If u can draw the tag( deer is draw only ) and it doesn't hurt your pocket then he'll yeah why not. If your not seeing lots of elk but some great bucks then change gears and belly crawl up to a great mulie!
I went to Colorado a couple years ago with just a deer tag in my pocket. Wish I had bought an elk tag because we saw a lot of elk, one being a really good bull. I wont make that mistake again. When opportunity knocks I want to make sure I'm ready with a tag in my pocket. So if you can afford the extra tag fee then by all means go for it.
The way I see It if you have only an elk tag you'll see big deer if you have only a deer tag you'll see big elk. If you have both this ensures you won't see any legal animal to take and will be plagued with bears and spikes Ha ha.
In all seriousness I say go for both that way you can be an opportunist and good luck