1988 Chevy Corvette engine stalls

Hi, I just recently bought a 1988 corvette V- 8 automatic with 14,000. It runs great, tranmission seems smooth however, just yesterday I noticed I had some difficulty starting the car. The car would run slightly then stall. To keep the car running I had to keep my foot on the accelerator. Once I had the car running when I stopped at a red light or stop sign I noticed it would not idol and stall again. Usually I noticed the car idols at about 8,000 however, the idol went down to 4,000 and lower until it stalled. I had to keep my foot on the gas to keep the idol up to 8000 and then I never had a problem. The next morning I went out to see if the car continued to do the same thing and found that the symptoms described were not there. The car started right up with no idoling problems. A second thing I noticed (has been rare but nonetheless there is upon starting some smoke coming from the exhaust. This lasts about five second and disappears. I do not see it all the time only on occassion. I hope there is nothing seriously wrong with the car since I just bought it and didn't notice this when I looked at it. If you can tell me what is wrong could you also give me an estimate on how much it would be to fix? Thank you for your time.