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ECON6001/6701 Microeconomic Analysis 1, S2 2022 Problem Set 6
发布时间：2022-11-22
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Problem Set 6
ECON6001/6701 Microeconomic Analysis 1, S2 2022
Exercise 1. Consider choice under risk. Assume there are three outcomes `; m; h . A lottery is therefore a probability distribution p = (p`; pm; ph). Under EUH, the utility of any lottery can be written as
U (p) = p` v` + pm vm + ph vh:
where v`; vm and vh denote the vNM utilities of the three outcomes. Assume that among the sure outcomes, h > m > `.
1) Fill in the blanks with choices from v`; vm and vh in the following: ”Without loss of generality, we may choose = 0, =1.“
”Moreover, the left-over value vi after filling in the above blanks, will be a number that lies strictly between and .“
2) In class, I have explained why indifference curves on the space of lotteries with three outcomes must be straight lines under EUH. Now draw some indifference curves of a DM in that space. Next, draw indifference curves of another individual DM who is more risk averse.
Exercise 2. Consider a simple portfolio decision, reproduced from the texbook Essential Micreconomics by Riley. The DM must decide how much to invest in a risky asset that has a gross yield of 1 + rs in state s K Money that is left out of the risky asset and carried forward in the form of money yields zero return. She begins with a wealth w. State s is known to occur with probability ps.
Let E[r] = µ and Var(r) = o2. Assume EUH holds.
1) 
Suppose the DM chooses to invest an amount q [0; w] in the risky asset. What would your wealth be in state s? Call this xs.
2) From the previous part then, the utility of choosing q units of investment is
n
U (q) = ps v(xs):
i=1
Assume that the vNM utility takes the form v(x) = a + b x + c x2. What must be the restrictions b; c and the range of x that is required to make him strictly risk averse. Write them down and assume all of these are met. Then show tha t
U (q) = a + b w + c w2 + (b + 2 c w )µ q + c (o2 + µ2) q2
3) Present the sufficient conditions (on µ; o2; b; c) needed for the above to have a unique maximum with respect to q. Call this q×, the demand for risky asset.
4) Do wealthy individuals invest (weakly) more in the risky asset? Could you have deduced this without explicit computation of q×?
Exercise 3. An individual begins with an initial wealth of w and satisfies EUH. Let u( ) denote his vNM utility. Now consider a lottery that pays tt with probability p and B with probability 1 ¡ p.
1) Suppose the individual owns the lottery. What price will he sell this for?
2) Next, suppose the same individual is looking to buy the above lottery. What is the maximum price she would be willing to pay?
3) In general Rb =/ Ra. Why? Can you give an economic interpretation? (It is enough to argue with a simple diagram.
Exercise 4. In the demand for insurance setup in Lec 9 class, we had set up the following maximization problem:
max p v(w ¡ q a ¡ L + a) + (1 ¡ p) v(w ¡ q a)
Let a× denote the optimal choice. Assume throughout that the agent is strictly risk averse. Let a× denote
1) Show that a× < L if q > p and a× > L if q < p.
2) Can you provide an economic reason by which q = p?
3) Without using the above derivation for a×, deduce that a× must be non-decreasing in L. (Assume q 2 (0; 1)).
