A consumer has quasilinear utility u(m, X1,X2. ....X,) = m+ V(X1,x2, ..,x,), 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 find it by taking first-order conditions). She has income I, and faces prices p1. P2. ..Pn- a) Find the first-order conditions that determine how much of each good is consumed, assuming that the solution is interior. b) What condition on / is needed for an interior solution? [Hint: m cannot be negative at the consumer's optimum.] Express your answer in terms of I, P1, P2. .... Pn and the quantities from the optimum determined in part a, which you may denote x], x', ...,14. For the remainder of this question, assume that the solution is always interior, so that your answer from part a is valid. c) Find -. [Hint: Do your first-order conditions depend on !?] Are goods 1, ..., n normal, inferior, or neither? Suppose the price of good I increases from p, to p;, and let the quantities consumed under the new prices be x]', x,',...,x,'. 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 -pixi - "-Paxi + v(xj,x), ....xi)) = (1 +y-pix;' - .- pax" + v(x x, ...,x; )). ii. The amount that the consumer would pay to avoid the price change, denoted z, is determined by the equation: (1 - z - pixi - - Paxi + v(xi, xi, ...x;)) = (I - pix, - ..- pax" + v(x,',x,",..x)).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 x], x), ..., x, and x]',x7,..., x, were obtained for an income of I (and not I + y or / - z). This is where part c comes in.] e) Solve for y and z. Show that they are equal. 2. X and Y are random variables. Their joint probability distribution is as follows: X - 0, Y = 0 with probability 0.2 X = 1, Y = 0 with probability 0.1 X - 3, Y - 2 with probability 0.2 X - 1, Y - 2 with probability 0.5 a) Find the expected value of X, denoted E[X]. b) For each possible value of Y, find the expected value of X conditional on that value. That is, find E[XJY=0] and E[XJY-2]- c) Find the probability that Y=0, denoted Pr(Y-0). d) For each possible value of X, find the probability that Y-2 conditional on that value. That is, find Pr(Y -2 X-0), Pr(Y=2 X=1) and Pr(Y-2 X-3). A third random variable, Z, is independent of X and has the following probability distribution: Z -5 with probability 0.6 - Z -7 with probability 0.4 e) Find the joint probability distribution of X and Z. f) Find the expected value of ZX, denoted E[Z* ]. 3. Show that whether an outcome x is Pareto efficient: a) cannot change when a strictly increasing function fi is applied to each agent i's utility over outcomes. b) can change when an increasing function fi is applied to each agent i's utility over outcomes. Hint: For part a, you need an argument, since your answer needs to be valid for all possible functions. You might want to proceed by contradiction: suppose x is Pareto efficient when each agent i's utility is up, but not Pareto efficient when it's f(u;) instead. Then show that this assumption leads to something that is impossible. For part b, you just need an example, since you're only asked to show that something can occur. 4. Consider a perfectly competitive market with demand Q 20-P and supply Q 2P-1 with a per-unit tax of $3. Find the quantity, the price paid by buyers, the price received by sellers, and the deadweight loss