Please answer b) only.
1. Oskar's preferences over lotteries can be represented by the VNM utility 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 pq then i=1npiU(xi)>i=1nqiU(xi) We want to show that this implies i=1npiV(xi)>i=1nqiV(xi) Substitute in V(xi)=a+bU(xi) 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 xi,Un(xi)=Wn(xi)=i where (1,xi)((1i),x1),(i,xn)) 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 B>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)? 1. Oskar's preferences over lotteries can be represented by the VNM utility 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 pq then i=1npiU(xi)>i=1nqiU(xi) We want to show that this implies i=1npiV(xi)>i=1nqiV(xi) Substitute in V(xi)=a+bU(xi) 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 xi,Un(xi)=Wn(xi)=i where (1,xi)((1i),x1),(i,xn)) 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 B>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)
