BACAP

We say that a BACAP exhibits Habit-Formation if [PSI](k) - [GAMMA](k) is increasing in k.

Proposition 8 .If a BACAP B exhibits Habit-Formation, its dual D(B) exhibits Habit-Formation.

We say that a BACAP presents Harmful Addiction if it exhibits Habit Formation and Negative Internalities.

If a BACAP presents Harmful (Beneficial) Addiction, its dual D(B) exhibits Beneficial (Harmful) Addiction.

Let [alfa]:[0, [k.sup.max]] [flecha diestra] {0,1} be a rule or strategy in a BACAP B, that is, for each state level k, a(k) prescribes an admissible action.

If a induces path A in a BACAP B, then [??] induces path [XI](A) in D(B).

The above proposition shows that, given a strategy [alfa] in a BACAP B, strategy [??] as given by expression (1) has the natural interpretation of being the strategy in -D(B) induced by applying the isomorphism [XI] to [alfa].

Consider a BACAP B = [[PSI], [GAMMA], U, [gamma], [K.sub.1]] presenting beneficial addiction and where the functional U takes the specific form

In O'Donoghue and Rabin (2002), it is shown that in a BACAP as D(B) the optimal decision rule of the agent is of the cut-off type.

In a BACAP B = [[PSI], [GAMMA], U, [gamma], [k.sub.1]] presenting beneficial addiction with [PSI] and [GAMMA] convex and intertemporal utility U given by (2), the optimal decision rule is of the cutoff type.

sophistication have been studied in O'Donoghue and Rabin (2002) and in Lopez (2006) for a BACAP presenting harmful addiction under time inconsistency as given by expression (4).

O'Donoghue and Rabin (2002) show that in a BACAP B presenting harmful addiction a naif follows a cutoff rule [[alfa].sup.n] that is, there is a critical state level [k.sup.n] such that