27 Public Goods
For more information on these topics, see Varian Chapter 36: Public Goods.
Public goods are goods that must be provided in the same amount to all affected consumers. Examples include national defense, streets and sidewalks, air quality, and pollution levels. The key characteristic is that everyone must “consume” the same amount of the good, though they may value it differently.
The fundamental challenge with public goods is determining:
- Whether they should be provided at all
- What quantity should be provided
- How to pay for them
- How to get people to truthfully reveal their preferences
27.1 Efficient Provision of Public Goods
Binary Case (Yes/No Decision)
Suppose you and your roommate are trying to decide whether or not to buy a TV for your shared living room. The efficiency condition is:
\[r_1 + r_2 \geq c\]
where:
- \(r_1\) and \(r_2\) are the reservation prices (maximum willingness to pay) of individuals
- \(c\) is the cost of providing the good
You want to be able to buy the TV if the sum of everyone’s willingness to pay exceeds the total cost of the TV.
If your maximum willingness to pay is $100, your roommate’s is $90, and the TV costs $150, then because \(100 + 90 = 190 \geq 150\), you want to be able to buy the TV. The challenge though: how to split the costs fairly, especially when you don’t know eachother’s maximum willingness to pay? We’ll get to this after discussing the continuous case.
Continuous Case
For public goods that can be provided in varying amounts (like streetlights or pollution control), the efficiency condition is:
\[MRS_1 + MRS_2 = MC(G)\]
where:
- \(MRS\) = Marginal Rate of Substitution between private consumption and public good
- \(MC(G)\) = Marginal Cost of providing the public good
- \(G\) = Amount of public good
This means the sum of everyone’s marginal willingness to pay should equal the marginal cost of provision.
Classwork 1) Streetlight Problem
Consider a neighborhood with two residents deciding on how many streetlights to put up:
- Total cost function: \(C(G) = G^2\), where G is number of lights
- Resident 1 has a marginal willingness to pay of: \(MRS_1 = 10 - G\)
- Resident 2 has a marginal willingness to pay of: \(MRS_2 = 8 - G\)
Write down the efficiency condition.
Solve: you should find that \(G = 4.5\).
27.2 The Free Rider Problem
The free rider problem occurs when individuals attempt to benefit from a public good without contributing to its cost, since they cannot be excluded from consuming it once it is provided. For example, if one roommate buys a TV that both can watch, the other roommate may refuse to contribute hoping their roommate will purchase it anyway. Since a public good must be consumed in the same amount by all affected parties, each person has an incentive to minimize their own contribution while still getting the full benefits. This typically leads to an undersupply of public goods relative to what would be Pareto efficient, as people try to “free ride” on others’ contributions rather than reveal their true preferences and willingness to pay.
Classwork 2) Free Riding Game
Two people are considering contributing to a public good that costs $150. Both people value the good at $100. They each have two options: contribute or don’t contribute.
- If both players contribute, they divide the costs in half, each paying a cost of $75 and each enjoying the benefit of $100, so on net, payoffs are (25, 25).
- If only one player contributes, they pay the entire cost of $150, and enjoys the $100 benefit for a net payoff of -50. The other player pays nothing and enjoys the $100 benefit for a net payoff of 100.
- If neither players contribute, the public good is not purchased and both players get payoffs of 0.
Draw the game matrix, label best responses with stars, report whether players have dominant strategies, and report on any Nash Equilibria if they exist.
27.3 Solutions to the Public Goods Problem
Voting
Voting provides one straightforward way to make decisions about public goods. Imagine a small town deciding how many streetlights to install along Main Street. When voters have what we call “single-peaked preferences” - meaning they have one most-preferred amount and like options less the further they get from this preferred amount - voting leads to a predictable outcome. For example:
Suppose 5 voters want to decide how many streetlights to install (0-10):
- Voter 1 prefers 3 lights
- Voter 2 prefers 4 lights
- Voter 3 prefers 6 lights
- Voter 4 prefers 7 lights
- Voter 5 prefers 8 lights
With single-peaked preferences, the median voter (Voter 3) who wants 6 lights will determine the outcome. Why?
Suppose you can vote to get 4 lights or 6 lights. Voters 1 and 2 will vote for 4 lights; everyone else votes for 6 and 6 wins. In this way, 6 lights will win against any alternative. This demonstrates why the median voter’s preference typically prevails in these situations: it represents the compromise position that can’t be defeated by any other proposal.
Voting Paradox
What happens when preferences aren’t so well oredered? Imagine three voters deciding on options A, B, or C.
- Voter 1 prefers A over B, and B over C: \(A \succ B \succ C\).
- Voter 2 prefers C over A, and A over B: \(C \succ A \succ B\).
- Voter 3 prefers B over C, and C over A: \(B \succ C \succ A\).
This creates what economists call a “cycling” problem. Here’s why:
Let’s start by having a vote between A and B. Voter 1 chooses A, Voter 2 chooses A, and Voter 3 chooses B. A wins with two votes. So we might think A is the group’s preferred choice.
But wait - let’s now vote between A and C. Voter 1 chooses A, but Voters 2 and 3 both prefer C over A. So C beats A with two votes.
Surely C must be the winner then? Not quite. If we vote between B and C, Voters 1 and 3 prefer B over C. So B beats C.
We’ve created a circle: A beats B, C beats A, and B beats C. There’s no clear winner because:
This means the final outcome depends entirely on which pair we vote on first. The outcome can be manipulated just by changing the order of votes.
Classwork 3) Voting
Consider again three voters with these preferences:
- Voter 1: \(A \succ B \succ C\)
- Voter 2: \(C \succ A \succ B\)
- Voter 3: \(B \succ C \succ A\)
You want option A to win. You can structure the voting in two rounds:
- First vote between two options (call them X and Y)
- Second vote between the winner and the remaining option (call it Z)
Which positions should you put A, B, and C in (i.e., what should X, Y, and Z be) to ensure A wins?
Seven voters need to decide how many public park benches to install. Each voter has a most preferred number, and likes other options less the further they get from this preference:
- Voter 1 most prefers 1 bench
- Voter 2 most prefers 2 benches
- Voter 3 most prefers 3 benches
- Voter 4 most prefers 4 benches
- Voter 5 most prefers 5 benches
- Voter 6 most prefers 6 benches
- Voter 7 most prefers 7 benches
What will be the voting outcome?
Vickrey-Clarke-Groves (VCG) Mechanism
The VCG mechanism is an economic tool designed to reveal how much people truly value public goods (like parks or infrastructure). Here’s how it works:
Instead of asking people directly how much they’d pay for something, the mechanism creates a system where being honest about their true willingness to pay becomes their dominant strategy. This leads to more efficient decisions about which public projects to pursue.
An important note: The payments collected through VCG aren’t meant to fund the projects themselves (you’ll never collect enough money). Rather, these payments serve as a tool to understand whether a project’s social benefits exceed its costs. Think of it like this: A government with sufficient funding uses VCG to determine which public projects would create the most value for society, not to raise money for those projects.
To see how it works, let’s consider a small town deciding whether to build a public park. There are three residents (A, B, and C) who would benefit differently from the park. The park costs $1200 to build.
Each resident’s true valuation of the park:
- A values it at $700
- B values it at $600
- C values it at $400
So in total, residents value the park at $1700. Because it only costs $1200 to build, the benefits outweigh the costs and it would be efficient to build the park. The key to VCG is that if the person was pivotal in deciding whether or not to build the park, they compensate society for that decision by paying a “pivotal payment”.
For resident A:
- Without A in the group, the social benefit is $1000 and the cost is $1200, so on net, the park has a -$200 value to society without A.
- Resident A was pivotal in making the decision to build the park, so resident A is responsible for that $200 social cost. They have a pivotal payment, and that is $200.
Now let’s consider resident B:
- Without B in the group, the social benefit is $1100 and the cost is $1200, so on net, the park has a -$100 value to society.
- Resident B was pivotal in making the decision to build the park, so resident B is responsible for that $100 social cost. They have a pivotal payment, and that is $100.
Finally, resident C:
- Without C in the group, the social benefit is $1300 and the cost is $1200, so the net benefit of the park is $100.
- Resident C’s valuation was not pivotal in making the decision to build the park, so C will pay no pivotal payment.
In summary, the park gets built because total value ($1700) exceeds the cost ($1200). Each person pays according to how much their participation “harms” others by affecting the group decision.
Classwork Extra Credit: VCG Mechanism (+5 points)
We claimed that the VCG Mechanism is a truth-telling mechanism. Let’s explore how this happens. Take the example above and show that if resident A lies and reports a much lower valuation of $100 (instead of their true value of $700), their utility falls compared to when they report accurately. Show your calculations for both scenarios.
Now calculate resident A’s utility if they lie and report a valuation of $300 (instead of their true value of $700). Why is resident A better off being truthful about their valuation?
Calculate the VCG payments for a public good with a cost of $90 given these residents’ valuations:
- A values the good at $30
- B values the good at $45
- C values the good at $50