Market Failures

Context

• Free markets of regulation? Ideal free markets result in economically efficient allocations on many occasions. Property rights and enforceable contracts are two necessary conditions for a free market structure to be viable.

• Nevertheless, property rights and enforceable contracts are not available in all situations, markets fail, and states intervene.
  • What characteristics lead to market failures?
  • What are the welfare implications of such failures?
  • Do policy interventions improve the welfare conditions of the agents?

Course Structure Overview

Lecture Structure and Learning Objectives

Structure

• The Aralkum Desert (Case Study)
• Basic Concepts
• An Externality Example
• Application: Predictions in the Aral Sea case
• A Public Good Example
• Welfare Comparisons
• Current Field Developments

Learning Objectives

• Illustrate production externalities using an ecological collapse case study.
• Describe the ecological collapse by an externality model.
• Compare model predictions with the case study events.
• Explain the difficulties arising in the provision of public goods.
• Illustrate the free-riding problem using a public good game.

The Aralkum Desert

The Desert of Forgotten Ships

The Aral Sea Aralkum Desert

The White Gold

• In the early 1960s, the Soviet government planned for cotton, or ‘white gold’, to become a major exporting industry.

• Amu Darya and Syr Darya rivers were diverted from feeding the Aral sea to irrigating the desert.
• The amount of water taken from the rivers doubled between 1960 and 2000.
• The plan was a success.
• In 1988, Uzbekistan was the largest exporter of cotton in the world.
• In 2006, \(17\%\) of Uzbekistan’s exports came from cotton.

The Ship Graveyards

Production Externalities

• On some occasions, agents’ actions outside of a production process affect the outcome of production.

• Environmental spillover cases are classic examples of negative effects.
  • E.g., a fishery and any number of consumer/producer polluters placed on the same river.
• The impact of external agents does not need to be negative.
  • E.g., an apple orchard and a beekeeper placed next to each other.

• A production technology exhibits a production externality when its production set (or production output) depends on the choices of economic agents other than the producing firm.

Types of Externalities

• A positive production externality is an externality for which a production process is boosted by the actions of external economic agents that are not beneficiaries of the profit derived from the process.
• A negative production externality is an externality for which a production process is hindered by actions of economic agents that are not beneficiaries of the profit derived from the process.

A Pollution Example

• Suppose there are two markets with one profit maximizing firm in each of them.
• The firm of market \(1\) produces cotton and uses a river for irrigation. The more water it draws from the river, the lower the production cost becomes.
• The firm of market \(2\) is a fishery. It does not control the level of water in the river. However, the less water there is in the river, the more costly it is to catch fish.
• Let \(p_{c}=5\) be the price of cotton and \(p_{f}=2\) be the price of fish. The water level is not priced in the market.
• Let \(\xi\in[0,1]\) be a variable that measures how much water of the river is used for irrigation as a percentage of its full capacity.

Costs with Externalities

• The cotton producer has production cost \[c_{c}(q_{c}, \xi) = q_{c}^{2} \left(5 + \left(1 - \xi\right)^{2}\right)\]
• The cost is decreasing in the percentage of water used in irrigation.
• The fishery cost function is \[c_{f}(q_{f}, \xi) = q_{f}^{2}\left(1 + \xi^{2}\right)\]
• The cost is increasing in the percentage of water used in irrigation (externality).

The Firms’ Problems

• The cotton producer solves \[\max_{q_{c}, \xi} \left\{ p_{c}q_{c} - q_{c}^{2} \left(5 + \left(1 - \xi\right)^{2}\right) \right\}\]
• The fishery problem is \[\max_{q_{f}} \left\{ p_{f}q_{f} - q_{f}^{2}\left(1 + \xi^{2}\right) \right\}.\]
• The fishery does not control the level of externality, but its cost function is increasing in it.

The Market Solution

• The cotton producer would like to choose \(q_{c}\) and \(\xi\) such that

  \begin{align*} 5 &= 2 q_{c} \left(5 + \left(1 - \xi\right)^{2}\right) \\ 0 &= -2 q_{c}^{2} \left(1 - \xi\right) \end{align*}

• This gives \(\xi = 1\) and \(q_{c}=\frac{1}{2}\).

• The fishery chooses \[2 = 2 q_{f} \left(1 + \xi^{2}\right)\]

• This gives (for \(\xi = 1\)) \(q_{f}=\frac{1}{2}\).

The Market Solution’s Welfare

• The river is dried out (\(\xi = 1\)).
• The cotton producer has profit \[\pi_{c} = 5 \frac{1}{2} - \frac{1}{4} 5 = \frac{5}{4}\]
• The fishery has profit \[\pi_{f} = 2 \frac{1}{2} - \frac{1}{4} 2 = \frac{1}{2} \]

The Merged Firm’s Problem

• Consider a merged, profit maximizing firm that produces in both markets.
• The merged firm solves \[\max_{q_{c}, q_{f}, \xi} \left\{ p_{c}q_{c} + p_{f}q_{f} - q_{c}^{2} \left(1 + \left(1 - \xi\right)^{2}\right) - q_{f}^{2} \xi \right\} .\]

The Merger Solution

• The merged firm chooses

  \begin{align*} 5 &= 2 q_{c} \left(5 + \left(1 - \xi\right)^{2}\right) \\ 2 &= 2 q_{f} \left(1 + \xi^{2}\right) \\ q_{c}^{2} 2 \left(1 - \xi\right) &= q_{f}^{2} 2 \xi \end{align*}

• This gives approximately \(\xi = 0.17\), \(q_{c}=0.44\), and \(q_{f}=0.97\).

The Merger Solution Welfare

• The externality is internalized.
• The river is not dried out (\(\xi \approx 17\%\)).
• Cotton production is reduced (\(0.44\) instead of \(\frac{1}{2}\)).
• Fish production is increased (\(0.97\) instead of \(\frac{1}{2}\)).
• The merged firm’s profit is \[\pi_{m} \approx 2.07 > \frac{5}{4} + \frac{1}{2} = 1.75\]
• The profit of the merged firm is greater than the sum of the profits of the market with two firms
• This shows that the market solution is not efficient.

Application: What Happened in the Aral Sea?

• The model shows that internalizing the externality is economically more efficient than the market solution.

• In the case of the Aral Sea, the externality was not internalized, despite that the Soviet Union’s economy was centrally planned.
• Why did this happen?
• Did we account for all the costs in the model? Did the Soviet regime do?

Public goods

• A public good is a commodity or service that is non-excludable and non-rivalrous. Non excludability means that the consumption of the public good cannot be limited to only paying customers. Non rivalry means that the consumption of the public good from an agent does not reduce the ability of others to consume it.

• Examples:
  • national defense,
  • Free and open-source software

Market collapse

• The combination of non-excludability and non-rivalry makes the market-based provision of public goods impossible.
• Non altruistic agents avoid contributing to the public good production and attempt to free-ride.
• This is typically Pareto inefficient.

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. \[q = 3\frac{x_{1} + x_{2}}{2}.\]
• The produced output (infrastructure capacity) is equally split.

The Firm’s Problem

• The firms’ profits are \[\pi_{i}\left(x_{1}, x_{2}\right) = 3\frac{x_{1} + x_{2}}{4} + 100 - x_{i}.\]

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 free-ride, and no production takes place.

Efficient Provision of the Public Good

• The total welfare in the market is determined by

  \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 \(W(200)>\pi_{1} + \pi_{2}\).

Current Field Developments

• Green growth and circular economy are very high on the European policy agenda.
  • In the “Europe 2020” strategy (published in 2010) it was discussed as sustainable, green growth.
  • In 2021, discussions for a new plan termed “A European Green Deal” had started.
• Many economists agree that carbon taxes are the most efficient way to tackle climate change (Council, 2019).
• There are many difficulties dealing with environmental externalities at an international level.
• The Paris agreement (2015) is an international treaty on climate change signed by 193 states.

Comprehensive Summary

• Market failures can lead to economically inefficient allocations or even complete market collapse.
• Two common reasons that markets fail are the presence of externalities or public goods.
• Externalities lead to non Pareto efficient allocations.
• They are very relevant in the analysis of environmental issues.
• If there is a way to restructure production so that the externality is internalized, the market solution can become efficient.
• Public goods can lead to complete production shutdown.
• State provision is the most common way of supplying public goods.

Further Reading

• Varian, 2010, secs. 35.3-7, 37.1-4
• CORE-Team, 2017, secs. 12.1, 12.5

Mathematical Details

Production Externalities

Suppose there are two markets with one profit maximizing firm in each of them. The emitter is a firm in the first market producing an externality that negatively affects the production cost of the second market. The receiver is a firm in the second market that cannot control the externality, yet this externality positively affects its production cost.

The emitter’s problem

The emitter solves \[\max_{q_{e}, \xi} \left\{ p_{e}q_{e} - c_{e}\left(q_{e}, \xi\right) \right\}.\] The externality, denoted by \(\xi\), is not priced in the market. The emitter controls the level of externality, and its cost function is decreasing in the externality.

The receiver’s problem

The receiver solves \[\max_{q_{r}} \left\{ p_{r}q_{r} - c_{r}\left(q_{r}, \xi\right) \right\}.\] The receiver does not control the level of externality, and its cost function is increasing in the externality.

The decentralized solution

For interior solutions, the emitter produces at the point that solves

\begin{align*} p_{e} &= \frac{\partial c_{e}}{\partial q_{e}}\left(q_{e}, \xi\right) \\ 0 &= \frac{\partial c_{e}}{\partial {\xi}}\left(q_{e}, \xi\right) \end{align*}

The receiver produces at the point that solves \[p_{r} = \frac{\partial c_{r}}{\partial q_{r}} \left(q_{r}, \xi\right)\] The emitter ignores the cost that the externality induces to the receiver.

The centralized solution

How can the externality be internalized? Consider a merged, profit maximizing firm that produces in both markets. The effect of the production externality of the emitting on the receiving production process is taken into account.

The merged firm solves \[\max_{q_{e}, q_{r}, \xi} \left\{ p_{e}q_{e} + p_{r}q_{r} - c_{e}\left(q_{e}, \xi\right) - c_{r}\left(q_{r}, \xi\right) \right\} .\]

For interior solutions, the merged firm produces at quantity levels that solve

\begin{align*} p_{e} &= \frac{\partial c_{e}}{\partial q_{e}}\left(q_{e}, \xi\right) \\ p_{r} &= \frac{\partial c_{r}}{\partial q_{r}}\left(q_{r}, \xi\right) \\ \frac{\partial c_{e}}{\partial {\xi}}\left(q_{e}, \xi\right) &= -\frac{\partial c_{r}}{\partial {\xi}}\left(q_{r}, \xi\right) \end{align*}

The first two conditions are also present in the non merged firms’ case. The third condition replaces the second optimization condition of the emitter’s problem in the decentralized solution. This condition incentivizes the merged firm to take into account the effects of the externality in the second production process.

Non efficiency

The market outcome is not Pareto efficient in the presence of externalities. The emitter tends to produce more than the efficient output at the cost of producing greater externalities. The receiver tends to produce less than the efficient output due to the presence of more than the efficient level of externalities in its production process.

Public goods

A public good is a good that is non-excludable and non-rivalrous. The consumption of a commodity is non-rivalrous when its consumption from an agent does not affect the availability of the commodity for other agents. The consumption of a commodity is non-excludable 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

1. Consider two monopolistic markets intermingled by an externality. In the first market, the firm’s profit as a function of output is \(\pi_{1}(q_{1}) = 48 q_{1} - q_{1}^{2}\). The output of the first monopolist affects the profit of the monopolist in the second market as an externality. The second firm’s profit as a function output is \(\pi_{2} = (60 - q_{1}) q_{2} - q_{2}^{2}\).

   1. Suppose that each firm independently maximizes its profit. Calculate the optimal quantity and profit of each market. Moreover, calculate the total profit for both markets.
   2. Suppose that the firm in market \(1\) is not allowed to produce any output. Calculate the optimal quantity and profit in market \(2\).
   3. Suppose that firm \(1\) has to pay a transfer to firm \(2\), equal to the damages caused by the production externality. How do the profit functions of the two firms change? Calculate the optimal quantity and profit of each market. Calculate also the total profit in both markets.
   4. Suppose that the two firms merge. What is the resulting profit function of the merged firm? Calculate the optimal quantity and profit.
   5. Compare the profits of the above cases. Which case is the most efficient?

   1. Firm 1 solves

      \begin{align*} \max_{q_{1}} \left\{ 48 q_{1} - q_{1}^2 \right\}, \end{align*}

      which results in \(q_{1} = 24\). Firm 2 solves

      \begin{align*} \max_{q_{2}} \left\{ (60 - q_{1}) q_{2} - q_{2}^2 \right\}, \end{align*}

      which gives

      \begin{align*} q_{2} = 30 - \frac{q_{1}}{2} = 30 - 12 = 18. \end{align*}

      Then, we can calculate the profits

      \begin{align*} \pi_{1} &= 48 \cdot 24 - 24^{2} = 576, \\ \pi_{2} &= (60 - 24) 18 - 18^{2} = 324, \\ \pi &= \pi_{1} + \pi_{2} = 900. \end{align*}

   2. In this case, \(q_{1}=0\) and the quantity that maximizes profit in the second market is

      \begin{align*} q_{2} = 30 - \frac{q_{1}}{2} = 30. \end{align*}

      The firm’s profit is

      \begin{align*} \pi = \pi_{2} = (60 - 0) 30 - 30^{2} = 900. \end{align*}

   3. With the transfer, firm 1 solves

      \begin{align*} \max_{q_{1}} \left\{ 48 q_{1} - q_{1}^2 - q_{1}q_{2}\right\}, \end{align*}

      which results in a best response function

      \begin{align*} q_{1} = 24 - \frac{q_{2}}{2}. \end{align*}

      Firm \(2\) solves

      \begin{align*} \max_{q_{2}} \left\{ 60 q_{2} - q_{2}^2 \right\}, \end{align*}

      which gives \(q_{2} = 30\). Substituting into the best response of the first firm, we get \(q_{1} = 9\). Then, the profits are given by

      \begin{align*} \pi_{1} &= (48 - 30) \cdot 9 - 9^{2} = 81, \\ \pi_{2} &= 60 \cdot 30 - 30^{2} = 900, \\ \pi &= \pi_{1} + \pi_{2} = 981. \end{align*}

   4. The merged firm maximizes

      \begin{align*} \max_{q_{1}, q_{2}} \left\{ 48 q_{1} - q_{1}^2 + 60 q_{2} - q_{2}^{2} - q_{1}q_{2}\right\}. \end{align*}

      The first order conditions of the problem are

      \begin{align*} 48 - 2 q_{1} - q_{2} &= 0, \\ 60 - 2 q_{2} - q_{1} &= 0. \end{align*}

      Solving the above system gives \(q_{1}=12\) and \(q_{2}=24\). The profit of the merged firm is

      \begin{align*} \pi = 48 \cdot 12 - 12^2 + 60 \cdot 24 - 24^{2} - 12 \cdot 24 = 1008. \end{align*}

   5. The merged case is the Pareto efficient structure because the externality is internalized.

      Case	Structure	\(\pi_{1}\)	\(\pi_{2}\)	\(\pi\)
      1	Externality in the second market	576	324	900
      2	First market shuts down	0	900	900
      3	Externality in the first market	81	900	981
      4	Internalized externality			1008

2. 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 Cobb-Douglas utility function \(u(x_{i}, G) = x_{i}^{\alpha}G^{\beta}\). Both players have a budget of \(B\).

   1. Explain why in this case, \(G\) can be considered a public good.
   2. Set up individual \(i\)’s budget constraint.
   3. Find individual \(i\)’s best response contribution \(g_{i}\).
   4. Compute the Nash equilibrium for the contributions \(g_{1}\) and \(g_{2}\).
   5. Does the Nash equilibrium allocation entail more or fewer resources directed to the public good than the Pareto efficient allocation?
   6. Argue that the Nash equilibrium is not Pareto-efficient.

   1. The good \(G\) is non-rivalrous, as both players consume the exact same good \(G\), and non-excludable as the good is not priced in the market.

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

   3. 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} &= \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*}

   4. 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*}

   5. 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.

   6. 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

1. 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 \(B_{2}\).

   1. What is the maximum amount that player 1 is willing to contribute to the public good?
   2. What is the maximum amount that player 2 is willing to contribute to the public good?
   3. 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 \(\hat x\).
   4. 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.

   1. 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*}

   2. 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*}

   3. 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*}

   4. 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 \(x_{2}>25\).

References

References

Topic's Concepts

• public good
• negative production externality
• positive production externality
• production externality