What is the Bertrand Model?
Last time, we discussed the basic concepts of Game Theory, including its elements and Nash equilibrium. Game Theory is widely applied in business management; it can be used to predict how companies int
eract with each other. This time, we will explore an interesting economic model of how companies that produce the same products compete through setting prices.
Here are some essential assumptions of Bertrand Model:
All companies are rational, and they assume the same for other companies.
The ultimate goal of companies is to maximize profit.
Companies make decisions simultaneously.
All companies know each other’s marginal cost.
Companies only compete by setting prices (no quantity, quality, or distance)
The company with the lowest price gains the entire market share.
There is no cooperation between companies.
Let’s assume there are only two companies competing for the entire market: company 1 and company 2. The variables can be defined as:
P₁: price of company 1
P₂: price of company 2
π₁: profit of company 1:
π₂: profit of company 2
C₁: marginal cost of company 1
C₂: marginal cost of company 2
D₁ (P₁, P₂): demand for company 1 when company 1 charges P₁ and company 2 charges P₂
D₂ (P₁, P₂): demand for company 2 when company 1 charges P₁ and company 2 charges P₂
D: demand for the whole market
For company 1, π₁ = (P₁ - C₁) x D₁ (P₁, P₂). π refers to the price chosen by the company.
When P₁ > P₂, all customers would buy from company 2. D₁ (P₁, P₂) = 0, therefore π₁ = 0. Company 1 makes no profit.
When P₁ = P₂, half of the customers would buy from company 1 and half would buy from company 2. D₁ (P₁, P₂) = 0.5D, therefore π₁ = (P₁ - C₁) x 0.5D.
When C₁ < P₁< P₂, all the customers would buy from company 1. D₁ (P₁, P₂) = D, therefore π₁ = (P₁ - C₁) x D. Company 1 captures the whole market and makes maximum profit.
When C₁ > P₁, P₁ - C₁ < 0, therefore π₁ < 0. Company 1 would lose money.
The circumstances are the same for company 2.
When the two companies share an identical marginal cost, they would slightly lower their price to appeal more to the customers. This would incentive the customers to choose their company over the other. At the end, the price would come infinitely close to the marginal cost, and both companies would make zero profit (when C = P, π = 0).
Diagram 1: The best strategy for company 1 in regards to the price of company 2 (same marginal cost)
Diagram 2: The best strategy for both companies in regards to the price of each other (same marginal cost)
Nash equilibrium is reached when neither of the companies have the motivation to change their strategies. According to diagram 2, the intersection of the best strategies of both sides occurs when both companies set the price at the marginal cost.
However, when the marginal cost of company 1 is lower than company 2, company 1 should set their price slightly lower than company 2. Company 2 would not have motivation to lower their price and produce more products, because C > P, meaning company 2 would lose money (circumstance (4)). Therefore, company 1 could capture the entire market and maximize profit (circumstance (3)).
The model can be used to determine the pricing and product strategy. If it is already known that the cost of the competing company's product is lower than mine, there is no need to compete with it. Instead, one should consider converting to alternative products.
Works cited:
Bertrand Competition (2023). Bertrand Competition: Model & Assumptions | StudySmarter. [online] StudySmarter UK. Available at: https://www.studysmarter.co.uk/explanations/microeconomics/imperfect-competition/bertrand-competition/ [Accessed 21 Oct. 2023].
INOMICS. (2021). Bertrand Competition | INOMICS. [online] Available at: https://inomics.com/terms/bertrand-competition-1504578 [Accessed 21 Oct. 2023].
Wikipedia Contributors (2023). Bertrand competition. [online] Wikipedia. Available at: https://en.wikipedia.org/wiki/Bertrand_competition [Accessed 21 Oct. 2023].
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