The competitive market of the good x, is populated by n > 0 consumers who have identical income m > 0 and identical preferences over bundles of goods x] and x2 (with prices p > 0 and > 0, respectively). The identical preferences are represented by the utility function u(x], X2) = BInx, + (1 - 8) Inx2, where BE (0,1). The market of the good x, is supplied by N > 0 firms who have the identical technology F(f, k) = ve + vk, where f denotes labour (with price w > 0) and k denotes capital (with price r > 0). None of the firms have a capacity constraint, there is no fixed cost for production, and new entry of a firm into the market is prohibited. a) Derive the individual demand function for x, including the corner solutions (if any). Clearly state the assumptions you used to calculate the individual demand function. (15 marks) b) Calculate the aggregate demand function for x1. Is the price of x2 relevant in the aggregate demand of x1 ? Explain. (5 marks) c) Derive the individual supply of a firm for x1 including the corner solutions (if any). Clearly state the assumptions and the type of profit maximisation problem you used to calculate the individual supply function. Calculate the aggregate supply function for x1. (25 marks) d) Derive the equilibrium price, p*, in the market for x1 . Derive the expressions for the consumer surplus and profits (do not calculate them). (15 marks) e) Using the mathematical model of equilibrium taught in the class, determine the wage (w) elasticity of the equilibrium price for x1, i.e., aw . Determine the sign of the elasticity and interpret. Explain which parameters of the model affect the elasticity and how? (45 marks)