Appendix
Derivation of Eq. 5 and Eq. 6
The instantaneous utility function can be approximated by:
$$ \begin{array}{*{20}l} {{U{\left( \bullet \right)}} \hfill} & { = \hfill} & {{\alpha _{1} c{\left[ t \right]} + \alpha _{2} s{\left[ t \right]} + \alpha _{3} G{\left[ t \right]} + \alpha _{4} H{\left[ t \right]} + \alpha _{5} y{\left[ t \right]} + } \hfill} \\ {{} \hfill} & {{} \hfill} & {{\frac{1} {2}u_{{cc}} c^{2} {\left[ t \right]} + \frac{1} {2}u_{{ss}} s^{2} {\left[ t \right]} + \frac{1} {2}u_{{GG}} G^{2} {\left[ t \right]} + \frac{1} {2}u_{{HH}} H^{2} {\left[ t \right]} + \frac{1} {2}u_{{yy}} y^{2} {\left[ t \right]} + } \hfill} \\ {{} \hfill} & {{} \hfill} & {{u_{{cs}} c{\left[ t \right]}s{\left[ t \right]} + u_{{cG}} c{\left[ t \right]}G{\left[ t \right]} + u_{{cH}} c{\left[ t \right]}H{\left[ t \right]} + u_{{sG}} s{\left[ t \right]}G{\left[ t \right]} + u_{{sH}} s{\left[ t \right]}H{\left[ t \right]} + u_{{GH}} G{\left[ t \right]}H{\left[ t \right]},} \hfill} \\ \end{array} $$
(9)
where the αi values are positive parameters, and the uij values are parameters with the same sign as their respective derivatives, for example, ucc <0 since Ucc <0. The maximization problem can be transformed to be a function of cigarettes and snus only:
$$ V^{*}{\left( \bullet \right)} = \gamma W - \frac{{{\left( {\gamma - \alpha _{5} } \right)}^{2} }} {{2ru_{{yy}} }} + {\mathop {\max }\limits_{c{\left[ t \right]},s{\left[ t \right]}} }{\sum\limits_{t = 1}^\infty {{\left( {1 + r} \right)}^{{ - t}} F{\left( {c{\left[ t \right]},s{\left[ t \right]},G{\left[ t \right]},H{\left[ t \right]}} \right)},} } $$
(10)
given c[0] and s[0], where:
$$ \begin{array}{*{20}l} {{F{\left( \bullet \right)}} \hfill} & { = \hfill} & {{\alpha _{1} c{\left[ t \right]} + \alpha _{2} s{\left[ t \right]} + \alpha _{3} {\left( {c{\left[ {t - 1} \right]}} + \delta s{\left[ {t - 1} \right]}\right)} + \alpha _{4} {\left( {1 - \delta } \right)}s{\left[ {t - 1} \right]} + } \hfill} \\ {{} \hfill} & {{} \hfill} & {{\frac{1} {2}u_{{cc}} c^{2} {\left[ t \right]} + \frac{1} {2}u_{{ss}} s^{2} {\left[ t \right]} + \frac{1} {2}u_{{GG}} {\left( {c{\left[ {t - 1} \right]}} + \delta s{\left[ {t - 1} \right]} \right)}^{2} + \frac{1} {2}u_{{HH}} {\left( {1 - \delta } \right)}^{2} s^{2} {\left[ {t - 1} \right]} + } \hfill} \\ {{} \hfill} & {{} \hfill} & {{u_{{cs}} c{\left[ t \right]}s{\left[ t \right]} + u_{{cG}} c{\left[ t \right]}{\left( {c{\left[ {t - 1} \right]} + \delta s{\left[ {t - 1} \right]}} \right)} + u_{{cH}} c{\left[ t \right]}{\left( {1 - \delta } \right)}s{\left[ {t - 1} \right]} + } \hfill} \\ {{} \hfill} & {{} \hfill} & {{u_{{sG}} s{\left[ t \right]}{\left( {c{\left[ {t - 1} \right]} + \delta s{\left[ {t - 1} \right]}} \right)} + u_{{sH}} s{\left[ t \right]}{\left( {1 - \delta } \right)}s{\left[ {t - 1} \right]} + } \hfill} \\ {{} \hfill} & {{} \hfill} & {{u_{{GH}} {\left( {c{\left[ {t - 1} \right]} + \delta s{\left[ {t - 1} \right]}} \right)}{\left( {1 - \delta } \right)}s{\left[ {t - 1} \right]} - \gamma {\left( {p_{c} {\left[ t \right]}c{\left[ t \right]} + p_{s} {\left[ t \right]}s{\left[ t \right]}} \right)}.} \hfill} \\ \end{array} $$
(11)
The first-order condition with respect to c[t] is:
$$ \frac{{\partial V^{*} {\left( \bullet \right)}}} {{\partial c{\left[ t \right]}}} = \frac{{\partial F{\left( {c{\left[ t \right]},s{\left[ t \right]},G{\left[ t \right]},H{\left[ t \right]}} \right)}}} {{\partial c{\left[ t \right]}}} + \frac{1} {{1 + r}}\frac{{\partial F{\left( {c{\left[ {t + 1} \right]},s{\left[ {t + 1} \right]},G{\left[ {t + 1} \right]},H{\left[ {t + 1} \right]}} \right)}}} {{\partial c{\left[ t \right]}}} = 0, $$
(12)
where:
$$ \frac{{\partial F{\left( {c{\left[ t \right]},s{\left[ t \right]},G{\left[ t \right]},H{\left[ t \right]}} \right)}}} {{\partial c{\left[ t \right]}}} = \alpha _{1} + u_{{cc}} c{\left[ t \right]} + u_{{cs}} s{\left[ t \right]} + u_{{cG}} {\left( {c{\left[ {t - 1} \right]} + \delta s{\left[ {t - 1} \right]}} \right)} + u_{{cH}} {\left( {1 - \delta } \right)}s{\left[ {t - 1} \right]} - \gamma p_{c} {\left[ t \right]}, $$
(13)
and:
$$ \frac{{\partial F{\left( {c{\left[ {t + 1} \right]},s{\left[ {t + 1} \right]},G{\left[ {t + 1} \right]},H{\left[ {t + 1} \right]}} \right)}}} {{\partial c{\left[ t \right]}}} = \alpha _{3} + u_{{GG}} {\left( {c{\left[ t \right]} + \delta s{\left[ t \right]}} \right)} + u_{{cG}} c{\left[ {t + 1} \right]} + u_{{sG}} s{\left[ {t + 1} \right]} + u_{{GH}} {\left( {1 - \delta } \right)}s{\left[ t \right]}. $$
(14)
Solve these equations for c[t]:
$$ c{\left[ t \right]} = \beta _{{10}} + {\left( {1 + r} \right)}\beta _{{11}} c{\left[ {t - 1} \right]} + \beta _{{11}} c{\left[ {t + 1} \right]} + \beta _{{12}} s{\left[ {t - 1} \right]} + \beta _{{13}} s{\left[ t \right]} + \beta _{{14}} s{\left[ {t + 1} \right]} + \beta _{{15}} p_{c} {\left[ t \right]}\;, $$
where:
$$ \left\{ {\begin{array}{*{20}l} {{\beta _{{10}} = - \frac{{{\left( {1 + r} \right)}\alpha _{1} + \alpha _{3} }} {{{\left( {1 + r} \right)}u_{{cc}} + u_{{GG}} }} > 0,} \hfill} & {{\beta _{{11}} = - \frac{{u_{{cG}} }} {{{\left( {1 + r} \right)}u_{{cc}} + u_{{GG}} }} > 0,} \hfill} \\ {{\beta _{{12}} = - \frac{{{\left( {1 + r} \right)}{\left( {\delta u_{{cG}} + {\left( {1 - \delta } \right)}u_{{cH}} } \right)}}} {{{\left( {1 + r} \right)}u_{{cc}} + u_{{GG}} }} > 0,} \hfill} & {{\beta _{{13}} = - \frac{{{\left( {1 + r} \right)}u_{{cs}} + \delta u_{{GG}} + {\left( {1 - \delta } \right)}u_{{GH}} }} {{{\left( {1 + r} \right)}u_{{cc}} + u_{{GG}} }} < 0,} \hfill} \\ {{\beta _{{14}} = - \frac{{u_{{sG}} }} {{{\left( {1 + r} \right)}u_{{cc}} + u_{{GG}} }} > 0,} \hfill} & {{\beta _{{15}} = \frac{{{\left( {1 + r} \right)}\gamma }} {{{\left( {1 + r} \right)}U_{{cc}} + U_{{GG}} }} < 0.} \hfill} \\ \end{array} } \right. $$
(15)
Finally, derive the first order condition with respect to s[t], i.e.: \( \frac{{\partial {\text{V}}^{ * } ( \cdot )}} {{\partial s{\left[ t \right]}}} = 0 \), and solve for s[t]:
$$ s{\left[ t \right]} = \beta _{{20}} + {\left( {1 + r} \right)}\beta _{{21}} s{\left[ {t - 1} \right]} + \beta _{{21}} s{\left[ {t + 1} \right]} + \beta _{{22}} c{\left[ {t - 1} \right]} + \beta _{{23}} c{\left[ t \right]} + \beta _{{24}} c{\left[ {t + 1} \right]} + \beta _{{25}} p_{s} {\left[ t \right]}\;, $$
where:
$$ \left\{ {\begin{array}{*{20}l} {{\beta _{{20}} = - \frac{{{\left( {1 + r} \right)}\alpha _{2} + \delta \alpha _{3} + {\left( {1 - \delta } \right)}\alpha _{4} }} {{{\left( {1 + r} \right)}u_{{ss}} + \delta ^{2} u_{{GG}} + {\left( {1 - \delta } \right)}^{2} u_{{HH}} + 2\delta {\left( {1 - \delta } \right)}u_{{GH}} }} > 0} \hfill} \\ {{\beta _{{21}} = - \frac{{ \delta u_{{sG}} + {\left( {1 - \delta } \right)}u_{{sH}}}} {{{\left( {1 + r} \right)}u_{{ss}} + \delta ^{2} u_{{GG}} + {\left( {1 - \delta } \right)}^{2} u_{{HH}} + 2\delta {\left( {1 - \delta } \right)}u_{{GH}} }} > 0} \hfill} \\ {{\beta _{{22}} = - \frac{{{\left( {1 + r} \right)}u_{{sG}} }} {{{\left( {1 + r} \right)}u_{{ss}} + \delta ^{2} u_{{GG}} + {\left( {1 - \delta } \right)}^{2} u_{{HH}} + 2\delta {\left( {1 - \delta } \right)}u_{{GH}} }} > 0} \hfill} \\ {{\beta _{{23}} = - \frac{{{\left( {1 + r} \right)}u_{{cs}} + \delta u_{{GG}} + {\left( {1 - \delta } \right)}u_{{GH}} }} {{{\left( {1 + r} \right)}u_{{ss}} + \delta ^{2} u_{{GG}} + {\left( {1 - \delta } \right)}^{2} u_{{HH}} + 2\delta {\left( {1 - \delta } \right)}u_{{GH}} }} < 0} \hfill} \\ {{\beta _{{24}} = - \frac{{\delta u_{{cG}} + {\left( {1 - \delta } \right)}u_{{cH}} }} {{{\left( {1 + r} \right)}u_{{ss}} + \delta ^{2} u_{{GG}} + {\left( {1 - \delta } \right)}^{2} u_{{HH}} + 2\delta {\left( {1 - \delta } \right)}u_{{GH}} }} > 0} \hfill} \\ {{\beta _{{25}} = \frac{{{\left( {1 + r} \right)}\gamma }} {{{\left( {1 + r} \right)}u_{{ss}} + \delta ^{2} u_{{GG}} + {\left( {1 - \delta } \right)}^{2} u_{{HH}} + 2\delta {\left( {1 - \delta } \right)}u_{{GH}} }} < 0} \hfill} \\ \end{array} } \right. \;.$$
(16)
Parameters in Eq. 7 and Eq. 8
The parameters are:
$$ \left\{ {\begin{array}{*{20}l} {{\beta _{{30}} = \frac{{\beta _{{10}} + \beta _{{13}} \beta _{{20}} }} {{1 - \beta _{{13}} \beta _{{23}} }},} \hfill} & {{\beta _{{31}} = \frac{{{\left( {1 + r} \right)}\beta _{{11}} + \beta _{{13}} \beta _{{22}} }} {{1 - \beta _{{13}} \beta _{{23}} }},} \hfill} & {{\beta _{{32}} = \frac{{\beta _{{11}} + \beta _{{13}} \beta _{{24}} }} {{1 - \beta _{{13}} \beta _{{23}} }}} \hfill} & {{\beta _{{33}} = \frac{{\beta _{{12}} + {\left( {1 + r} \right)}\beta _{{13}} \beta _{{21}} }} {{1 - \beta _{{13}} \beta _{{23}} }}} \hfill} \\ {{\beta _{{34}} = \frac{{\beta _{{13}} \beta _{{21}} + \beta _{{14}} }} {{1 - \beta _{{13}} \beta _{{23}} }},} \hfill} & {{\beta _{{35}} = \frac{{\beta _{{15}} }} {{1 - \beta _{{13}} \beta _{{23}} }},} \hfill} & {{\beta _{{36}} = \frac{{\beta _{{13}} \beta _{{25}} }} {{1 - \beta _{{13}} \beta _{{23}} }},} \hfill} & {{} \hfill} \\ \end{array} } \right. $$
(17)
and:
$$ \left\{ {\begin{array}{*{20}l} {{\beta _{{40}} = \frac{{\beta _{{10}} \beta _{{23}} + \beta _{{20}} }} {{1 - \beta _{{13}} \beta _{{23}} }},} \hfill} & {{\beta _{{41}} = \frac{{\beta _{{12}} \beta _{{23}} + {\left( {1 + r} \right)}\beta _{{21}} }} {{1 - \beta _{{13}} \beta _{{23}} }},} \hfill} & {{\beta _{{42}} = \frac{{\beta _{{14}} \beta _{{23}} + \beta _{{21}} }} {{1 - \beta _{{13}} \beta _{{23}} }},} \hfill} & {{\beta _{{43}} = \frac{{{\left( {1 + r} \right)}\beta _{{11}} \beta _{{23}} + \beta _{{22}} }} {{1 - \beta _{{13}} \beta _{{23}} }},} \hfill} \\ {{\beta _{{44}} = \frac{{\beta _{{11}} \beta _{{23}} + \beta _{{24}} }} {{1 - \beta _{{13}} \beta _{{23}} }},} \hfill} & {{\beta _{{45}} = \frac{{\beta _{{25}} }} {{1 - \beta _{{13}} \beta _{{23}} }},} \hfill} & {{\beta _{{46}} = \frac{{\beta _{{15}} \beta _{{23}} }} {{1 - \beta _{{13}} \beta _{{23}} }}.} \hfill} & {{} \hfill} \\ \end{array} } \right. $$
(18)
Long-run demand elasticities
All the price and quantity variables presented below, for example, pc and c, are prices and quantities in steady state. In the estimations of the long-run demand elasticities, we assume that these steady-state values equalize the mean values in the dataset. The long-run demand elasticities are:
$$ \left\{ {\begin{array}{*{20}l} {{\frac{{\partial c}} {{\partial p_{c} }}\frac{{p_{c} }} {c} = \frac{{\beta _{{15}} {\left( {1 - \beta _{{21}} - \beta ^{*}_{{21}} } \right)}}} {{{\left( {1 - \beta _{{11}} - \beta ^{*}_{{11}} } \right)}{\left( {1 - \beta _{{21}} - \beta ^{*}_{{21}} } \right)} - {\left( {\beta _{{12}} + \beta _{{13}} + \beta _{{14}} } \right)}{\left( {\beta _{{22}} + \beta _{{23}} + \beta _{{24}} } \right)}}}\frac{{p_{c} }} {c}} \hfill} \\ {{\frac{{\partial s}} {{\partial p_{s} }}\frac{{p_{s} }} {s} = \frac{{{\left( {1 - \beta _{{11}} - \beta ^{*}_{{11}} } \right)}\beta _{{25}} }} {{{\left( {1 - \beta _{{11}} - \beta ^{*}_{{11}} } \right)}{\left( {1 - \beta _{{21}} - \beta ^{*}_{{21}} } \right)} - {\left( {\beta _{{12}} + \beta _{{13}} + \beta _{{14}} } \right)}{\left( {\beta _{{22}} + \beta _{{23}} + \beta _{{24}} } \right)}}}\frac{{p_{s} }} {s}} \hfill} \\ {{\frac{{\partial c}} {{\partial p_{s} }}\frac{{p_{s} }} {c} = \frac{{{\left( {\beta _{{12}} + \beta _{{13}} + \beta _{{14}} } \right)}\beta _{{25}} }} {{{\left( {1 - \beta _{{11}} - \beta ^{*}_{{11}} } \right)}{\left( {1 - \beta _{{21}} - \beta ^{*}_{{21}} } \right)} - {\left( {\beta _{{12}} + \beta _{{13}} + \beta _{{14}} } \right)}{\left( {\beta _{{22}} + \beta _{{23}} + \beta _{{24}} } \right)}}}\frac{{p_{s} }} {c}} \hfill} \\ {{\frac{{\partial s}} {{\partial p_{c} }}\frac{{p_{c} }} {s} = \frac{{\beta _{{15}} {\left( {\beta _{{22}} + \beta _{{23}} + \beta _{{24}} } \right)}}} {{{\left( {1 - \beta _{{11}} - \beta ^{*}_{{11}} } \right)}{\left( {1 - \beta _{{21}} - \beta ^{*}_{{21}} } \right)} - {\left( {\beta _{{12}} + \beta _{{13}} + \beta _{{14}} } \right)}{\left( {\beta _{{22}} + \beta _{{23}} + \beta _{{24}} } \right)}}}\frac{{p_{c} }} {s}} \hfill} \\ \end{array} } \right., $$
(19)
where: \( \beta ^{ * }_{{11}} = {\left( {1 + r} \right)}\beta _{{11}} \;\;{\text{and}}\;\;\beta ^{ * }_{{21}} = {\left( {1 + r} \right)}\beta _{{21}} , \) i.e., we relax the parameter restriction in the model. The long-run demand elasticities, based on the “semireduced” system, are:
$$ \left\{ {\begin{array}{*{20}l} {{\frac{{\partial c}} {{\partial p_{c} }}\frac{{p_{c} }} {c} = \frac{{{\left( {\beta _{{33}} + \beta _{{34}} } \right)}\beta _{{46}} + \beta _{{35}} {\left( {1 - \beta _{{41}} - \beta _{{42}} } \right)}}} {{{\left( {1 - \beta _{{31}} - \beta _{{32}} } \right)}{\left( {1 - \beta _{{41}} - \beta _{{42}} } \right)} - {\left( {\beta _{{33}} + \beta _{{34}} } \right)}{\left( {\beta _{{43}} + \beta _{{44}} } \right)}}}\frac{{p_{c} }} {c}} \hfill} \\ {{\frac{{\partial s}} {{\partial p_{s} }}\frac{{p_{s} }} {s} = \frac{{{\left( {1 - \beta _{{31}} - \beta _{{32}} } \right)}\beta _{{45}} + \beta _{{36}} {\left( {\beta _{{43}} + \beta _{{44}} } \right)}}} {{{\left( {1 - \beta _{{31}} - \beta _{{32}} } \right)}{\left( {1 - \beta _{{41}} - \beta _{{42}} } \right)} - {\left( {\beta _{{33}} + \beta _{{34}} } \right)}{\left( {\beta _{{43}} + \beta _{{44}} } \right)}}}\frac{{p_{s} }} {s}} \hfill} \\ {{\frac{{\partial c}} {{\partial p_{s} }}\frac{{p_{s} }} {c} = \frac{{{\left( {\beta _{{33}} + \beta _{{34}} } \right)}\beta _{{45}} + \beta _{{36}} {\left( {1 - \beta _{{41}} - \beta _{{42}} } \right)}}} {{{\left( {1 - \beta _{{31}} - \beta _{{32}} } \right)}{\left( {1 - \beta _{{41}} - \beta _{{42}} } \right)} - {\left( {\beta _{{33}} + \beta _{{34}} } \right)}{\left( {\beta _{{43}} + \beta _{{44}} } \right)}}}\frac{{p_{s} }} {c}} \hfill} \\ {{\frac{{\partial s}} {{\partial p_{c} }}\frac{{p_{c} }} {s} = \frac{{{\left( {1 - \beta _{{31}} - \beta _{{32}} } \right)}\beta _{{46}} + \beta _{{35}} {\left( {\beta _{{43}} + \beta _{{44}} } \right)}}} {{{\left( {1 - \beta _{{31}} - \beta _{{32}} } \right)}{\left( {1 - \beta _{{41}} - \beta _{{42}} } \right)} - {\left( {\beta _{{33}} + \beta _{{34}} } \right)}{\left( {\beta _{{43}} + \beta _{{44}} } \right)}}}\frac{{p_{c} }} {s}} \hfill} \\ \end{array} } \right.. $$
(20)