 So, we will continue our discussion on non-collisive oligopoly model also in this session. So, if you remember in the last class, we discussed about the connoit models and connoit models typically a duopoly market situation, where the basic characteristic is that it operates in zero cost of production and that is why we get a marginal cost which is equal to zero. And here, even if the firm they knows that they are interdependent on each other, but when they fix their output, when they fix their price, they are not taking into consideration the what will be the rivals plan on the basis of their revised plan. And that is the reason if you remember in the last class, we discussed a situation when in the period they one-fourth of the market remain unexplored, no one produce in that one-fourth market, the rest three-fourth market is got produced by both the firms A and B. So, we will continue our discussion on the connoit model, we will see that how the equilibrium solution can be achieved or how the equilibrium can be achieved with the reaction and action patterns of the reaction and action pattern of the two firms. So, in the last class, if you remember, we talked about a situation where the three-fourth of the market generally. So, in that case, if you remember, then a, b together they were just producing three-fourth of the market and one-fourth was not produced either by firm A or firm B at least in the first period. Now, what will happen in period two? So, this in the first case, three-fourth was produced together by a and b and one-fourth was not. Now, how it will happen in period two? From A, assume that B is just going to produce one-fourth of it and he will feel that three-fourth is his own market typically for firm A. So, if three-fourth is his own market, now generally he is producing only half of it, so he will produce half of three-fourth in the market and that will come to three-eighth of total market. So, A is going to produce three-eighth of total market. A, assume that B is just going to produce one-fourth of the market. So, A will think that he is just going to produce only three-fourth, the rest three-fourth is his own market and since A produce only half of the total market, in this case the half of total market is half multiplied by three-fourth and that comes to three-eighth as the total market. Now, we will see how B is going to react to this. So, B will consider that again when it comes to B, what will be the assumption for B? The assumption for B is A will continue to produce three-eighth of the market means what is the market available for B, the remaining market that is less by three-eighth, so this is one by three-eighth is the market available to B. Now, B will produce again half of this because half of one by one minus three-eighth, so this is five by sixteen, so A is going to produce three-eighth and B is going to produce five by sixteen. Now, this action-reaction pattern continues from always assume that A always assume that B is going to produce half and from B is always going to assume that A is going to produce half. So, they take out that half and they feel that the rest of the market whatever they could produce and again it comes the half of the total market demand. So, this action-reaction pattern for form A and B will continue. Now, what is the outcome of this action-reaction pattern of form A and B? So, it is just goes on with the share that is half of share, half of the total market except the share of A for B and half of the total market except the share of B for A. So, this action-reaction will continue. So, in this action-reaction generally share of A goes on decreasing and share of B goes on increasing and this will lead to a situation where each form supplies to an equal market that is one-third of the total market. So, this if you look at initially the share of A was higher than share of B. So, this action-reaction pattern will continue and with the action-reaction pattern finally each form will reach to a situation where they are just producing one-third of the total market and remaining this one-third of total market is produced by A, one-third of total market is produced by B and remaining one-third is not getting produced either by A or by B. So, we will just take the graph to understand this that how this reach to a situation where they just produce one-third, one-third and leads to a case where rest one-third is remain on produce either by A and B. So, if you remember your reaction function R 1, R 1 dash is the reaction function of form A, R 2, R 2 dash is the reaction function of form B. So, ideally this should be the situation where this is the amount of Q 1, this is the amount of Q 2. Q 1 is produced by A, Q 2 is produced by B. Now, how they reach to this Q 1 A to Q 1 B? It is not they start from here, rather they start from here and what is this R 1 dash here? At this point the total market demand is we can say O R 2 dash is the total market demand. So, out of this initially A will just produce this much and from there actually this action reaction pattern happens. So, if this A is producing this corresponding to this B will produce here that is the reaction curve of B. R 2, R 2 dash is the reaction curve function of B, R 1, R 1 is the reaction curve function of A. So, if you look at if initially A produce this much, now with reaction to this B will produce take a point corresponding to this in the reaction function of B that is R 2, R 2 dash corresponding to this again A will react to this and choose a combination where in the reaction function of A. And if you remember what is reaction function? Reaction function gives the different quantity of Q 1 and Q 2 combination of Q 1 and Q 2 where profit is maximum. So, whether B chooses any combination in reaction function 2, A chooses any combination on reaction function 1 ideally they maximize the profit. So, to start with they will do this with A start with this point corresponding to this B choose a combination over here. Now, suppose this is combination 1, then this is combination 2 again chosen by A with respect to the reaction to a combination B by this 1, then corresponding to this again B will choose a combination here. And corresponding to this A will choose a combination here, then corresponding to this again B will choose a combination here and here A will choose a combination here. This action reaction will continue till the time they are not reaching this point E. And after reaching the point E, the generally they reach the equilibrium solution where A produce Q 1 unit of output and B produce Q 2 unit of output. So, graphically how we reach the Carnot's equilibrium? We reach the Carnot equilibrium through the reaction function approach. We take the reaction function 1, reaction function 2 and the reaction of both the forms get captured in their own reaction function. And finally, eventually this reaction action pattern leads time to the equilibrium where they this is generally stable and after that at least in that time period again the reaction action pattern never continue. Then we will see the detailed description that how this one-third of this total market output comes to form 1. We will just take period wise that how this final yield comes to one-third for form A and one-third for form B. So, if you generally we will just take a form of equation in order to understand this. So, period 1 we will see what is form A, what is form B. So, form A is half of total market. So, this is half, form B is half of half market that is A share that comes to one-fourth. Then period 2, what will be for form A? Half of 1 minus 1 4, 1 4 is the B share that comes to 3 by 8. Then what will be for form B? That is half of 1 minus 3 by 8 that comes to 5 by 16. Now, what will be in period 3 form A? That is half of 1 minus 5 by 16. This is B share that comes to 11 by 32. So, this is B share 1 minus this half of it going to be produced by form A, form B again half of 1 minus 11 by 32 this is share of A. So, this comes to 21 by 64. Then we will take out period 4 output of form A and form B. So, this is nothing but just taking the share and making a half of it, but eventually we will see how they just produce one-third of the total output. So, for period 4 this is again half of 1 minus 11 by 32. So, that comes to sorry 1 minus 21 by 64. So, that comes to 43 divided by 128 and form B again half by 1 minus 43 by 128. So, that comes to 85 by 256. Now, this continue till the time period n. So, we will see how what will be the value in period n because n takes any number. So, it is 1 to n. So, in period n what will be the share of form A and form B? Eventually it is form A half by 1 minus 1 by 3. So, that comes to 1 by 3 and form B half of 1 by 1 minus 3 that is coming to 1 by 3. Now, what is A's equilibrium output now? A's equilibrium output will be half minus 1 by 8 by 1 by 1 minus 4. So, that comes to half minus 1 by 8 by 3 by 4 which comes to 8 by 24 and 1 by 3. This is form A equilibrium output. Similarly, we will find form B's equilibrium output. So, form B's equilibrium output 1 4 by 1 minus 1 4. So, this is 1 by 4 divided by 3 by 4 which comes to 1 by 3. And for n number of form what will be the industry output and what will be the firms output? Industry output will be n by n plus 1 or to again describe this. This is e is equal to 1 summation e 1 to n 1 by n plus 1. And what will be the individual firms output and 1 by n plus 1? This will be the individual firms output both for A and B. So, ideally what we want to check over here? We want to check over here is that when the action reaction pattern happens they assume the same behavior from other firms. And that is why they if you find some of this is remain on utilize that is not being produced either by firm A or firm B. And that is why they are just producing one-third of the total output of the market and remaining one-third is not being produced either by A and B. Next we will say we will just take an example to understand this Connard model. We will just take a numerical to understand the Connard's model and then we will move into the next model that is Stackelberg model and Paul Swigic King demand curve model. So, P is equal to 100 minus 0.5 x this is the demand function. This is cost function of firm A which is it is a constant cost function. For firm B it is the increasing cost function. We need to find out the profit maximizing level of output for both the firm A and B. And how we will find out this? We generally take the profit maximizing rule that is marginal revenue is equal to marginal cost. So, what is this P? P we can simplify this as 100 minus 0.5 x A plus x B because this x is the total output is the summation of output of A and B. And what will be the profit of firm A? That is total revenue minus total cost. And what is total revenue over here? So, total revenue minus total cost will give us the profit of A. So, this is P x A this is the output this is the price minus 5 x A this is the cost function of the firm A. So, what is P x A that is 100 minus 0.5 that is x. So, this is x A plus x B multiplied by x A minus 5 x A. So, this is cost function this is P and this is x A. So, if you simplify this then it comes to 100 x A minus 0.5 x A square plus sorry if you open the bracket this is minus plus minus 0.5 x A x B minus 5 x A. Simplifying this again this is 95 x A because this 5 x A will get deducted from here minus 0.5 x A square minus 0.5 x A and x B. Simplifying this 95 minus x A minus 0.5 x B is equal to 0. So, this is our marginal that is our this profit. So, this comes to marginal revenue equal to minus marginal cost has to be equal to 0. So, this is 95 if you take the derivative then this comes to 95 this comes to x A this comes to 0.5 x B. So, if you simplify this in term of x A this comes as a 95 minus 0.5 x B and what is this x A in term of x B this is the reaction function of A this is the reaction function of A. Similarly, we will find the reaction function of B and if you remember what is the reaction function of A this combination gives the maximum level of profit to the firm A. Similarly, we will now find it for B. So, if you simplify this for x B we get its 50 minus 0.25 x A. So, I am not just doing a detailed calculation for B you need to follow the same formula to find out the what we did for A the same formula to find out the reaction curve function for B. Basically, you need to find out the pi then you to maximize it pi is the difference between total revenue total cost you need to maximize this and then solve for the value in term of x B in term of x A and that will give us the reaction curve function for reaction curve function for B. Now, to find this value of x A and x B we need to need to put the reaction curve of reaction curve function of B in A and reaction and we can solve the value for x A. So, in this case you can find out x B is equal to 50 minus 0.25 x A. So, now what we will do we will put the value of x A in order to solve for x B. So, this is equal to 50 minus 0.25 by 95 minus 0.5 x B. So, that comes to 50 minus 23.75 plus 0.125 x B. So, solving for this x B will be equal to 30, x A is equal to 80 and putting a value of x A and x B that is 80 plus 30 that comes to 120 and putting the value of this will get the value of B which is equal to 45. So, this is the output of the B, this is the output of A and this is the profit maximizing level of output of A and B using the cornered model or generally using the reaction curve approach. So, what we discuss in case of cornered model this is a typical situation where two firms engage with each other they know the interdependence, but they are not considering the fact that they are interdependent to each other in order to decide the output plan. And that is the reason when they are revising their plan they are not considering the what will be the rivals reaction to the revised plan and ultimately they are reaching to a equilibrium which is stable, but in that case they are not exploring the output for the entire the market. And here we take the assumption that there is zero cost of production and we also discuss in case of cornered model that when we assume the zero cost and there is a linear demand curve the output of perfect the output of monopoly is half of your competitive output and the duopoly output is the two-third of competitive output. So, to summarize this H4 cornered solution equilibrium is stable and each firm will be maximizing profit by selling equal amount of output at the same price. So, price same they are selling the equal amount of output like one-third one-third of the total market and equilibrium is reached when both the firm earns maximum profit and have no tendency to change their output.