Public Good Optimum
© Allen C. Goodman 2015
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).
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.
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.
MB
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
Look at Adam
MB = 30 - Q
MB - Adam
35
30
25
Marginal Benefits
Q
What does this mean?
20
MB
15
10
5
0
0
5
10
15
20
Quantity
25
30
35
MB
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
20
18
16
14
12
10
8
6
4
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Look at Bert
MB = 20 - 2Q
MB - Bert
35
30
25
Marginal Benefits
Q
20
MB
15
10
5
0
0
5
10
15
20
Quantity
25
30
35
MB
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
20
19.5
19
18.5
18
17.5
17
16.5
16
15.5
15
14.5
14
13.5
13
12.5
12
11.5
11
10.5
10
9.5
9
8.5
8
7.5
7
6.5
6
5.5
5
Look at Charlie
MB = 20 – 0.5Q
MB - Charlie
35
30
25
Marginal Benefits
Q
20
MB
15
10
5
0
0
5
10
15
20
Quantity
25
30
35
Building up the Curve
MB
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
70
66.5
63
59.5
56
52.5
49
45.5
42
38.5
35
33.5
32
30.5
29
27.5
26
24.5
23
21.5
20
18.5
17
15.5
14
12.5
11
9.5
8
6.5
5
MB - A, B, C
80
Look at spreadsheet
70
60
Marginal Benefits (Total)
Q
This isn’t what
we usually do!
A+B+C
50
40
MB
30
Why are we
doing it this
way?
20
10
A+C
0
A
0
5
10
15
20
Quantity
25
30
35
Sum of Marginal Benefits – Calculation
MB
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
70
66.5
63
59.5
56
52.5
49
45.5
42
38.5
35
33.5
32
30.5
29
27.5
26
24.5
23
21.5
20
18.5
17
15.5
14
12.5
11
9.5
8
6.5
5
MB - A, B, C
80
Efficient Amount
at Q = 12
70
60
Marginal Benefits (Total)
Q
50
Why?
40
MB
30
20
At Q = 12, they are worth
10
0
0
5
10
15
20
Quantity
25
30
35
$1800 to Adam
$0 to Bert
$1400 to Charlie
What
happens if demand changes? – Calculation
MB
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
70
66.5
63
59.5
56
52.5
49
45.5
42
38.5
35
33.5
32
30.5
29
27.5
26
24.5
23
21.5
20
18.5
17
15.5
14
12.5
11
9.5
8
6.5
5
MB - A, B, C
80
Look at spreadsheet
70
60
Marginal Benefits (Total)
Q
50
40
MB
30
20
10
0
0
5
10
15
20
Quantity
25
30
35
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!
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.
Problems w/ Lindahl Eq’m
1. 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?
2. 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.
3. 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 (!?).
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.
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.
Public Goods
w1 = -/U1x
w2 = -/U2x
x
w
=

/U
3
(1)3
Etc.
So, optimize
W = wiUi (xi, G) +  [ xi - f(G)]
w.r.t. xi
W/xi = wiUix +  = 0.
(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.
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.