I have used a binomial sample calculation to conclude that we need a sample size of 150, which for a response rate of 80% there is 80% power that the confidence limits between 71% and 89% cover our target. But, for a sample size of 100 and a response rate of 80% there is 80% power that the confidence limits between 70% and 90% cover our target, which does not change the results significantly at all. I'm just wondering what effect missing the optimal sample size gives if this doesn't change the CI?
If you are comparing confidence intervals with the same confidence level then having less data will make the interval wider (i.e., less accurate). The only way you could be getting confidence intervals of the same width with less data is if you have simultaneously dropped your confidence level.