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In the last Game Theory article, we explored an important concept in economics, the Bertrand Model, through the lens of mathematics. This time, we will delve into another interesting model similar to the Bertrand Model except that the firms compete by setting production quantities rather than price: the Cournot Model.
Setting assumptions is essential in economics studies. Here I listed some critical assumptions in the Cournot 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 independently and simultaneously.
Companies make homogenous products.
A market price is settled and can not be changed .
Companies only compete by setting quantities of products that they produce.
The company with the lowest price gains the entire market share.
There is no cooperation between companies. (INOMICS, 2022)
How does the Cournot Model work?
Let’s assume two companies are competing for the entire market: Company 1 and Company 2. The variables can be defined as:
π₁: profit of Company 1
π₂: profit of Company 2 q₁: quantity of production of Company 1
q₂: quantity of production of Company 2
c₁: marginal cost of Company 1
c₂: marginal cost of Company 2
C₁: total production cost of Company 1
C₂: total production cost of Company 2 Q(P): the market demand function in regards to the market price
Given demand and market price have an inverse relationship, let’s assume: Q(P) = 10-P
Therefore, P(Q) = 10-Q
Given total production is Q = q₁+ q₂
Therefore, P(Q) = 10 - (q₁+ q₂)
Given total cost equals the amount of production times the marginal cost, we have:
Substitute P(Q), C₁ and C₂ with the equation we derived:
Assuming the other company’s quantity of production acts as a coefficient in each equation, we can derive the vortex of the two quadratic equations (we can also use partial derivatives to find the maximum, you can explore this method here):
The two companies can achieve the maximum profit when they produce:
Therefore, q₂ =
Given c₁ and c₂ are two coefficients, we can easily find a solution to this equation. For example, if c₁ = 2, c₂ = 4:
To check,
are both true.
Therefore, the equilibrium quantity is q₁ = 10/3 and q₂ = 4/3.
Under this condition, we can calculate:
(SebastianWaiEcon, 2019)
Citations
INOMICS. (2022). Cournot Competition | INOMICS. [online] Available at: https://inomics.com/terms/cournot-competition-1525473 [Accessed 7 Oct. 2024].
Michel van Biezen (2016). Calculus 3: Partial Derivative (19 of 50) Finding the Max and Min. YouTube. Available at: https://www.youtube.com/watch?v=W0QTrTibuUk [Accessed 7 Oct. 2024].
SebastianWaiEcon (2019). Cournot Model Exercise. YouTube. Available at: https://www.youtube.com/watch?v=UdPhF0PcdxQ [Accessed 7 Oct. 2024].
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