I will now show you an application, a very simple application of the Bertrand model with differentiation. It's going to be really simple. But we are going to derive numbers now because we don't want equations anymore, we want numbers. And then, in the segment after this one, we're going to plug these numbers into the game and I will show you amazing things about the interaction and competition of firms. We'll get right into the heart of the competition and collusion problems in today's competition in business. So, let's start with the application. I'm giving you two demand functions that resemble the demand functions I gave you before, totally symmetric. Now, instead of a, we have 10, and instead of d, we have minus two, and as I told you, d has to be more than one there in order for the model to make sense. We have the semantic functions. And then, assume that the cost is just one, simple number we can get. So, we have this setting now which is exactly the previous setting. As I said, with simple numbers, and let's solve it. We have to calculate the reaction function, so if you want, you can go to the previous segment and take the formulas and replace the numbers. If you want for practice, to solve it from the beginning, that will also be a good thing to do. So, if you do that, you're going to take these reaction functions. Again, as you can see, they are solved with the choice variable of the second firm because soon, we're going to graph them and who will need that. And as I have these two reaction functions, I can solve them with respect to each other as a two-by-two simultaneous linear system. This would be very easy for you to do, I guess. And once you do that, you get that the prices should be equal. That's reasonable because the model is symmetric, you shouldn't expect to get different prices. So, price is four, if they compete according to Bertrand, and then the quantities will be six, and then the profits will be 18. So, these are the numbers that you are going to receive. After the end of the segment, you will solve it, and you will verify that these numbers are correct: four, six and 18. Now, because we would like to compare in the end, let's assume what would happen if those two firms do not compete to each other but they act as a single firm. They act as a unit. They collude, or in other words, they form a cartel. This means that once the firms collude, they have to set a common price. So p_1 will be equal to p_2, and let's denote that by p dropping the indexes because we do not need them. The firm will produce now only one brand because it's a monopolist. It's one brand. It will go to the market and people will not really want to understand the differences between the two goods. So, the two demand curves that we had before, if we set p_1 equal p and p_2 equal p also, they will collapse into a single demand curve which here is q equal to 20 minus 2p. It will be a little more convenient for me to solve this demand curve with respect to p. This could make my life easier. So, if I solved it with respect to p and I take this inverse form, p equals to 10 minus half the quantity, and we are going to use that in order to form the profit function. To form the profit function, we need cost. Cost is one, again, like before, so the maximization of the profit function now for joined two firms, jointly, both together as a cartel, so we maximize the profits and we will yield that we will get that the quantity is 4.5. Of course, they will limit quantity because now it's a monopoly, monopolies limit quantities. Also, increase prices, so we observe now a price of 5.5 more than before, and the profit is 20.25 which also is more than before because the reason that firms want to become monopolies is because they have more profits. You shouldn't expect anything different here. So, of course, firms benefit when they collude. Now, let's see the graph again. The choice variable of the second firm is on the vertical axis, choice variable of the first firm is on the horizontal axis, and from the reaction from one's reaction curve will look like that, and firm two's reaction curve will look like this. They will intersect at the positive quartile, you get the Nash-Bertrand equilibrium at the intersection, and the two prices will be equal to four as we saw a few moments ago when we talked about the Bertrand with differentiation. Now, the collusive outcome will be outside of the reaction functions. If you recall from the previous lecture in game theory, we said that the collusion outcome is not best response, is not the best the two firms can do, because there is also a cheating case. If you remember that, this is exactly what appears in this graph here. The collusive outcome is not on the optimal response curves. It's better for both of them but both of them might have an incentive to deviate. This incentive for deviation, this cheating potential that there is in such corporations, is the topic of our last segment for today. So, stay with us. It's going to be very, very interesting, and also, it's going to show you something that we will keep using in every lecture from now on. Stay with us.