Consider a community of two families A and B, who contribute EA and EB 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 and all other consumption spending X. Family utility functions are given by UA = XA^1/2 E^1/2 and UB = XB^1/2E^1/2. (a) What event budget would maximize the social welfare function W = UA + UB? (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.)
Consider a community of two families A and B, who contribute EA and EB 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 and all other consumption spending X. Family utility functions are
given by UA = XA^1/2 E^1/2 and UB = XB^1/2E^1/2.
(a) What event budget would maximize the social welfare function W = UA + UB?
(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.)
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