Yes, by all means, keep calling it an A and play away. You move the capo to find the place where that A chord is in a good range for your voice.

If you need to know what chord you are actually playing, like if you need to communicate with other musicians, just count up the scale. A on the 1st fret becomes Bb. On the 2nd fret it's B. On so forth: C, C#, D, Eb, F, F#, G, Ab, and back to A (if you could capo on the 12th fret!). If it is a major chord without a capo, it will stay that way with a capo.

The other big reason to use a capo other than pitching the guitar to your voice is to play with other musicians. Say a fiddler plays a tune in the key of D and you really want to play the chords in A. Just count up the neck from A until you get to D and put the capo there (5th fret). Now you are playing A chords, but the guitar is producing D chords for the fiddler.

John