PLEASE DO NUMBER 12-14

Problem 1: MRS and the allocation of leisure John and Maria are siblings and they are both in high school. Their father is concerned about their school performance and has decided to strictly monitor the time that they spend playing videogames on the family computer and chatting on social networks using their shared smart- phone. Thus, he has decided that each of them should spend no more than 1/2 hour per day playing videogames (VG), and 1 hour per day on social networks (SN). He can enforce this policy strictly by careful supervision. Denote by VG and VGM* the number of VG hours \"consumed\" by John and Maria, respectively. Similarly, denote by SN and SN2 their SN consumption. Assume that John and Maria each have their own preferences over the number of hours of VG and SN they spend during their leisure time. 1. Explicitly taking into account their father's policy, what is John and Maria's consumption set over VG and SN? Represent it in a graph with VG on the horizontal axis and SN on the vertical axis. Please draw one graph for John (labeled Figure 1) and another one for Maria (labeled Figure 2). (You will use these graphs later in the problem so make them large enough.) 2. If John and Maria's preferences both satisfy the monotonicity property, how many hours of VG and SN will they consume? Now assume that John's preferences can be represented by the utility function 1 ey SNY = VGISNG while Maria's preferences can be represented by the utility function UMy G SN) = VGISNT . You can take for granted that John and Maria's preferences each satisfy balance in consumption. 3. Do John and Maria have the same preferences over VG and SN? Justify your answer by considering how each of the two siblings feels about the following two (V G, SN) bundles: (%,%]and(%,}l . 4. Compute the marginal rate of substitution of SN for VG for John at his consumption point identified in part 2. What is his utility level at that point? On Figure 1, draw the indifference curve that goes through his consumption point, paying attention to its slope. 5. Compute the marginal rate of substitution of SN for VG for Maria at her consumption point identified in part 2. What is her utility level at that point? On Figure 2, draw the indifference curve that goes through her consumption point, paying attention to its slope. 6. Can it be said that Maria is happier than John under their father's policy? Justify your answer in words. John and Maria's father just got a new job that requires him to be away from home most evenings and weekends. He decides to control how much time his children spend on VG and 5N by using an application that tells him how much ime the family computer has been on cach day, and using parental control on his children's shared smart phone. Unfortunately, he cannot tell who turned the computer on, or who used the smart phone. John and Maria decide to exploit this weakness in his surveillance o make themselves better off. 7 110 What is the maximum total amount of VG that can be consumed by the siblings? (Assume that the siblings do not play VG together, maybe because they like different games.) . What is the maximum total amount of SN that can be consumed by the siblings? (Assume that the siblings do not spend time on SN together, maybe because they have different friends.) . How much SN would John be willing to give up, starting at the initial point identified in guestion 2, for an extra 5 minutes (1/12 of an hour) of VG? (Hint: Give an approximation using the MRSy e for Johind) How much SN would Maria require Lo give up 5 minutes (1/12 of an hour) of VG, starting at the initial point identified in question 27 (Hint: Give an approximation using the MRSsy-ve for Maria.) . Based on your answers o parts 9 and 10, explain in words why John and Maria can exchange VG and SN hours, making themselves better off while still abiding, by their fathers's rule (in aggregate). (Hink: Argue using the monotonicily property.) Mengfei, who has taken graduate microcconomics, tells you that an optimal allocation of VG and SN time between John and Maria (from their point of view, not their father's) would be: 12, Check that if implemented, Mengfei's allocation would not raise the father's suspicion. 13, Calculate the utility levels of John and Maria at this allocation. Represent the new consumption points in Figures 1 and 2. 14, Calculate the marginal rates of substitution of SN for VG for John and Maria at their new consumption points. Do you agree that the new allocation is optimal? (Hint: Use the inter pretation of the MRS} Now explain how you think Menglei found that allocation