_{\mathrm{H}} }}{{{\mathrm{n}}(1 - \tau {\mathrm{b}}_{\mathrm{H}} )}}\left( {\frac{{{\mathrm{nL}}_{\mathrm{w}} [\chi ( {\mathrm{E*)}}]^{{ - \frac{1}{\phi }}} }}{{({\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}} )}} + 1} \right) + \frac{{{\mathrm{L}}_{\mathrm{w}} [\chi ( {\mathrm{E*)}}]^{{ - \frac{1}{\phi }}} }}{{({\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}} )}}} \right]\frac{{\alpha ({\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}} )^{\phi } (1 - \phi )(1 - \tau {\mathrm{b}}_{\mathrm{K}} )[g({\mathrm{E}}* )- 1+ \delta_{\mathrm{H}} ]}}{{[(1 - \tau )(1 - \eta )\left( {\mathrm{nE*}} \right)^{\eta } - (1 - \tau {\mathrm{b}}_{\mathrm{K}} )(\delta_{\mathrm{K}} - \delta_{\mathrm{H}} )]}} \\ & \quad + \frac{{\zeta \alpha \phi (\xi {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}} )\left[ {(1 - \tau )\left( {\mathrm{nE*}} \right)^{\eta } (1 - \eta ) - [\chi ({\mathrm{E*}})]^{{\frac{1 - \phi }{\phi }}} (1 - \tau )(1 - {\mathrm{b}}_{\mathrm{K}} \tau ) ( 1- \delta_{\mathrm{H}} )+ ( 1- \delta_{\mathrm{K}} )(1 - {\mathrm{b}}_{\mathrm{K}} \tau )} \right]}}{{ ( 1- \zeta ) {\mathrm{n}}[\chi ({\mathrm{E*}})]^{{\frac{ 1}{\phi }}} [(1 - \tau )(1 - \eta )\left( {\mathrm{nE*}} \right)^{\eta } - (1 - \tau {\mathrm{b}}_{\mathrm{K}} )(\delta_{\mathrm{K}} - \delta_{\mathrm{H}} )][[\chi ({\mathrm{E*}})]^{ - 1} (1 - \tau ) - 1]({\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}} )^{1 - \phi } }} \\ & \quad \left. { - \frac{{\zeta \alpha \phi (\xi {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}} )({\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}} )^{\phi - 1} }}{{ ( 1- \zeta ) {\mathrm{n}}[[\chi ({\mathrm{E*}})]^{ - 1} (1 - \tau ) - 1]}}} \right\}\left\{ {\frac{{\xi \tau \alpha \gamma [\left( {\mathrm{nE*}} \right)^{\eta } - {\mathrm{b}}_{\mathrm{K}} g({\mathrm{E*}}) + {\mathrm{b}}_{\mathrm{K}} (1 - \delta_{\mathrm{K}} )]}}{{{\mathrm{n}}(1 - \tau )^{2} \beta \eta \left( {\mathrm{nE*}} \right)^{\eta - 1} (1 - {\mathrm{E*}})}} + \frac{\xi \tau \alpha }{(1 - \tau )} - \frac{{\alpha ( 1- \gamma - {\mathrm{E*)}}}}{{\beta (1 - \tau ) ( 1- {\mathrm{E*)}}}}} \right\}^{ - 1} . \\ \end{aligned} $$

• B.4 By setting ${\mathrm{E}}_{{{\mathrm{t}} + 1}} = {\mathrm{E}}_{\mathrm{t}} = {\mathrm{E}}$, ${\mathrm{F}}_{{{\mathrm{t}} + 1}} = {\mathrm{F}}_{\mathrm{t}} = {\mathrm{F}}$, ${\mathrm{V}}_{{{\mathrm{t}} + 1}} = {\mathrm{V}}_{\mathrm{t}} = {\mathrm{V}}$ and ${\mathrm{L}}_{{{\mathrm{wt}} + 1}} = {\mathrm{L}}_{\mathrm{wt}} = {\mathrm{L}}_{\mathrm{w}}$, Eq. (A.25) becomes

$$ \begin{aligned} & \frac{\mathrm{nF}}{(1 - \alpha )} + \frac{{(1 - \xi )\tau \alpha \gamma ({\mathrm{L}}_{\mathrm{w}}^{\phi } {\mathrm{V}}^{\phi - 1} + {\mathrm{F}})[\left( {\mathrm{nE}} \right)^{\eta } - {\mathrm{b}}_{\mathrm{K}} g({\mathrm{E}}) + {\mathrm{b}}_{\mathrm{K}} (1 - \delta_{\mathrm{K}} )]}}{{(1{ - }\alpha )(1{ - }\tau )^{2} \beta \eta \left( {\mathrm{nE}} \right)^{{\eta { - 1}}} (1- {\mathrm{E}})}} \\ & \quad + \frac{{( 1- \xi ){\mathrm{n}}\tau \alpha {\mathrm{F}}}}{(1 - \alpha )(1 - \tau )} - ({\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}} )^{\phi } - \left( {\frac{\mathrm{n}}{\mathrm{V}} + 1} \right) \\ & \quad \times \frac{{[g({\mathrm{E}}) - 1+ \delta_{\mathrm{H}} ](1 - \xi )\tau \alpha ({\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}} )^{\phi } (1 - \phi )(1 - \tau {\mathrm{b}}_{\mathrm{K}} ){\mathrm{b}}_{\mathrm{H}} }}{{(1 - \alpha )(1 - \tau {\mathrm{b}}_{\mathrm{H}} )[(1 - \tau )(1 - \eta )\left( {\mathrm{nE}} \right)^{\eta } - (1 - \tau {\mathrm{b}}_{\mathrm{K}} )(\delta_{\mathrm{K}} - \delta_{\mathrm{H}} )]}} \\ & \quad - \frac{{\zeta \alpha ({\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}} )^{\phi } (1 - \phi )(1 - \tau {\mathrm{b}}_{\mathrm{K}} )(\xi {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}} )q ( {\mathrm{V,L}}_{\mathrm{w}} , {\mathrm{E*)}}}}{{(1 - \zeta )(1 - \tau {\mathrm{b}}_{\mathrm{H}} )(1 - \alpha )[(1 - \tau )(1 - \eta )\left( {\mathrm{nE}} \right)^{\eta } - (1 - \tau {\mathrm{b}}_{\mathrm{K}} )(\delta_{\mathrm{K}} - \delta_{\mathrm{H}} )]}} \\ & \quad - \frac{{\alpha (1 - \tau {\mathrm{b}}_{\mathrm{K}} )[g({\mathrm{E}}) - 1+ \delta_{\mathrm{H}} ]({\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}} )^{\phi } (1 - \phi )}}{{(1 - \alpha )[(1 - \tau )(1 - \eta )\left( {\mathrm{nE}} \right)^{\eta } - (1 - \tau {\mathrm{b}}_{\mathrm{K}} )(\delta_{\mathrm{K}} - \delta_{\mathrm{H}} )]}} \\ & \quad - \frac{{({\mathrm{L}}_{\mathrm{w}}^{\phi } {\mathrm{V}}^{\phi - 1} + {\mathrm{F}})(1 - \tau {\mathrm{b}}_{\mathrm{K}} )[g({\mathrm{E}}) - 1+ \delta_{\mathrm{K}} ]}}{{(\alpha \gamma )^{ - 1} \beta \eta (1 - \alpha )(1 - \tau )^{2} (1 - {\mathrm{E}})\left( {\mathrm{nE}} \right)^{{\eta { - 1}}} }} + \frac{{\alpha \gamma ({\mathrm{L}}_{\mathrm{w}}^{\phi } {\mathrm{V}}^{\phi - 1} + {\mathrm{F}})(1 - \eta ){\mathrm{nE}}}}{{(1 - \alpha )(1 - \tau )\eta \beta (1 - {\mathrm{E}})}} = 0, \\ \end{aligned} $$

from which (by using (B.1) to substitute for V and by setting E = E*) one can obtain

$$ {\mathrm{F}} = p({\mathrm{L}}_{\mathrm{w}} ,{\mathrm{E*}}), $$

(B.4)

where

$$ \begin{aligned} p(.) & = \left\{ {\left[ {\frac{{(1 - \xi )\tau {\mathrm{b}}_{\mathrm{H}} }}{{(1 - \tau {\mathrm{b}}_{\mathrm{H}} )}}\left( {\frac{{{\mathrm{nL}}_{\mathrm{w}} [\chi ( {\mathrm{E*)}}]^{{ - \frac{1}{\phi }}} }}{{({\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}} )}} + 1} \right) + 1} \right] \frac{{({\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}} )^{\phi } \alpha (1 - \phi )(1 - \tau {\mathrm{b}}_{\mathrm{K}} )[g({\mathrm{E}}* )- 1+ \delta_{\mathrm{H}} ]}}{{[(1 - \tau )(1 - \eta )\left( {\mathrm{nE}} \right)^{\eta } - (1 - \tau {\mathrm{b}}_{\mathrm{K}} )(\delta_{\mathrm{K}} - \delta_{\mathrm{H}} )]}}} \right. \\ & \quad - \frac{{\zeta \alpha \phi (\xi {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}} )\left[ {(1 - \tau )\left( {\mathrm{nE*}} \right)^{\eta } (1 - \eta ) - [\chi ({\mathrm{E*}})]^{{\frac{1 - \phi }{\phi }}} (1 - \tau )(1 - {\mathrm{b}}_{\mathrm{K}} \tau ) ( 1- \delta_{\mathrm{H}} )+ ( 1- \delta_{\mathrm{K}} )(1 - {\mathrm{b}}_{\mathrm{K}} \tau )} \right]}}{{ ( 1- \zeta )[\chi ({\mathrm{E*}})]^{{\frac{ 1}{\phi }}} [(1 - \tau )(1 - \eta )\left( {\mathrm{nE*}} \right)^{\eta } - (1 - \tau {\mathrm{b}}_{\mathrm{K}} )(\delta_{\mathrm{K}} - \delta_{\mathrm{H}} )][[\chi ({\mathrm{E*}})]^{ - 1} (1 - \tau ) - 1]({\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}} )^{1 - \phi } }} \\ & \quad + \frac{{\zeta \alpha \phi (\xi {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}} )({\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}} )^{\phi - 1} }}{{ ( 1- \zeta )[[\chi ({\mathrm{E*}})]^{ - 1} (1 - \tau ) - 1]}} - \frac{{(1 - \xi )\tau \alpha \gamma {\mathrm{L}}_{\mathrm{w}} [\chi ({\mathrm{E*}})]^{{\frac{\phi - 1}{\phi }}} [\left( {\mathrm{nE*}} \right)^{\eta } - {\mathrm{b}}_{\mathrm{K}} g({\mathrm{E*}}) + {\mathrm{b}}_{\mathrm{K}} (1 - \delta_{\mathrm{K}} )]}}{{(1 - \tau )^{2} \beta \eta \left( {\mathrm{nE*}} \right)^{\eta - 1} (1 - {\mathrm{E*}})({\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}} )^{1 - \phi } }} \\ & \quad + \left. {\frac{{\alpha \gamma {\mathrm{L}}_{\mathrm{w}} (1 - \tau {\mathrm{b}}_{\mathrm{K}} )[\chi ({\mathrm{E}})]^{{\frac{\phi - 1}{\phi }}} [g({\mathrm{E*}}) - 1+ \delta_{\mathrm{K}} ]}}{{(1 - \tau )^{2} \beta \eta \left( {\mathrm{nE*}} \right)^{{\eta { - 1}}} (1 - {\mathrm{E*}})({\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}} )^{1 - \phi } }} + \frac{(1 - \alpha )}{{({\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}} )^{ - \phi } }} - \frac{{\alpha \gamma (1 - \eta ){\mathrm{L}}_{\mathrm{w}} {\mathrm{nE*}}[\chi ({\mathrm{E*}})]^{{\frac{\phi - 1}{\phi }}} }}{{\beta \eta (1 - {\mathrm{E*}})(1 - \tau )({\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}} )^{1 - \phi } }}} \right\} \\ & \left\{ \frac{{(1 - \xi ){\mathrm{n}}\tau \alpha }}{(1 - \tau )} + \frac{{(1 - \xi )\tau \alpha \gamma [\left( {\mathrm{nE*}} \right)^{\eta } - {\mathrm{b}}_{\mathrm{K}} g({\mathrm{E*}}) + {\mathrm{b}}_{\mathrm{K}} (1 - \delta_{\mathrm{K}} )]}}{{(1 - \tau )^{2} \beta \eta \left( {\mathrm{nE*}} \right)^{\eta - 1} (1 - {\mathrm{E*}})}} + \frac{{\alpha \gamma {\mathrm{nE*(1}} - \eta )}}{{\eta \beta ( 1- {\mathrm{E*)(1}} - \tau )}} \right. \\ & \quad + \left. {{\mathrm{n}} - \frac{{\alpha \gamma (1 - \tau {\mathrm{b}}_{\mathrm{K}} )[g({\mathrm{E*)}} - 1+ \delta_{\mathrm{K}} ]}}{{\beta \eta (1 - \tau )^{ 2} (1 - {\mathrm{E*}})({\mathrm{nE*}})^{\eta - 1} }}} \right\}^{ - 1} . \\ \end{aligned} $$

• B.5 By setting z(L_w, E*) = p(L_w, E*), one can solve for the BGP value of L_wt:

$$ {\mathrm{L}}_{\mathrm{w}}^{ *} = {\mathrm{L}}l ( {\mathrm{E*),}} $$

(B.5)

where

$$ \begin{aligned} l({\mathrm{E*)}} & = \left\{ {\left\{ {\frac{{\alpha (1 - \phi )(1 - \tau {\mathrm{b}}_{\mathrm{K}} )[g({\mathrm{E}}* )- 1+ \delta_{\mathrm{H}} ]}}{{[(1 - \tau )(1 - \eta )\left( {\mathrm{nE}} \right)^{\eta } - (1 - \tau {\mathrm{b}}_{\mathrm{K}} )(\delta_{\mathrm{K}} - \delta_{\mathrm{H}} )]}}\left[ {\frac{{\tau (1 - \xi ) {\mathrm{b}}_{\mathrm{H}} }}{{(1 - \tau {\mathrm{b}}_{\mathrm{H}} )}} + 1} \right] + \frac{\zeta \alpha \phi \xi }{{ ( 1- \zeta )[[\chi ({\mathrm{E*}})]^{ - 1} (1 - \tau ) - 1]}}} \right.} \right. \\ & \quad \left. { - \frac{{\zeta \alpha \phi \xi \left[ {(1 - \tau )\left( {\mathrm{nE*}} \right)^{\eta } (1 - \eta ) - [\chi ({\mathrm{E*}})]^{{\frac{1 - \phi }{\phi }}} (1 - \tau )(1 - {\mathrm{b}}_{\mathrm{K}} \tau ) ( 1- \delta_{\mathrm{H}} )+ ( 1- \delta_{\mathrm{K}} )(1 - {\mathrm{b}}_{\mathrm{K}} \tau )} \right]}}{{ ( 1- \zeta )[\chi ({\mathrm{E*}})]^{{\frac{ 1}{\phi }}} [(1 - \tau )(1 - \eta )\left( {\mathrm{nE*}} \right)^{\eta } - (1 - \tau {\mathrm{b}}_{\mathrm{K}} )(\delta_{\mathrm{K}} - \delta_{\mathrm{H}} )][[\chi ({\mathrm{E*}})]^{ - 1} (1 - \tau ) - 1]}} + 1 - \alpha } \right\} \\ & \qquad \left\{ \frac{{\alpha \gamma {\mathrm{nE*(1}} - \eta )}}{{\eta \beta ( 1- {\mathrm{E*)(1}} - \tau )}} + \frac{{(1 - \xi )\tau \alpha \gamma [\left( {\mathrm{nE*}} \right)^{\eta } - {\mathrm{b}}_{\mathrm{K}} g({\mathrm{E*}}) + {\mathrm{b}}_{\mathrm{K}} (1 - \delta_{\mathrm{K}} )]}}{{(1 - \tau )^{2} \beta \eta \left( {\mathrm{nE*}} \right)^{\eta - 1} (1 - {\mathrm{E*}})}}\right. \\ & \qquad \left. { + \frac{{(1 - \xi ){\mathrm{n}}\tau \alpha }}{ ( 1- \tau )} + {\mathrm{n}} - \frac{{\alpha \gamma (1 - \tau {\mathrm{b}}_{\mathrm{K}} )[g({\mathrm{E*)}} - 1+ \delta_{\mathrm{K}} ]}}{{\beta \eta (1 - \tau )^{ 2} (1 - {\mathrm{E*}})({\mathrm{nE*}})^{\eta - 1} }}} \right\}^{ - 1} \\ \end{aligned} $$

$$\begin{aligned} &- \left\{ {\frac{{\zeta \alpha \phi \xi \left[ {(1 - \tau )\left( {\mathrm{nE*}} \right)^{\eta } (1 - \eta ) - [\chi ({\mathrm{E*}})]^{{\frac{1 - \phi }{\phi }}} (1 - \tau )(1 - {\mathrm{b}}_{\mathrm{K}} \tau ) ( 1- \delta_{\mathrm{H}} )+ ( 1- \delta_{\mathrm{K}} )(1 - {\mathrm{b}}_{\mathrm{K}} \tau )} \right]}}{{ ( 1- \zeta ) {\mathrm{n}}[\chi ({\mathrm{E*}})]^{{\frac{ 1}{\phi }}} [(1 - \tau )(1 - \eta )\left( {\mathrm{nE*}} \right)^{\eta } - (1 - \tau {\mathrm{b}}_{\mathrm{K}} )(\delta_{\mathrm{K}} - \delta_{\mathrm{H}} )][[\chi ({\mathrm{E*}})]^{ - 1} (1 - \tau ) - 1]}}} \right.\\ & - \frac{{\zeta \alpha - \xi [ ( 1- \zeta ) {\mathrm{n]}}^{ - 1} }}{{[[\chi ({\mathrm{E*}})]^{ - 1} (1 - \tau ) - 1]}} + \left. {\frac{{\xi \tau {\mathrm{b}}_{\mathrm{H}} \alpha (1 - \phi )(1 - \tau {\mathrm{b}}_{\mathrm{K}} )[g({\mathrm{E}}* )- 1+ \delta_{\mathrm{H}} ](1 - \tau {\mathrm{b}}_{\mathrm{H}} )^{ - 1} }}{{{\mathrm{n}}[(1 - \tau )(1 - \eta )\left( {\mathrm{nE*}} \right)^{\eta } - (1 - \tau {\mathrm{b}}_{\mathrm{K}} )(\delta_{\mathrm{K}} - \delta_{\mathrm{H}} )]}}} \right\}\\ &\quad\left\{ {\frac{{[\left( {\mathrm{nE*}} \right)^{\eta } - {\mathrm{b}}_{\mathrm{K}} g({\mathrm{E*}}) + {\mathrm{b}}_{\mathrm{K}} (1 - \delta_{\mathrm{K}} )]}}{{{\mathrm{n}}(1 - \tau )^{2} \beta \eta \left( {\mathrm{nE*}} \right)^{\eta - 1} (1 - {\mathrm{E*}})(\xi \tau \alpha \gamma )^{ - 1} }} - } \right.\frac{{\alpha ( 1- \gamma - {\mathrm{E*)}}}}{{\beta ( 1- {\mathrm{E*)(1}} - \tau )}} \left. { + \left. {\frac{\xi \tau \alpha }{(1 - \tau )}} \right\}^{ - 1} } \right\} \\ &\quad \left\{ {\left\{ {\frac{{\zeta \alpha \phi ( 1- \zeta )^{ - 1} }}{{[[\chi ({\mathrm{E*}})]^{ - 1} (1 - \tau ) - 1]}} + \left[ {\frac{{\xi \tau {\mathrm{b}}_{\mathrm{H}} }}{{(1 - \tau {\mathrm{b}}_{\mathrm{H}} )}}\left( {[\chi ( {\mathrm{E*)}}]^{{\frac{ - 1}{\phi }}} - 1} \right) + [\chi ( {\mathrm{E*)}}]^{{\frac{ - 1}{\phi }}} } \right] } \right.} \right. \\ & \qquad\qquad\frac{{(1 - \phi )\alpha (1 - \tau {\mathrm{b}}_{\mathrm{K}} )[g({\mathrm{E}}* )- 1+ \delta_{\mathrm{H}} ]}}{{[(1 - \tau )(1 - \eta )\left( {\mathrm{nE*}} \right)^{\eta } - (1 - \tau {\mathrm{b}}_{\mathrm{K}} )(\delta_{\mathrm{K}} - \delta_{\mathrm{H}} )]}}\\ \end{aligned} $$

$$ \begin{aligned} & \quad - \frac{{\zeta \alpha \phi \left[ {(1 - \tau )\left( {\mathrm{nE*}} \right)^{\eta } (1 - \eta ) - [\chi ({\mathrm{E*}})]^{{\frac{1 - \phi }{\phi }}} (1 - \tau )(1 - {\mathrm{b}}_{\mathrm{K}} \tau ) ( 1- \delta_{\mathrm{H}} )+ ( 1- \delta_{\mathrm{K}} )(1 - {\mathrm{b}}_{\mathrm{K}} \tau )} \right]}}{{ ( 1- \zeta )[\chi ({\mathrm{E*}})]^{{\frac{ 1}{\phi }}} [(1 - \tau )(1 - \eta )\left( {\mathrm{nE*}} \right)^{\eta } - (1 - \tau {\mathrm{b}}_{\mathrm{K}} )(\delta_{\mathrm{K}} - \delta_{\mathrm{H}} )][[\chi ({\mathrm{E*}})]^{ - 1} (1 - \tau ) - 1]}} \\ &\quad \left. + \frac{{\alpha [ ( 1- \beta )(1 - {\mathrm{E*}}) - \gamma ]\chi ({\mathrm{E*}})}}{{[\chi ({\mathrm{E*}})]^{{\frac{ 1}{\phi }}} \beta (1 - \tau )(1 - {\mathrm{E*}})}} - \frac{{\xi \tau \alpha \gamma [\left( {\mathrm{nE*}} \right)^{\eta } - {\mathrm{b}}_{\mathrm{K}} g({\mathrm{E*}}) + {\mathrm{b}}_{\mathrm{K}} (1 - \delta_{\mathrm{K}} )]}}{{[\chi ({\mathrm{E*}})]^{{\frac{{ 1 { - }\varphi }}{\varphi }}} {\mathrm{n}}(1 - \tau )^{2} \beta \eta \left( {\mathrm{nE*}} \right)^{\eta - 1} (1 - {\mathrm{E*}})}} \right\} \\ &\left\{ {\frac{{\xi \tau \alpha \gamma [\left( {\mathrm{nE*}} \right)^{\eta } - {\mathrm{b}}_{\mathrm{K}} g({\mathrm{E*}}) + {\mathrm{b}}_{\mathrm{K}} (1 - \delta_{\mathrm{K}} )]}}{{{\mathrm{n}}(1 - \tau )^{2} \beta \eta \left( {\mathrm{nE*}} \right)^{{\eta { - 1}}} (1 - {\mathrm{E*}})}} + \frac{\xi \tau \alpha }{(1 - \tau )} - \frac{{\alpha ( 1- \gamma - {\mathrm{E*)}}}}{{\beta ( 1- {\mathrm{E*)}}(1 - \tau )}}} \right\}^{ - 1} \\ & + \bigg\{{\mathrm{n}}(1 - \alpha ) +\left[ {\frac{{(1 - \xi )\tau {\mathrm{b}}_{\mathrm{H}} }}{{(1 - \tau {\mathrm{b}}_{\mathrm{H}} )}}\left({1 - [x ( {\mathrm{E*)}}]^{{ - \frac{1}{\phi }}} } \right) + 1} \right]\\&\frac{\mathrm{n}\alpha (1 - \phi )(1 - \tau {\mathrm{b}}_{\mathrm{K}})[g({\mathrm{E}}* )- 1+ \delta_{\mathrm{H}}]}{[(1 - \tau )(1 - \eta )\left( {\mathrm{nE}} \right)^{\eta } - (1 - \tau {\mathrm{b}}_{\mathrm{K}} )(\delta_{\mathrm{K}} - \delta_{\mathrm{H}} )]} + \frac{{\alpha \gamma (1 - \eta )\mathrm{nE*}[x({\mathrm{E*}})]^{\frac{\phi - 1}{\phi }}}}{{\beta \eta (1 - {\mathrm{E*}})(1 - \tau )}} \\ \end{aligned} $$

$$ \begin{aligned} & \quad - \frac{{\zeta \alpha \phi {\mathrm{n}}\left[ {(1 - \tau )\left( {{\mathrm{nE*}}} \right)^{\eta } (1 - \eta ) - [\chi ({\mathrm{E*}})]^{{\frac{{1 - \phi }}{\phi }}} (1 - \tau )(1 - {\mathrm{b}}_{{\mathrm{K}}} \tau ){\mathrm{(1}} - \delta _{{\mathrm{H}}} {\mathrm{)}} + {\mathrm{(1}} - \delta _{{\mathrm{K}}} {\mathrm{)}}(1 - {\mathrm{b}}_{{\mathrm{K}}} \tau )} \right]}}{{{\mathrm{(1}} - \zeta {\mathrm{)}}[\chi ({\mathrm{E*}})]^{{\frac{{\mathrm{1}}}{\phi }}} [(1 - \tau )(1 - \eta )\left( {{\mathrm{nE*}}} \right)^{\eta } - (1 - \tau {\mathrm{b}}_{{\mathrm{K}}} )(\delta _{{\mathrm{K}}} - \delta _{{\mathrm{H}}} )][[\chi ({\mathrm{E*}})]^{{ - {\mathrm{1}}}} (1 - \tau ) - {\mathrm{1}}]}} \\ & \quad + \frac{{\zeta \alpha \phi {\mathrm{n(1}} - \zeta {\mathrm{)}}^{{ - {\mathrm{1}}}} }}{{[[\chi ({\mathrm{E*}})]^{{ - {\mathrm{1}}}} (1 - \tau ) - {\mathrm{1}}]}}- \frac{{\alpha \gamma (1 - \tau {\mathrm{b}}_{{\mathrm{K}}} )[g({\mathrm{E*)}} - {\mathrm{1}} + \delta _{{\mathrm{K}}} ][\chi ({\mathrm{E*}})]^{{\frac{{\phi - {\mathrm{1}}}}{\phi }}} }}{{(1 - \tau )^{2} \beta \eta ({\mathrm{nE*)}}^{{\eta - {\mathrm{1}}}} (1 - {\mathrm{E*}})}} \\ & \quad \left. + \frac{{(1 - \xi )\tau \alpha \gamma [\left( {{\mathrm{nE*}}} \right)^{\eta } - {\mathrm{b}}_{{\mathrm{K}}} g({\mathrm{E*}}) + {\mathrm{b}}_{{\mathrm{K}}} (1 - \delta _{{\mathrm{K}}} )][\chi ({\mathrm{E*}})]^{{\frac{{\phi - {\mathrm{1}}}}{\phi }}} }}{{(1 - \tau )^{2} \beta \eta \left( {{\mathrm{nE*}}} \right)^{{\eta - {\mathrm{1}}}} (1 - {\mathrm{E*}})}} \right\}\left\{ {\frac{{(1 - \xi ){\mathrm{n}}\tau \alpha }}{{(1 - \tau )}}} \right. \\ & \quad +\frac{{(1 - \xi )\tau \alpha \gamma [\left( {{\mathrm{nE*}}} \right)^{\eta } - {\mathrm{b}}_{{\mathrm{K}}} g({\mathrm{E*}}) + {\mathrm{b}}_{{\mathrm{K}}} (1 - \delta _{{\mathrm{K}}} )]}}{{(1 - \tau )^{2} \beta \eta \left( {{\mathrm{nE*}}} \right)^{{\eta - {\mathrm{1}}}} (1 - {\mathrm{E*}})}} + \frac{{\alpha \gamma {\mathrm{nE*(1}} - \eta {\mathrm{)}}}}{{\eta \beta {\mathrm{(1}} - {\mathrm{E*)}}(1 - \tau )}} \\ &\quad \left. {\left. +\, {\mathrm{n}}- \frac{{\alpha \gamma (1 - \tau {\mathrm{b}}_{{\mathrm{K}}} )[g({\mathrm{E*)}} - {\mathrm{1}} + \delta _{{\mathrm{K}}} ]}}{{\beta \eta (1 - \tau {\mathrm{)}}^{{\mathrm{2}}} (1 - {\mathrm{E*}})({\mathrm{nE*}})^{{\eta - {\mathrm{1}}}} }} \right\}^{{ - {\mathrm{1}}}} } \right\}^{{ - {\mathrm{1}}}} \\ \end{aligned} $$

Notice that the BGP value of V_t and F_t are given, respectively, by

$$ {\mathrm{V*}} = [\chi ( {\mathrm{E*)]}}^{{ \, \frac{ 1}{\phi }}} \frac{{({\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}}^{ *} )}}{{{\mathrm{L}}_{\mathrm{w}}^{ *} }} $$

(B.6)

and

$$ {\mathrm{F*}} = z({\mathrm{L}}_{\mathrm{w}}^{*} ,{\mathrm{E*}}) = p({\mathrm{L}}_{\mathrm{w}}^{*} ,{\mathrm{E*}}). $$

(B.7)

• B.6 Along a BGP, total income is given by

$$ \begin{aligned} & {\mathrm{K}}_{\mathrm{t}} ({\mathrm{nE*)}}^{\eta } + {\mathrm{P}}_{\mathrm{t}} [ ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}}^{*} )^{\phi } {\mathrm{H}}_{\mathrm{ct}}^{1 - \phi } + {\mathrm{n(L}}_{\mathrm{w}}^{*} )^{\phi } {\mathrm{H}}_{\mathrm{wt}}^{1 - \phi } ] \\ \quad & = \frac{{{\mathrm{H}}_{\mathrm{ct}} (1 - \tau {\mathrm{b}}_{\mathrm{H}} )[(1 - \eta )(1 - \tau )({\mathrm{nE*)}}^{\eta } - (1 - \tau {\mathrm{b}}_{\mathrm{K}} )(\delta_{\mathrm{K}} - \delta_{\mathrm{H}} )]}}{{(1 - \tau {\mathrm{b}}_{\mathrm{K}} )(1 - \phi )(1 - \tau ) ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}}^{*} )^{\phi } }} \\ & \quad \times \left\{ {\frac{{\gamma {\mathrm{nE*[(L}}_{\mathrm{w}}^{ *} )^{\phi } ({\mathrm{V*)}}^{\phi - 1} + {\mathrm{F*]}}}}{{ ( 1- {\mathrm{E*)(1}} - \tau )\beta \eta }} + ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}}^{*} )^{\phi } + {\mathrm{n(L}}_{\mathrm{w}}^{ *} )^{\phi } ({\mathrm{V*)}}^{\phi - 1} } \right\}. \\ \end{aligned} $$

(B.8)

Notice that (B.8) includes the imputed rent for self-owned housing.

Along a BGP, capitalists’ (pre-tax and pre-government transfers) income is given by

$$ \begin{aligned} \frac{{{\mathrm{K}}_{\mathrm{t}} ({\mathrm{nE*)}}^{\eta } }}{{ ( 1- \eta )^{ - 1} }} + \frac{{{\mathrm{P}}_{\mathrm{t}} ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}}^{*} )^{\phi } }}{{{\mathrm{H}}_{\mathrm{ct}}^{\phi - 1} }} & = \frac{{{\mathrm{H}}_{\mathrm{ct}} [(1 - \eta )(1 - \tau )({\mathrm{nE*)}}^{\eta } - (1 - \tau {\mathrm{b}}_{\mathrm{K}} )(\delta_{\mathrm{K}} - \delta_{\mathrm{H}} )]}}{{(1 - \tau {\mathrm{b}}_{\mathrm{H}} )^{ - 1} (1 - \tau {\mathrm{b}}_{\mathrm{K}} )(1 - \phi )(1 - \tau ) ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}}^{*} )^{\phi } }} \\ & \quad \times \left\{ {\frac{{\gamma {\mathrm{nE*[(L}}_{\mathrm{w}}^{ *} )^{\phi } ({\mathrm{V*)}}^{\phi - 1} + {\mathrm{F*]}}}}{{ ( 1- \eta )^{ - 1} ( 1- {\mathrm{E*)(1}} - \tau )\beta \eta }} + ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}}^{*} )^{\phi } } \right\}. \\ \end{aligned} $$

(B.9)

Notice that (B.9) includes the imputed rent for self-owned housing.

Along a BGP, capitalists’ (post-tax and post-government transfers) income is given by

$$ \begin{aligned} & \frac{{{\mathrm{H}}_{\mathrm{ct}} [(1 - \eta )(1 - \tau )({\mathrm{nE*)}}^{\eta } - (1 - \tau {\mathrm{b}}_{\mathrm{K}} )(\delta_{\mathrm{K}} - \delta_{\mathrm{H}} )]}}{{(1 - \tau {\mathrm{b}}_{\mathrm{H}} )^{ - 1} (1 - \tau {\mathrm{b}}_{\mathrm{K}} )(1 - \phi )(1 - \tau ) ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}}^{*} )^{\phi } }} \\ & \quad \left\{ {\frac{{ [ ( {\mathrm{L}}_{\mathrm{w}}^{ *} )^{\phi } ({\mathrm{V*)}}^{\phi - 1} + {\mathrm{F*]}}}}{{[\gamma {\mathrm{nE*(1}} - \eta ) ]^{ - 1} ( 1- {\mathrm{E*)}}\beta \eta }} + ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}}^{*} )^{\phi } - \tau \xi {\mathrm{nF*}}} \right\} \\ & \quad + {\mathrm{H}}_{\mathrm{ct}} {\mathrm{b}}_{\mathrm{H}} \tau \xi [g({\mathrm{E*)}} - 1+ \delta_{\mathrm{H}} ] \\ & \quad + \frac{{{\mathrm{H}}_{\mathrm{ct}} {\mathrm{b}}_{\mathrm{K}} \tau \xi [(1 - \eta ) ( 1- \tau ) ( {\mathrm{nE*)}}^{\eta } - (1 - \tau {\mathrm{b}}_{\mathrm{K}} )(\delta_{\mathrm{K}} - \delta_{\mathrm{H}} )]\gamma (1 - \tau {\mathrm{b}}_{\mathrm{H}} ) [g ( {\mathrm{E*)}} - 1+ \delta_{\mathrm{k}} ]}}{{(1 - \tau {\mathrm{b}}_{\mathrm{K}} )(1 - \phi ) ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}}^{*} )^{\phi } ( 1- {\mathrm{E*)(nE*)}}^{\eta - 1} ( 1- \tau )^{ 2} \beta \eta [ ( {\mathrm{L}}_{\mathrm{w}}^{ *} )^{\phi } ({\mathrm{V*)}}^{\phi - 1} + {\mathrm{F*]}}^{ - 1} }} \\ & \quad - \frac{{\zeta {\mathrm{H}}_{\mathrm{ct}} \phi }}{(1 - \zeta )}\left\{ \frac{{[(1 - \eta ) ( 1- \tau )({\mathrm{nE*)}}^{\eta } - (1 - \tau {\mathrm{b}}_{\mathrm{K}} )(\delta_{\mathrm{K}} - \delta_{\mathrm{H}} )]}}{{ ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}}^{*} ) ( {\mathrm{L}}_{\mathrm{w}}^{ *} )^{ - \phi } ({\mathrm{V*)}}^{ - 1} }} + \frac{{(1 - \tau {\mathrm{b}}_{\mathrm{K}} ) ( 1- \tau ) ( 1- \delta_{\mathrm{H}} )}}{{ ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}}^{*} )^{1 - \phi } ( {\mathrm{V*}})^{\phi - 1} }}\right. \\ & \quad \left. - \frac{(1 - \eta ) ( 1- \tau )}{{ ( {\mathrm{L}}_{\mathrm{w}}^{ *} )^{1 - \phi } ({\mathrm{nE*)}}^{ - \eta } }}{ - }\frac{{ ( 1- \delta_{\mathrm{K}} ) ( 1- \tau {\mathrm{b}}_{\mathrm{K}} )}}{{ ( {\mathrm{L}}_{\mathrm{w}}^{ *} )^{1 - \phi } }} \right\} \\ & \quad \times \frac{{(1 - \tau {\mathrm{b}}_{\mathrm{H}} )(\xi {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}}^{*} )[ (1 - \phi )(1 - \tau {\mathrm{b}}_{\mathrm{K}} )]^{ - 1} }}{{[ ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}}^{*} )^{\phi } ({\mathrm{V*)}}^{1 - \phi } ( 1- \tau )- {\mathrm{V*(L}}_{\mathrm{w}}^{*} )^{\phi } ]}} - ( 1- \xi )\tau {\mathrm{H}}_{\mathrm{ct}} \left\{ {\frac{{[g({\mathrm{E*)}} - 1+ \delta_{\mathrm{H}} ]}}{{ ( {\mathrm{nb}}_{\mathrm{H}} )^{ - 1} {\mathrm{V*}}}}} \right. \\ & \quad \left. { - \frac{{(1 - \tau {\mathrm{b}}_{\mathrm{H}} )[(1 - \eta ) ( 1- \tau )({\mathrm{nE*)}}^{\eta } - (1 - \tau {\mathrm{b}}_{\mathrm{K}} )(\delta_{\mathrm{K}} - \delta_{\mathrm{H}} )]{\mathrm{nE*}}\gamma [ ( {\mathrm{L}}_{\mathrm{w}}^{ *} )^{\phi } ({\mathrm{V*)}}^{\phi - 1} + {\mathrm{F*]}}}}{{(1 - \tau {\mathrm{b}}_{\mathrm{K}} )(1 - \phi ) ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}}^{*} )^{\phi } ( 1- {\mathrm{E*)(1}} - \tau )^{ 2} \beta \eta }}} \right\}. \\ \end{aligned} $$

(B.10)

Notice that (B.10) includes the imputed rent for self-owned housing.

• B.7 Along a BGP, the stock of productive capital, total wealth and capitalists’ wealth are given, respectively, by

$$ {\mathrm{K}}_{\mathrm{t}} = \frac{{{\mathrm{H}}_{\mathrm{ct}} (1 - \tau {\mathrm{b}}_{\mathrm{H}} )\gamma [(1 - \eta )(1 - \tau )({\mathrm{nE*)}}^{\eta } - (1 - \tau {\mathrm{b}}_{\mathrm{K}} )(\delta_{\mathrm{K}} - \delta_{\mathrm{H}} )] [ ( {\mathrm{L}}_{\mathrm{w}}^{ *} )^{\phi } ({\mathrm{V*)}}^{\phi - 1} + {\mathrm{F*]}}}}{{(1 - \tau {\mathrm{b}}_{\mathrm{K}} )(1 - \phi )({\mathrm{nE*)}}^{\eta - 1} ( 1- {\mathrm{E*)(1}} - \tau )^{ 2} \beta \eta ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}}^{*} )^{\phi } }}, $$

(B.11)

$$ \begin{aligned}& {\mathrm{K}}_{\mathrm{t}} + {\mathrm{Q}}_{\mathrm{t}} {\mathrm{L}} + {\mathrm{H}}_{\mathrm{ct}} + {\mathrm{nH}}_{\mathrm{wt}} \\ &\quad = \frac{{{\mathrm{H}}_{\mathrm{ct}} (1 - \tau {\mathrm{b}}_{\mathrm{H}} )\gamma [(1 - \eta )(1 - \tau )({\mathrm{nE*)}}^{\eta } - (1 - \tau {\mathrm{b}}_{\mathrm{K}} )(\delta_{\mathrm{K}} - \delta_{\mathrm{H}} )] [ ( {\mathrm{L}}_{\mathrm{w}}^{ *} )^{\phi } ({\mathrm{V*)}}^{\phi - 1} + {\mathrm{F*]}}}}{{(1 - \tau {\mathrm{b}}_{\mathrm{K}} )(1 - \phi )({\mathrm{nE*)}}^{\eta - 1} ( 1- {\mathrm{E*)(1}} - \tau )^{ 2} \beta \eta ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}}^{*} )^{\phi } }} \\ & \quad + {\mathrm{H}}_{\mathrm{ct}} + \frac{{{\mathrm{nH}}_{\mathrm{ct}} }}{\mathrm{V*}} + \frac{{{\mathrm{H}}_{\mathrm{ct}} \phi (1 - \tau {\mathrm{b}}_{\mathrm{H}} )[(1 - \zeta ) (1 - \phi )(1 - \tau {\mathrm{b}}_{\mathrm{K}} )]^{ - 1} {\mathrm{L}}}}{{[ ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}}^{*} )^{\phi } ({\mathrm{V*)}}^{1 - \phi } ( 1- \tau )- {\mathrm{V*(L}}_{\mathrm{w}}^{*} )^{\phi } ]}} \\ & \quad \times \left\{ \frac{{[(1 - \eta )(1 - \tau )({\mathrm{nE*)}}^{\eta } - (1 - \tau {\mathrm{b}}_{\mathrm{K}} )(\delta_{\mathrm{K}} - \delta_{\mathrm{H}} )]}}{{ ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}}^{*} ) ( {\mathrm{L}}_{\mathrm{w}}^{ *} )^{ - \phi } ({\mathrm{V*)}}^{ - 1} }} + \frac{{(1 - \tau {\mathrm{b}}_{\mathrm{K}} ) ( 1- \tau ) ( 1- \delta_{\mathrm{H}} )}}{{ ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}}^{*} )^{1 - \phi } ( {\mathrm{V*}})^{\phi - 1} }} \right. \\ &\quad \quad \left. - \frac{(1 - \eta ) ( 1- \tau )}{{({\mathrm{nE*)}}^{ - \eta } ( {\mathrm{L}}_{\mathrm{w}}^{ *} )^{1 - \phi } }} - \frac{{ ( 1- \delta_{\mathrm{K}} ) ( 1- \tau {\mathrm{b}}_{\mathrm{K}} )}}{{ ( {\mathrm{L}}_{\mathrm{w}}^{ *} )^{1 - \phi } }} \right\} \\ \end{aligned} $$

(B.12)

and

$$\begin{aligned} & {\mathrm{K}}_{\mathrm{t}} + {\mathrm{Q}}_{\mathrm{t}} ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}}^{ *} ) + {\mathrm{H}}_{\mathrm{ct}}\\ &\quad = \frac{{{\mathrm{H}}_{\mathrm{ct}} (1 - \tau {\mathrm{b}}_{\mathrm{H}} )\gamma [(1 - \eta )(1 - \tau )({\mathrm{nE*)}}^{\eta } - (1 - \tau {\mathrm{b}}_{\mathrm{K}} )(\delta_{\mathrm{K}} - \delta_{\mathrm{H}} )] [ ( {\mathrm{L}}_{\mathrm{w}}^{ *} )^{\phi } ({\mathrm{V*)}}^{\phi - 1} + {\mathrm{F*]}}}}{{(1 - \tau {\mathrm{b}}_{\mathrm{K}} )(1 - \phi )({\mathrm{nE*)}}^{\eta - 1} ( 1- {\mathrm{E*)(1}} - \tau )^{ 2} \beta \eta ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}}^{*} )^{\phi } }} \\ & \quad + \left\{ \frac{{[(1 - \eta )(1 - \tau )({\mathrm{nE*)}}^{\eta } - (1 - \tau {\mathrm{b}}_{\mathrm{K}} )(\delta_{\mathrm{K}} - \delta_{\mathrm{H}} )]}}{{ ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}}^{*} ) ( {\mathrm{L}}_{\mathrm{w}}^{ *} )^{ - \phi } ({\mathrm{V*)}}^{ - 1} }} + \frac{{(1 - \tau {\mathrm{b}}_{\mathrm{K}} ) ( 1- \tau ) ( 1- \delta_{\mathrm{H}} )}}{{ ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}}^{*} )^{1 - \phi } ( {\mathrm{V*}})^{\phi - 1} }} \right. \\ & \quad \left. - \frac{(1 - \eta ) ( 1- \tau )}{{({\mathrm{nE*)}}^{{{ - }\eta }} ( {\mathrm{L}}_{\mathrm{w}}^{ *} )^{1 - \phi } }} - \frac{{ ( 1- \delta_{\mathrm{K}} ) ( 1- \tau {\mathrm{b}}_{\mathrm{K}} )}}{{ ( {\mathrm{L}}_{\mathrm{w}}^{ *} )^{1 - \phi } }} \right\} \\ & \quad \times \frac{{{\mathrm{H}}_{\mathrm{ct}} \phi (1 - \tau {\mathrm{b}}_{\mathrm{H}} )[(1 - \zeta ) (1 - \phi )(1 - \tau {\mathrm{b}}_{\mathrm{K}} )]^{ - 1} ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}}^{ *} )}}{{[ ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}}^{*} )^{\phi } ({\mathrm{V*)}}^{1 - \phi } ( 1- \tau )- {\mathrm{V*(L}}_{\mathrm{w}}^{*} )^{\phi } ]}} + {\mathrm{H}}_{\mathrm{ct}} \\ \end{aligned} $$

(B.13)

• B.8 Along a BGP, the discounted sequence of utilities of the representative worker is given by

$$\begin{aligned} \sum\limits_{{{\mathrm{v}} = {\mathrm{0}}}}^{\infty } {\rho _{{\mathrm{w}}}^{{\mathrm{v}}} } & \left\{ {\beta {\mathrm{ln}}\left[ {{\mathrm{H}}_{{{\mathrm{ct}} + {\mathrm{v}}}}^{{{\mathrm{1}} - \phi }} \left( {\frac{{{\mathrm{(L}}_{{\mathrm{w}}}^{*} {\mathrm{)}}^{\phi } }}{{{\mathrm{(V*)}}^{{{\mathrm{1}} - \phi }} }} + {\mathrm{F*}}} \right)} \right] + \gamma {\mathrm{ln(1}} - {\mathrm{E*)}}} \right. \\ & \quad + {\mathrm{(1}} - \beta - \gamma {\mathrm{)ln}}\left[ {\frac{{{\mathrm{H}}_{{{\mathrm{ct}} + {\mathrm{v}}}} {\mathrm{((L}}_{{\mathrm{w}}}^{*} {\mathrm{)}}^{\phi } {\mathrm{(V*)}}^{{\phi - {\mathrm{1}}}} + {\mathrm{F*)(1}} - \tau {\mathrm{b}}_{{\mathrm{H}}} {\mathrm{)}}}}{{{\mathrm{(1}} - \phi {\mathrm{)(1}} - \tau {\mathrm{)}}\beta {\mathrm{(1}} - \beta - \gamma {\mathrm{)}}^{{{\mathrm{ - 1}}}} {\mathrm{(1}} - \tau {\mathrm{b}}_{{\mathrm{K}}} {\mathrm{)}}}}} \right] + \\ & \quad + {\mathrm{(1}} - \beta - \gamma {\mathrm{)ln }}\left. {\left[ {\frac{{(1 - \eta )(1 - \tau )({\mathrm{nE*)}}^{\eta } - (1 - \tau {\mathrm{b}}_{{\mathrm{K}}} )(\delta _{{\mathrm{K}}} - \delta _{{\mathrm{H}}} )}}{{{\mathrm{(L}} - {\mathrm{nL}}_{{\mathrm{w}}}^{{\mathrm{*}}} )^{\phi } }}} \right]} \right\}, \\ \end{aligned}$$

which can be rewritten as

$$ \begin{aligned} & \frac{{\gamma \,{\mathrm{ln(1}} - {\mathrm{E*)}}}}{{ 1- \rho_{\mathrm{w}} }} + \frac{(1 - \gamma )}{{ 1- \rho_{\mathrm{w}} }}{ \ln }\left( {\frac{{ ( {\mathrm{L}}_{\mathrm{w}}^{*} )^{\phi } }}{{ ( {\mathrm{V*)}}^{1 - \phi } }} + {\mathrm{F*}}} \right) \\ & \quad + \frac{(1 - \beta - \gamma )}{{ 1- \rho_{\mathrm{w}} }}{ \ln }\left[ {\frac{{ ( 1- \tau {\mathrm{b}}_{\mathrm{H}} ) ( 1- \beta - \gamma )}}{{ (1 - \phi ) ( 1- \tau )\beta ( 1- \tau {\mathrm{b}}_{\mathrm{K}} )}}} \right] \\ & \quad + \frac{ ( 1- \beta - \gamma )}{{ 1- \rho_{\mathrm{w}} }}{ \ln }\left[ {\frac{{(1 - \eta )(1 - \tau )({\mathrm{nE*)}}^{\eta } - (1 - \tau {\mathrm{b}}_{\mathrm{K}} )(\delta_{\mathrm{K}} - \delta_{\mathrm{H}} )}}{{ ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}}^{ *} )^{\phi } }}} \right] \, \\ & \quad + \frac{(1 - \beta \phi - \gamma )}{{ 1- \rho_{\mathrm{w}} }}\left\{ {{\mathrm{ln(H}}_{\mathrm{ct}} ) + \frac{{\rho_{\mathrm{w}} \ln [g({\mathrm{E*)}}]}}{{ 1- \rho_{\mathrm{w}} }}} \right\}. \\ \end{aligned} $$

(B.14)

Along a BGP, the discounted sequence of utilities of the representative capitalist is given by

$$ \begin{aligned} \sum\limits_{{{\mathrm{v}} = {\mathrm{0}}}}^{\infty } {\rho _{{\mathrm{c}}}^{{\mathrm{v}}} } & \left\{ {\alpha {\mathrm{ln}}\left[ {{\mathrm{H}}_{{{\mathrm{ct}} + {\mathrm{v}}}}^{{{\mathrm{1 }} - \phi }} {\mathrm{((L}} - {\mathrm{nL}}_{{\mathrm{w}}}^{{\mathrm{*}}} )^{\phi } - {\mathrm{nF*)}}} \right]} \right. \\ & \quad + (1 - \alpha {\mathrm{)ln}}\left[ {\frac{{{\mathrm{H}}_{{{\mathrm{ct}} + {\mathrm{v}}}} {\mathrm{((L}} - {\mathrm{nL}}_{{\mathrm{w}}}^{{\mathrm{*}}} )^{\phi } - {\mathrm{nF*)(1}} - \tau {\mathrm{b}}_{{\mathrm{H}}} {\mathrm{)(1}} - \alpha {\mathrm{)}}}}{{\alpha {\mathrm{(1}} - \phi {\mathrm{)(1}} - \tau {\mathrm{b}}_{{\mathrm{K}}} {\mathrm{)}}}}} \right] + \\ {\mathrm{ }} & \quad \left. { + (1 - \alpha {\mathrm{)ln}}\left[ {\frac{{(1 - \eta )(1 - \tau )({\mathrm{nE*)}}^{\eta } - (1 - \tau {\mathrm{b}}_{{\mathrm{K}}} )(\delta _{{\mathrm{K}}} - \delta _{{\mathrm{H}}} )}}{{{\mathrm{(L}} - {\mathrm{nL}}_{{\mathrm{w}}}^{{\mathrm{*}}} )^{\phi } }}} \right]} \right\}, \\ \end{aligned} $$

which can be rewritten as

$$ \begin{aligned} & \frac{{{\mathrm{ln((L}} - {\mathrm{nL}}_{\mathrm{w}}^{ *} )^{\phi } - {\mathrm{nF*)}}}}{{ 1- \rho_{\mathrm{c}} }} + \frac{(1 - \alpha )}{{ 1- \rho_{\mathrm{c}} }}{ \ln }\left[ {\frac{{ ( 1- \tau {\mathrm{b}}_{\mathrm{H}} ) ( 1- \alpha )}}{{\alpha ( 1- \phi ) ( 1- \tau {\mathrm{b}}_{\mathrm{K}} )}}} \right] \\ & \quad+ \frac{ ( 1- \alpha )}{{ 1- \rho_{\mathrm{c}} }}{ \ln }\left[ {\frac{{(1 - \eta )(1 - \tau )({\mathrm{nE*)}}^{\eta } - (1 - \tau {\mathrm{b}}_{\mathrm{K}} )(\delta_{\mathrm{K}} - \delta_{\mathrm{H}} )}}{{ ( {\mathrm{L}} - {\mathrm{nL}}_{\mathrm{w}}^{ *} )^{\phi } }}} \right] \\ & \quad + \frac{(1 - \alpha \phi )}{{ 1- \rho_{\mathrm{c}} }}\left\{ {{\mathrm{ln(H}}_{\mathrm{ct}} ) + \frac{{\rho {}_{\mathrm{c}}\ln [g({\mathrm{E*)}}]}}{{ 1- \rho_{\mathrm{c}} }}} \right\}. \\ \end{aligned} $$

(B.15)

1.3 C Transitional Path

By linearizing the system (A.24)–(A.27) around (E*, V*, F*, ${\mathrm{L}}_{\mathrm{w}}^{ *}$), one obtains the following linearized system:

$$ \left[ \begin{aligned} {\mathrm{E}}_{{{\mathrm{t}} + 1}} - {\mathrm{E*}} \hfill \\ {\mathrm{V}}_{{{\mathrm{t}} + 1}} - {\mathrm{V*}} \hfill \\ {\mathrm{F}}_{{{\mathrm{t}} + 1}} - {\mathrm{F*}} \hfill \\ {\mathrm{L}}_{{{\mathrm{wt}} + 1}} - {\mathrm{L}}_{\mathrm{w}}^{ *} \hfill \\ \end{aligned} \right] = \left[ \begin{aligned} {\mathrm{d}}_{ 1 1} {\mathrm{ d}}_{ 1 2} {\mathrm{ d}}_{ 1 3} {\mathrm{ d}}_{ 1 4} \, \hfill \\ {\mathrm{d}}_{ 2 1} {\mathrm{ d}}_{ 2 2} {\mathrm{ d}}_{ 2 3} {\mathrm{ d}}_{ 2 4} \hfill \\ {\mathrm{d}}_{ 3 1} {\mathrm{ d}}_{ 3 2} {\mathrm{ d}}_{ 3 3} {\mathrm{ d}}_{ 3 4} \hfill \\ {\mathrm{d}}_{ 4 1} {\mathrm{ d}}_{ 4 2} {\mathrm{ d}}_{ 4 3} {\mathrm{ d}}_{ 4 4} \hfill \\ \end{aligned} \right]\left[ \begin{aligned} {\mathrm{E}}_{\mathrm{t}} - {\mathrm{E*}} \hfill \\ {\mathrm{V}}_{\mathrm{t}} - {\mathrm{V*}} \hfill \\ {\mathrm{F}}_{\mathrm{t}} - {\mathrm{F*}} \hfill \\ {\mathrm{L}}_{\mathrm{wt}} - {\mathrm{L}}_{\mathrm{w}}^{ *} \hfill \\ \end{aligned} \right], $$

(C.1)

where

$$ \begin{aligned} {\mathrm{d}}_{ 1 1} & \equiv - \frac{{ [{\Gamma}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} ) + {\Gamma}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} ({\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ) ] {\mathrm{d}}_{ 4 1} }}{{{\Gamma}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} - {\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} )}} \\ & \quad - \frac{{ [{\Gamma}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{E}}_{\mathrm{t}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{E}}_{\mathrm{t}} }} ) + {\Gamma}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{E}}_{\mathrm{t}} }} {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{E}}_{\mathrm{t}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{E}}_{\mathrm{t}} }} ({\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ) ]}}{{{\Gamma}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} - {\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} )}}, \\ \end{aligned} $$

$$ \begin{aligned} {\mathrm{d}}_{ 1 2} & \equiv - \frac{{ [{\Gamma}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} ) + {\Gamma}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} ({\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ) ] {\mathrm{d}}_{ 4 2} }}{{{\Gamma}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} - {\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} )}} \\ & \quad - \frac{{[{\Gamma}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{V}}_{\mathrm{t}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{\mathrm{t}} }} )+ {\Gamma}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{V}}_{\mathrm{t}} }} {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{V}}_{\mathrm{t}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{V}}_{\mathrm{t}} }} ({\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} )]}}{{{\Gamma}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} - {\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} )}}, \\ \end{aligned} $$

$$ \begin{aligned} {\mathrm{d}}_{ 1 3} & \equiv - \frac{{ [{\Gamma}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} ) + {\Gamma}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} ({\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ) ] {\mathrm{d}}_{ 4 3} }}{{{\Gamma}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} - {\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} )}} \\ & \quad - \frac{{[{\Gamma}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{F}}_{\mathrm{t}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{\mathrm{t}} }} )+ {\Gamma}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{F}}_{\mathrm{t}} }} {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{F}}_{\mathrm{t}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{F}}_{\mathrm{t}} }} ({\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} )]}}{{{\Gamma}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} - {\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} )}} , { } \\ \end{aligned} $$

$$ \begin{aligned} {\mathrm{d}}_{ 1 4} & \equiv - \frac{{ [{\Gamma}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} ) + {\Gamma}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} ({\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ) ] {\mathrm{d}}_{ 4 4} }}{{{\Gamma}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} - {\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} )}} \\ & \quad - \frac{{[{\Gamma}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{L}}_{\mathrm{wt}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{L}}_{\mathrm{wt}} }} )+ {\Gamma}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{L}}_{\mathrm{wt}} }} {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{L}}_{\mathrm{wt}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{L}}_{\mathrm{wt}} }} ({\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} )]}}{{{\Gamma}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} - {\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} )}}, \\ \end{aligned} $$

$$ \begin{aligned} {\mathrm{d}}_{ 2 1} & \equiv - \frac{{ [{\Gamma}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} - {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} )+ {\Gamma}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} - {\Omega}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} )+ {\Gamma}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} ({\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} - {\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ) ] {\mathrm{d}}_{ 4 1} }}{{{\Gamma}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} - {\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} )}} \\ & \quad - \frac{{ [{\Gamma}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{E}}_{\mathrm{t}} }} - {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{E}}_{\mathrm{t}} }} )+ {\Gamma}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{E}}_{\mathrm{t}} }} - {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{E}}_{\mathrm{t}} }} ) + {\Gamma}_{{{\mathrm{E}}_{\mathrm{t}} }} ({\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} - {\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} )]}}{{{\Gamma}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} - {\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} )}}, \\ \end{aligned} $$

$$ \begin{aligned} {\mathrm{d}}_{ 2 2} & \equiv - \frac{{ [{\Gamma}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} - {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} )+ {\Gamma}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} - {\Omega}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} )+ {\Gamma}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} ({\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} - {\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ) ] {\mathrm{d}}_{ 4 2} }}{{{\Gamma}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} - {\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} )}} \\ & \quad - \frac{{[{\Gamma}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{V}}_{\mathrm{t}} }} - {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{\mathrm{t}} }} )+ {\Gamma}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{\mathrm{t}} }} - {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{V}}_{\mathrm{t}} }} )+ {\Gamma}_{{{\mathrm{V}}_{\mathrm{t}} }} ({\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} - {\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ) ]}}{{{\Gamma}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} - {\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} )}}, \\ \end{aligned} $$

$$ \begin{aligned} {\mathrm{d}}_{ 2 3} & \equiv - \frac{{ [{\Gamma}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} - {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} )+ {\Gamma}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} - {\Omega}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} )+ {\Gamma}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} ({\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} - {\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ) ] {\mathrm{d}}_{ 4 3} }}{{{\Gamma}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} - {\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} )}} \\ & \quad - \frac{{[{\Gamma}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{F}}_{\mathrm{t}} }} - {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{\mathrm{t}} }} ) + {\Gamma}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{\mathrm{t}} }} - {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{F}}_{\mathrm{t}} }} ) + {\Gamma}_{{{\mathrm{F}}_{\mathrm{t}} }} ({\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} - {\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} )]}}{{{\Gamma}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} - {\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} )}}, \\ \end{aligned} $$

$$ \begin{aligned} {\mathrm{d}}_{ 2 4} & \equiv - \frac{{ [{\Gamma}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} - {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} )+ {\Gamma}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} - {\Omega}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} )+ {\Gamma}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} ({\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} - {\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ) ] {\mathrm{d}}_{ 4 4} }}{{{\Gamma}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} - {\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} )}} \\ & \quad - \frac{{[{\Gamma}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{L}}_{\mathrm{wt}} }} - {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{L}}_{\mathrm{wt}} }} ) + {\Gamma}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{L}}_{\mathrm{wt}} }} - {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{L}}_{\mathrm{wt}} }} ) + {\Gamma}_{{{\mathrm{L}}_{\mathrm{wt}} }} ({\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} - {\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} )]}}{{{\Gamma}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} - {\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} )}}, \\ \end{aligned} $$

$$ \begin{aligned} {\mathrm{d}}_{ 3 1} & \equiv - \frac{{ [{\Gamma}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} - {\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} )+ {\Gamma}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} )+ {\Gamma}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} ({\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ) ] {\mathrm{d}}_{ 4 1} }}{{{\Gamma}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} { - }{\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} { - }{\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} { - }{\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} )}} \\ & \quad - \frac{{[{\Gamma}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{E}}_{\mathrm{t}} }} - {\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{E}}_{\mathrm{t}} }} )+ {\Gamma}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{E}}_{\mathrm{t}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{E}}_{\mathrm{t}} }} ) + {\Gamma}_{{{\mathrm{E}}_{\mathrm{t}} }} ({\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} )]}}{{{\Gamma}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} - {\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} )}}, \\ \end{aligned} $$

$$ \begin{aligned} {\mathrm{d}}_{ 3 2} & \equiv - \frac{{ [{\Gamma}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} - {\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} )+ {\Gamma}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} )+ {\Gamma}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} ({\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ) ] {\mathrm{d}}_{ 4 2} }}{{{\Gamma}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} - {\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} )}} \\ & \quad - \frac{{[{\Gamma}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{\mathrm{t}} }} - {\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{V}}_{\mathrm{t}} }} )+ {\Gamma}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{V}}_{\mathrm{t}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{\mathrm{t}} }} ) + {\Gamma}_{{{\mathrm{V}}_{\mathrm{t}} }} ({\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} )]}}{{{\Gamma}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} - {\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} )}}, \\ \end{aligned} $$

$$ \begin{aligned} {\mathrm{d}}_{ 3 3} & \equiv - \frac{{ [{\Gamma}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} - {\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} )+ {\Gamma}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} )+ {\Gamma}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} ({\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ) ] {\mathrm{d}}_{ 4 3} }}{{{\Gamma}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} - {\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} )}} \\ & \quad - \frac{{[{\Gamma}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{\mathrm{t}} }} - {\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{F}}_{\mathrm{t}} }} )+ {\Gamma}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{F}}_{\mathrm{t}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{\mathrm{t}} }} ) + {\Gamma}_{{{\mathrm{F}}_{\mathrm{t}} }} ({\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} )]}}{{{\Gamma}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} - {\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} )}}, \\ \end{aligned} $$

$$ \begin{aligned} {\mathrm{d}}_{ 3 4} & \equiv - \frac{{ [{\Gamma}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} - {\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} )+ {\Gamma}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} )+ {\Gamma}_{{{\mathrm{L}}_{{{\mathrm{wt}} + 1}} }} ({\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ) ] {\mathrm{d}}_{ 4 4} }}{{{\Gamma}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} { - }{\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} { - }{\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} { - }{\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} )}} \\ & \quad - \frac{{[{\Gamma}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{_{{{\mathrm{L}}_{\mathrm{wt}} }} }} - {\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{_{{{\mathrm{L}}_{\mathrm{wt}} }} }} )+ {\Gamma}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{L}}_{\mathrm{wt}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{L}}_{\mathrm{wt}} }} ) + {\Gamma}_{{{\mathrm{L}}_{\mathrm{wt}} }} ({\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} )]}}{{{\Gamma}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} - {\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} ({\Psi}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} ) + {\Gamma}_{{{\mathrm{E}}_{{{\mathrm{t}} + 1}} }} ({\Omega}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} - {\Omega}_{{{\mathrm{V}}_{{{\mathrm{t}} + 1}} }} {\Psi}_{{{\mathrm{F}}_{{{\mathrm{t}} + 1}} }} )}}, \\ \end{aligned} $$

$$ \begin{aligned} {\text{d}}{}_{ 41} & \equiv \left\{ {{\Lambda}_{{{\text{E}}_{{{\text{t}} + 1}} }} [{\Gamma}_{{{\text{V}}_{{{\text{t}} + 1}} }} ({\Omega}_{{{\text{E}}_{\text{t}} }} {\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{E}}_{\text{t}} }} ) + {\Gamma}_{{{\text{F}}_{{{\text{t}} + 1}} }} ({\Psi}_{{{\text{E}}_{\text{t}} }} {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{E}}_{\text{t}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} ) + {\Gamma}_{{{\text{E}}_{\text{t}} }} ({\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} ) ]} \right. \\ & \quad + {\Lambda}_{{{\text{F}}_{{{\text{t}} + 1}} }} [{\Gamma}_{{{\text{V}}_{{{\text{t}} + 1}} }} ({\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{E}}_{\text{t}} }} - {\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{E}}_{\text{t}} }} )+ {\Gamma}_{{{\text{E}}_{{{\text{t}} + 1}} }} ({\Omega}_{{{\text{E}}_{\text{t}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{E}}_{\text{t}} }} ) + {\Gamma}_{{{\text{E}}_{\text{t}} }} ({\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} )] \\ & \quad + {\Lambda}_{{{\text{V}}_{{{\text{t}} + 1}} }} [{\Gamma}_{{{\text{F}}_{{{\text{t}} + 1}} }} ({\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{E}}_{\text{t}} }} - {\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{E}}_{\text{t}} }} )+ {\Gamma}_{{{\text{E}}_{{{\text{t}} + 1}} }} ({\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{E}}_{\text{t}} }} - {\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{E}}_{\text{t}} }} ) + {\Gamma}_{{{\text{E}}_{\text{t}} }} ({\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} - {\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} )\left. ] \right\} \\ & \quad \left\{ {{\Lambda}_{{{\text{L}}_{{{\text{wt}} + 1}} }} [{\Gamma}_{{{\text{V}}_{{{\text{t}} + 1}} }} ({\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} - {\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} ) + {\Gamma}_{{{\text{F}}_{{{\text{t}} + 1}} }} ({\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} ) + {\Gamma}_{{{\text{E}}_{{{\text{t}} + 1}} }} ({\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} )]} \right. \\ & \quad - {\Lambda}_{{{\text{E}}_{{{\text{t}} + 1}} }} [{\Gamma}_{{{\text{V}}_{{{\text{t}} + 1}} }} ({\Omega}_{{{\text{L}}_{{{\text{wt}} + 1}} }} {\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{L}}_{{{\text{wt}} + 1}} }} ) + {\Gamma}_{{{\text{F}}_{{{\text{t}} + 1}} }} ({\Psi}_{{{\text{L}}_{{{\text{wt}} + 1}} }} {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{L}}_{{{\text{wt}} + 1}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} ) + {\Gamma}_{{{\text{L}}_{{{\text{wt}} + 1}} }} ({\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} ) ]\\ & \quad - {\Lambda}_{{{\text{F}}_{{{\text{t}} + 1}} }} [{\Gamma}_{{{\text{V}}_{{{\text{t}} + 1}} }} ({\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{L}}_{{{\text{wt}} + 1}} }} - {\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{L}}_{{{\text{wt}} + 1}} }} )+ {\Gamma}_{{{\text{E}}_{{{\text{t}} + 1}} }} ({\Omega}_{{{\text{L}}_{{{\text{wt}} + 1}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{L}}_{{{\text{wt}} + 1}} }} )+ {\Gamma}_{{{\text{L}}_{{{\text{wt}} + 1}} }} ({\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} ) ]\\ & \quad - {\Lambda}_{{{\text{V}}_{{{\text{t}} + 1}} }} \left. { [{\Gamma}_{{{\text{F}}_{{{\text{t}} + 1}} }} ({\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{L}}_{{{\text{wt}} + 1}} }} - {\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{L}}_{{{\text{wt}} + 1}} }} )+ {\Gamma}_{{{\text{E}}_{{{\text{t}} + 1}} }} ({\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{L}}_{{{\text{wt}} + 1}} }} - {\Omega}_{{{\text{L}}_{{{\text{wt}} + 1}} }} {\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} )+ {\Gamma}_{{{\text{L}}_{{{\text{wt}} + 1}} }} ({\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} - {\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} ) ]} \right\}^{ - 1} , \\ \end{aligned} $$

$$ \begin{aligned} {\text{d}}_{ 42} & \equiv \left\{ {{\Lambda}_{{{\text{E}}_{{{\text{t}} + 1}} }} } \right.[{\Gamma}_{{{\text{V}}_{{{\text{t}} + 1}} }} ({\Omega}_{{{\text{V}}_{\text{t}} }} {\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{V}}_{\text{t}} }} )+ {\Gamma}_{{{\text{F}}_{{{\text{t}} + 1}} }} ({\Psi}_{{{\text{V}}_{\text{t}} }} {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{V}}_{\text{t}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} ) + {\Gamma}_{{{\text{V}}_{\text{t}} }} ({\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} )] \\ & \quad + {\Lambda}_{{{\text{F}}_{{{\text{t}} + 1}} }} [{\Gamma}_{{{\text{V}}_{{{\text{t}} + 1}} }} ({\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{V}}_{\text{t}} }} - {\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{V}}_{\text{t}} }} )+ {\Gamma}_{{{\text{E}}_{{{\text{t}} + 1}} }} ({\Omega}_{{{\text{V}}_{\text{t}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{V}}_{\text{t}} }} ) + {\Gamma}_{{{\text{V}}_{\text{t}} }} ({\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} )] \\ & \quad + {\Lambda}_{{{\text{V}}_{{{\text{t}} + 1}} }} [{\Gamma}_{{{\text{F}}_{{{\text{t}} + 1}} }} ({\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{V}}_{\text{t}} }} - {\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{V}}_{\text{t}} }} )+ {\Gamma}_{{{\text{E}}_{{{\text{t}} + 1}} }} ({\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{V}}_{\text{t}} }} - {\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{V}}_{\text{t}} }} )+ {\Gamma}_{{{\text{V}}_{\text{t}} }} ({\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} - {\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} ) ]\\ & \quad - {\Lambda}_{{{\text{V}}_{\text{t}} }} [\left. {{\Gamma}_{{{\text{V}}_{{{\text{t}} + 1}} }} ({\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} - {\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} ) + {\Gamma}_{{{\text{F}}_{{{\text{t}} + 1}} }} ({\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} ) + {\Gamma}_{{{\text{E}}_{{{\text{t}} + 1}} }} ({\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} )]} \right\} \\ & \quad \left\{ {{\Lambda}_{{{\text{L}}_{{{\text{wt}} + 1}} }} [{\Gamma}_{{{\text{V}}_{{{\text{t}} + 1}} }} ({\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} - {\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} ) + {\Gamma}_{{{\text{F}}_{{{\text{t}} + 1}} }} ({\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} ) + {\Gamma}_{{{\text{E}}_{{{\text{t}} + 1}} }} ({\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} )]} \right. \\ & \quad - {\Lambda}_{{{\text{E}}_{{{\text{t}} + 1}} }} [{\Gamma}_{{{\text{V}}_{{{\text{t}} + 1}} }} ({\Omega}_{{{\text{L}}_{{{\text{wt}} + 1}} }} {\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{L}}_{{{\text{wt}} + 1}} }} ) + {\Gamma}_{{{\text{F}}_{{{\text{t}} + 1}} }} ({\Psi}_{{{\text{L}}_{{{\text{wt}} + 1}} }} {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{L}}_{{{\text{wt}} + 1}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} ) + {\Gamma}_{{{\text{L}}_{{{\text{wt}} + 1}} }} ({\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} ) ]\\ & \quad - {\Lambda}_{{{\text{F}}_{{{\text{t}} + 1}} }} [{\Gamma}_{{{\text{V}}_{{{\text{t}} + 1}} }} ({\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{L}}_{{{\text{wt}} + 1}} }} - {\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{L}}_{{{\text{wt}} + 1}} }} )+ {\Gamma}_{{{\text{E}}_{{{\text{t}} + 1}} }} ({\Omega}_{{{\text{L}}_{{{\text{wt}} + 1}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{L}}_{{{\text{wt}} + 1}} }} )+ {\Gamma}_{{{\text{L}}_{{{\text{wt}} + 1}} }} ({\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} ) ]\\ & \quad - {\Lambda}_{{{\text{V}}_{{{\text{t}} + 1}} }} \left. { [{\Gamma}_{{{\text{F}}_{{{\text{t}} + 1}} }} ({\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{L}}_{{{\text{wt}} + 1}} }} - {\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{L}}_{{{\text{wt}} + 1}} }} )+ {\Gamma}_{{{\text{E}}_{{{\text{t}} + 1}} }} ({\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{L}}_{{{\text{wt}} + 1}} }} - {\Omega}_{{{\text{L}}_{{{\text{wt}} + 1}} }} {\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} )+ {\Gamma}_{{{\text{L}}_{{{\text{wt}} + 1}} }} ({\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} - {\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} ) ]} \right\}^{ - 1} , \\ \end{aligned} $$

$$ \begin{aligned} {\text{d}}_{ 43} & \equiv \left\{ {{\Lambda}_{{{\text{E}}_{{{\text{t}} + 1}} }} [{\Gamma}_{{{\text{V}}_{{{\text{t}} + 1}} }} ({\Omega}_{{{\text{F}}_{\text{t}} }} {\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{F}}_{\text{t}} }} )+ {\Gamma}_{{{\text{F}}_{{{\text{t}} + 1}} }} ({\Psi}_{{{\text{F}}_{\text{t}} }} {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{F}}_{\text{t}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} ) + {\Gamma}_{{{\text{F}}_{\text{t}} }} ({\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} ) ]} \right. \\ & \quad + {\Lambda}_{{{\text{F}}_{{{\text{t}} + 1}} }} [{\Gamma}_{{{\text{V}}_{{{\text{t}} + 1}} }} ({\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{F}}_{\text{t}} }} - {\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{F}}_{\text{t}} }} )+ {\Gamma}_{{{\text{E}}_{{{\text{t}} + 1}} }} ({\Omega}_{{{\text{F}}_{\text{t}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{F}}_{\text{t}} }} ) + {\Gamma}_{{{\text{F}}_{\text{t}} }} ({\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} )] \\ & \quad + {\Lambda}_{{{\text{V}}_{{{\text{t}} + 1}} }} [{\Gamma}_{{{\text{F}}_{{{\text{t}} + 1}} }} ({\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{F}}_{\text{t}} }} - {\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{F}}_{\text{t}} }} ) + {\Gamma}_{{{\text{E}}_{{{\text{t}} + 1}} }} ({\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{F}}_{\text{t}} }} - {\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{F}}_{\text{t}} }} ) + {\Gamma}_{{{\text{F}}_{\text{t}} }} ({\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} - {\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} )] \\ & \quad - {\Lambda}_{{{\text{F}}_{\text{t}} }} [\left. {{\Gamma}_{{{\text{V}}_{{{\text{t}} + 1}} }} ({\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} - {\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} ) + {\Gamma}_{{{\text{F}}_{{{\text{t}} + 1}} }} ({\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} ) + {\Gamma}_{{{\text{E}}_{{{\text{t}} + 1}} }} ({\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} )]} \right\} \\ & \quad \left\{ {{\Lambda}_{{{\text{L}}_{{{\text{wt}} + 1}} }} [{\Gamma}_{{{\text{V}}_{{{\text{t}} + 1}} }} ({\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} - {\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} ) + {\Gamma}_{{{\text{F}}_{{{\text{t}} + 1}} }} ({\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} ) + {\Gamma}_{{{\text{E}}_{{{\text{t}} + 1}} }} ({\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} )]} \right. \\ & \quad - {\Lambda}_{{{\text{E}}_{{{\text{t}} + 1}} }} [{\Gamma}_{{{\text{V}}_{{{\text{t}} + 1}} }} ({\Omega}_{{{\text{L}}_{{{\text{wt}} + 1}} }} {\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{L}}_{{{\text{wt}} + 1}} }} ) + {\Gamma}_{{{\text{F}}_{{{\text{t}} + 1}} }} ({\Psi}_{{{\text{L}}_{{{\text{wt}} + 1}} }} {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{L}}_{{{\text{wt}} + 1}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} ) + {\Gamma}_{{{\text{L}}_{{{\text{wt}} + 1}} }} ({\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} ) ]\\ & \quad - {\Lambda}_{{{\text{F}}_{{{\text{t}} + 1}} }} [{\Gamma}_{{{\text{V}}_{{{\text{t}} + 1}} }} ({\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{L}}_{{{\text{wt}} + 1}} }} - {\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{L}}_{{{\text{wt}} + 1}} }} )+ {\Gamma}_{{{\text{E}}_{{{\text{t}} + 1}} }} ({\Omega}_{{{\text{L}}_{{{\text{wt}} + 1}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{L}}_{{{\text{wt}} + 1}} }} )+ {\Gamma}_{{{\text{L}}_{{{\text{wt}} + 1}} }} ({\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} ) ]\\ & \quad - {\Lambda}_{{{\text{V}}_{{{\text{t}} + 1}} }} \left. { [{\Gamma}_{{{\text{F}}_{{{\text{t}} + 1}} }} ({\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{L}}_{{{\text{wt}} + 1}} }} - {\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{L}}_{{{\text{wt}} + 1}} }} )+ {\Gamma}_{{{\text{E}}_{{{\text{t}} + 1}} }} ({\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{L}}_{{{\text{wt}} + 1}} }} - {\Omega}_{{{\text{L}}_{{{\text{wt}} + 1}} }} {\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} )+ {\Gamma}_{{{\text{L}}_{{{\text{wt}} + 1}} }} ({\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} - {\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} ) ]} \right\}^{ - 1} , \\ \end{aligned} $$

$$ \begin{aligned} {\text{d}}_{ 44} & \equiv \left\{ {{\Lambda}_{{{\text{E}}_{{{\text{t}} + 1}} }} } \right.[{\Gamma}_{{{\text{V}}_{{{\text{t}} + 1}} }} ({\Omega}_{{{\text{L}}_{\text{wt}} }} {\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{L}}_{\text{wt}} }} )+ {\Gamma}_{{{\text{F}}_{{{\text{t}} + 1}} }} ({\Psi}_{{{\text{L}}_{\text{wt}} }} {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{L}}_{\text{wt}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} ) + {\Gamma}_{{{\text{L}}_{\text{wt}} }} ({\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} )] \\ & \quad + {\Lambda}_{{{\text{F}}_{{{\text{t}} + 1}} }} [{\Gamma}_{{{\text{V}}_{{{\text{t}} + 1}} }} ({\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Psi}_{{_{{{\text{L}}_{\text{wt}} }} }} - {\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{_{{{\text{L}}_{\text{wt}} }} }} )+ {\Gamma}_{{{\text{E}}_{{{\text{t}} + 1}} }} ({\Omega}_{{{\text{L}}_{\text{wt}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{L}}_{\text{wt}} }} ) + {\Gamma}_{{{\text{L}}_{\text{wt}} }} ({\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} )] \\ & \quad + {\Lambda}_{{{\text{V}}_{{{\text{t}} + 1}} }} [{\Gamma}_{{{\text{F}}_{{{\text{t}} + 1}} }} ({\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{L}}_{\text{wt}} }} - {\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{L}}_{\text{wt}} }} ) + {\Gamma}_{{{\text{E}}_{{{\text{t}} + 1}} }} ({\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{L}}_{\text{wt}} }} - {\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{L}}_{\text{wt}} }} ) + {\Gamma}_{{{\text{L}}_{\text{wt}} }} ({\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} - {\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} )] \\ & \quad - {\Lambda}_{{{\text{L}}_{\text{wt}} }} [\left. {{\Gamma}_{{{\text{V}}_{{{\text{t}} + 1}} }} ({\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} - {\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} ) + {\Gamma}_{{{\text{F}}_{{{\text{t}} + 1}} }} ({\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} ) + {\Gamma}_{{{\text{E}}_{{{\text{t}} + 1}} }} ({\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} )]} \right\} \\ & \quad \left\{ {{\Lambda}_{{{\text{L}}_{{{\text{wt}} + 1}} }} [{\Gamma}_{{{\text{V}}_{{{\text{t}} + 1}} }} ({\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} - {\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} ) + {\Gamma}_{{{\text{F}}_{{{\text{t}} + 1}} }} ({\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} ) + {\Gamma}_{{{\text{E}}_{{{\text{t}} + 1}} }} ({\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} )]} \right. \\ & \quad - {\Lambda}_{{{\text{E}}_{{{\text{t}} + 1}} }} [{\Gamma}_{{{\text{V}}_{{{\text{t}} + 1}} }} ({\Omega}_{{{\text{L}}_{{{\text{wt}} + 1}} }} {\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{L}}_{{{\text{wt}} + 1}} }} ) + {\Gamma}_{{{\text{F}}_{{{\text{t}} + 1}} }} ({\Psi}_{{{\text{L}}_{{{\text{wt}} + 1}} }} {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{L}}_{{{\text{wt}} + 1}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} ) + {\Gamma}_{{{\text{L}}_{{{\text{wt}} + 1}} }} ({\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} ) ]\\ & \quad - {\Lambda}_{{{\text{F}}_{{{\text{t}} + 1}} }} [{\Gamma}_{{{\text{V}}_{{{\text{t}} + 1}} }} ({\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{L}}_{{{\text{wt}} + 1}} }} - {\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{L}}_{{{\text{wt}} + 1}} }} )+ {\Gamma}_{{{\text{E}}_{{{\text{t}} + 1}} }} ({\Omega}_{{{\text{L}}_{{{\text{wt}} + 1}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{L}}_{{{\text{wt}} + 1}} }} )+ {\Gamma}_{{{\text{L}}_{{{\text{wt}} + 1}} }} ({\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{V}}_{{{\text{t}} + 1}} }} - {\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{V}}_{{{\text{t}} + 1}} }} ) ]\\ & \quad - {\Lambda}_{{{\text{V}}_{{{\text{t}} + 1}} }} \left. { [{\Gamma}_{{{\text{F}}_{{{\text{t}} + 1}} }} ({\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{L}}_{{{\text{wt}} + 1}} }} - {\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{L}}_{{{\text{wt}} + 1}} }} )+ {\Gamma}_{{{\text{E}}_{{{\text{t}} + 1}} }} ({\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Psi}_{{{\text{L}}_{{{\text{wt}} + 1}} }} - {\Omega}_{{{\text{L}}_{{{\text{wt}} + 1}} }} {\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} )+ {\Gamma}_{{{\text{L}}_{{{\text{wt}} + 1}} }} ({\Psi}_{{{\text{F}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{E}}_{{{\text{t}} + 1}} }} - {\Psi}_{{{\text{E}}_{{{\text{t}} + 1}} }} {\Omega}_{{{\text{F}}_{{{\text{t}} + 1}} }} ) ]} \right\}^{ - 1} . \\ \end{aligned} $$

Notice that all partial derivatives are evaluated at (E*, V*, F*, ${\mathrm{L}}_{\mathrm{w}}^{ *}$).

The solution to (C.1) is

$$ {\mathrm{E}}_{\mathrm{t}} - {\mathrm{E* }} = {\mathrm{ G}}_{ 1} {\mathrm{j}}_{ 1 1} \mu_{ 1}^{\mathrm{t}} + {\mathrm{G}}_{ 2} {\mathrm{j}}_{ 1 2} \mu_{ 2}^{\mathrm{t}} + {\mathrm{G}}_{ 3} {\mathrm{j}}_{ 1 3} \mu_{ 3}^{\mathrm{t}} + {\mathrm{G}}_{ 4} {\mathrm{j}}_{ 1 4} \mu_{ 4}^{\mathrm{t}} , $$

(C.2)

$$ {\mathrm{V}}_{\mathrm{t}} - {\mathrm{V*}} = {\mathrm{G}}_{ 1} {\mathrm{j}}_{ 2 1} \mu_{ 1}^{\mathrm{t}} + {\mathrm{G}}_{ 2} {\mathrm{j}}_{ 2 2} \mu_{ 2}^{\mathrm{t}} + {\mathrm{G}}_{ 3} {\mathrm{j}}_{ 2 3} \mu_{ 3}^{\mathrm{t}} + {\mathrm{G}}_{ 4} {\mathrm{j}}_{ 2 4} \mu_{ 4}^{\mathrm{t}} , $$

(C.3)

$$ {\mathrm{F}}_{\mathrm{t}} - {\mathrm{F*}} = {\mathrm{G}}_{ 1} {\mathrm{j}}_{ 3 1} \mu_{ 1}^{\mathrm{t}} + {\mathrm{G}}_{ 2} {\mathrm{j}}_{ 3 2} \mu_{ 2}^{\mathrm{t}} + {\mathrm{G}}_{ 3} {\mathrm{j}}_{ 3 3} \mu_{ 3}^{\mathrm{t}} + {\mathrm{G}}_{ 4} {\mathrm{j}}_{ 3 4} \mu_{ 4}^{\mathrm{t}} , $$

(C.4)

$$ {\mathrm{L}}_{\mathrm{wt}} - {\mathrm{L}}_{\mathrm{w}}^{ *} = {\mathrm{G}}_{ 1} {\mathrm{j}}_{ 4 1} \mu_{ 1}^{\mathrm{t}} + {\mathrm{G}}_{ 2} {\mathrm{j}}_{ 4 2} \mu_{ 2}^{\mathrm{t}} + {\mathrm{G}}_{ 3} {\mathrm{j}}_{ 4 3} \mu_{ 3}^{\mathrm{t}} + {\mathrm{G}}_{ 4} {\mathrm{j}}_{ 4 4} \mu_{ 4}^{\mathrm{t}} , $$

(C.5)

where $\left[ \begin{aligned} \mu_{ 1} \, \hfill \\ \mu_{ 2} \hfill \\ \mu_{ 3} \hfill \\ \mu_{ 4} \hfill \\ \end{aligned} \right]$ and $J \equiv \left[ \begin{aligned} {\mathrm{j}}_{ 1 1} {\mathrm{ j}}_{ 1 2} {\mathrm{ j}}_{ 1 3} {\mathrm{ j}}_{ 1 4} \hfill \\ {\mathrm{j}}_{ 2 1} {\mathrm{ j}}_{ 2 2} {\mathrm{ j}}_{ 2 3} {\mathrm{ j}}_{ 2 4} \hfill \\ {\mathrm{j}}_{ 3 1} {\mathrm{ j}}_{ 3 2} {\mathrm{ j}}_{ 3 3} {\mathrm{ j}}_{ 3 4} \hfill \\ {\mathrm{j}}_{ 4 1} {\mathrm{ j}}_{ 4 2} {\mathrm{ j}}_{ 4 3} {\mathrm{ j}}_{ 4 4} \hfill \\ \end{aligned} \right]$ are, respectively, the eigenvalues and the eigenvectors of the matrix $D \equiv \left[ \begin{aligned} {\mathrm{d}}_{ 1 1} {\mathrm{ d}}_{ 1 2} {\mathrm{ d}}_{ 1 3} {\mathrm{ d}}_{ 1 4} \, \hfill \\ {\mathrm{d}}_{ 2 1} {\mathrm{ d}}_{ 2 2} {\mathrm{ d}}_{ 2 3} {\mathrm{ d}}_{ 2 4} \hfill \\ {\mathrm{d}}_{ 3 1} {\mathrm{ d}}_{ 3 2} {\mathrm{ d}}_{ 3 3} {\mathrm{ d}}_{ 3 4} \hfill \\ {\mathrm{d}}_{ 4 1} {\mathrm{ d}}_{ 4 2} {\mathrm{ d}}_{ 4 3} {\mathrm{ d}}_{ 4 4} \hfill \\ \end{aligned} \right]$, and $\left[ \begin{aligned} {\mathrm{G}}_{ 1} \, \hfill \\ {\mathrm{G}}_{ 2} \hfill \\ {\mathrm{G}}_{ 3} \hfill \\ {\mathrm{G}}_{ 4} \hfill \\ \end{aligned} \right]$ are constants whose values are determined by using the initial conditions. The eigenvalues and the eigenvectors can be found by solving, respectively, ${\mathrm{Det}}(D - {\mathrm{M}}) = 0$ and DJ = JM, where ${\mathrm{M}} \equiv \left[ {\begin{array}{*{20}l} {\mu _{{\mathrm{1}}} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & {\mu _{{\mathrm{2}}} } \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & {\mu _{{\mathrm{3}}} } \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & {\mu _{{\mathrm{4}}} } \hfill \\ \end{array} } \right] .$