QUESTION 3 Throughout this question X], Xys ..., Xmy ... will be a sequence of random variables that are defined on the same sample space !. (a) Explain what is meant by each of the following, stating which (if any) implies the other: (i) X., converges to X in L', (ii) X. converges to X in distribution. [6 marks] (b) Assume, in addition, that X1, X2, .... X., ... are independent and identically dis- tributed, with each having finite mean = p. Thus, as a consequence of the weak law of large numbers, the random variable M. = (X, + X2 + ... + X,) converges in dis- tribution to u as n + co. Use this fact to sketch the graph of lim, + V.(x), where V. is the distribution function of M.. [5 marks] (c) For the remainder of this question X1, X2, ..., Na, .. . will be a sequence of independent, identically distributed, exponential random variables with parameter A > 0. That is, each X, has density function f(x) = de As if x 2 0 and zero otherwise. If F. denotes the distribution function of the sum X, + X2 + ... + X,, do the following: (i) Fix r E R. By conditioning the event {Xi + X2+ ... + Xn+1 5 x} by the values of Xmel, or otherwise, show that Fat(x) := P[X, + X2+ ... + Xn+ Sx] = / F.(x - v)f(u)dy: [6 marks] (ii) Use the change of variable u(y) = (x -y) to rewrite the formula for F.+1(r) given in part (i) ; [3 marks] (iii) Prove by induction on n that F.(x) = [1 - eds Er , (Ar)/j!] if x 2 0 and zero otherwise. Hint: For the induction step, use the formula for F.+1(x) that you obtained in part (ii) ; [8 marks] (iv) Explain how the function . (defined in part (b)) is related to F, and, hence using part (b), determine lim, . "Am Ey , (Anr)'/j!, for all non-negative r # 1/1. [5 marks]5. Indifference curves. Assume that the consumer purchases only two goods, x and y. Based on the information in each of the following parts, sketch a plausible set of indifference curves (that is, draw at least two curves, and indicate the direction of higher utility). (a) Nomis enjoys muffins (x) and chai latte (y) if they are consumed together. (b) Tra likes chocolate (x), but he hates broccoli (y). (c) Aras likes steaks (x), but she doesn't care one way or the other about apples (y). (d) Fernando always buys five Polo-shirts (x) with every pair of jeans (y). (e) Paolo is happiest when he consumes exactly two brownies (x) and four glasses of milk (y); any more or less is just not as good.four players, who gets the highest equilibrium utility? 3. Consider an ultimatum game where two players can share a "cake" of size one. Let (x,y) denote a proposal, where x is the proposer's share and y the responder's share. The cake is perfectly divisible, so a and y can be any non-negative real numbers such that a + y = 1. We will consider the effect of different preferences. In parts a and b, these preferences are commonly known: each player is sure about the other player's preferences. In part c, we consider the effect of uncertainty about the opponent's preferences. We always assume, for simplicity, that if the responder is indifferent between accepting and rejecting, then she accepts. Note that if the responder rejects, both players get utility 0. (a) Suppose they have altruistic preferences. Specifically, the proposer's utility function is up(x,y) = vi + vy and the responders utility function is up(r,y) = = + y. Use backward induction to find the subgame perfect Nash equilibrium. For example, suppose each player i keeps 7 and contributes 2; =3 dollars. The total contribution is c =4x3 = 12, so the public goods level is y =24. This $24 is distributed equally among the four players, i.e., $6 each. Thus, in this case each player i earns r, = # + (10 -3) = 13 dollars. For example, if on = 0.1 then player I's utility function is an + 0.1 (12 + 23 + 24). If each player i keeps 7 and contributes z, =3 dollars, then player I's utility is 13 + 0.1 x (13 + 13 + 13) = 16.9. (b) Again, the proposer's utility function is up(r, y) = vr + : vy. But now the responder is envious: her utility function is up(r, y) = 3/y - VI. Use backward induction to find the subgame perfect Nash equilibrium. (c) Now suppose the proposer is selfish, with the utility function up(x, y) = r. But he is uncertain about the responder's utility function. He thinks there is a probability q that the responder also is selfish, up(r,y) = y. But there is a probability 1 - q that the responder is envious, with utility function up(r, y) = 3 /y - vx (as in part (b)). The proposer is an expected utility maximizer. Explain exactly how his optimal proposal depends on q. Is there q such that the proposer is indifferent between two different proposals? If so, what q ?dow Help 100% Tue 10:10 ds x L Refunds | x Homewor X Wisconsin x WisconsinEco 11 02/assignments/1938646 tudentprinting.U. air Rodan + Fields Lif... I Public Health and. Sign In TIVING WOIR IT'S Instructions: Answer the following questions as completely and concisely as possible. Students must submit electronic versions of their answers via Canvas. Electronic versions can be created by either scanning hand-written answers and converting them to a .pdf file or by taking pictures of hand-written answers and combining the picture files into a single .pdf file. Review the syllabus for other suggestions on how to create .pdf files. Consult the course schedule or Canvas calendar for the due date/time. Late submissions will not be accepted. Each question is worth six points each, giving a total of 60 points. Earn as many of them as possible. 1. An early freeze in California sours the lemon crop. Explain what happens to consumer surplus in the market for lemons. Explain what happens to consumer surplus in the market for lemonade. Illustrate your answers with diagrams. 2. Suppose the demand for French bread rises. Explain what happens to producer surplus in the market for French bread. Explain what happens to producer surplus in the market for flour. Illustrate your answers with diagrams. 3. Suppose a technological advance reduces the cost of making computers. a. Draw a supply-demand diagram to show what happens to price, quantity, consumer surplus, and producer surplus in the market for computers. b. Computers and typewriters are substitutes. Use a supply-demand diagram to a show what happens to price, quantity, consumer surplus, and producer surplus in the market for typewriters. Should typewriter producers be happy or sad about the technological advance in computers? c. Computers and software are complements. Draw a supply-demand diagram to show what happens to price, quantity, consumer surplus and producer surplus in the market for software. Should software producers be happy or sad about the technological advance in computers? D A X