Cheddar equivalents as far as amount of flavor
Cheese isn't just used for flavor; it generally contributes greatly to the texture, whether via a crispy browned layer on top, or thick goopy texture inside a dish, or even just background richness. If you change the total amount of cheese, even if you have the same amount of cheddar flavor, you won't have the same dish.

So I would suggest that if you like cheddar, just use the quantity that's called for, and replace mild with medium or sharp. You'll get more flavor, and you'll probably like it.

Otherwise, if you want to keep the flavor of the dish roughly the same intensity but replace mild with sharp, then make up the rest with another mild cheese. Obviously it won't be the same flavor but it'll be roughly similar strength; if all you have is sharp cheddar, this will be better than using all sharp cheddar (too strong a flavor) or using half as much (upsetting the balance of the dish). For example, you might replace a cup of mild cheddar with half a cup of sharp cheddar and half a cup of mozzarella.  (See rumtscho's answer for a good explanation of exactly how not the same it will be.)

But the ratio depends on the particular cheddar; you can get a pretty good variety of things marketed as "sharp cheddar". I think the half and half ratio will be a good starting point for nice strong, sharp cheddar, and I personally would just use medium instead of mild without changing any quantities. So experiment with the cheese you have and see what you like. 

And remember, it'll never be the same, it'll just keep the sharp flavor from being overpowering. I'm not saying this is an exact substitute, just what I'd do if I only had sharp cheddar around.

Finally if you're asking this because you're planning on making and aging cheddar yourself... don't get ahead of yourself. See how well you do at the basic cheesemaking before you worry about aging it.