Question 1 Consider an individual with the quasi-linear utility function, U = In(x) + y. Assume an interior solution to her utility maximization problem and answer the following questions: (i) Derive her Marshallian demand functions for x and y. 4 Marks] (i) Derive her Hicksian demand functions for x and y. 4 Marks] (iii) Using your answers in (i) and (ii), verify that the Slutsky equation holds for good vy. [4 Marks] (iv) Verify that Roy's identity holds. 4 Marks] (v) Without solving his dual problem (i.e., minimising expenditure subject to a given utility), find the expenditure function and show that it is homogenous of degree of 1 in prices. [4 Marks] Consider a Cournot duopoly with the standard linear demand curve P=a - b@ where Q is a market supply Pis market price and a and b are the standard intercept and slope coefficients. Let be the marginal cost of production and let fixed costs be zero a) Obtain the Cournot (Mash) equilibrium quantities for each firm, profits and the market price [5 Marks]. b) Now assume that the two firms decide to collude. Obtain the equilibrium guantities price and profits, and compare them to your answers in a). [4Marks] c) s the equilibrium in b) a Nash Equilibrium? Why or why not? Demonstrate [4 Marks] d) Suppose the market (Cournot) game above is played sequentially, with firm 1 as the leader. Assume that firm 2 observes firm 1's move. Obtain the optimal outputs and profits for the two firms. Compare your findings to (a) above and comment. [7Marks] Question three Consider the general case of n firms. Suppose the firms compete a /a Bertrand. The market demand is 0(p)=1-p and marginal costs are equal and constant ( =c ). Assume that the firms use a grim trigger strategy. (a) Write down the trigger strategy, and derive the condition on the discount factor which must be satisfied for the price p 2(c. p\"), where p\" is the monopoly price, to be sustained in equilibrium when the game is infinitely repeated. [BMarks] (b) How does the number of firms affect the possibility of sustaining collusion in equilibrium? [4 Marks] (c) Explain how the following factors affect the sustainability of collusion: (i) detection lags, (i) product differentiation and (iii) capacity constraints. [8Marks]