Question 3. [25 total points) It is the morning and Harold has just arrived at the office. Before starting work he contemplates what activity he plans to do this evening after he returns from the office and before he goes out to dinner. He may choose to work out in his local gym, to read a literary prize-winning) novel, or to watch (trashy) TV. In the morning when he is full of vigor and good intentions, he prefers to work-out in the gym to reading his novel which in turn he prefers to watching TV. But in the evening after having spent a hard day at the office, slumping in front of the TV is more tempting than reading the novel which in turn is more tempting than the work-out in the gym. To avoid such temptations, he may try to restrict the options available to him in the evening. For example, he might solicit a ride from a colleague at work who is also going to the gym, or he might tell his room mate to hide the TV remote. In lieu of taking any preventive measures such as these, he may exercise or he may read the book, but only by incurring the self-control cost required to resist the temptation to watch trashy TV. To model, Harold's situation let us employ Gul and Pesendorfer's (2001) framework of costly self-control preferences which are defined over menus of alternatives. That is, first let us take X := { gym, book, TV } to be the universe" of possible evening activities for Harold. His preference relation is defined over the seven non-empty subsets of X: {gym, book, TV}, {gym, book}, {gym, TV}, {book, TV}, {gym}, {book}, {TV}. Let u: X + R and v: X + R denote respectively, Harold's commitment utility and his temptation utility that he associates with each alternative in X. Their values are given in the following table: utility Alt. u(1) () gym 12 0 book 8 1 TV 0 8 1: u V (a) (10 marks) Rank the seven menus from most to least preferred and for each of the seven menus identify which alternative Harold will choose. (6) (5 marks] Show that his choice of alternative from each of the seven possible menus satisfies choice coherence Question 3. [25 total points) It is the morning and Harold has just arrived at the office. Before starting work he contemplates what activity he plans to do this evening after he returns from the office and before he goes out to dinner. He may choose to work out in his local gym, to read a literary prize-winning) novel, or to watch (trashy) TV. In the morning when he is full of vigor and good intentions, he prefers to work-out in the gym to reading his novel which in turn he prefers to watching TV. But in the evening after having spent a hard day at the office, slumping in front of the TV is more tempting than reading the novel which in turn is more tempting than the work-out in the gym. To avoid such temptations, he may try to restrict the options available to him in the evening. For example, he might solicit a ride from a colleague at work who is also going to the gym, or he might tell his room mate to hide the TV remote. In lieu of taking any preventive measures such as these, he may exercise or he may read the book, but only by incurring the self-control cost required to resist the temptation to watch trashy TV. To model, Harold's situation let us employ Gul and Pesendorfer's (2001) framework of costly self-control preferences which are defined over menus of alternatives. That is, first let us take X := { gym, book, TV } to be the universe" of possible evening activities for Harold. His preference relation is defined over the seven non-empty subsets of X: {gym, book, TV}, {gym, book}, {gym, TV}, {book, TV}, {gym}, {book}, {TV}. Let u: X + R and v: X + R denote respectively, Harold's commitment utility and his temptation utility that he associates with each alternative in X. Their values are given in the following table: utility Alt. u(1) () gym 12 0 book 8 1 TV 0 8 1: u V (a) (10 marks) Rank the seven menus from most to least preferred and for each of the seven menus identify which alternative Harold will choose. (6) (5 marks] Show that his choice of alternative from each of the seven possible menus satisfies choice coherence