Cournot Model in a Duopoly Market

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Discussion Overview

The discussion revolves around the Cournot model in a duopoly market, focusing on the price competition between two retailers. Participants are attempting to derive equilibrium price and profit expressions based on given demand functions and marginal costs, while addressing the lack of specific numerical values for parameters.

Discussion Character

  • Homework-related
  • Mathematical reasoning

Main Points Raised

  • One participant presents the demand functions for two firms and seeks to find equilibrium price and profit expressions based on parameters a, b, and c.
  • Another participant requests relevant equations related to the Cournot model, indicating a lack of clarity on the topic.
  • A third participant outlines the profit function for each firm and discusses the objective of maximizing profit, noting the need to derive equilibrium price from quantities.
  • One participant questions the concern about the absence of specific values for a, b, and c, suggesting that the solution should be expressed in terms of these parameters instead.

Areas of Agreement / Disagreement

Participants appear to agree that the solution should be expressed in terms of the parameters a, b, and c. However, there is some uncertainty regarding the approach to deriving the equilibrium price and profit without specific numerical values.

Contextual Notes

Participants note the absence of specific values for parameters a, b, and c, which may limit the ability to derive concrete numerical solutions. The discussion remains focused on theoretical expressions rather than numerical outcomes.

Who May Find This Useful

Students or individuals studying game theory, particularly those interested in economic models of competition and pricing strategies in duopoly markets.

samchan5167
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Homework Statement


Two retailers compete on price in a market. Firm 1’s demand depends on both its own price and firm 2’s price as follows: q1 = b – p1 + ap2. Similarly, firm 2’s demand depends on its own price and firm 1’s price: q2 = b – p2 + ap1. Their marginal costs of producing one unit of product are both c.
a. Find the expression of firm 1’s equilibrium price (as a function of a, b and c).
b. Find the expression of firm 1’s equilibrium profit (as a function of a, b and c).

Homework Equations

The Attempt at a Solution


I can't seem to figure this out as normally the price function would be given...
 
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Can you help us out with some Relevant Equations here? I have vaguely fond memories of the Cournot model from game theory, but emphasis on vague at the moment...
 
Two firms producing the same kind of product in quantities of q1 and q2, respectively:
Market clearing price p = a - b (q1 + q2)

Profit for firm i:
πi = (p – c) qi = [a - b (q1 + q2) – c] qi ,where c is the unit production cost.

Define B = (a – c)/b,
πi = b (B – q1 – q2) qi

Objective: choose qi to maximize profit,
maxqi b(B – q1 – q2) qi

Normally, after you find q1 and q2, you sub them into the price equation and then you can get the equilibrium price. However, in this question, there’s no value given for a or b or c and price and profit are to be found rather than finding q1 or q2. Thanks for replying.
 
samchan5167 said:
However, in this question, there’s no value given for a or b or c
Why are you concerned? You're supposed to solve in terms of a, b & c, not some set of actual numbers.
 

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