I highly recommend reading Inspired - How to Create Products Customers Love. In Part 1 Marty Cagan describes, in general terms, the typically responsibilities for Product Management, Product Marketing, Project Management and, of course, Design and Engineering.
The product manager has two key responsibilities: assessing product opportunities, and defining the product to be built. - pg 2
To answer your question, I believe identifying the "true needs" of the customer, as opposed to the "wants" or "think they wants," to be an important aspect of "assessing product opportunities." This is no easy task (in many cases even the customers don't know their true needs), but great PMs are in touch with customers and potential customers, and constantly have their fingers on the pulse of the market. They collect as much data as possible and then use experience and judgement to determine if an opportunity exists.
That being said, however, working backwards (go find a product that'll increase sales by 20%) might be a challenging mandate. For this to happen, you need to start with an idea. Again, according to Mr. Cagan:
Typically, new ideas can come from anywhere - company executives, discussions with customers, usability testing, your own product team, your sales or marketing staff, industry analysts, to name a few. - pg 2
A lot of that is under the PM's purview, but putting the entire responsibility of generating ideas for consideration onto the PM doesn't sound reasonable, nor does it seem like a good idea. I hope that helps.