Microeconomics Study Guide: Oligopoly, Game Theory, Externalities, and Public Goods

Oligopoly and Monopolistic Competition

Key Concepts

• Oligopoly: A market structure with a few firms whose decisions are interdependent.

• Monopolistic Competition: Many firms, differentiated products, free entry and exit.

Oligopoly Characteristics

• Few firms in the market

• Strategic interdependence (each firm's actions affect others)

• Barriers to entry

• Products may be identical or differentiated

Monopolistic Competition

• Many firms

• Free entry and exit

• Differentiated products

• Each firm earns zero economic profit in the long run

Mathematical Models of Oligopoly

1. Cournot Competition (Quantity Competition)

• Firms choose quantities simultaneously; each firm's best response function shows optimal output given rivals' quantity.

Key Equations:

• Market demand: $P = a - bQ$ where $Q = q_1 + q_2$

• Firm profit (net of fixed cost): $ext{profit}_i = P imes q_i - MC_i imes q_i$

• Best response function: $q_i = (a - MC_i - bq_j)/2b$

• Nash equilibrium: Solve both best response functions simultaneously

Example:

• $P = 120 - Q$, $MC_1 = MC_2 = 30$

• Best response: $q_1 = (120 - 30 - q_2)/2 = 45 - 0.5q_2$

• Symmetric solution: $q_1 = q_2 = 30$, $Q = 60$, $P = 60$

2. Stackelberg Competition (Sequential Quantity)

• Leader moves first, follower observes and responds. Solve by backward induction.

Solution Method:

1. Stage 1: Find follower's best response function

2. Stage 2: Leader maximizes profit knowing follower will use best response

Example:

• Same demand/costs as above

• Follower: $q_2 = (120 - q_1 - 30)/2 = 45 - 0.5q_1$

• Leader substitutes into profit function, solves for $q_1$

• Leader earns more than in Cournot

3. Bertrand Competition (Price Competition)

• Firms undercut each other until $P = MC$

• Result: zero economic profit (Bertrand paradox)

• Just 2 firms enough to drive profit to zero

Differentiated Products

• Firms can charge above MC due to brand loyalty

4. Cartel Formation

• Firms collude to act like monopoly

• Divide output equally among $n$ firms

• Problem: Each firm has incentive to cheat

Example:

• $P = 500 - 2Q$, $MC = 20$, $n = 4$ firms

• Find monopoly $Q$ where $MR = MC$

• Divide output among firms

• Each firm profits unless it cheats

Comparison of Oligopoly Models

Model	Market Output	Price	Total Profit	Key Feature
Bertrand (identical)	Highest	Lowest ($=MC$)	Zero	Most competitive
Cournout	Medium	Medium	Medium	Simultaneous quantity
Stackelberg	Medium-High	Medium-Low	Medium-High	First-mover advantage
Cartel/Monopoly	Lowest	Highest	Highest	Collusion

Game Theory

Key Concepts

• Players: Decision-makers in the game

• Strategies: Complete plans of action

• Payoffs: Outcomes (usually profits) from strategy combinations

• Nash Equilibrium: No player can improve by changing strategy alone

Types of Games

• Simultaneous-move games: Players choose simultaneously (payoff matrix)

• Sequential games: Players move in order (game tree)

• Repeated games: Same game played multiple times

Finding Nash Equilibrium

1. Best Response Analysis: Mark each player's best response to each rival strategy; Nash equilibrium occurs where both players are playing best responses.

2. Underline Method: For each row, underline column player's highest payoff; for each column, underline row player's highest payoff. Nash equilibrium is where both payoffs are underlined.

Dominant Strategy

• Strategy that is best regardless of what others do

• If dominant strategy exists, use it

• Dominant strategy equilibrium: Everyone plays dominant strategy

Example: Advertising Game

• Payoff matrix shows both firms have dominant strategy to advertise; Nash equilibrium is (Advertise, Advertise)

Sequential Games and Backward Induction

• Start at final stage (terminal nodes)

• Determine optimal action at each decision node in that stage

• Work backward to earlier stages

• Confirm used reaching initial node

Subgame Perfect Nash Equilibrium

• Strategy profile that is Nash equilibrium in every subgame

• Found by backward induction

• Rules out non-credible threats

Example: Entry Deterrence

• Incumbent decides: Build Capacity or Don't Build

• Rival decides: Enter or Stay Out

• Backward induction used to find equilibrium

Repeated Games

• Games with all known time $T$, backward induction applies

• By backward induction, play Nash in every period

• If infinite/uncertain horizon, cooperation possible

Grim Trigger

• Punish forever after cheating

• Cooperation sustained if future factor (patience) high enough

Externalities and Public Goods

Key Concepts

• Externality: Direct effect of one person's action on another's well-being, not through market prices

Types of Externalities

• Negative production externality: Pollution from factory

• Negative consumption externality: Loud music from neighbor

• Positive production externality: R&D spillovers, learning-by-doing

• Positive consumption externality: Vaccination (herd immunity)

Market Failure with Externalities

• Competitive market ignores external costs/benefits

• Results in overproduction (negative) or underproduction (positive)

Key Relationships

• Private Marginal Cost (PMC): Firm's production cost

• External Marginal Cost (EMC): Harm imposed on others

• Social Marginal Cost (SMC): $PMC + EMC$

• Marginal Benefit (MB): Value to consumers (demand)

Mathematical Example

• Inverse demand: $P = 212 - Q$

• $EMC = 80$ (marginal harm from pollution)

Competitive Equilibrium:

• $PMC = 80$, $Q = 132$, $P = 80$

Social Optimum:

• $SMC = PMC + EMC = 80 + 80 = 160$

• $Q = 52$, $P = 160$

Efficient Tax

• Tax should equal EMC at social optimum

• $t^* = EMC(Q^*)$

Solutions to Externality Problems

1. Pigovian Tax/Subsidy: Tax = external cost (negative externality); Subsidy = external benefit (positive externality)

2. Cap-and-Trade (Quota Trading): Set overall cap, issue permits, market determines permit price

3. Coase Theorem: If property rights are well-defined and transaction costs are low, private bargaining achieves efficient outcome

Example: Airplane Seat

• Trade occurs to price between $150$-$400$ depending on values and bargaining

When Coase Theorem Fails

• High transaction costs

• Many parties involved

• Asymmetric information

• Strategic behavior

Open-Access and Public Goods

Key Concepts

• Rival: One person's consumption reduces availability for others

• Non-rival: One person's consumption doesn't reduce availability

• Excludable: Can prevent non-payers from consuming

• Non-excludable: Cannot prevent non-payers from consuming

Four Types of Goods

	Rival	Non-Rival
Excludable	Private goods (food, clothing)	Club goods (cable TV, toll road)
Non-excludable	Common resources (fish in ocean)	Public goods (national defense)

Public Good Characteristics

• Non-rival and non-excludable

• Examples: National defense, lighthouse, clean air, basic research

Free Rider Problem

• Individuals don't voluntarily pay for public goods

• Each wants others to pay while they benefit for free

• Leads to underprovision of public goods

• Justifies government provision/subsidies

Optimal Provision of Public Goods

• Private Good: Horizontal summation of demand

• Public Good: Vertical summation of demand

Mathematical Example: Mall Security

• Store 1 demand: $P_1 = 45 - Q$

• Store 2 demand: $P_2 = 10 - 0.25Q$

• $MC = 25$ per guard

Social Demand (Vertical Sum):

• For $Q ext{ guards}$: $P_1 + P_2 = 45 - Q + 10 - 0.25Q = 55 - 1.25Q$

• For $Q = 24$, $WTP = P_1 + P_2 = 25$

Competitive Outcome (Private Provision):

• Only Store 1 hires (higher WTP)

• $P_1 = MC: 45 - Q = 25 ightarrow Q = 20$

Social Optimum:

• $P_1 + P_2 = 55 - 1.25Q = 25 ightarrow Q = 24$

• Subsidy needed: Difference between social and private equilibrium shows underprovision

Common Pool Resources

• Rival but non-excludable

• Examples: Open fishery, grazing land, groundwater

• Problem: Overuse (tragedy of the commons)

• Each user ignores cost imposed on others

Solutions

• Assign property rights

• Government regulation (quotas, taxes)

• Community management

• Cap-and-trade systems

Problem-Solving Strategies

Oligopoly Problems

1. Identify the model (Cournot, Stackelberg, Bertrand, Cartel)

2. Write down profit functions

3. Take first-order conditions (derivatives)

4. Solve for best response functions

5. Find equilibrium (intersection of best responses or backward induction)

Game Theory Problems

• Draw payoff matrix clearly

• Find each player's best response to each rival strategy

• Mark best responses (stars, underlines, circles)

• Nash equilibrium: where both play best responses

• Check for dominant strategies first

Externality Problems

• Identify type: production/consumption, positive/negative

• Draw graph with PMC, SMC (or PMB, SMB), and Demand/Supply

• Find competitive equilibrium: PMC = PMB

• Find social optimum: SMC = SMB

• Calculate optimal tax/subsidy = external cost/benefit at $Q_{social}$

• Compare to monopoly if asked

Public Goods Problems

• Identify if private or public good

• For public goods: vertically sum individual demands

• Find where social MB = MC

• Compare to private provision (highest individual demand = MC)

• Calculate required subsidy if asked

Vertical Summation

• At each $Q$, add individual WTP (prices)

• Be careful with ranges where some individuals have zero WTP

• Social demand curves may have kinks

Key Formulas Summary

• Cournot: Best response: $q_i = (a - bq_j - MC_i)/2b$

• Stackelberg: Follower: $q_2 = (a - q_1 - MC_2)/2$; Leader: $q_1 = (a - MC_1)/2$

• Cartel: Total $Q$ where $MR = MC$; Each firm produces $Q/n$

• Externalities: $SMC = PMC + EMC$; Social benefit: $SMB = PMB + EMB$; Optimal tax = $EMC(Q_{social})$; Optimal subsidy = $EMB(Q_{social})$

Study Tips

• Draw graphs for every problem

• Understand the logic, not just formulas

• Compare market structures

• Practice backward induction for sequential games

• Distinguish horizontal vs. vertical summation (private vs. public goods)

• Check your algebra and logic

• Write out logic for game theory

• Practice with interactive tools if available