Is Academy City really an educational city? What is its purpose? Academy City seems like a really weird city to me. At first I thought it was some super advanced city that was created with the sole purpose of filling it out with educational institutions for science and research, but a lot of things don't seem to make sense.

I watched the first season and it seemed really weird that the government was supporting the experiments for a level 6 esper. Something like that sounds like something the military would want. Why in Academy City?

Another thing that seems kind of weird is the Enforcement body "Judgment". Why is there such an enforcing body with such high level espers? Doesn't that imply that such weird and dangerous things are normal in Academy City?
Imagine 5 years old Misaka Mikoto living not in Academy City, but in "normal" city.


Misaka touched her friend.
Friend got electrified.
Misaka was surprised.
Friend still electrified.
Misaka don't now how to stop.
Friend got fried.


That is just one example. Another would be Shirai Kuroko using her teleport power but left her hair and thus shows up bald at the destination point.

The main purpose of Academy City, is to educate the students with esper powers so that they can control it well. It also educates people that want to develop esper powers as can be seen when Kamijou Touma attended school with fellow level 0 or Saten Ruiko's school. As those students can control their powers better, their level goes up. This is also one of the goals of academy city, researching esper powers. At the very least, this is how academy city shows itself to the world.

As things currently are, the highest level known for esper powers is level 5 with only 7 people managed to achieve that. Therefore, for research purpose, they did the experiment to see whether it is possible to achieve higher level or not. Of course Accelerator, being the best among the 7 level 5s is chosen as the candidate for the experiment. Railgun was chosen as the mob for Accelerator to kill as they hypothesized that by killing Railgun 20k times (not sure if that is the correct figure) Accelerator would be able to achieve level 6. This is the dark part of academy city that it doesn't show to the world.

Judgement

The use of Judgement is mainly because one can't be very sure whether the criminal would be a level 0 or a leveled esper. Suppose that it is not a level 0, then normal means might not be possible to arrest them. Therefore to catch those criminals another esper would be needed. This use the principal of fighting fire with fire. Also to be noted, not all members of judgement are high leveled esper. In fact, in Railgun and Railgun S anime there were no level 5 member of judgement. Misaka indeed illegally used Shirai's judgement armband and posed to be one before but that was illegal. As far as I know Shirai Kuroko is the highest level member of judgment, being a level 4. Another member, Kazari Uiharu was a level 1. The chief of the 177th branch (Shirai and Kazari's boss) is a level 3.