Need a moderate hotel in LA central to Studios, shopping, Beverly Hills, Hollywood.

Your title is misleading, since it would seem to indicate the center of LA or downtown, which is NOT want you want, apparently.
How much do you want to spend? All the Beverly Hills places will be convenient, but I suspect you won't consider them "moderate". The big newish Renaissance right by the Hollywood Walk of Fame is really "central" to what you mention. There are often really good "deals" or packages there.
A lot of people are recommending The Farmer's Daughter, a sort of strip motel turned into a cutesy B&B, which is right by Farmer's Market.

You have already received some good suggestions in your other thread. Really, the West Hollywood/Beverly Hills areas will be most centrally-located (although the Holiday Inn Bel-Air isn't a bad choice either) to your points of interest. However-- what do you consider "moderate"? To me, that's $90-150 in LA Metro. How about for you?

The old Biltmore Hotel is the most central I can think of (it's basically located where the dot on the map is for Los Angeles). But it's not moderate or particularly close to the things you are mentioning.

We stayed during new year's at the Beverly Hilton hotel. The rooms are all remodeled and it was a great price for Beverly Hills at $155/night. We were very happy and thought the location was great.