Imagine a world with two goods: good x and a cash m. There are 1,000 identical consumers in this economy, each with an income of y and all having the following preferences: UJ-['X#}'TI'-] = ].J.DDDXJ_ lxzi'f m". Suppose that the price of the good x is p and that the price of cash mis 1. a) Mathematically calculate the Walrasian demand, x,(p,y), from a consumer i? Assume an interior solution (i.e. that given by the first-order conditions). b) Explain in words in which case(s) we would not have an internal solution? c) Calculate mathematically, the quantity of good x consumed in such a case? d) Mathematically calculate the market demand, X"{p] for the good x? Mow suppose that the good x is produced by an industrial sector with firms all having identical technology. The cost function of each of these firms is: c(q;) = g + 400 Let us first assume that the sector is made up of firms in pure and perfect competition. e) Mathematically calculate the long-term equilibrium price on the market? f) Calculate mathematically the total quantity produced and consumed? g) Calculate mathematically the number of firms there are in this industrial sector? Mow suppose that the market is made up of only one firm which behaves as a monopolist. h) Calculate mathematically the price charged by this firm? i) Calculate mathematically the total quantity produced and consumed? Mow suppose that the market consists only of n firms that compete with Cournot. j) Calculate mathematically the price charged by these firms? k) Calculate mathematically the total quantity produced and consumed? ) If we assume free entry and exit from this oligopolistic industrial sector, calculate mathematically the number of firms, the price and total production at long-term equilibrium