3. Altruistic and Warm Glow Preferences (35 points). Consider two roommates. Suppose that each gets utility from private consumption, c, and a house that is in a good state, H = h + h2, where h and h is the contribution that each roommate makes to the house. That is, for agent i utility is u(ci, H) = ln(ci) + yln(H). The price of repairing services is p in terms of the consumption good (that is, the price of consumption is normalized to 1). Each person has income I. Let &; = ph;/I denote the income share of house expenditures for person i. (a) Find the social optimal income share of house repair services, . (10 points) (b) Find the private optimal income share of house repair services, 8*. (10 points) (c) Compare & and 8*. Explain intuitively why they are different. (5 points) (d) Now suppose that agents are altruistic, that is: u(ci, H) = ln(ci) + yln(H) + ln(H), where y. Find the private optimal income with altruism A. Compare SA and 8. What happens if = y? (5 points) (e) Now, consider the case with warm glow, where each person cares about their contribution to the house rather than their utility from the house. Specifically, u(ci) = ln(ci) + yln(hi). Find the private optimal income share with warm glow W. Compare 6W and 8. Explain briefly. (5 points)