Appendix 1: Derivation of Eqs. (4)–(6)
The quadratic utility function is:
$$\begin{aligned} U= & {} \frac{1}{2}u_{\textit{CC}}C_{t}^{2}+ \frac{1}{2}u_{\textit{RR}}R_{t}^{2}+ \frac{1}{2}u_{\textit{HH}}H_{t}^{2}+ \frac{1}{2}u_{\textit{NN}}N_{t}^{2} +\frac{1}{2}u_{\textit{SS}}S_{t}^{2}+ \frac{1}{2}u_{\textit{DD}}D_{t}^{2}+ \frac{1}{2}u_{LL}L_{t}^{2}\\&+\,u_{\textit{CR}}C_{t}R_{t}+ u_{CH}C_{t}H_{t}+ u_{\textit{CS}}C_{t}S_{t}+ u_{\textit{CD}}C_{t}D_{t}+ u_{CL}C_{t}L_{t}+ u_{RH}R_{t}H_{t}\\&+\, u_{\textit{RS}}R_{t}S_{t}+{ u}_{\textit{RD}}R_{t}D_{t}+ u_{RL}R_{t}L_{t}+{ u_{\textit{HS}}H_{t}S_{t}+u}_{HD}H_{t}D_{t}+{{ u_{HL}H_{t}L_{t}+u}_{\textit{SD}}S}_{t}D_{t}\\&+\, u_{\textit{SL}}S_{t}L_{t}+ { u}_{\textit{DL}}D_{t}L_{t}+{ u}_{C}C_{t}+ u_{R}R_{t}+ u_{H}H_{t}+{{ u_{N}N_{t}+u}_{S}S_{t}+u}_{D}D_{t}+{ u}_{L}L_{t} \end{aligned}$$
The \(u_{ij}\)’s are parameters carrying the sign of their respective derivatives (e.g., \(u_{\textit{CC}}<0\) because \(U_{\textit{CC}}<0)\).
$$\begin{aligned} \max L= & {} \sum \limits _{t=1}^\infty \beta ^{t-1} U\left( C_{t },R_{t},{H_{t},S}_{t},D_{t}, {L_{t},N}_{t} \right) \\&+\,\lambda \left( W-\sum _{t=1}^\infty \beta ^{t-1} \left( P_{\textit{Ct}}C_{t}+P_{\textit{Rt}}R_{t}+{P_{\textit{Ht}}H_{t}+N}_{t } \right) \right) \end{aligned}$$
Derive the first-order condition (FOC) with respect to \(C_{t}\):
$$\begin{aligned} \frac{\partial L}{\partial C_{t}}= & {} \frac{\partial U\left( C_{t },R_{t},{H_{t},S}_{t},D_{t}, {L_{t},N}_{t} \right) }{\partial C_{t}}\\&+\,\beta \frac{\partial U\left( C_{t+1 },R_{t+1},{H_{t+1},S}_{t+1},D_{t+1}, {L_{t+1},N}_{t+1} \right) }{\partial C_{t}}-\lambda P_{C_{t}}\\= & {} u_{\textit{CC}}C_{t}+u_{\textit{CR}}R_{t}+u_{CH}H_{t}+u_{\textit{CS}}C_{t-1}+u_{\textit{CD}}\left( R_{t-1}+\delta H_{t-1} \right) \\&+\,u_{CL}\left( 1-\delta \right) H_{t-1}+u_{C}+\beta (u_{\textit{SS}}C_{t}+u_{\textit{CS}}C_{t+1}+ u_{\textit{RS}}R_{t+1}\\&+\, u_{\textit{HS}}H_{t+1}+u_{\textit{SD}}(R_{t}+\delta H_{t})+\,u_{\textit{SL}}(1-\delta ) H_{t}+u_{S})- \lambda P_{\textit{Ct}} =\quad 0 \end{aligned}$$
Solving the FOC for \(C_{t}\):
$$\begin{aligned} C_{t}= & {} \beta _{10}+\beta _{11 }C_{t-1}+\beta _{12}C_{t+1}+\beta _{13}R_{t-1}+\beta _{14}R_{t}+\beta _{15}R_{t+1}+\beta _{16}H_{t-1}\\&+\,{ \beta }_{17}H_{t}+\beta _{18}H_{t+1}+\beta _{19} P_{\textit{Ct}} \end{aligned}$$
where
$$\begin{aligned} \beta _{10}= & {} -\frac{u_{C}+\beta u_{S}}{u_{\textit{CC}}+\beta u_{\textit{SS}}} \qquad \qquad \beta _{14}=-\frac{u_{\textit{CR}}+\beta u_{\textit{SD}}}{u_{\textit{CC}}+\beta u_{\textit{SS}}}\qquad \beta _{18}=-\frac{\beta u_{\textit{HS}}}{u_{\textit{CC}}+\beta u_{\textit{SS}}}\\ \beta _{11}= & {} -\frac{u_{\textit{CS}}}{u_{\textit{CC}}+\beta u_{\textit{SS}}}>0 \qquad \beta _{15}=-\frac{\beta u_{\textit{RS}}}{u_{\textit{CC}}+\beta u_{\textit{SS}}}\qquad \beta _{19}=\frac{\lambda }{u_{\textit{CC}}+\beta u_{\textit{SS}}}<0\\ \beta _{12}= & {} -\frac{\beta u_{\textit{CS}}}{u_{\textit{CC}}+\beta u_{\textit{SS}}}>0\qquad \beta _{16}=-\frac{{u_{\textit{CD}}\delta +u}_{CL}(1-\delta )}{u_{\textit{CC}}+\beta u_{\textit{SS}}}\\ \beta _{13}= & {} -\frac{u_{\textit{CD}}}{u_{\textit{CC}}+\beta u_{\textit{SS}}} \qquad \qquad \beta _{17}=-\frac{u_{CH}+\beta u_{\textit{SD}}\delta +\beta u_{\textit{SL}}(1-\delta )}{u_{\textit{CC}}+\beta u_{\textit{SS}}}\\ \frac{\partial L}{\partial R_{t}}= & {} \frac{\partial U\left( C_{t },R_{t},{H_{t},S}_{t},D_{t}, {L_{t},N}_{t} \right) }{\partial R_{t}}\\&+\,\beta \frac{\partial U\left( C_{t+1 },R_{t+1},{H_{t+1},S}_{t+1},D_{t+1}, {L_{t+1},N}_{t+1} \right) }{\partial R_{t}}-\lambda P_{R_{t}}\\= & {} u_{\textit{RR}}R_{t}+u_{\textit{CR}}C_{t}+u_{RH}H_{t}+u_{\textit{RS}}C_{t-1}+u_{\textit{RD}}\left( R_{t-1}+\delta H_{t-1} \right) \\&+\,u_{RL}\left( 1-\delta \right) H_{t-1}+u_{R}+\,\beta ({u_{\textit{DD}}{(R}_{t}+\delta H_{t})+u}_{\textit{CD}}C_{t+1}+u_{\textit{RD}}R_{t+1}\\&+\,u_{HD}H_{t+1}+u_{\textit{SD}}C_{t}+ u_{\textit{DL}}(1-\delta )H_{t} +u_{D}) -\lambda P_{\textit{Rt}}= 0\\ R_{t}= & {} \beta _{20}+\beta _{21 }R_{t-1} +\beta _{22}R_{t+1}+\beta _{23}C_{t-1}+\beta _{24}C_{t}+\beta _{25}C_{t+1}+\beta _{26}H_{t-1}\\&+\,{ \beta }_{27}H_{t}+\,\beta _{28}H_{t+1}+\beta _{29} P_{\textit{Rt}} \end{aligned}$$
where
$$\begin{aligned} \beta _{20}= & {} -\frac{u_{R}+\beta u_{D}}{u_{\textit{RR}}+\beta u_{\textit{DD}}}\qquad \qquad \beta _{24}=-\frac{u_{\textit{CR}}+\beta u_{\textit{SD}}}{u_{\textit{RR}}+\beta u_{\textit{DD}}}\qquad \beta _{28}=-\frac{\beta u_{HD}}{u_{\textit{RR}}+\beta u_{\textit{DD}}}\\ \beta _{21}= & {} -\frac{u_{\textit{RD}}}{u_{\textit{RR}}+\beta u_{\textit{DD}}}>0\qquad \beta _{25}=-\frac{\beta u_{\textit{CD}}}{u_{\textit{RR}}+\beta u_{\textit{DD}}}\qquad \beta _{29}=\frac{\lambda }{u_{\textit{RR}}+\beta u_{\textit{DD}}}<0\\ \beta _{22}= & {} -\frac{\beta u_{\textit{RD}}}{u_{\textit{RR}}+\beta u_{\textit{DD}}}>0\qquad \beta _{26}=-\frac{u_{\textit{RD}}\delta +u_{RL}\left( 1-\delta \right) }{u_{\textit{RR}}+\beta u_{\textit{DD}}}\\ \beta _{23}= & {} -\frac{u_{\textit{RS}}}{u_{\textit{RR}}+\beta u_{\textit{DD}}}\qquad \qquad \beta _{27}=-\frac{u_{RH}+{\beta u}_{\textit{DD}}\delta +\beta u_{\textit{DL}}\left( 1-\delta \right) }{u_{\textit{RR}}+\beta u_{\textit{DD}}}\\ \frac{\partial L}{\partial H_{t}}= & {} \frac{\partial U\left( C_{t },R_{t},{H_{t},S}_{t},D_{t}, {L_{t},N}_{t} \right) }{\partial H_{t}}\\&+\,\beta \frac{\partial U\left( C_{t+1},R_{t+1},{H_{t+1},S}_{t+1},D_{t+1}, {L_{t+1},N}_{t+1} \right) }{\partial H_{t}}-\lambda P_{H_{t}}\\= & {} u_{\textit{HH}}H_{t}+u_{CH}C_{t}+u_{RH}R_{t}+u_{\textit{HS}}C_{t-1}+u_{HD}\left( R_{t-1}+\delta H_{t-1} \right) \\&+\,u_{HL}\left( 1-\delta \right) H_{t-1}+u_{H}\\&+\,\beta (\delta {u_{\textit{DD}}{(R}_{t}+\delta H_{t})+u}_{LL}\left( 1-\delta \right) ^{2}H_{t}+{\delta u}_{\textit{CD}}C_{t+1}+{\left( 1-\delta \right) u}_{CL}C_{t+1}\\&+\,{\delta u}_{\textit{RD}}R_{t+1}+{\left( 1{-}\delta \right) u}_{\textit{RL}}R_{t+1}+{\delta u}_{HD}H_{t+1}+{\left( 1-\delta \right) u}_{HL}H_{t+1}+{\delta u}_{\textit{SD}}C_{t}\\&+\,{\left( 1-\delta \right) u}_{\textit{SL}}C_{t}+u_{\textit{DL}}\left( 1-\delta \right) R_{t}+{2u}_{\textit{DL}}(1-\delta ) {\delta H}_{t}\\&+\,{\delta u}_{D}+{(1-\delta )u}_{L}) -\lambda P_{\textit{Rt}}= 0\\ H_{t}= & {} \beta _{30}+\beta _{31 }H_{t-1}+\beta _{32}H_{t+1}+\beta _{33}C_{t-1}+\beta _{34}C_{t}+\beta _{35}C_{t+1}+\beta _{36}R_{t-1}\\&+\,{\beta }_{37}R_{t}+\,\beta _{38}R_{t+1}+\beta _{39} P_{\textit{Ht}} \end{aligned}$$
where
$$\begin{aligned} \beta _{30}= & {} -\frac{u_{H}+\beta \delta u_{D}+\beta u_{L}\left( 1-\delta \right) }{u_{\textit{HH}}+\beta u_{\textit{DD}}\delta ^{2}+{\beta u}_{LL}\left( 1-\delta \right) ^{2}+2{\beta u}_{\textit{DL}}\delta \left( 1-\delta \right) }\\ \beta _{31}= & {} -\frac{u_{HD}\delta +u_{HL}\left( 1-\delta \right) }{u_{\textit{HH}}+\beta u_{\textit{DD}}\delta ^{2}+{\beta u}_{LL}\left( 1-\delta \right) ^{2}+2{\beta u}_{\textit{DL}}\delta \left( 1-\delta \right) }\\ \beta _{32}= & {} -\frac{{\beta u}_{HD}\delta +\beta u_{HL}\left( 1-\delta \right) }{u_{\textit{HH}}+\beta u_{\textit{DD}}\delta ^{2}+{\beta u}_{LL}\left( 1-\delta \right) ^{2}+2{\beta u}_{\textit{DL}}\delta \left( 1-\delta \right) }\\ \beta _{33}= & {} -\frac{u_{\textit{HS}}}{u_{\textit{HH}}+\beta u_{\textit{DD}}\delta ^{2}+{\beta u}_{LL}\left( 1-\delta \right) ^{2}+2{\beta u}_{\textit{DL}}\delta \left( 1-\delta \right) }\\ \beta _{34}= & {} -\frac{u_{CH}+\beta u_{\textit{SD}}\delta +\beta u_{\textit{SL}}\left( 1-\delta \right) }{u_{\textit{HH}}+\beta u_{\textit{DD}}\delta ^{2}+{\beta u}_{LL}\left( 1-\delta \right) ^{2}+2{\beta u}_{\textit{DL}}\delta \left( 1-\delta \right) }\\ \beta _{35}= & {} -\frac{\beta u_{\textit{CD}}\delta +\beta u_{CL}\left( 1-\delta \right) }{u_{\textit{HH}}+\beta u_{\textit{DD}}\delta ^{2}+{\beta u}_{LL}\left( 1-\delta \right) ^{2}+2{\beta u}_{\textit{DL}}\delta \left( 1-\delta \right) }\\ \beta _{36}= & {} -\frac{u_{HD}}{u_{\textit{HH}}+\beta u_{\textit{DD}}\delta ^{2}+{\beta u}_{LL}\left( 1-\delta \right) ^{2}+2{\beta u}_{\textit{DL}}\delta \left( 1-\delta \right) }\\ \beta _{37}= & {} -\frac{u_{RH}+\beta u_{\textit{DD}}\delta +\beta u_{\textit{DL}}\left( 1-\delta \right) }{u_{\textit{HH}}+\beta u_{\textit{DD}}\delta ^{2}+{\beta u}_{LL}\left( 1-\delta \right) ^{2}+2{\beta u}_{\textit{DL}}\delta \left( 1-\delta \right) }\\ \beta _{38}= & {} -\frac{\beta u_{\textit{RD}}\delta +\beta u_{\textit{RL}}(1-\delta )}{u_{\textit{HH}}+\beta u_{\textit{DD}}\delta ^{2}+{\beta u}_{LL}\left( 1-\delta \right) ^{2}+2{\beta u}_{\textit{DL}}\delta \left( 1-\delta \right) }\\ \beta _{39}= & {} \frac{\lambda }{u_{\textit{HH}}+\beta u_{\textit{DD}}\delta ^{2}+{\beta u}_{LL}\left( 1-\delta \right) ^{2}+2{\beta u}_{\textit{DL}}\delta \left( 1-\delta \right) }<0 \end{aligned}$$
For \({k}=1,2,3\): \(\beta _{k2}=\beta *\beta _{k1}\) or \(\beta _{k1}=(1+r)\beta _{k2}\) since \(\beta =\frac{1}{(1+r)}\) with r being discount rate.
Appendix 2: Calculation of Lewbel price indices for alcoholic beverages
Lewbel price indices allow heterogeneity in preferences within a given bundle of goods. Within bundle, Cobb Douglas preferences are assumed, while among different bundles any specification is allowed. See Lewbel (1989) for details. Following Lewbel (1989) and Hoderlein and Mihaleva (2008), we construct Lewbel price indices as:
$$\begin{aligned} { v}_{i}=\frac{1}{k_{i}}\prod \nolimits _{j=1}^{n_{i}} \left( \frac{p_{ij}}{w_{ij}} \right) ^{w_{ij}} \end{aligned}$$
where \(w_{ij}\) is the household’s budget share of good j in group i. \({k}_{i}\) is a scaling factor with \(k_{i}=\prod \nolimits _{j=1}^{n_{i}} \bar{w}_{ij}^{{-\bar{w}}_{ij}} \) and \(\bar{w}_{ij}\) is the budget share of the reference household.
Let \(p_{ij}=P_{i}\) where \(P_{i}\) is the price index for group i which is set to 1 in the first time period. Because there are zero expenditures for some subcategories, Lewbel price index cannot be used in levels. In the empirical analysis, Hoderlein and Mihaleva (2008) used log prices instead of prices in levels using the result that \(\lim \log _{x\rightarrow 0}{x\log {\left( x \right) =0}}\). In our economic model, prices are in levels, so we first took the log of the Lewbel price index and then took the anti-log of it to obtain price indices.
In the current study, zero alcohol consumption might be due to so many different reasons such as quitting, abstention, corner solution, and infrequency of purchase. For non-consumers, the Lewbel price index is assigned to be equal to 1 which means if the consumption took place, the expenditure shares would have been identical to that of reference household. To determine the expenditure shares of the reference household, we took the average of the expenditure shares for each consumer unit in the whole sample in the whole sample period.
Appendix 3: Long-run elasticity equations
$$\begin{aligned} E_{CC}= & {} \{[ \beta _{68}\left( \beta _{43}+\beta _{44} \right) \left( \beta _{55}+\beta _{56} \right) +\beta _{58}\left( \beta _{43}+\beta _{44} \right) \left( 1-\beta _{61}-\beta _{62} \right) \\&+\,\beta _{58}\left( \beta _{45}+\beta _{46} \right) \left( \beta _{65}+\beta _{66} \right) +\beta _{68}\left( \beta _{45}+\beta _{46} \right) \left( 1-\beta _{51}-\beta _{52} \right) \\&+\,\beta _{47}\left( 1-\beta _{51}-\beta _{52} \right) \left( 1-\beta _{61}-\beta _{62} \right) {-\beta }_{47}\left( \beta _{55}+\beta _{56} \right) \left( \beta _{65}+\,\beta _{66} \right) ]/ \\&[ \left( 1-\beta _{41}-\beta _{42} \right) \left( 1-\beta _{51}-\beta _{52} \right) \left( 1-\beta _{61}-\beta _{62} \right) -\left( 1-\beta _{41}-\,\beta _{42} \right) \\&\times \,\left( \beta _{55}+\beta _{56} \right) \left( \beta _{65}+\beta _{66} \right) -\left( 1-\beta _{51}-\beta _{52} \right) \left( \beta _{45}+\beta _{46} \right) \left( \beta _{63}+\beta _{64} \right) \\&-\,\left( 1-\beta _{61}-\beta _{62} \right) \left( \beta _{43}+\beta _{44} \right) \left( \beta _{53}+\beta _{54} \right) -\left( \beta _{43}+\beta _{44} \right) \left( \beta _{55}+\beta _{56} \right) \\&\times \left( \beta _{63}+\beta _{64} \right) -\,\left( \beta _{45}+\beta _{46} \right) \left( \beta _{65}+\beta _{66} \right) \left( \beta _{53}+\beta _{54} \right) ] \}P_{C}/C\\ \\ E_{RR}= & {} \{ [ \beta _{69}\left( \beta _{53}+\beta _{54} \right) \left( \beta _{45}+\beta _{46} \right) +\beta _{48}\left( \beta _{53}+\beta _{54} \right) \left( 1-\beta _{61}-\beta _{62} \right) \\&+\,\beta _{48}\left( \beta _{55}+\beta _{56} \right) \left( \beta _{63}+\beta _{64} \right) +\beta _{69}\left( \beta _{55}+\beta _{56} \right) \left( 1-\beta _{41}-\beta _{42} \right) \\&+\,\beta _{57}\left( 1-\beta _{41}-\beta _{42} \right) \left( 1-\beta _{61}-\beta _{62} \right) {-\beta }_{57}\left( \beta _{45}+\beta _{46} \right) \left( \beta _{63}\right. \\&\left. +\,\beta _{64} \right) ]/ [ \left( 1-\beta _{41}{-}\beta _{42} \right) \left( 1-\beta _{51}-\beta _{52} \right) \left( 1{-}\beta _{61}-\beta _{62} \right) -\left( 1-\beta _{41}-\,\beta _{42} \right) \\&\times \,\left( \beta _{55}+\beta _{56} \right) \left( \beta _{65}+\beta _{66} \right) -\left( 1-\beta _{51}-\beta _{52} \right) \left( \beta _{45}+\beta _{46} \right) \left( \beta _{63}+\beta _{64} \right) \\&-\,\left( 1-\beta _{61}-\beta _{62} \right) \left( \beta _{43}+\beta _{44} \right) \left( \beta _{53}+\beta _{54} \right) -\left( \beta _{43}+\beta _{44} \right) \left( \beta _{55}+\,\beta _{56} \right) \\&\times \,\left( \beta _{63}+\beta _{64} \right) -\left( \beta _{45}+\beta _{46} \right) \left( \beta _{65}+\beta _{66} \right) \left( \beta _{53}+\beta _{54} \right) ] \}P_{R}/R\\ \\ E_{\textit{HH}}= & {} \{ [ \beta _{59}\left( \beta _{63}+\beta _{64} \right) \left( \beta _{43}+\beta _{44} \right) +\beta _{49}\left( \beta _{63}+\beta _{64} \right) \left( 1-\beta _{51}-\beta _{52} \right) \\&+\,\beta _{49}\left( \beta _{65}+\beta _{66} \right) \left( \beta _{53}+\beta _{54} \right) +\beta _{59}\left( \beta _{65}+\beta _{66} \right) \left( 1-\beta _{41}-\beta _{42} \right) \\&+\,\beta _{67}\left( 1-\beta _{41}-\beta _{42} \right) \left( 1-\beta _{51}-\beta _{52} \right) {-\beta }_{67}\left( \beta _{43}+\beta _{44} \right) \left( \beta _{53}\right. \\&\left. +\,\beta _{54} \right) ]/ [ \left( 1-\beta _{41}-\beta _{42} \right) \left( 1-\beta _{51}-\beta _{52} \right) \left( 1-\beta _{61}-\beta _{62} \right) -\left( 1-\beta _{41}\right. \\&\left. -\,\beta _{42} \right) \left( \beta _{55}+\beta _{56}\right) \left( \beta _{65}+\beta _{66} \right) -\left( 1-\beta _{51}-\beta _{52} \right) \left( \beta _{45}+\beta _{46} \right) \left( \beta _{63}+\beta _{64} \right) \\&-\,\left( 1-\beta _{61}-\beta _{62} \right) \left( \beta _{43}+\beta _{44} \right) \left( \beta _{53}+\beta _{54} \right) -\left( \beta _{43}+\beta _{44} \right) \left( \beta _{55}+\,\beta _{56} \right) \\&\times \,\left( \beta _{63}+\beta _{64} \right) -\left( \beta _{45}+\beta _{46} \right) \left( \beta _{65}+\beta _{66} \right) \left( \beta _{53}+\beta _{54} \right) ] \}P_{H}/H\\ \\ E_{\textit{CR}}= & {} \{ [ \beta _{69}\left( \beta _{43}+\beta _{44} \right) \left( \beta _{55}+\beta _{56} \right) +\beta _{57}\left( \beta _{43}+\beta _{44} \right) \left( 1-\beta _{61}-\beta _{62} \right) \\&+\,\beta _{57}\left( \beta _{45}+\beta _{46} \right) \left( \beta _{65}+\beta _{66} \right) +\beta _{69}\left( \beta _{45}+\beta _{46} \right) \left( 1-\beta _{51}-\beta _{52} \right) \\&+\,\beta _{48}\left( 1-\beta _{51}-\beta _{52} \right) \left( 1-\beta _{61}-\beta _{62} \right) {-\beta }_{48}\left( \beta _{55}+\beta _{56} \right) \left( \beta _{65}\right. \\&\left. +\,\beta _{66} \right) ]/ [ \left( 1-\beta _{41}-\beta _{42} \right) \left( 1-\beta _{51}-\beta _{52} \right) \left( 1-\beta _{61}-\beta _{62} \right) -\left( 1-\beta _{41}\right. \\&\left. -\,\beta _{42} \right) \left( \beta _{55}+\beta _{56} \right) \left( \beta _{65}+\beta _{66} \right) -\left( 1-\beta _{51}-\beta _{52} \right) \left( \beta _{45}+\beta _{46} \right) \left( \beta _{63}+\beta _{64} \right) \\&-\,\left( 1-\beta _{61}-\beta _{62} \right) \left( \beta _{43}+\beta _{44} \right) \left( \beta _{53}+\beta _{54} \right) -\left( \beta _{43}+\beta _{44} \right) \left( \beta _{55}\right. \\&\left. +\,\beta _{56} \right) \left( \beta _{63}+\beta _{64} \right) -\left( \beta _{45}+\beta _{46} \right) \left( \beta _{65}+\beta _{66} \right) \left( \beta _{53}+\beta _{54} \right) ] \}P_{R}/C\\ \end{aligned}$$
$$\begin{aligned} E_{{\textit{CH}}}= & {} \{[ \beta _{67}\left( \beta _{43}+\beta _{44} \right) \left( \beta _{55}+\beta _{56} \right) +\beta _{59}\left( \beta _{43}+\beta _{44} \right) \left( 1-\beta _{61}-\beta _{62} \right) \\&+\,\beta _{59}\left( \beta _{45}+\beta _{46} \right) \left( \beta _{65}+\beta _{66} \right) +\beta _{67}\left( \beta _{45}+\beta _{46} \right) \left( 1-\beta _{51}-\beta _{52} \right) \\&+\,\beta _{49}\left( 1-\beta _{51}-\beta _{52} \right) \left( 1-\beta _{61}-\beta _{62} \right) {-\beta }_{49}\left( \beta _{55}+\beta _{56} \right) \left( \beta _{65}\right. \\&\left. +\,\beta _{66} \right) ]/ [ \left( 1-\beta _{41}-\beta _{42} \right) \left( 1-\beta _{51}-\beta _{52} \right) \left( 1-\beta _{61}-\beta _{62} \right) -\left( 1-\beta _{41}\right. \\&\left. -\,\beta _{42} \right) \left( \beta _{55}+\beta _{56} \right) \left( \beta _{65}+\beta _{66} \right) -\left( 1-\beta _{51}-\beta _{52} \right) \left( \beta _{45}+\beta _{46} \right) \left( \beta _{63}+\beta _{64} \right) \\&-\,\left( 1-\beta _{61}-\beta _{62} \right) \left( \beta _{43}+\beta _{44} \right) \left( \beta _{53}+\beta _{54} \right) -\left( \beta _{43}+\beta _{44} \right) \left( \beta _{55}\right. \\&\left. +\,\beta _{56} \right) \left( \beta _{63}+\beta _{64} \right) -\left( \beta _{45}+\beta _{46} \right) \left( \beta _{65}+\beta _{66} \right) \left( \beta _{53}+\beta _{54} \right) ] \}P_{H}/C\\ \\ E_{\textit{RC}}= & {} \{ [ \beta _{68}\left( \beta _{53}+\beta _{54} \right) \left( \beta _{45}+\beta _{46} \right) +\beta _{47}\left( \beta _{53}+\beta _{54} \right) \left( 1-\beta _{61}-\beta _{62} \right) \\&+\,\beta _{47}\left( \beta _{63}+\beta _{64} \right) \left( \beta _{55}+\beta _{56} \right) +\beta _{68}\left( \beta _{55}+\beta _{56} \right) \left( 1-\beta _{41}-\beta _{42} \right) \\&+\,\beta _{58}\left( 1-\beta _{41}-\beta _{42} \right) \left( 1-\beta _{61}-\beta _{62} \right) {-\beta }_{58}\left( \beta _{45}+\beta _{46} \right) \left( \beta _{63}\right. \\&\left. +\,\beta _{64} \right) ]/ [ \left( 1-\beta _{41}-\beta _{42} \right) \left( 1-\beta _{51}-\beta _{52} \right) \left( 1-\beta _{61}-\beta _{62} \right) -\left( 1-\beta _{41}\right. \\&\left. -\,\beta _{42} \right) \left( \beta _{55}+\beta _{56} \right) \left( \beta _{65}+\beta _{66} \right) -\left( 1-\beta _{51}-\beta _{52} \right) \left( \beta _{45}+\beta _{46} \right) \left( \beta _{63}+\beta _{64} \right) \\&-\,\left( 1-\beta _{61}-\beta _{62} \right) \left( \beta _{43}+\beta _{44} \right) \left( \beta _{53}+\beta _{54} \right) -\left( \beta _{43}+\beta _{44} \right) \left( \beta _{55}\right. \\&\left. +\,\beta _{56} \right) \left( \beta _{63}+\beta _{64} \right) -\left( \beta _{45}+\beta _{46} \right) \left( \beta _{65}+\beta _{66} \right) \left( \beta _{53}+\beta _{54} \right) ] \} P_{C}/R\\ {} E_{{\textit{RH}}}= & {} \{ [ \beta _{67}\left( \beta _{53}+\beta _{54} \right) \left( \beta _{45}+\beta _{46} \right) +\beta _{49}\left( \beta _{53}+\beta _{54} \right) \left( 1-\beta _{61}-\beta _{62} \right) \\&+\,\beta _{49}\left( \beta _{55}+\beta _{56} \right) \left( \beta _{63}+\beta _{64} \right) +\beta _{67}\left( \beta _{55}+\beta _{56} \right) \left( 1-\beta _{41}-\beta _{42} \right) \\&+\,\beta _{59}\left( 1-\beta _{41}-\beta _{42} \right) \left( 1-\beta _{61}-\beta _{62} \right) {-\beta }_{59}\left( \beta _{45}+\beta _{46} \right) \left( \beta _{63}\right. \\&\left. +\,\beta _{64} \right) ]/ [ \left( 1-\beta _{41}-\beta _{42} \right) \left( 1-\beta _{51}-\beta _{52} \right) \left( 1-\beta _{61}-\beta _{62} \right) -\left( 1-\beta _{41}\right. \\&\left. -\,\beta _{42} \right) \left( \beta _{55}+\beta _{56} \right) \left( \beta _{65}+\beta _{66} \right) -\left( 1-\beta _{51}-\beta _{52} \right) \left( \beta _{45}+\beta _{46} \right) \left( \beta _{63}+\beta _{64} \right) \\&-\,\left( 1-\beta _{61}-\beta _{62} \right) \left( \beta _{43}+\beta _{44} \right) \left( \beta _{53}+\beta _{54} \right) -\left( \beta _{43}+\beta _{44} \right) \left( \beta _{55}\right. \\&\left. +\,\beta _{56} \right) \left( \beta _{63}+\beta _{64} \right) -\left( \beta _{45}+\beta _{46} \right) \left( \beta _{65}+\beta _{66} \right) \left( \beta _{53}+\beta _{54} \right) ] \}P_{H}/R\\ \end{aligned}$$
$$\begin{aligned} E_{{\textit{HC}}}= & {} \{ [ \beta _{58}\left( \beta _{43}+\beta _{44} \right) \left( \beta _{63}+\beta _{64} \right) +\beta _{47}\left( \beta _{63}+\beta _{64} \right) \left( 1-\beta _{51}-\beta _{52} \right) \\&+\,\beta _{47}\left( \beta _{65}+\beta _{66} \right) \left( \beta _{53}+\beta _{54} \right) +\beta _{58}\left( \beta _{65}+\beta _{66} \right) \left( 1-\beta _{41}-\beta _{42} \right) \\&+\,\beta _{68}\left( 1-\beta _{41}-\beta _{42} \right) \left( 1-\beta _{51}-\beta _{52} \right) {-\beta }_{68}\left( \beta _{43}+\beta _{44} \right) \left( \beta _{53}\right. \\&\left. +\,\beta _{54} \right) ]/ [ \left( 1-\beta _{41}-\beta _{42} \right) \left( 1-\beta _{51}-\beta _{52} \right) \left( 1-\beta _{61}-\beta _{62} \right) -\left( 1-\beta _{41}\right. \\&\left. -\,\beta _{42} \right) \left( \beta _{55}+\beta _{56} \right) \left( \beta _{65}+\beta _{66} \right) -\left( 1-\beta _{51}-\beta _{52} \right) \left( \beta _{45}+\beta _{46} \right) \left( \beta _{63}+\beta _{64} \right) \\&-\,\left( 1-\beta _{61}-\beta _{62} \right) \left( \beta _{43}+\beta _{44} \right) \left( \beta _{53}+\beta _{54} \right) -\left( \beta _{43}+\beta _{44} \right) \left( \beta _{55}\right. \\&\left. +\,\beta _{56} \right) \left( \beta _{63}+\beta _{64} \right) -\left( \beta _{45}+\beta _{46} \right) \left( \beta _{65}+\beta _{66} \right) \left( \beta _{53}+\beta _{54} \right) ] \}P_{C}/H\\ \\ E_{HR}= & {} \{ [ \beta _{57}\left( \beta _{43}+\beta _{44} \right) \left( \beta _{63}+\beta _{64} \right) +\beta _{48}\left( \beta _{63}+\beta _{64} \right) \left( 1-\beta _{51}-\beta _{52} \right) \\&+\,\beta _{48}\left( \beta _{65}+\beta _{66} \right) \left( \beta _{53}+\beta _{54} \right) +\beta _{57}\left( \beta _{65}+\beta _{66} \right) \left( 1-\beta _{41}-\beta _{42} \right) \\&+\,\beta _{69}\left( 1-\beta _{41}-\beta _{42} \right) \left( 1-\beta _{51}-\beta _{52} \right) {-\beta }_{69}\left( \beta _{43}+\beta _{44} \right) \left( \beta _{53}\right. \\&\left. +\,\beta _{54} \right) ]/ [ \left( 1-\beta _{41}-\beta _{42} \right) \left( 1-\beta _{51}-\beta _{52} \right) \left( 1-\beta _{61}-\beta _{62} \right) -\left( 1-\beta _{41}\right. \\&\left. -\,\beta _{42} \right) \left( \beta _{55}+\beta _{56}\right) \left( \beta _{65}+\beta _{66} \right) -\left( 1-\beta _{51}-\beta _{52} \right) \left( \beta _{45}+\beta _{46} \right) \left( \beta _{63}+\beta _{64} \right) \\&-\,\left( 1-\beta _{61}-\beta _{62} \right) \left( \beta _{43}+\beta _{44} \right) \left( \beta _{53}+\beta _{54} \right) -\left( \beta _{43}+\beta _{44} \right) \left( \beta _{55}\right. \\&\left. +\,\beta _{56} \right) \left( \beta _{63}+\beta _{64} \right) -\left( \beta _{45}+\beta _{46} \right) \left( \beta _{65}+\beta _{66} \right) \left( \beta _{53}+\beta _{54} \right) ]\}P_{R}/H\\ \\ \end{aligned}$$
$$\begin{aligned} E_{CI}= & {} \{ [ \gamma _{61}\left( \beta _{43}+\beta _{44} \right) \left( \beta _{55}+\beta _{56} \right) +\gamma _{51}\left( \beta _{43}+\beta _{44} \right) \left( 1-\beta _{61}-\beta _{62} \right) \\&+\,\gamma _{51}\left( \beta _{45}+\beta _{46} \right) \left( \beta _{65}+\beta _{66} \right) +\gamma _{61}\left( \beta _{45}+\beta _{46} \right) \left( 1-\beta _{51}-\beta _{52} \right) \\&+\,\gamma _{41}\left( 1-\beta _{51}-\beta _{52} \right) \left( 1-\beta _{61}-\beta _{62} \right) -\gamma _{41}\left( \beta _{55}+\beta _{56} \right) \left( \beta _{65}\right. \\&\left. +\,\beta _{66} \right) ]/ [ \left( 1-\beta _{41}-\beta _{42} \right) \left( 1-\beta _{51}-\beta _{52} \right) \left( 1-\beta _{61}-\beta _{62} \right) -\left( 1-\beta _{41}\right. \\&\left. -\,\beta _{42} \right) \left( \beta _{55}+\beta _{56} \right) \left( \beta _{65}+\beta _{66} \right) -\left( 1-\beta _{51}-\beta _{52} \right) \left( \beta _{45}+\beta _{46} \right) \left( \beta _{63}+\beta _{64} \right) \\&-\,\left( 1-\beta _{61}-\beta _{62} \right) \left( \beta _{43}+\beta _{44} \right) \left( \beta _{53}+\beta _{54} \right) -\left( \beta _{43}+\beta _{44} \right) \left( \beta _{55}\right. \\&\left. +\,\beta _{56} \right) \left( \beta _{63}+\beta _{64} \right) -\left( \beta _{45}+\beta _{46} \right) \left( \beta _{65}+\beta _{66} \right) \left( \beta _{53}+\beta _{54} \right) ] \}I/C\\ \\ E_{RI}= & {} \{ [ \gamma _{61}\left( \beta _{53}+\beta _{54} \right) \left( \beta _{45}+\beta _{46} \right) +\gamma _{41}\left( \beta _{53}+\beta _{54} \right) \left( 1-\beta _{61}-\beta _{62} \right) \\&+\,\gamma _{41}\left( \beta _{55}+\beta _{56} \right) \left( \beta _{63}+\beta _{64} \right) +\gamma _{61}\left( \beta _{55}+\beta _{56} \right) \left( 1-\beta _{41}-\beta _{42} \right) \\&+\,\gamma _{51}\left( 1-\beta _{41}-\beta _{42} \right) \left( 1-\beta _{61}-\beta _{62} \right) -\gamma _{51}\left( \beta _{45}+\beta _{46} \right) \left( \beta _{63}\right. \\&\left. +\,\beta _{64} \right) ]/ [ \left( 1-\beta _{41}{-}\beta _{42} \right) \left( 1-\beta _{51}{-}\beta _{52} \right) \left( 1-\beta _{61}-\beta _{62} \right) -\left( 1-\beta _{41}-\,\beta _{42} \right) \\&\times \,\left( \beta _{55}+\beta _{56} \right) \left( \beta _{65}+\beta _{66} \right) -\left( 1-\beta _{51}-\beta _{52} \right) \left( \beta _{45}+\beta _{46} \right) \left( \beta _{63}+\beta _{64} \right) \\&-\,\left( 1-\beta _{61}-\beta _{62} \right) \left( \beta _{43}+\beta _{44} \right) \left( \beta _{53}+\beta _{54} \right) -\left( \beta _{43}+\beta _{44} \right) \left( \beta _{55}+\,\beta _{56} \right) \\&\times \,\left( \beta _{63}+\beta _{64} \right) -\left( \beta _{45}+\beta _{46} \right) \left( \beta _{65}+\beta _{66} \right) \left( \beta _{53}+\beta _{54} \right) ] \}I/R\\ \\ E_{HI}= & {} \{[ \gamma _{51}\left( \beta _{63}+\beta _{64} \right) \left( \beta _{43}+\beta _{44} \right) +\gamma _{41}\left( \beta _{63}+\beta _{64} \right) \left( 1-\beta _{51}-\beta _{52} \right) \\&+\,\gamma _{41}\left( \beta _{65}+\beta _{66} \right) \left( \beta _{53}+\beta _{54} \right) +\gamma _{51}\left( \beta _{65}+\beta _{66} \right) \left( 1-\beta _{41}-\beta _{42} \right) \\&+\,\gamma _{61}\left( 1-\beta _{41}-\beta _{42} \right) \left( 1-\beta _{51}-\beta _{52} \right) -\gamma _{61}\left( \beta _{43}+\beta _{44} \right) \left( \beta _{53}\right. \\&\left. +\,\beta _{54} \right) ]/ [ \left( 1-\beta _{41}-\beta _{42} \right) \left( 1-\beta _{51}-\beta _{52} \right) \left( 1-\beta _{61}-\beta _{62} \right) -\left( 1-\beta _{41}\right. \\&\left. -\,\beta _{42} \right) \left( \beta _{55}+\beta _{56} \right) \left( \beta _{65}+\beta _{66} \right) -\left( 1-\beta _{51}-\beta _{52} \right) \left( \beta _{45}+\beta _{46} \right) \left( \beta _{63}+\beta _{64} \right) \\&-\,\left( 1-\beta _{61}-\beta _{62} \right) \left( \beta _{43}+\beta _{44} \right) \left( \beta _{53}+\beta _{54} \right) -\left( \beta _{43}+\beta _{44} \right) \left( \beta _{55}\right. \\&\left. +\,\beta _{56} \right) \left( \beta _{63}+\beta _{64} \right) -\left( \beta _{45}+\beta _{46} \right) \left( \beta _{65}+\beta _{66} \right) \left( \beta _{53}+\beta _{54} \right) ] \}I/H \end{aligned}$$
