There are two residents in a town. They both have demand for a public good. The price of the public good is $6 each. Resident A's demand is Q=40-8p, Resident B's demand is Q=36-4P. What is the private optimum and public optimum? What is the difference between private and public optimum?

In a private equilibrium, each consumer takes the price as given, thus resident A's quantity demanded is:

• Q = max(0, 40 - 8*6)
• Q = max(0, -8)
• Q = 0

The quantity demanded by resident B is:

• Q = max(0, 36 - 4 * 6)
• Q = 12

Thus, the total quantity demanded is 12 + 0 = 12.

In a public equilibrium, each resident recognizes that a public good is non-excludable and non-rivalry. Thus willingness to pay is the smaller of that of the two residents, i.e.,

• p = min((40 - Q) / 8, (36 - Q) / 4)
• p = min(5 - Q/8, 9 - Q/4)

The public equilibrium is where social willingness to pay is equal to the price, i.e.,

• min(5 - Q/8, 9 - Q/4) = 6
• Q = 0

That is, in the public equilibrium, the provision of the public good is zero, i.e., none.