4KQ "hot" wire and digidash escapade

[Kneale Brownson]

I finally finished up the wiring work on the wife's 4kq, I think.  That
included the replacement of the ground wire that got hot when I put a
non-fuel-pump relay in the fuel pump relay spot while trying to determine
whether the fuel pump would continue working and the installation of the
digidash from my old rustbucket 4kq that had become a yard fixture waiting
to be used for parts.   I got everything for the dash hooked up and, after
a couple times in and out,  got all the wires connected to various parts of
the center console.  

So now we have the digidash AND something I don't recall:  A permanent red
light around the cigarette lighter.  Is that thing supposed to glow when
the key and lights are off?  Will that little bulb run down the battery any
faster than the clock?