You guys are lightweights. The OP needs to pull a Ben Franklin for maximum effect.
(just kidding obviously)
I really can’t imagine the cops would give two flying figs about this, unless you live in Mayberry and they have nothing else to do. Your landlord and/or the power company might care, but then again they might not.
Just unplug the damn cord. Remove any tape that’s on your part of the property and toss the cord aside. Your neighbor will get the message. In case he turns out to be brazen enough to plug it back in, flip the breaker off too. You can flip it back on if you ever use that outlet; not that big an inconvenience.
I don’t suppose mine is the most creative solution, but it would surely be effective. Nobody can steal electricity from a non-powered outlet.
Every once in a while, the OP could unplug the cord, wait five seconds and plug it in again. Just in case the thief has a powered-up PC plugged in. But I really like the idea of running his cord back to his own outlet. How long will he go using his own power while thinking that he’s stealing it? And aren’t there locking plugs one can put into outlets to prevent this thing in the future?
Not if you discount the whole “possibly growing pot thing”. And I suspect cops actually like the WTF? “interesting” calls. Being a cop is probably pretty boring with random moments of terror thrown in. Even if there is no pot it would be a nice diversion for them IMO.
Are you sure it’s your outlet, one that you are paying power for?
Are you sure it’s your neighbor who plugged that cord in? Could it be building management for some legit reason or even part of some service you ordered - perhaps even a external light fixture that is for your apartment?
If you decide to send 1.21 jigawatts into that line first consider if there is a flux capacitor in whatever is hooked up to it and also consider if you set the thing on fire inside your apartment complex.
Take some scotch magic transparent tape and put a thin layer on each side of one blade of the plug on the end of the cord. Trim it with an exacto knife, so it doesn’t show at all.
In a couple of days, you will find the cord has been replaced. Wash, rinse, repeat. Eventually he will figure out he is spending more on extension cords than he is saving on power.
Yeah, I was thinking along the lines of wrapping the cord around your trailer hitch and driving around the block.
Years ago, when we were buying a house in a new development, we went out to see how it was coming along and found that our neighbor (whose house had been completed a month or so before) was using our hose outlet to water his lawn. I just detached his hose and kept it. He didn’t say anything.
Problems:
#2 has a serious chance of backfiring - just ask O.J. Simpson how well “self-administered justice” works.
#3 might result in, while you are away, your neighbor removing his cord, taking a picture of your cord, and complaining to the management that you are the one stealing his power - and he has the visual evidence to prove it.
In Philly, the police instituted a policy of intervening in disputes between neighbors before they become violent. Further, theft of services is a crime.
Any decent landlord would definitely care that one tenant is stealing electricity from another and has violated the fire code. The power company would absolutely be interested in theft of services.
Heartless bastard! I would come to your office and use my fat, puffy, emotionally and hormonally unstable hands to wring your neck! 
Better way, IMHO: clear fingernail polish.
I like this idea very, very much.
Why should SenorBeef give two shits about whether or not the thief has any PC much less a powered-up one plugged into it. He’s a goddamn thief and doesn’t deserve any, much less that kind of, consideration.
He’s suggesting SenorBeef unplug the cord at random intervals so the guy’s computer shuts down and he loses whatever he’s working on. A lot of these ideas are really funny, but calling the landlord and the electric company sounds like the smartest and the most effective way to go here.
Heartless bitch.
And just try it, honey. 
A few years ago at my business, a neighbor saw someone put stuff into my dumpster when i was closed. He got the license plate number. A call to the local police led to the guy being charged with theft of services.
The cop brought him to the dumpster and showed him things with his address on it (at first he denied the crime).
Oh crap—it’s the Voice of Reason, and it’s making sense.
Another vote for notifying your landlord and taking pictures/video. Always cover your ass. Then introduce your landlord to some of the suggestions listed here. He might enjoy them. 
I came here to suggest supplying 220V to the extension cord, but I see two others have mentioned it already. I would do it.
On a serious note, the utility companies do not fuck around. I had a landlord that had a small light above our garages that was getting power before the meter. This light had run for almost twenty years like this. Long story short, total fines, penalties, etc. cost her $18,000.
Call your management company, document on video, and call the utility company.
Don’t forget the 220V.
One thing. If you want to get an idea of the power he is drawing but don’t want to tip him off by unplugging the cord to use your meter then carefully feel the cord with your hand. If it is not a really heavy duty cord and he is drawing lots of power the cord and or plug itself will be warm.
And if he is drawing LOTS of power, then some shenagins are probably about.