Do everything the guys have suggested to rule out carburetion, manifold leaks, or other causes but don't fall into the endless "87 trap" of chasing a fuel problem, multiple carb rebuilds, a new carb, etc. before at least taking another look at the CDI.
Bear in mind that it's an '87. They're prone to CDI failures and they fail in a number of different ways. Sometimes they misfire, sometimes they trigger no signal at all, sometimes they have no spark advance and feel like the bike has a "governor", etc. Sometimes just squeezing their casings or turning them upside down will either correct the problem or make it worse.
Regardless of whatever your dealer told you there is no way to "check" a deteriorating CDI on a bench or sitting in the shop. The only way I've ever found to test an intermittent CDI is with a timing light capable of being operated on a moving bike. Take Ft. Payne's suggestion and try to locate a known good CDI to try before you go buying a new carb. If you can't locate one I'll tell you how to rig a timing light for bike use.