Two consulting companies reached out to me for the same job position. I agreed to the contract pay and signed the exclusive right to interview with consulting company A, but later I learned that company B would offer higher pay. After interviewing with company B, I think I will get the offer. Should I renegotiate the rate with company A that I interviewed to first?
Simple answer to your question: No, you should not renegotiate - at least not yet.
You reached an agreement, you gave your word and probably your signature an you should honor it. Take it as a learning for you the next time when it is appropriate to (re-) negotiate. Start working and get to know the client. You´ll have a solid foundation and with the added knowledge of your market worth will probably succeed if you renegotiate in some month.
Part of why older Professionals tend to earn more, is because they made the exact same ... experiences as you just did, and learned from them.
Also try to see the other side: Maybe the first consulting company got the client through lower rates, which they where able to provide because the also offered you lower rates.
Disclaimer: Not a Lawyer.
I think you said it yourself there:
I agreed the contract rate and signed the exclusive right to present with the first consulting company
Unless your contract allows for a 'cooling-off' period, or probationary clause, whether or not you signed a Non-Compete depending on your location (as some countries, States in the US don't enforce non-competes), you might be SOL.
Now, agreeing to interview is different from agreeing to be employed. Unfortunately, I am not well versed in this distinction. But from what I understand, the contract has you interviewing with and only with Company A for a set amount of time where you will not interview with any other company. You broke this agreement by interviewing with Company B and learning that you will receive a higher wage. If you use this information with Company A, it would be clear that you broke the agreement! If you don't state that this was an offer based off of Company B, this is more an ethics question than anything else, but considering that you've already broken the agreement with Company A to no interview with other companies. This area is already grey enough.
Instead, approach this situation beyond money, ask yourself "what is your reputation worth to you"? If Company B's offer is X+Y, then clearly your reputation is worth Y. I am often told that the world is a small place and some industries even more so. If you develop a reputation where you are willing to break agreements for more money, what's to say that your next company will hear this through the grapevine and decline to offer you a job?
Your question is a little vague, but I'll try to answer it as best as I can.
1 - Obviously the most important thing here is whether you are bound by law to fulfill whatever contractural obligation you signed with the first company. Basically are you allowed to just leave.
2 - If you are able to just leave, (once again I am not sure of your exact position with the contract you signed) then just ask the first company if they can match or make a better offer than the second company. If they say no, then leave, business is business there is nothing wrong at all with taking a better offer.
3 - If you aren't able to leave, there is nothing wrong with simply asking for a better rate from the first company, if they say no, well, you signed the contract, not much else you can do. I can't see why they would be offended by you just asking for a better rate.
One thing that is extremely important here, which you didn't mention in the above post, is TIME. Did you sign the contract with the first company 5 minutes before you noticed the offer from the second. Or was it 5 weeks, there is quite a major difference here. It would be unfair to ask the first company for more money weeks after they gave you the contract. And I could definitely see them getting a little offended over that.