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Suppose the Phillips curve equation is given as:

u= 6 - 1.5 ( - e)

which is equivalent to:

Unemployment rate = Natural rate of unemployment - a (actual inflation - expected inflation)

Based on the above Phillips curve, what is the natural rate of unemployment? Suppose = 6% and e = 2%. What is the actual rate of unemployment Continue from part b) and given than Okun's Law holds which states that 1% increase in unemployment rate reduces output by 2% of the full employment output, how much is the actual GDP compared to the full employment GDP?all is well answer all If the Phillips curve equation is given by u= 7 - 0.5 ( - e), repeat a) b) and c) using this new equation.

substantially modified from number 14 on Problem Set 1) According to Paul Samuelson, the mathematician Stanislaw Ulam defined a coward as someone who will not bet even when you offer him two-to-one odds and let him choose his side. (A gamble with two-to-one odds is one in which the individual wins $2x if an event A occurs and loses $x if A does not occur. Letting the individual choose his side means letting him choose between winning $2x if A occurs and losing $x if A does not occur, or winning $2x if A does not occur and $x if A occurs.) (a) Assuming for simplicity that the events that A occurs and A does not occur are equally likely, and letting initial wealth be y, draw a graph for an expected-utility maximizer who likes money, with a differentiable von Neumann-Morgenstern utility function, and who is a coward according to Ulam's definition. (Put von NeumannMorgenstern utility u(y) on the vertical axis and final wealth y on the horizontal axis.) (b) Use your graph to show that he cwith a piecewise linear value function with a kink at winning or losing 0, and who is a coward according to Ulam's definition. (Put value v(y) on the vertical axis and gains or losses on the horizontal axis.) (d) Use your graph to show that such a Prospect Theory expected-value maximizer is a Ulam-coward for large x if and only if he is a Ulam-coward for small x.

3. (30 points) Consider a Prospect Theory expected-value maximizer with value function defined over gains and losses relative to a reference point 0, defined as no gains or losses: v(x) = {x for x 0 {2x for x < 0. He has some money invested, and each day the value of his investments goes up by $3000 with probability or down by $1000 with probability , and the probability of "up" or "down" on the second day is independent of what happened on the first day. Suppose first that he has the choice of checking his portfolio's performance either at the end of each day, or only at the end of the second day. However, even if he chooses to check at the end of each day, he still cannot change his portfolio after the first day. His expected value is additive across days, so that if he checks at the end of each day, his total expected value equals his expected value from the first day plus his expected value from the second day. But if he checks only at the end of the second day, his total expected value is just his expected value from the sum of both days' outcomes. (That is, he experiences his gains or losses whenever he checks, whether it is at the end of each day or only at the end of both days.) (a) Which will he prefer, to check his portfolio's performance at the end of each day or to check only at the end of the second day? Explain, both algebraically and intuitively. Now suppose that he faces the same choice, but that if he decides to check at the end of each day, he can pull all of his money out of the stock market (the only option) at the end of the first day is he wishes. Further suppose that even if he decides to check at the end of each day, his value is still determined by his total gains or losses over both days, with reference point 0. (b) What would his investment decision be at the end of the first day, if he finds that his stocks have gone up by $3000? Explain, both algebraically and intuitively. (c) What would his investment decision be at the end of the first day, if he finds that his stocks have gone down by $1000? (d) Given the investment decisions in (b) and (c), and assuming that he breaks any ties in his optimal decisions by leaving his money in the investment, which will he prefer, to check at the end of each day or to check only at the end of the second day? 4. (20 points; slightly modified from number 27 on Problem Set 1) Consider the following hypothetical facts: "1% of people in the world population are rational and 99% are irrational. We have a test for rationality. If someone is rational, they have a 60% chance of passing. If someone is irrational, they have a 40% chance of passing. JJ was just given the test, and she passed." (a) Assume that JJ was drawn randomly from the world population. What is the probability that she is rational? Don't bother with the long division. Expressing your answer as a ratio without simplifying it is fine. (b) Predict the responses of a population of normal people who have not studied either probability theory/statistics or behavioral economics, who are asked to estimate the probability that JJ is rational, given the information in the question. Justify your answer. Describe the kinds of errors that they are likely to make.

1. You are risk neutral, and care only about your income. With probability p, you will catch a disease that reduces your income from y, its level when you are healthy, to y - k, where k > 0. A vaccine is available, at cost c, that reduces the probability of your catching the disease from p to q < p. (a) Suppose that you know the values of p, q, y, k, and c, so that the only thing about which you are uncertain is whether you will catch the disease. Write the condition that determines whether or not you should buy the vaccine. (b) Now suppose that you know y, k, and c, but neither p nor q. Which is more relevant to your decision, the percentage amount by which the vaccine reduces the probability of catching the disease (what is usually reported in the press), or the absolute amount? Explain. (c) How do your answers to (a) and (b) change if you are a risk-averse expected-utility maximizer? 2. In the game Former Soviet Union Roulette, a number of bullets are loaded into a revolver with six chambers; an individual then points the revolver at his head, pulls the trigger, and is killed if and only if the revolver goes off. Assume the individual must play this game; that he is an expected-utility maximizer; and that each chamber is equally likely to be in firing position, so if the number of bullets is b his probability of being killed is b/6. Suppose further that the maximum amount he is willing to pay to have one bullet removed from a gun initially containing only one bullet is $x, and the maximum amount he is willing to pay to have one bullet removed from a gun initially containing 4 bullets is $y, where x and y are both finite. Finally, suppose that he prefers more money to less and that he prefers life (even after paying $x or $y) to death. Let UD denote his von Neumann-Morgenstern utility when dead, which is assumed to be independent of how much he paid (as suggested by empirical studies of the demand for money); and let UA0, UAx, and UAy denote his von Neumann-Morgenstern utilities when alive after paying $0, $x, or $y respectively. (a) What restrictions are placed on UD, UA0, UAx, and UAy by the assumption that he prefers more money to less when alive? (b) What restrictions are placed on UD, UA0, UAx, and UAy by the assumption that he prefers life (even after paying $x or $y) to death? (c) Is it possible to tell from the information given above whether x > y for an expected utility maximizer? Does it matter whether he is risk-averse? Explain. 3. Suppose that there are two states of the world, s1 and s2, and that an individual who knows the probabilities, p1 and p2, of the two states chooses among state-contingent consumption bundles to maximize the expectation of a state-independent, strictly increasing von Neumann-Morgenstern utility function. (a) Suppose that the individual is risk-neutral, and that he is indifferent between (8, 2) and (4, 4). What must the value of p1 be? (b) Now suppose that the individual may be either risk-averse or risk-loving. What is the lowest possible value of p1 for which the individual could weakly (or strictly) prefer the state-contingent consumption bundle (6, 2) to the bundle (2, 6)? (c) Now suppose that the individual is risk-averse, and that he is indifferent between (6, 2) and (2, 6). Show (graphically or algebraically) that he must weakly prefer (4, 4) to either of these bundles. 4. Consider an expected utility-maximizing student, who cares only about his income. Cheating on his 142 exam adds a given amount to his income, whether or not he is caught at it. Suppose, however, that a student who is caught cheating is fined a given amount. It is observed that a 1% increase in the probability of being caught lowers the student's expected utility of cheating by more than a 1% increase in the amount of the fine. (a) Is the student a risk-averter or a risk-lover? Explain. 5. An expected utility-maximizing person has von NeumannMorgenstern utility function u(), with u'() > 0, and deterministic initial wealth w. He is just indifferent between losing x > 0 for certain, and losing y > x with probability p > 0 and losing nothing with probability 1 - p. (In other words, x is the most he will pay to be insured against a random loss of y with probability p.) (a) Prove that for any given values of w and y, x is an increasing function of p. (b) Prove that for any given values of w and p, x is an increasing function of y. (c) Prove that if the person is risk-averse, then x > py. (d) How does x vary with w when u(w) a - be-cw with b, c > 0, so that the person has constant absolute risk aversion? (Here, e is the base of natural logarithms.) 6. An individual has initial wealth w and holds a lottery ticket that will be worth zA with probability p and -zB with probability 1 - p. Here, A, B, and z are positive, with pA (1-p)B. Let X be the maximum amount the individual would pay someone to take this ticket off his hands. Prove that if the individual is risk-averse, then X is an increasing function of z.

7. Consider a risk-averse, expected-utility maximizing agent with von NeumannMorgenstern utility function u() and initial wealth y. (a) Show how to determine (by giving an expression that implicitly defines it) the minimum probability of winning, p, needed to get the agent to accept a binary bet in which the outcomes are winning or losing z > 0. (Assume he will accept if indifferent.). (b) Use Taylor's Theorem to derive an approximate expression for p when z is small, and use your expression to show that p is then an increasing function of z. (c) What aspect of the agent's risk preferences determines whether p is an increasing function of y for small z? Explain. 8. A rich uncle gives you a gift certificate that entitles you to an insurance contract, of your own choosing, with an actuarial value (an expected return) of $1 million. The only uncertainty you face us about whether you will be involved in an accident that leaves you paralyzed; this will happen with probability . You may allocate the $1 million in actuarial value however you wish between the two states, paralyzed and not paralyzed. It has been observed that, when faced with this choice, some people choose to receive more money in the paralyzed state, some choose to receive less money in the paralyzed state, and some choose to receive equal amounts in both states. (a) Which of these choices is/are consistent with expected-utility maximization with a strictly concave, differentiable, state-independent von NeumannMorgenstern utility function? Explain. (b) Which of these choices is/are consistent with expected-utility maximization with a strictly concave, differentiable state-dependent von NeumannMorgenstern utility function? Explain. 9. Suppose that there are three money outcomes, x1, x2, and x3, with x1 < x2 < x3, and that you can observe which values of p make a person prefer getting x2 for certain to getting a random outcome {x1 with probability p, x3 with probability (1-p)}. Is this enough to determine a person's preferences over arbitrary probability distributions over x1, x2, and x3: (a) if he is an expected-utility maximizer? Explain. (b) if he chooses among distributions to maximize some differentiable preference function, not necessarily consistent with expected-utility maximization? Explain. 10. An individual chooses among lotteries according to preferences that are complete, transitive, and continuous. He cares only about money, and all the lotteries he faces have the same three possible outcomes, $1, $2, and $3; the probabilities of these outcomes are denoted p1, p2, and p3, respectively. Labeling your diagrams carefully so that I can tell how they were constructed, draw an indifference map in (p1, p3)-space for an individual who: (a) is risk-loving, likes money (always prefers first-order stochastically dominating shifts in the distribution of money outcomes), and satisfies the independence axiom