Suppose 10 people each have the demand Q = 10 – P for streetlights, and 5 people

Question

Suppose 10 people each have the demand Q = 10 – P for streetlights, and 5 people have the demand Q = 10 – P/2 for streetlights. The cost of building each streetlight is 40. How many streetlights are socially optimal?

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8. Suppose 10 people each have the demand Q 10 -P for streetlights, and 5 people have the demand Q 10 - P/2 for streetlights. The cost of building each streetlight is 40 How many streetlights are socially optimal?

Explanation / Answer

The social optimum can be obtained by inverting the demand curves and summing to get the social demand:

Thus,

Q=10-P

P=10-Q-----------(1)

Q= 10-P/2

P=20-2Q -----------(2)

thus by using eq. 1 and eq. 2 for 10 and 5 individual respectively, the price can be given as below:-

P=10(10-Q) +5(20-2Q)=100-10Q+100-10Q =200-20Q

Setting the result equal to the marginal cost of 40 and solving for Q gives

40=200-20Q

20Q=160

Q=160/20=8

So socially optimal quantity is 8 units

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