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Gateway Theory and Positive Reinforcement through Bayesian Updating

Bert Grider September 27th, 2007 Abstract

Empirical evidence suggests that a causal mechanism operates through progressive consumption of particular drugs to produce a common adolescent substance use pattern. We theorize that a Bayesian updating mechanism underlies observed use patterns, and incorporate a positive reinforcement component into a standard gateway model of adolescent drug use to explain such patterns. We argue that the failure to experience immediate negative consequences

(in the form of arrest by the police) increases the likelihood of future drug use by decreasing the adolescent user’s estimate of the probability of being caught. In contrast to standard gateway theory arguments, results from the 1997 National Longitudinal Survey of Youth (NLSY97) provide support for something other than a purely pharmacological explanation of teenage drug use patterns.

CHAPTER 1: Introduction and Background

A. Introduction

Gateway theory postulates that the use of cigarettes and alcohol in adolescence causes later use of illegal drugs such as marijuana, cocaine, or heroin, and that consumption of a lesser drug such as marijuana leads to subsequent hard drug consumption. Gateway theorists, and most empirical work, suggest that causation flows either pharmacologically or mechanically from the physical act of using one of the gateway drugs (O’Donnell and Clayton, 1982). Yet, others argue that use of a gateway drug simply predicts progression to more powerful substances but does not actually cause the progression (Baumrind, 1983; Beenstock and Rahav, 2002). Cause and effect in the latter argument rest on unobserved, individual-level heterogeneity, such as differences in risk aversion. Proponents of the individual-level heterogeneity explanation argue that the sheer numbers of users who follow common use patterns make their counter argument difficult to support empirically (cite).

2 We look for evidence of unobserved heterogeneity by focusing on a failure to experience immediate negative consequences from substance use during the initiation stage. We theorize that most episodes of adolescent drug consumption do not impose anything other than a monetary cost on the user, and investigate the implications of a standard gateway model that incorporates an individual’s estimate of the likelihood that use will result in a negative outcome. Results may or may not contradict standard gateway theory arguments about the causal mechanisms underlying adolescent substance use, and may or may not indicate that early experimentation, and consequently positive reinforcement from that experimentation, occurs most often for the less risk-averse individuals most likely to progress to illicit drug consumption in the first place.

B. Gateway Theory

While gateway theory arose in the fields of psychology and sociology, Pacula (2001) describes two possible gateway mechanisms based in economic theory. The first emphasizes the addictive nature of substance use in a rational addiction framework. Since initiation into use of a particular drug only occurs when the marginal expected utility of use exceeds the marginal expected cost, continued use of a gateway drug might increase the expected marginal benefit from use of a stronger substance and cause new experimentation. This marginal approach emphasizes the need to account for the full cost of consumption, including access restrictions, possible legal or punitive repercussions, and (properly discounted) future health consequences.

The second causal process, mentioned by Pacula but discounted, suggests a household production framework. It states that the use of lesser drugs and the learning process that occurs with such use lower the subsequent price of achieving the same level of intoxication with harder substances.1 Under this hypothesis, the marginal utility of use stays constant while the learning process decreases the marginal cost, increasing the likelihood that first use of a harder drug takes place. This hypothesis builds on social learning theories, which postulate that the learning

1 Here, learning describes the information learned about intoxication through consumption of the lesser substance, and assumes the user knows these effects would be magnified through use of the harder drug.

3 process for drug use begins prior to initiation, and that societal and individual surroundings greatly influence both the likelihood of experimentation and the post-initiation pattern of use

(Kandel and Maloff, 1982; Pacula, 1995).

The alternative to the gateway idea argues that while drinking, smoking, and even marijuana consumption predict later use of harder drugs, the lower tier substances play no causal role in this progression. Instead, the observed gateway phenomenon reflects individual-level heterogeneity present both before and after substance use initiation. Under this argument, any findings in support of gateway theory result mainly from omitted variables. Fergusson and Horwood (2000) mention possible omissions as social or family measures, especially parental factors, and other individual-level factors like differential risk preferences. Decker and Schwartz (2000) outline other possible explanations, such as differences in general preferences or in psychological measures like the addictiveness of personality type.

Here, we propose an alternative theory – that the observed gateway progressions reflect a combined process of unobserved heterogeneity, access differentials, and simple Bayesian updating. The postulated model combines elements of all three explanations presented above.

Rather than reflecting a natural progression from lesser to harder substances, however, this study theorizes that the observed patterns result from the fact that the so-called gateway drugs, though illegal for youth to use or possess, are relatively easy to obtain. First, and as noted in Kenkel,

Mathios, and Pacula (2001), the lower potency gateways have lower marginal use costs, encouraging gateway experimentation prior to experimentation with harder drugs. Second, and similar to a household production framework, we allow differences in parental and familial backgrounds to influence adolescent drug use decisions. Based on the observation that some individuals progress from gateway drugs to harder substances while others do not, we also theorize that individual-level heterogeneity must play a role in the initiation process into harder drug use. We investigate the hypothesis that a Bayesian updating mechanism can explain why any drug progression occurs at all.

4 The main contribution of this study is the idea that a Bayesian updating process influences observed consumption patterns. Adolescents receive strong anti-smoking, anti-drinking, and anti- drug messages from the time they are very young. These messages warn of the dire consequences of substance use. The first time a teenager smokes a cigarette or has a drink, though, the likelihood of any dire consequence or punishment is remote. Moreover, this action likely proves to be enjoyable or exciting (cite), so that the entire prior-posterior belief set changes after initiation. The change in beliefs (the updating) may then encourage use of stronger substances along the commonly observed progression pattern.

The updating process changes the information set used to make initiation decisions, increasing the likelihood not only that the initiating individual smokes another cigarette or has another drink in the future, but also that the likelihood of using a stronger substance increases once a subsequent opportunity to use such a substance presents itself.2 The decision to smoke the first cigarette or take the first drink remains important, largely driven by unobserved factors and marginal cost considerations. Here, however, changes in the information set and the continued presence of individual-level heterogeneity drive the observed gateway process, not the nicotine or the alcohol. Given the access situation, this study predicts that the gateway phenomenon results from the very same experimentation process described by the theory’s proponents, but that the reasoning behind the causal interpretation attributed to the particular substances in this process is flawed.

C. Why We Care

Smoking, drinking, and illicit drug use impose substantial costs on individual users and society as a whole. Smoking contributes to numerous health problems and consequently leads to larger health care costs for both individuals and governments.3 Alcohol, while perhaps providing

2 Further initiations into more potent substances will also differ based on unobserved heterogeneity 3 Second-hand cigarette smoke likely affects the health and thus the pocketbooks of nonsmokers who regularly inhale it. For an examination of the medical consequences of smoking, see USDHHS (2004a); see CDC (2000) for an estimation of the economic costs.

5 some health benefits when used moderately, negatively affects health when used in excess.4

Drinking also contributes to socioeconomic problems such as drunken driving, dangerous sexual behaviors, violence, and alcoholism.5 Consumption of drugs such as marijuana, cocaine, and heroin worsens health almost by assumption. The socioeconomic costs of illegal drugs (basically the same as those listed above for alcohol) are difficult to quantify but substantial. It is well established, for example, that drug use often occurs in conjunction with other criminal activity. 6

Empirically, the first use of cigarettes, alcohol, and illicit drugs and the consequent consumption patterns for them almost always emerge during adolescence (Kandel, 1978a). It is therefore widely believed that the best way to deal with the societal problems created by use of nicotine, alcohol, and harder drugs is initiation prevention. Substantial efforts and expenditures from both the public and private sectors attempt to dissuade youth from substance use by educating them about the problems and costs associated with these activities. Similarly substantial efforts and expenditures (as well as laws) attempt to restrict youth access to cigarettes and alcohol and to restrict the access of the entire society to marijuana, cocaine, and other illicit substances.

A problem with current prevention efforts reveals itself in the commonplace observation that tobacco, alcohol, and illegal drugs are not usually consumed in isolation, so that a policy targeting the use of a particular substance potentially has either negative or positive effects on the use of one or more of the others. Numerous studies provide conflicting findings on the direction of such effects.7 Econometric analyses, for example, yield divergent results about whether alcohol and marijuana are economic substitutes or complements for youth and young adults.8

Such results make it difficult to determine if the usual policy prescriptions for preventing initial use of a particular substance actually work.

4 “In excess” typically means more than 2 drinks per day for women and more than 4 per day for men. 5 Refer to NIAAA (2000) for an in-depth analysis of economic, health, and other issues involving alcohol. 6 See EOOTP (2003) for estimates of the broad range of costs associated with drug use. 7 For a review, see Kenkel, Mathios, and Pacula (2001). 8 These studies include Chaloupka and Laixuthai (1997), Pacula (1998b), Cameron and Williams (2001), DiNardo and Lemieux (2001), Williams et al (2001), and Williams and Mahmoudi (2004).

6 To combat the multiple substance use problem, the typical policy prescription seeks to prevent the first use of tobacco and alcohol, taking as given the idea that these lesser drugs serve as gateways leading adolescents to later begin harder drug use.9 Yet, the gateway assumption runs into problems if substances are substitutes in consumption, since nothing in the theory suggests that use of a lesser substance ceases, even periodically, after initiation into the harder one. A possible explanation for the conflicting evidence about substitutability found thus far in the literature and detailed later is that the existing literature inadequately models the individual decision to use a given drug. The model outlined addresses the individual decision problem, and also allows the choice to use a substance to depend on messages received prior to initiation and the consequences of consumption decisions after they occur.

CHAPTER 2: A Review of the Literature

For convenience, we categorize the relevant literature on adolescent substance use into psychosocial and economic studies. This stratification simplifies the four-category approach to substance use employed by Pacula (1995), combining theories of social interaction/learning, information, and development into one empirical approach labeled as psychosocial. Early work on gateway theory (indeed, the theory itself) originates in the fields of sociology and psychology, using data from various epidemiological studies of U.S. youth and young adults in the late 1960’s and early 1970’s.10 Kandel (1978b) lists the general empirical observations underlying the theory:11

1. The period for risk of initiation into illicit drug use is over by the mid 20’s. 2. A high proportion of youths who have tried marijuana will eventually go on to experiment with other illicit drugs. 3. Later age of onset is associated with lesser involvement and greater probability of stopping. 4. There are clear-cut developmental steps and sequences in drug behavior, so that use of one of the legal drugs almost always precedes use of illegal drugs.

9 Typically, marijuana also receives gateway drug status as a cause of later use of cocaine and other more powerful drugs. 10 For a review of such studies, see Kandel (1978b) and Clausen (1978). 11 Kandel actually lists six “patterns of involvement,” but the last two apply specifically to heroin addiction.

7 Data bear out that drug use follows a remarkably consistent pattern. Smoking and drinking

precede marijuana use, which precedes use of more dangerous drugs. Empirically, the data

also demonstrate a key concept of gateway theory. The theory does not necessarily apply in

specificity to any one drug or substance such as cigarettes. Alcohol and marijuana also serve

as gateways, though cigarettes usually receive attention as being the most important causal

path. Importantly, and as mentioned above, the mechanisms behind causation are not

important in the early literature (O’Donnell and Clayton, 1982) - the important point is that a

causal process exists in the sense that prior use of one drug biologically causes later use of

another.

As Pacula (1995) describes in Table 1, later psychosocial theories of substance use seek to explain the causal mechanisms at work in the gateway process. The alternative (to the gateway

Table 1 - Adolescent Substance Use Theories (from Pacula, 1995) Theory Role of Psychosocial Development in Substance Use Social Interaction & Learning Socialization outweighs individual development in predicting drug use. Information Poorly processed information, likely from bad sources, leads to drug use. Development Substance use is simply a stage of adolescent development. idea) psychosocial explanation for observed use patterns is individual–level heterogeneity, a causal mechanism picked up and expanded on by economists searching for gateway explanations grounded in economic theory. Heterogeneity between study participants obviously exists prior to and after initiation into use of any of the substances in question. Though it is a given that individual drug users differ from one another in preferences for use, the specific explanation as to how individual-level differences provide a viable alternative to gateway theory rests on the assumption that these differences affect the likelihood that a particular individual uses a particular drug in isolation or uses a number of drugs in combination. Based on the innocuous assumption that heterogeneity exists, then, to the extent that it remains unobservable to the researcher, heterogeneity implies that unobservable, confounding factors largely explain why one person who drinks and smokes later uses marijuana and cocaine, while another person abstains from the use of all four products (Clausen, 1978). More importantly, though, heterogeneity might explain

8 why many drinkers and smokers never make a further progression, and why many of those who do never progress beyond cannabis use (Kleiman, 1992).

While the early non-economic literature on gateway theory typically operates under the assumption that gateway theory actually exists, investigators nearly universally base this assumption on observed correlations in the data instead of any robust statistical analysis.12 An overview of the existing economic literature indicates that economists, though relative latecomers to work on gateway theory, are more likely than authors from other fields to critically examine the posited causal role of the gateway drugs. Economists’ main contribution, along with other investigators from fields such as public policy, is a critical examination of the actual methods through which causation may operate.

DeSimone (1998) and Kleiman (1992) generalize two avenues, based in economic theory, through which a gateway effect might operate. The first avenue reflects a reduced uneasiness or enhanced curiosity about trying progressively harder drugs after initiation into the gateway drug, where the initial experimentation provides evidence of enjoyment. Consumption of the gateway drug thereby alters the marginal benefit-marginal cost comparison for initial use of other drugs, increasing the probability of initiation into those substances. The second pathway describes a rational addiction process, where the “high” from using a lower tier substance gradually diminishes and the user experiments with more powerful substances in order to replicate the lost effect on overall utility. For instance, Morral, McCaffrey, and Paddock (2002) find that gateway effects disappear once they control for an individual’s propensity to use drugs, which they theorize to be correlated both with the opportunity to use drugs and the probability of use given such an opportunity.

Stepping back, however, it is important to recognize that the relevant (to gateway theory) economic literature on substance use can be divided into three categories: use of one substance in

12 O’Donnell and Clayton (1982) state that a “widely accepted” conception of gateway causality in sociology is that use measures are non-spuriously correlated and that use of the gateway drug precedes the harder drug.

9 isolation, single-period use of two or more substances, and multi-period use of more than one

drug. Kenkel, Mathios, and Pacula (2001) provide an excellent and concise outline of the single-

substance studies, and Table 2 summarizes the single-period multiple-substance studies. Though

all three types of investigation provide insight into gateway theory, only the latter studies claim to

examine the gateway hypothesis explicitly.13

Table 2 – Economic Studies of Contemporaneous Cross-Price Elasticities In Support of Gateway Theory Researchers (Year) Findings Moore and Cook (1995) The persistence into adulthood of drinking patterns developed in adolescence arises mainly from habit formation instead of unobserved variables. Pacula (1998b) Higher beer taxes lead to lower marijuana use, implying that beer and marijuana consumption are complementary. Chaloupka et al (1999) Cigarette prices are inversely related to marijuana use, both for initial use as well as the amount of consumption by current users. Dee (1999) Increasing the legal drinking age from 18 to 21 decreased youth smoking, and higher tobacco taxes are associated with lower levels of teenage drinking. Farelly et al (2001) An increase in the tax on cigarettes induces a decrease in the level of marijuana use among current users. Williams et al (2001) College students use alcohol and marijuana in complement. Williams and Mahmoudi (2004) The price of alcohol and the fine for a higher blood alcohol content than legally allowed (for driving) are both negatively related to marijuana use. Casting Doubts on Gateway Theory Researchers (Year) Findings Model (1993) Between 1975 and 1978, marijuana decriminalization was correlated with a “significant reduction” in hospital emergency room visits associated with harder drugs. Goel and Morey (1995) Cigarettes and liquor are substitutes, though both appear to be complementary to leisure. Chaloupka and Laixuthai (1997) Youth living in states where marijuana possession was decriminalized to some extent during the 1980’s were less likely to drink during that time frame than their counterparts in criminalized states. DiNardo and Lemieux (2001) The standardization of the legal drinking age in the U.S. to 21 in 1984* led to an increase in marijuana consumption. Other Findings Researchers (Year) Findings Grossman and Chaloupka (1998) Cocaine is a complement to marijuana but a substitute for alcohol. Decker and Schwartz (2000) Alcohol prices are inversely related to smoking participation (suggesting cigarettes and alcohol are complements) while cigarette prices are positively correlated with alcohol use (suggesting they are substitutes) Cameron and Williams (2001) Marijuana substitutes for alcohol but is complementary to cigarette use; drinking and tobacco smoking are complementary. DeSimone and Farrelly (2003) Higher cocaine prices depress marijuana demand, but youthful marijuana use is unaffected by cocaine prices. * - Standardization increased the legal drinking age in most states (Toomey, Rosenfeld, and Wagenaar, 1996).

A true gateway study looks at intertemporal cross-price elasticities by examining individual-

level demand equations. Prices need not always be monetary, however. As typically studied, a

13 Most of the single-period, multi-substance studies simply look for cross-price elasticity estimates and do not claim to explicitly study gateway theory. Among those that do claim gateway implications, findings of contemporaneous complementarity are typically viewed as supportive of the gateway hypothesis, since such complementarity does not contradict the observed gateway progression. Under this view, evidence of substitutability would support arguments for individual heterogeneity, since a reversal of the gateway progression or a movement back and forth between two substances at different reference points on the progression scale is inconsistent with the theory. Such a view is not universal, however. Saffer and Chaloupka (1999), for example, view complementarity as evidence against gateway theory, since it “suggests that drug users prefer to use various drugs together rather than to substitute one for the other.”

10 true gateway exists if current policy or a monetary price affects current drug use as well as future

use; no gateway implies that current factors influence current consumption but not use in the

future. A summary of the relevant economic literature appears in Table 3,14 which also shows

that few published economic studies of this type exist and indicates that work remains undone.

Examination of Table 3 shows utilization of a variety of methods to examine gateway theory,

and also indicates conflicting findings, sometimes even within a single study. Thies and Register

Table 3 – Economic Studies of Gateway Theory In Support of Gateway Theory Researchers (Year) Findings DeSimone (1998) Prior marijuana participation increases the chances of current cocaine use, and alcohol and marijuana are complements. Pacula (1998a) Higher alcohol prices in the past lead to lower current marijuana use, though previous period marijuana prices have no effect on current alcohol consumption. Beenstock and Rahav (2002) A natural experiment among Israelis aged 18 to 40 yields evidence of a “causal effect” of smoking on later marijuana use, but no such effect for marijuana use progressing to harder drugs. Abdel-Ghany and Wang (2003) Cigarettes are a gateway to marijuana, though alcohol is not. Casting Doubts on Gateway Theory Researchers (Year) Findings Thies and Register (1993) No evidence of strong gateway effects from marijuana decriminalization, though decriminalization might have led to some experimentation with harder drugs and is weakly associated with an increase in heavy drinking (six or more drinks in one sitting). Saffer and Chaloupka (1999) Alcohol and marijuana are substitutes, though alcohol, marijuana, cocaine, and heroin are all complements with the noted exception. Marijuana decriminalization results provide weak evidence for complementarity between marijuana and alcohol, but measured effects are too small to be meaningful. van Ours (2003) Sensitivity analysis suggests that any gateway from marijuana to harder drugs is limited and that results showing complementarity between marijuana and harder drug use are driven by unobserved heterogeneity.

(1993), for example, study the effects of marijuana decriminalization on alcohol, marijuana and

cocaine consumption using the Monitoring the Future (MTF) data, finding that results using the

1984 sample differ from those using the 1988 sample. Decriminalization significantly increases

cocaine consumption in 1984 but not 1988, and substantially decreases heavy drinking in 1988

but not 1984. They take their results to be unsupportive of the gateway idea that marijuana use

leads to “compulsive” use of harder drugs, but that continued decriminalized status for marijuana

correlates with more occasions of consuming six or more drinks in a single sitting.

DeSimone (1998), using the 1979 National Longitudinal Survey of Youth (NLSY79), finds

support for gateway effects in that prior marijuana use is significantly associated with the current

14 The categorization of findings in Table 2 and Table 3 is based purely on the author’s reading of the literature and does not necessarily reflect the investigators’ views on gateway theory. See footnote 9 for the example of Saffer and Chaloupka (1999).

11 probability of using cocaine. Also employing the NLSY, Pacula (1998a) finds evidence that prior smoking and drinking increase the probability an individual currently smokes marijuana.

Utilizing the National Household Surveys on Drug Abuse (NHSDA), Saffer and Chaloupka

(1999) state that although statistically significant results show that alcohol, marijuana, cocaine, and heroin are generally complements in consumption, these results actually argue against gateway theory due to the small magnitude of the effects.

More recently, Beenstock and Rahav (2002) use Israeli data to show that cohorts growing up in periods of cheaper cigarette prices are more likely to smoke and to start smoking at a younger age than cohorts growing up in times when cigarette prices are relatively higher, while Abdel-

Ghany and Wang (2003) determine that prior smoking significantly increases the likelihood an individual currently consumes marijuana.15 Finally, work by van Ours (2003) uses data from

Amsterdam to suggest the importance of unobserved heterogeneity between substance users. The author uses sensitivity analysis to raise questions about his own initial findings, which indicate that marijuana serves as a “stepping-stone” for cocaine use.

Indeed, unobserved heterogeneity possibly explains the observation by Abdel-Ghany and

Wang (2003) that “the earlier an individual starts using these gateway substances (cigarettes and alcohol), the more likely such an individual is to progress to smoking marijuana.” That statement suggests the presence of bias in findings that do not control for individual-level variables that influence both early experimentation and eventual progression. A new study controlling for some individual-level factor like risk preference might find that a previously observed gateway effect disappears, or alternatively indicate that prior results supportive of gateway theory are robust to sensitivity analysis. The model presented below attempts to deal with one aspect of differential risk preferences through the introduction of a Bayesian information updating process.

CHAPTER 3: Data and Methodology

15 Alcohol use, however, does not affect the likelihood of marijuana use. The authors use the 1999 National School-based Youth Risk Behavior Survey as the basis of their study.

12 Data

To investigate joint substance use decisions as described above, we employ the first seven annual panels of the NLSY97. NLSY97 is a continuing national probability sample of 8,984 men and women aged 12-16 as of December 31, 1996 (and thus born from 1980 to 1984), and is designed to gather data on these individuals as they move out of school and into the labor force and adult life. The survey oversamples non-military Hispanic, black, and economically disadvantaged non-black/non-Hispanic youth.16 A more detailed description of the respective datasets employed from each panel, and the process of trimming those datasets down from the original 12,686 survey members, occurs in the sections below for which the respective panels are relevant.

Other data sources include various years’ versions of the Brewer’s Almanac, The Tax Burden on Tobacco, The Book of the States, and 1997-2004 System to Retrieve Information from Drug

Evidence (STRIDE) data collected by the Drug Enforcement Agency (DEA). State beer tax rates and corresponding cigarette tax rates come from the Brewer’s Almanac and The Tax Burden on

Tobacco, respectively. STRIDE contains marijuana price information that necessitates a more extensive description below, and The Book of the States lists minimum drinking ages in each state.

STRIDE

A New Theoretical Model

To expand upon the replicated paper, the theoretical model proposed here extends (through the addition of a learning component) an explicit gateway model outlined in Kenkel, Mathios, and

Pacula (1998). Additionally, the proposed model adheres to a Bayesian information updating formula similar to that presented in Orphanides and Zervos (1995).17 For context, the constrained optimization problem of the original Kenkel, Mathios, and Pacula model is presented first; then, 16 For a more detailed summary of the data, see Pacula (1998), BLS (1999a), and BLS (2002). 17 As detailed later, the Bayesian process here, which focuses on punitive consequences of use, differs from that of Orphanides and Zervos, which centers on the process surrounding addiction. Also, the model presented here is similar to the Pacula (1998b) model presented in the previous section.

13 we outline, expand, and solve a simplified, single-substance version of the revised model. The unique contribution in the revised model, the learning process, is also described. Finally, we expand the new model into a multi-substance framework.

In the Kenkel, Mathios, and Pacula framework, then, an adolescent maximizes

t   [U (Yt)  btV (At,Tt, Mt,Ct, St)] (1) t subject to

t  R It  (Yt  PAtAt PTtTt  PMtMt  PCtCt) . (2) t t

Choice variables are Y (the composite consumption good), A (alcohol), T (cigarettes), M

(marijuana), and C (cocaine), and the price of the composite good Y is normalized to one each period. I is earned income.18 Rt = 1/(1 + r)t-1, where r is the market interest rate and R thus serves as the discount rate for earned income. Factors influencing the marginal utility of drug

consumption appear in the vector b: bt = G(Zt,it,t), where Z includes observables,  represents

legal risks (specific to substance i), and  captures individual-level heterogeneity. Lastly, St is a stock variable that accounts for past use of any and all of the addictive goods and puts the model in a rational addiction setting. St = G(At-j,Tt-j,Mt-j,Ct-j), and allows for reinforcement across substances, though such reinforcement can differ from one substance to the next.

The idea that the common pattern of adolescent drug progression depends upon society-wide legality issues, mentioned above, also appears in the Kenkel, Mathios, and Pacula paper. That paper focuses on the higher marginal cost of obtaining the harder substances. From the point of view of the current study,19 however, the key component of the model is b, the marginal utility of drug consumption. Prior to initiation, an adolescent does not know much about b, so in addition to any marginal benefit and marginal cost comparisons, experimentation occurs in order to learn about b. Experimentation potentially increases overall utility by moving the adolescent from a

18 For adolescents, we might envision an unearned income component as well. 19 The insights about b are original. They are not taken from Kenkel, Mathios, and Pacula.

14 world of unknowns to a world of knowledge about the individual welfare effects of substance use.

If b turns out to be relatively high, an adolescent might decide to increase use of the experimental substance or progress to use of a harder one, due to the increased likelihood that the marginal benefit of drug use exceeds the marginal cost. Alternatively (again, with a high b), a teenager may decide to stop using – if the individual foresees the possibility of future legal, parental, or school-based punishment problems, for instance.20 Cessation more likely occurs if b turns out to be relatively low. The learning model outlined below is consistent with any of these outcomes, depending on the specific structure of the V function.

We now present a simplified version of the revised model. To focus first on the contribution of Bayesian updating to the model, assume that the world includes only two consumption goods, alcohol and the composite good. To simplify the notation needed to solve the model, assume a static model. Finally, restrict b so that b = b(), where  {L,H} captures the probability of being caught, one measure of a negative consequence of drug use.21 This restriction implies that only the likelihood of being caught affects the marginal utility of drug use, and that in any period the likelihood itself only takes on a value of “low” or “high.” The constrained optimization problem each period becomes:

MAXY , A{U (Y)  b( )V (A)  (I -Y - PAA)} (3)

20 The possibility of future addiction (something else b provides potential knowledge about) could also influence an individual to not progress beyond an experimental use level (Orphanides and Zervos, 1995). 21 As mentioned in the prior footnote, see Orphanides and Zervos (1995) for a similar model of an addiction-based learning process. The authors also delineate the assumptions necessary for such a model.

15 While the adolescent does not know , he believes that with probability p,  = H, and that with

probability (1-p),  = L. If  = H, the probability of getting caught is high; if  = L, the

probability is low. Also, note that if  = H, the probability of not being caught is (1-H), and that

this case occurs with likelihood p. Similarly, if  = L, the probability of not being caught is (1-

L). The key point is that H + L  1. We restrict the b function so that b(L) > b(H), implying that drug use adds more to total utility in an environment where the probability of punishment for substance use is low (when local enforcement of drug laws and underage alcohol and tobacco

22 regulations is less strict) than when it is high. Accordingly, we assume that H > L. Finally,

note that the probability of not being caught is p(1-H) + (1-p) (1-L).

Pr(not caught |H )  p Thus by Bayes’ Theorem, Pr(  H | not caught)  (4) Pr(not caught)

(1H )  p = (5) (1H )  p  (1L)  (1 p)

(Greene, 2000). The ratio on the right-hand side of Equation (5) is shown in Appendix B to be

less than p, based on the assumption that H > L. In words, Pr(  H | not caught)  p says that the posterior, updated perception of the probability that being caught is high, given that the individual is not caught, is less than the prior perceived probability that being caught is high.

Thus, not being caught provides positive reinforcement to the individual about future use risk and increases the likelihood that subsequent use occurs.

(1L)  (1 p) Similarly, Pr(  L | not caught)  (6) (1H )  p  (1L)  (1 p)

Given the uncertainty about b,23 the constrained maximization problem becomes

MAXY , A{U (Y)  pb(H )V (A)  (1 p)b(L)V (A)  (I -Y - PAA)} (7)

1 22 An example of a simple b function that satisfies these assumptions is b( )  .  23 Other than the uncertainty regarding b, the analysis here follows that of Pacula (1998b).

16 The first order conditions are:

U   (8) Y

V [ pb(H )  (1 p)b(L)]  PA (9) A

Y + PAA = I (10)

Substituting (8) into (9) and dividing through by the marginal utility of consumption with respect to the composite good, we end up with

V [ pb(H )  (1 p)b(L)] A  PA . (11) U Y

When the left-hand side of (10), evaluated at A = 0, is greater than PA, initiation occurs. The bracketed term in (11) represents the belief set about the marginal utility of use, which is a function of the likelihood of being caught. Given no prior alcohol consumption, this belief set multiplied by the ratio of the marginal utility of alcohol use to the marginal utility of consumption of the composite good yields the expected addition to utility from drinking initiation. In other words, initiation occurs when the expectation of the additional weighted utility provided by drinking exceeds the price of alcohol, or, more simply stated, when the expected marginal utility gained by drinking outweighs the marginal cost incurred. To see this relationship more

V U explicitly, note that at A = 0, the ratio / is only a function of Y and can therefore be A Y represented as the function f(Y). Initiation occurs when the expected marginal benefit obtained by trying alcohol for the first time exceeds the expected marginal cost, as explicitly shown in equation (12):

f(Y)[pb(θH) + (1-p)b(θL)] > PA. (12)

17 The above analysis further allows for the postulation of a generic demand function for alcohol, based on the assumption that the first order conditions from (8), (9), and (10) hold:

* A = A(p,PA,I). (13)

The simplified world presented here amplifies the key point of the model proposed in this study: adolescent demand for alcohol or any other drug of this type depends in large part on p, a youth’s

A perception that the probability of being caught is high. By assumption,  0 , since it seems p logical to conclude that a negative relationship exists between youth drinking and the likelihood of being caught, holding income and the price of alcohol constant. If the Bayesian updating process described above serves to decrease an adolescent’s posterior estimate of p, the simple act of drinking today and not getting caught increases the likelihood that drinking occurs again in the future.24

To explore the possibility of gateway effects, however, the model must include at least two substances. Here, the model focuses on the two-drug case, with marijuana as the second drug, leaving a more complicated model for future work. The constrained optimization problem each period thus becomes

MAXY , A, M{U (Y )  pb(H )V (A, M )  (1- p)b(L)V (A, M )  (I -Y - PAA - PMM )}. (15)

Though this specification assumes that the two drugs exhibit exactly the same effects on the

utility maximization problem in terms of the probability of being caught, a less restrictive

assumption that allows the probability to differ across substances does not affect the

general conclusions of the model. The additional first order condition yields:

V [ pb(H )  (1 p)b(L)]  PM , (16) M

24 Given the vast amount of substance experimentation that occurs among youth (Fishburne, Abelson, and

Cisin, 1980), it is likely that the true likelihood of getting caught in a particular period,t, is low for most adolescents, regardless of the substance in question. Moreover, if the true t is low, the probability that a positive reinforcement occurs (assuming such an effect exists) is correspondingly high.

18 so that for marijuana, initiation occurs when

V [ pb(H )  (1 p)b(L)] M  PM , (17) U Y

V U where / only depends on Y and A, through a function g(Y,A), when M = 0. Similar to M Y the single-drug situation described above, we have initiation when g(Y,A)[pb(θH) + (1-p)b(θL)] > PM , (18) or when the expected marginal benefit provided by initiation exceeds the expected marginal cost.

Since marijuana use is presumably zero (in accordance with the typically observed progression that gives rise to standard gateway theory) prior to alcohol initiation, the alcohol initiation equation remains f(Y)[pb(θH) + (1-p)b(θL)] >PA. (19)

Equations (18) and (19) again show the importance of the belief distribution about the probability of being caught. In a multi-period model using pt terms instead of simply p, the Bayesian updating process appears more explicitly as pt evolves (presumably downward, on average) over time. Also, a more comprehensive model would introduce factors other than information updating, such as individual-level heterogeneity and/or an addictive-stock variable, that affect the marginal utility of use for the substances under study. The steps just outlined, however, sufficiently describe the model’s original contribution to the literature; we next turn to empirical work, seeking evidence that the model might play a role in the gateway process.

The Empirical Framework

19 References

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21 Appendix - Proof

(1H )  p Assuming H > L, it is now shown that < p. (1H )  p  (1L)  (1 p)

(1H )  p < p (1H )  p  (1L)  (1 p)

(1H ) < 1 (1H )  p  (1L)  (1 p)

(1H )  (1H )  p  (1L)  (1 p)

(1L) 1  p   (1 p) (1H )

(1L) 1 p   (1 p) (1H )

(1L) 1  (1H )

(1H )  (1L) ,which is true based on the assumption that H > L. Q.E.D.

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