Reduce Medium-Hair Kitty's Poopy Butt?

I'm asking this on the behalf of my friend's cat - Ellie. My friend who I've seen her cat's um... dirty bum, and its rather unhygenic and nauseating for me to see. I've only met the cat once as I've only been in her house one time before - which was a few months ago. Yet, I want to make it better for her family to reduce the poo stains on Ellie's bum so that way they dont have to constantly shave her down back there. 

Ellie is a medium haired cat - I forget what my friend's mom said the breed was but I know she's a mix. The cat's super duper sweet, and she's declawed (not that we're pro-declawing but thats how she came!) so she's never allowed outside. I'm unaware of what diet she's currently on, I could ask my friend. . . And the chances of changing it to a higher quality food are least likely as they don't have a whole lot of money to go around for things like that. 

So, in an inexpensive way, do you think there's any way to reduce the dirty bum so its easier on the family's fundages? If not, they can put up with it. I was just asking on behalf of my friend. 

It may help to change the diet within the price range they can afford.  Just do that SLOW by adding the new food to the old.  I have a long hair who needs to be shaved as he has stomach issues and poop is not always the best.  Is this cat overweight?  Slimming down may help it maintain a shave if it can properly clean itself.

Many people call any long or medium hair a persian mix just cause persians are popular.

Hiya CAA - thank you for the advice. But, she actually *Ellie* is on a really good food. I looked it up on the website that they get it from. It's Chicken Soup for the Cat Lover's Soul. So you might want to look that up if your worried about the ingredients or anything like that because I saw it and they actually look really good quality. But I'm no cat fancier as I've already said! 

My friend's parents want to get another cat, do you think that poopy but can be a problem health-wise if another cat or kitten enters the house?  

But some cats do better on other diets then others.  Though it is a good food it just may not be the right fit.  I have fed it to some of my cats in past but at this point that is to expensive for the cats I have.

A new cat may work but I would make sure there is a plan to make sure the cat can be spay or neutered if they are tight on funds/vet visit if ever needed.  Also have to take into account a new cat with claws may be to much depending on Ellies temperment.