How to calculate payback period of the project? In order to finish a technological product, it was spent 50000 dollars (for materials, salary). 500 dollars per month is needed for support of the project (will be spend for servers). This product automated some process that required manual work for 10000 dollars per month. So, how to calculate payback period of the project? 

My solution would be 50000:10000 = 5 month, but I don't know what to do with the cost of support per month.
This is more finance than economics but here it is:

The payback period is pretty much how much time you need to recover the initial investment. 

Each month, you are saving 10000 - 500 = 9500.  Then you simply just divide 50000 by 9500. This comes out to be around 5. 

I made some simplifying assumptions: Discount rate is 0. This is usually not true, so you would probably have to discount the cash flows. Note: This will probably involve monthly compounding. Check out the formula for annuity. 

Also, just FYI: payback method often results in wrong decisions. When you have a choice, always use NPV.