Public Goods
and Common Resources

Christopher Makler

Stanford University Department of Economics

Econ 51: Lecture 17

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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

When the benefits of a good accrue to everyone,
everyone has an incentive to shirk their contribution 

Model 2: Voluntary Contribution to a Public Good, and the Free Rider Problem

Two people, 1 and 2, can contribute to a public good. Each has income \(m = 12\).

g_i = \text{Player i's contribution}, i \in \{1,2\}
G = g_1 + g_2 = \text{total contribution}
u_i(G,x_i) = Gx_i = \text{Player i's utility}
x_i = 12 - g_i = \text{Player i's private consumption}
u_1(g_1,g_2) = (g_1 + g_2)(12 - g_1)

Write payoffs in terms of strategies:

u_2(g_1,g_2) = (g_1 + g_2)(12 - g_2)
u_1(g_1,g_2) = (g_1 + g_2)(12 - g_1)

Write payoffs in terms of strategies:

u_2(g_1,g_2) = (g_1 + g_2)(12 - g_2)

We could start to write out a payoff matrix...

but the strategy space is continuous so we could never list out every possible (real number) contribution.

Instead, let's think about a best response function

What is player 1's best response to player 2 choosing to contribute some amount \(g_2\)?

u_1(g_1,g_2) = (g_1 + g_2)(12 - g_1)

Write payoffs in terms of strategies:

u_2(g_1,g_2) = (g_1 + g_2)(12 - g_2)

Solve for each player's best response function

What is player 1's best response to player 2 choosing to contribute some amount \(g_2\)?

u_1(g_1 | g_2) = 12g_1 + 12g_2 - g_1^2 - g_1g_2
u_2(g_2|g_1) = 12g_1 + 12g_2 - g_1g_2 -g_2^2
u_1^\prime(g_1 | g_2) = 12 - 2g_1 - g_2 = 0
12 - g_2 = 2g_1

To maximize give the other's strategy, take the derivative of your payoff function
with respect to your own strategy and set it equal to zero:

g_1^*(g_2) = 6 - {1 \over 2}g_2
u_2^\prime(g_2 | g_1) = 12 - g_1 - 2g_2 = 0
12 - g_1 = 2g_2
g_2^*(g_1) = 6 - {1 \over 2}g_1

BEST RESPONSE FUNCTIONS

g_1^*(g_2) = 6 - {1 \over 2}g_2
g_2^*(g_1) = 6 - {1 \over 2}g_1

BEST RESPONSE FUNCTIONS

In a Nash equilibrium, each player is choosing a strategy which is a best response to the other's strategy.

To solve, plug one's BR into the other:

g_1 = 6 - {1 \over 2}
(6 - {1 \over 2}g_1)
g_1 = 6 - 3 + {1 \over 4}g_1
{3 \over 4}g_1 = 3
g_1 = 4
g_2^*(4) = 6 - {1 \over 2}\times 4 = 4
g_1^*(g_2) = 6 - {1 \over 2}g_2
g_2^*(g_1) = 6 - {1 \over 2}g_1

BEST RESPONSE FUNCTIONS

In a Nash equilibrium, each player is choosing a strategy which is a best response to the other's strategy.

g_1
g_2
g_1^*(g_2) = 6 - {1 \over 2}g_2
g_2^*(g_1) = 6 - {1 \over 2}g_1

To solve, plug one's BR into the other:

g_1 = 6 - {1 \over 2}
(6 - {1 \over 2}g_1)
g_1 = 6 - 3 + {1 \over 4}g_1
{3 \over 4}g_1 = 3
g_1 = 4
g_2^*(4) = 6 - {1 \over 2}\times 4 = 4
u_1(g_1,g_2) = (g_1 + g_2)(12 - g_1)
u_2(g_1,g_2) = (g_1 + g_2)(12 - g_2)

Could they both do better?

 

What would a social planner do?

U(g) = 2(g+g)(12-g) = 2 \times 2g \times (12-g) = 48g - 4g^2
U'(g) = 48 - 8g
g^* = 6
\text{In Nash Equilibrium: }u(4,4) = (4 + 4)(12 - 4) = 8 \times 8 = 64
\text{Social Optimum: }u(6,6) = (6 + 6)(12 - 6) = 12 \times 6 = 72

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 3: 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 3: 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

\(G\) = 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: \(G = \${300g \over \$200/\text{min}} = {3 \over 2}g \text{ minutes}\)

Thoughts on the Last 20 Weeks: What is a microeconomic model,
and what is it good for?

Thank You