Suppose 20 people live on a street and that each of them is willing to pay $2 for each extra streetlight, regardless of the number of streetlights provided. If the cost of providing x streetlight is given by c(x) = x2, what is the Pareto efficient number of streetlights to provide?
Pareto efficiency refers to an allocation of resources where the marginal rate of substitution is the same for all consumers. The allocation should be such that if there are two entities being considered, any change in the allocation makes one better off at the expense of the other.
Each of the twenty people that live on the street is willing to pay $2 for each extra light irrespective of how many are provided. The cost of x streetlights is given by c(x) = x^2.
The marginal utility of each streetlight is $2*20 = $40.
If 19 lights are already present on the street, the cost of installing one more is equal to 20^2 - 19^2 = $39 while the people on the street are willing to pay $40 for this. On the other hand, the cost of the 21st light is 21^2 - 20^2 = $41 while the people are only willing to pay $40 for this. The optimal number streetlights that should be provided is 20.