Externalities

Context

• Free markets of regulation? Ideal free markets result in economically efficient allocations. 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 always available and the presence of externalities can lead to market failures in such situations.
  • What do economists mean when they talk about externalities?
  • What are the welfare implications of such externalities?
  • Can 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
• Welfare Analysis and Improvements.
• Current Field Developments

Learning Objectives

• Illustrate production externalities using an ecological collapse case study.
• Explain the various types of externalities.
• Describe the ecological collapse by an externality model.
• Compare model predictions with the case study events.
• Explain how economic efficiency can be obtained in the presence of externalities.

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 Production 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.

Consumption Externalities

• We are focusing on production, but similar concepts are applicable on the consumption side of the economy (e.g., the decision problem studied in Microeconomics I).
  • Listening to loud music at 3:00 AM can (negatively) affect your neighbors.
  • Maintaining a beautiful garden can (positively) affect your neighbors.

• A commodity or a service exhibits a consumption externality when its consumption from one agent affects the utility of other agents.

Types of Consumption Externalities

• A positive consumption externality is an externality for which the utility of other agents is positively affected.
• A negative consumption externality is an externality for which the utility of other agents is negatively affected.

A Pollution Externality 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 out 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

• Since \(\frac{\partial c_{c}(q_{c}, \xi)}{\partial \xi} < 0\), the cotton producer chooses \(\xi=1\).

• Moreover, the cotton producer would like to choose \(q_{c}\) such that

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

• This gives \(q_{c}=\frac{1}{2}\).

• 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 \[\max_{q_{c}, q_{f}, \xi} \left\{ p_{c}q_{c} + p_{f}q_{f} - q_{c}^{2} \left(1 + \left(1 - \xi\right)^{2}\right) - q_{f}^{2} \left(1 + \xi^{2}\right) \right\} .\]

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 so?

Other Ways to Obtain Efficient Allocations

• We have seen that internalizing the production externality gives Pareto efficient allocations.
• Any property rights issues are solved when the firms merge.

• Are there other ways we can reach efficient allocations?
• Two alternative ideas proposed in the literature are
  • (Pigouvian) taxation
  • Enforcing rights for the missing market.

Efficient Production through Taxation

• Suppose that the government introduces a tax that the cotton producer has to pay for each percentage point of water it draws from the river.

• The modified profit of the cotton producer is \[\max_{q_{c}, \xi} \left\{ p_{c}q_{c} - q_{c}^{2} \left(5 + \left(1 - \xi\right)^{2}\right) - t \xi \right\}\]

• Now, 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) \\ t &= q_{c}^{2} 2 \left(1 - \xi\right) \end{align*}

• How should the government choose \(t\) to make production economically efficient?

Choosing the Right Tax

• In principle, the government can calculate the optimal \(q_{f}^{\star}=0.97\) and \(\xi^{\star}=0.17\) by solving merged firm’s problem, and set \[t = \left(q_{f}^{\star}\right)^{2} 2 \xi^{\star}.\]
• This enforces \[ q_{c}^{2} 2 \left(1 - \xi\right) = t = \left(q_{f}^{\star}\right)^{2} 2 \xi^{\star}, \] and leads to Pareto efficient allocations.

• In practice, calculating the \(q_{f}^{\star}\) and \(\xi^{\star}\) is very difficult.
• It requires information that governments typically do not possess.
• The Pigouvian taxation solution is either impossible or at best impractical.

Enforcing a Market for the Externality

• Another approach is to enforce property rights for the missing market.

• The government legislates and supervises a market in which the firm(s) receiving the externality can sell the rights to produce the externality to the firm(s) emitting it.

• E.g., in the fishery-cotton example, the government creates a water drain market and gives the right to the fishery to sell a draining percentage to the cotton firm.

• Let \(r\) be the price per percentage unit of drained water. The modified profit of the cotton producer is \[\max_{q_{c}, \xi} \left\{ p_{c}q_{c} - q_{c}^{2} \left(5 + \left(1 - \xi\right)^{2}\right) - r \xi \right\}\]

• With first order conditions

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

How Much Water Should the Fishery Sell?

• With the new market, the fishery solves \[\max_{q_{f}, \xi} \left\{ p_{f}q_{f} - q_{f}^{2}\left(1 + \xi^{2} \right) + r \xi \right\}.\]

• The fishery chooses

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

Feasibility of Efficiency

• Combining the fishery and cotton necessary conditions for \(\xi\) gives \[ q_{c}^{2} 2 \left(1 - \xi\right) = q_{f}^{2} 2 \xi .\]
• This is the same condition as in the merged firm’s problem.
• Enforcing the missing market is economically efficient.

• However, practically it is not always feasible to regulate the missing market.
• E.g., emitting and receiving firms may operate in different countries.

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 (Climate Leadership 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.

Concise Summary

• Externalities lead to economically inefficient allocations when property rights cannot be enforced.
• Externalities 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.
• Essentially internalizing the externality bypasses the problem of non-enforceable property rights.
• The economic literature also proposes other solutions based on
  • taxation (see e.g., carbon tax discussion) and
  • enforcing property rights for the missing market.

Further Reading

• Varian (2010, secs. 35.3-7)
• CORE Team (2017, secs. 12.1, 12.4)

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 whenever \(\frac{\partial c_{r}}{\partial {\xi}}\left(q_{r}, \xi\right)\neq 0\) (why?). 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.

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. When firm \(1\) produces \(q_{1}\), the damage caused by the externality to firm \(2\) is equal to \(q_{1}q_{2}\). 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 essentially moves the externality to the first market. This results in the necessary condition

      \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

References

References

Topic's Concepts

• negative consumption externality
• positive consumption externality
• consumption externality
• negative production externality
• positive production externality
• production externality