Question I In Machina {1987) it is demonstrated that the VNM expected utility lnction is linear in probabilities. (a) Show that ifxl > x; > it} then the indifference curves in a Machina Triangle are negatively slopped and linear. {6) (b) Draw the triangle and show the direction of increasing preferences. {4) (c) Discuss the main weaknesses of the expected utility theory (BUT). Also discuss the Prospect Theory and why it has been suggested as an alternative to the BUT. (IS) QuEStion 2 (a) With the aid of diagrams and some calculations, show that diminishing marginal utility of wealth implies risk aversion. (10) {13} Use the variance-covariance matrix below to calculate the variance of an equally weighted portfolio. (5) 0'\" 0'... 0'13 24 10 9 or. : 10 1'5 3 23 011 012 0'3. 9 3 12 J (c) The table below shows the probability distributions for risky payoffs from alternatives X and Z. Use the information in the table to answer questions (1') and (if). (i) (ii) X Probability Z Probability -10 0.1 2 0.2 5 0.4 3 0.5 10 0.3 4 0.2 12 0.2 30 0.1 If the only available choice is 100% of your wealth in X or 100% in Z and you choose on the basis of mean and variance, which asset is preferred? (4) Use second-order stochastic dominance criterion to determine whether one should choose X over Z. (6)