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 