Question

Suppose 1 person has the demand Q = 20-4P for streelights, and 1 person has the demand Q = 18-2P for streetlights. The cost of building each streetlight is 3. If it is impossible to purchase a fractional number of streetlights, how many streetlights are socially optimal?

Answer 1

Total demand is:

P = (5 − 0.25Q) + (9 − 0.5Q) = 14 − 75Q
Setting the result equal to the marginal cost, we get

14 - 0.75Q = 3

0.75Q = 11

Q = 11 / 0.75 = 14.7

This indicates that the social optimum is somewhere between 14 and 15. We need to figure out whether society is better off with 14 or 15 units. For this, we have to compare the social benefits versus the social costs (i.e. find the area below the social demand and above the social cost curve.
To do so, first compute the total social benefit (ignoring costs) from 18 and 19 units. This is
the area underneath the social demand curve to the left of 18 and 19 units.

Total social benefit of 14 units is:

14 - 0.75(14) = 3.5

Total social benefit of 15 units is:

14 - 0.75(15) = 2.75

The area under social demand to the left of 14 units :
= 0.5(14 − 3.5)(14) + 3.5(14)

= 73.5 + 49 = 122.5
Area under social demand to the left of 15 units :

= 0.5(14 − 2.75)(15) + 2.75(15)

= 84.375 + 41.25 = 125.63

The total social cost of 18 units = 14(3) = 42.

The total social cost of 15 units = 15(3) = 45.

Hence, the total social surplus from 14 units is 122.5 − 42 = 80.5

The total social surplus from 15 units =125.63 - 45 = 80.63
For social welfare, 15 units is little better than 14 units.