Question

W 2 Assume you have the following von Neumann-Morgenstern utility function: U(w) = 200 – (12 – 1000) where w denotes wealth measured in pounds. Your friend believes that it will rain in Reading tomorrow while you think that there is a 75% probability that it will not rain. You are going to take a bet. Your friend will pay you x pounds if it does not rain and you pay your friend x pounds if it rains. Your current wealth is £6000, and therefore the maximum bet is £6000. a. According to your beliefs and expected utilities, what is the better choice, don't bet or bet £2000? (5 marks) b. What is the optimal amount that you should bet in order to maximise your expected utility? and what is the maximum expected utility? (10 marks)

Answer 1

Nat lain Rain Probability- 15% or 25 25701 a) of 9 bit faovo, Expected utility = Expectang 12 - 2000 4 1 1 +1 1200-/12-400071 41 [ loud) - 2 [2002 (1-)*7 +4 200-12-1997 = { [amo-16+4 (02-647 = 3 x 184 + 1 X 136 = 138 + 34 = 172 Original utility = 200 - (12-6000 12 = 200- (12-6)2 1000 = 200-36=164 Since expicted utility of the bet is greater than the utility without bet, hence betting £ 200 will be a better choice. b) of het a be the optimal amount of bet. Elu)=3/200-/12-6000+x) 27+ 1 200-/12-6000-u 1000 E 150-2 (1000 -x) P + 50 - 4( 12000-600+ x)? = 20 - (bouw )- sorento) ligerentiating E(u) with respect to x . 9 no JUD 7 (10-) Giowe) - (opment *)(ibu) = 0 3 a 1600o-n loud X 1070 ) = 116000 tn Toor X 100 )

3(6000-x) = 6000 + 18000 – 370 = 6000 + 2 18000 - 6000 =404 12000 = 400 - = 12000 z 3000 y optimal bet amount = 3000 600D +3067 loco 6000 - 2012 1000 Elu) = 7(200- (12– 600 42000 ) 7+420vec (12- saisons)}} 200-9]+4 [200- 81] = 143.25 +29.75 = 173 :Maximum expected utility = 173