If a client decides to buy your product only if you give him/her a discount. However, you don't have the authority to do so. What would you do?

I would try to contact the one who has this authority & request his/her permission to do so, however if the customer was willing to buy a great quantity of the product, I may risk and give the minimal discount that could be granted to make the biggest return that could be made. at the end, selling more would mostly justify bypassing the lines :)