In their famous 'common ratio effect experiment, Kahneman and Tversky found that most people preferred $3000 with certainty over an 80% chance of $4000, but preferred a 20% chance of $4000 over a 25% chance of $3000. These gambles can be written as: Choice 1: A: ($3000, 1) B: ($4000, 0.80; $0, 0.20) Choice 2: C: ($3000, 0.25; $0, 0.75) D: ($4000, 0.20; $0,0.80) A) Prove mathematically that the preference pattern observed by Kahneman and Tversky (A in choice 1, D in choice 2) violates expected utility theory (i.e., show that the preference pattern observed by Kahneman and Tversky cannot hold under expected utility theory for any utility function). B) Show that the following value function can explain the choice pattern observed by Kahneman and Tversky. In your calculations, let e = 0. 75,a = 0. 10, u(x) = x, and u(y) = y. V(x, p; y, 1 - p) = 0[pu(x) + (1 plu(y)] + (1 - 0)[amax{u(x), uly)} + (1 a)min{u(x), u(y)}] (Note: The value of lottery B is calculated as follows. V(4000,0.80; 0, 0.20) = e[0.8(4000) + 0.2(0)] + (1 - 0)[a4000 + (1 - a)0}] To show the choice pattern, you need to plug in the numbers for a and 0, and show that the value of lottery A is greater than the value of lottery B and that the value of lottery D is greater than the value of lottery C):