Suppose Chen has demand Q = 40 – 2P for streetlights and Abby has the demand Q =

Question

Suppose Chen has demand Q = 40 – 2P for streetlights and Abby has the demand Q = 10 – (1/2)P for streetlights. The cost of building each streetlight is $19. How many streetlights are socially optimal? Compare this socially optimal quantity with the quantity that would be privately purchased.

A. How many streetlights would be privately purchased?
B. Now suppose that streetlights are a public good. Write an equation for the market demand curve of streetlight consumption.
C. How many streetlights are socially optimal?
D. Compare this socially optimal quantity with the quantity that would be privately purchased
E. If the socially optimal quantity is not equal to the quantity that would be privately purchased, how could government intervene?


Can someone help me solve these please?

Explanation / Answer

a. Demand of Chen

Q1 = 40 – 2P

abby's demand , Q2 = 10 – (1/2)P

Assuming that Marginal cost of providing the street light is equal to marginal social cost of provding the street light= $19

So, substituting this price in indivdual demand curves, we get

Q1 = 40 – 2P

=40-2(19)

=2

Q2 = 10 – (1/2)P

=10-(1/2)19

0.5

Total demand Q= 2.5 , or we can say that the toatl demand is less than 3.

B. When streetlights are a public good, the market demand curve will be the vertical summation of the individual demand curves.

Writing the demand curves in the terms of price we get,

P=20-(1/2)Q

P=20-2Q

adding these two equations, we get the market demand.

P=40-(3/2)Q

C. To find socially optimal quantity of streetlight, we have to equate market demand curve with the marginal social cost,that is

40-(3/2)Q=19

21=(3/2)Q

Q=14

Thus, the socially optimal quantity is 14.

d. The private demand of the street light is less than the socially desired quantity because people are not willing to reveal thier true demand for the public good and want to free ride.

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