3. Altruistic and Warm Glow Preferences (35 points). Consider two roommates. Suppose that each gets utility from private consumption, (2, and a house that is in a good state, H = In + hg, where In and hg is the contribution that each roommate makes to the house. That is, for agent 15 utility is \"605,, H) = ln(q) + '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 6, E ph, / I denote the income share of house expenditures for person 2'. (a) Find the social optimal income share of house repair services, 6. (10 points) (b) Find the private optimal income share of house repair services, 6*. (10 points) (c) Compare 6 and 6*. Explain intuitively why they are different. (5 points) (d) Now suppose that agents are altruistic, that is: u(c,;, H) = ln(c,-) + *yln(H2 + R/ln(H), where 3/ g 7. Find the private optimal income with altruism 6\". Compare 6'4 and 6. What happens if 3/ = 'y? (5 points) (6) 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(hz-). Find the private optimal income share with warm glow 6 W. Compare 6W and 6. Explain briey. (5 points)