Air taxi flight crew with C90 TRACON experienced TCAS RA event at 5000 FT with issued traffic; subsequent descent resulted in an ATC loss of separation.

Narrative

Received a traffic advisory from TRACON regarding traffic identified at 400 FT above and converging from approximately 1130-1200 o'clock position. The Controller also issued a vector upon reaching the next fix on our route. About 10 seconds later TCAS generated a 'TRAFFIC' alert followed which was soon followed by a 'DESCEND' command. I did not visually acquire the target prior to the 'DESCEND' command. Upon receiving the 'DESCEND' RA I disconnected the autopilot and simultaneously began a descent as commanded on my VSI. The descent resulted in an approximate 300-400 foot altitude loss. I then received a 'CLEAR OF CONFLICT' and I returned the aircraft to the assigned altitude. In the meanwhile; the First Officer; as soon as there was a break in other ATC radio communications announced that our aircraft was descending for TCAS. Upon arrival into MDW I was asked to call TRACON. The Controller that I spoke to at TRACON asked for my pilot certificate information and address and stated that they understood we responded to a TCAS RA; but had to document the situation since there was a loss of separation. Being aware of the potential conflict and the busy radio I would like to have had the vector prior to the next fix; I believe this would have remedied the RA in advance. We were going to make that request when it appeared evident that the traffic would be a conflict; but were not able to get a break in the radio in those few moments.