[Music] [Music] so let us talk about cournot model of duopoly what we have here is let us say that the market demand is given by 10 minus Q this is linear demand function that we have been using now what happens that we have two firms operating in the market now both the forms independently decide how much quantity to produce and price automatically gets determined through the quantity so if firm 1 decides to produce Q 1 and firm 2 decides to produce Q 2 the market price is going to be 10 minus Q 1 plus Q 2 as long as Q 1 plus Q 2 is less than 10 otherwise the market price is going to be equal to zero okay so let us say we will talk about residual demand function what is residual demand function firm 1 and firm 2 they make their decisions simultaneously what it means that they don't know how much the other firm has decided to produce firm 1 is going to produce 8 units then the price will become equal to 0 okay hold on I will talk about it once again so edit Carnegie 0 to V shikaku lokomat he has a fair shake out there ok so let us look at the market demand function okay this is 10 this is 10 it means this is P is equal to 10 minus Q here we have Q here we have P now let us say firm one thinks that firm 2 is going to produce two units okay again firm one doesn't know how much firm 2 is actually going to produce this is what firm let's say if this is what form one believes in that firm 2 is going to produce two units then if it produces zero unit then the price in the market is going to be eight and if it's if you one happens to be zero and if Q one happens to be eight P is going to be equal to zero so what I mean to say that this particular form would see the residual demand in the market which is nothing but this particular line which is given by eight minus Q how did we obtain this line we sifted this line by two units everywhere and that's how we obtain the residual demand function so let us say what is the profit maximizing problem for this particular form now so TR minus T C this is what this form would like to maximize so TR is going to be 8 minus Q 1 multiplied by Q 1 remember this is not the market demand this is the residual demand that this form for C for the future here we will have the total cost and let us say just for example the cost is 2q 1 okay so we will see how much this form should produce the form would maximize this particular equation and from here we can get this is 6 minus Q 1 multiplied by Q 1 so when the firm maximizes Q 1 will come out to be 3 so what we figured out when firm one thinks that firm 2 is going to produce 2 unit then firm 1 should produce 3 units and what we can do we have to calculate this for all the different values of Q 2 so rather than proceeding it like this we will do it for the general case so let us say that market demand in this general case is a minus B Q which we are familiar with we have done it many times this is the linear demand function in our previous problem a is equal to 10 and B is equal to 1 and let us say from one thinks that form 2 is going to produce q2 and let us say the cost function are both the firm's our C Qi so they have exactly the same cost Qi is the amount this particular firm I produces ok so let us look at how much the firm one should produce for firm 1 a minus B Q 1 plus Q 2 this is not actual Q 2 this is the Q 2 that from one thinks that firm 2 is going to produce multiplied by Q 1 because this is the price which is determined by Q 1 plus Q 2 amount so even before this production has taken place the firm were one would think this is what would be my revenue okay and minus C Q 1 now the form one would maximize this particular equation as a function of Q 1 so what we are going to get if we do this maximization we are going to get our first-order condition as a minus B Q 1 plus Q 2 hat minus V Q 1 minus C Q 1 it has to be equal to 0 so from here firm one can decide how much Q 1 to produce and it's going to be the function of from once belief about level of production by firm 2 so we can write a minus C minus BQ 2 hat divided by 2 B or we can write it like this a minus C 2 B minus Q 2 hat by 2 in other word we can draw here let us say we have q1 and here we have Q - okay let's say q1 and q2 hat so we can draw a firm one thinks that firm 2 is not going to produce anything the optimal amount for firm 1 is e minus C divided by 2 B so if Q 2 is equal to 0 this is a minus C divided by 2 B so q2 hat has to be equal to a minus C divided by B perform 1 to not to produce anything so if we draw this line and this is a minus C by 2 sorry a minus C by B this is what is called form 1 ones reaction function this is reaction or response function we can say this is the response that from one would give according to its belief about the label of production from form 2 similarly from 2 would calculate remember that firms are not deciding these things simultaneously so form 2 will decide how much to produce which would come out to be if you do it it's really going to come out to be this particular equation also you should understand the problem is symmetric so how did we do it that instead of using Q 2 here we are using Q 1 hat here and then we can plot it and it's going to look like here we have a minus C by B and here we have a minus C by 2 B if we want to change the color I think it's good idea to understand this is for for Mature ok and this is the special point that we would pay attention to okay what is this point equal to in these two lines they intersect here okay so at this is equal to a minus C divided by 3 B and here also a minus C divided by 3 B what it says that if farm one thinks that farm 2 is going to produce a minus C divided by 3 B then farm one would produce a minus C divided by 3 B ok and same is true for the farm too and you should think you should understand that this is the Nash equilibrium in the problem we cannot have any other equilibrium because let's see that firms are producing here so if let us say the production is taking place here to begin with it's taking place here what it means that sorry this violet color is the reaction or the response function of firm 2 and the red color here gives the response function of firm 1 so what it says that given that firm 2 is producing this much how much is the optimal for form 1 to produce more so firm 1 should increase its level of production and if firm 1 increases the level of production to here how much is the optimal for form 2 to do we can go back to this and this much is the optimal for form 2 to do so we see that in this zone from 1 and firm 2 both have incentive to increase their level of production and here when they reach to this particular point they will not have any incentive to change the level of production and therefore this is the equilibrium level so in cournot competition in this particular setting they will produce a minus C 3 B and a minus C 3 B how much is the total production to buy 3 B multiplied by a minus C what would be the price the price is going to be a minus B this is what is Q a minus B Q so from here we can get a minus B multiplied by 2 by 3 B multiplied by a minus C so from here we get 1 by 3 a plus 2 by 3 C so this is going to be the price in the cournot competition so that's it about the cournot competition we would come back to the cournot competition when we compare cournot competition with Stackelberg competition thank you [Music] [Music]