3. (Housing economy with Gold) This exercise is a modication of our housing economy. Consider a world with a set H consisting of n houses held by an agent (the chief) who is interested only in gold and N merchants who own gold but are 1 interested only in houses. Denote the chief as agent N + 1 and gold as good N +1. Gold is divisible but the houses are not. The initial bundle of the chief is (1,1,...,1,0) containing 0 units of gold and 1 unit of each house. His consumption set contains all bundles of the form (0,0,...,0,m). Each merchant i owns an initial amount mi > 0 of gold (suppose mi > m2 > ... > mn > 0), can consume only one house, and has a strict ordering over the houses >i. a. Define the competitive equilibrium for this economy first in words and then formally. b. Find a competitive equilibria for this economy, if it exists. c. Check if First Welfare Theorem holds for this economy, i.e. whether each competitive equilibrium allocation is Pareto efficient or not. d. Check if Second Welfare Theorem holds for this economy, i.e. if each Pareto efficient allocation can be obtained as a competitive equilibrium allocation by reallo- cating the gold among the merchants