Question Sheet 2 Answers Topics in Microeconomics 1. (a) Consider any two lotteries,  and . If  Â  then  P   ( )  =1  P   ( ) =1 We want to show that this implies  P   ( )  =1  ( +  ( ))  =1   P  +  =1 +  P =1  P =1  P   ( ) =1 Substituting in for  ()  P  P  P  ( +  ( )) =1  P   ( )    +  =1   ( )   +   P   ( ) =1  P   ( ) =1   ( )  =1  P   ( ) as required =1 i. Take away 20 to make the range (0 80) and then divide by 80 to make it (0 1)  () = 1 1  () − 20 = − +  () 80 4 80 ii. Take away 8 to make the range (0 7) and then divide by 7 to make it (0 1)  () = 8 1  () − 8 = − +  () 7 7 7 iii. From continuity, for any  there is a  such that (1  ) ∼ ((1 −  ) 1 ) (   ) and on a scale from 0 to 1, the utility we give to  is  so  ( ) =  ( ) =  1 iv.  () =  () 1 1 8 1 − +  () = − +  () 4 80 7 7 80 8 1  () = 80(− + ) +  ()) 7 4 7 v. In general  () =  ()  +   () =  +   () () =  −   +  ()   (b) If in part a) and only if in part b). 2. This question just requires you to go through the steps with  = 098 to highlight the inconsistency. (a) Substitution (b) Reduction. 0.11x0.98=0.1078 (c) Monotonicity 3. (((1 − ) 1 ) (  )) = (((1 − ) 1 ) ( −   ) (  )) (((1 − ) 1 ) (  )) = (((1 − ) 1 ) ( −  1 ) (  )) as  Â 1 by IIA (((1 − ) 1 ) ( −   ) (  )) Â (((1 − ) 1 ) ( −  1 ) (  )) which means (((1 − ) 1 ) (  )) Â (((1 − ) 1 ) (  )) 2 4. (a) Yes, eg  () = () = 10  () = 9  () = 8  () = 7 (b) Best outcome is £8000 and worst outcome is 0 so they get utilities of 1 and 0 respectively. For £6000 we ask which value  gives the indifference (1 £6000) ∼ (((1 − ) 0) ( £8000)) which we can see from  ∼  is 09 However, the expected utility of  is 02 and the expected utility of  is 0225 so expected utility does not work as we are told that  Â  (c) Before the coin is tossed, there is a 075 chance of getting zero and a 025 chance of choosing between  and . So choosing  gives a 025 chance of getting £6000 and choosing  gives a 025x08 = 02 chance of getting £8000 To relate it to IIA since Â ((075 0) (025 )) Â ((075 0) (025 )) and using reduction on the right hand lottery ((075 0) (025 )) Â ((08 0) (02 £8000))  Â  (d) Explain gut feeling with certainty effect plus similarity of probability of winning prize in  and  but  has a bigger prize. Discussion of prospect theory plus discussion from your own research into alternatives. 3 1. Oskar's preferences over lotteries can be represented by the VNM util- ity function U(x). (a) Show that the utility function V(x) = a +bU (x), b>0, will also represent his preferences over lotteries. (Hint: Consider any two lotteries, p and q. If p > q then ΣPU(T) > QU(T₂) i=1 We want to show that this implies (1) ÊP;V(xi) > Ê giV(x;) i=1 (2) Substitute in V(x₁) = a+bU (x;) into 2 and show that 2 will hold when 1 holds) (b) Assume that his preferences over lotteries can also be represented by the VNM utility function W(x). Suppose also that the utility range for U(x) is (20, 100) (that is U (x₁) = 20 and U(xn) = 100) and for W(x) is (8, 15). i. From part a) a linear transformation of U(x) will represent the same preferences. What linear transformation Un(x) au + buu (x) will give a representation with range (0, 1)? ii. Similarly, what linear transformation Wn(x) = aw+bwW(x) will give a representation with range (0, 1)? iii. Explain why for each x₁, Un(x₁) = Wn(x₁) = αi where (1, i) ~ ((1 − 0i), 21), (@i,n)) iv. Use biii) to show that there is a linear relationship between U(x) and W (x) = v. Use this to prove in general that if there are two functions U(x) and W(x) that represent the same preferences then there exist constants A and > 0 such that W(x) = A + BU (x)(c) Two utility functions represent the same VNM preferences if and only if there is a linear relationship between them. How have we proved this in parts a) and b)? 2. Consider Allais' Paradox from the notes. J K L M = = (1, £16) ((0.01, £0), (0.89, £1b), (0.1, £5b)) ((0.89, £0), (0.11, £1b)) ((0.9, £0), (0.1, £5b)) In a choice between J and K David chooses J and in a choice between L and M he chooses M. He is also asked which value of x makes him indifferent between £1b for sure and the lottery (((1-x), 0), (x, £5b)). After reflecting on this, he decides that it is x = 0.98. So David is indifferent between £1b for sure and the lottery B B = ((0.02, 0), (0.98, £5b)) Use the VNM axioms and the following steps to try and persuade David that he is making a mistake and that he should choose L over M. In each case, be clear about which axiom is being used. (a) Why does rationality imply that David is indifferent between the lottery L and the compound lottery L' L = ((0.89, £0), (0.11, £1b)) ~ ((0.89, £0), (0.11, B)) = L' (b) Why does rationality imply that David is indifferent between the compound lottery L' and the simple lottery L"? L'~L" = (0.8922, £0), (0.1078, £5b)) (c) If David agrees with all this then L~L~L" and comparing L and M should be the same as comparing L" and M. L" M = = (0.8922, £0), (0.1078, £5b)) ((0.9, £0), (0.1, £5b)) Why should David choose L"? 3. *Note: This is a difficult question but the important thing to take away from it is that with expected utility theory, independence of irrelevant alternatives (IIA) holds and as you will see in question 4c it helps us to understand the mechanics of the theory. Recall the substitutionaxiom from the notes which can be written as follows, if you have 3 lotteries I,m and n and m~ n then for any qe[0,1] ((1-q), l), (q, m)) ~ ((1 — q), l), (q, n)) We could also have used a stronger version of this axiom called in- dependence of irrelevant alternatives (IIA) which adds that if m > n then ((1 — q), l), (q, m)) > ((1 − q), l), (q, n)) which is saying that as I is common in both lotteries, the preference between these lotteries should depend only on the preference between n and m. We will be seeing versions of this axiom in bargaining theory and social choice theory. It is easy to see that together with the other axioms, substitution implies IIA (you can just follow the proof in the notes and find the reduced lottery that includes only the best and worst outcomes). Show that if we assume IIA from the outset, then we do not need to assume monotonicity as it follows from IIA. (Hint: You want to show that if a > ß then (((1-B), x1), (B, xn)) C and D > E but he un- derstands the inconsistency. How can you explain his gut feeling. More generally, use prospect theory and/or other alternatives to expected utility theory to discuss the limitations of expected util- ity theory.