How To Find Cournot Reaction Function
We all take a friend or a relative that is always late. Imagine you are planning to meet them this calendar week. You lot've known them for and then long that you are sure they will be at least one hour late. Would you however come to run into them at the agreed time? If you lot were to maximize your personal benefit, you lot would probably utilize this time to practice something productive. Time is precious, and then why waste it? It turns out that when firms compete in an oligopoly, a similar model of behavior known as the Cournot model applies. Firms also want to put their resource to employ in the all-time way given the other party's action! Interested in learning about what this model predicts? So chop chop and read on!
Cournot Model of Oligopoly
Augustin Cournot came up with the model of oligopoly in 1838. Simply we won't dwell on history for too long and instead bound straight into the definition and the details.
Cournot Model Definition
The definition of Cournot's model in economic science is that it is a model of oligopoly where firms producing homogeneous products compete in quantities. Let's spring right into it and look at which characteristics this model has!
- Cournot model has several characteristics:
- It is a static i-period model;
- It describes the beliefs of firms in an oligopoly;
- There is no consideration of dynamics or adjustment.
That doesn't look so intimidating, does it? Let's then take a look at some of the model'southward assumptions.
Cournot Model Assumptions
Permit's become over the assumptions in the Cournot model!
- In that location are several assumptions in Cournot's model:
- Firms are rational, and their objective is to maximize their profits;
- Firms produce homogeneous products;
- Firms compete by setting output quantities;
- Firms make decisions simultaneously;
- Firms treat their competitor'southward output equally fixed;
- There is no cooperation between the firms;
- Firms have enough market place power such that their output decision can affect the market price.
Proceed these at the back of your head, as everything will become more credible in the next section, where we will look at the model mathematically!
Cournot's model in economics is a model of oligopoly where firms produce homogeneous products and compete in quantities.
Cournot Model of Duopoly
Permit's wait at the Cournot model of a duopoly in terms of some mathematical equations and graphs!Equally economists love to take fun, let's give our firms names: 'The Happy Firm' and 'The Lucky Firm.'We assume that the products that the firms produce are homogeneous. The 2 firms volition determine to set up their quantities simultaneously. Each firm volition first consider what its competitor would do and then set up its own output to maximize its profits.The Happy Firm is thinking about how to tackle this challenge and decides to create a schedule of all the possible quantities that the Lucky Firm could produce.The Happy Firm had plotted a line representing how much output it should produce given the Lucky Firm's conclusion. This part is chosen Happy Firm's reaction function in a duopoly.
The reaction part, or the reaction curve, depicts the relationship betwixt the quantity the business firm should produce to maximize profit and the amount information technology presumes the other firm will brand.
Imagine that the Lucky Business firm goes through the same practice and finds its reaction function. Nosotros tin can at present plot these two reaction functions on 1 graph, as shown in Effigy 1 below.
Fig. ane - Reaction functions example
Figure 1 above shows the 2 reaction functions; one for the Happy Firm and i for the Lucky Firm. The two curves have the same course because the two firms in our example are the same. The reaction curves await different because they show one firm'due south profit-maximizing output given the other firm'southward output. Where the two reaction functions intersect is known every bit Cournot equilibrium. Why is this an equilibrium?Think about it more more often than not from the Nash equilibrium point of view. It is an equilibrium considering, at this point, no firm has an incentive to deviate from its strategy. Or in other words, each firm is doing the best it possibly can considering what the other firm is doing.
Cournot equilibrium is an equilibrium in a duopoly where each firm sets its output quantities, having correctly anticipated the amount that its competitor chooses to produce.
Nash equilibrium is an equilibrium in which no firm has an incentive to deviate from its strategy. Each firm exercises its most profitable strategy, given its competitor's selection.
Suppose the firms initially start producing quantities that differ from the Cournot equilibrium. In that example, the model cannot predict any of the dynamics of quantity adjustments, which is the limitation of this model.
Cournot Model Example
Let's look at an instance of a Cournot model with equations and graphs!
Let's revisit our Happy Firm and Lucky Firm. Imagine the market need curve is:\(P=300-Q=300-(Q_1+Q_2)\)
Where:\(Q=Q_1+Q_2\)\(Q_1 - \hbox{the production of the Happy Firm}\)\(Q_2 - \hbox{the product of the Lucky Firm}\)\(Q - \hbox{the total production of both firms}\)Let's set the marginal costs to zip for simplicity:\(MC_1=MC_2=0\)
How can we find the reaction function of the Happy Firm?Remember the turn a profit-maximizing rule:\(MC=MR\)
Nosotros demand to find the total revenue of the Happy Firm:
\(TR_1=P\times Q_1=(300-Q)\times Q_1=\)\(=300Q_1-(Q_1+Q_2)Q_1=\)\(=300Q_1-Q_1^two-Q_2Q_1\)Marginal revenue is and then the first derivative with respect to Q1:
\(MR_1=\frac{\Delta TR_1}{\Delta Q_1}=300-2Q_1-Q_2\)
We know that:
\(MC_1=0\)
For the profit-maximizing rule to hold:\(MC_1=MR_1=0\)\(MR_1=300-2Q_1-Q_2=0\)
Rearrange to find Q1:\(2Q_1=300-Q_2\)\(Q_1=150-\frac{1}{2}Q_2\) (1)We found the reaction function for the Happy House!
We don't demand to become over all these calculations for the Lucky Firm every bit we know that its reaction role is symmetric and is:
\(Q_2=150-\frac{one}{2}Q_1\) (2)
Nosotros know that the Cournot equilibrium occurs when the ii functions intersect. We can then plug the value of Q2 into the equation for Q1 (1) to get:
\(Q_1=150-\frac{one}{2}\times(150-\frac{1}{two}Q_1)\)\(Q_1=150-75+\frac{ane}{four}Q_1\)
\(\frac{3}{4}Q_1=75\)
\(Q_1=100\)
We have found Q1! Now we can plug the value of Q1 into (2):
\(Q_2=150-\frac{1}{two}Q_1=150-\frac{100}{2}=100\)
Nosotros have now found Q2 equally well!
The Happy House and the Lucky Firm happen to produce the same quantities, just this doesn't have to be the case.
The total quantity produced in the market is:
\(Q=Q_1+Q_2=100+100=200\)
We tin can now find the equilibrium market price from the original demand equation:
\(P=300-Q=300-200=100\)
This means that each of the two firms earns a profit equivalent to their total revenue, as the marginal costs are naught:
\(\pi_1=\pi_2=TR_1=TR_2=(300-Q)\times Q_i=(300-200)\times 100=x,000\)
We can now plot our Cournot equilibrium on a diagram! Accept a expect at Figure 2 beneath.
Fig. 2 - Cournot equilibrium example
Figure 2 shows a Cournot equilibrium for the duopoly consisting of the Happy and Lucky firms. Annotation that this equilibrium occurs at the intersection of the two reaction functions. The reaction function of each firm represents its output given its competitor's output.
Cournot model collusion
Let's imagine for a moment that the two firms decided to collude. How would the Cournot equilibrium wait, so? Information technology would be rational for the Happy Business firm and the Lucky Firm to maximize their total profits and then split those however they agree.
C ollusion occurs when 2 or more than firms cooperate to set up either prices or outputs for common advantages, such as higher profits.
Recall the market demand equation:
\(P=300-Q\)
The full combined revenue for the two firms is then:
\(TR=P \times Q=(300-Q) \times Q =300Q-Q^2\)
Let'due south discover the marginal revenue of the joint production:
\(MR=\frac{\Delta TR}{\Delta Q}=300-2Q\)
Setting MR equal to zero and solving for Q yields:
\(Q=150\)
At present the two firms can produce any quantities they desire. Still, to jointly profit-maximize, they demand the full quantities to add upwardly to 150.
The Happy Firm and the Lucky Firm owners are friends, and then they decide to divide the profit evenly. Therefore, they produce the aforementioned quantities:
\(Q_1=Q_2=75\)
What is interesting to run across is something called a standoff curve. A collision curve would show all the possible output combinations that the firms can produce. These outputs would inevitably add up to 150 and thus maximize joint profits.
The equation of the collusion curve is:
\(Q_1=Q_2=75\)
Accept a look at Figure 3 below for a visualization.
Fig. 3 - The collision curve
Figure 3 shows the bunco curve in yellow, which has some very important insights. Without cooperation, firms can brand less profit and have to produce higher output. With cooperation, they can restrict their joint output and enjoy higher profits.
The amount of profit that the firms were making jointly before cooperation was:
\(\pi_1+\pi_2=10,000+10,000=xx,000\)
By colluding, they can enjoy higher profits of:
\(\pi_1+\pi_2=P \times Q = (300-150) \times 150 = 22,500\)
A collusion bend shows all the possible output combinations the colluding firms can produce to maximize joint profits.
Cournot Model vs. Bertrand Model
What is the difference between the Cournot model vs. the Bertrand model? The master divergence is that in the Cournot model, firms compete in quantities. In contrast, in the Bertrand model, firms compete in prices. This has a few significant implications.
Cournot saw a colluding duopoly acting akin to a monopoly in terms of price and quantity setting. In contrast, Bertrand saw price competition in a duopoly leading to a similar outcome as in the perfect competition.
A Cournot equilibrium is stable, and in that location is no incentive for the two firms to engage in price wars. However, in the Bertrand model, firms are likely to become through a toll war, bidding down prices to their marginal costs until no firm has an incentive to deviate. That is, raising the price either higher up or lowering it below the marginal cost would exist worse for the business firm. This is an event that similarly occurs in the perfect competition model.
Cournot Model - Key takeaways
- The Cournot model in economics is a model of oligopoly where firms produce homogeneous products and compete in quantities.
- The reaction function depicts the human relationship between the quantity the firm should produce to maximize profit and the amount it presumes the other house will make.
- Cournot equilibrium is an equilibrium in a duopoly where each firm sets its output quantities, having correctly anticipated the amount that its competitor chooses to produce.
- Nash equilibrium is an equilibrium in which no firm has an incentive to deviate from its strategy. Each firm exercises its near profitable strategy, given its competitor'southward option.
- Bunco occurs when two or more firms cooperate to fix either prices or outputs for mutual advantages, such equally higher profits. A collusion curve shows all the possible output combinations the colluding firms tin can produce to maximize articulation profits.
Source: https://www.studysmarter.co.uk/explanations/microeconomics/imperfect-competition/cournot-model/
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