3. (a) Consider a market for homogeneous goods with the price of the product given by P=aX,(a>0), where X is the industry output. There are only two firms, 1 and 2 . The cost of production for the i-th firm is Cl(xi)=cxl,(a>c>0), where xi is output of the i-th firm. Now consider the following three stage game. Stage 1: Both firms simultaneously choose whether to delegate (D) or not (ND). In case of delegation a manager is hired. There is a fixed cost of hiring a manager given by Z which is the same for both firms. Stage 2: The incentive contract that is offered in case of delegation is to make the manager of firm i maximise the following expression while choosing quantity: maxxiii+(1i)Si where i and Si are respectively profit and sale revenue of firm i, and i[0,1]. Stage 3: the quantities are chosen by the manager(s) or owner(s) simultaneously. (i) Determine the range of values of Z for which both firms will choose delegation. What would be the optimal 1 s for both firms? (ii) Suppose only one firm delegates and the other one does not delegate. What would be the value of 1 and output levels in that case? Can there exist a range of Z for which only one firm delegating is an equilibrium? (iii) Find the condition under which both firms choose not to delegate