11. a. Suppose David spends his income M on goods x1 and x2, which are priced p1 and p2, respectively. Davids preference is given by the utility function( 1, 2) = ? 1 + ? 2.(i) Derive the Marshallian (ordinary) demand functions for x1 and x2. (25 marks) (ii) Show that the sum of all income and (own and cross) price elasticity of demandfor x1 is equal to zero. (25 marks) b. For Jimmy both current and future consumption are normal goods. He has strictly convex and strictly monotonic preferences. The initial real interest rate is positive. If the real interest rate falls, in each of the following cases, argue what will happen to his period 2 consumption level? Clearly illustrate your argument on a graph.(i) He is initially a borrower. (25 marks)(ii) He is initially a lender. (25 marks)12. a. Peter is contemplating to sell a risky asset that gives £625 with probability 0.60 and £100 with probability 0.40. Peters utility from income/wealth is given by = ? where y refers to his income/wealth.(i) If Peter currently does not have insurance for the risky asset, how much will he be willing to accept to sell this asset? Also, illustrate your answer graphically.(20 marks)(ii) If Peter currently has an insurance with full cover from a perfectly competitive insurance market, how much will he be willing to accept to sell the asset? Explain your answer. (25 marks)b. (i) For Jimmy both current and future consumption are normal goods. He has strictly convex and strictly monotonic preferences. The initial real interest rate is positive. He is initially a lender. If the real interest rate rises, using the substitution and income effects, argue what will happen to his period 1 consumption level? Clearly illustrate your argument on a graph. (25 marks) (ii) Mr. Kandle will live for only two periods. In the first period he will earn £101,000. In the second period he will earn £44,000. Mr. Kandle has a utility function ( 1, 2) = 12 2, where c1 and c2 are his period 1 and 2 consumption levels, respectively. The real interest rate is r = 0.1. What are his optimal consumption levels c1 and c2. Does he borrow or lend? How much? (30 marks)13. Consider a firm that has a production function = ( , ) = . The prices of labour and capital are denoted by and respectively. Assume that = .a. Calculate the firms long run contingent input demand functions. Do you needto assume that 0 < < 0.5 for these functions to be valid? Justify youranswer. (25 marks)b. Denote output price by . Assume that the firm operates in a perfectlycompetitive market and that = = . (i) Find the firms long run total cost function. Derive the firms (long run) profit maximising output supply. Do you need to assume that 0 < < 0.5 for this to be valid? Justify your answer.(ii) From your answer in part (i), derive the firms profit maximising long runlabour and capital demand functions. c. Now suppose that capital is fixed in the short run at and that = = .Calculate the average variable cost function. Verify that the average variable cost curve is upward sloping for a decreasing returns to scale production function. Derive the short run average cost (SAC) function. Determine the shape of the SAC curve for a decreasing returns to scale production function.(30marks)14. Consider the following game. An incumbent decides whether to advertise at a cost or not. This action is observed by a challenger who has the option of entering the market at a cost or staying out. If the challenger stays out of the market, the incumbent firm remains a monopolist. If the challenger enters the market, the two firms compete a la Cournot. Assume that both firms have zero marginal cost. Advertisement increases demand at any given price. The inverse demand functions when the incumbent advertises and when not correspondingly are( )=60?( ) = 48 ?where denotes the total industry output.1. (i) Calculate the profits of each firm when the challenger enters depending onwhether the incumbent advertises and not. (20 marks)(ii) Draw the extensive form representation of the game. (20 marks)2. Suppose = 350. Determine the values of for which the incumbent decides to advertise? (20 marks)3. Determine how the range of values of that you calculated in part (b) would change if, upon entry of the challenger the two firms compete a la Stackelberg where the incumbent is the leader and the challenger is the follower. As before, if the challenger stays out of the market, the incumbent firm remains a monopolist. (40 marks)