If I plan to do a purely internet based startup, could I outsource myself to the cheapest place on Earth, and manage my startup from there?
This way, my cost of living would be minimal, while my income would essentially stay the same, because the internet is a global market.
The startup would do algorithmic trading over the internet based on artificial intelligence. As for the moment, I'm doing a PhD in this area that I'll soon finish. If I plan to recruit addition people, I'd most likely do that through the internet and have a remote work force. I would do all research and further education through the internet too, i.e. reading papers in Google Scholars or watching top uni lectures in AI and finance.
Basically, I'm on my laptop most of the time, about 95% of hours I'm awake. Does it really matter where I'm located?
Do you really want "the cheapest place on Earth"? Perhaps you would be better off picking a place where you would enjoy living and that is safe.
At a minimum I would think you would want a place with sufficient infrastructure to reliably provide electricity and internet connections.
Those of us living in the first world get spoiled and often don't realize what life is like in the third world. In the US one can order something on the internet and have it delivered the next day at low cost. In many parts of the world ordering something can involve extremely high shipping costs, delays and corruption in customs, long transit times and many firms that simply refuse to ship to those countries.
What will you do if your laptop breaks and you need a spare part?
Hopefully with a PhD you will be able to make enough money to be able to afford to live where you want, not simply the cheapest.
Assuming you don't ever need to meet face to face with clients then yes sure you can. Lots of professional poker players for example play from Thailand, spend much less on living expenses and get to explore a new country.
If your sole reason for going is to save money then I would strongly advise against it. What would you spend they money on that's more important than living where you want?
It sounds like you're trying to get in to some high frequency trading. You can certainly do that from anywhere, as the quality of your algorithms are the most important thing.
And, if you're recruiting people to help you look at the data and write code with you, that can certainly be done remotely too.
However (and you probably know this already), the speed of your connection is super important and having your servers close to the exchange is key. If you're in a good data center and have a reliable connection and machine and good ways of collaborating with your associates, then there shouldn't be any reason why you can't do this out of any random park.
A startup isn't about making good software, putting it online and watching money roll in.
What about biz development? Who are your clients? How do you plan to grow your customer base?