Public Goods
and Common Resources
Christopher Makler
Stanford University Department of Economics
Econ 50: Lecture 24
pollev.com/chrismakler
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Plan for the Week
- Wednesday lecture: review
- Wednesday Q section: review
- No other sections
- Office hours (rotating cast of TA's, me): 10am-6pm Thursday, Lathrop 292
- SEA study dinner: Thursday night, time TBA
Today's Agenda
- Classifying goods: rival/excludable
- Common Resources
- When to provide a public good
- How much of a public good to provide
- Voting over public goods
- Concluding thoughts
Classifying Goods
Rivalry in Consumption
One agent's consumption affects others' ability to consume it the same good.
Excludability
Agents may be prevented from consuming the good
Rival
Nonrival
Excludable
Nonexcludable
Private Goods
Common Resources
Public Goods
Club Goods
These aren't hard-and-fast categories
- Interstate 280 at 4am is a public good (nonrival, nonexcludable)
- Interstate 280 at 4pm is a common resource (rival, nonexcludable)
- "Public housing" and "public education" are publicly provided/funded but the "goods" are private (like housing and private education)
Tragedy of the Commons
- Each individual, acting in their own best interest, overuses the common resource
- Possible solutions: regulation (issue permits); taxation (charge for use); privatization (avoid problem by making them not a commons at all)
Tragedy of the Commons
Village of 35 people who can choose to fish or hunt.
Each fish is worth $10; each deer is worth $100. Every hunter gets one deer.
If \(L\) people fish, (and \(35 - L\) people hunt), total fish caught: \(f(L) = 40L - L^2\)
Total revenue from fishing:
Total revenue from hunting:
Average revenue per fisher:
Average revenue per hunter:
Marginal revenue from additional fish:
Marginal cost of having that person not hunt:
What's the effect of an increase in \(L\)?
10f(L)
400L-10L^2
100(35-L)
3500 - 100L
400-10L
100
100
400-20L
Fees
Suppose you needed to buy a fishing permit for a fee F.
What value of F would result in the optimal L*?
Taxes
Suppose the village levied a tax of t per fish caught.
What value of t would result in the optimal L*?
When to Provide a Public Good
| Roommate | Willingness to Pay |
|---|---|
| Amy | $350 |
| Boris | $150 |
| Carlos | $800 |
Three roommates are deciding whether to get a washer/dryer:
If the washer/dryer costs $1200, is it efficient for them to buy it?
If they each have to pay an equal share ($1200/3 = $400), would a majority of them agree to buy the washer/dryer?
✅
❌
| Roommate | Willingness to Pay |
|---|---|
| Amy | $450 |
| Boris | $150 |
| Carlos | $500 |
Now let's change up the valuations
just a bit:
If the washer/dryer costs $1200, is it efficient for them to buy it?
If they each have to pay an equal share ($1200/3 = $400), would a majority of them agree to buy the washer/dryer?
✅
❌
| Agent | WTP |
|---|---|
| Amy | $450 |
| Boris | $150 |
| Carlos | $500 |
If the washer/dryer costs $1200, is it efficient for them to buy it?
If they each have to pay an equal share ($1200/3 = $400), would a majority of them agree to buy the washer/dryer?
✅
❌
| Agent | WTP |
|---|---|
| Amy | $350 |
| Boris | $150 |
| Carlos | $800 |
✅
❌
Median Voter Theorem
If voters and candidates are distributed along a one-dimensional spectrum and voters have single-peaked preferences, any voting method that is compatible with majority-rule will elect the candidate preferred by the median voter.
| Agent | WTP |
|---|---|
| Amy | $450 |
| Boris | $150 |
| Carlos | $500 |
| Agent | WTP |
|---|---|
| Amy | $350 |
| Boris | $150 |
| Carlos | $800 |
Median Voter Theorem
If voters and candidates are distributed along a one-dimensional spectrum and voters have single-peaked preferences, any voting method that is compatible with majority-rule will elect the candidate preferred by the median voter.
WILLINGNESS TO PAY
$400
B
A
$350
$150
C
$800
| Agent | WTP |
|---|---|
| Amy | $450 |
| Boris | $150 |
| Carlos | $500 |
| Agent | WTP |
|---|---|
| Amy | $350 |
| Boris | $150 |
| Carlos | $800 |
Median Voter Theorem
If voters and candidates are distributed along a one-dimensional spectrum and voters have single-peaked preferences, any voting method that is compatible with majority-rule will elect the candidate preferred by the median voter.
WILLINGNESS TO PAY
$400
B
A
$450
$150
C
$500
How Much Public Good to Provide
Private goods: aggregate preferences by summing demand curves horizontally.
Public goods: aggregate preferences by summing demand curves vertically.
Since the demand curve represents the MRS, this means the efficient amount to produce is where the sum of the MRS's equals the cost.
Model 2: A Fireworks Show
A town is preparing to have a fireworks show.
Let \(G\) be the total number of minutes of fireworks
Each minute of fireworks costs $200.
Model 2: A Fireworks Show
A town is preparing to have a fireworks show.
Let \(G\) be the total number of minutes of fireworks
Let \(g_i \ge 0\) be the amount citizen \(i\) contributes to the show;
each person has income \(m_i\), so their private consumption is \(x_i = m_i - g_i\).
Each minute of fireworks costs $200.
There are two types of citizen:
200 kinda love fireworks
u(G, x_i) = 12\sqrt G + x_i
u(G, x_i) = 4\sqrt G + x_i
Note: \(x_i\) is private consumption (dollars left over after contributing \(g_i\)) so we can also write
u(G, g_i) = 12\sqrt G + m_i - g_i
u(G, g_i) = 4\sqrt G + m_i - g_i
100 really love fireworks
\(G\) = total number of minutes of fireworks
\(g_i \ge 0\) = amount citizen \(i\) contributes
Each minute of fireworks costs $200
There are two types of citizen:
100 really love fireworks
200 kinda love fireworks
u(G, g_i) = 12\sqrt G + m_i - g_i
u(G, g_i) = 4\sqrt G + m_i - g_i
Questions:
What is the efficient length of a fireworks show?
What is the fairest way to pay for the fireworks?
How can the town decide how long of a show to have?
\(G\) = total number of minutes of fireworks
\(g_i \ge 0\) = amount citizen \(i\) contributes
100 really love fireworks
200 kinda love fireworks
u(G, g_i) = 12\sqrt G + m_i - g_i
u(G, g_i) = 4\sqrt G + m_i - g_i
What is the efficient length of a fireworks show?
MB = {6 \over \sqrt{G}}
MB = {2 \over \sqrt{G}}
Cost
200G
MC = 200
"Samuelson Condition" : set the sum of the MB's equal to the MC:
100 \times {6 \over \sqrt{G}}
+
=
200 \times {2 \over \sqrt{G}}
200
G^* = 25
\(G\) = total number of minutes of fireworks
\(g_i \ge 0\) = amount citizen \(i\) contributes
100 really love fireworks
200 kinda love fireworks
What is the efficient length of a fireworks show?
MB = {6 \over \sqrt{G}}
MB = {2 \over \sqrt{G}}
Cost
200G
MC = 200
Suppose everyone else had contributed some amount \(G\).
How much would each type voluntarily contribute?
For example, suppose everyone else had contributed enough for a 4-minute-long show.
What is the marginal benefit of each type?
This is called the free rider problem.
u(G, g_i) = 12\sqrt G + m_i - g_i
u(G, g_i) = 4\sqrt G + m_i - g_i
When the benefits of a good accrue to everyone,
everyone has an incentive to shirk their contribution
The Free Rider Problem
\(G\) = total number of minutes of fireworks
\(g_i \ge 0\) = amount citizen \(i\) contributes
100 really love fireworks
200 kinda love fireworks
What mechanism can achieve the efficient quantity of \(G^* = 25\)?
Cost
200G
Suppose you knew everyone's type, and could charge them differently.
You could charge everyone half their valuation:
$30 for people who love fireworks, $10 for people who kinda love fireworks.
u(25, g_i) = 60 + m_i - g_i
u(25, g_i) = 20 + m_i - g_i
5000
What's the problem with this?
u(G, g_i) = 12\sqrt G + m_i - g_i
u(G, g_i) = 4\sqrt G + m_i - g_i
\(G\) = total number of minutes of fireworks
\(g_i \ge 0\) = amount citizen \(i\) contributes
each minute costs $200
100 really love fireworks
200 kinda love fireworks
Suppose people voted on a common amount \(g\) for everyone to contribute.
u(g_i) = 12\sqrt{{3 \over 2}g_i} + m_i - g_i
u(g_i) = 4\sqrt{{3 \over 2}g_i} + m_i - g_i
g_i^* = 54
g_i^* = 6
How many minutes would a majority of citizens vote for?
u(G, g_i) = 12\sqrt G + m_i - g_i
u(G, g_i) = 4\sqrt G + m_i - g_i
u(g_i) = 12\sqrt{{3 \over 2}g_i} + m_i - g_i
u'(g_i) = {6 \sqrt{3 \over 2} \over \sqrt{g_i}} - 1 = 0
u'(g_i) = 2\sqrt{{3 \over 2} \over \sqrt{g_i}} - 1 = 0
Total contributions: $\(300g\)
Amount of public good: \(G = \${300g \over \$200/\text{min}} = {3 \over 2}g \text{ minutes}\)
Concluding thoughts
Econ 50 | Spring 25 | Lecture 24
By Chris Makler
Econ 50 | Spring 25 | Lecture 24
Public Goods and Common Resources
