public good

public good

A residence with 10 identical households is trying to build a public pool. Each household's preference is represented by a utility function, u{(x,P) = x?/3 p1/3 where x; is household i's consumption of private good x and P is the size of the public pool, i = 1, 2, ..., 10. Each household has an income of $60. The price of x is $2 and the cost in dollars of building the pool of size P is c(P) = 8P. i) How much money will each household contribute to the construction of the pool in a Nash equilibrium? (6 marks) ii) Is there a 'Free Riding' problem under the Nash equilibrium in i)? Is it optimal? Explain briefly. (2 marks) iii) Find via explanation, the Pareto optimal size of the pool where all households contribute the same amount of money and consume the same amount of x. (6 marks)