 Then, we will take the case of another non-colonial model that is generally known as the Stackelberg model and this is the extension of the Carnot model. And this is popularly known as the leader follower model and here one player is sufficiently sophisticated to recognize that the rival firms act according to Carnot assumption. So, here one player they feel that they recognize that rival firms what is the rival firms reaction and what assumption they are taking when they are revising their output plan. So, this sophisticated firm when they recognize that what will be the rival's plan or what is the reaction curve of the rival, they also able to incorporate that in their own profit function because they know now what is the reaction curve function of the rival. They act as a monopolist, the native firm will act as a follower. So, here how it is different from the Carnot model, here in the case of Carnot model, when the firm they were deciding about the output plan they are not considering the reaction function of the other firm. But in case of this Stackelberg model, they consider this reaction at least one of the firm who is sophisticated enough to understand or identify the reaction function of the other firm and they incorporate that in the profit function. And that firm who has identified this, they generally act as the leader and the other firms act as a follower and that is why it is also known as the leader follower model, this typical Stackelberg model. Both the firms is in equilibrium because they are maximizing the profit. Before going into this equilibrium, we will see that ideally graphically how they reach to this equilibrium. So, this is the reaction curve of firm A and this is the reaction curve function of firm B. Now, this is the point where both the, so this is the isoprofit and this is another isoprofit for B. So, this is x dash B and this is x dash A. Similarly, this is the isoprofit function for A. Now, this is also, this is isoprofit function of B. So, this is also the isoprofit function of A. This is A, this is B. So, R B is the firms B reaction function, R A is the firm A reaction function. Now, corresponding to this, here we have x A and here we have x B. E is the equilibrium output. Now, let us assume that firm A is sophisticated enough and they operate in typically this R A, R A, E, R A because this is their reaction curve function and they will produce A which is profit maximizing because the isoprofit curve is also in the reaction curve function now. Now, in this case, A will produce O X A and B will produce O X B. In this case, A is the leader and B is the follower. Now, suppose we will take the term, suppose B is the sophisticated firm over here. What is their reaction function? Approach the reaction function is again R B E is the reaction function for firm B. It is equilibrium at the point B. So, B will produce O X B dash and firm A will produce O X dash A. So, if you look at whoever is sophisticated, they are producing more, who is they are the leader, whoever is not sophisticated, their follower, they are generally paying, they are producing less. Like in case of, in the first case, B is producing less, A is producing more and in the second case, B is producing more since B is the leader and A is producing less. Now, till the time the situation in one of them is leader, the other is follower or the reverse may happen that B is the leader and A is the follower, they will just the output will change because they are sophisticated. Now, what happens when both of them they become sophisticated? Price war will continue. What is the outcome? Price war will continue, but price war is also not beneficial for the oligopolies that also they know. So, initially when both of them they will be trying to sophisticated, they will be trying to be the leader in the market, initially price war will continue, but when they realize that price war is not going to benefit them, rather price war is going to benefit the consumer, they will stop over there or they will stabilize price over there and then finally, they get into the cartel. So, Steckelberg model says, what Steckelberg model says that it is always profitable if one of them is leader, the other follower, still the time the follower is also getting their share of profit and their share of output. But the question is that it will not continue for long run because if one firm is getting more profit because he knows that what is the reaction pattern of the other firm, the other firms will also try to do that in the long run and eventually both of them they were trying to be the leader and that will lead to the price war and finally, it is a cartel. So, the end outcome when you think about the end outcome of a Steckelberg model, still it is not determined fully that what should be the end outcome where they should stop. So, when it comes to monopolies, both the firm this typically the steopolis, both the firms in equilibrium because they are maximizing their profit and have no tendency to change the output, typical in the graph if you have seen at the point E and equilibrium is reached when each firm is able to assess the other output correctly. And this is achieved after a series of change in the output by each firm in the anticipation of other outputs remaining unchanged. Like in the previous case, in case of Karnat model we are discussing the action reaction pattern of both the firms finally, take them to the equilibrium and the same thing happen in case of Steckelberg model also, equilibrium is reached when each firm is able to assess the other output correctly and this is not happened once this generally happens after the action reaction pattern and in the anticipation that the other outputs is remaining unchanged. Then before going into the next model, we will just take a small numerical to understand this Steckelberg model. So, we have a demand function which P is equal to 200 minus Q, then cost is fixed so MCA and MCB is equal to 80. So, P also we can say this is 200 minus, 200 minus QA plus QB and what will be the revenue function of A? So, total revenue of A is P QA. So, this is 200 minus Q by QA which is 200 QA, this is no minus here 200 QA minus Q square A minus QA QB because this Q is again QA plus QB. So, total revenue of A is 200 QA minus Q square A minus QA QB and for marginal revenue of A this is DTRA with respect to DQA. So, this is 200 minus 2 QA minus QB, marginal cost of A is equal to 80. So, if marginal profit maximizing rule says that marginal revenue of A should be equal to marginal cost of A. So, 200 minus 2 QA minus QB should be equal to 80. So, QA is equal to 60 minus half QB and this is generally the reaction function of A. Similarly, for B we can find out QB is equal to I am not just getting into the detail of the derivation. So, QB is equal to 60 minus half QA. This is the reaction function of B. So, now to solve the value of QA and QB we can just put the value of QA in equation of QB or QB in equation of QA. So, QA is equal to 60 minus half QB. So, 60 minus half 60 minus half QA this is the value of QB. So, simplifying this we will get QA is equal to 40 and QB also equal to 40. So, QA is equal to 40, QB is equal to 40 and Q has to be equal to 80 since it is QA plus QB and price is equal to 200 minus Q. So, 200 minus 80. So, 120. So, Q is equal to 80, B is equal to 120, QA is equal to 40 and QB is equal to 40 with this demand function and cost using the Stackelberg model. Then generally how to solve this numerical or how to find out this profit maximizing level of output for both the firms. Using the profit maximizing rule we need to find out the reaction curve function for both the firms that is for firm 1 and firm 2 and from there we can solve the value of output that is QA, QB or Q1, Q2 just putting the value of the others and that gives us the total output in the market and also the output specific to the firm. So, here typically the reaction function generally we say that this is a reaction function approach through which generally we get the individual firms output and the total market output. Then we will get into the discussion of king demand curve model and king demand curve model is also one form of the non-collusive oligopoly model where it assume that there is no cooperation or no collusion among the firm in case of a king demand curve model and this model generally explain us that why the price is rigid for the firms and at least in the oligopoly market why it changes very slowly over time. So, individual firms basically afraid to change their price because of what other firms might do. So, if one firm change the possibility that the other firm may not to change and that is the reason they afraid to change the price and that is the reason in case of oligopoly market may be decreasing price is not so slow, but increasing price is slow because others may not follow to this. There are certain assumption we take in order to understand the king demand curve model. The first one if a firm raises price other firms own follow and firm loses a lot of business. So, whenever there is a increase in the price the other firm will not follow it automatically and that is why that firm who has raised the price they generally lose lot of business. So, demand is very responsive or elastic to price increase and if a firm lower the price other firm follow, but the firm does not gain much business. So, in this case if you look at this part of the demand curve is inelastic because whenever one firm lower the price the other firm also lower the price in order to get more market share or more demand and that is why in this case the change in the price is not affecting the quantity demanded of the firm much and that is why we get a inelastic demand curve. So, demand is very responsive or the elastic to increase in the price and demand is fairly unresponsive and inelastic to price decrease. So, if you look at this graph if reduced price and competitor match the price typically if you look at now we are getting two set of the demand curve. One is the elastic demand curve another is the inelastic demand curve. If reduced price and competitor match the price cut then move along the inelastic demand curve that is segment DI and if increase in the price and competitor is not following that then we get in the segment of the elastic. So, we have two kind of demand curve now. One we have a inelastic demand curve and we have a elastic demand curve. So, this inelastic demand curve is when price decreases other firm also decreases the price and elastic is the basis is that whenever there is a increase in the price other firm keep the price constant. So, ideally what will be the demand curve for the firm there are two segment one and one segment of the elastic demand curve and one segment of the inelastic demand curve. So, this segment of the elastic demand curve because of the fact that whenever there is a increase in the price the competitor they are not matching to it and this part of the demand curve is one whenever the price cut happens the other firms or the competitor also decreasing the price. So, increase in the price competitor price remain constant decrease in the price the competitor also decreasing the price. That is why the demand curve of the firm has two segment one is the elastic segment with respect to increase in the price and other is the inelastic segment that is the with respect to decrease in the price. Here to remember that decrease in the price is generally followed by the competitor, but increase in the price is not followed by the competitor and that is why we get two separate segment in the demand curve one is the elastic segment and other is the inelastic segment. So, this is generally the shape of the king demand curve where the upper portion is elastic and the lower portion is inelastic. The upper portion is come from the elastic demand curve and in this segment whenever there is a increase in the price because increase in the price the competitors they are not going to follow it and the downward segment is the part of the inelastic segment where whenever there is a increase in the decrease in the price competitor generally follows this. Now, how this marginal revenue curve here we get this the first case this the margin we get also two marginal revenue curve here because since we have two demand curve we have two marginal revenue curve. The first segment of the marginal revenue curve comes from the top part of the demand curve which is the elastic demand curve. So here this marginal revenue curve with respect to the elastic segment of the demand curve. Then we will see what is the marginal revenue curve for the bottom segment and for the marginal revenue curve for the bottom segment is again it is a part of the inelastic demand curve. So, if you will find there are two marginal revenue curve with respect to two demand curve, because it is one demand curve, but it has two segment. One is the elastic segment other is other one is the inelastic segment. So, one marginal revenue curve with respect to the elastic segment and the other marginal revenue curve with respect to the inelastic segment. So, this is for the bottom part of the demand curve and previously it was the top part for the demand curve. So, this is the inelastic part of the demand curve. So, this is generally the king demand curve and we have two marginal revenue curve. If you notice here, there is a gap between the two marginal revenue curve. Why there is a gap between two marginal revenue curve? Because our demand curve has a kink and at the point of kink we are not able to decide which one is the marginal revenue curve. So, if you look at demand curve is generally known as a king demand curve because it has a kink between the two segment of the demand curve that is between the elastic segment and inelastic segment of the demand curve. Corresponding to the elastic segment, we have one marginal revenue curve corresponding to the inelastic segment, we have another marginal revenue curve. At the corresponding to the point of kink, there is a gap between the marginal revenue curve 1 and marginal revenue curve 2. So, that is why in case of a king demand curve, there is a gap between the marginal revenue curve 1 and marginal revenue curve 2. Now, the question comes how the marginal cost should be because we need to get the equality between the marginal revenue and marginal cost to get the profit maximizing level of output. And the marginal cost should cut which segment of the marginal revenue curve whether the segment related to the elastic demand curve or whether the segment related to the inelastic demand curve. So, marginal cost generally intersect the marginal revenue curve in the gap in the vertical segment in the gap between the marginal revenue 1 and marginal revenue 2. And whenever there is a increase in the marginal cost, if the cost shift up slightly, but marginal cost still intersect the marginal revenue in the vertical segment, there will be no change in the price. Because if any point of time if marginal cost goes to marginal revenue 1 that is elastic segment or to the inelastic segment, still it will not consider as for the whole demand curve whole king demand curve. And that is why you will find there is a price rigidity and this is the outcome of the price rigidity that we get to level of marginal revenue curve and the marginal cost curve is not going for the marginal revenue 1 or marginal revenue 2 rather it is in the gap. So, even if there is a increase in the cost till the firm is not changing the price because if it is changing the price again it may lead to a situation that the other firm will not follow it and they will get into the loss. So, king demand model generally it provides the explanation, detailed description of firm under oligopoly and it explains the various characteristics such as price rigidity, why there is a indeterminate demand curve, non-price competition and independent decision making. But this model fails to explain a basic question how price is determined. Because it is really a fuzzy when the price is decided in the gap in the vertical segment between marginal revenue 1 and marginal revenue 2. And that is why this model has a criticise on the ground that it fails to explain the basic question of any model that how the price and output is determined. Because we have a king demand curve we have two level of marginal revenue curve that is marginal revenue 1 and marginal revenue 2. So, we will just take a numerical to understand that when you take a numerical, when you take a real production function, when we take a real demand function, cost function, whether we get the gap between the marginal revenue 1 and marginal revenue 2 with respect to two different demand and whether the marginal cost also pass through the vertical segment or the gap between the marginal revenue 1 and marginal revenue 2. So, we will take two demand function, we will take as Q 1 and we will take as Q 2. So, we have two demand function, 1 is 28 minus 4 P 1 and second is 10 minus P 2. We will take a total cost function that is 1 4 Q square plus Q plus 50 and we need to find out the marginal revenue for both the firms, marginal cost, price, output and we need to say what is the upper and lower limit of MR because that will tell us whether there is a vertical segment or gap between marginal revenue or not and whether MC falls in the gap of 2 MR or not. So, to start with we will find out since we have Q 1 is equal to 28 minus 4 P 1, we will find P 1 is equal to 7 minus 1 by 4 Q 1, Q 2 is equal to 10 minus P 2. So, P 2 is equal to 10 minus Q 2, total revenue 1 is P 1 Q 1. So, that comes to 7 Q 1 minus 1 by 4 Q 1 square and total revenue on corresponding to this we will get the marginal revenue 1 that is 7 minus half Q 1. Similarly, total revenue 2 is P 2 Q 2, so this is 10 Q 2 minus Q 2 square, marginal revenue 2 is 10 minus 2 Q 2 Q 2 and we have now marginal revenue 1, we have now marginal revenue 2. We will find out the marginal cost from our total cost function. So, total cost function is 1 4 Q square plus Q plus 50. So, marginal cost will come as 1 plus half Q and if you look at the king at the point of king both the demand curve should intersect and to get this intersection we have to do is equal to Q is equal to Q 1 plus Q 1 equal to Q 2, so 7 minus 1 4 Q is equal to 10 Q, so Q is equal to 4 and P is equal to 6. So, taking the value of Q and P, marginal revenue 1 is equal to 7 minus half into 4 that is equal to 5, marginal revenue 2 is equal to 10 minus 2.4 that is 2, so marginal revenue 1 is 5, marginal revenue 2 is 2. Now, we need to find out m c, so m c is equal to 1 plus half Q, so this comes to 1 plus half multiplied by 4, so this comes to 3. So, we can say that Q is equal to 4 this is the output P is equal to 6, we have first segment of m r is equal to 5, the value of second segment of m r is equal to 2 and m c is equal to 3. So, we can also prove that the m c falls in the gap of 2 level of m r that is marginal revenue 1 and marginal revenue 2. So, today we discuss about typically in this session we discuss about the Carnot model, Stackelberg model and King demand curve model and all these three models are part of non-collusive oligopoly model. In the next session we will discuss about the collusive oligopoly model typically the cartels and the price leadership model.