Lecture 19: Introduction to Oligopoly and Bertrand Price Competition

The primary objectives of this lecture are to understand the characteristics that define an oligopoly market structure and to examine the profit-maximising behaviour of firms within that structure.
The analysis of firm pricing behaviour follows a specific pedagogical sequence in the course:
- Week 8: Firms as price takers in a perfectly competitive market.
- Week 9: A single firm with market power (monopolist) charging a single uniform price.
- Previous Lecture: A single firm with market power charging different prices (price discrimination).
- Today: Multiple firms with market power determining prices through strategic interaction.

Pricing decisions are no longer made in isolation; firms must consider the moves of their competitors.
Retail Petrol Market Case Study:
- Typically composed of a small number of firms.
- The limited number of retailers means each possesses a degree of market power.
- Firms engage in strategic behaviour by mimicking the pricing of their rivals.
- Specific source: Byrne, David P., and Nicolas de Roos. 2019. ‘Learning to Coordinate: A Study in Retail Gasoline.’ American Economic Review 109 (2): 591–619.
The Information Paradox:
- Retailers often exploit high consumer search costs to increase mark-ups.
- Governments have attempted to mitigate this by introducing real-time price platforms.
- These platforms are considered a "double-edged sword": While intended to lower search costs, evidence suggests they may actually lead to higher prices. This is because transparency allows firms to coordinate or monitor each other's prices more effectively, potentially facilitating higher mark-ups.

Definition: An oligopoly is a market characterized by a small number of firms interacting strategically.
Key Features:
- Firms do not take prices as given; they have the power to set them.
- The market equilibrium is worse for consumers than perfect competition but generally better than a pure monopoly.
- Strategic interaction dictates that every firm must anticipate and consider its rivals' likely reactions before choosing an optimal strategy.
Special Case (Duopoly):
- A duopoly is a market with exactly two firms.
- Firms must decide whether to compete based on prices (Bertrand competition) or quantities (Cournot competition).

Assumptions for the Differentiated Product Model:
- The products are imperfect substitutes.
- The market consists of two firms: BP (Firm B) and Ampol (Firm A).
- Both firms have identical cost functions where marginal cost ( $MC$ ) equals average cost ( $AC$ ): $MC = AC = 25$ .
Demand and Profit Functions for BP:
- Demand for BP is represented by: $QB = 50 - PB + \frac{PA}{2}$ .
- In this function, $QB$ is BP’s output, $PB$ is its price, and $PA$ is the price set by Ampol.
- The positive sign for the term $PA$ indicates the products are substitutes; as Ampol raises its price, demand for BP's product increases.
- Profit function for BP: $\pi_B = (PB - AC) \times QB$ .
- By substituting the demand equation into the profit equation, $\pi_B$ becomes a function related only to the prices $PB$ and $PA$ .

BP’s Reaction Function ( $RB$ ):
- Profit maximisation is found where $\frac{d\pi_B}{dPB} = 0$ .
- Note that the standard $MR = MC$ condition is not used in this specific pricing derivation as we are solving for price reactions directly.
- BP's Reaction Function: $PB = \frac{75}{2} + \frac{PA}{4}$ .
- This function represents BP’s profit-maximising price for any given price set by Ampol.
- $PB$ is increasing in $PA$ (upward sloping), meaning if the rival raises their price, it is optimal for BP to raise its price as well.
Ampol’s Reaction Function ( $RA$ ):
- Symmetrically, Ampol’s reaction function is: $PA = \frac{75}{2} + \frac{PB}{4}$ .
- To plot this on the same axes as BP (with $PB$ on the vertical axis), we rearrange the equation: $PB = 4PA - 150$ .
- This curve is also upward sloping with a slope of 4.

Definition: The Bertrand equilibrium occurs at the intersection of the two reaction functions ( $RA$ and $RB$ ). At this point, neither firm has an incentive to unilaterally change its price once it knows the rival's price.
Stability and Adjustment:
- If BP unilaterally increases its price to a point ( $X$ ), Ampol will respond by adjusting its own price, moving the state to a point ( $Y$ ).
- This cycle of responses continues until the market settles at the equilibrium point ( $W$ ).
Numerical Solution:
- Substitute Ampol's function into BP's function: $PB = \frac{75}{2} + (\frac{1}{4}) \times (\frac{75}{2} + \frac{PB}{4})$ .
- Simplify the expression: $PB = \frac{75}{2} + \frac{75}{8} + \frac{PB}{16}$ .
- Group the variables: $(1 - \frac{1}{16}) PB = \frac{375}{8}$ .
- Calculate the values: $PB = 50$ and $PA = 50$ .
Quantity and Profit Results:
- Substituting the prices into the demand function: $QB = 50 - 50 + \frac{50}{2} = 25$ .
- Substitution for Ampol due to symmetry: $QA = 25$ .
- BP's Profits: $\pi_B = (50 - 25) \times 25 = 625$ .
- Ampol's Profits: $\pi_A = 625$ .
Welfare Implications: Because there are only two firms, the price remains above marginal cost (P > MC), resulting in a deadweight loss to society.

Monopoly Scenario:
- Suppose BP and Ampol merge or act as a single monopolist, charging the same price $P$ for both products.
- The combined demand function for the monopolist: $Q = QB + QA = (50 - P + \frac{P}{2}) + (50 - P + \frac{P}{2}) = 100 - P$ .
Solving the Monopolist’s Problem:
- Total Revenue ( $TR$ ): $TR(Q) = (100 - Q)Q = 100Q - Q^2$ .
- Marginal Revenue ( $MR$ ): $MR(Q) = 100 - 2Q$ .
- Setting $MR = MC$ : $100 - 2Q = 25$ .
- Monopoly equilibrium: $QL = 37.5$ and $PL = 62.5$ .
- Joint monopoly profits: $\pi = (62.5 - 25) \times 37.5 = 1406.25$ .
Comparison Analysis:
- The monopoly price ( $62.5$ ) is higher than the Bertrand equilibrium price ( $50$ ).
- The monopoly quantity ( $37.5$ ) is lower than the total Bertrand quantity ( $25 + 25 = 50$ ).
- Monopoly profits ( $1406.25$ ) are higher than the sum of individual Bertrand profits ( $625 + 625 = 1250$ ).
The Externality Explanation:
- In Bertrand competition, when one firm lowers its price, it reduces the demand for its rival's product. This is an "externality" imposed on the rival.
- A monopolist internalises this externality, leading them to charge a higher price to maximise total industry profits.
Policy/Regulatory Implications:
- Regulators often block mergers to prevent firms from internalising these externalities at the expense of consumers.
- Example: The Perth petrol study was used around 2018 as a justification to block the proposed merger between BP and Woolworths.