Only question (b) need to be answered. Thanks a lot. Andrew, Beth and Cathy live

Question

Only question (b) need to be answered.

Thanks a lot.

Andrew, Beth and Cathy live in Lindhville. Andrew's demand for bike paths, a public good, is given by Q = 12 - 2 P. Beth's demand is Q = 18 - P, and Cathy's is 8 - P/3. The marginal cost of another bike path is MC = 21. The town decides to use the following process for deciding on the final number of bike paths. Ask each resident how many they want, and build the largest amount someone asks for. To pay for this, it taxes A, B, and C prices a, b, and c, where a + b + c= MC. (The residents know these tax rates before saying how many paths they want.) If the taxes are set so all 3 pay equally a = b = c, how many paths are build? Show that the government can get to the social optimum with the right tax prices a, b, and c. What are these right prices?

Explanation / Answer

b Here we do the vertical summation to find the total demand

P = 6 - Q/2

P = 18 - Q

P = 24 - 3Q

So the total demand for public good

P = 48 - 9/2Q

Putting it equal to MC

48 - 9/2Q = 21

So Q = 27*2/9 = 6

S0 a = 6 -Q/2 = 6 -6/2 = $3

b =  18 - Q = 18 - 6 = $12

c =  24 - 3Q =  24 - 3*6 = $6

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