Andrew, Beth, and Cathy live in Lindhville. Andrew's demand for bike paths, a public good, is given by Q = 12 - 2P.
Beth's demand is Q = 18 - P, and Cathy's is Q = 8 - P/3.
The marginal cost of building a bike path is MC = 21.
The town government decides to use the following procedure for deciding how many paths to build. It asks each resident how many paths they want, and it builds the largest number asked for by any resident.
To pay for these paths, it then taxes Andrew, Beth, and Cathy the prices a, b, and c per path, respectively, where a + b + c = MC. (The residents know these tax rates before stating how many paths they want.)
a. If the taxes are set so that each resident shares the cost evenly (a = b = c), how many paths will get built?
b. Show that the government can achieve the social optimum by setting the correct tax prices a, b, and c. What prices should it set?