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, ...,*n), where x; 2 0 for all i = 1,. .., n. Each good i is sold in a competitive market at price pi > 0. A consumer has preferences represented by a continuous utility function u : IR" - IR. The consumer is endowed with wealth w > 0 and maximises her utility subject to a budget constraint: max u(x) subject to [pixi S w. xER' 1 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 iceberg lettuce). Given the current crisis 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. Show all X Set 3 .pdf