Corporate jet First Officer reported ATC failed to provide separation between traffic resulting in an NMAC.

Narrative

On takeoff we were climbing out on our initial climb following our climb via clearance. I was flying and saw Aircraft Y cross the windscreen above us from left to right less than 500 feet. I reacted by pulling back power as we received a Resolution Advisory to descend which I started to follow. That Resolution Advisory went away after a couple seconds. CA informed Tower that we responded to a Resolution Advisory and asked where the traffic was since the Resolution Advisory disappeared so fast. I had told him I saw Aircraft Y pass above and ATC confirmed they had cleared traffic to cross overhead. We continued the flight as normal. I had heard ATC clear traffic to cross overhead thinking they would be over the approach end and we had a slightly steeper climb since we were light. After landing we called the ZZZ Tower to ask what happened and they informed us that the Aircraft Y had crossed further west than they anticipated. After the fact; pulling power and lowering the nose after traffic passed and following the Resolution Advisory for a couple seconds.