This paper employs the AIDS model, developed by Deaton and Muellbauer (1980), which empirically uses Zellner (1962) seemingly unrelated regression (SUR), estimating a system of four share equations; one for each drug.

The AIDS model takes the form ${w}_{i}={\alpha}_{i}+{\mathrm{\Sigma}}_{j=1}^{n}\phantom{\rule{thinmathspace}{0ex}}{\beta}_{ij}ln\left({p}_{j}\right)+{\gamma}_{i}ln\left(x/P\right)+{\psi}_{i}ln\left(pmc\right)+{\epsilon}_{i}$(1)

where *w*_{i} is the share of the *i*th drug. Drugs in the study include cocaine, methamphetamines, heroin, and marijuana. Shares are calculated by dividing expenditures per drug by total drug expenditures. The coefficient *α*_{i} is the constant in the *i*^{th} share equation, *p*_{j} is the price of the *j*^{th} drug, *x* is total drug expenditures, *pmc* is past marijuana consumption, and *β*_{ij}, γ_{i,} and *ψ*_{i} are coefficients to be estimated. All variables are in natural logs (ln). *P* is an aggregate price index specified as $ln\left(P\right)={\lambda}_{0}+{\mathrm{\Sigma}}_{i}{\mu}_{i}ln\left({p}_{i}\right)+.5{\mathrm{\Sigma}}_{i}{\mathrm{\Sigma}}_{j}{\phi}_{ij}ln\left({p}_{i}\right)ln{\left({p}_{j}\right)}_{}$(2)

As the aggregated price index is difficult to estimate, most empirical studies adopt Stone (1953) Price Index, ln (P*) = *Σ*_{i}w_{i}ln(*p*_{i}), the linear approximation of the price index. Eales and Unnevehr (1988) and Taljaard, Alemu, and Van Schalkwyk (2004) discuss simultaneity issues that may arise since *w*_{i} is the dependent variable but is also included in Stone (1953) Price Index as a left-hand side variable. Eales and Unnevehr (1988) suggest lagging *w*_{i} to avoid simultaneity issues, the technique in this paper. Thus, Stone (1953) Price Index takes the form, ${\mathrm{\Sigma}}_{i}{w}_{{i}_{t-1}}ln\left({p}_{i}\right)$.

Hunt-McCool, Kiker, and Ng (1994) discuss the shortcomings of the more common linear or log-linear approach to estimating illegal drug demand. Unlike these approaches, the AIDS approach is consistent with consumer demand theory (Deaton and Muellbauer 1980). Homogeniety and symmetry can be test for or imposed upon the system of equations. Despite its consistency with theory, to the authors’ knowledge, the AIDS model has not been used very often in drug demand studies. Jofre-Bonet and Petrey (2008) use the AIDS model to estimate own and cross-price elasticities between various illegal drugs while controlling for several variables including age and gender. This appears to be the first study that uses the AIDS approach while specifically attempting to see if previous marijuana consumption increases current consumption of other drugs.

Homogeneity and symmetry restrictions are imposed on the system of equations. The homogeneity restriction follows ${\mathrm{\Sigma}}_{j}{\beta}_{ij}=0$

implying quantity demanded of the good stays the same if prices and expenditures change by same proportion.

Symmetry follows ${\beta}_{ij}={\beta}_{ji}$(3)

suggesting consistency among drug choices.

Adding up requires budget shares to sum to unity therefore $\sum _{i=1}^{n}{a}_{}=1,\phantom{\rule{thinmathspace}{0ex}}\sum _{i=1}^{n}{\gamma}_{i}=0,\sum _{i=1}^{n}{b}_{ij}=0,\sum _{i=1}^{n}{\psi}_{i}=0$

To avoid singularity, the marijuana share equation is dropped and later recovered with the adding up restriction. shows the coefficient estimates from share equations and recovered estimates for the marijuana equation. Standard errors for the dropped equation approximated in Stata with the delta method.

Coefficients from the share equations are used to derive income and price elasticity estimates in . Before moving to , results from have some interesting policy implications.^{2} Previous marijuana consumption (*pmc*) does not increase current consumption of harder drugs, and actually decreases current cocaine consumption. This substitution effect implies that a benefit from marijuana legalization, which would most likely increase marijuana consumption (Saffer and Chaloupka 1995), would be a decrease of cocaine consumption. Marijuana does appear to be addictive as previous marijuana consumption increases the current marijuana budget share. However, society may as a whole may benefit from relatively less cocaine consumption, especially since the ongoing drug wars, which can be violent, typically involves the cocaine trade.

Table 2: Elasticity Estimates

Own price coefficients are positive and all statistically significant, suggesting all drugs are price inelastic. An increase in the price of a price inelastic good increases expenditures on the good as consumers are relatively insensitive to price changes. This result may be expected for addictive goods such as the drugs included in this study.

To the authors’ knowledge, only a few papers have used the AIDS model to study illegal drug substitution. Some focus solely on the addiction of alcohol and/or cigarettes including Fanelli and Mazzocchi (2004) and Aepli (2014). Jofre-Bonet and Petrey (2008) include data both on legal and illegal drugs in their drug-related AIDS empirical study. Our paper resembles this study with a few important exceptions. Most importantly, our paper focuses on the effect past marijuana consumption has on the shares of other drugs. Jofre-Bonet and Petrey (2008) include other explanatory variables but are all related to demographics (race, age, etc.). In addition, the authors obtain data using a classroom experiment with cocaine and heroin addicts as participants. The authors caution that participants may not behave in the same manner in a classroom setting as in the real world. Data from STRIDE may more closely reflect actual street-level prices. Both papers employ Stone (1953) Price Index. In our paper the share of the ${i}^{th}$ drug, ${w}_{i}$ is lagged one period.

The estimated coefficients and budget shares from the each SUR model in reported are used to derive compensated (Hicksian) elasticities. The derivation of compensated elasticities is from Taljaard, Alemu, and Van Schalkwyk (2004). Income elasticities measure the percentage change in quantity demanded given a one percent increase in income and follow $1+({\gamma}_{i}/{w}_{i})$

where *γ*_{i} is the estimated coefficient on the expenditure term in eq. 1 and *w*_{i} is the budget share of the ${i}^{th}$ drug.

Own price elasticities measure the percentage change in quantity demanded given a one percent change in price and follow $-1+\frac{{\beta}_{ij}}{{w}_{i}}+{w}_{j}$

where ${\beta}_{ij}$ is the own price coefficient with *j = i, w*_{i} is the budget share of the ${i}^{th}$ drug and *w*_{j} is the budget share of the${i}^{th}$ drug.

Cross-price elasticities measure the percentage change in quantity demanded of *i*^{th} drug given a one percentage increase in the price of *j*^{th} drug and follow $\frac{{\beta}_{ij}}{{w}_{i}}+{w}_{j}$

where ${\beta}_{ij}$ represents the cross-price coefficient of ${j}^{th}$ drug on the ${i}^{th}$ drug. A positive cross-price elasticity implies the drugs are substitutes and a negative cross-price elastic indicates the drugs are compliments. reports derived compensated price and expenditure elasticities. All standard errors are estimated with the delta method.

With respect to income elasticities, we find cocaine is a luxury good. A 1 % increase in income increases quantity demanded of cocaine by 1.08 %. The data suggests heroin and marijuana are normal goods. A 1 % increase in income increases quantity demanded of heroin and marijuana by 0.64 % and 0.75 %, respectively.

Cocaine has a negative own price elasticity. A 1 % increase in the price of cocaine decreases quantity demanded of cocaine by 0.10 %. We find meth to be a giffen good. A 1 % increase in the price of meth increases quantity demanded by 0.51 %. In our study, this own price effect could be the result of solely focusing on addictive drugs and not including other goods (such as food). In addition this positive own price elasticity could reflect better quality. Heroin has a positive own price elasticity although it is not statistically significant. Marijuana is relatively inelastic, which is consistent with previous literature (Pacula and Lundberg 2014; Van Ours and Williams 2007) although the own price coefficient is not statistically significant in this study.

Negative cross-price elasticities indicate drugs are complements, while positive cross-price elasticities indicate drugs are substitutes. Heroin and meth are complements. A 1 % decrease in the price of heroin increases quantity demanded of meth by 0.89 %. A 1 % decrease in the price of meth increases quantity demanded of heroin by 0.40 %. None of the other cross-price elasticities are statistically significant. In general, our results indicate that there is not too much interaction across drug markets, which are consistent with previous studies (Cunningham and Finlay 2015).

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