Suppose 20 people each have the demand Q = 20 - P for streetlights, and 5 people have the demand Q = 18 - 2P for streetlights

Streetlights are public goods because there is both non-rivalry and non-excludability in consumption. Non-rivalry means that one person's consumption of the good does not diminish the benefit that others derive from it. Non-excludability means that it is not feasible to exclude some people from benefitting from the good once it is provided. The optimal provision of a public good is determined by the point where the marginal social benefit (MSB) from the good equals the marginal social cost (MSC) of the good. To obtain the MSB for a public good, we sum vertically the individual inverse demand curves for the goods. Solving for the two inverse demand curves for street lights gives the following:

• Q = 20 - P
• P = 20 - Q

and

• Q = 18 - 2P
• P = 9 - Q/2.

Summing the two curves vertically gives an aggregate inverse demand curve that shows the total willingness-to-pay of all the beneficiaries at any particular level of provision of the public good. Bearing in mind that there are 20 people with the first inverse demand curve and 5 people with the second, the aggregate demand curve is determined as follows:

• P = (20)(20 - Q) + (5)(9 - Q/2) = 400 - 20Q + 45 - 5Q/2 = 445 - 45Q/2.

P at any given output reflects the MSB of the streetlights. Since no evidence is given to the contrary in the question, we will assume that MSC is equal to the cost of building each street light, which is 10. Equating MSB and MSC and solving for Q gives:

• 10 = 445 - 45Q/2
• 45Q/2 = 435
• Q = 19.33.

So, after rounding down, the socially optimal number of streetlights is 19.