# Marijuana on Main Streets? The Story Continues in Colombia
An Endogenous Three-part Model††thanks: We acknowledge the supercomputing
resources that were made available by the Centro de Computación Científica
Apolo at Universidad EAFIT (http://www.eafit.edu.co/apolo) to conduct the
research reported in this scientific product. All errors are our own.

Andrés Ramírez–Hassan, Catalina Gómez-Toro
Katherin Tangarife-Londoño, Santiago Velásquez OMEGA research group,
Universidad EAFIT, Medellín, Colombia. E-mail<EMAIL_ADDRESS>Carrera
49 7 Sur-50, phone: + 57 2619500.OMEGA research group, Universidad EAFIT,
Medellín, Colombia. E-mail<EMAIL_ADDRESS>research group,
Universidad EAFIT, Medellín, Colombia. E-mail<EMAIL_ADDRESS>research group, Universidad EAFIT, Medellín, Colombia. E-mail:
<EMAIL_ADDRESS>
###### Abstract

Cannabis is the most common illicit drug, and understanding its demand is
relevant to analyze the potential implications of its legalization. This paper
proposes an endogenous three-part model taking into account incidental
truncation and access restrictions to study demand for marijuana in Colombia,
and analyze the potential effects of its legalization. Our application
suggests that modeling simultaneously access, intensive and extensive margin
is relevant, and that selection into access is important for the intensive
margin. We find that younger men that have consumed alcohol and cigarettes,
living in a neighborhood with drug suppliers, and friends that consume
marijuana face higher probability of having access and using this drug. In
addition, we find that marijuana is an inelastic good (-0.45 elasticity). Our
results are robust to different specifications and definitions. If marijuana
were legalized, younger individuals with a medium or low risk perception about
marijuana use would increase the probability of use in 3.8 percentage points,
from 13.6% to 17.4%. Overall, legalization would increase the probability of
consumption in 0.7 p.p. (2.3% to 3.0%). Different price settings suggest that
annual tax revenues fluctuate between USD 11.0 million and USD 54.2 million, a
potential benchmark is USD 32 million.

JEL: D12, H25, K14, K42

Keywords: Marijuana demand, Marijuana legalization, Three-part model,
Truncation.

##  1 Introduction

Understanding the demand for illicit drugs is relevant, as it is a necessary
analysis to comprehend the potential effects of their legalization. Cannabis
remains the most commonly used illicit drug worldwide, with an estimated 209
million consumers in 2020, reflecting a 23% increase over the last decade.
This is followed by opioids (61 million), amphetamines (34 million), cocaine
(21 million), and ecstasy (20 million) (UNODC, , 2022). For decades, several
countries have entered the debate on the legalization of marijuana. In this
paper, we provide a new methodology for examining the consequences of
legalizing marijuana, which helps to answer questions regarding its access,
and extensive and intensive margins. Therefore, we can have a better
understanding on: how much the prevalence and intensity of marijuana use rise
under legalization?, is there heterogeneity in response to legalization among
different age groups?, and could government policies based on taxation and
campaigns about risk perception be effective in curbing use?

We extend Jacobi and Sovinsky,  (2016)’s proposal, who proposed two
independent two-part models to study the extensive and intensive margins of
marijuana demand taken into account access restrictions when modeling demand
for illicit drugs in Australia. In particular, we propose an endogenous three-
part model that simultaneously takes into account access, extensive and
intensive margins, as well as incidental truncation. To the best of our
knowledge, this is the first paper to estimate demand for an illicit drug that
considers simultaneously these features. Taking into account access
restrictions is relevant as illicit drugs are not as easy to find, thus non-
users have little information about how to get marijuana, which is a necessary
condition to becoming a user (Jacobi and Sovinsky, , 2016). Truncation is also
relevant when modeling demand for illicit drugs as individuals tend to
withhold information about their use (Lloyd, , 2013). We apply our
methodological proposal to model the demand for marijuana in Colombia, and
study the potential implications of the legalization of marijuana in this
market, where marijuana is decriminalized, making different counterfactual
exercises trying to respond relevant inquiries regarding marijuana
legalization. Colombian market is significant as this is one of the world’s
main producers of marijuana (UNODC, , 2021), resulting in relatively easy
access and low prices.

For decades, countries have faced pressures regarding the decision to
decriminalize or legalize the marijuana market. It is important to note that
countries can implement a policy of liberalizing the marijuana market through
either full legalization or decriminalization. According to Nkansah-Amankra
and Minelli,  (2016), legalization occurs when authorities approve the use of
a substance previously prohibited by law, thereby eliminating the risk of
arrest or fines. Decriminalization suspends criminal sanctions for using or
possessing a particular substance; however, it maintains the substance’s
illegal status, allowing for potential punishment through civil fines,
education, social work, and other measures. Uruguay was the first country to
legalize recreational marijuana in 2013; some US states have also done so, as
well as Canada in 2018 (Jacobi and Sovinsky, , 2016).

The literature highlights that the use of illicit drugs entails high costs for
society, such as pressure on health systems, productivity loss (Van Ours, ,
2006), delinquency, violence (Norström and Rossow, , 2014), incarceration and
costs of the criminal justice system, educational performance, child abuse,
and corruption, among others (MacCoun et al., , 1996; Wen et al., , 2014).
There is also evidence that fewer young people consider cannabis consumption
dangerous, which leads to the normalization of consumption behavior (Järvinen
and Demant, , 2011). Additionally, there is the hypothesis that the
legalization or decriminalization of marijuana serves as an incentive for
consumption for younger populations since the early onset of cannabis use
substantially increases subsequent consumption rates (Pudney, , 2004). This
scenario implies that it is not easy for countries to decide to liberalize the
market. Studies that show post-legalization effects, such as those of Rubin-
Kahana et al.,  (2022) and Roffman,  (2016), suggest that some negative
repercussions may not manifest until 5 or 10 years later, and the same applies
to positive impacts.

Those in favor of marijuana legalization argue for a reduction in violence
generated by the illegal market (Donohue III et al., , 2010), as well as a
decrease in the cost of law enforcement (police and judiciary), a reduction in
arrests (Roffman, , 2016). Mace et al.,  (2020); Irvine and Light,  (2020);
Jacobi and Sovinsky,  (2016); Caputo and Ostrom,  (1994), also speak about tax
revenue that could benefit from legalization for the United States, Canada,
and Australia, with funds allocated to education, sports, and other addiction
programs. On the issue of taxation, Miron,  (2005) also provides insights.
Those who oppose legalization refer to the negative aspects of legalization,
such as the possibility of a price decrease due to the elimination of
transaction costs associated with illegality (Becker et al., , 2006). Another
relevant aspect is the potency of cannabis and its psychoactive component,
which can have increasingly harmful consequences for health, including mental
health problems (ElSohly et al., , 2016; Van Ours, , 2006).

Thus, there is evidence of increased hospitalizations and intoxications due to
excessive marijuana consumption post-legalization (Roffman, , 2016; Rubin-
Kahana et al., , 2022), as well as psychotic and mental health effects (Moran
et al., , 2022). Studies show an increase in consumption due to legalization
in the United States. The most recent ones, such as that of Mennis et al.,
(2023), show that after the legalization of recreational marijuana, there was
an increase in consumption, especially among teenagers. Additionally,
researchers have found an increased frequency of marijuana use and a higher
prevalence of symptoms associated with marijuana use disorder (Kilmer et al.,
, 2022). Barker and Moreno,  (2021) found that legalization had more
significant direct effects on those who were already marijuana consumers,
surrounded by an increasingly favorable climate for consumption.

The Canadian case (Rotermann, , 2020) shows similar results, where the
increase in consumption is mainly associated with men over 25 years old.
However, it is essential to note that individuals between 15 and 17 years old
are experiencing a decrease. At the same time, the prevalence of consumption
remains stable, and there is a decrease in the acquisition of marijuana from
illegal sources. Rubin-Kahana et al.,  (2022) found different sources with
conflicting data for Canadian teenagers, some studies report an increase, but
most do not show a pronounced increase, demonstrating ambiguity in the
analysis. In the case of Uruguay, Laqueur et al.,  (2020) sought to study the
effect of legalization on high school students in Montevideo and regions
within the country after legalization, but found no evidence of an impact on
cannabis consumption or perceived consumption risk. However, they did find an
increase in student perception of cannabis availability after legalization.

In addition to the above, polydrug use, which involves the combination of
several substances, has become more visible. Regarding this, the most common
combination in the Americas is that of cannabis with stimulants (such as
cocaine and ecstasy), followed by opioids with stimulants, and finally,
cannabis with opioids. The growing trend of polydrug use poses significant
risks to consumers due to the interaction between substances (UNODC, , 2022),
constituting an essential topic on the global public agenda. Furthermore,
since the 1970s, the question has arisen as to whether access to marijuana
leads to an increase in the use of other more problematic substances, with the
so-called “gateway hypothesis” (DeSimone, , 1998; Kandel, , 1975). Regarding
this, Kandel et al.,  (1992) argue that the progression to more toxic drugs
depends on prior use of cigarettes or alcohol, then marijuana, and then more
harmful substances. Among the new trends in consumption are an expansion in
the forms of use and more potent products (Hammond et al., , 2021; Rubin-
Kahana et al., , 2022). Furthermore, researchers (Guttmannova et al., , 2021;
Kim et al., , 2021) discovered a positive association between the frequency of
cannabis and alcohol consumption. Additionally, (Weinberger et al., , 2022)
established a positive relationship between marijuana consumption and
frequency among tobacco users.

Finally, different studies have attempted to estimate the price elasticity of
marijuana demand in the US, finding that it is inelastic (Davis et al., ,
2016; Kilmer et al., , 2014; Grossman, , 2005; Nisbet and Vakil, , 1972),
ranging from -0.69 to -0.26. Researchers conducted the same exercise in other
countries, such as South Africa, Thailand and Australia, and found that demand
in those countries is also in an inelastic range (Riley et al., , 2020;
Sukharomana and Chang, , 2017; Van Ours and Williams, , 2007). Gallet,  (2014)
observed that marijuana demand shows less responsiveness to prices than other
drugs.

Our results suggests that selection into access is relevant for the intensive
margin conditional in the extensive margin, and that modeling simultaneously
the three stages is important. We found that marijuana is an inelastic good
(the average elasticity is -0.45), there is not statistically significance
regarding price heterogeneity among age groups, and the risk perception about
its use is also not relevant on the intensive margin, but it is very relevant
in the access and extensive margin. Moreover, we did not find heterogeneity
regarding age splines in the intensive margin, but there is also relevant
heterogeneity regarding the access and extensive margin. In general, we found
that demographic and socioeconomic features are important to explain the three
stages of marijuana demand in Colombia, all parameters estimates give
intuitive results, for instance, women have less probability of having access
and using marijuana, and their consumption is 45.5% lower than consumption of
men.

Regarding the counterfactual exercises, we found that legalization of
marijuana implies an overall increase of the probability of use in 0.7
percentage points (p.p.), from 2.3% pre-legalization to 3.0% under
legalization. The population group that faces the higher increase in the
probability of consumption is younger individuals (20s age spline) with a low
or medium risk perception about using marijuana, this is 3.8 p.p., from 13.6%
to 17.4%. In addition, tax revenues from taxation to marijuana may fluctuate
between USD 11.0 million to USD 54.2 million, this depends on the tax setting.
A potential benchmark is USD 32 million, where marijuana tax is US¢37.8, which
implies a price equal to US¢39.1, half the price of actual marijuana for
individuals with access.

After this introduction, we show our econometric framework in Section 2.
Section 3 shows the results of the demand of marijuana in Colombia using
information from the National Survey on the Consumption of Psychoactive
Substances in 2019. Section 4 shows results of tests regarding exclusionary
restrictions and some robustness checks. Section 5 shows results of potential
implications of marijuana legalization on the extensive and intensive margins,
and potential revenues that the government would have from this public policy.
Concluding remarks are shown in Section 6.

##  2 Econometric approach

###  2.1 Drug access

We set $A_{im}$ indicating if individual $i$ in market $m$ has access to
marijuana,

$\displaystyle A_{im}=\begin{Bmatrix}1,&U_{im}^{a}>0\\\ 0,&U_{im}^{a}\leq 0\\\
\end{Bmatrix},$ (1)

where
$U_{im}^{a}=\bm{w}_{i}^{\top}\bm{\alpha}_{a}+o_{i}\tau_{a}+\bm{\omega}_{a}+d_{im}\beta_{a}+V_{im}$
is the access latent variable, $i=1,\dots,N$, $m=1,\dots,M$.

Equation 1 defines if individual $i$ has access to marijuana in market $m$
($A_{im}=1$), or not ($A_{im}=0$), if she/he has a net positive utility from
it. The latter is a function of socioeconomic and demographic characteristics
($\bm{w}_{i}$) such as age splines (teenager or 20’s, 30’s, 40’s, and 50’s or
older), socioeconomic strata (low, medium and high), years of education,
gender, mental and physical health status (good or bad), a dummy variable
indicating if friends or family members use marijuana, and risk perception
about using marijuana (low, medium and high). The latter variable is
associated with mental, physical or/and social risks. There is no legal risk
for marijuana users in Colombia as it is legally allowed the personal dose of
regular cannabis (up to 20 grams). The risk perception is potentially
influenced by the public policy, as marketing campaigns may affect risk
perception about marijuana use. We also control for a dummy variable
indicating previous consumption of alcohol and cigarette ($o_{i}$),
characteristics of the market such regional-fixed effects ($\bm{\omega}_{a}$),
and presence of drug dealers in the neighborhood ($d_{im}$). The latter is a
supply side variable that should affect access to marijuana. The location
parameters are $\bm{\alpha}_{a}$, $\bm{\omega}_{a}$ and $\beta_{a}$, and we
assume $V_{im}\sim N(0,1)$, where the variance is set to 1 due to scale
identification issues.

###  2.2 Drug extensive margin

The latent variable
$U^{c}_{im}=\bm{w}^{\top}_{i}\bm{\alpha}_{c}+\omega_{c}+o_{i}\tau_{c}+E_{im}$
defines drug use (extensive margin),

$\displaystyle C_{im}=\begin{Bmatrix}1,&U_{im}^{c}>0|A_{im}=1\\\
0,&U_{im}^{c}\leq 0|A_{im}=1\\\ -,&A_{im}=0\end{Bmatrix}.$ (2)

We observe if individual $i$ uses marijuana in market $m$ ($C_{im}=1$), or not
($C_{im}=0$). Individuals use marijuana if they have a net positive utility
from it, conditional on having access ($A_{im}=1$). Otherwise, individuals do
not use marijuana if they do not have a net positive utility from it
conditional on having access. Observe that we have missing values regarding
consumption when individuals do not have access to marijuana. This is due to
individuals without access may or may not have net positive utility from
marijuana use, therefore, we do not have information about their potential
preferences, and consequently, this set of individuals do not contribute to
identify these parameters.

The net indirect utility defining drug use depends on demographic and
socioeconomic characteristics ($\bm{w}_{i}$), regional-fixed effects
($\omega_{c}$), and a dummy indicating previous consumption of alcohol and
cigarette ($o_{i}$). The latter variable is due to the gateway hypothesis
which indicates that previous consumption of these legal substances precedes
use of illicit drugs (Kandel, , 1975; Kandel et al., , 1992). Observe that we
do not use price of marijuana in the extensive margin equation as due to its
low price in Colombia (US¢83 per joint), this variable should not affect the
extensive margin. We assume that $E_{im}\sim N(0,1)$ due to scale
identification issues, and the location parameters are $\bm{\alpha}_{c}$,
$\omega_{c}$ and $\tau_{c}$.

###  2.3 Drug intensive margin

We model the consumption quantity (intensive margin), as a function of
socioeconomic and demographic characteristics ($\bm{w}_{i}$), regional-fixed
effects ($\omega_{y}$), price of marijuana ($p_{i}$), and the interaction
between price and age brackets ($w_{i}^{\text{Age(j)}}\times p_{i}$, where
Age(j) refers to j-th age bracket). The latter due to potential heterogeneity
regarding price sensitivity among age splines. Observe that we have prices at
individual level weighted by quality (see Appendix A sections A.1 and A.2),
this implies that endogeneity would not be a concern due to there is no
demand-supply simultaneity neither endogeneous variation in the stochastic
error due to quality of marijuana (Jacobi and Sovinsky, , 2016).

We consider in the intensive margin equation incidental truncation, that is,
there are missing values for consumption quantity when individuals do not have
access, or having access reporting not to use marijuana.

$\displaystyle
Y_{im}=\begin{Bmatrix}\bm{w}^{\top}_{i}\bm{\alpha}_{y}+\omega_{y}+p_{i}\gamma_{y}+\sum_{j=1}^{J}w_{i}^{\text{Age(j)}}\times
p_{i}\gamma_{yj}+W_{im},&C_{im}=1\\\ -&C_{im}=0\ \text{or}\
A_{im}=0\end{Bmatrix}.$ (3)

Observe that the truncation setting given in equation 3, as we are not taking
into account zero consumption. This is because some marijuana users reporting
not consumption due to social stigma. The location parameters related to
consumption level are $\bm{\alpha}_{y}$, $\omega_{y}$, $\gamma_{y}$ and
$\gamma_{yj}$, and we assume $W_{im}\sim N(0,\sigma^{2}_{y})$.

###  2.4 Correlation on unobservable variables

We model simultaneously the three stages (access, extensive and intensive
margins) due to there should be unobservable variables that drive these
stages. Thus, we assume $\Xi_{im}=\left[V_{im}\ E_{im}\
W_{im}\right]^{\top}\sim N_{3}(\bm{0},\bm{\Sigma})$, where

$\displaystyle\bm{\Sigma}=\begin{bmatrix}1&\sigma_{ac}&\sigma_{ay}\\\
\sigma_{ca}&1&\sigma_{cy}\\\
\sigma_{ya}&\sigma_{yc}&\sigma_{y}^{2}\end{bmatrix}.$ (4)

Observe that if $\bm{\Sigma}$ is a diagonal matrix, the three stages are
independent, and we can perform inference estimating each equation separately.
This maybe a situation where there is no strategic search of drug dealers by
users, and that more intense consumers do not make a greater effort to get
drug dealers when arriving to a new market. On the other hand, if there is
endogenous access, which means $\sigma_{ca}\neq 0$ and/or $\sigma_{ya}\neq 0$,
we should model simultaneously these equations to get good sampling properties
of our estimators. In addition, $\sigma_{yc}$ takes into account potential
unobserved dependence between extensive and intensive margins conditional on
selection into access.

###  2.5 Estimation strategy

Observe that modeling the joint distribution of access, extensive and
intensive margin implies to integrate over a multivariate space to recover the
likelihood function. In addition, this is not a standard likelihood as
incidental truncation implies that different sets of individuals contribute to
different sets of parameters. Particularly, there are three different groups
of individuals when taking the model setting given by equations 1, 2 and 3:
all individuals ($G_{1}$) contribute to estimate the location parameters in
the access equation, individuals who report having access ($G_{2}$) contribute
to estimate the location parameters in the extensive margin equation and
$\sigma_{ac}$, and individuals who report to use marijuana ($G_{3}$)
contribute to estimate the parameters of the intensive margin equation, and
$\sigma_{ay}$, $\sigma_{cy}$ and $\sigma_{y}^{2}$.

Thus, we use data augmenting (Tanner and Wong, , 1987) to facilitate
inference, we treat latent variables as parameters, such that the augmented
model is,

$\displaystyle\underbrace{\begin{bmatrix}{U}_{im}^{a}\\\ {U}_{im}^{c}\\\
{Y}_{im}\end{bmatrix}}_{\bm{T}_{im}}$
$\displaystyle=\underbrace{\begin{bmatrix}\bm{x}_{im}^{a\top}&\bm{0}&\bm{0}\\\
\bm{0}&\bm{x}_{im}^{c\top}&\bm{0}\\\
\bm{0}&\bm{0}&\bm{x}_{im}^{y\top}\end{bmatrix}}_{\bm{X}_{im}}\underbrace{\begin{bmatrix}\bm{\theta^{a}}\\\
\bm{\theta^{c}}\\\
\bm{\theta^{y}}\end{bmatrix}}_{\bm{\theta}}+\underbrace{\begin{bmatrix}{V}_{im}\\\
{U}_{im}\\\ {E}_{im}\end{bmatrix}}_{\bm{\Xi}_{im}},$ (5)

where $\bm{x}_{im}^{s\top}$ is the vector of regressors associated with
individual $i$ in market $m$ in stage $s=\left\\{\text{access, consumption,
quantity}\right\\}$, and $\bm{\theta}^{s}$ is the vector of location
parameters in stage s.

The likelihood function is

$\displaystyle
p(\tilde{\bm{T}}_{1},\dots,\tilde{\bm{T}}_{n}|\tilde{\bm{X}}_{1},\dots,\tilde{\bm{X}}_{n},\bm{\theta},\bm{\Sigma})$
$\displaystyle=\prod_{s=1}^{G}\prod_{i\in
G_{s}}\left\\{\mathds{1}(A_{im}=0)\mathds{1}(U_{im}^{a}\leq
0)+\mathds{1}(A_{im}=1)\mathds{1}(U_{im}^{a}>0)\right.$
$\displaystyle\left.\times\left[\mathds{1}(C_{im}=0)\mathds{1}(U_{im}^{c}\leq
0)+\mathds{1}(C_{im}=1)\mathds{1}(C_{im}^{c}>0)\right]\right\\}$
$\displaystyle\times\phi(\tilde{\bm{T}}_{i}|\tilde{\bm{X}}_{im}\tilde{\bm{\theta}}_{G_{s}},\tilde{\bm{\Sigma}}_{G_{s}}),$
(6)

where $G=3$,
$\phi(\cdot|\tilde{\bm{X}}_{im}\tilde{\bm{\theta}}_{G_{s}},\tilde{\bm{\Sigma}}_{G_{s}})$
is the density function of a normal distribution with mean
$\tilde{\bm{X}}_{im}\tilde{\bm{\theta}}_{G_{s}}$ and variance
$\tilde{\bm{\Sigma}}_{G_{s}}$, and $\tilde{M}=f(G_{s},M)$, where $f(G_{s},M)$
is a function that takes as inputs a state ($G_{s}$) and a matrix ($M$), and
returns as output the appropriate subset of rows and columns of $M$. For
instance,

$\tilde{\bm{T}}_{im}\equiv f(G_{2},\bm{T}_{im})=\begin{bmatrix}U_{im}^{a}\\\
U_{im}^{c}\\\ \end{bmatrix},\ \tilde{\bm{X}}_{im}\equiv
f(G_{2},\bm{X}_{im})=\begin{bmatrix}\bm{x}_{im}^{a\top}&\bm{0}\\\
\bm{0}&\bm{x}_{im}^{c\top}\\\ \end{bmatrix},$
$\tilde{\bm{\theta}}_{G_{2}}\equiv
f(G_{2},\bm{\theta})=\begin{bmatrix}\bm{\theta}^{a}\\\ \bm{\theta}^{c}\\\
\end{bmatrix}\text{and}\ \tilde{\bm{\Sigma}}_{G_{2}}\equiv
f(G_{2},\bm{\Sigma})=\begin{bmatrix}1&\sigma_{ac}\\\ \sigma_{ca}&1\\\
\end{bmatrix}.$

We use the Bayes’ rule to perform inference in our model. However, we
implement our inferential algorithm in the unidentified parameter space as
getting draws from the posterior distribution in the identified parameter
space has a high computational cost and inferior mixing properties than a more
straightforward Gibbs sampler traversing over the unidentified space (Rossi et
al., , 2005, p. 118). Thus, we set

$\displaystyle\bm{\Omega}=\begin{bmatrix}\omega_{a}^{2}&\omega_{ac}&\omega_{ay}\\\
\omega_{ca}&\omega_{c}^{2}&\omega_{cy}\\\
\omega_{ya}&\omega_{yc}&\omega_{y}^{2}\end{bmatrix}.$ (7)

We implement our Gibbs sampling algorithm using standard conjugate independent
priors to obtain standard conditional posterior distributions that facilitate
computation. In particular, we assume that
$\pi(\bm{\theta},\bm{\Omega})=N(\bm{\theta}|\bm{\theta}_{0},\bm{\Theta}_{0})\times\pi(\bm{\Omega}|\bm{R}_{0},r_{0})$,
that is, a multivariate normal distribution for the location parameters and an
inverse Wishart distribution for the unidentified covariance matrix. We use
non-informative hyperparameters in all our exercises, that is,
$\bm{\theta}_{0}=\bm{0}$, $\bm{\Theta}_{0}=\text{diag}\left\\{1,000\right\\}$,
$\bm{R}_{0}=\bm{I}_{3}$ and $r_{0}=3+2$.

Reparameterizing the data-augmented likelihood function from equation 2.5 in
terms of $\bm{\Omega}$, and using the previous prior density function, the
Bayes’ rule implies that the posterior conditional distribution for the
location parameters is $N(\bm{\theta}|\bm{\theta}_{n},\bm{\Theta}_{n})$ where
$\bm{\Theta}_{n}=\left[\sum_{s=1}^{G}\sum_{i\in
G_{s}}\bm{J}_{G_{s}}\tilde{\bm{X}}_{im}^{\top}\tilde{\bm{\Omega}}_{G_{s}}^{-1}\tilde{\bm{X}}_{im}\bm{J}_{G_{s}}^{\top}+\bm{\Theta}_{0}^{-1}\right]^{-1}$
and
$\bm{\theta}_{n}=\bm{\Theta}_{n}\left[\sum_{s=1}^{G}\sum_{i\in{G_{s}}}\bm{J}_{G_{s}}\tilde{\bm{X}}_{im}^{\top}\tilde{\bm{\Omega}}_{G_{s}}^{-1}\tilde{\bm{T}}_{im}+\bm{\Theta}_{0}^{-1}\bm{\theta}_{0}\right]$,
and

$\bm{J}_{G_{1}}=\begin{bmatrix}\bm{I}_{\left\\{H\right\\}}\\\ \bm{0}\\\
\bm{0}\\\
\end{bmatrix},\bm{J}_{G_{2}}=\begin{bmatrix}\bm{I}_{\left\\{H\right\\}}&\bm{0}\\\
\bm{0}&\bm{I}_{\left\\{K\right\\}}\\\ \bm{0}&\bm{0}\\\ \end{bmatrix}\
\text{and}\
\bm{J}_{G_{3}}=\begin{bmatrix}\bm{I}_{\left\\{H\right\\}}&\bm{0}&\bm{0}\\\
\bm{0}&\bm{I}_{\left\\{K\right\\}}&\bm{0}\\\
\bm{0}&\bm{0}&\bm{I}_{\left\\{L\right\\}}\\\ \end{bmatrix},$

where $\bm{I}_{\left\\{d\right\\}}$ is a $d\times d$ identity matrix, $H$, $K$
and $L$ are the dimensions of $\bm{\theta}_{s}$.

We follow a sequential approach (Chib et al., , 2009; Li, , 2011) to get
standard conditional posterior distributions that allow to recover
$\bm{\Omega}$. In particular, all observations contribute to estimate
$\omega_{a}^{2}$, and given the prior distribution of $\bm{\Omega}$, which
implies that the prior distribution of $\omega_{a}^{2}$ is inverse gamma with
parameters $r_{11,0}$ and $r_{0}-2$, where $r_{11,0}$ is the element 1,1-th of
$\bm{R}_{0}$ (Greenberg, , 2012, p. 190), then the posterior conditional
distribution of $\omega_{a}^{2}$ is $IG(r_{11,n},r_{0}-2+n)$ where
$r_{11,n}=\sum_{i=1}^{n}(T_{1,im}-X_{1,im}\bm{\theta})^{2}+r_{11,0}$,
$T_{1,im}$ is the first element of $\bm{T}_{im}(U^{a}_{im})$ and $X_{1,im}$ is
the first row of $\bm{X}_{im}$.

In the next stage, we set

$\displaystyle\bm{\Omega}_{22}=\begin{bmatrix}\omega_{a}^{2}&\omega_{ac}\\\
\omega_{ca}&\omega_{c}^{2}\end{bmatrix},$

and

$\omega_{c.1}^{2}=\omega_{c}^{2}-\frac{\omega_{ac}^{2}}{\omega_{a}^{2}}.$ (8)

Given the prior distribution of $\bm{\Omega}$, and a consistent partition of
$\bm{R}_{0}$,

$\displaystyle\bm{R}_{22,0}=\begin{bmatrix}r_{11,0}^{2}&r_{12,0}\\\
r_{21,0}&r_{22,0}^{2}\end{bmatrix},$

the prior distribution of $\omega_{c.1}^{2}$ is inverse gamma with parameters
$r_{22.1,0}^{2}$ and $r_{0}$, where
$r_{22.1,0}^{2}=r_{22,0}^{2}-r_{12,0}^{2}/r_{11,0}^{2}$.

In addition, we set

$\omega_{ca.1}=\frac{\omega_{ca}}{\omega_{a}^{2}},$ (9)

and given the prior distribution of $\bm{\Omega}$, the prior distribution of
$\omega_{ca.1}|\omega_{c.1}^{2}$ is normal with mean $r_{21,0}/r_{11,0}^{2}$
and variance $\omega_{c.1}^{2}/r_{11,0}^{2}$.

We calculate

$r_{22,n}=\sum_{i\in
G_{2}}\left\\{(\bm{T}_{1:2,im}-\bm{X}_{1:2,im}\bm{\theta})(\bm{T}_{1:2,im}-\bm{X}_{1:2,im}\bm{\theta})^{\top}\right\\}+\bm{R}_{22,0}=\begin{bmatrix}r_{11,n}^{2}&r_{12,n}\\\
r_{21,n}&r_{22,n}^{2}\end{bmatrix},$

where $\bm{T}_{1:2,im}$ and $\bm{X}_{1:2,im}$ are the first and second rows of
$\bm{T}_{im}$ and $\bm{X}_{im}$, respectively.

The posterior distribution of $\omega_{c.1}^{2}$ is inverse gamma with
parameters $r_{22.1,n}^{2}$ and $r_{0}+|G_{2}|$, where
$r_{22.1,n}^{2}=r_{22,n}^{2}-r_{12,n}^{2}/r_{11,n}^{2}$, and $|G_{2}|$ is the
number of individuals in group two.

The posterior distribution of $\omega_{ca.1}$ conditional on
$\omega_{c.1}^{2}$ is normal with mean $r_{21,n}/r_{11,n}^{2}$ and variance
$\omega_{c.1}^{2}/r_{11,n}^{2}$.

We can recover $\bm{\Omega}_{22}$ using equation 9, such that
$\omega_{ca}=\omega_{ca.1}\omega_{a}^{2}$, and equation 8, where we have that
$\omega_{c}^{2}=\omega_{c.1}^{2}+\omega_{ca}^{2}/\omega_{a}^{2}$.

We set

$\displaystyle\bm{\Omega}=\begin{bmatrix}\bm{\Omega}_{22}&\bm{\Omega}_{23}\\\
\bm{\Omega}_{32}&\omega_{y}^{2}\end{bmatrix},$

where $\bm{\Omega}_{32}=\left[\omega_{ya}\ \omega_{yc}\right]$, and

$\omega_{y.1}^{2}=\omega_{y}^{2}-\bm{\Omega}_{32}\bm{\Omega}_{22}^{-1}\bm{\Omega}_{23},$
(10)

and given the prior distribution of $\bm{\Omega}$, and a consistent partition
of $\bm{R}_{0}$,

$\displaystyle\bm{R}_{0}=\begin{bmatrix}\bm{R}{22,0}&\bm{R}_{23,0}\\\
\bm{R}_{32,0}&r_{33,0}^{2}\end{bmatrix},$

the prior distribution of $\omega_{y.1}^{2}$ is inverse gamma with parameters
$r_{33.1,0}^{2}$ and $r_{0}$, where
$r_{33.1,0}^{2}=r_{33,0}^{2}-\bm{R}_{32,0}\bm{R}_{22,0}^{-1}\bm{R}_{23,0}$.

Given

$\bm{\Omega}_{32.1}=\bm{\Omega}_{32}\bm{\Omega}^{-1}_{22},$ (11)

where the prior distribution of $\bm{\Omega}_{32.1}|\omega_{y.1}^{2}$ is
matrix normal with mean $\bm{R}_{32,0}\bm{R}_{22,0}^{-1}$ and scale matrices
$\bm{R}^{-1}_{22,0}$ and $\omega^{2}_{y.1}$.

Given

$\bm{R}_{n}=\sum_{i\in
G_{3}}\left\\{(\bm{T}_{im}-\bm{X}_{im}\bm{\theta})(\bm{T}_{im}-\bm{X}_{im}\bm{\theta})^{\top}\right\\}+\bm{R}_{0}=\begin{bmatrix}\bm{R}_{22,n}^{2}&\bm{R}_{23,n}\\\
\bm{R}_{32,n}&r_{33,n}^{2}\end{bmatrix},$

The posterior distribution of $\bm{\Omega}_{32.1}$ conditional on
$\omega^{2}_{y.1}$ is matrix normal with mean
$\bm{R}_{32,n}\bm{R}^{-1}_{22,n}$ and scale matrices $\bm{R}^{-1}_{22,n}$ and
$\omega^{2}_{y.1}$. We can recover $\bm{\Omega}$ using equation 11, such that
$\bm{\Omega}_{32}=\bm{\Omega}_{32.1}\bm{\Omega}_{22}$, and equation 10, where
we have that
$\omega^{2}_{y}=\omega^{2}_{y.1}+\bm{\Omega}_{32}\bm{\Omega}^{-1}\bm{\Omega}_{23}$.

The posterior conditional distributions of $U_{im}^{a}$ and $U_{im}^{c}$ are
truncated normal in the interval $(-\infty,0]$ if $U_{im}^{l}=0$, and
$(0,\infty)$ if $U_{ij}^{l}=1$, respectively, $l=\left\\{a,c\right\\}$. Their
conditional means are
$m_{l,im}=\tilde{\bm{X}}_{l,im}\tilde{\bm{\theta}}+\tilde{\bm{\Omega}}_{l,-l}\tilde{\bm{\Omega}}_{-l,-l}^{-1}(\tilde{\bm{T}}_{-l,im}-\tilde{\bm{X}}_{-l,im}\tilde{\bm{\theta}})$,
and conditional variances
$\tau_{l}^{2}=\tilde{\omega}_{ll}^{2}-\tilde{\bm{\Omega}}_{l,-l}\tilde{\bm{\Omega}}_{-l,-l}^{-1}\tilde{\bm{\Omega}}_{-l,l}$,
where $\tilde{\bm{T}}_{-l,im}$ is the vector $\tilde{\bm{T}}_{im}$ excluding
the $l$-th component, $\tilde{\bm{X}}_{l,im}$ is the $l$-th row of matrix
$\tilde{\bm{X}}_{im}$, $\tilde{\bm{X}}_{-l,im}$ is the matrix
$\tilde{\bm{X}}_{im}$ without the $l$-th row, $\tilde{\bm{\Omega}}_{l,-l}$ is
the $l$-th row of $\tilde{\bm{\Omega}}$ excluding the $l$-th element,
$\tilde{\bm{\Omega}}_{-l,-l}$ is equal to $\tilde{\bm{\Omega}}$ excluding the
$l$-th row and $l$-th column, and $\tilde{\omega}^{2}_{ll}$ is the $ll$
element of $\tilde{\bm{\Omega}}$.

Our econometric framework differs from other literature regarding modeling
marijuana consumption in a few fundamental ways. In particular, we follow a
simultaneous three-part modeling approach that takes into account that access
is a necessary condition for use, such that the latter is endogenously
determined with extensive and intensive margins. We also incorporate that use
is a necessary condition for the intensive margin, and allow for unobserved
correlation between these stages. In addition, we consider the incidental
truncation issue due to missing reports when individuals report no to have
access to marijuana, or when reporting access, they report not to use it.
Omitting access restrictions, potential correlation on unobservable variables
and/or incidental truncation may generate inconsistent estimators. There is
also a clear link between our modeling strategy and the policy that we want to
analyze, as marijuana legalization implies basically free access for all
potential users, such that the “breaking the law” hindrance will disappear,
this is formally $P(A_{imM}=1)=1$ in our econometric framework, and will be
the basis for our counterfactual exercises.

##  3 Marijuana demand in Colombia

###  3.1 Data

We apply our approach to the problem of estimating the demand for marijuana in
Colombia. We leveraged individual-level data on the consumption of
psychoactive substances representative of the entire country in 2019. In
particular, our data come from the National Survey on the Consumption of
Psychoactive Substances (ENCSPA 2019), which is a national representative
survey, performed by the government statistical department of Colombia (DANE)
aiming to measure both legal and illegal substance abuse within the
population. The survey randomly sampled households from several municipalities
in Colombia. It targeted individuals between 12 and 65 years of age, who were
selected randomly from all the household members that met the age criterion.
The enumerators privately performed the survey. If the chosen person was
absent during the survey, the enumerator should return later but was not
allowed to change the individual to be interviewed.

We are especially interested in variables describing marijuana consumption
patterns based on the literature review and data availability. Our measure of
marijuana consumption takes into account quality based on the
tetrahydrocannabinol (THC) content (see Appendix A, subsection A.1). In
addition, we use a nearest neighbor algorithm to impute prices for those
individuals who report not to consume marijuana, and consequently, who do not
report average price (see Appendix A, subsection A.2),111Observe that
posterior estimates of our model composed by equation 1, 2 and 3 does not
require these prices. and a basic counting algorithm to construct risk
perception about drug use (see Appendix A, subsection A.3).

Table 1 presents the summary statistics for the key variables in our sample.
We consider three different outcome variables, one for each part of the three-
part model: First, whether the individual has access or not to marijuana by
reporting that it would be easy for her/him to get it. Second, whether the
individual is a consumer, specifically if they had consumed marijuana during
the last 12 months, and finally, the quantity of marijuana consumed on average
per month. The quantity consumed is measured in the number of cigarettes or
joints of marijuana; although it is not specified the number of grams, there
is some consistency in the sizes of common joints, which are usually about 1
gram.

Table 1: Summary Statistics.

|  | All |  | Access to marijuana |  | Marijuana consumer
---|---|---|---|---|---|---
|  |  |  | No |  | Yes |  | No |  | Yes
Variable |  | (1) |  | (2) |  | (3) |  | (4) |  | (5)
Access to marijuana |  | 0.58 |  | 0.00 |  | 1.00 |  | 0.57 |  | 1.00
| (0.49) |  | (0.00) |  | (0.00) |  | (0.5) |  | (0.00)
Marijuana consumer |  | 0.02 |  | 0.00 |  | 0.04 |  | 0.00 |  | 1.00
| (0.15) |  | (0.00) |  | (0.20) |  | (0.00) |  | (0.00)
Quantity consumed |  | 1.01 |  | 0.00 |  | 1.75 |  | 0.00 |  | 43.08
| (13.61) |  | (0.00) |  | (17.89) |  | (0.00) |  | (78.07)
Drug dealer in neighborhood |  | 0.38 |  | 0.26 |  | 0.47 |  | 0.38 |  | 0.50
| (0.49) |  | (0.44) |  | (0.50) |  | (0.48) |  | (0.50)
Alcohol and tobacco user |  | 0.32 |  | 0.22 |  | 0.40 |  | 0.31 |  | 0.86
| (0.47) |  | (0.41) |  | (0.49) |  | (0.46) |  | (0.35)
Price of marijuana |  | 0.83 |  | 0.86 |  | 0.80 |  | 0.83 |  | 0.84
| (0.44) |  | (0.48) |  | (0.40) |  | (0.43) |  | (0.54)
Female |  | 0.58 |  | 0.67 |  | 0.52 |  | 0.59 |  | 0.25
| (0.49) |  | (0.47) |  | (0.50) |  | (0.49) |  | (0.44)
Years of education |  | 11.8 |  | 11.48 |  | 12.04 |  | 11.79 |  | 12.34
| (4.24) |  | (4.46) |  | (4.04) |  | (4.25) |  | (3.75)
Worker |  | 0.58 |  | 0.53 |  | 0.61 |  | 0.58 |  | 0.59
| (0.49) |  | (0.5) |  | (0.49) |  | (0.49) |  | (0.49)
Good mental health |  | 0.79 |  | 0.81 |  | 0.77 |  | 0.79 |  | 0.70
| (0.41) |  | (0.39) |  | (0.42) |  | (0.41) |  | (0.46)
Good physical health |  | 0.77 |  | 0.75 |  | 0.77 |  | 0.76 |  | 0.79
| (0.42) |  | (0.43) |  | (0.42) |  | (0.42) |  | (0.41)
Marijuana users in network |  | 0.36 |  | 0.19 |  | 0.49 |  | 0.35 |  | 0.94
|  | (0.48) |  | (0.39) |  | (0.50) |  | (0.48) |  | (0.24)
Age 20’s or younger |  | 0.34 |  | 0.30 |  | 0.38 |  | 0.34 |  | 0.67
|  | (0.47) |  | (0.46) |  | (0.48) |  | (0.47) |  | (0.47)
Age 30’s |  | 0.21 |  | 0.19 |  | 0.23 |  | 0.21 |  | 0.20
|  | (0.41) |  | (0.4) |  | (0.42) |  | (0.41) |  | (0.40)
Age 40’s |  | 0.17 |  | 0.17 |  | 0.16 |  | 0.17 |  | 0.08
|  | (0.37) |  | (0.38) |  | (0.37) |  | (0.38) |  | (0.27)
Age 50’s or older |  | 0.28 |  | 0.34 |  | 0.23 |  | 0.28 |  | 0.05
|  | (0.45) |  | (0.47) |  | (0.42) |  | (0.45) |  | (0.23)
Risk perception of usage: |  |  |  |  |  |  |  |  |  |
Low |  | 0.04 |  | 0.05 |  | 0.03 |  | 0.03 |  | 0.09
|  | (0.19) |  | (0.21) |  | (0.16) |  | (0.18) |  | (0.29)
Medium |  | 0.05 |  | 0.03 |  | 0.07 |  | 0.05 |  | 0.30
|  | (0.22) |  | (0.17) |  | (0.25) |  | (0.21) |  | (0.46)
High |  | 0.91 |  | 0.92 |  | 0.90 |  | 0.92 |  | 0.61
|  | (0.28) |  | (0.26) |  | (0.30) |  | (0.27) |  | (0.49)
Socio-economic Strata: |  |  |  |  |  |  |  |  |  |
Low |  | 0.64 |  | 0.65 |  | 0.64 |  | 0.65 |  | 0.56
|  | (0.48) |  | (0.48) |  | (0.48) |  | (0.48) |  | (0.5)
Medium |  | 0.32 |  | 0.31 |  | 0.32 |  | 0.32 |  | 0.38
|  | (0.47) |  | (0.46) |  | (0.47) |  | (0.47) |  | (0.49)
High |  | 0.08 |  | 0.08 |  | 0.08 |  | 0.08 |  | 0.13
|  | (0.28) |  | (0.28) |  | (0.28) |  | (0.28) |  | (0.33)
Sample size |  | 49,414 |  | 20,909 |  | 28,505 |  | 48,255 |  | 1,159

  * Notes: Standard deviations in parenthesis. This table presents descriptive statistics for ENCSPA 2019 regarding access and consumption of Marijuana in Colombia, as well as control variables. Column 1 shows the information for the entire sample. Columns 2 and 3 show the information for individuals without and with access to marijuana. Columns 4 and 5 show the information for not consumers and consumers, respectively. Individuals who are consumers should have access; this is internally consistent for more than 98% of the population and inputted for the remaining 2%. Prices are shown in 2019 USD and were converted using the average exchange rate in 2019 (3,274 USD/COP).
Source: Authors’ construction using ENCSPA data.

Our sample population consists of 49,414 individuals. Less than 60% report
having access (row 1, column 1); of those reporting having access,
approximately 4% report consuming marijuana during the last 12 months (row 2,
column 3). Finally, the consumers take 23 joints of marijuana per month on
average, with an average price of USD 0.84 (rows 3 and 6, column 5). By
definition, and aiming to achieve consistency in the model, individuals who
are consumers should have access; this is internally consistent in the data
for more than 98% of the population and manually inputted for the remaining
2%222We replace access equal to one if the person is a consumer for two main
reasons. First, it is not intuitive for people who are current users to report
that they cannot get marijuana so it is likely an error in responding on the
part of these individuals or misreporting due to stigma. Second, for the model
to be internally consistent by construction, all users must have access. To
review possible changes to the definition of the access measure, we conducted
multiple exercises to validate the robustness of the results. The
representative individual in the survey is a female with complete secondary
education in her 20s living in a low socioeconomic stratum. More than half of
the people are workers, almost 80% report having good mental health and 77%
report having good physical health, 91% have a high-risk perception of the use
of marijuana, and 36% report having a peer (family or friend) who is a
consumer of marijuana. Approximately, 38% of the sample declares to have a
drug dealer in the neighborhood, and 32% report to have consumed alcohol and
cigarette.

###  3.2 Model results

We perform inference of the model in system 5 running 6,000 iterations with a
burn-in equal to 1,000 and a thin parameter equal to 5, thus we have 1,000
effective posterior draws. We compute several diagnostics to assess the
convergence and stationarity of the posterior chains. In general, the
posterior chains look good. Particularly, all location parameters have
dependence factors that are less than 5, actually most of them less than 2,
using Raftery et al.,  (1992)’s diagnostic, with a 95% probability of
obtaining an estimate in the interval $2.5\%\pm 1.0\%$. Regarding Heidelberger
and Welch,  (1983)’s and Geweke,  (1992)’s tests at 5% significance level, all
parameter estimates pass the former test, and 147 out of 163 location
parameters passed the latter test. The former uses the Cramer-von-Mises
statistic to test the null hypothesis that the sampled values come from a
stationary distribution, and the latter tests for equality of the posterior
means using the first 10% and the last 50% of the Markov chains. Regarding the
scale parameters, we have that all parameters have a dependence factor less
than 5, and one out of 4 does not passed the conventional values of the
Heidelberger and Welch,  (1983) and Geweke,  (1992) diagnostics.

Table 2: Posterior results of location parameters: Marijuana demand in
Colombia.

Variable | Univariate |  | Multivariate
---|---|---|---
Access | Margin |  | Access | Margin
| Extensive | Intensive |  |  | Extensive | Intensive
(1) | (2) | (3) |  | (4) | (5) | (6)
Exclusionary restrictions |  |  |  |  |  |  |
Drug dealer in neighborhood | 0.438 |  |  |  | 0.433 |  |
(0.014) |  |  |  | (0.014) |  |
Alcohol and cigarette use | 0.374 | 0.949 |  |  | 0.369 | 0.931 |
(0.014) | (0.041) |  |  | (0.014) | (0.045) |
Age |  |  |  |  |  |  |
30s | -0.095 | -0.389 | 0.241 |  | -0.094 | -0.378 | 0.457
(0.018) | (0.044) | (1.294) |  | (0.018) | (0.044) | (1.302)
40s | -0.217 | -0.611 | 0.608 |  | -0.216 | -0.617 | 0.534
(0.019) | (0.061) | (1.982) |  | (0.019) | (0.054) | (1.899)
50s and older | -0.404 | -1.050 | -3.743 |  | -0.403 | -1.019 | -2.995
(0.017) | (0.064) | (2.421) |  | (0.017) | (0.066) | (2.197)
Strata |  |  |  |  |  |  |
Medium | -0.048 | 0.086 | -0.045 |  | -0.049 | 0.090 | -0.067
(0.015) | (0.039) | (0.098) |  | (0.014) | (0.039) | (0.099)
High | -0.049 | 0.134 | -0.198 |  | -0.051 | 0.130 | -0.199
(0.024) | (0.056) | (0.140) |  | (0.023) | (0.063) | (0.149)
Risk perception marijuana use |  |  |  |  |  |  |
Medium | 0.536 | -0.112 | -0.573 |  | 0.520 | -0.112 | -0.204
(0.043) | (0.078) | (0.162) |  | (0.044) | (0.081) | (0.300)
High | 0.288 | -0.793 | -0.644 |  | 0.279 | -0.790 | -0.327
(0.032) | (0.070) | (0.152) |  | (0.034) | (0.072) | (0.247)
Years of education | 0.017 | -0.012 | -0.084 |  | 0.017 | -0.012 | -0.069
(0.002) | (0.005) | (0.014) |  | (0.002) | (0.005) | (0.015)
Female | -0.308 | -0.459 | -0.524 |  | -0.310 | -0.449 | -0.607
(0.013) | (0.040) | (0.104) |  | (0.013) | (0.037) | (0.146)
Good mental health | -0.127 | -0.193 | 0.086 |  | -0.129 | -0.192 | 0.052
(0.017) | (0.041) | (0.103) |  | (0.016) | (0.044) | (0.107)
Good physical health | 0.010 | -0.037 | -0.152 |  | 0.009 | -0.033 | -0.110
(0.016) | (0.044) | (0.114) |  | (0.016) | (0.051) | (0.105)
Marijuana users in network | 0.694 | 0.981 | 0.384 |  | 0.695 | 0.971 | 0.619
(0.014) | (0.050) | (0.183) |  | (0.013) | (0.052) | (0.306)
Worker | 0.125 | -0.098 | 0.160 |  | 0.127 | -0.094 | 0.151
(0.014) | (0.037) | (0.092) |  | (0.014) | (0.037) | (0.096)
Price |  |  |  |  |  |  |
$\log\left\\{\text{price of marijuana}\right\\}$ |  |  | -0.496 |  |  |  | -0.445
|  | (0.094) |  |  |  | (0.101)
Age 30s $\times\log\left\\{\text{price of marijuana}\right\\}$ |  |  | 0.002 |  |  |  | -0.033
|  | (0.168) |  |  |  | (0.167)
Age 40s $\times\log\left\\{\text{price of marijuana}\right\\}$ |  |  | -0.051 |  |  |  | -0.046
|  | (0.257) |  |  |  | (0.245)
Age 50s and older $\times\log\left\\{\text{price of marijuana}\right\\}$ |  |  | 0.447 |  |  |  | 0.337
|  | (0.305) |  |  |  | (0.283)
Constant | -0.540 | -1.693 | 6.326 |  | -0.527 | -1.675 | 5.043
| (0.057) | (0.166) | (0.888) |  | (0.061) | (0.159) | (1.352)
Regional-fixed effects | $\checkmark$ | $\checkmark$ | $\checkmark$ |  | $\checkmark$ | $\checkmark$ | $\checkmark$
Sample size | 49,414 | 28,505 | 1,159 |  | 49,414 | 49,414 | 49,414

  * Notes: Bold font indicates statistically significant variables. Columns (1), (2) and (3) show posterior results of univariate models, and columns (4), (5) and (6) show posterior results of multivariate model. Columns (1) and (4) show results of the access equation, columns (2) and (5) show results of the extensive margin equations, and columns (3) and (6) show results of the intensive margin equation. There are 38 region fixed effects in each equation. There are meaningful differences in the intensive margin equation due to a statistically significant unobserved co-variation between the access and intensive margin equations (see Table 3).
Source: Authors’ construction.

Table 2 reports the posterior estimates. Columns labeled univariate show the
posterior results of estimating univariate models, that is, probit models for
the access and extensive margin equations, and a linear model for the
logarithm of marijuana consumption (intensive margin). Columns labeled
multivariate show the posterior results of modeling simultaneously equations
1, 2 and 3 taking into account truncation.

We observe that univariate and multivariate models give similar results
regarding the access and extensive margin equations. This is due to Table 3
suggesting that these two equations are exogenous. However, we observe in
Table 2 that posterior estimates of the univariate and multivariate models are
different regarding the intensive margin (see columns (3) and (6)). This is
explained by the fact that there is endogeneity between access and intensive
margin. In particular, Table 3 indicates that the unobserved co-variation
between these equations is statistically significant. This suggests that
conditional on the extensive margin, more frequent marijuana consumers make
higher effort to have access to this drug.

Columns (1) and (4) in Table 2 show that having a drug dealer in the
neighborhood increases the probability of having access to marijuana. In
particular, the probability of having access to marijuana for the
representative individual based on the descriptive statistics (see the last
paragraph of the previous subsection) increases by 16.9 percentage points due
to presence of drug dealer in the neighborhood, that is, from 36.5% to 53.4%,
given the results in column (4). In addition, the probability of having access
is lower for women, individuals who declare to have good mental health status,
and are older. On the other hand, individuals who live in a low socioeconomic
stratum, have a high or medium risk perception about marijuana use, who have
more years of education, friends or family members that consume marijuana, and
work, have a higher probability of having access to this drug.

We observe in columns (2) and (5) in Table 2 that previous consumption of
alcohol and cigarette increases the probability of consumption. This is
evidence for the gateway drug hypothesis. In addition, the probability of use
increases with socioeconomic strata and having friends or family members who
also consume. On the other hand, the probability of use decreases with age,
risk perception, being female, having good mental health and being a worker.
The results in Table 2 allows to predict the potential significant effects of
a public policy that increases the risk perception about using marijuana. For
instance, the posterior estimates in column (5) indicates that the probability
of using marijuana is equal to 40.5% for a man in his 20s that works, who has
12 years of education, consumes alcohol and cigarettes, lives in a low
socioeconomic stratum, whose mental and physical health status is good, has
friends who consume marijuana and has a low risk perception about using
marijuana. On the other hand, this probability is equal to 15.2% for the same
individual, except that his risk perception about marijuana use is high, that
is, there is a decrease of 25.3 percentage points due to changing the risk
perception.

Columns (3) and (6) in Table 2 show that the marijuana is an inelastic good.
Particularly, column (6) shows that this elasticity is on average equal to
-0.45, and is statistically significant; this agrees with previous literature
(Gallet, , 2014; Davis et al., , 2016; Sukharomana and Chang, , 2017; Riley et
al., , 2020). We also observe in this column that there is not statistically
significant heterogeneity regarding price sensitivity between age groups. This
is relevant from a public policy perspective as a valid concern regarding
marijuana legalization is the implications of price variations on young
individuals. The multivariate setting also suggests that the effect of risk
perception is through the extensive margin, rather than directly on the
intensive margin, as is suggested by the univariate modeling framework. This
is, conditional on the extensive margin, the risk perception does not have any
effect on the intensive margin. Finally, we see that one additional year of
education decreases marijuana consumption by 6.9%, and women consume 45.5%
($\exp(-0.607)-1$) less than men, whereas having a network where some
individuals consume marijuana increase marijuana consumption by 85.7%
($\exp(0.619)-1$). All these variables are statistically significant.

Table 3: Posterior results of scale parameters: Marijuana demand in Colombia.

Posterior measures | Univariate | Multivariate
---|---|---
| $\sigma_{y}^{2}$ | $\sigma_{ca}$ | $\sigma_{ya}$ | $\sigma_{yc}$ | $\sigma_{y}^{2}$
Mean | 2.103 | 8.21.E-04 | 7.182 | -0.103 | 2.845
Standart deviation | (0.092) | (0.010) | (0.407) | (0.129) | (0.131)

  * Notes: Bold font indicates statistically significant variables. Columns labeled multivariate show posterior estimates of the identified covariance matrix. Column labeled univariate shows the posterior results of the variance of the intensive margin equation. The sample size in the univariate model is 1,159, and the sample size in the multivariate model is 49,414.
The unobserved co-variation between the access and the intensive margin is
statistically significant. The univariate model of the intensive margin shows
a lower variance than the multivariate model.
Source: Authors’ construction.

##  4 Statistical checks

###  4.1 Exclusionary restrictions

It is well known that we can achieve identification of causal effects in
nonlinear models without exclusion restrictions (McManus, , 1992). However,
exclusion restrictions improve inference due to reducing variability of
estimates because of data variability (Munkin and Trivedi, , 2003). We
exclusively use presence of drug dealers in the neighborhood in the access
equation as this variable affects drug supply which helps to identify demand
parameters. Particularly, we would expect that presence of drug dealers would
positively affect the probability of an individual having access to marijuana.
We argue that the effect of this variable on the extensive and intensive
margins should be just through the access.

However, if this supply side variable affects directly the net utility of
using marijuana, then our exclusion restriction would not be valid. We try to
test this restriction using the subset of individuals who were offered
marijuana, as a consequence, they do not have to search for this drug, which
means that they are relatively free of the selection issue. We check the
statistical significance of this supply side variable running a probit model
on the access equation in this subsample. Column 1 of Table 4 shows the
posterior results using a non-informative normal prior distribution, the
number of iterations of the Gibbs sampler is 6,000, a burn-in equal to 1,000,
and a thin parameter equal to 5. We observe that the presence of drug dealer
is not statistically significant in the extensive margin equation. Hence, this
result suggests that “drug dealer in neighborhood” has validity as an
exclusionary restriction.

We do not use presence of drug dealer in the neighborhood neither if an
individual has consumed alcohol and cigarettes any time in her/his life in the
intensive margin equation. The gateway drug hypothesis would support the
latter variable as pattern of legal substance use would precede the use of
illicit substances (Kandel, , 1975; Kandel et al., , 1992). This means that
individuals who have consumed substances like alcohol and nicotine would have
a higher probability of accessing and using marijuana. However, past or
present consumption of these substances should not directly affect the
intensive margin. Column 2 of Table 4 shows the posterior results of
estimating the intensive margin of marijuana, that is, the logarithm of
quantity, as function of these two variables as well as all regressors in
equations 3 using the subset of individuals that were offered marijuana, that
is, the set of individuals that presumably is exogenous to the access
equation. Given that we still have endogeneity due to the extensive margin in
this set of individual, this is not a formal test of exclusionary
restrictions. However, the fact that presence of drug dealer in the
neighborhood, and consumption of alcohol and cigarettes are not statistical
significant to explain the intensive margin, suggests that these variables
have some validity as exclusionary restrictions.

Finally, we have included marijuana price in the intensive margin equation,
but not in the access and extensive margin equations. Although this is not
necessary for identification, we consider that excluding price from these
equations make sense. First, the argument to exclude marijuana price from the
access equation is that individuals who do not have access are unlikely to
know the price of the marijuana they would obtain. Second, exclusion of
marijuana price from the extensive margin equation is due to marijuana being
very cheap in Colombia (US¢83), consequently, price is not a barrier to define
the extensive margin.

Table 4: Exclusionary restrictions validation: Marijuana demand in Colombia.

Variable | Margin
---|---
Extensive | Intensive
(1) | (2)
Exclusionary restrictions |  |
Drug dealer in neighborhood | 0.065 | 0.180
(0.041) | (0.098)
Alcohol and cigarette use | 0.837 | 0.089
(0.045) | (0.133)
Age |  |
30s | -0.339 | 0.020
(0.050) | (1.473)
40s | -0.498 | 0.334
(0.063) | (2.122)
50s and older | -0.973 | -4.280
(0.074) | (2.400)
Strata |  |
Medium | 0.098 | -0.023
(0.044) | (0.105)
High | 0.066 | -0.047
(0.060) | (0.153)
Risk perception marijuana use |  |
Medium | -0.169 | -0.665
(0.088) | (0.176)
High | -0.817 | -0.707
(0.080) | (0.165)
Years of education | -0.014 | -0.085
(0.006) | (0.015)
Female | -0.422 | -0.520
(0.043) | (0.108)
Good mental health | -0.190 | 0.203
(0.046) | (0.109)
Good physical health | -0.031 | -0.270
(0.049) | (0.122)
Marijuana users in network | 0.944 | 0.230
(0.062) | (0.208)
Worker | -0.103 | 0.180
(0.041) | (0.097)
Price |  |
$\log\left\\{\text{price of marijuana}\right\\}$ |  | -0.526
| (0.100)
Age 30s $\times\log\left\\{\text{price of marijuana}\right\\}$ |  | 0.028
| (0.191)
Age 40s $\times\log\left\\{\text{price of marijuana}\right\\}$ |  | -0.012
| (0.275)
Age 50s and older $\times\log\left\\{\text{price of marijuana}\right\\}$ |  | 0.526
| (0.303)
Constant | -1.551 | 6.637
| (0.195) | (0.931)
Regional-fixed effects | $\checkmark$ | $\checkmark$
Sample size | 12,994 | 1,043

  * Notes: Bold font indicates statistically significant variables. This table shows the posterior means and standard deviation (in parenthesis) of parameters of the extensive margin (column 1) and intensive margin (column 2) using the subset of individuals that were offered marijuana. These results suggest plausibility of the exclusionary restrictions.
Source: Authors’ construction.

###  4.2 Robustness checks

We perform several exercises to check the robustness of the posterior
estimates of our baseline specification. First, as the definition of access is
very relevant in our analysis, we check our results with another definition of
access. In particular, our baseline definition of access is equal to one if an
individual responds that is easy to get marijuana, and zero in case that
responds that it is difficult, impossible or does not know how to get it. This
implies that 58% of our sample has access to marijuana. In this exercise, we
relax the access definition by including within the individuals who have
access those who report that it would be easy or difficult to get marijuana.
This new definition of access implies that 67% of individuals in the sample
have access (see Table 11 in the Appendix). We reran our baseline model using
this new definition of the accessibility, the results can be seen in the
second column of tables 5, 6 and 7, where we show the posteriors estimates for
the access, extensive and intensive equations, respectively. In general, we
obtain qualitatively similar results in the alternative scenario relaxing the
access definition (column 2) compared to the baseline estimates (column 1) in
the three stages. In most of the posterior estimates there are not
statistically significant differences in both exercises, except that
individuals in high strata do not have statistically significant differences
compared with individuals in low strata regarding access in this new set of
estimates (see Table 5).

We also estimate our baseline model using the subset of uni-personal
households. This is because lying is a valid concern when modeling demand for
illicit drugs due to, for instance, social stigma (Lloyd, , 2013). We guess
that individuals who live alone have less incentives to lie regarding
marijuana use. Column (3) in tables 5, 6 and 7 show the results using this
sub-sample. We observe again that there are not statistically significant
differences compared to the baseline exercise in the access and extensive
margin, except that now there are not statistically significant differences
regarding strata or being a worker. In addition, years of education is not
statistically relevant in the extensive margin. However, we observe some
intriguing results in the intensive margin estimates. In particular, three
very robust regressors in all the estimations are not statistically
significant in this subset: female, marijuana users in the network and price.
Although their coefficients have the expected sign. We suspect that can be a
power issue due to being just 233 marijuana consumers in this sub-sample.

We also include interaction effects between age splines and risk perception
about marijuana consumption in our main specification. We perform this to
identify potential heterogeneous effects in this variable among age groups,
thus thinking about marketing campaigns targeting young adults to curve
marijuana consumption through risk perception. We observe in column (4) of
Table 7 that these variables are not statistically significant, and in
general, we get very similar results in this alternative specification
compared to the baseline exercise.

We get our price measure in the baseline estimation calculating expenditure in
marijuana over quantity. The advantage of this measure is that takes
implicitly quality into account when an individual buys different types of
marijuana. However, the survey asks directly individuals about price, there is
the question “Do you know how much a marijuana cigarette or joint costs? We
estimate our model using this alternative measure of price, which implies
calculating again the quantity weighted by THC. Column (5) in tables 5, 6 and
7 show the results. It seems that our results are robust to the price measure,
the price elasticity is numerically lower under the alternative price compared
to the baseline exercise, but there is not statistically significant
differences.

A potential endogeneity issue that we did not take into account in our main
specification is self-perception about health status. This variable may be
considered endogenous (Jacobi and Sovinsky, , 2016), thus we estimate the
baseline specification without the mental and physical self-perception of
health status. The results can be seen in column (6) of tables 5, 6 and 7. The
results are very similar comparing this exercise with the baseline
specification, except that medium strata do not have statistically significant
differences with the low strata in the extensive margin in this new setting.

Table 5: Robustness checks: Posterior results of location parameters in access
of marijuana demand in Colombia.

Variable | Baseline | Access | Uni-personal | Interaction | Price | Perception
---|---|---|---|---|---|---
Specification | definition | household | Age and risk | definition | health status
(1) | (2) | (3) | (4) | (5) | (6)
Exclusionary restrictions |  |  |  |  |  |
Drug dealer in neighborhood | 0.433 | 0.366 | 0.441 | 0.433 | 0.432 | 0.433
(0.014) | (0.014) | (0.043) | (0.014) | (0.014) | (0.013)
Alcohol and cigarette use | 0.369 | 0.330 | 0.316 | 0.369 | 0.365 | 0.374
(0.014) | (0.015) | (0.042) | (0.015) | (0.015) | (0.014)
Age |  |  |  |  |  |
30s | -0.094 | -0.093 | -0.214 | -0.093 | -0.092 | -0.086
(0.018) | (0.019) | (0.065) | (0.018) | (0.018) | (0.018)
40s | -0.216 | -0.195 | -0.386 | -0.216 | -0.216 | -0.205
(0.019) | (0.019) | (0.066) | (0.019) | (0.020) | (0.019)
50s and older | -0.403 | -0.375 | -0.542 | -0.403 | -0.403 | -0.392
(0.017) | (0.017) | (0.054) | (0.017) | (0.016) | (0.016)
Strata |  |  |  |  |  |
Medium | -0.049 | -0.034 | -0.031 | -0.050 | -0.050 | -0.051
(0.014) | (0.015) | (0.045) | (0.014) | (0.015) | (0.014)
High | -0.051 | -0.032 | -0.012 | -0.049 | -0.050 | -0.053
(0.023) | (0.024) | (0.063) | (0.025) | (0.023) | (0.024)
Risk perception marijuana use |  |  |  |  |  |
Medium | 0.520 | 0.559 | 0.553 | 0.520 | 0.515 | 0.518
(0.044) | (0.045) | (0.127) | (0.045) | (0.045) | (0.045)
High | 0.279 | 0.365 | 0.214 | 0.275 | 0.271 | 0.270
(0.034) | (0.033) | (0.095) | (0.033) | (0.033) | (0.032)
Years of education | 0.017 | 0.013 | 0.014 | 0.017 | 0.017 | 0.016
(0.002) | (0.002) | (0.004) | (0.002) | (0.002) | (0.002)
Female | -0.310 | -0.281 | -0.471 | -0.310 | -0.312 | -0.300
(0.013) | (0.014) | (0.040) | (0.014) | (0.013) | (0.013)
Good mental health | -0.129 | -0.113 | -0.113 | -0.128 | -0.129 |
(0.016) | (0.017) | (0.050) | (0.016) | (0.016) |
Good physical health | 0.009 | 0.015 | -0.017 | 0.010 | 0.009 |
(0.016) | (0.017) | (0.049) | (0.016) | (0.016) |
Marijuana users in network | 0.695 | 0.617 | 0.785 | 0.696 | 0.696 | 0.705
(0.013) | (0.015) | (0.041) | (0.013) | (0.013) | (0.013)
Worker | 0.127 | 0.123 | 0.081 | 0.127 | 0.127 | 0.121
(0.014) | (0.013) | (0.047) | (0.014) | (0.014) | (0.014)
Constant | -0.527 | -0.164 | -0.225 | -0.526 | -0.518 | -0.599
| (0.061) | (0.057) | (0.173) | (0.058) | (0.058) | (0.054)
Regional-fixed effects | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$
Sample size | 49,414 | 49,414 | 5,574 | 49,414 | 49,414 | 49,414

  * Notes: Bold font indicates statistically significant variables. Robustness checks under different specifications and measures of relevant variables. Column (1) shows the baseline setting to facilitate comparisons.
Source: Authors’ construction.

Table 6: Robustness checks: Posterior results of location parameters in the
extensive margin of marijuana demand in Colombia.

Variable | Baseline | Access | Uni-personal | Interaction | Price | Perception
---|---|---|---|---|---|---
Specification | definition | household | Age and risk | definition | health status
(1) | (2) | (3) | (4) | (5) | (6)
Exclusionary restrictions |  |  |  |  |  |
Alcohol and cigarette use | 0.931 | 0.953 | 0.711 | 0.942 | 0.942 | 0.956
(0.045) | (0.042) | (0.098) | (0.039) | ’(0.042) | (0.042)
Age |  |  |  |  |  |
30s | -0.378 | -0.385 | -0.385 | -0.385 | -0.384 | -0.379
(0.044) | (0.047) | (0.117) | (0.044) | (0.043) | (0.044)
40s | -0.617 | -0.613 | -0.548 | -0.603 | -0.608 | -0.591
(0.054) | (0.061) | (0.134) | (0.055) | (0.060) | (0.058)
50s and older | -1.019 | -1.041 | -1.025 | -1.026 | -1.028 | -1.015
(0.066) | (0.063) | (0.131) | (0.064) | (0.063) | (0.063)
Strata |  |  |  |  |  |
Medium | 0.090 | 0.081 | 0.082 | 0.084 | 0.084 | 0.077
(0.039) | (0.037) | (0.097) | (0.039) | (0.039) | (0.041)
High | 0.130 | 0.131 | -0.006 | 0.133 | 0.132 | 0.132
(0.063) | (0.06) | (0.128) | (0.058) | (0.057) | (0.061)
Risk perception marijuana use |  |  |  |  |  |
Medium | -0.112 | -0.092 | -0.043 | -0.106 | -0.105 | -0.106
(0.081) | (0.073) | (0.176) | (0.076) | (0.080) | (0.077)
High | -0.790 | -0.773 | -0.951 | -0.786 | -0.786 | -0.788
(0.072) | (0.067) | (0.160) | (0.067) | (0.071) | ’(0.071)
Years of education | -0.012 | -0.010 | -0.002 | -0.012 | -0.012 | -0.014
(0.005) | (0.005) | (0.011) | (0.005) | (0.005) | (0.005)
Female | -0.449 | -0.447 | -0.344 | -0.452 | -0.452 | -0.424
(0.037) | (0.039) | (0.099) | (0.039) | (0.041) | (0.038)
Good mental health | -0.192 | -0.192 | -0.212 | -0.189 | -0.192 |
(0.044) | (0.040) | (0.103) | (0.042) | (0.042) |
Good physical health | -0.033 | -0.038 | -0.061 | -0.035 | -0.034 |
(0.051) | (0.047) | (0.109) | (0.043) | (0.046) |
Marijuana users in network | 0.971 | 1.004 | 1.175 | 0.972 | 0.979 | 0.981
(0.052) | (0.048) | (0.143) | (0.053) | (0.052) | (0.055)
Worker | -0.094 | -0.093 | -0.166 | -0.100 | -0.095 | -0.115
(0.037) | (0.037) | (0.101) | (0.039) | (0.039) | (0.037)
Constant | -1.675 | -1.813 | -1.516 | -1.681 | -1.694 | -1.830
| (0.159) | (0.163) | 0.465 | (0.167) | (0.177) | (0.162)
Regional-fixed effects | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$
Sample size | 49,414 | 49,414 | 5,574 | 49,414 | 49,414 | 49,414

  * Notes: Bold font indicates statistically significant variables. Robustness checks under different specifications and measures of relevant variables. Column (1) shows the baseline setting to facilitate comparisons.
Source: Authors’ construction.

Table 7: Robustness checks: Posterior results of location parameters in the
intensive margin of marijuana demand in Colombia.

Variable | Baseline | Access | Uni-personal | Interaction | Price | Perception
---|---|---|---|---|---|---
Specification | definition | household | Age and risk | definition | health status
(1) | (2) | (3) | (4) | (5) | (6)
Age |  |  |  |  |  |
30s | 0.457 | 0.593 | 3.192 | 0.859 | 0.562 | 0.648
(1.302) | (1.278) | (2.951) | (1.391) | (1.962) | (1.313)
40s | 0.534 | 0.518 | 0.521 | -0.041 | -2.732 | 0.633
(1.899) | (1.941) | (3.785) | (1.974) | (3.111) | (2.008)
50s and older | -2.995 | -2.449 | -5.609 | -2.568 | 1.585 | -2.384
(2.197) | (2.090) | (4.361) | (2.360) | (2.675) | (2.128)
Strata |  |  |  |  |  |
Medium | -0.067 | -0.067 | 0.116 | -0.074 | -0.112 | -0.084
(0.099) | (0.091) | (0.224) | (0.091) | (0.093) | (0.091)
High | -0.199 | -0.168 | -0.340 | -0.208 | -0.149 | -0.214
(0.149) | (0.132) | (0.280) | (0.135) | (0.129) | (0.127)
Risk perception marijuana use |  |  |  |  |  |
Medium | -0.204 | -0.040 | -0.701 | -0.144 | -0.064 | -0.036
(0.300) | ’(0.147) | (0.459) | (0.301) | (0.161) | (0.152)
Age 30s $\times$ Medium |  |  |  | -0.328 |  |
|  |  | (0.384) |  |
Age 40s $\times$ Medium |  |  |  | 0.544 |  |
|  |  | (0.628) |  |
Age 50s and older $\times$ Medium |  |  |  | -0.516 |  |
|  |  | (0.674) |  |
High | -0.327 | -0.162 | -0.818 | -0.302 | -0.208 | -0.201
(0.247) | (0.137) | (0.413) | (0.256) | (0.142) | (0.134)
Age 30s $\times$ High |  |  |  | -0.319 |  |
|  |  | (0.355) |  |
Age 40s $\times$ High |  |  |  | 0.558 |  |
|  |  | (0.550) |  |
Age 50s and older $\times$ High |  |  |  | -0.273 |  |
|  |  | (0.551) |  |
Years of education | -0.069 | -0.065 | -0.081 | -0.068 | -0.060 | -0.065
(0.015) | (0.013) | (0.031) | (0.016) | (0.012) | (0.013)
Female | -0.607 | -0.582 | -0.421 | -0.599 | -0.640 | -0.627
(0.146) | (0.097) | (0.336) | (0.115) | (0.100) | (0.094)
Good mental health | 0.052 | 0.050 | 0.201 | 0.057 | -0.005 |
(0.107) | (0.093) | (0.279) | (0.103) | (0.094) |
Good physical health | -0.110 | -0.115 | -0.304 | -0.128 | -0.115 |
(0.105) | (0.110) | (0.277) | (0.112) | (0.108) |
Marijuana users in network | 0.619 | 0.652 | 0.364 | 0.663 | 0.844 | 0.827
(0.306) | (0.144) | (0.606) | (0.271) | (0.151) | (0.157)
Worker | 0.151 | 0.153 | 0.140 | 0.143 | 0.135 | 0.157
(0.096) | (0.092) | (0.233) | (0.088) | (0.085) | (0.086)
Price |  |  |  |  |  |
$\log\left\\{\text{price of marijuana}\right\\}$ | -0.445 | -0.436 | -0.312 | -0.463 | -0.363 | -0.437
(0.101) | (0.097) | (0.251) | (0.096) | (0.128) | (0.092)
Age 30s $\times\log\left\\{\text{price of marijuana}\right\\}$ | -0.033 | -0.058 | -0.325 | -0.050 | -0.049 | -0.063
(0.167) | (0.166) | (0.376) | (0.174) | (0.249) | (0.171)
Age 40s $\times\log\left\\{\text{price of marijuana}\right\\}$ | -0.046 | -0.055 | 0.030 | -0.039 | 0.370 | -0.065
(0.245) | (0.252) | (0.485) | (0.244) | (0.400) | -0.261
Age 50s and older $\times\log\left\\{\text{price of marijuana}\right\\}$ | 0.337 | 0.258 | 0.653 | 0.309 | -0.269 | 0.252
(0.283) | (0.265) | (0.553) | (0.291) | (0.343) | -0.269
Constant | 5.043 | 4.798 | 4.899 | 5.107 | 3.666 | 4.411
| (1.352) | (0.884) | (2.374) | (1.196) | (1.119) | (0.836)
Regional-fixed effects | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$
Sample size | 49,414 |  | 5,574 | 49,414 | 49,414 | 49,414

  * Notes: Bold font indicates statistically significant variables. Robustness checks under different specifications and measures of relevant variables. Column (1) shows the baseline setting to facilitate comparisons.
Source: Authors’ construction.

In general, it seems that the results of the baseline specification are
robust, there are not statistically significant differences in most of the
cases compared to the alternative measures of relevant variables or model
specifications. We observe that the same variables are statistically
significant in all three stages of demand for marijuana, and the numeric
values of the posterior estimates are relatively similar. However, the sub-
sample of uni-personal households present some intriguing results, potentially
due to power issues.

##  5 Policy analysis

###  5.1 Marijuana legalization

We perform some counterfactual experiments to estimate the potential effects
of the legalization of marijuana in Colombia for different representative
individuals. Particularly, we estimate the posterior predictive probability
for individual 0,

$\displaystyle p(A_{0},C_{0},Y_{0}|\bm{T},\bm{X})$
$\displaystyle=\int_{\bm{S}}\int_{\bm{\Theta}}\left\\{\mathds{1}(A_{0}=0)\mathds{1}(U_{0}^{a}\leq
0)+\mathds{1}(A_{0}=1)\mathds{1}(U_{0}^{a}>0)\right.$
$\displaystyle\left.\times\left[\mathds{1}(C_{0}=0)\mathds{1}(U_{0}^{c}\leq
0)+\mathds{1}(C_{0}=1)\mathds{1}(C_{0}^{c}>0)\right]\right\\}$
$\displaystyle\times p(A_{0}|\bm{T},\bm{X})\times
p(C_{0}|A_{0},\bm{T},\bm{X})\times
p(Y_{0}|C_{0},A_{0},\bm{T},\bm{X})\pi(\bm{\theta},\bm{\Sigma}|\bm{T},\bm{X})d\bm{\theta}d\bm{\Sigma},$

where $\bm{S}$ is the support of integration of $\bm{\Sigma}$,
$p(A_{0}=1|\bm{T},\bm{X})=1-\Phi(\bm{x}_{0}^{a\top}\bm{\theta}^{a},1)$,
$p(C_{0}=1|A_{0}=1,\bm{T},\bm{X})=1-\Phi(\bm{x}_{0}^{c\top}\bm{\theta}^{c}+\sigma_{ac}(U_{0}^{a}-\bm{x}_{0}^{a\top}\bm{\theta}^{a}),1-\sigma_{ac}^{2})$
and $p(Y_{0}|C_{0}=1,A_{0}=1,\bm{T},\bm{X})\sim
N(\mu_{y|ac},\sigma^{2}_{y|ac})$, where

$\displaystyle\mu_{y|ac}$
$\displaystyle=\bm{x}_{0}^{y\top}\bm{\theta}^{y}+[\sigma_{ya}\
\sigma_{yc}]\begin{bmatrix}1&\sigma_{ac}\\\
\sigma_{ac}&1\end{bmatrix}^{-1}\begin{bmatrix}U_{0}^{a}-\bm{x}_{0}^{a\top}\bm{\theta}^{a}\\\
U_{0}^{c}-\bm{x}_{0}^{c\top}\bm{\theta}^{c}\end{bmatrix},$

and

$\displaystyle\sigma^{2}_{y|ac}$ $\displaystyle=\sigma^{2}_{y}-[\sigma_{ya}\
\sigma_{yc}]\begin{bmatrix}1&\sigma_{ac}\\\
\sigma_{ac}&1\end{bmatrix}^{-1}\begin{bmatrix}\sigma_{ya}\\\
\sigma_{yc}\end{bmatrix}.$

Observe the relevance of the selection parameters, $\sigma_{ac}$,
$\sigma_{ya}$ and $\sigma_{yc}$ in the previous expressions. These account for
unobserved dependence between the three stages of the demand for marijuana.

The above integral can be estimated in a straight forward way using the draws
from the posterior distribution. Therefore, we use simulation to estimate the
effects of the legalization of marijuana on the probability of use, given that
under legalization the probability of access is equal to 1, that is,
$p(A_{0}=1|\bm{T},\bm{X})=1$, and the amount of consumption, conditional on
use, where we take into account that we model $\log(Y_{it})$, so we get by
simulation $Y_{it}$, that is, the amount of joints per month.

We show in tables 8 and 9 the results of these exercises for the
representative individual who has access to marijuana. In particular, this is
an individual with 12 years of education, working, good self-perception of
health status, family members and friends who do not consume marijuana, but
she/he has consumed alcohol and cigarettes, lives in Medellín in a low
socioeconomic stratum, and there is a drug dealer in the neighborhood. We have
in these tables seven scenarios, rows one to three in each panel show results
under different scenarios about risk perceptions of marijuana use, these
experiments are based on the fact that marketing campaigns warning about bad
consequences of consuming some products may curve demand. For instance,
warning labeling explains why consumers become more health-conscious, and
consequently, more risk-averse (Barahona et al., , 2023; Berg et al., , 2023;
Cannoy et al., , 2023; Kaai et al., , 2023; Nguyen et al., , 2023; Nian et
al., , 2023; Brennan et al., , 2022). Rows four to seven show results under
different price scenarios, the ones that we analyze in the next subsection for
potential tax revenues, taxes contribute to raise revenues for public policy,
and help to curve the demand function (Allcott et al., , 2019).

We can see in Table 8 the results for the representative woman, where each
panel shows results by age spline. For instance, the first row in the first
panel shows the results under a the baseline price for this representative
woman (US¢78.2), who has a high risk perception about marijuana use. We
observe that the predicted probability of having access to marijuana is 74.5%,
and the predicted probabilities of marijuana use are 1.30% and 1.75%, overall
women with these features, and those with access, respectively. This means
that the probability of use overall these representative women increases 0.44
percentage points given a policy of legalization of marijuana. Conditional on
access and use, the predicted consumption for this representative woman is 5.9
joints per month. All these estimates has the standard errors that are
calculated by simulation using repeated sampling.

We observe from Table 8 that under legalization of marijuana, risk perception
has a higher effect among women in decreasing the probability of use than
price. For instance, the second and third rows show that given access, the
probability of use increases 9.37 and 4.58 percentage points for women who
have a medium and low risk perceptions about marijuana use, compared with just
0.44 p.p. for women with high risk perception. However, the effect of risk
perception decreases with age. We observe the same pattern among men (see
Table 9). Overall, taking into account these three risk scenarios, age splines
and gender, and using the expansion factors of the survey, we find that the
probability of use increases from 2.3% pre-legalization to 3.0% under
legalization.

Given the effect of risk perception, and the patterns of this by age splines,
we can deduce that marketing campaigns targeting young individuals will be an
effective way to curve demand for marijuana under a legal setting. First,
young have a lower risk perception about marijuana use than other age splines,
the sample average for a high risk is 88% for the former, whereas this
percentage is equal to 92%, 93% and 94% for 30s, 40s and 50s age splines, this
means that there is a higher gap among young individuals. And second, reducing
more the demand of this group implies that the cumulative effect on
consumption through their life span is higher, with potentially more good
externalities. However, we should take into account that the effect of risk
perception has a limit, this is, on average 91% of the individuals already
have a high risk perception about marijuana use. This fact motivates to
perform experiments with different prices (taxes), which also means different
scenarios regarding government revenues.

Thus, the second set of experiments consider the effect of legalization of
marijuana on price, and as a consequence, on access, extensive and intensive
margins. Particularly, marijuana legalization implies that the inherent extra
cost due to illegality would disappear. However, we should take a potential
tax into account. A first benchmark is the average cost of marijuana in
Colombia, Vélez-Torres et al.,  (2021) found that the average production cost
is US¢1 per gram, and taking into account that the average percentage of
distribution cost in Colombia is 15%, the cost of one joint of marijuana is
approximately US¢1.15. In addition, the average return of capital in Colombia
is around 15.25% according to Corficolombiana, a prestigious financial
institution in this country, fluctuating between 13.9% y 16.6%,333See
Rentabilidad esperada del capital propio. thus a gram of marijuana has a base
price without taxes of approximately US¢1.33. The first counterfactual
exercise uses as reference the tax on cigarettes, which is US¢5.9 in
Colombia,444Real price 2019, see CERTIFICACIÓN 4, 2021 of Dirección General de
Apoyo Fiscal del Ministerio de Hacienda. then the potential price of one gram
of marijuana, tax included, is approximately US¢7.3. This price can be
considered as a potential lower bound due to being less expensive that the
legal price of cigarettes in Colombia, which is on average US¢11.5. Although,
this price is higher than the price of illegal cigarettes, US¢5.3.555Real
price 2019, see Estudio de incidencia del consumo de cigarrillos en colombia
2022. The second exercise assumes that the price of marijuana is equal to the
legal price of cigarettes (US¢11.5). Finally, the potential lower bound and
the actual price of marijuana offer a spectrum of possibilities for tax
scenarios. Thus, we perform two more experiments, 50% decrease and 25%
increase with respect to the actual price of marijuana. The former implies a
tax of US¢37.8 per joint, which is more than 6 times the tax on cigarettes.
The latter is based on Jacobi and Sovinsky,  (2016), who propose a 25% tax on
the actual price of marijuana in Australia. In the Colombian case, this tax
would be more than 16 times the tax of cigarettes. We think about the this
scenario as a potential upper bound price in the case of legalization of
marijuana due to a relatively high price may imply a huge black market of
marijuana in this country. For instance, the size of the black market of
cigarettes in Colombia is 34%.666See Estudio de incidencia del consumo de
cigarrillos en colombia 2022.

The fourth to seventh rows in each panel of tables 8 and 9 show the results.
We observe that there are not remarkable differences regarding the probability
of access and use under different price settings; however, the intensity of
consumption decreases with price, as expected. The shape of the intensity
margin as a function of age splines has an “inverted U-shape”, that is, this
is low for 20s and 50s, increases in 30s, and has a peak in 40s. This latter
group has a very high level of consumption, however, we should take with
caution this result due to the also high volatility level. Observe that price
helps to curve the intensity of consumption under a legalization policy;
however, there are not significant changes in the extensive margin due to
different price regimes.

Other patterns that we observe from tables 8 and 9 are that access is lower
for individuals who have a low risk perception regarding marijuana use;
however, the probability of marijuana use is higher for this group. In
addition, access also decreases with age, and is higher for men, who in turn
have a higher probability of use, and given use, have a substantially higher
level of consumption, approximately two times the level of women.

We perform ceteris paribus exercises in order to isolate the effects of
different control variables, and get a better understanding of the situation.
However, Manthey et al.,  (2023) demonstrated that warning information about
product use that potentially curve consumption must be complemented with taxes
(pricing). Both approaches should be integrated as part of a comprehensive
strategy aimed at mitigating the burdens on public health, social well-being,
and economy, of marijuana use.

Table 8: Counterfactual experiments: Effects of legalization of marijuana in
Colombia for a representative woman.

|  |  |  | Pred. Prob. access |  | Pred. prob use |  | Pred. consumption
---|---|---|---|---|---|---|---|---
Scenario |  | All |  | All |  | Access |  | Change |  | Consumption
Price scenario | Price | Risk perception |  | Mean | Std. Error |  | Mean | Std. Error |  | Mean | Std. Error |  | Mean |  | Mean | Std. Error
Woman in 20s or younger |
Baseline | US¢78.2 | High |  | 74.50% | 0.0137 |  | 1.30% | 0.0036 |  | 1.74% | 0.0048 |  | 0.44 p.p. |  | 5.90 | 0.30
Baseline | US¢78.2 | Medium |  | 77.70% | 0.0131 |  | 8.30% | 0.0087 |  | 10.68% | 0.0110 |  | 9.37 p.p. |  | 6.58 | 0.33
Baseline | US¢78.2 | Low |  | 66.50% | 0.0149 |  | 9.10% | 0.0091 |  | 13.68% | 0.0130 |  | 4.58 p.p. |  | 9.06 | 0.50
Lower bound | US¢7.3 | High |  | 70.80% | 0.0143 |  | 2.30% | 0.0047 |  | 3.24% | 0.0067 |  | 0.94 p.p. |  | 17.44 | 0.87
Cigarette | US¢11.5 | High |  | 72.70% | 0.0142 |  | 1.50% | 0.0038 |  | 2.08% | 0.0053 |  | 0.58 p.p. |  | 13.74 | 0.64
50% decrease | US¢39.1 | High |  | 72.60% | 0.0141 |  | 1.60% | 0.0040 |  | 2.20% | 0.0054 |  | 0.60 p.p. |  | 8.22 | 0.39
25% increase | US¢97.8 | High |  | 72.00% | 0.0141 |  | 2.60% | 0.0050 |  | 3.58% | 0.0069 |  | 0.98 p.p. |  | 5.18 | 0.31
Woman in 30s or younger |
Baseline | US¢78.2 | High |  | 69.40% | 0.0146 |  | 0.50% | 0.0022 |  | 0.72% | 0.0032 |  | 0.22 p.p. |  | 22.20 | 2.64
Baseline | US¢78.2 | Medium |  | 76.70% | 0.0133 |  | 2.50% | 0.0049 |  | 3.25% | 0.0064 |  | 0.75 p.p. |  | 20.80 | 1.65
Baseline | US¢78.2 | Low |  | 62.80% | 0.0152 |  | 3.80% | 0.0061 |  | 6.05% | 0.0095 |  | 2.25 p.p. |  | 30.12 | 2.80
Lower bound | US¢7.3 | High |  | 70.50% | 0.0144 |  | 0.01% | 0.0030 |  | 1.28% | 0.1123 |  | 1.27 p.p. |  | 48.21 | 3.87
Cigarette | US¢11.5 | High |  | 67.50% | 0.0148 |  | 0.05% | 0.0022 |  | 0.74% | 0.0033 |  | 0.69 p.p. |  | 47.54 | 7.40
50% decrease | US¢39.1 | High |  | 71.80% | 0.0142 |  | 0.40% | 0.0020 |  | 0.55% | 0.0028 |  | 0.15 p.p. |  | 23.29 | 1.87
25% increase | US¢97.8 | High |  | 65.50% | 0.0150 |  | 0.40% | 0.0019 |  | 0.61% | 0.0030 |  | 0.21 p.p. |  | 17.72 | 1.62
Woman in 40s or younger |
Baseline | US¢78.2 | High |  | 67.10% | 0.0150 |  | 0.30% | 0.0013 |  | 0.44% | 0.0025 |  | 0.14 p.p. |  | 57.68 | 10.86
Baseline | US¢78.2 | Medium |  | 73.00% | 0.0140 |  | 2.00% | 0.0044 |  | 2.74% | 0.0060 |  | 0.74 p.p. |  | 69.14 | 15.44
Baseline | US¢78.2 | Low |  | 60.80% | 0.0154 |  | 1.90% | 0.0043 |  | 3.12% | 0.0070 |  | 1.24 p.p. |  | 133.25 | 34.22
Lower bound | US¢7.3 | High |  | 66.30% | 0.0150 |  | 0.30% | 0.0017 |  | 0.45% | 0.0026 |  | 0.15 p.p. |  | 313.13 | 110.55
Cigarette | US¢11.5 | High |  | 62.20% | 0.0153 |  | 0.60% | 0.0024 |  | 0.96% | 0.0039 |  | 0.36 p.p. |  | 172.62 | 40.93
50% decrease | US¢39.1 | High |  | 64.30% | 0.0151 |  | 0.20% | 0.0014 |  | 0.31% | 0.0022 |  | 0.08 p.p. |  | 76.03 | 16.98
25% increase | US¢97.8 | High |  | 68.90% | 0.0146 |  | 0.20% | 0.0014 |  | 0.29% | 0.0021 |  | 0.09 p.p. |  | 63.72 | 11.05
Woman in 50s or younger |
Baseline | US¢78.2 | High |  | 64.00% | 0.0152 |  | 0.10% | 0.0010 |  | 0.16% | 0.0015 |  | 0.06 p.p. |  | 1.52 | 0.17
Baseline | US¢78.2 | Medium |  | 68.50% | 0.0147 |  | 0.90% | 0.0020 |  | 1.31% | 0.0043 |  | 0.41 p.p. |  | 1.72 | 0.22
Baseline | US¢78.2 | Low |  | 56.70% | 0.0157 |  | 0.40% | 0.0020 |  | 0.70% | 0.0035 |  | 0.30 p.p. |  | 2.47 | 0.32
Lower bound | US¢7.3 | High |  | 59.90% | 0.0155 |  | 0.00% | 0.0000 |  | 0.00% | 0.0000 |  | 0.00 p.p. |  |  5.40∗ | 1.27
Cigarette | US¢11.5 | High |  | 60.30% | 0.0155 |  | 0.30% | 0.0017 |  | 0.50% | 0.0029 |  | 0.20 p.p. |  | 6.18 | 1.64
50% decrease | US¢39.1 | High |  | 61.90% | 0.0154 |  | 0.00% | 0.0000 |  | 0.00% | 0.0000 |  | 0.00 p.p. |  |  2.51∗ | 0.50
25% increase | US¢97.8 | High |  | 60.70% | 0.0155 |  | 0.20% | 0.0014 |  | 0.33% | 0.0023 |  | 0.13 p.p. |  | 1.63 | 0.29

  * Notes: ∗ Up to four digits the probability of marijuana use is zero for this individual, but in the highly unlikely case of consumption, this is the expected amount. Predicted measures for the representative woman with access to marijuana. Her level of education is high school (12 years of education), works, has a good self-perception of mental and physical health, no marijuana users in her network, consumes alcohol and cigarettes, lives in Medellín in a low socioeconomic stratum, and there is a drug dealer in her neighborhood. Pred. stands for prediction, and prob. is probability.
Source: Authors’ construction.

Table 9: Counterfactual experiments: Effects of legalization of marijuana in
Colombia for a representative man.

|  |  |  | Pred. Prob. access |  | Pred. prob use |  | Pred. consumption
---|---|---|---|---|---|---|---|---
Scenario |  | All |  | All |  | Access |  | Change |  | Consumption
Price scenario | Price | Risk perception |  | Mean | Std. Error |  | Mean | Std. Error |  | Mean | Std. Error |  | Mean |  | Mean | Std. Error
Man in 20s or younger |
Baseline | US¢78.2 | High |  | 79.30% | 0.0128 |  | 4.90% | 0.0068 |  | 6.10% | 0.0085 |  | 1.30 p.p. |  | 10.45 | 0.55
Baseline | US¢78.2 | Medium |  | 83.70% | 0.0117 |  | 17.80% | 0.0121 |  | 21.26% | 0.0142 |  | 3.46 p.p. |  | 10.44 | 0.55
Baseline | US¢78.2 | Low |  | 71.40% | 0.0143 |  | 15.30% | 0.0114 |  | 21.43% | 0.0154 |  | 6.13 p.p. |  | 15.17 | 0.76
Lower bound | US¢7.3 | High |  | 80.40% | 0.0126 |  | 5.60% | 0.0073 |  | 6.96% | 0.0089 |  | 1.36 p.p. |  | 29.38 | 1.47
Cigarette | US¢11.5 | High |  | 77.50% | 0.0132 |  | 5.20% | 0.007 |  | 6.70% | 0.0090 |  | 1.50 p.p. |  | 25.29 | 1.41
50% decrease | US¢39.1 | High |  | 72.20% | 0.0132 |  | 4.80% | 0.0067 |  | 6.21% | 0.0087 |  | 0.60 p.p. |  | 13.88 | 0.62
25% increase | US¢97.8 | High |  | 79.60% | 0.0127 |  | 4.40% | 0.0065 |  | 5.53% | 0.0081 |  | 1.13 p.p. |  | 9.98 | 0.46
Man in 30s or younger |
Baseline | US¢78.2 | High |  | 76.70% | 0.0133 |  | 2.70% | 0.0051 |  | 3.52% | 0.0066 |  | 0.82 p.p. |  | 34.46 | 3.09
Baseline | US¢78.2 | Medium |  | 82.30% | 0.0120 |  | 9.50% | 0.0092 |  | 11.54% | 0.0111 |  | 2.04 p.p. |  | 40.53 | 4.50
Baseline | US¢78.2 | Low |  | 71.70% | 0.0142 |  | 6.80% | 0.0079 |  | 9.48% | 0.0109 |  | 2.68 p.p. |  | 48.52 | 4.64
Lower bound | US¢7.3 | High |  | 76.40% | 0.0134 |  | 2.20% | 0.0046 |  | 2.88% | 0.0061 |  | 0.68 p.p. |  | 94.66 | 8.77
Cigarette | US¢11.5 | High |  | 77.90% | 0.0131 |  | 2.30% | 0.0047 |  | 2.95% | 0.0061 |  | 0.65 p.p. |  | 74.21 | 6.24
50% decrease | US¢39.1 | High |  | 77.90% | 0.0131 |  | 1.90% | 0.0043 |  | 2.44% | 0.0055 |  | 0.54 p.p. |  | 44.01 | 3.63
25% increase | US¢97.8 | High |  | 74.70% | 0.0137 |  | 1.00% | 0.0031 |  | 1.34% | 0.0042 |  | 0.34 p.p. |  | 33.54 | 4.44
Man in 40s or younger |
Baseline | US¢78.2 | High |  | 75.20% | 0.0137 |  | 0.70% | 0.0026 |  | 0.93% | 0.0035 |  | 0.23 p.p. |  | 123.18 | 43.60
Baseline | US¢78.2 | Medium |  | 80.20% | 0.0126 |  | 4.20% | 0.0063 |  | 5.23% | 0.0078 |  | 1.03 p.p. |  | 106.25 | 26.12
Baseline | US¢78.2 | Low |  | 67.00% | 0.0148 |  | 6.20% | 0.0076 |  | 9.25% | 0.0112 |  | 3.05 p.p. |  | 197.54 | 65.80
Lower bound | US¢7.3 | High |  | 74.80% | 0.0137 |  | 1.20% | 0.0034 |  | 1.60% | 0.0045 |  | 0.40 p.p. |  | 405.79 | 128.76
Cigarette | US¢11.5 | High |  | 75.10% | 0.0136 |  | 0.90% | 0.0030 |  | 1.99% | 0.0039 |  | 1.09 p.p. |  | 266.60 | 67.18
50% decrease | US¢39.1 | High |  | 73.60% | 0.0139 |  | 1.10% | 0.0033 |  | 1.49% | 0.0044 |  | 0.39 p.p. |  | 272.48 | 135.76
25% increase | US¢97.8 | High |  | 72.20% | 0.0141 |  | 1.70% | 0.0040 |  | 2.35% | 0.0056 |  | 0.65 p.p. |  | 109.08 | 19.89
Man in 50s or younger |
Baseline | US¢78.2 | High |  | 70.30% | 0.0144 |  | 0.40% | 0.0019 |  | 0.57% | 0.0028 |  | 0.17 p.p. |  | 3.13 | 0.43
Baseline | US¢78.2 | Medium |  | 73.80% | 0.0139 |  | 1.60% | 0.0039 |  | 2.17% | 0.0053 |  | 0.57 p.p. |  | 4.21 | 0.73
Baseline | US¢78.2 | Low |  | 61.00% | 0.0154 |  | 1.70% | 0.0040 |  | 2.79% | 0.0067 |  | 1.09 p.p. |  | 4.65 | 0.77
Lower bound | US¢7.3 | High |  | 70.80% | 0.0143 |  | 0.50% | 0.0023 |  | 0.70% | 0.0031 |  | 0.20 p.p. |  | 10.51 | 2.22
Cigarette | US¢11.5 | High |  | 70.10% | 0.0145 |  | 0.60% | 0.0024 |  | 0.85% | 0.0034 |  | 0.15 p.p. |  | 6.90 | 0.89
50% decrease | US¢39.1 | High |  | 70.00% | 0.0150 |  | 0.40% | 0.0020 |  | 0.57% | 0.0029 |  | 0.17 p.p. |  | 3.96 | 0.57
25% increase | US¢97.8 | High |  | 68.40% | 0.0147 |  | 0.30% | 0.0017 |  | 0.44% | 0.0025 |  | 0.14 p.p. |  | 2.87 | 0.43

  * Notes: Predicted measures for the representative man with access to marijuana. His level of education is high school (12 years of education), works, has a good self-perception of mental and physical health, no marijuana users in his network, consumes alcohol and cigarettes, lives in Medellín in a low socioeconomic stratum, and there is a drug dealer in her neighborhood. Pred. stands for prediction, and prob. is probability.
Source: Authors’ construction.

###  5.2 Tax revenues

We perform some simulation exercises regarding the potential tax revenue that
the government could collect from a tax on marijuana consumption under
legalization. In particular, we use the posterior draws to simulate the model
for all the individuals in the survey using the predictive framework of the
previous subsection, assuming that the probability of access is equal to one
for every one under a legal framework. We estimate the probability of
consumption given access for each individual, and use it to sample from a
Bernoulli distribution, if the realization of this experiment is 0, then the
associated consumption is 0, if the realization is equal to 1, then, we
predict their consumption conditional on access and use. Then, we use the
expansion factors of the survey to estimate the potential yearly tax revenues.
The survey represents 23.6 million individuals between 12 and 65 years-old in
2019, this is approximately 75% of the total Colombian population in this age
range. Thus, we should consider these predictions as underestimating the
potential revenue. Although, the missing 25% of the population is located in
relatively isolated rural areas where potentially would have not legal
marijuana suppliers. We also take into account that approximately 34% of the
demand of marijuana under a legal framework would be in the black market. This
figure is based on the situation in the cigarettes market.777See Estudio de
incidencia del consumo de cigarrillos en colombia 2022.

We set fourth potential scenarios where all of them assume that the difference
between the average cost per gram (US¢1.33) and the price is equal to the tax.
The first is a lower bound where the price is equal to the cost (US¢1.33) plus
a tax that is equal to the cigarette tax (US¢5.9). The second scenario assumes
that the price of marijuana is equal to the average price of a legal
cigarette, the third assumes a price that is the 50% of approximately the
actual average price that a representative individual with access pays for a
joint, and the fourth scenario uses a price that is 25% more expensive than
the latter.

We can see in Table 10 the results. We observe that the annually average tax
revenue under legalization of marijuana in Colombia would be between USD 11.0
million and USD 54.2 million, depending on which taxation scheme is used. This
is between 2.9% to 14.4% of the tax revenues of cigarettes, according to the
Ministry of Finance and Public Credit, that reports tax revenues of cigarettes
are equal to USD 374 million in 2019.888See Ministerio de hacienda y crédito
público We should take into account that in Colombia, the amount of cigarettes
per month of a smoker is 208, whereas the predicted average consumption of
marijuana under the counterfactual of cigarette price is 57 per month, almost
four times less. In addition, the predicted probability of use of marijuana is
2.5% in this price setting under legalization, whereas the probability of
cigarette is 9.5% (reporting smoking in the last month), according to the
ENCSPA survey, that is, approximately 4 times higher.

Table 10: Tax income experiments: Legalization of marijuana in Colombia.

Price scenario | Price | Tax revenue | Tax income
---|---|---|---
per gram | per gram | USD million
Lower bound | US¢7.3 | US¢6.0 | USD 11.0
Cigarette | US¢11.5 | US¢10.2 | USD 14.9
50% decrease | US¢39.1 | US¢37.8 | USD 32.0
25% increase | US¢97.8 | US¢96.5 | USD 54.2

  * Notes: Potential scenarios of tax income under different prices and taxes. This is yearly tax income in million USD using the average exchange rate in 2019 (COP/USD 3.274), taking into account that 34% of marijuana demand would be in the black market, and using the expansion factors that account for 75% of the Colombian population in 2019 between 12 and 65 years-old. The reference cost is US¢1.33.
Source: Authors’ construction.

##  6 Concluding remarks

We present an endogenous three-part model to estimate the demand of marijuana
in Colombia that allows to infer the potential effects of its legalization
finding heterogeneous effects among age groups and gender. Thus, we extend
Jacobi and Sovinsky,  (2016)’s proposal modeling simultaneously the three
stages of marijuana demand (access, extensive and intensive margins), taking
truncation into account.

The main estimation findings indicate that women have a lower probability of
access, use, and quantity of consumption than men. Individuals over 30 also
have a lower probability of access and use than younger individuals (20s and
below), and individuals with good mental health also have a lower probability
of access and use. Overall, we find that the demand for marijuana exhibits
inelasticity (-0.45); moreover, there is no statistically significant
difference in this elasticity in prices across age groups. These results are
robust to different specifications, and access, price and intensive margin
definitions.

We also find that a legalization policy would increase the probability of use
from 2.3% to 3.0%, particularly affecting young individuals, from 4.3% to
5.5%, where risk perception is a relevant driver to curve marijuana demand.
Therefore, under a legalization policy, marketing warning the potential bad
consequences of consuming marijuana targeting younger individuals would be
mandatory. Therefore, part of the potential tax revenues, which under a
realistic setting would be approximately USD 32 million, should be invested in
these warning campaigns. In addition, we have that given the relatively low
price of marijuana in Colombia, this variable can be use to drive the
intensive margin, rather than the extensive margin. In any case, warning
campaigns should be complemented with taxes as a comprehensive strategy to
mitigate negative effects associated with marijuana consumption.

We consider in this study the short-term effects of a legalization policy on
the extensive and intensive margins of marijuana demand, and potential tax
revenues from this activity. However, future research should consider long-
term effects of this policy. In this point, the “gateway hypothesis” is
relevant as a legal framework for marijuana consumption may imply consequences
on consumption of more toxic drugs due to the polydrug use. In addition, other
aspects of a policy of legalization should be considered, for instance,
effects on public health, labor and crime. The latter is particularly relevant
in Colombia due to the relevance of marijuana in the micro traffic business.

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##  Appendix A Variable construction

###  A.1 Quantity consumed weighted by marijuana type

There are different types of marijuana, among which the ENCSPA highlights
regular, corinto, creepy, and others. All of these have different intensities
of active compounds (tetrahydrocannabinol \- THC). It is crucial to weigh the
amounts consumed by the proportion of active compounds, thus converting the
different varieties into one in common. Taking into account the studies of
Castaño-Pérez et al.,  (2017) and Pachón,  (2012), we learn that the corinto
and regular varieties cannot be considered statistically different in terms of
their THC composition (therefore, for this study, they will be considered as
the same variety); While the creepy variety can contain on average up to 4
times more THC than the regular one. With this in mind, the nearest neighbor
algorithm is estimated to find the price of each variety for all individuals
consuming marijuana. Taking advantage of the fact that there are individuals
who only consume one class, a Euclidean distance is calculated from observable
characteristics between individuals who consume more than one variety and
individuals who only consume a single one, and the nearest neighbor price is
associated.
Once we have the price for each marijuana type consumed by all marijuana
consumers, we assume that the price reported in the survey corresponds to a
weighted average of the consumption of each class; therefore, using the
following equations, we can obtain the amounts per variety for creepy and
regular:

$\text{Avg.
price}=\frac{\text{price}_{i}\times\text{quantity}_{i}+\text{price}_{j}\times\text{quantity}_{j}}{\text{Total
consumption}}$ $\text{Total
consumption}=\text{quantity}_{i}+\text{quantity}_{j}$

Finally, after having the amounts per variety, the quantities of creepy are
multiplied by 4 (since this is the proportion of more THC among the strains),
leaving everything in terms of regular marijuana. As for users of the “other”
class (0.3% of the total population and 7% of marijuana users), it is not
possible to know precisely the intensity of the active compound. Therefore,
including them in the analysis is not possible, and they are not taken into
account in this analysis.

###  A.2 Prices

We can find a direct measure of price by dividing the monthly expenditure by
the average number of units consumed by individuals who report marijuana
consumption in the last year. However, the price is not directly observed for
individuals who report no consumption of marijuana. Thus, we follow the
nearest neighbor algorithm to impute price: First, prices below 10% and above
95% of the specific drug price distribution are left out of consideration as
we want to avoid a bias toward the imputed distribution due to extreme values.
Second, we impute the price of marijuana for those individuals who report no
marijuana consumption using the average price for the same municipality and
stratum. Third, if there is not a marijuana consumer in the same municipality
and stratum, we use the average price at the municipality level, then, the
average price at the stratum level, and finally, the unconditional average
price, in case there is not any match.

###  A.3 Risks perceptions

We created a single measure of risk perception from the questions asked in the
survey about self-perceived risk according to the use of marijuana at
different frequencies (rarely, sometimes, and frequently). Each of these three
variables takes the ordered values of 1 if you consider it to be no risk, 2
slight risks, 3 moderate risks, and 4 high risks. To create a single measure
we take the average between the three variables and divide it into three broad
categories of low risk, medium risk, and finally high risk (independently of
the frequency).

###  A.4 Different access, prices, and quantities measures

We use different measures of access, prices, and quantities consumed of
marijuana weighted by THC content to test the robustness of the estimates to
small changes in the definitions of each variable from the survey data. In
particular, we made variations in three key variables. The changes made are as
follows.

Table 11 shows the descriptive statistics of these three alternative measures
compared to descriptive statistics in Table 1. We observe that the new
definition of access implies overall more individuals having access to
marijuana as expected. This increases 9 percentage points compared to the
baseline setting. Note that columns 2 and 3 in Table 11 correspond to the data
filtered by the access variable under the main definition used in the primary
analysis. The access variable under the definition used for heterogeneous
effects is less strict. It differs from the original definition so that 22%
more of the people with access originally end up having access under this
definition (Table 11, Column 2, Row 4). Regarding prices and quantities, we
observe that the baseline and alternative settings are very similar. This
means that these measures are both relatively consistent.

Table 11: Summary statistics different measures of access, prices, and
quantities.

|  | All |  | Access (main) to marijuana |  | Marijuana consumer
---|---|---|---|---|---|---
|  |  |  | No |  | Yes |  | No |  | Yes
Variable |  | (1) |  | (2) |  | (3) |  | (4) |  | (5)
Access to marijuana |  |  |  |  |  |  |  |  |  |
Main |  | 0.58 |  | 0.00 |  | 1.00 |  | 0.57 |  | 1.00
|  | (0.49) |  | (0.00) |  | (0.00) |  | (0.5) |  | (0.00)
Robustness |  | 0.67 |  | 0.22 |  | 1.00 |  | 0.66 |  | 1.00
|  | (0.47) |  | (0.41) |  | (0.00) |  | (0.47) |  | (0.00)
Quantity consumed |  |  |  |  |  |  |  |  |  |
Main |  | 1.01 |  | 0.00 |  | 1.75 |  | 0.00 |  | 43.08
|  | (13.61) |  | (0.00) |  | (17.89) |  | (0.00) |  | (78.07)
Robustness |  | 1.05 |  | 0.00 |  | 1.82 |  | 0.00 |  | 44.67
|  | (14.04) |  | (0.00) |  | (18.44) |  | (0.00) |  | (80.36)
Price of marijuana |  |  |  |  |  |  |  |  |  |
Main |  | 0.83 |  | 0.86 |  | 0.80 |  | 0.83 |  | 0.84
|  | (0.44) |  | (0.48) |  | (0.40) |  | (0.43) |  | (0.54)
Robustness |  | 0.81 |  | 0.84 |  | 0.79 |  | 0.81 |  | 0.84
|  | (0.3) |  | (0.31) |  | (0.28) |  | (0.29) |  | (0.37)
Sample size |  | 49,414 |  | 20,909 |  | 28,505 |  | 48,255 |  | 1,159

  * Notes: Standard deviations in parenthesis. This table presents descriptive statistics for ENCSPA 2019 regarding access and consumption of Marijuana in Colombia, as well as control variables. Column 1 shows the information for the entire sample. Columns 2 and 3 show the information for individuals without and with access to marijuana. Columns 4 and 5 show the information for not consumers and consumers, respectively. Individuals who are consumers should have access; this is internally consistent for more than 98% of the population and inputted for the remaining 2%. Prices are shown in 2019 USD and were converted using the average exchange rate in 2019 (3,274 USD/COP).
Source: Authors’ construction using ENCSPA data.