So it seems like you have two problems. First, you need to decide how to compensate a contractor with equity given that you already issued all of the equity in your articles of incorporation. Then, once you decide how to pay, your next problem is deciding how much equity to pay to the contractor.
First, since you have already issued all the stock from your articles of incorporation, you will have to issue more equity. This is not necessarily a bad thing, it is pretty natural for a growing company to issue more equity. Then, you will need to decide what type of equity to issue, common stock, stock options, etc. This is totally up to you and your company. Different forms of equity will have various pros and cons and you should decide which one makes the most sense for your company. In general, it is probably a good idea to allow the contractor to earn shares over time. This will give an incentive to the contractor to stay on and do a good job for you as he works to earn more shares.
Next, you need to decide how much equity you will pay the contractor. Like I said above, it is probably best to not just give the contractor a lump sum upfront, but instead let him work his way towards the total amount with which you are comfortable paying him. The amount of equity that you are comfortable paying the contractor will obviously depend on how much work you plan on giving him and how much value he will add, just like any other employment decision. You should also keep in mind at this point that this is probably not the only contractor you will be paying through the startup process, so coming up with a total contractor equity budget would be helpful.
However, your question is a pretty deep one and this is answer is only a general overview. An equity transaction at the startup stage that is not done properly can lead to big financial problems down the road. My suggestion is to hire an attorney that has experience drawing up these types of equity transactions to ensure that you are protected down the road.
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