This week’s problem asks you to dig into the problem of public goods. In ordinary markets with private goods, it’s easy to show that the market solution where supply and demand intersect is also the most efficient solution. However, in markets where goods are “public” in the economic sense of non-rivalry and non-excludability, it is hard for producers and consumers to come naturally to the efficient quantity through the price system.

Consider the town of Springfield with only three residents, Sophia, Amber, and Cedric. The three residents are trying to determine how large, in acres, they should build the public park. The table below shows each resident’s willingness to pay for each acre of the park.

Acres Sophia Amber Cedric
1 $10 $24 $6
2 8 18 5
3 6 14 4
4 3 8 3
5 1 6 2
6 0 4 1
7 0 2 0

We’ll start by searching for the efficient outcome in one scenario (I’ll walk you through it), and then you’ll be asked to think about how things change when our assumptions change.

Example 1: Suppose the cost to build the park is $9 per acre. How many acres should the park be to maximize total surplus from the park in Springfield?

To do this question, consider how much value the first acre would provide. Sophia gets $10 in benefits, Amber gets $24 in benefits, and Cedric Gets $6. That’s $40, so the benefit exceeds the $9 cost of building the first acre. The first acre should get built. Keep checking this for each acre.
For the second acre, Sophia gets $8, Amber gets $18, and Cedric gets $5, for a total of $31. Again, $31>$9, so keep building.
If you do this, you’ll see that five acres is the total that should get built. (You can assume that if the costs of an acre are exactly equal to the benefits, that acre will get built.)
Also, think about this from the perspective of each of the residents. When 5 acres are built, Sophia gets $28 in benefits. She gets $10 in benefits from the first acre, $8 from the second, and so on. ($10+8+6+3+1=$28). If she was asked to vote “Yes” or “No” for building a 5-acre park in exchange for a $15 increase in property taxes, say, she would vote YES.

QUESTIONS: Refer to the table above. Show your work for each of your answers.

1. Suppose the cost to build the park is $24 per acre. How large should the park be to maximize total surplus from the park in Springfield?

2. Suppose the cost to build the park is $24 per acre and that the residents have agreed to split the cost of building the park equally. That is, they each have to pay $8 per acre. To maximize his own surplus, how many acres would Cedric like Springfield to build? HINT: To find Cedric’s surplus, add up how much benefit he gets from each acre that is built, then compare it to the amount he has to pay.

3. Suppose the cost to build the park is $24 per acre and that the residents have agreed to split the cost of building the park equally. If the residents vote to determine the size of park to build, basing their decision solely on their own willingness to pay (and trying to maximize their own surplus), what is the largest park size for which the majority of residents would vote “yes?”

4. Suppose the cost to build the park is $24 per acre and that the residents have agreed to split the cost of building the park equally. If the residents decide to build a park with size equal to the number of acres that maximizes total surplus from the park, how much total surplus will Sophia receive?

This week’s problem asks you to dig into the problem of public good...