A consumer has quasilinear utility u(m, x1, x2, , x11) = m + 12051, x2, , x11), where: x,- is the quantity of good i, m is money (left over after paying for goods 1, , n), and v is a differentiable strictly concave function (this just means that if the optimum is interior, you can nd it by taking rst-order conditions). She has income I, and faces prices p1, p2, , p11. a) Find the rst-order conditions that determine how much of each good is consumed, assuming that the solution is interior. b) What condition on I is needed for an interior solution? [Hintz m cannot be negative at the consumer's optimum] Express your answer in terms of 1, p1, p2, , p11 and the quantities from the optimum determined in part a, which you may denote x2, x2, , 2:13;. For the remainder of this question, assume that the solution is always interior, so that your answer from part a is valid. c) Find %. [Hintz Do your rst-order conditions depend on 1?] Are goods 1, , 11 normal, inferior, or neither? Suppose the price of good 1 increases from p1 to p{, and let the quantities consumed under the new prices be x1 ,x2 , ... 26'\" Consider the following claims: i. The amount that you need to pay the consumer to compensate her for the price change, denoted y, is determined by the equation: (I - P117; - - :9an + 17061". 965. --- x9) = (I + Zy pix;r_ ' _ pnxn + 17051 1x2 I m 3530)- ii. The amount that the consumer would pay to avoid the price change, denoted z, is determined by the equation: (I z p1x2' 211195}; + 170:2, x2, ..., 2:12)) = I ilk! (I p1x1 pnxn + 17051 ,x2 ,...,x1';')). d) Explain why the claims are true by stating what each side of each equation represents, and by using your answer to part c. [Hint Note that x2, x2, ... ,x11 and x1 ,x2 , ... ,x' were obtained for an income of I (and not I + y or I z). This is where part c comes in.] e) Solve for y and z. Show that they are equal