Consider a community of two families A and B, who contribute E_A and E_B dollars to the community's events budget. Community events are a public good and the total quality of events provided E = EA + EB .

Assume each family has an annual income of $60 which it splits between the events budget budget and all other consumption spending X.

Family utility functions are given by U_A =X_A^1/2E^1/2 andU_B =X^1/2_BE^1/2.

(a)What event budget would maximize the social welfare function W = U_A + U_B ? (Study tip: You can either set up and solve the social planner's problem, or recall the necessary condition for optimal public good provision. I recommend the latter.)

(b) If households privately decide how much to contribute to the event budget, what will the total budget be? (Study tip: Begin by finding each household's best response to the other, then solve. You can use the fact that the problem is symmetric to save time in this step.)

(c) If the an outsider gives a $6 gift to the community event budget, do you predict the event spending each family contributes will increase, decrease, or remain the same? Justify your prediction.

(d) With the $6 gift to the school, do you predict that total event funding (including the donation) will increase, decrease of remain the same? Justify your prediction.

(e) Assume there is no gift but instead the city government takes $3 in taxes from each family and adds the entire $6 in revenue to the event budget. How much will each family voluntarily contribute to the event budget now?