Should one use smokeless tobacco in smoking cessation programs?

Abstract

The rational addiction model is often used for empirical analysis of the demand for addictive goods. We propose an extension of the model to include two goods, cigarettes and Swedish moist snuff, locally known as snus. Demand equations are estimated using aggregated annual time series data (in first differences) for the period 1964–1997. The findings from the dataset used give some support to the rational addiction hypothesis. The cross-price elasticities are negative, which indicates that taking snus contributes to increased smoking. Thus it is not advisable to encourage the use of the less harmful snus in smoking cessation programs.

Appendix

Derivation of Eq. 5 and Eq. 6

The instantaneous utility function can be approximated by:

$$ \begin{array}{*{20}l} {{U{\left( \bullet \right)}} \hfill} & { = \hfill} & {{\alpha _{1} c{\left[ t \right]} + \alpha _{2} s{\left[ t \right]} + \alpha _{3} G{\left[ t \right]} + \alpha _{4} H{\left[ t \right]} + \alpha _{5} y{\left[ t \right]} + } \hfill} \\ {{} \hfill} & {{} \hfill} & {{\frac{1} {2}u_{{cc}} c^{2} {\left[ t \right]} + \frac{1} {2}u_{{ss}} s^{2} {\left[ t \right]} + \frac{1} {2}u_{{GG}} G^{2} {\left[ t \right]} + \frac{1} {2}u_{{HH}} H^{2} {\left[ t \right]} + \frac{1} {2}u_{{yy}} y^{2} {\left[ t \right]} + } \hfill} \\ {{} \hfill} & {{} \hfill} & {{u_{{cs}} c{\left[ t \right]}s{\left[ t \right]} + u_{{cG}} c{\left[ t \right]}G{\left[ t \right]} + u_{{cH}} c{\left[ t \right]}H{\left[ t \right]} + u_{{sG}} s{\left[ t \right]}G{\left[ t \right]} + u_{{sH}} s{\left[ t \right]}H{\left[ t \right]} + u_{{GH}} G{\left[ t \right]}H{\left[ t \right]},} \hfill} \\ \end{array} $$

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where the α_i values are positive parameters, and the u_ij values are parameters with the same sign as their respective derivatives, for example, u_cc <0 since U_cc <0. The maximization problem can be transformed to be a function of cigarettes and snus only:

$$ V^{*}{\left( \bullet \right)} = \gamma W - \frac{{{\left( {\gamma - \alpha _{5} } \right)}^{2} }} {{2ru_{{yy}} }} + {\mathop {\max }\limits_{c{\left[ t \right]},s{\left[ t \right]}} }{\sum\limits_{t = 1}^\infty {{\left( {1 + r} \right)}^{{ - t}} F{\left( {c{\left[ t \right]},s{\left[ t \right]},G{\left[ t \right]},H{\left[ t \right]}} \right)},} } $$

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given c[0] and s[0], where:

$$ \begin{array}{*{20}l} {{F{\left( \bullet \right)}} \hfill} & { = \hfill} & {{\alpha _{1} c{\left[ t \right]} + \alpha _{2} s{\left[ t \right]} + \alpha _{3} {\left( {c{\left[ {t - 1} \right]}} + \delta s{\left[ {t - 1} \right]}\right)} + \alpha _{4} {\left( {1 - \delta } \right)}s{\left[ {t - 1} \right]} + } \hfill} \\ {{} \hfill} & {{} \hfill} & {{\frac{1} {2}u_{{cc}} c^{2} {\left[ t \right]} + \frac{1} {2}u_{{ss}} s^{2} {\left[ t \right]} + \frac{1} {2}u_{{GG}} {\left( {c{\left[ {t - 1} \right]}} + \delta s{\left[ {t - 1} \right]} \right)}^{2} + \frac{1} {2}u_{{HH}} {\left( {1 - \delta } \right)}^{2} s^{2} {\left[ {t - 1} \right]} + } \hfill} \\ {{} \hfill} & {{} \hfill} & {{u_{{cs}} c{\left[ t \right]}s{\left[ t \right]} + u_{{cG}} c{\left[ t \right]}{\left( {c{\left[ {t - 1} \right]} + \delta s{\left[ {t - 1} \right]}} \right)} + u_{{cH}} c{\left[ t \right]}{\left( {1 - \delta } \right)}s{\left[ {t - 1} \right]} + } \hfill} \\ {{} \hfill} & {{} \hfill} & {{u_{{sG}} s{\left[ t \right]}{\left( {c{\left[ {t - 1} \right]} + \delta s{\left[ {t - 1} \right]}} \right)} + u_{{sH}} s{\left[ t \right]}{\left( {1 - \delta } \right)}s{\left[ {t - 1} \right]} + } \hfill} \\ {{} \hfill} & {{} \hfill} & {{u_{{GH}} {\left( {c{\left[ {t - 1} \right]} + \delta s{\left[ {t - 1} \right]}} \right)}{\left( {1 - \delta } \right)}s{\left[ {t - 1} \right]} - \gamma {\left( {p_{c} {\left[ t \right]}c{\left[ t \right]} + p_{s} {\left[ t \right]}s{\left[ t \right]}} \right)}.} \hfill} \\ \end{array} $$

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The first-order condition with respect to c[t] is:

$$ \frac{{\partial V^{*} {\left( \bullet \right)}}} {{\partial c{\left[ t \right]}}} = \frac{{\partial F{\left( {c{\left[ t \right]},s{\left[ t \right]},G{\left[ t \right]},H{\left[ t \right]}} \right)}}} {{\partial c{\left[ t \right]}}} + \frac{1} {{1 + r}}\frac{{\partial F{\left( {c{\left[ {t + 1} \right]},s{\left[ {t + 1} \right]},G{\left[ {t + 1} \right]},H{\left[ {t + 1} \right]}} \right)}}} {{\partial c{\left[ t \right]}}} = 0, $$

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where:

$$ \frac{{\partial F{\left( {c{\left[ t \right]},s{\left[ t \right]},G{\left[ t \right]},H{\left[ t \right]}} \right)}}} {{\partial c{\left[ t \right]}}} = \alpha _{1} + u_{{cc}} c{\left[ t \right]} + u_{{cs}} s{\left[ t \right]} + u_{{cG}} {\left( {c{\left[ {t - 1} \right]} + \delta s{\left[ {t - 1} \right]}} \right)} + u_{{cH}} {\left( {1 - \delta } \right)}s{\left[ {t - 1} \right]} - \gamma p_{c} {\left[ t \right]}, $$

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and:

$$ \frac{{\partial F{\left( {c{\left[ {t + 1} \right]},s{\left[ {t + 1} \right]},G{\left[ {t + 1} \right]},H{\left[ {t + 1} \right]}} \right)}}} {{\partial c{\left[ t \right]}}} = \alpha _{3} + u_{{GG}} {\left( {c{\left[ t \right]} + \delta s{\left[ t \right]}} \right)} + u_{{cG}} c{\left[ {t + 1} \right]} + u_{{sG}} s{\left[ {t + 1} \right]} + u_{{GH}} {\left( {1 - \delta } \right)}s{\left[ t \right]}. $$

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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:

$$ \left\{ {\begin{array}{*{20}l} {{\beta _{{10}} = - \frac{{{\left( {1 + r} \right)}\alpha _{1} + \alpha _{3} }} {{{\left( {1 + r} \right)}u_{{cc}} + u_{{GG}} }} > 0,} \hfill} & {{\beta _{{11}} = - \frac{{u_{{cG}} }} {{{\left( {1 + r} \right)}u_{{cc}} + u_{{GG}} }} > 0,} \hfill} \\ {{\beta _{{12}} = - \frac{{{\left( {1 + r} \right)}{\left( {\delta u_{{cG}} + {\left( {1 - \delta } \right)}u_{{cH}} } \right)}}} {{{\left( {1 + r} \right)}u_{{cc}} + u_{{GG}} }} > 0,} \hfill} & {{\beta _{{13}} = - \frac{{{\left( {1 + r} \right)}u_{{cs}} + \delta u_{{GG}} + {\left( {1 - \delta } \right)}u_{{GH}} }} {{{\left( {1 + r} \right)}u_{{cc}} + u_{{GG}} }} < 0,} \hfill} \\ {{\beta _{{14}} = - \frac{{u_{{sG}} }} {{{\left( {1 + r} \right)}u_{{cc}} + u_{{GG}} }} > 0,} \hfill} & {{\beta _{{15}} = \frac{{{\left( {1 + r} \right)}\gamma }} {{{\left( {1 + r} \right)}U_{{cc}} + U_{{GG}} }} < 0.} \hfill} \\ \end{array} } \right. $$

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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:

$$ \left\{ {\begin{array}{*{20}l} {{\beta _{{20}} = - \frac{{{\left( {1 + r} \right)}\alpha _{2} + \delta \alpha _{3} + {\left( {1 - \delta } \right)}\alpha _{4} }} {{{\left( {1 + r} \right)}u_{{ss}} + \delta ^{2} u_{{GG}} + {\left( {1 - \delta } \right)}^{2} u_{{HH}} + 2\delta {\left( {1 - \delta } \right)}u_{{GH}} }} > 0} \hfill} \\ {{\beta _{{21}} = - \frac{{ \delta u_{{sG}} + {\left( {1 - \delta } \right)}u_{{sH}}}} {{{\left( {1 + r} \right)}u_{{ss}} + \delta ^{2} u_{{GG}} + {\left( {1 - \delta } \right)}^{2} u_{{HH}} + 2\delta {\left( {1 - \delta } \right)}u_{{GH}} }} > 0} \hfill} \\ {{\beta _{{22}} = - \frac{{{\left( {1 + r} \right)}u_{{sG}} }} {{{\left( {1 + r} \right)}u_{{ss}} + \delta ^{2} u_{{GG}} + {\left( {1 - \delta } \right)}^{2} u_{{HH}} + 2\delta {\left( {1 - \delta } \right)}u_{{GH}} }} > 0} \hfill} \\ {{\beta _{{23}} = - \frac{{{\left( {1 + r} \right)}u_{{cs}} + \delta u_{{GG}} + {\left( {1 - \delta } \right)}u_{{GH}} }} {{{\left( {1 + r} \right)}u_{{ss}} + \delta ^{2} u_{{GG}} + {\left( {1 - \delta } \right)}^{2} u_{{HH}} + 2\delta {\left( {1 - \delta } \right)}u_{{GH}} }} < 0} \hfill} \\ {{\beta _{{24}} = - \frac{{\delta u_{{cG}} + {\left( {1 - \delta } \right)}u_{{cH}} }} {{{\left( {1 + r} \right)}u_{{ss}} + \delta ^{2} u_{{GG}} + {\left( {1 - \delta } \right)}^{2} u_{{HH}} + 2\delta {\left( {1 - \delta } \right)}u_{{GH}} }} > 0} \hfill} \\ {{\beta _{{25}} = \frac{{{\left( {1 + r} \right)}\gamma }} {{{\left( {1 + r} \right)}u_{{ss}} + \delta ^{2} u_{{GG}} + {\left( {1 - \delta } \right)}^{2} u_{{HH}} + 2\delta {\left( {1 - \delta } \right)}u_{{GH}} }} < 0} \hfill} \\ \end{array} } \right. \;.$$

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Parameters in Eq. 7 and Eq. 8

The parameters are:

$$ \left\{ {\begin{array}{*{20}l} {{\beta _{{30}} = \frac{{\beta _{{10}} + \beta _{{13}} \beta _{{20}} }} {{1 - \beta _{{13}} \beta _{{23}} }},} \hfill} & {{\beta _{{31}} = \frac{{{\left( {1 + r} \right)}\beta _{{11}} + \beta _{{13}} \beta _{{22}} }} {{1 - \beta _{{13}} \beta _{{23}} }},} \hfill} & {{\beta _{{32}} = \frac{{\beta _{{11}} + \beta _{{13}} \beta _{{24}} }} {{1 - \beta _{{13}} \beta _{{23}} }}} \hfill} & {{\beta _{{33}} = \frac{{\beta _{{12}} + {\left( {1 + r} \right)}\beta _{{13}} \beta _{{21}} }} {{1 - \beta _{{13}} \beta _{{23}} }}} \hfill} \\ {{\beta _{{34}} = \frac{{\beta _{{13}} \beta _{{21}} + \beta _{{14}} }} {{1 - \beta _{{13}} \beta _{{23}} }},} \hfill} & {{\beta _{{35}} = \frac{{\beta _{{15}} }} {{1 - \beta _{{13}} \beta _{{23}} }},} \hfill} & {{\beta _{{36}} = \frac{{\beta _{{13}} \beta _{{25}} }} {{1 - \beta _{{13}} \beta _{{23}} }},} \hfill} & {{} \hfill} \\ \end{array} } \right. $$

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and:

$$ \left\{ {\begin{array}{*{20}l} {{\beta _{{40}} = \frac{{\beta _{{10}} \beta _{{23}} + \beta _{{20}} }} {{1 - \beta _{{13}} \beta _{{23}} }},} \hfill} & {{\beta _{{41}} = \frac{{\beta _{{12}} \beta _{{23}} + {\left( {1 + r} \right)}\beta _{{21}} }} {{1 - \beta _{{13}} \beta _{{23}} }},} \hfill} & {{\beta _{{42}} = \frac{{\beta _{{14}} \beta _{{23}} + \beta _{{21}} }} {{1 - \beta _{{13}} \beta _{{23}} }},} \hfill} & {{\beta _{{43}} = \frac{{{\left( {1 + r} \right)}\beta _{{11}} \beta _{{23}} + \beta _{{22}} }} {{1 - \beta _{{13}} \beta _{{23}} }},} \hfill} \\ {{\beta _{{44}} = \frac{{\beta _{{11}} \beta _{{23}} + \beta _{{24}} }} {{1 - \beta _{{13}} \beta _{{23}} }},} \hfill} & {{\beta _{{45}} = \frac{{\beta _{{25}} }} {{1 - \beta _{{13}} \beta _{{23}} }},} \hfill} & {{\beta _{{46}} = \frac{{\beta _{{15}} \beta _{{23}} }} {{1 - \beta _{{13}} \beta _{{23}} }}.} \hfill} & {{} \hfill} \\ \end{array} } \right. $$

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Long-run demand elasticities

All the price and quantity variables presented below, for example, p_c 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:

$$ \left\{ {\begin{array}{*{20}l} {{\frac{{\partial c}} {{\partial p_{c} }}\frac{{p_{c} }} {c} = \frac{{\beta _{{15}} {\left( {1 - \beta _{{21}} - \beta ^{*}_{{21}} } \right)}}} {{{\left( {1 - \beta _{{11}} - \beta ^{*}_{{11}} } \right)}{\left( {1 - \beta _{{21}} - \beta ^{*}_{{21}} } \right)} - {\left( {\beta _{{12}} + \beta _{{13}} + \beta _{{14}} } \right)}{\left( {\beta _{{22}} + \beta _{{23}} + \beta _{{24}} } \right)}}}\frac{{p_{c} }} {c}} \hfill} \\ {{\frac{{\partial s}} {{\partial p_{s} }}\frac{{p_{s} }} {s} = \frac{{{\left( {1 - \beta _{{11}} - \beta ^{*}_{{11}} } \right)}\beta _{{25}} }} {{{\left( {1 - \beta _{{11}} - \beta ^{*}_{{11}} } \right)}{\left( {1 - \beta _{{21}} - \beta ^{*}_{{21}} } \right)} - {\left( {\beta _{{12}} + \beta _{{13}} + \beta _{{14}} } \right)}{\left( {\beta _{{22}} + \beta _{{23}} + \beta _{{24}} } \right)}}}\frac{{p_{s} }} {s}} \hfill} \\ {{\frac{{\partial c}} {{\partial p_{s} }}\frac{{p_{s} }} {c} = \frac{{{\left( {\beta _{{12}} + \beta _{{13}} + \beta _{{14}} } \right)}\beta _{{25}} }} {{{\left( {1 - \beta _{{11}} - \beta ^{*}_{{11}} } \right)}{\left( {1 - \beta _{{21}} - \beta ^{*}_{{21}} } \right)} - {\left( {\beta _{{12}} + \beta _{{13}} + \beta _{{14}} } \right)}{\left( {\beta _{{22}} + \beta _{{23}} + \beta _{{24}} } \right)}}}\frac{{p_{s} }} {c}} \hfill} \\ {{\frac{{\partial s}} {{\partial p_{c} }}\frac{{p_{c} }} {s} = \frac{{\beta _{{15}} {\left( {\beta _{{22}} + \beta _{{23}} + \beta _{{24}} } \right)}}} {{{\left( {1 - \beta _{{11}} - \beta ^{*}_{{11}} } \right)}{\left( {1 - \beta _{{21}} - \beta ^{*}_{{21}} } \right)} - {\left( {\beta _{{12}} + \beta _{{13}} + \beta _{{14}} } \right)}{\left( {\beta _{{22}} + \beta _{{23}} + \beta _{{24}} } \right)}}}\frac{{p_{c} }} {s}} \hfill} \\ \end{array} } \right., $$

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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:

$$ \left\{ {\begin{array}{*{20}l} {{\frac{{\partial c}} {{\partial p_{c} }}\frac{{p_{c} }} {c} = \frac{{{\left( {\beta _{{33}} + \beta _{{34}} } \right)}\beta _{{46}} + \beta _{{35}} {\left( {1 - \beta _{{41}} - \beta _{{42}} } \right)}}} {{{\left( {1 - \beta _{{31}} - \beta _{{32}} } \right)}{\left( {1 - \beta _{{41}} - \beta _{{42}} } \right)} - {\left( {\beta _{{33}} + \beta _{{34}} } \right)}{\left( {\beta _{{43}} + \beta _{{44}} } \right)}}}\frac{{p_{c} }} {c}} \hfill} \\ {{\frac{{\partial s}} {{\partial p_{s} }}\frac{{p_{s} }} {s} = \frac{{{\left( {1 - \beta _{{31}} - \beta _{{32}} } \right)}\beta _{{45}} + \beta _{{36}} {\left( {\beta _{{43}} + \beta _{{44}} } \right)}}} {{{\left( {1 - \beta _{{31}} - \beta _{{32}} } \right)}{\left( {1 - \beta _{{41}} - \beta _{{42}} } \right)} - {\left( {\beta _{{33}} + \beta _{{34}} } \right)}{\left( {\beta _{{43}} + \beta _{{44}} } \right)}}}\frac{{p_{s} }} {s}} \hfill} \\ {{\frac{{\partial c}} {{\partial p_{s} }}\frac{{p_{s} }} {c} = \frac{{{\left( {\beta _{{33}} + \beta _{{34}} } \right)}\beta _{{45}} + \beta _{{36}} {\left( {1 - \beta _{{41}} - \beta _{{42}} } \right)}}} {{{\left( {1 - \beta _{{31}} - \beta _{{32}} } \right)}{\left( {1 - \beta _{{41}} - \beta _{{42}} } \right)} - {\left( {\beta _{{33}} + \beta _{{34}} } \right)}{\left( {\beta _{{43}} + \beta _{{44}} } \right)}}}\frac{{p_{s} }} {c}} \hfill} \\ {{\frac{{\partial s}} {{\partial p_{c} }}\frac{{p_{c} }} {s} = \frac{{{\left( {1 - \beta _{{31}} - \beta _{{32}} } \right)}\beta _{{46}} + \beta _{{35}} {\left( {\beta _{{43}} + \beta _{{44}} } \right)}}} {{{\left( {1 - \beta _{{31}} - \beta _{{32}} } \right)}{\left( {1 - \beta _{{41}} - \beta _{{42}} } \right)} - {\left( {\beta _{{33}} + \beta _{{34}} } \right)}{\left( {\beta _{{43}} + \beta _{{44}} } \right)}}}\frac{{p_{c} }} {s}} \hfill} \\ \end{array} } \right.. $$

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