Bounded Rationality, Taxation, and Prohibition 1. Suren Basov 2 and Svetlana Danilkina 3

Transcription

1 Bounded Rationality, Taxation, and Prohibition 1 Suren Basov 2 and Svetlana Danilkina 3 Keywords and Phrases: addiction, bounded rationality, taxation, prohibition. 1 The second author s work was supported by ARC Discovery grant on Harm Minimization Policies and the Economics of Controlling Illegal Drug Use. 2 Department of Economics and Finance, La Trobe University, Bundoora, Victoria 3086, Australia ( S.Basov@latrobe.edu.au) 3 Department of Economics, Melbourne University, Victoria 3010, Australia ( sdanilki@unimelb.edu.au) 1

2 Summary. In this paper we develop a model were consumers of illicit drug start as rational addicts but face a risk of loosing control and being carried by the momentum. We show that if this happens, they end up consuming higher amounts of the addictive substance than a rational addict would. We argue that the groups that are at a particular risk of such a runaway behavior are those who have high genetic predisposition for drug use. Using stylized facts about the e ects of prohibition summarized in Basov, Jacobson, and Miron (2001) we argue that prohibition is unlikely to diminish the incident of the runaway behavior. The taxation of the addictive substance, on the other hand, can be quite an e ective policy. 2

3 1 Introduction The most popular economic model of addiction was proposed by Becker and Murphy (1988). The model states that an individual when deciding on the usage of an addictive substance is fully capable of anticipating all the consequences and selects the consumption path to maximize her discounted life-time utility. Practitioners in the eld, on the contrary, often argue that many addicts want to stop the consumption of the substance, but are unable to do so. They also argue that the consumption usually starts at a relatively early age, when the prefrontal cortex, the area of the brain responsible for the forward-looking behavior, is still not fully developed. Therefore, the individuals are unable to make fully rational decisions concerning future events. What is more, the ability for the forward looking behavior is further deteriorated by the drug use, since most drugs cause brain damage. It is this idea that we develop in the current paper. In this paper we assume that all individuals start as rational addicts, described by the Becker and Murphy model. Once the consumption of the addictive substance passes certain level, the individual looses the ability for a forward looking behavior and is carried on by the momentum. This phase is described by the boundedly rational model of addiction (Basov and Danilk- 3

4 ina, 2006). If the switching level is crossed the individual enters a runaway regime and ends up consuming more addictive substance, than she herself nds optimal. Note that the individual in this model cannot be viewed as fully rational even in the Becker and Murphy s phase, since she does not foresee the possibility of loosing the control. We assume that the consumption level at which the loss of control occurs depends on the age of the individual: a teenager with an underdeveloped prefrontal cortex is more likely to lose control for a given consumption level than an adult over thirty years old. The last observation has important policy implications. Note that the individual will cross the switching level if and only if it is below the rational level of consumption as calculated from the Becker and Murphy model. If we assume that the switching level of consumption is below the rational level for young individuals but below it for the mature ones, we will derive at a conclusion that the group in danger are the individuals who have strong genetic predisposition for the drug use. An e ective policy to decrease the incidence of the runaway behavior should either decrease the rational level of addiction or decrease the time-derivative of the consumption. We will see that a tax on the addictive substance have both of those e ects. Prohibition, on the contrary has little e ect on the black market price (see, Basov, Jacobson, 4

5 and Miron, 2001). It simply imposes a xed cost of illegal behavior, which is likely to deter only those users, who have low genetic predisposition and would have likely behaved rationality. Therefore, prohibition is associated with high welfare costs: loss of utility from consumption to the rational users, high costs of enforcement, criminalization of the society, the loss of the tax dollars that could have been used on the rehabilitation of recovering drug addicts, with little or none actual bene ts. 4 2 The model In this Section we develop a model of addiction that blends together the model of rational addiction by Becker and Murphy (1988) and the model of boundedly rational addiction developed by Basov and Danilkina (2006).We assume that the utility of a consumer depends on the level of consumption of the addictive good, c, and the stock of the consumption capital, S. In 4 Miron (2005) argued that replacing marijuana prohibition with a system of legal regulation would save approximately $7.7 billion in government expenditures on prohibition enforcement $2.4 billion at the federal level and $5.3 billion at the state and local levels. Revenue from taxation of marijuana sales would range from $2.4 billion per year if marijuana were taxed like ordinary consumer goods to $6.2 billion if it were taxed like alcohol or tobacco. 5

6 particular, it is given by v(; ); where v(c; S) = u(c; S) p c c; (1) u(; ) is continuously twice di erentiable, weakly increasing in the rst argument, and concave function, > 0 is the marginal utility of money, and p c is the price of the addictive good. The evolution of the consumption capital is given by: S = c S; (2) so. the consumption capital depreciates at rate ; but can be increased by consumption. The individual starts with maximizing the discounted present value of the life-time utility subject to (2), i.e. she solves: 8 >< >: Z 1 max v(c; S) exp( t)dt 0 ; (3) s:t: S = c S; S(0) = S 0 where S 0 is the initial stock of the addictive capital, which we interpret as the genetic predisposition to addiction. Let c op (t) be the optimal consumption 6

7 path and S op (t) the corresponding stock of the addictive capital, i.e.(c op (); S op ()) solve problem (3). If c op (t) < c (t) for all t 0 then c(t) = c op (t); i.e. the actual consumption follows the optimal path and the addict is rational. Function c (t) is an exogenously given switching level, which depends on the individual s age. If at t = t 0 the consumption of the addictive good reaches level c (t 0 ) the individual looses the control over her consumption decisions and gets carried by the momentum, i.e. her consumption and addiction levels evolve according to: 8 >< >: c = (u c p c ) S = c S ; (4) for some > 0: The rst equation in the system (4) stipulates that the rate of change of the consumption of addictive good is proportional to its marginal utility, while the second equation is the addictive stock evolution equation (2). System (4) is a system of the rst order di erential equations for c and S, which should be solved subject to the initial conditions c(t 0 ) = c op (t 0 ) = c (t 0 ); S(t 0 ) = S op (t 0 ). The last model is a boundedly rational model of addiction proposed by Basov and Danilkina (2006). Let us for a moment forget about switching and 7

8 compare the behavior of the consumption of the addictive good in the Becker and Murphy model versus Basov and Danilkina model. For the purpose of this analysis we will assume that u(c; S) = c c + S S + ccc cs cs + SSS 2 : (5) 2 The standard assumptions are: u c = c + cc c + cs S > 0 (6) u s = S + SS S + cs c < 0 (7) u cc = cc < 0; u SS = SS < 0; u cc u ss u 2 cs = cc SS 2 cs > 0 (8) u cs = cs > 0: (9) The rst assumption states that the consumption of the addictive good increases the individuals utility, the second states that the increase in the consumption capital decreases utility, i.e. the consumption is harmful, the third states that the utility function is concave and is made for technical reasons, nally the fourth states that the good is indeed addictive, increase in the consumption capital increases the marginal utility of consumption. It 8

9 is clear that assumptions (6) and (7) cannot hold for all possible (c; S). We will assume that the hold for all economically relevant values. In particular, the hold for c = S = 0; which implies S < 0 and c > 0. On the top of usual assumptions we will assume that cs + cc < 0 (10) cs + ss < 0: (11) Conditions (10)-(11) essentially state that the good is not too addictive. We will see below that condition (10) is necessary and su cient for stability of the steady state in the boundedly rational model of addiction. Together with (11) it ensures the saddle path stability of the Becker and Murphy s steady state, provided the time discount factor is su ciently small (Clarke and Danilkina, 2006). 3 The Becker and Murphy regime In the Becker and Murphy regime the individual, whose utility is given by (5), solves the optimal control problem (3). The present value Hamiltonian 9

10 is: H = c c + S S + ccc cs cs + SSS 2 2 p c c + (c S): (12) The rst order conditions are: 8 S = c H c = 0 S : (13) >: S(0) = S 0 ; lim t!1 (t) exp( t) = 0: The standard calculation shows that the solution to system (13) (c ra: (t); S ra (t)) follows the unique stable saddle path and the consumption of the addictive good converges to c p c c BM = max( ; 0): (14) j cs + cc + cs+ss j + The solution to system (13) will describe the actual evolution of the consumption as long as c ra (t) < c (t) (15) 10

11 for all t. Note that since c ra () is increasing in S 0 this condition is more likely to hold for individuals with low genetic predisposition for drug consumption. Let us assume that the initial value of the consumption capital is not too high, so consumption converges to c BM from below. Then the su cient condition for the solution to system (13) to describe the actual evolution of the consumption is c BM c (T ); (16) where T is the age at which the individual starts to consume the good. Note that the sign in inequality (16) is rather than <; since actual consumption always stays below the asymptotic level. 4 The boundedly rational regime In the boundedly rational regime the evolution of consumption and the consumption capital is given by system (4), with utility given by (5), subject to the initial conditions S(0) = S 0 and c(0) = c 0. The rst of the equations in system (4)) represents gradient dynamics that was rst used to model boundedly rational behavior by Arrow and Hurwicz (1960) and subsequently utilized by Anderson, Goeree, Holt (1997), Basov (2003), and Friedman and 11

12 Yellin (1997) among others. Note that in this model the consumers are myopic. However, unlike Orphanides and Zevros (1998) the agents are not even myopically rational. Performing the calculations one obtains that the consumption of the addictive good converges to the level c BR = max( c p c j cs + cc j ; 0) > c BM; (17) i.e. the boundedly rational individuals will overconsume the addictive good in the long-run. 5 Regime switching and policy implications The complete model allows for regime switching, i.e. an individual can switch from the Becker and Murphy s regime to the boundedly rational one. Such a switch is harmful to the individual, since it will lead to higher consumption level and lower terminal utility, than the individual initially planned. The government may try to design policies to minimize incidents of such switches. 5 We will consider two approaches: taxation and prohibition. 5 Minimizing incidence of the switches is consistent with the ideology of harm minimization favored by health care professionals. Full economic analysis should also include 12

13 Let us consider the taxation rst. E ectively, per unit tax will simply increase the price paid by the consumers for the drug. This will lead to a decrease in c BM and make condition (16) more likely to be satis ed. In addition, tax policy can generate a considerable tax revenue (see footnote 2), which can be used for rehabilitation programmes. Establishing minimal legal age can also be e ective. Prohibition, on the contrary, does not have signi cant e ect on prices of most illegal drugs (Basov, Jacobson, and Miron, 2001). It rather presents the potential consumers with a xed cost of breaking the law and therefore deters the consumers whose life-time utility from the drug consumption is low, i.e. those with low S 0. But these are exactly the consumers for whom condition (15) is likely to be satis ed and who therefore remain rational addicts though their lives. Therefore, prohibition leads to the loss of utility from consumption to the rational users without signi cant prevention of run-away consumption on top of being associated with high costs of enforcement, criminalization of the society, and the loss of the tax revenues. the nancial costs of the policy and the loss of utility to the rational consumers. In this particular case recommendations based on a fully edged welfare function will di er quantitatively, but not qualitatively from ones based on the idea of harm minimization. 13

14 References Anderson S. P., Goeree, J. K., Holt C. A.: Stochastic game theory: adjustment to equilibrium under bounded rationality. Working Paper, University of Virginia, (1999). Arrow, K. J.,. Hurwicz, L. : Stability of gradient process in n-person games. Journal of Society of Industrial and Applied Mathematics, 8, (1960). Basov, S.: Incentives for Boundedly Rational Agents. Topics in Theoretical Economics, 3,. article 2, (2003). Basov, S., Danilkina, S.: A Theory of Rational Addiction, (2006) Basov, S., Jacobson, M., Miron, J. : Prohibition and the Market for Illegal Drugs. World Economics, 2, 1-25, (2001). Becker, G. S., and Murphy, K. M.: A theory of rational addiction. Journal of Political Economy, 96, , (1988). Clarke, H., Danilkina, S: Talking Rationally about Rational Addiction, (2006). Friedman, D., Yellin, J.: Evolving landscapes for population games, University of California Santa Cruz, mimeo (1997). Fudenberg, D., Harris, C.: Evolutionary dynamics with aggregate shocks. Journal of Economic Theory, 57, (1992). 14

15 Orphanides, A., and Zervos, D.: Myopia and addictive behaviour, The Economic Journal, 108, 75-91, (1998). 15