Basically my cofounders and I have an idea that we think is really good and while we have all some background in programming our knowledge is not specific or deep enough to allow us to build our own MVP. We do not want to give up equity and have cash from previous projects that were not tech firms. Answer Request
I run services at Gun.io, and as such I spend most of my time on the phone with businesses who are doing software projects. Though we don't tend to do a lot of MVPs (just because of the budget hurdles you identify), I speak to a lot of entrepreneurs in your position. The main thing you want to ask yourself is how comfortable do you feel managing the outsourced partner to actually do what you need them to do. Be wary of the bait-and-switch where the guy on the phone seems amazing and then dishes you off to the lower level people who can't communicate and don't have a clue what you are trying to build. I see this all the time. I'm not going to say there aren't amazing engineers and shops everywhere in the world (there are). However, you have to be aware that 80% (generally) are not very good. The same 20% exists everywhere, and even with good prices because of geographic arbitrage, but you need to search for them.
If you really need to go offshore I'd check Eastern EU, Russia, Ukraine, and strong recommendation for Costa Rica and other Central/South American countries. Nearshoring is awesome because they are on the same timezone. You can actually work with them instead of trying to Skype in the middle of the night (assuming you are US).
There's a huge proliferation of shops who "specialize" in doing MVPs for startups. I'm sure 20% of them are awesome, and the same 80% are not, so you really need to do your diligence. I speak to people every single day who have blown their entire budget on $30/hour shops and on $300/hour shops. There seems to be an equal number of lousy performers at every level.
Also, be careful to understand that an hourly rate as a comparison assumes productivity of an hour is the same for everyone, which is clearly nuts. I also have some issues with the economic incentives that are created by fixed bids and other agency models. Not to say anyone of these things are bad in their own right, but when you study the models it's clear that certain incentives are built in and you need to be armed with those things in mind when searching and negotiating.
If you want an impartial subjective party to help you choose I'd be happy to consult with you.