1. Public Good Optimum © Allen C. Goodman 2015
7/1/2018
Public Good Optimum
© Allen C. Goodman 2015

2. Public Goods
Most important factor is that everyone gets the same amount.
We have to get some agreement as to how much we’ll want (we’ll discuss that a lot).
We’ll have to get some agreement as to how to pay for it (we’ll discuss that a lot, also).

3. What do you think about libraries?
Go around class.
What do you think about tennis courts?
Go around class.
What do you think about sidewalks?
Go around class.

4. Query: Why, necessarily
should tennis courts be
provided publicly?
Why shouldn’t people
join private tennis clubs?
Consider a Town
Trying to decide how many tennis courts should be provided in public parks.
Comes out to be a number per 10,000 people.
Marginal benefit is in hundreds of dollars.
Cost is in hundreds of dollars. Let’s assume MC is 32.
We have people like Adam, Bert, and Charlie. Assume we have enough people to collect money to build what they might want.

5. Look at Adam
What does this mean?
MB = 30 - Q

6. Look at Bert
MB = Q

7. Look at Charlie
MB = 20 – 0.5Q

8. Building up the Curve Look at spreadsheet
This isn’t what we usually do!
A+B+C
Why are we doing it this way?
A+C A

9. Sum of Marginal Benefits – Calculation
Efficient Amount
at Q = 12
Why?
At Q = 12, they are worth
$1800 to Adam
$0 to Bert
$1400 to Charlie

10. What happens if demand changes? – Calculation
Look at spreadsheet

11. Tricky Issue How do we pay for these. Why do it publicly?
Suppose we say “How many tennis courts should we build?” assuming it costs $3,200 to build them.
What will happen?
A> No one here values a single court at $3,200, BUT collectively, they value 12!

12. Lindahl equilibrium With 12 courts
Adam values them at 18(00)
Bert values them at 0
Charlie values them at 14(00)
If we “know” these values, we can charge the people accordingly. This is sometimes called a “Lindahl” equilibrium.

13. Problems w/ Lindahl Eq’m
With private goods, people pay … it means that they value the goods at least that much. How do we get people to reveal preferences if we’re not withholding services from those who won’t pay?
What if MC is close to 0? If we charge where MB = MC, we get close to 0 price, and may not be able to afford the good.
It may be hard to exclude those who won’t pay, although for tennis court you could charge an hourly fee. Can’t do the same for police protection. Can’t do the same for religious services (!?).

14. Who provides?
Some public goods are provided at the national level. National defense for example (at least in US) – although in some other (generally less developed) countries you often have local militias.
Others at the state, county, or lower levels.
Some amounts of goods are also provided privately. Ford, or GM do not depend on public police to guard their property, for example.

15. More Advanced – Public Goods
It's helpful to derive a public good equilibrium. My favorite way is a simple one where we have a social welfare function in which:
W = W(U1, U2, U3, ...) = Weighted Σ Ui for a community of individuals.
We must decide how much public good to make. In the pure sense, the public good is nonexcludable and nonrival.
Ui = Ui (xi , G)
X =  xi = f (G) Constraint (why)
f' < 0.

16. Public Goods So, optimize W = wiUi (xi, G) +  [ xi - f(G)] (1)
w1 = -/U1x
w2 = -/U2x
w3 = -/U3x
Etc.
So, optimize
W = wiUi (xi, G) +  [ xi - f(G)] (1)
w.r.t. xi
W/xi = wiUix +  = (2)
w.r.t. G
W/G =  wiUiG - f' = 0 (3)
From (2)
wi = -/Uix
Insert into (3)
W/G =  UiG/UiX + f' = 0 (3)
Factor out , and we get:
 UiG/UiX + f' = 0   MRSGX = -f' = MRT
Well-known Samuelson condition.

17. KEY Point
With private goods, we can exclude others. We add demand horizontally.
With public goods, we cannot exclude others. We add demand vertically.
If everyone gets the same amount, then the appropriate benefit measure is “how much (together) they value the amount that they get.” This is a vertical summation.