Thank you MiyoLinux, and everyone else.

So should I install Xfce's power manager? Or something else?

Another more disturbing question: if Mate's power manager has systemd as a dependency, will the whole Mate desktop one day have it as a dependency?

But according to Synaptic it is installed.

I get bash: mate-power-manager: command not found

If I put sudo in front I get: sudo: mate-power-manager: command not found

It doesn't help if I put in mate-power-manager-common either.

EDIT:
I also tried to reinstall it in hopes the same error message would pop up again, but it did not.

Thanks Rolfie for the suggestion to search askUbuntu. I found something there that seems to be working:

[Disable suspend (upower)]
Identity=unix-user:*
Action=org.freedesktop.upower.suspend
ResultActive=no
ResultInactive=no
ResultAny=no

[Disable suspend (logind)]
Identity=unix-user:*
Action=org.freedesktop.login1.suspend
ResultActive=no
ResultInactive=no
ResultAny=no

[Disable suspend when others are logged in (logind)]
Identity=unix-user:*
Action=org.freedesktop.login1.suspend-multiple-sessions
ResultActive=no
ResultInactive=no
ResultAny=no

I put this where you suggested (/etc/polkit-1/localauthority/50-local.d) with your file name (enable-shutdown-reboot.pkla), and it seems to be working, even without a reboot. I will continue to monitor this throughout the day (with multiple reboots) and report back.

EDIT:
Didn't notice at first that this only dropped suspend from the shut down menu. So I went back and made this file look like this and that also removed hibernate:

[Disable suspend (upower)]
Identity=unix-user:*
Action=org.freedesktop.upower.suspend
ResultActive=no
ResultInactive=no
ResultAny=no

[Disable suspend (logind)]
Identity=unix-user:*
Action=org.freedesktop.login1.suspend
ResultActive=no
ResultInactive=no
ResultAny=no

[Disable suspend when others are logged in (logind)]
Identity=unix-user:*
Action=org.freedesktop.login1.suspend-multiple-sessions
ResultActive=no
ResultInactive=no
ResultAny=no

[Disable hibernate (upower)]
Identity=unix-user:*
Action=org.freedesktop.upower.hibernate
ResultActive=no
ResultInactive=no
ResultAny=no

[Disable hibernate (logind)]
Identity=unix-user:*
Action=org.freedesktop.login1.hibernate
ResultActive=no
ResultInactive=no
ResultAny=no

[Disable hibernate when others are logged in (logind)]
Identity=unix-user:*
Action=org.freedesktop.login1.hibernate-multiple-sessions
ResultActive=no
ResultInactive=no
ResultAny=no

Read that thread and followed GNUser's advice in installing lightdm and libpolkit-backend-1-0-elogind and libpolkit-gobject-1-0-elogind. I'm still getting suspend and hibernate in the shut down menu. Did I forget to do something else?

BTW, last night I removed hibernate & uswsusp as MiyoLinux wrote. Then I looked up upower in Synaptic to see what would happen if I deleted that also. It would have removed the Mate desktop as well! That doesn't seem right. And I still had suspend and hibernate in the shut down menu.

Also, this reminded me of another thing. In the Mate control center, I don't have any link to Power Management. I downloaded the mate-power-manager-common app, but still no dice. I do remember after installing it some kind of error message popped up which unfortunately I didn't write down.

I have a feeling that all of this is somehow related. Anyone have any advice?

Thanks MiyoLinux. I already had the "ugly" installed but I needed the "bad." So it's now working. If you can remember how you found out it needed both these plugins, I'd appreciate knowing. I searched last night and came up empty. I did look at the dependencies listed in Synaptic, and I had them all. But gstreamer1.0-plugins-bad wasn't listed in Synaptic. Why would that be? It lists gstreamer1.0-plugins-base & gstreamer1.0-plugins-good. (Is there a gstreamer package that includes all this stuff so a user wouldn't have to download a bunch of different ones?

Thanks again.

Yes, that's it. It doesn't work on my system, on either Jessie or ASCII.

Never mind about the login sound. I found out how to fix it. I just changed paplay to aplay. Lucky guess!