Now, we will take the linear city model and we will try to make it dynamic. More specifically, let's add an earlier stage in this model, so that firms are now able to choose also, their locations simultaneously. In other words, we're going to drop, get rid of the assumption that firms randomly select their locations. And we will assume now, that firms choose the best location that they can, in the first stage of the game. So, therefore, we will have a two-stage dynamic game. At stage one, firms choose where they will locate, that is they choose the distances from the each end a and b. So, this will be variables that the firms will have to optimize in our game. And then, at stage two, firms choose simultaneously prices, as they did it in the previous version of the static model that I showed you in the previous segment. The timing of the game captures the essence that we know from the very introductory courses in microeconomics about the distinction of the short run and the long run period. That is, in the short run period, you have to make some decisions that they refer to production factors which are flexible and you can adjust, adjustable. But there are some other production factors that you cannot adjust and that's why this is called the short run period. So, this is the essence. In the beginning, when you are in the long run, you make a choice about where you will be locating. Now, unless your firm is in a trailer and you move it around all the time, location is not a short run decision. Is a long run decision, you make it in the beginning and then you stick with it for many many periods. You cannot change it from period to period according to the short run needs of the market. As usual, we will apply the backward induction method, to solve this model, that is, we will start from stage two. We will see how the firms compete with respect to prices, and then we will get back to stage one and we will see how firms will locate. We have already solved stage two, in the previous segment when we had the static model which was exactly what firms did, just competed with respect to prices, and therefore, now, what is remaining is to go back to stage one. So, when we were in the previous segment, I told you plug the prices, the equilibrium prices that we calculated into the profit function, to receive the profit function, to calculate the profit function. I hope you have done that, if you didn't, I have done it for you anyways but you can check me if I'm correct, I think I am. So, this is the profit that we will have at stage two, and now, what do we need to do is to take this profit and maximize it with respect to location. Find out which location in this model will maximize our product. So, I take the derivative with respect to a and I have to set this equal to zero, in order to derive what is the optimal position, optimal location that this firm has to go into that straight line. So, I'm looking for the optimal a in other words. Once I do that, I can realize that this first order condition, can never be equal to zero, it will always be negative and if you look at this situation, you're going to realize that giving meaningful values to my parameter l and t, then, you will understand that this cannot be positive. This gives us the solution that as you probably know from your mathematics of optimization, your calculus classes, this is called the bang-bang solution. And therefore, in order to be able to solve it, we have to set a to it's minimum possible value, so therefore, we're going to make this expression as close to zero as possible. So, this means that the firm one, must locate at the West end of the line, has to go as far as it can towards the west. Similarly, because this problem is symmetric, firm two will locate at the East end. So, if we do the same for firm two, we'll derive an optimal b to be equal to zero, the minimum possible value, because you cannot get negative values, this would be outside of the city and will give us a result that the model will not know how to treat. So, hence, what we obtain from this solution is maximal differentiation. Those firms, they want to be as far from each other as possible. They go to the sides, each firm picks a corner and they go to the two corners to be as far as possible they can. This happens in this particular model. There are other specifications of the Hotelling model of the linear city model, which will give you either minimal differentiation, meaning that the two firms will go and locate exactly next to each other or there are other specifications that they will give you no equilibrium at all. Firms, indeed, they will locate at random by just mixing strategies there. What does this depend on? This depends on the exponent of the transportation costs and we have research that shows that if the exponent is above one and two thirds in our case, it's two, so it's above one and two thirds, then firms will decide to differentiate at the maximum point. So, what happens with differentiation? What is important to see in this model, the intuition that we will understand now, and it's really worth it to look at that because to me, it's one of the most interesting parts that you will see ever in differentiation. Is that, there are two opposite effects in our model. Two effects that we can only see if we indeed solve this model and we understand how it works. There is first of all, the demand effect, that is moving towards your rival, increases your market share. This is what I showed you before. If firm two steps closer one step to the other firm, then this will make its market share to go up. So, your market share is increasing, if you move towards your opponent and this is the demand effect. But this demand effect is not alone, is contradicted by the market power effect. Once you move closer to your opponent, what happens is that you decrease your market power. So, moving towards the opposite side of your opponent, increases your market power. Why? Because you are in a different location, much different location than your opponent. So, it means that the people that are around you, they have a higher incentive to shop from you so you can increase your price much higher. So, you have a smaller market share, which you can exploit more intensively in this case. So, you have two opposite effect that they contradict each other. In the previous setting, the second effect dominated, so the market power effects was dominant and therefore, firms tried to move as far from each other as possible. Therefore, we got maximal differentiation. In other models, the demand effect dominates therefore, firms try to move together. In reality, we see this effect several times. As I told you, clothing stores and malls, they try to locate far apart from each other. On the other hand, car dealerships in most of the cities, they locate next to each other. This has to do with a demand effect and the market power effect and how they contradict each other in each case. What is really important is, how is this location? Is it differentiation method. How do we actually understand location to be the differentiation? There are several other methods of differentiation. Why do we insist so much with location? Let me show you something that is really cool. The linear model that we just examined, can be easily interpreted as a product differentiation model rather than, a model of location, a spatial model. City, what we called city, can be interpreted as the preference space of consumers. Firm location can be interpreted as the product specification. And then, consumer location represents individual preferences for each consumer. Transportation cost represent the dis-utility from the consumer not being able to buy the exact variety that they prefer. What exactly do all of these mean? Everything of the above, will come into place, once I tell you the following example. Imagine, when you go to the store and you want to buy a pair of jeans. Some people they like their jeans to be totally distressed almost destroyed. They look in very bad condition like as if they are 100 years old. And there are some other people that they like their jeans to be totally neat, that they have no problem at all into how they look. They look like slacks pants. So, in this case, you have preferences and you have linear city where jeans can be located in the city. In the one end, the genes will be entirely distressed and the other side they will be entirely neat. And you can have all the combinations between these two possible situations of your jeans. Now, firms will decide where to locate. For example, Donna Karan New York, they usually locate towards the side of the neat jeans, not totally neat but towards that side, and Diesel will decide to locate towards the other side and you have many other firms that they are between the two or even, towards the two extremes. You as a consumer, depending on how you like your jeans, you're going to locate yourself in a specific part of the curve also. And if you are in the part that your favorite jeans are not served by a company, you will have a dis-utility. You will have something like a transportation cost, to go to the closest firm, if you like them too stressed and the most distressed jeans out there are let's say, they are Guess Jeans. Then you will have a cost. Say, "Yes, I like those Guess Jeans, but they're not as distressed as I want them." So, this is something like your transportation cost. So, instead of having a driving distance from going from one firm to another, now, you have a utility cost, a utility distance, from going from one firm to another. And this gives you how differentiation works in a location model. Everything that we said in the model, can be interpreted in the space of the variables that we show in this slide. City, firm location, consumer location and transportation costs can also be interpreted in the context of differentiation without requiring location to be a factor. In the next segment, we're going to talk about a different model, that will show us very good intuition about entry consideration and deterrence, as we said two lectures ago.