Cournot Model in a Duopoly Market

In summary, two firms compete in a market for the sale of the same product. Each firm has a demand function that depends on the price of the other firm and on its own price. The marginal costs of producing one unit of product are both c. The equilibrium price and profit for each firm are functions of a, b, and c.
  • #1

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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  • #2
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...
 
  • #3
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.
 
  • #4
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.
 

1. What is the Cournot Model in a Duopoly Market?

The Cournot Model in a Duopoly Market is an economic model that describes the behavior of two firms in a market with limited competition. The model assumes that the two firms make their production decisions simultaneously and independently, taking into account the other firm's production level.

2. How does the Cournot Model differ from other economic models?

The Cournot Model differs from other economic models, such as the perfect competition model or the monopoly model, in that it takes into account the interdependence of the two firms in the market. This means that each firm's decision affects the other firm's profits, leading to a more realistic representation of market behavior.

3. What are the assumptions of the Cournot Model?

The Cournot Model assumes that the two firms have identical production costs, that they produce a homogeneous product, and that they have complete information about the market. It also assumes that the firms are profit-maximizing and that they act independently of each other.

4. How does the Cournot Model determine equilibrium in a duopoly market?

The Cournot Model determines equilibrium in a duopoly market by finding the point where the two firms' production levels intersect, known as the Cournot-Nash equilibrium. At this point, neither firm has an incentive to change their production level, as it would not increase their profits. This equilibrium results in a non-competitive market with higher prices and lower quantities produced compared to a perfectly competitive market.

5. What are some limitations of the Cournot Model?

Some limitations of the Cournot Model include the assumption of identical production costs and the assumption of complete information. In reality, firms may have different production costs and may not have perfect information about the market. Additionally, the model does not take into account factors such as advertising, product differentiation, and economies of scale, which can also impact market behavior.

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