answer the following, it micro and please answer if you know how to go about it

Problem 1. Relative and Absolute Risk aversion (6 points) In class we introduced the concepts of relative and absolute risk aversion, but we have not used them. This exercise introduces you to two useful classes of utility functions. 1. Consider the exponential utiliy function - exp (-pc) . Show that it is increasing (u' > 0) and concave (u" 0, that is, as long as the agent is risk-averse. Show that this function has constant absolute risk aversion coefficient ry given by p. (2 points) 2. Consider the power utiliy function . for p # 1. Show that it is increasing (u' > 0) and concave (u" 0. Show that this function has constant relative risk aversion coefficient FR given by p. (2 points) 3. Consider the log utility function In (c) . Show that it is increasing (u' > 0) and concave (u" 0. Show that this function has constant relative risk aversion coefficient ra equal to 1. (in fact, it is possibile to show lim, 1 "-1 = In (c) - you are not required to prove this) (2 points).Question 1: Suppose someone offers you the opportunity to play the following game. This person will toss a fair coin repeatedly until heads appear. If this happens on the very first toss, you get $2. If on the second (that is, tails on the first toss is followed by heads on the second toss), you get $4. And so on; if (n - 1) tails in succession are followed by heads on the nuh toss, you get $2". a ) Calculate the mathematical expectation of your (random variable) winnings W. (b) What sure sum would you be willing to pay in advance to be given the opportunity of participating in this game? (c) Suppose your von Neumann-Morgenstern utility function is vW. Calculate your expected utility. What sure sum, if offered to you instead of the game, would give you the same utility? (d) Suppose your von Neumann-Morgenstern utility function is In W. Calculate your expected utility. What sure sum, if offered to you instead of the game, would give you the same utility? (e) Suppose your von Neumann-Morgenstern utility function is - 1/W. Calculate your expected utility. What sure sum, if offered to you instead of the game, would give you the same utility? (f) What is the coefficient of relative risk aversion for each of the utility functions in (c)-(e) ? Question 2: Consider the following simplified version of "Who Wants To Be A Millionaire". You have reached the $32,000 level. (Any Princeton student should be able to do that.) Now you face a succession of questions, each with two possible answers. If you answer a question correctly, whether because you know the correct answer or because you make a lucky guess, you proceed to the next higher prize level. We call your first question (the one that if correctly answered takes you to the $64,000 level) "the $64,000 question" for short; similar abbreviations apply to the following levels. The prize doubles at each level. If you answer the $64,000 question correctly and so reach the $64,000 level, you face the $128,000 question, and so on to $256,000, and $512,000. If you reach the $512,000 level, you face one final question, and the correct answer to it will win you $1,024,000. At any level, when you get the question that can take you to the next level, you may choose not to answer it, and leave with the prize of the level you have already reached. At any level, if you answer the next question and your answer is wrong, you will leave with only $32,000. At each level, before you have seen the question that can take you to the next level, there is a probability that you know the correct answer. The questions get successively harder, so these probabilities decline from one level to the next. Before you see the $64000 question, the probability that you know the answer is 40 percent. If you answer this question successfully to reach the $64,000 level, and before you see the $128,000 question, the probability that you know the answer to that is 35 percent. Similarly, the probabilities are 30 percent for the $256,000 question, 25 percent for the $512,000 question, and 20 percent for the $1,024,000 question. At each level, once you have seen the question, you will know for sure whether you know the correct answer. Thus there is no possibility that you are confident but wrong, or that you know the correct answer but fail to realize that you know. At each level, after you see the question, if you know the correct answer, of course you will give it. If you don't, you have to decide whether to make a guess, which at any level has a 50 percent chance of being correct, or to walk, that is, leave with the amount of the level you have reached. Remember that if you choose to guess and are lucky, you proceed to the next level, but if you choose to guess and are unlucky, you will have to leave with only $32,000, no matter what level you had reached. Your von Neumann-Morgenstern utility function is log,(W/32000), where W is the amount of dollars of your prize. (Note that logs are to base 2, not 10 or e.) (a ) What are the utility numbers corresponding to the various possible levels of prizes? (b) Find your optimal strategy, namely your plan of action that prescribes, at each level, whether to make a guess if you don't know the answer to the question, or to walk with the prize of the level you have reached. You have to begin at the end (where you have already reached the $512,00 level) and work your way backward. At each stage you will find it useful to draw a mini "decision tree" like the one shown here, with the appropriate probability and utility values filled where ... is shown: Know; Probability = ... Utility = ... Utility = ... Walk Don't know; Right Probability = ... Probability 0.5 Utility = ... Guess Wrong: Probability 0.5 Utility = ... (c) What is your expected utility at the initial situation where you have reached the $32000 level and are about to receive the $64,000 question? What amount of sure dollars would give you the same utility