Abstract
We extend the Becker-Murphy rational addiction model to account for a period before the onset of addiction. While during the first stage of recreational consumption of the addictive good does not imply negative effects, the second stage is analogous to the classical Becker-Murphy model. In line with neurological research, the onset of addiction is a random event positively related to the past consumption of the addictive good. The resulting multistage optimal control model with random switching time is analyzed by way of a transformation into an age-structured deterministic optimal control model. This enables us to analyze in detail the anticipation of the second stage, including the possible emergence of a Skiba point. A numerical example demonstrates that it is optimal to stop consuming the addictive good in case of an early onset (i.e. at a low level of cumulative consumption) of addiction. A late onset tends to lead into long-run addiction.
This research was supported by the Austrian Science Fund (FWF) under Grant P 30665-G27.
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Notes
- 1.
Indeed, Volkow et al. [23] liken the onset of addiction to the onset of a chronic brain disease and provide evidence on some of the factors that affect susceptibility to acquiring “addiction”.
- 2.
See Volkow et al. [23] for how these mechanisms are grounded in neurobiological changes within the brain’s stimulus-and-reward system.
- 3.
Koob [15] argues that long-term neurological changes aimed at maintaining the stability of the stress-and-reward system in the presence of addiction may ultimately be responsible for a former addict’s permanent vulnerability to relapse.
- 4.
In other contributions Skiba points are also referred to as Dechert-Nishimura-Skiba (DNS), Dechert-Nishimura-Sethi-Skiba (DNSS) or Sethi-Skiba points.
- 5.
- 6.
Note that state and control variables corresponding to stage i (\(i=1,2\)) are denoted by subscript i.
- 7.
This is in line with the typical observation that individuals who have shed their addiction will typically refrain from any (future) consumption of the addictive good, such that \(c_{2}\equiv 0\) from some point onward. Any small amount \(c_{2}>0\) would immediately retrigger addiction, implying that, technically speaking, such an individual remains in stage 2 even if \( c_{2}\equiv 0\) and \(S_{2}=0\) from some point. As we will see below, our model allows for such an allocation.
- 8.
The value of addictive capital at time t within the pre-addiction stage 1 consists of the discounted stream of (i) the stage-2 value of addictive capital, \(\xi _{S}(\tau ,\tau )\) if the transition occurs at \(\tau \) plus (ii) the expected change in the value of the addictive capital for an incrase in the switching rate, \(\eta _{S_{1}}>0,\) due to the accumulation of addictive capital in period \(\tau \).
- 9.
The value of remaining without addiction at time t consists of the discounted stream of (i) the value of pre-addiction utility, \(u^{1}\), within period \(\tau \) of the expected remaining lifetime without addiction plus (ii) the expected discounted stream of continuation utility in addiction, \( u^{2},\) should a switch occur in period \(\tau \).
- 10.
Intuitively, the value of being addicted at some time \(t\ge s\) consists of the discounted stream of stage-2 utility over the remaining lifetime (within addiction).
- 11.
The value of addictive capital at time t, conditional on a transition into addiction at \(s\le t,\) consists of the discounted stream of the utility loss from addcitive capital, \(u_{S}^{2}<0,\) over the remaining lifetime \( \left[ t,\infty \right) ,\) weighted with the unconditional/ex-ante probability \(z_{1}(s)\eta (S_{1}(s))\) of having had a transition into addiction at time s.
- 12.
Steady state values are denoted by a hat.
- 13.
The Eigenvalue corresponding to \( \frac{d\xi _{A}}{dt}\) is always equal to zero (since the dynamics is always zero). Since \(A_{2}\) does not enter the equations for \(\frac{dS_{2}}{dt}\) and \(\frac{d\xi _{S}}{dt}\), the Eigenvalue corresponding to \(\frac{dA_{2}}{dt }\) can be isolated as \(r=\rho \).
- 14.
Formally, this implies that \(u_{y}^{2}>u_{y}^{1}\) in (4.15).
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The authors are grateful for very helpful and constructive comments by two anonymous referees.
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A Derivation of Equation (4.20)
A Derivation of Equation (4.20)
Starting point is \(\left. \frac{\partial \mathcal {H}}{\partial c_{1}} \right| _{c_{1}=0} \le 0\), i.e.
Using (4.9) we obtain
Using the functional specification and \(\hat{c}_{1} =0\) we arrive (after rearrangement) at
Solving and inserting the steady state value of \(\lambda _{S}\), which is \(\hat{\lambda }_{S} = \frac{az_{1}}{\rho ( \rho + \delta )} \left( \hat{u}^{2}- \hat{u}^{1} \right) \), we obtain (4.20). For the derivation of \(\lambda _{S}\) we used the result by Caulkins et al. [4] stating that the long-run optimal solution converges to the non-addictive consumption steady state if S is sufficiently low.
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Kuhn, M., Wrzaczek, S. (2021). Rationally Risking Addiction: A Two-Stage Approach. In: Haunschmied, J.L., Kovacevic, R.M., Semmler, W., Veliov, V.M. (eds) Dynamic Economic Problems with Regime Switches. Dynamic Modeling and Econometrics in Economics and Finance, vol 25. Springer, Cham. https://doi.org/10.1007/978-3-030-54576-5_4
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