Microeconomics: Externalities and the Coase Theorem Study Guide

Course Context and Preliminary Review Questions

Course Details: These notes cover Lectures 25 & 26 on April 20 & 24, 2017, focusing on Microeconomics and Specifically Externalities as discussed in Chapter 34 of Varian.
General Assessment Review: The session included a review of several microeconomic quiz questions with specific adjustments to the grading: * Question 1: Discussed a textile firm where labor is the only short-term variable input and the manager observes a constant marginal product of labor. The correct answer was B (though the slide notes indicated this question was to be deleted). * Question 2: Concerned expressions for the cost-minimizing combination of inputs. * Question 3: Addressed the implications when Total Revenue (TR) and Total Cost (TC) curves have the same slope. The correct answers were (A, B). * Question 4: Regarding a competitive firm with a U-shaped marginal cost curve. The correct answers were (E, B). * Question 5: Focused on the losses caused by an effective price ceiling. The correct answers were (A, C). * Question 6: Regarding when government intervention can increase total welfare. The correct answer was (D). * Question 7: Focused on how changes in demand affect a monopolist. The correct answer was (D) (slated for deletion). * Question 8: Focused on demand at the profit-maximizing level of output.
Grading Corrections: * Two questions are to be deleted entirely from the assessment. Full marks are granted to students who did not receive points for these initially. * For the remaining 6 questions, answers were updated. Students providing the corrected answers will receive marks.

Fundamentals of Externalities

Definition of Externality: An externality is a cost or a benefit imposed upon someone by actions taken by others. Crucially, the cost or benefit is generated externally to the person affected.
Types of Externalities: * Positive Externality: An externally imposed benefit. * Negative Externality: An externally imposed cost.
Examples of Negative Externalities: * Air pollution. * Water pollution. * Loud parties next door. * Traffic congestion. * Second-hand cigarette smoke. * Increased insurance premiums resulting from alcohol or tobacco consumption by others.
Examples of Positive Externalities: * A well-maintained property next door that raises the market value of your own property. * A pleasant cologne or scent worn by a person seated next to you. * Improved driving habits that reduce accident risks for everyone. * A scientific advance.
Externalities and Efficiency: * Externalities impact a third party—someone who is not a participant in the activity producing the cost or benefit. * Externalities cause Pareto inefficiency. * Negative Externalities: Typically lead to too much of a scarce resource being allocated to the activity. * Positive Externalities: Typically lead to too little resource being allocated to the activity.

Modeling Negative Externalities: The Smoke Example

Agent Profiles: * Agent A: Consumption consists of money and smoke. Both are viewed as "goods" for Agent A. * Agent B: Consumption consists of money and smoke. Money is a "good" but smoke is a "bad."
Characteristics of the Commodity: Smoke is treated as a purely public commodity. * Nonexcludability: It is consumed by everyone. * Nonrivalry: Everyone consumes the entire amount of the commodity (similar to a broadcast television program).
Parameters: * Agent A is endowed with wealth $y_A$ . * Agent B is endowed with wealth $y_B$ . * Smoke intensity resides on a scale from $0$ (no smoke) to $1$ (maximum concentration).
Inefficiency Without Trade: * If there is no mechanism to exchange money for changes in smoke levels: * Agent A’s most preferred allocation is the maximum level of smoke ( $1$ ). * Agent B’s most preferred allocation is the minimum level of smoke ( $0$ ). * Individual choices without trade are socially inefficient. The outcome settles at either extreme, whereas efficient allocations lie somewhere in between where marginal rates of substitution can be balanced.

Property Rights and Internalization

Coase’s Insight: Ronald Coase argued that most externality problems arise from the inadequate specification of property rights. This absence of property rights prevents markets from trading and "internalizing" the external costs or benefits.
Internalization: This occurs when a producer of an externality is made to bear the full external cost or allowed to enjoy the full external benefit of their actions.
Scenario 1: Agent B Owns the Air: * If Agent B has the right to clean air, Agent A must pay B to smoke. * Let $p(s_A)$ be the price paid by Agent A to Agent B to create smoke intensity $s_A$ . * Through trading rights to smoke, an efficient allocation is achieved where both agents gain.
Scenario 2: Agent A Owns the Air: * If Agent A has the right to smoke, Agent B must pay A to reduce the smoke intensity. * Let $p(s_B)$ be the compensation paid by Agent B to Agent A for reduced smoke $s_B$ . * A market for trading rights to reduce smoke also results in an efficient allocation.
Distributional Effects: * The agent assigned the property right is better off than they would be in the absence of that right. * The total equilibrium amount of smoking typically depends on which agent was assigned the property right, appearing at different points ( $s_A$ vs. $s_B$ ) on the contract curve.

The Coase Theorem

Quasilinear Preferences: If preferences are quasilinear in money, given by the utility function $U(m,s) = m + f(s)$ , the Marginal Rate of Substitution (MRS) between smoke and money is independent of the amount of money.
The Theorem: If all agents' preferences are quasilinear in money, the efficient level of the externality-generating commodity is produced regardless of which agent is assigned the property right.

Production Externalities: Steel Mill and Fishery

The Scenario: A steel mill produces steel ( $s$ ) and pollution ( $x$ ). The pollution negatively affects a nearby fishery producing fish ( $f$ ). Both firms are price-takers in their respective markets. * Steel Price: $p_S$ * Fish Price: $p_F$
Steel Firm's Independent Problem: * Cost Function: $c_S(s,x)$ . Profit: $ext{\pi}_S = p_S s - c_S(s,x)$ . * First-Order Conditions (FOC): 1. $p_S - rac{\delta c_S(s,x)}{\delta s} = 0$ (Price equals marginal production cost). 2. $- rac{\delta c_S(s,x)}{\delta x} = 0$ (The marginal cost of pollution reduction to the firm is set to zero). * Numerical Example: Let $c_S(s,x) = s^2 + (x-4)^2$ and $p_S = 12$ . * $12 - 2s = 0 ightarrow s^* = 6$ . * $-2(x - 4) = 0 ightarrow x^* = 4$ . * Steel Profit: $ext{\pi}_S = 12(6) - [6^2 + (4-4)^2] = 72 - 36 = 36$ .
Fishery's Problem: * Cost Function: $c_F(f,x)$ . Given $f$ , cost increases with $x$ . * Profit: $ext{\pi}_F = p_F f - c_F(f,x)$ . * FOC: $p_F - rac{\delta c_F(f,x)}{\delta f} = 0$ . * Numerical Example: Let $c_F(f,x) = f^2 + xf$ and $p_F = 10$ . * External cost is $xf$ . * $10 - 2f - x = 0 ightarrow f^* = 5 - rac{x}{2}$ . * Taking the steel firm's pollution $x^* = 4$ , the fishery produces $f^* = 5 - rac{4}{2} = 3$ . * Fishery Profit: $ext{\pi}_F = 10(3) - [3^2 + 4(3)] = 30 - 21 = 9$ .
Inefficiency Summary: * Total Profit (unmerged): $36 + 9 = 45$ . * External cost inflicted on the fishery: $x^<em>f^</em> = 4 imes 3 = 12$ .

Merger and Internalization Analysis

Merged Firm Profit: $ext{\pi}_m = p_S s + p_F f - c_S(s,x) - c_F(f,x)$ . * $ext{\pi}_m = 12s + 10f - s^2 - (x-4)^2 - f^2 - xf$ .
Merged FOCs: 1. $rac{\delta \pi_m}{\delta s} = 12 - 2s = 0 ightarrow s = 6$ . 2. $rac{\delta \pi_m}{\delta f} = 10 - 2f - x = 0$ . 3. $rac{\delta \pi_m}{\delta x} = -2(x - 4) - f = 0$ .
Solution for Merged Firm: * Solving the system: $10 - 2f - x = 0$ and $8 - 2x - f = 0$ . * $x_m = 2, f_m = 4, s_m = 6$ . * Merged Profit: $12(6) + 10(4) - 6^2 - (2-4)^2 - 4^2 - (2)(4) = 72 + 40 - 36 - 4 - 16 - 8 = 48$ .
Comparison: The merged profit ( $48$ ) is higher than the sum of solo profits ( $45$ ). Merger improves efficiency and reduces pollution from $4$ to $2$ .
Efficiency Logic: The merged firm faces the "Marginal External Pollution Cost" ( $MC_E = f$ ) in addition to the steel firm's private marginal cost of pollution reduction ( $MC_s = 2(x-4)$ ). Efficiency requires $MC_E + MC_s = 0$ .

Market for Pollution Rights

Fishery Ownership Scenario: * Fishery sells pollution rights at price $p_x$ . * Fishery Profit: $ext{\pi}_F = p_F f - f^2 - xf + p_x x$ . * Fishery Supply FOCs: $10 - 2f - x = 0$ and $-f + p_x = 0 ightarrow f = p_x$ . * Combining these: $10 - 2p_x - x = 0 ightarrow x_S = 10 - 2p_x$ .
Steel Firm Demand Scenario: * Steel firm buys rights: $ext{\pi}_S = p_S s - s^2 - (x-4)^2 - p_x x$ . * Steel Demand FOC: $-2(x-4) - p_x = 0 ightarrow 8 - 2x = p_x \rightarrow x_D = 4 - rac{p_x}{2}$ .
Market Equilibrium: * Set $x_S = x_D$ : * $10 - 2p_x = 4 - rac{p_x}{2}$ . * $6 = 1.5p_x ightarrow p_x = 4$ . * Equilibrium Quantities: $x^* = 2, f^* = 4, s^* = 6$ . * This result is identical to the merged firm's efficient outcome. Per the Coase Theorem, the result is the same even if the steel firm owns the water rights.

The Tragedy of the Commons

Scenario: A grazing area owned "in common" by a village for grazing cows ( $c$ ).
Production Function: Total milk production is $f(c)$ where f'(c) > 0 and f''(c) < 0.
Social Optimum (Income Maximization): * Village Profit: $ext{\pi}(c) = f(c) - p_c c$ (assuming milk price is $1$ ). * Condition: $f'(c^*) = p_c$ (Marginal income gain = marginal cost).
Common Use Outcome: * Since entry is unrestricted, individuals add cows as long as the average gain is positive: $\frac{f(c)}{c} - p_c \geq 0$ . * Equilibrium occurs at $\hat{c}$ where $\frac{f(\hat{c})}{\hat{c}} = p_c$ .
The Tragedy: Because f'(c) < \frac{f(c)}{c} for concave functions, \hat{c} > c^. The commons are over-grazed. * Each villager ignores the cost their additional cow inflicts on others by reducing the average output per cow for everyone else.
Modern Examples: * Over-fishing in the high seas. * Over-logging forests on public lands. * Over-intensive use of public parks (e.g., Yellowstone). * Urban traffic congestion.