Public Goods
 11 minutes read  2271 wordsContext
 Free markets of regulation? Externalities can lead to inefficient economic allocations. One remedy in such occasions is for the government to intervene and define a clear structure (new market) in which property rights can be traded.
 Nevertheless, when property rights are nonexcludable and should be split among more than one interested party, coordination difficulties become severe and attempts to organize (missing) markets can completely fail.
 What characteristics lead to market provision failures?
 What are the welfare implications of such failures?
 Are there alternative production plans that can be used to produce such products and services?
Course Structure Overview
Lecture Structure and Learning Objectives
Structure
 Open Source Software (Case Study)
 Basic Concepts
 A Public Good Game Example
 Welfare Comparisons
 Application: Predictions in the Open Source Software case
 Current Field Developments
Learning Objectives
 Explain the concept of public goods and illustrate why they are relevant.
 Introduce and explain the freeriding problem.
 Explain the difficulties arising in the provision of public goods.
 Illustrate the freeriding problem using a public good game.
 Explain why opensource software is so prevalent today.
Open Source Software
 Much of the modern software is free and/or open source.
 Everyone can download and use the software without paying for it.
 In many cases, the source code is also distributed using permissive licenses that
 allow users to modify the source according to their needs,
 redistribute (and even sell) the code.
 Usually, there is no direct payment for the developers of the code.
Enterprises Using Open Source for It Infrastructure Modernization Worldwide in 2020 & 2021, by Region
 Most of the digital software infrastructures use at least to some extent open source software.
 The time trend of using open software is positive.
Distribution of Permissive & Copyleft Open Source Licenses Worldwide from 2012 to 2021
 Companies distribute software products with permissive licenses.
 The share of permissive products almost doubled during the last ten years.
Most Important Open Source Skills for Professionals According to Hiring Managers, as of 2022
 Managers' have endorsed the opensource software model today.
 Although might be permitted by its license, setting a price for opensource cannot exclude nonpaying consumers from using it.
 Opensource usage from one company does not reduce the capacity of other companies to use the same software.
Reasons why Professionals Worldwide Choose Careers in Open Source as of 2022
 Industry culture (status quo) affects developers' decisions.
 Ideals and beliefs play a significant role.
 There is no market system driving development and collaboration.
Excludability and Rivalry
 A public good is a commodity or service that is nonexcludable and nonrivalrous.
 A good (or a service) is non excludable if the consumption of the good cannot be limited to only paying customers.
 A good (or a service) is non rivalrous if the consumption of the good from an agent does not reduce the ability of others to consume it.
Excludability and Rivalry Examples
Excludable  Non Excludable  

Rivalrous  apples, oranges  natural resources 
Non Rivalrous  theaters, concerts  opensource software, public television 
Public and Other Goods Examples
Excludable  Non Excludable  

Rivalrous  private goods  common goods 
Non Rivalrous  club goods  public goods 
 A private good is a commodity or service that is excludable and rivalrous.
 A common good is a commodity or service that is nonexcludable and rivalrous.
 A club good is a commodity or service that is excludable and nonrivalrous.
 Bonus: What is the difference between public goods and externalities?
Market collapse
 The combination of nonexcludability and nonrivalry can make the marketbased provision of public goods impossible.
 Non altruistic agents avoid contributing to the public good production and attempt to freeride.
 This is typically Pareto inefficient.
Free Riding
 The free riding problem occurs whenever there are economic agents who benefit from using resources either without paying at all or by underpaying for acquiring them.
 Examples:
 All citizens consume the same level of national defense, irrespective of their individual contributions to the national budget.
 All ships benefit from a lighthouse, irrespective of paying for its construction.
Free Riding Game
 Two roommates consider subscribing to Netflix (say \(15\) Euros per subscription).
 Each roommate values the service \(10\) Euros.
 No one ends up buying the subscription.
 Bonus: What is the difference with prisoners' dilemma?
The Public Good Game
 Suppose that there are \(2\) firms in a market.
 Each firm has a budget equal to \(100\) Euros.
 They decide how much to contribute to the production of a network infrastructure (say \(x_{1}\) and \(x_{2}\)).
 The sum of the contributions is used in the production of the infrastructure. \[q = f(x_{1}, x_{2}) = 3\frac{x_{1} + x_{2}}{2}.\]
 The produced output (infrastructure capacity) is equally split.
The Firm’s Problem
 The firms maximize their profits \[\pi_{i}\left(x_{1}, x_{2}\right) = 3\frac{x_{1} + x_{2}}{4} + 100  x_{i}.\]
The Market Solution
 The profit of firm \(1\) is strictly decreasing in its contribution. \[\frac{\partial \pi_{1}}{\partial {x_{1}}}\left(x_{1}, x_{2}\right) = \frac{3}{4}  1 < 0.\]
 Therefore, \(x_{1} = 0\) and \[\pi_{1}\left(0,x_{2}\right) = 3\frac{x_{2}}{4} + 100.\]
 Since the game is symmetric, the same argument is valid for agent \(2\).
 Therefore the Nash equilibrium of the game is \(x_{i} = 0\), with payoffs \(\pi_{i} = 100\) for every agent \(i\).
 The total welfare, in this case, is equal to \(\pi_{1} + \pi_{2} = 200\).
 Both firms try to freeride, and no production takes place.
Efficient Provision of the Public Good

The total welfare in the market is determined by (why?)
\begin{align*} W\left(x\right) = 3\frac{x}{2} + 200  x. \end{align*}

The total welfare is increasing in \(x\). Therefore, it is maximized for \(x = 200\).

The maximized total welfare is \(W(200) = 300\).

This solution is Pareto dominates the market solution \(W(200)>\pi_{1} + \pi_{2}\).
Discussion: Free and Open Source Software
 Is opensource software a public good?
 Can opensource software be priced?
 Why do people contribute and use opensource software?
Current Field Developments
 Public goods games are used by economists to study cooperation and coordination (Takeuchi and Seki 2023)
 Digital public goods have a central role in the discussion of achieving a sustainable economy.
 At the same time, they create a series of new challenges:
 Should the AI models be democratized?
 What are the security implications of using Open Source Software in the governments' infrastructures?
Concise Summary
 Free riding can lead to economically inefficient allocations.
 A common reason that free riding arises is the presence of public goods.
 Public goods can lead to a complete production shutdown.
 State provision is the most common way of supplying public goods.
Further Reading
 Varian (2010, secs. 37.14)
 CORE Team (2017, sec. 12.5)
Mathematical Details
Public goods
A public good is a good that is nonexcludable and nonrivalrous. The consumption of a commodity is nonrivalrous when its consumption from an agent does not affect the availability of the commodity for other agents. The consumption of a commodity is nonexcludable when it is impossible to exclude agents from its consumption.
The public good game
Suppose that there are \(n\) agents in a market. Each agent has a budget equal to \(B\). She decides how much she contributes to the production of a public good. The sum of the contributions of all players is used as input in a linear production technology that scales them by \(1 < A < n\). The production output is split equally among all agents. The agent payoff is determined by \[\pi_{i}\left(x_{1}, …, x_{n}\right) = \frac{A \sum_{j=1}^n x_j}{n} + B  x_{i}.\]
The market solution
The payoff of each agent is strictly decreasing in her own contribution, i.e., \[\frac{\partial \pi_{i}}{\partial {x_{i}}}\left(x_{1}, …, x_{n}\right) = \frac{A}{n}  1 < 0.\] Therefore, the payoff is maximized for \(x_{i} = 0\), resulting in \[\pi_{i}\left(0, …, x_{n}\right) = \frac{A \sum_{j=i}^n x_j}{n} + B.\] Since the game is symmetric, the same argument is valid for all agents. Therefore, the Nash equilibrium of the game is \(x_{i} = 0\), with payoffs \(\pi_{i} = B\) for every agent \(i\). The total welfare, in this case, is equal to \(nB\).
The efficient provision of the public good
The total welfare in the market is determined by \[W\left(x\right) = A x + n B  x\] The total welfare is increasing in \(x\). Therefore, it is maximized for \(x = nB\). Thus, the total welfare is \(\pi = AnB\) at the optimum. Since \(A > 1\), this solution Pareto dominates the market solution.
Exercises
Group A

Suppose there are two individuals, \(1\) and \(2\), each consuming one private good in the amounts \(x_{1}\) and \(x_{2}\), respectively. The price of this private good is \(p\). In addition, they consume a public good \(G\). The marginal cost of production of \(G\) is equal to one. The amount of the public good is determined by \(G = g_1 + g_2\), where \(g_{1}\) and \(g_{2}\) are the individual contributions of the players. Individual \(i\)’s preferences are represented by a CobbDouglas utility function \(u(x_{i}, G) = x_{i}^{\alpha}G^{\beta}\). Both players have a budget of \(B\).
 Explain why in this case, \(G\) can be considered a public good.
 Set up individual \(i\)’s budget constraint.
 Find individual \(i\)’s best response contribution \(g_{i}\).
 Compute the Nash equilibrium for the contributions \(g_{1}\) and \(g_{2}\).
 Does the Nash equilibrium allocation entail more or fewer resources directed to the public good than the Pareto efficient allocation?
 Argue that the Nash equilibrium is not Paretoefficient.

The good \(G\) is nonrivalrous, as both players consume the exact same good \(G\), and nonexcludable as the good is not priced in the market.

The budget of individual \(i\) is \(p x_{i} + g_{i} \le B\).

Individual \(i\)’s optimization problem is
\begin{align*} \max_{x_{i}, g_{i}} \left\{ u(x_{i}, G) + \lambda(B  p x_{i}  g_{i}) \right\}, \end{align*}
where \(\lambda\) is the Lagrange multiplier. The first order conditions are
\begin{align*} \alpha \frac{u(x_{i}, G)}{x_{i}} &= \lambda p, \\ \beta \frac{u(x_{i}, G)}{g_{i}} &= \lambda, \end{align*}
which imply
\begin{align*} \frac{\alpha}{\beta}G &= px_{i}. \end{align*}
Combining the last condition with the budget constant, we get
\begin{align*} \frac{\alpha}{\beta}G &= B  g_{i}, \end{align*}
from which we conclude that the best response is
\begin{align*} g_{i} &= \frac{\beta}{\alpha + \beta}B  \frac{\alpha}{\alpha + \beta}g_{j}. \end{align*}

Due to symmetry, the Nash equilibrium can be easily calculated by setting \(g_{i}=g_{j}\), which results in
\begin{align*} g_{n} &= \frac{\beta}{2\alpha + \beta}B. \end{align*}

The Pareto efficient solutions are obtained by solving the joint maximization problem
\begin{align*} \max_{x_{1}, x_{2}, G} \left\{ u(x_{1}, G) + u(x_{2}, G) + \mu(2B  p (x_{1} + x_{2})  G) \right\}, \end{align*}
where \(\mu\) is the Lagrange multiplier of this problem. The first order conditions are
\begin{align*} \alpha \frac{u(x_{1}, G)}{x_{1}} &= \mu p, \\ \alpha \frac{u(x_{2}, G)}{x_{2}} &= \mu p, \\ \beta \frac{u(x_{1}, G) +u(x_{2}, G)}{G} &= \mu. \end{align*}
The first two conditions and the budget constraint imply
\begin{align*} \alpha \left( u(x_{1}, G) + u(x_{2}, G) \right) &= \mu p \left( x_{1} + x_{2} \right) = \mu \left( 2B  G \right). \end{align*}
Combining with the third first order condition gives
\begin{align*} G_{p} &= \frac{\beta}{\alpha + \beta}2B. \end{align*}
For \(\alpha \neq 0\), \(G_{p} \neq 2 g_{n}\), from which we conclude that the Nash equilibrium is not Pareto efficient.

In the Nash equilibrium, fewer than the Pareto efficient resources are allocated in the production of the public good. Specifically, for all \(\alpha>0\), we have \(G_{p} > 2 g_{n}\).
Group B

Consider a public good game with two players whose payoffs are determined by \(\pi_{1}(q) = (1 + q) (B_{1}  q x_{1})\) and \(\pi_{2}(q) = (2 + q)(B_{2}  q x_{2})\). In these expressions, \(x_{1}\) and \(x_{2}\) represent the player specific, fixed contributions to producing the public good, and \(q\) the public good output. Players have a binary choice concerning the public good; they can commonly choose \(q = 1\) and purchase the public good, or \(q=0\) and abstain from it. Player 1 has budget \(B_{1}\), while player 2 has budget \(B_{2}\).
 What is the maximum amount that player 1 is willing to contribute to the public good?
 What is the maximum amount that player 2 is willing to contribute to the public good?
 Suppose that \(B_{1} = 100\) and \(B_{2} = 75\). By producing the public good, the players have a Pareto improvement over not producing it, as long as the production cost of the public good is no greater than \(\hat x\). Find \(\hat x\).
 Find a combination of specific contributions \(x_{1}\) and \(x_{2}\) whose sum is less than \(\hat x\), but for which letting the players individually maximize their payoffs does not result in a Pareto efficient allocation.

Player 1 prefers consuming the public good if and only if
\begin{align*} \pi_{1}(1) \ge \pi_{1}(0) \iff 2 (B_{1}  x_{1}) \ge B_{1} \iff x_{1} \le \frac{B_{1}}{2}. \end{align*}

For player 2, an analogous approach gives
\begin{align*} \pi_{2}(1) \ge \pi_{2}(0) \iff 3 (B_{2}  x_{2}) \ge 2 B_{2} \iff x_{2} \le \frac{B_{2}}{3}. \end{align*}

The total production cost of the public good, say \(x\), has to be covered by the sum of the individual contributions, i.e., \(x_{1} + x_{2} \ge x\). Both players prefer covering the cost if
\begin{align*} x &\le x_{1} + x_{2} \\ &\le \frac{B_{1}}{2} + \frac{B_{2}}{3} \\ &\le \frac{3B_{1} + 2B_{2}}{6} \\ &= \frac{300 + 150}{6} = 75 =: \hat x \end{align*}

Suppose that \(x_{2}=50\) and \(x_{1}=20\). We then have \(x_{1} + x_{2} = 70 < 75 = \hat x\), but player 2 prefers not to contribute to the public good production because \(x_{2}>25\).