2. (20 points) A consumer maximises their utility by choosing how many tshirts to buy and how many limited-edition sweaters to buy. If they buy m t-shirts and n sweaters, then their utility will be u(m, n) = m 1 5 n 4 5 . Each t-shirt costs 1 and each sweater costs p. The consumer aims to spend exactly 150 on these t-shirts and sweaters. Given the limited-edition nature of the sweaters, the store allows each shopper to buy a maximum of 5 sweaters. a) (4 points) Formally write the consumer's maximisation problem. b) (4 points) Write the problem's Lagrangian and first-order condition(s). c) (4 points) Identify the cases from complementary slackness, along with their associated equation and check. d) (6 points) Sketch the consumer's demand function for sweaters as a function of their price p, labelling any key points. Justify your answer mathematically. e) (2 points) Interpret your previous answers with an economic rationalisation