High addiction capital lowers average utility but raises the marginal utility of smoking. In this way, smoking lowers future utility but also increases the craving for another cigarette. The key feature of any addiction model is on how people deal with this intertemporal problem. In the original Becker-Murphy formulation individuals discounted the future exponentially, meaning that they discount k-periods forward by 5k, where 5 is the per-period time discount factor. Since an exponential individual makes a time-consistent choice to smoke, a rise in taxes can only lower discounted utility today. If it were to raise it, then the rational addict could raise utility by simply reducing smoking by the amount that the tax does, i.e. by emulating the tax. So cigarette taxes should reduce the happiness of time consistent rational addicts.
Gruber and Koszegi develop an alternative to the Becker and Murphy model where smokers can actually be made better off by cigarette taxation. Their alternative embeds within the Becker-Murphy stock addiction framework preferences that are time inconsistent, following Laibson and O’Donoghue and Rabin. In this quasi-hyperbolic formulation, next period is discounted by P5, the following period by P52, and k periods in the future by P5k, where P<1 is an extra discount factor that changes the discounting of this period relative to the entire future. The key feature of such a hyperbolic model is that individuals will have self-control problems.
Specifically, a sophisticated hyperbolic individual (one who knows that he discounts hyperbolically) would like to smoke less in the future than he actually can. The problem arises because he is patient about the future (the relative discount rate between future periods is 5), but impatient about the present (the relative discount rate between today and tomorrow is P5<5). This means that when the future arrives he will end up making more impatient choices (i.e. smoke more) than he would like to from today's vantage point. natural asthma inhaler
As Gruber and Koszegi show, the discounted utility of a sophisticated hyperbolic consumer can rise if a tax is imposed. The reason is that the tax serves as a self-commitment device. By forcing a reduction in the smoking in the future, the tax allows the sophisticated hyperbolic agent to do something they would not be otherwise be able to do. This is the essence of the empirical test carried out below: a positive impact of cigarette taxation on the present discounted value of happiness is the direct implication of a sophisticated time inconsistent model.
Existing empirical evidence on smoking, reviewed in Chaloupka and Warner and Gruber, does not distinguish between these models. There is a strong consensus that smoking is moderately price elastic, but agents are price elastic under either of these models.