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Homework answers / question archive / Suppose that 20 people each have the demand Q = 20 - P for streetlights, and 5 people have the demand Q = 18 - 2P for streetlights
Suppose that 20 people each have the demand Q = 20 - P for streetlights, and 5 people have the demand Q = 18 - 2P for streetlights. The cost of building each streetlight is 10. If it is impossible to purchase a fractional number of streetlights, how many streetlights are socially optimal?
We can solve this by first rewriting the demand functions as inverse demand functions (price in terms of quantity). For the group of 20 consumers, this is P = 20 - Q and for the group of 5, P = 9 - 0.5Q. We can now add up each individual's inverse demand function to create a societal inverse demand function. In this case we get P = 10 (20 - Q) + 5 (9- 0.5Q) = 200 - 10Q + 45 - 2.5Q = 245 - 12.5Q. Since we know that in a competitive market marginal cost equals price, we can set P to be 10. This yields the equation 10 = 245 - 12.5Q. Solving for Q set get 18.8. Therefore the socially optimal number of streetlights is between 18 and 19.
