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=h1+h2, where h1 and h2 is the contribution that each roommate makes to the house. That is, for agent i utility is u(ci,H)=ln(ci)+ln(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 iphi/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, . (10 points) (c) Compare ^ and . Explain intuitively why they are different. (5 points) (d) Now suppose that agents are altruistic, that is: u(ci,H)=ln(ci)+ln(H)+^ln(H), where ^. Find the private optimal income with altruism A. Compare A and ^. What happens if ^=?(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)+ln(hi). Find the private optimal income share with warm glow W. Compare W and ^. Explain briefly. (5 points) 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=h1+h2, where h1 and h2 is the contribution that each roommate makes to the house. That is, for agent i utility is u(ci,H)=ln(ci)+ln(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 iphi/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, . (10 points) (c) Compare ^ and . Explain intuitively why they are different. (5 points) (d) Now suppose that agents are altruistic, that is: u(ci,H)=ln(ci)+ln(H)+^ln(H), where ^. Find the private optimal income with altruism A. Compare A and ^. What happens if ^=?(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)+ln(hi). Find the private optimal income share with warm glow W. Compare W and ^. Explain briefly. (5 points)
