Andrew, Beth, and Cathy live in Lindhville. Andrew’s demand function for bike pa
ID: 1163721 • Letter: A
Question
Andrew, Beth, and Cathy live in Lindhville. Andrew’s demand function for bike paths, a public good, is given by Q = 16 – 2P. Beth’s demand is Q = 9 – P/2, and Cathy’s is Q = 17 – P.
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9. Andrew, Beth, and Cathy live in Lindhville. Andrew's demand function for bike paths, a public good, is given by Q-16-2P. Beth's demand is Q-9-P/2, and Cathy's is Q 17 - P. The marginal cost of building a bike path is MC 15. 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+btc- 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?Explanation / Answer
Firstly, we will arrange the demand curves in the form of f(p) = Q
Andrew's demand curve - Q = 16 - 2P
In the original form, P = 8 - Q/2
Beth's demand curve - Q = 9 - P/2
P = 18 - 2Q
Cathy's Demand curve - Q 17 - P
P = 17 - Q
1) If the taxes are set so that each of them share cost evenly it means a = b = c = MC/3
a = b= c = 15/3 = 5, so each one will cost $5 for the path.
NOw at $5 how many path each want
Andrew wants,
P = 8 - Q/2
Put P = $5
5 = 8 - Q/2
solving this, we get
Q = 6, SO at price $5 Andrew wants 6 paths to built.
Beth wants,
P = 18 - 2Q
put P = $5
5 = 18 - 2Q
Q = 6.5, So at Price $5, Beth wants 6.5 path to built.
Cathy wants,
P = Q - 17
5 = Q - 17
Q = 12, so at Price $5, Cathy wants 12 paths to built.
Government will builds 12 paths because it is the largest number asked for by the residents.
2. The social optimum level is where MSB = MC
where MSB = marginal social benefit which is equal to the sum of the demand fuctions of all 3 residents.
MSB = 8 - Q/2 + 18 - 2Q + 17 - Q
MSB = 43 - 7Q/2
Put MSB = MC
43 - 7Q/2 = 15
7Q/2 = 28
Q = 8
So we need to tax from all the three residents so that nobody will wants more than 8 paths to built or want exactly 8 paths.
So we put Q = 8 in the demand curves to get this tax prices.
As the good here is a public good the price P is the tax prices it means for Andrew price P = a, for Beth Price P = b, and Cathy price P = c
Andrew -
a = 8 - Q/2
Put Q = 8
a = 8 - 8/2
a = 8 - 4
a = $4, so the tax price for Andrew is $4 at social optimum level.
Beth -
b = 18 - 2Q
Put Q = 8
b = 18 - 16
b = 2, so the tax price for Beth is $2 at social optimum level.
Cathy -
c = 17 - Q
Put Q = 8
c = $9, so the tax price for Cathy is $9 at social optimum level.
Since, 4 + 2 +9 = 15, so it is enough to cover MC, also social optimum level is achieved.
So 8 paths will be built at a = 4, b = 2 and c = 9