Now, we will examine a model that looks very similar but also has a lot of important differences from the previous model. So let's talk about Salop's circular city. Now, our city is not going to be linear anymore, it's going to be a circle. And let's see how this is going to work. So we have a circular city that the perimeter around it is one and the density is S. Now, we have S consumers in each single point of this circle. Here is the circle, in order for you to see. And entry now, if you decide to enter into this market, is not free. It's open but it cost a fee lower case f. So in order to enter, you have to pay lowercase f, in order to set up a firm and locate at specific point into this circle. What is really important in this model is that once a firm decides to entry, picks a point on the circle, a location on the circle, and settles to that location, that's it. The firm afterwards cannot relocate. The cost of relocation will be much higher. This model has two stages. In Stage 1, firms enter and they scatter around the city in equal distances from each other. So we decided, for example, that eight firms will enter and then they will look like that. So they scatter to equal positions around the circle. And then, in the second stage, once they have settled and they have located, in the second stage, they will compete with respect to price exactly like before, we have Burton competition in the second stage. Some very useful properties of this model. The distance between firms is equal to 1/N. So this assumes that we have N firms to enter. So once we have N firms entering, then the circle has a perimeter of one, the distance between the firms, since it will be equal, it will be for all of them 1/N. So firms have a distance 1/N from each other. Competition in this model is localized. That is firm i that you see there in the middle, the red firm, will compete only with light blue one and the green one. That is, because the model is symmetric, every consumer that is between those two firms, like this guy that we have here, will decide to buy from one of the two neighbouring firms. So for example, this little blue guy will not want to go to the pink firm across the circle just because can buy either from the red or from the green at usually symmetric price, or prices that they are the same. For any distance X on this circle, on the perimeter of the circle, like for example the distance from our guy here towards the red firm, there is a total demand equal to S times X. X is a fraction. This is because it is a part of the circle and the whole circle is one. So X has to be a fraction, and then, S is the density, how many consumers are per point. So multiplying that by the fraction of the circle that we consider, we take the demand for this particular part of the circle. We also have transportation costs here. Our transportation cost are not quadratic anymore. We do not need to have increasing cost here. So our cost is t, which is a parameter times x. We're going to use one more time the backward induction method. We will start from stage two, solve stage two and once we have the solution for the terminal stage, we are going to go back to that first stage and examine what happens there with the location of the firms. So we assume that somehow N firms have entered from the previous stage. Because of symmetry, we can assume that at equilibrium, all firms will set a uniform price. In other words, Salop did a trick in this model. Said that, 'Okay, so my firms are the same. They have the same cost. They have the same distance from each other. They are the same mass of consumers around them, why should they charge different prices?' So, assumed a correct assumption that firms will price equally at equilibrium. So let's call this price p_upper bar. This is the price that all firms will set. And now, what we have to see here, we have to check if indeed this is a Nash equilibrium. How do we check Nash equilibria? In general, we check if there is tendency for deviation. So we want to see if any firm i, let's say our firm, wants to deviate from these uniform price to another price, let's say p_i. So we want to see if our firm wants to set a different price than p_upper bar, say price p_i, and see if this solution will be a good idea for our firm. Recall that competition is not as hard as it seems. Competition is localized. You only compete with your neighbors. You do not compete with the entire circle. Consumers will have no reason to consider someone who's across the entire circle in order to buy the product from them. They will always try to buy from the one that is in the neighborhood, unless you have some anomaly in this model and some of the firms, for some mysterious reason, charges extremely low prices, which is not an equilibrium solution as Salop has assumed from the beginning in this model. So a consumer located from a distance x from our firm i, that will try to deviate as we said, then has to consider to buy either from us or from the firm on the other side of this consumer. This can be either firm j or firm h. And what we are looking now is, again like in the Houghtaling model, we are looking for the indifferent consumer. Now, we'll have two sides to each firm because we have more competitors than in the Houghtaling model, so we'll have one indifferent consumer on one side and one indifferent consumer on the other side. But again, because of symmetry, we can understand everything by just considering one consumer. So a consumer is indifferent margin now between the two firms if the price for buying for our firm, which don't forget, we charge p_i and the overall price for buying from the other firm, don't forget that the others charge p_upper bar, plus the transportation cost, because from our firm, the distance is X as we have said before, and from the other firm, the distance is what? Is 1/N minus X. So this is how much they are away from the previous from the other firm because they are x away from us and we are 1/N away from the other firm, so they are 1/N minus X away from the other firm. Again, we will continue what we did in the Houghtaling model. We will solve these equation with respect to x and this will give us the exact location of the marginal consumer from that particular site. So in this case, I'm giving you the solution right here. And demand for each firm therefore is q_i. The amount of products that you will sell is equal to two, because you have two sides, one side, the other side, times s, which is the density per point in this case, times X, the distance of the marginal consumer. Because up to the marginal consumer, you will sell. After the marginal consumer, the other firm will start selling. So where is the marginal consumer? Is given by the function above our demand that we tried to calculate here and this parenthesis there p_upper bar minus p_i means that, as you can see from the equation above, that X is a function of p_upper bar minus p_i, is a function of the price differential between the two firms. So firm i problem at stage two is that it will maximize the profit. So we said that this is the average profit p_i minus c, because we are charging p_i and it's cost C to produce its given unit times our demand, which is two times s times x, which I'm giving you here in terms of how we have calculated it. I take the first order condition in order to maximize respect to p_i. And this is a well-behaved problem. It will give me a solution that at equilibrium, symmetry will prevail and therefore, we'll have the p_i equal to p_ upper bar and this will be equal to C, my cost, plus t/N. That is, we are going to sell above cost again also in this model because of location differentiation. Profit in this case will be p_upper bar and it will be a decreasing function of N. The more firms that they are in, the lower the profits will be. And this can be seen very easily because I have calculated it. It's S times t/N square. So it's decreasing with respect to the number of firms that they're in. Let's go back to stage one and see what happens in stage one. What is important for you to understand here is that we will use an optimal stopping condition, meaning that firms will keep entering till the point that the profit that they acquire by entering is equal to the entry fee that they have to pay in order to set up a firm in this market. Since we said that profit is a decreasing function, as the number of firms is increasing, p is decreasing. And therefore, we will have a situation in which as N is increasing, p or pi, the Greek P, is decreasing and when it meets f, firms will stop entering. This means that we can replace what's p of N is and we can get that this is equal to s times t/N squared if this is equal to f, then firms at that point will stop entering. We can solve this with respect to N and we can figure out the equilibrium number of firms that they will enter into these industry. And this is going to be the square root of s times t/f. So we know now what is the maximum amount of firms that these industry can accommodate. And this will be very important. Note it down because we will use it in the next segment to understand how deterrence works. So the maximum number of firms is increasing with market size. Higher market size means more consumers, so you can accommodate more firms into this industry. Transportation cost t also play a role of increasing the maximum number of firms that can be accommodated. If you have higher transportation cost, it means that you want more firms to be close to consumers. So therefore, transportation distances will be staying low. And then, market can accommodate less firms if the entry cost is higher, meaning that the more expensive it is to set up, the less firms will decide to enter into this market. You can plug all these to the price function, to the demand function above, and you can get that equilibrium price is, again, above marginal cost C plus the square root of t times f/s. This is a positive quantity. So again, we have more than normal profit here. We'll have more than perfectly competitive profit for the firms. So again, because of differentiation, we do solve the Burton paradox. In the next stage, we will see how we can use these findings of our model to understand what happens with firms proliferating into industries. Stay with us.