Problem 5 (3 marks). The following is the standard formulation of the utility maximi- sation problem (UMP) in classic demand theory. There are n goods in the economy. A bundle of goods is x = (x1, . .., X), where x; 2 0 for all i = 1, ..., n. Each good i is sold in a competitive market at price p; > 0. A consumer has preferences represented by a continuous utility function u: R" - R. The consumer is endowed with wealth w > 0 and maximises her utility subject to a budget constraint: max u(x) subject to Epixi S w. TER' i=1 (a) Show that the UMP has an optimal solution. (b) What can go wrong when p; = 0 for some i = 1, . .., n? (c) Go back to the case where all prices are strictly positive. Suppose that good 1 represents a 'necessity' (like toilet paper). Given the current pandemic situation, the government has imposed a limit on the amount of good 1 that can be bought. Each consumer can spend at most half her wealth on good 1. Argue whether or not the UMP has an optimal solution in this case