The streets around the common of a small isolated upstate New York village are c

Question

The streets around the common of a small isolated upstate New York village are currently unlit. The villagers must decide whether to install streetlights and how many to install. There are 70 households in the village. If Q streetlights are installed, 50 households have a private marginal benefit from an additional streetlight of PMB1 = 20 – 0.1Q each and the other 20 households have private marginal benefit from an additional streetlight of PMB2 = 10 – 0.1Q each. The cost of a streetlight is $570. (10 points) a) How many streetlights should be installed? b) Suppose the village council knows the value of the PMB of every villager. How much does the council recommend that each household should pay? c) Suppose the council decides that all the householders will share the cost of a streetlight equally but the number of the streetlights installed will be decided by simple majority voting. What is the result? d) Is this result efficient? Explain your answer.

Explanation / Answer

Number of streetlights will be installed till Marginal cost to put extra stretlight is equal to marginal benefit to society

So, MC or Price of Street Light = Marginal Social benefit

Marginal Social benefit = 50*PMB1 + 20*MPB2

Now , Equating MC = MSB

570 = 50[20 – 0.1Q] + 20[10 – 0.1Q]

570 = 1000 - 5Q + 200 - 2Q

7Q = 630

Q = 90

b.

Each household should pay as per their marginal Private Benefit

So, PMB1 = 20 – 0.1Q = 20 - 0.1*90 = 11

  PMB2 = 10 – 0.1Q = 10 - 0.1*90 = 1

So, 50 households with MPB= 11 will Pay 11 , and 20 households with MSB= 1, Will be 1.

c.

Since  the number of the streetlights installed will be decided by simple majority voting, So 50 people with MPB = 20 – 0.1Q , will decide no. of streetlights

Gqoup with MPB1 will decide Streetlights, Such that Sum of there MPB1 is equal to MC of Streetlight

MC = 50*MPB1

570 =  50(20 – 0.1Q)

430 = 5Q

Q = 86

Cost paid by everyone = 570/70 = 8.14

d. No, this result isn't efficient as group of 20 with MPB2 pays a price of 8.14 where as its MPB2 < 8.14 . So they would be better off with the decrase in the number of stretlights as there MPB will increase with it.

Price Paid for Streetlight > MPB2

8.14 > 10 – 0.1Q

8.14 > 10 - 0.1*86

8.14 > 1.4

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