Public Goods
and Common Resources
Christopher Makler
Stanford University Department of Economics
Econ 51: Lecture 17
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Section this Week
- Q section, tonight at 6:30 only:
public choice and the VCG mechanism - Kyra's sections, Friday at 10:30 and 11:30: workshopping exam question writing
- Everyone else's section: old exam questions 10.7 and 10.9 from the homework
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−L2
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)
10f(L)
400L−10L2
400L-10L^2
100(35−L)
100(35-L)
3500−100L
3500 - 100L
400−10L
400-10L
100
100
100
100
400−20L
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
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.
gi=Player i’s contribution,i∈{1,2}
g_i = \text{Player i's contribution}, i \in \{1,2\}
G=g1+g2=total contribution
G = g_1 + g_2 = \text{total contribution}
ui(G,xi)=Gxi=Player i’s utility
u_i(G,x_i) = Gx_i = \text{Player i's utility}
xi=12−gi=Player i’s private consumption
x_i = 12 - g_i = \text{Player i's private consumption}
u1(g1,g2)=(g1+g2)(12−g1)
u_1(g_1,g_2) = (g_1 + g_2)(12 - g_1)
Write payoffs in terms of strategies:
u2(g1,g2)=(g1+g2)(12−g2)
u_2(g_1,g_2) = (g_1 + g_2)(12 - g_2)
u1(g1,g2)=(g1+g2)(12−g1)
u_1(g_1,g_2) = (g_1 + g_2)(12 - g_1)
Write payoffs in terms of strategies:
u2(g1,g2)=(g1+g2)(12−g2)
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 g2?
u1(g1,g2)=(g1+g2)(12−g1)
u_1(g_1,g_2) = (g_1 + g_2)(12 - g_1)
Write payoffs in terms of strategies:
u2(g1,g2)=(g1+g2)(12−g2)
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 g2?
u1(g1∣g2)=12g1+12g2−g12−g1g2
u_1(g_1 | g_2) = 12g_1 + 12g_2 - g_1^2 - g_1g_2
u2(g2∣g1)=12g1+12g2−g1g2−g22
u_2(g_2|g_1) = 12g_1 + 12g_2 - g_1g_2 -g_2^2
u1′(g1∣g2)=12−2g1−g2=0
u_1^\prime(g_1 | g_2) = 12 - 2g_1 - g_2 = 0
12−g2=2g1
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:
g1∗(g2)=6−21g2
g_1^*(g_2) = 6 - {1 \over 2}g_2
u2′(g2∣g1)=12−g1−2g2=0
u_2^\prime(g_2 | g_1) = 12 - g_1 - 2g_2 = 0
12−g1=2g2
12 - g_1 = 2g_2
g2∗(g1)=6−21g1
g_2^*(g_1) = 6 - {1 \over 2}g_1
BEST RESPONSE FUNCTIONS
g1∗(g2)=6−21g2
g_1^*(g_2) = 6 - {1 \over 2}g_2
g2∗(g1)=6−21g1
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:
g1=6−21
g_1 = 6 - {1 \over 2}
(6−21g1)
(6 - {1 \over 2}g_1)
g1=6−3+41g1
g_1 = 6 - 3 + {1 \over 4}g_1
43g1=3
{3 \over 4}g_1 = 3
g1=4
g_1 = 4
g2∗(4)=6−21×4=4
g_2^*(4) = 6 - {1 \over 2}\times 4 = 4
g1∗(g2)=6−21g2
g_1^*(g_2) = 6 - {1 \over 2}g_2
g2∗(g1)=6−21g1
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.
g1
g_1
g2
g_2
g1∗(g2)=6−21g2
g_1^*(g_2) = 6 - {1 \over 2}g_2
g2∗(g1)=6−21g1
g_2^*(g_1) = 6 - {1 \over 2}g_1
To solve, plug one's BR into the other:
g1=6−21
g_1 = 6 - {1 \over 2}
(6−21g1)
(6 - {1 \over 2}g_1)
g1=6−3+41g1
g_1 = 6 - 3 + {1 \over 4}g_1
43g1=3
{3 \over 4}g_1 = 3
g1=4
g_1 = 4
g2∗(4)=6−21×4=4
g_2^*(4) = 6 - {1 \over 2}\times 4 = 4
u1(g1,g2)=(g1+g2)(12−g1)
u_1(g_1,g_2) = (g_1 + g_2)(12 - g_1)
u2(g1,g2)=(g1+g2)(12−g2)
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)=2×2g×(12−g)=48g−4g2
U(g) = 2(g+g)(12-g) = 2 \times 2g \times (12-g) = 48g - 4g^2
U′(g)=48−8g
U'(g) = 48 - 8g
g∗=6
g^* = 6
In Nash Equilibrium: u(4,4)=(4+4)(12−4)=8×8=64
\text{In Nash Equilibrium: }u(4,4) = (4 + 4)(12 - 4) = 8 \times 8 = 64
Social Optimum: u(6,6)=(6+6)(12−6)=12×6=72
\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 gi≥0 be the amount citizen i contributes to the show;
each person has income mi, so their private consumption is xi=mi−gi.
Each minute of fireworks costs $200.
There are two types of citizen:
200 kinda love fireworks
u(G,xi)=12G+xi
u(G, x_i) = 12\sqrt G + x_i
u(G,xi)=4G+xi
u(G, x_i) = 4\sqrt G + x_i
Note: xi is private consumption (dollars left over after contributing gi) so we can also write
u(G,gi)=12G+mi−gi
u(G, g_i) = 12\sqrt G + m_i - g_i
u(G,gi)=4G+mi−gi
u(G, g_i) = 4\sqrt G + m_i - g_i
100 really love fireworks
G = total number of minutes of fireworks
gi≥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,gi)=12G+mi−gi
u(G, g_i) = 12\sqrt G + m_i - g_i
u(G,gi)=4G+mi−gi
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
gi≥0 = amount citizen i contributes
100 really love fireworks
200 kinda love fireworks
u(G,gi)=12G+mi−gi
u(G, g_i) = 12\sqrt G + m_i - g_i
u(G,gi)=4G+mi−gi
u(G, g_i) = 4\sqrt G + m_i - g_i
What is the efficient length of a fireworks show?
MB=G6
MB = {6 \over \sqrt{G}}
MB=G2
MB = {2 \over \sqrt{G}}
Cost
200G
200G
MC=200
MC = 200
"Samuelson Condition" : set the sum of the MB's equal to the MC:
100×G6
100 \times {6 \over \sqrt{G}}
+
+
=
=
200×G2
200 \times {2 \over \sqrt{G}}
200
200
G∗=25
G^* = 25
G = total number of minutes of fireworks
gi≥0 = amount citizen i contributes
100 really love fireworks
200 kinda love fireworks
What is the efficient length of a fireworks show?
MB=G6
MB = {6 \over \sqrt{G}}
MB=G2
MB = {2 \over \sqrt{G}}
Cost
200G
200G
MC=200
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,gi)=12G+mi−gi
u(G, g_i) = 12\sqrt G + m_i - g_i
u(G,gi)=4G+mi−gi
u(G, g_i) = 4\sqrt G + m_i - g_i
G = total number of minutes of fireworks
gi≥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
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,gi)=60+mi−gi
u(25, g_i) = 60 + m_i - g_i
u(25,gi)=20+mi−gi
u(25, g_i) = 20 + m_i - g_i
5000
5000
What's the problem with this?
u(G,gi)=12G+mi−gi
u(G, g_i) = 12\sqrt G + m_i - g_i
u(G,gi)=4G+mi−gi
u(G, g_i) = 4\sqrt G + m_i - g_i
G = total number of minutes of fireworks
gi≥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(gi)=1223gi+mi−gi
u(g_i) = 12\sqrt{{3 \over 2}g_i} + m_i - g_i
u(gi)=423gi+mi−gi
u(g_i) = 4\sqrt{{3 \over 2}g_i} + m_i - g_i
gi∗=54
g_i^* = 54
gi∗=6
g_i^* = 6
How many minutes would a majority of citizens vote for?
u(G,gi)=12G+mi−gi
u(G, g_i) = 12\sqrt G + m_i - g_i
u(G,gi)=4G+mi−gi
u(G, g_i) = 4\sqrt G + m_i - g_i
u(gi)=1223gi+mi−gi
u(g_i) = 12\sqrt{{3 \over 2}g_i} + m_i - g_i
u′(gi)=gi623−1=0
u'(g_i) = {6 \sqrt{3 \over 2} \over \sqrt{g_i}} - 1 = 0
u′(gi)=2gi23−1=0
u'(g_i) = 2\sqrt{{3 \over 2} \over \sqrt{g_i}} - 1 = 0
Total contributions: $300g
Amount of public good: G=$$200/min300g=23g minutes
Q Section: Social Choice
Up to now we've had an individual agent choosing something.
Societies also make choices:
how much to tax and spend, who to lead them, etc.
How do we evaluate mechanisms for aggregating preferences?
Big Ideas
We're going from individual choice to social choice.
Up to now we've had an individual agent choosing something.
Societies also make choices: how much to tax and spend, who to lead them, etc.
How do we evaluate mechanisms for aggregating preferences?
It's "impossible" to find an aggregation method that works in general.
However, under certain more restrictive assumptions on preferences,
some mechanisms will work better than others, with predictable outcomes.
From Social Preferences to Social Choice
Terminology
A SWF ranks all possible alternatives:
e.g., has indifference curves within the Edgeworth Box
Just needs to pick a winner;
sometimes called a Social Choice Function (SCF)
Arrow's Impossibility Theorem
Desirable Properties of a Social Welfare Function
- Non-dictatorship: it isn't just simply the preferences of one dictator.
- Unanimity: if everyone likes A better then B, then society cannot prefer B to A.
- Independence of Irrelevant Alternatives: if society prefers A to B when C isn't an option, it cannot prefer B to A when C is an option.
Arrow's Theorem: There is no complete and transitive SWF that satisfies all three of these.
Example: Failure of Majority Rule
Ana, Beth, and Claudia are buying a television.
They can buy an RCA (R), Sony (S), or Toshiba (T).
Their preference rankings are given by the following:
Suppose we were to create a social preference ordering by pairwise ranking each alternative, choosing the preferred one by majority rule.
Then society prefers R to S...
and S to T...
and T to R...
which isn't transitive!
Another Mechanism: Borda Count
Let's assign a point value to your choice ranking:
Now we see that this is a tie: each gets 6 points.
But now let's look to see what happens if there's a fourth option, a Zenith:
Now R has 6 points, but S and T have 7. So the introduction of Z changes the preferences over
R, S, T => violates the independence of irrelevant alternatives axiom.
The Condorcet Paradox
As in the TV question, a CW or CL may not always exist, which can result in a voting cycle!
If you set the voting rule,
you can choose the winner.
| Voting Rule | Winner |
|---|---|
| S vs T, then winner vs. R | R |
| R vs T, then winner vs. S | S |
| S vs R, then winner vs. T | T |
I do not expect you to solve a VCG problem on the final exam; however, it is fair game to ask what problem it is trying to solve,
and its approach to solving that problem.
Note
The VCG Mechanism
General class of problems: a group of people with
private information has to make a collective choice.
For example: suppose three roommates are moving into a new apartment,
so there are 6 possible ways they could be assigned to rooms.
They each have preferences over all 6 possible combinations.
How can they decide who gets which room?
General Formulation
There are people i=1,...,N and possible decisions k=1,...,K.
Person i would get payoff vik if option k is chosen.
The goal is to collect accurate information about individual values and pick the decision k∗ that maximizes the total value:
A direct mechanism asks each person to report their values for each outcome and select the one that maximizes total welfare.
W(k)=v1k+⋯+vNk
Each player can, by its report, force the selection of any outcome, so why would it always choose k∗?
Goal: align payoffs of individuals with the total social payoff.
Mechanism: define a series of transfer payments based on stated valuations that encourage truthful revelation of value.
Vickrey-Clarke-Groves Mechanism
1. Each player reports their vector of values for each of the K alternatives: vi=(vi1,...,viK). They can lie if they want.
3. When the "center" chooses k, it pays player i:
Analysis of incentives: player i wants to induce the center to choose the k that maximizes:
2. The "center" selects the k∗ that maximizes the total reported value.
pi(k)=j=i∑vjk+fn(v−i)
\displaystyle{p_i(k) = \sum_{j \ne i} v_{jk} + f_n(v_{-i})}
vik+pi(k,v−i)
v_{ik} + p_i(k, v_{-i})
=vik+j=i∑vjk+fn(v−i)
\displaystyle{= v_{ik} + \sum_{j \ne i} v_{jk} + f_n(v_{-i})}
They do this by reporting truthfully!
A player is said to be pivotal when its reported vector of values
changes the outcome of the mechanism,
compared to the case in which they didn't participate
(i.e., reported just a vector of zeroes).
In a pivot mechanism, only the pivotal players have a side payment.
In particular, they have to pay the amount
by which they change other people's (reported) utility.
Thoughts on the Last 20 Weeks: What is a microeconomic model,
and what is it good for?
Thank You
Public Goods and Common Resources Christopher Makler Stanford University Department of Economics Econ 51: Lecture 17
Econ 51 | 17 | Public Goods
By Chris Makler
Econ 51 | 17 | Public Goods
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