Are Marijuana and Cocaine Complements or Substitutes? Evidence from the Silk Road

Transcription

1 Are Marijuana and Cocaine Complements or Substitutes? Evidence from the Silk Road Scott Orr University of Toronto June 3, 2016

2 Introduction Policy makers increasingly willing to experiment with marijuana legalization. Colorado, Washington, Oregon and Alaska have already legalized. Trudeau government aiming to implement marijuana legalization in Canada in early A question that repeatedly pops up in decriminalization and legalization discussions: Will marijuana legalization lead to greater hard drug (cocaine, heroin, etc...) use? Basically a question of whether different drugs are complements or substitutes.

3 Previous Research Small literature on estimating cross-price elasticities for various illegal drugs. Grossman and Chaloupka (1998), Saffer and Chaloupka (1999), DeSimone and Farrelly (2003), Williams et al. (2006), Jofre-Bonet and Petry (2008). Mixed results concerning the cross-price elasticity of marijuana and cocaine. Limitations: Most studies rely on STRIDE data to measure drug prices, which has been criticized by Horowitz (2001). This paper: Estimate cross-price elasticity for marijuana and cocaine using data from an online marketplace where I observe prices and quantities sold by individual sellers. Less price measurement error Can actually obtain competition relevant elasticities

4 Methodology Lots of individual items in my data (more than 3000) Estimation in product space as in Deaton and Muellbauer (1980) not feasible. Instead estimate using characteristic space approach of Berry (1994) Problem: This approach assumes ex-ante that all products are substitutes I consider a version of discrete choice model developed in Gentzkow (2007) that allows for complementarity I show how (and when) one can estimate this type of model only with aggregate (i.e. sales) data. Considering when the complementarity parameter is identified distinguishes this paper from Song and Chintagunta (2006).

5 Methodology and Preview of Results I propose two different estimators: A two-step and a one-step approach. One-step approach requires a set on instrumental variables that satisfy more stringent exclusion restriction However, easier asymptotic inference Preview of Results: Point estimates indicate that marijuana and cocaine are substitutes in this marketplace. Inference complicated by the fact that the corner case of the standard logit substitutes fits the data best.

6 Data: Silk Road Online Marketplace The Silk Road: darkweb marketplace that specialized in the sale of illegal commodities (primarily drugs). Top Categories? Opened February Shut down by FBI October Dataset: Daily scrape of all item pages from February - July Item Page Information: Price, Number of Feedbacks, Item Description, Ships From, Ships To, Seller Id. Leaving feedback a required part of finalizing a sale- Good proxy for overall sales.

7 Silk Road: Item Page Figure: Example Item Page (Heroin)

8 Summary Statistics: Marijuana Table: Summary statistics item-level: Marijuana Variable Obs Mean Std. Dev. Min Max Median Price (USD) Price per gram (USD) Weight (grams) Total Feedback Comparable to street? Typical Canadian price in 2013: $ 23.4 USD per gram Median U.S. price in (STRIDE) : $ 10 USD per gram Mean price : $ 18 USD per gram

9 Summary Statistics: Cocaine Table: Summary statistics item-level: Cocaine Variable Obs Mean Std. Dev. Min Max Median Price (USD) Price per gram (USD) Weight (grams) Total Feedback Comparable to street? Typical Canadian price in 2013: $93.7 USD per gram Median U.S. price from (STRIDE): $100 USD per gram Mean price: $113 USD per gram

10 Competition Table: Competition on the Silk Road Drug Total sellers Avg. sellers per week HHI CR2 CR4 Marijuana Cocaine

11 Price Variation Figure: Marijuana price variation conditional on a sale Trim Outliers?

12 Price Variation Figure: Cocaine price variation conditional on a sale Trim Outliers?

13 Weight Variation Figure: Marijuana weight variation conditional on a sale

14 Weight Variation Figure: Cocaine weight variation conditional on a sale

15 Demand Estimation: Standard Approach Starting Point: A simple logit model of drug demand Each consumer i {1, 2,...S} who chooses to consume a particular variety j Mt d of drug d {M, C} earns indirect utility given by: V d ijt = δ d jt + ɛ d ijt = (β d 0 + X d j β d 1 αp d jt + ξ d jt) + ɛ d ijt β0 d d mean utility of drug d, Xj drug variety j s observable characteristics, ξjt d unobservable characteristics, pd jt price, ɛd ijt idiosyncratic taste term. Assume ɛ d ijt iid draw from Type-1 Extreme value distribution, normalize outside option to zero Standard Logit Demand System Qjt d = S 1 + b Ω exp ( ) δjt d k M d t ( ) exp δjt d

16 Demand Estimation: Allowing for Complementarity Based on the discrete choice model developed in Gentzkow (2007) that allows for complementarity Assume a nesting structure: Now a given consumer can also choose to consume one variety of a drug, or one variety of of two different drugs. Indirect utility of consuming variety j of marijuana and variety k of cocaine: V db ijkt = δ M jt + δ C kt + Γ MC + ɛ MC ijkt Leads to the following demand system: exp(δ d Qjt d = S jt (1 ) + exp ( Γ MC ) ) k Ω bt exp(δb kt ) 1 + ( ) j Ω M exp δ M t jt + k Ω C exp ( δ C ( t kt) ) + exp Γ MC (j,k) Ω MC t ( exp δjt M + δ C kt )

17 Complements versus Substitutes The sign of Γ MC determines whether marijuana and cocaine are aggregate demand complements or substitutes. Where I d t AD d t Qjt d = S j Ω d t 1 + I M t I d t + exp(γ MC )It b It d + I C t + exp (Γ MC ) I M t is the (exponentiated) inclusive value: It d = j Ω d exp ( ) δjt d t Straightforward to verify: AD d t I b t = S ( 1 + I M t I d t ( exp(γ MC ) 1 ) + I C t + exp (Γ MC ) I M t ) It C 2 Exogenous variation in prices and choice sets should identify Γ MC I C t

18 Estimation: First Stage Logit structure means that we can difference out the impact of marijuana prices of the variety level cocaine demand systems. log Q d jt log Q d t = ( ) ( ) Xj d X d t β1 d α pjt d p d t + ξjt d ξ d t Basically standard logit estimating equation from Berry (1994) To deal with price endogeneity, use BLP IVs Details

19 Identifying Γ MC Previous regression gives me estimates of α and β1 d, but not β0 d or Γ db To identify the remaining parameters, I invert the aggregate demand system st d j Ω d Qd It jt = d +exp(γmc )It b I t d t Proposition 1 1 S 1+I M t +I C t +exp(γmc )I M t I C t If both st M and st C lie between 0 and 1, then there is a unique set of inclusive 2 values ( ) It M, It C R 2 + that perfectly rationalize the observed aggregate sales for all values of Γ MC, log 1 + max( s M t 1 s M t 1 s, t C ) 1 s t C Call Î t d (Γ g, s t) the inclusive value that rationalizes the observed market shares Suppose ξjt d = ξ t d + ξ jt. d Straightforward to show that: ) ξ t d d = log (Ît (Γ g, s t) β0 d log j Ω d t exp ( X d j β d 1 αp d jt + ξ d jt)

20 Second Stage Estimator Let Ĝ d t = log j Ω d t ( exp X d j β d 1 αp d jt + ξ d jt) Now we can write ξ t d as a function of parameters to be estimated: ) ξ t d d = log (Î t (Γ g, s t ) β0 d Ĝ t d Second stage GMM estimator based on moment conditions of the form: [ ] E ξt d Zt d = 0 Where Z d t is a vector of instrumental variables

21 Second Stage Instruments Z d t Need three instruments for three parameters Drug dummies function as instruments for the β d 0 intercepts. Need a time varying instrument for Γ MC Since variation in inclusive values pins down Γ MC, using the part of the inclusive value estimated in the first stage seems like a promising candidate. Problem: Endogenous. Construct a second stage instrument that harnesses exogenous variation in prices (via bitcoin exchange rate), and products (exit of big players) log j ActiveBigPlayer dt exp ( X d j β d 1 α(bitcoin price t )p d j + ξ d jt)

22 Alternative One Step Approach Can also estimate model as in Berry (1994), where one inverts the market shares item-by-item. Allowing for complementarity means that we cannot obtain a closed-form expression for mean utilities as a function of market shares. Still possible to numerically invert demand system after finding inclusive values that rationalize market shares: ( ) 1 + δ jt d (Γg, st) = log s Î M jt d t (Γ g, s t) + Î t C (Γ g, s t) + exp(γ g )Î t M (Γ g, s t) Î t C (Γ g, s t) 1 + exp(γ g )Î t b (Γg, st) Outer loop 2SLS regression: ) δ jt (Γ d MC, s t = β0 d + X j d β1 d αpd jt + ξd jt Note: Cannot include drug/week fixed effects, since that would absorb all the variation that identifies Γ MC BLP instruments only valid if entry and exit completely exogenous.

23 First Stage Results (1) (2) (3) (4) OLS IV OLS IV Price (USD) *** *** ( ) ( ) Lagged Price (USD) *** *** ( ) ( ) Weight (grams) *** *** ( ) (0.0424) ( ) (0.0364) Weight (Drug=Cocaine) 0.109* 0.999*** 0.122* 0.805** (0.0620) (0.372) (0.0670) (0.318) Observations 5,862 5,862 4,979 4,979 Item clustered standard errors in parentheses *** p<0.01, ** p<0.05, * p<0.1 Specification (2) estimates consistent with item-level own-price elasticities of approximately for marijuana and for cocaine.

24 Second Stage Results (1) (2) (3) (4) β0 M *** *** *** *** (0.0308) (0.0310) (0.0325) (0.0364) β0 C *** *** *** *** (0.0439) (0.0542) (0.0420) (0.0689) Γ MC ( ) ( ) ( ) ( ) Observations Standard errors in parentheses *** p<0.01, ** p<0.05, * p<0.1 Large standard errors likely due to the fact that criterion function is minimized at the corner case where Γ MC More details

25 Second Stage Criterion Functions Figure: GMM criterion functions

26 One-Step Results Full Sample Limited Sample Price (USD) *** *** (0.0024) (0.0023) Weight (grams) *** *** (0.0231) (0.0222) Weight (Drug=Cocaine) *** *** (0.1968) (0.1915) β0 M *** *** (0.0620) (0.0562) β0 C *** *** (0.0728) (0.0730) Γ MC ( ) ( ) Observations Standard errors in parentheses *** p<0.01, ** p<0.05, * p<0.1

27 One-Step Criterion Functions Figure: GMM criterion function

28 Conclusion This paper: Proposes two different estimators for a discrete choice demand system that allows broad product categories to either be complements of substitutes. One-step estimator more straightforward for inference, but requires instruments that require stronger exclusion restrictions Estimated the demand system using item-level sales data from the Silk Road. Point estimates consistent with marijuana and cocaine being substitutes. Cross-price elasticities of (cocaine demand with respect to median marijuana price) and (marijuana demand with respect to median cocaine price) Currently some inference issues associated with these estimates due to corner case of logit substitutes.

29 Top Silk Road Categories Figure: Popular Item Categories

30 Top Silk Road Categories Figure: Popular Item Categories Back

31 BLP Instruments Standard instruments form BLP with a minor modification. Zjst1 d = Xi d Xj d Z d jst2 = ( i Ω d i Ω d st X d t i ) ( ) i Ω d X d st i i Ω d t X d i I normalize the second instrument by the total size of the market so the instrument has more item level variation. Back

32 Is the criterion function hitting a corner? Cross-price elasticities given by: p M t AD M t S ( 1 + I M t + I C t αit M It C + exp (Γ MC ) I M t ( ) ) It C 2 exp(γ MC ) 1 Could redefine parameter of interest as exp(γ MC ) [0, ) exp ( 30) 1 = Smaller values of exp(γ MC ) pretty much observationally equivalent exp(γ MC ) is numerically zero If the true parameter value is at the corner where exp(γ MC ) = 0, GMM estimator is no longer asymptotically normal (Newey and McFadden 1994) Bootstrap does not help (Andrews 2000) Need to derive the appropriate asymptotic distribution to get standard errors. Back

33 Figure: Marijuana price variation conditional on a sale & price 50 Back

34 Figure: Cocaine price variation conditional on a sale & price 200 Back