Problem V. Shared Driveway (55 points) Recall the textbook example where two neigh- bors share a driveway. Suppose each person gets utility on the private consumption good, c, and the driveway, D = d + d2, such that wc,D) = ln(c) + viln(D). Price of snowplow- ing/repairing services is p ir terms of the consumption good. Each person has income I,. Let 8 = pd /I, denote the income share of driveway expenditures for person i. First, assume that y1 = 72 = 7 and I1 = 12 = 1. 2 1. Find the social optimal income share of driveway services, 8. (5pts) 2. Find the private optimal income share of driveway services, *. (5 pts) 3. Compare 8 and 8. Explain intuitively why they are different. (5 pts) 4. Now suppose there is some extent of altruism. l.e. (G.D)=In() + n(D) + In(D). Find the private optimal income share with altruism 54. Compare 84 and 6. What happens if 9 = 7. (10 pts) 5. Now, consider the case with warm glow where each person cares about his contribution to the driveway rather than his consumption of driveway. Specifically, ui(c,d) = ln(G) + yIn(d.). Find the private optimal income share with warm glow,5". Compare 5W and 8. Explain intuitively. (10 pts) 6. Now suppose 11 > 12. Specifically, 11 = MI, but 91 = 72 = Find the social and private optimal level of income share: and 81. What happens if M+. Explain intuitively. (10 pts) 7. Now suppose I1 = I. but 71 = My. Find the social and private optimal level of income share: & and 6. What happens if M+. Explain intuitively. (10 pts) Problem V. Shared Driveway (55 points) Recall the textbook example where two neigh- bors share a driveway. Suppose each person gets utility on the private consumption good, c, and the driveway, D = d + d2, such that wc,D) = ln(c) + viln(D). Price of snowplow- ing/repairing services is p ir terms of the consumption good. Each person has income I,. Let 8 = pd /I, denote the income share of driveway expenditures for person i. First, assume that y1 = 72 = 7 and I1 = 12 = 1. 2 1. Find the social optimal income share of driveway services, 8. (5pts) 2. Find the private optimal income share of driveway services, *. (5 pts) 3. Compare 8 and 8. Explain intuitively why they are different. (5 pts) 4. Now suppose there is some extent of altruism. l.e. (G.D)=In() + n(D) + In(D). Find the private optimal income share with altruism 54. Compare 84 and 6. What happens if 9 = 7. (10 pts) 5. Now, consider the case with warm glow where each person cares about his contribution to the driveway rather than his consumption of driveway. Specifically, ui(c,d) = ln(G) + yIn(d.). Find the private optimal income share with warm glow,5". Compare 5W and 8. Explain intuitively. (10 pts) 6. Now suppose 11 > 12. Specifically, 11 = MI, but 91 = 72 = Find the social and private optimal level of income share: and 81. What happens if M+. Explain intuitively. (10 pts) 7. Now suppose I1 = I. but 71 = My. Find the social and private optimal level of income share: & and 6. What happens if M+. Explain intuitively. (10 pts)