Consider an energy market with exactly two producers, i = 1; 2 who decide individually their nonnegative, production levels qi based on a production cost function ci(qi), and production capacities qmaxi > 0. In addition, the market price x is determined by market-clearing conditions balancing supply and demand where the demand function is given by D(x) = a - bx where a; b are two positive constants.
a. Assume that these producers have no market power. Write down their two prof-maximization problems and the market-clearing conditions that determine x.
b. Under what conditions will the resulting Karush-Kuhn-Tucker conditions for these two producer optimization problems be necessary and/or sufficient? Provide mathematical justication.