4. The market for crude oil has two producers - A and B - only. The preceding sentence and what follows is common knowledge to both producers. Each producer decides how much to produce and bring to the market, taking the quantity chosen by the other producer as a given, and seeks to maximize her own profit. Let and denote the quantities chosen by A and B, respectively. Producer A faces the cost function ( ) = + while Producer B faces the cost function () = + The market price P is determined by the total quantity of oil = + that the two producers bring to the market and is given by = where A and K are positive constants. (a) Characterize each producer's profit as a function of her own quantity, the other producer's quantity and the parameters. (b) Characterize each producer's optimal production as a function of the other producer's quantity and the parameters. (c) Determine the equilibrium production of the two producers as a function of the parameters only. Also determine the equilibrium profits of the two producers. (d) How does the equilibrium production of the two producers vary with the parameters?