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 = hi + h2, where hi and hy is the contribution that each roommate makes to the house. That is, for agent , utility is a(c, H) = In(c) + win( 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 /. Let & = ph,/ I denote the income share of house expenditures for person . (a) Find the social optimal income share of house repair services, S. (10 points) (b) Find the private optimal income share of house repair services, 8*. (10 points) (c) Compare & and 5". Explain intuitively why they are different. (5 points) (d) Now suppose that agents are altruistic, that is: u(c, H) = In(c.) + yin( H) + Win( H), where