If there are problems with updating or with the permissions then best is to download the full version and trash the currently installed version to do a clean install of the new version.

Download a new copy of the Firefox application and save the disk image file to the desktop

• Firefox 27.0: http://www.mozilla.org/en-US/firefox/all.html

• Trash the current Firefox application (open the Applications folder in the Finder and drag the Firefox application to the Trash) to do a clean (re-)install
• Install the new version that you have downloaded
• https://support.mozilla.org/kb/Installing+Firefox+on+Mac

Your personal data is stored elsewhere in the Firefox profile folder, so you won't lose your bookmarks and other personal data when you uninstall and (re)install Firefox.

• http://kb.mozillazine.org/Profile_folder_-_Firefox

Turns out that Firefox.app was not in the Applications folder. I think that since I dropped Firefox.app into the Applications folder updates will work properly now.

Thanks for leading me to the issue.