Alternate Title: A Simple Model of Drug Smuggling
Happy New Year! Having re-binged Narcos Colombia and Narcos Mexico during the holidays, I was inspired to develop a simple model of drug smuggling.
First off, assume that p is the probability of interdiction on any cross-border smuggling operation; as a result, the expected number of trips across the border until a drug shipment is captured is 1/p.
Next, assume that the profits are $X for each successful smuggling trip, and further assume that the value of the drugs and the value of the truck, airplane, or other vessel transporting the drugs totals $Y. (In reality, the values of X and Y may vary per trip; I, however, am holding both values constant for simplicity.)
On the last trip, the transport vehicle is captured and no profits are made. Therefore, in expectation, we will have [(1/p) – 1] smuggling operations earning $X per trip and one trip losing $Y.
According to standard price theory in economics, the ex ante expected profit in equilibrium of a rational, risk-neutral smuggler should be zero. This logic can be stated formally as follows:
Y = [(1/p) – 1]X
which can be further simplified as follows:
Y = X(1 – p)/p
Given these super-simplifying assumptions, if p = ½ (i.e. a 50% probability of interdiction), then Y = X. In other words, the smuggler’s initial investment is recouped in just one smuggling operation! Put another way, if we want to combat smuggling, the probability of detection must be greater ½.
If, however, p is below ½, smuggling will always be profitable. For example, if p = 1/10, then Y = 9X. That is to say, the smuggler will recoup nine times his initial investment when the probability of interdiction is just 10%. More generally, this simple model shows that the lower the probability of detection, the more profitable smuggling will be.
So, what is the actual rate of detection? Your guess is as good as the feds’!
Bonus (1/2): I have included an acoustic version of the Narcos theme song “Tuyo” by Rodrigo Amarante below.
prior probability 
Books I liked in this genre. You may already know them:
Narconomics: How to Run a Drug Cartel by Tom Wainwright
https://www.goodreads.com/book/show/25159062-narconomics?ac=1&from_search=true&qid=yLCVXIydAY&rank=1
The Alchemy of Meth: A Decomposition by Jason Pine
https://www.goodreads.com/book/show/45038259-the-alchemy-of-meth?from_search=true&from_srp=true&qid=U98Dwm7W0U&rank=1
The Least of Us: True Tales of America and Hope in the Time of Fentanyl and Meth by Sam Quinones
https://www.goodreads.com/book/show/44452952-the-least-of-us?from_search=true&from_srp=true&qid=mtApGK8fNk&rank=1
thanks for the pointers!
What about a model that includes bribery?
That would certainly impact the likelihood of drug parcels being intercepted by authorities.
It reminds of Tullock’s insights on bribery, the costs of bribery are generally minuscule when you consider the colossal benefits.
Excellent suggestion! Bribery will reduce the frequency of interdiction and can be added to the Y variable as one of the costs of doing smuggling!
I am glad that I made keen observation on that one. I would assume that effective bribery attempts would longrun improve profitability by lowering the interdiction rate.
It is small price to pay on variable Y, when the profits from fentanyl, meth, heroin, cocaine, marijuana (?)*** being so high.
**** I question the continued profitability of black-market Marijuana as more states continue to legalize creational pot. Then again this takes me back to white paper I read a while back from the Reason Foundation. Legalization can backfire if licensed Marijuana sales are taxed too heavily. The standard Laffer Curver argument, in my humble opinion has some veracity.
Excellent observations. Another interesting puzzle is why do so many people want to consume drugs in the first place? What if we paid people *not* to take drugs?
There is a multitude of different reasons for people taking “drugs”. Technically, caffeine, alcohol, and nicotine all fall into the category of being psychoactive.
I would surmise a causal Marijuana or causal Cocaine user would gladly take the money. A hardcore junkie wouldn’t; as they value the drug more than life itself.
A hardened drug addict has a vastly different utility function than even casual”drug” users.
As a fellow economics student! I understood the model quite well! But I do have a question here. How can risk be incorporated here?
That is an excellent question, and not an easy one to answer. In theory, we could make the actor’s “expected utility” a function of his level of risk aversion, but here is a critique of that approach: https://www.aeaweb.org/articles?id=10.1257/jep.32.2.91
P.S.: Thanks for reading my stuff.
P.P.S.: In a previous paper, I modelled risk aversion with respect to litigation risk in legal cases; see here: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1891918