This really depends on your contract with them and how its worded. At cost or for a budgeted price are different. Contrary to what a couple others are saying here overhead can be included as cost as it is something you would have to cover and can be captured. You would also most likely be able to get industry standard admin fees (part of overhead) and pay for your time. Although debatable, net profit would most likely be everything above that and you may or may not be able to say you deserve a little profit on the construction as well above the overhead. You getting pay for your time is not profit. Imagine if as a group you just hired a different GC. The GC isn't running a charity and neither should you be.
So if you told them or had a contract with them to do it at cost (without any stipulation as to what cost truly is) you would probably be able to argue that you should get the actual true expenses plus some admin fees and your time on the job without profit. Yes your time is a cost. Assuming you were all equal outside of the construction you would most likely get an equal split on the profit of the deal and enough to cover the costs plus normal fees. This really isn't fair to you since you truly should get profit on the construction side but at cost would mean you didn't agree to profit so you would be out.
If your contract said an agreed budget and you didn't go over and have to worry about change orders like it seemed in your first post you would most likely get the entire budget plus your 1/3 of the profit on deal.
It would not be an equal deal if you put in the same money and ran the project but only received the same compensation.
Running the project is worth something so if its not paid in the construction contract then you would be able to fairly say you should get more than 1/3 of the deal.
OF course you can lose any court case even if the paperwork is completely solid if the other side can prove you deceived them or any of a million other things.
Basically I agree with @Matthew Olszak I think he's spot on, as well as @Jay Hinrichs as no one will win now.
@J Scott I agree with for the most part as it does seem deceptive to some degree (could be an easy misunderstanding). I do however think a contractor that does work on an equity deal deserves more than an equal percent of the deal (assuming money is the same) they have resources they are using that they could be using to profit on another job so they are losing money by doing the flip if they don't make profit on that side. Same concept as people that use their own money of flips that don't account for what their money could be making just invested somewhere else. Also not mentioned is what happens when someone has a warrantee claim. I bet the investors aren't going to help you then. But you will be on the hook. At least in many states you would be for 12 months. Often not a problem but it is your problem when it happens.
Really like J and some others are hinting at, just show your books to them. Be reasonable with it and argue that you deserve a little extra plus your time. If you do a deal like that again make it clear that you will get cost plus a percent, and make sure your hourly time is spelled out in the agreement and you will work x amount of hours a week managing it.
The issue with a fixed price in these scenarios is that if you don't feel comfortable with change orders then you get screwed. So Time and Material work better if you give them a rom and set your fees in advance. Only charge for the paperwork per change order don't profit on them (you could).
