Now, we will examine a model for analyzing networks. This model is not difficult. It's not complicated it will be a simple model but however, it will have a slightly different logic from the models that we have done so far. I choose to show you this model because of the amazing things that we can discover through this model. It can show us things that you don't even imagine how networks work in that depth. So, let's see. And I promise that when we finish this model, you will see some truly nice intuition. So, consumers can have access to a service let's say a service from which they are no fixed substitutes and the service just charges a fixed fee, like you pay a fee and you have a monthly access to this service, unlimited. OK imagine whichever service you want. N is the maximum number of potential consumers. N is number of those people in the population that think that these network can offer something to them. These service can give them something more than zero. Therefore these are the people who would access the service if the price was zero. If there was no charge to join the service and people would join. The other people would say oh this service I don't even understand what it does they will not join even if the prize was zero but N is the amount of number of potential consumers that they will join if the price is zero. v_i is the valuation of a consumer let's say i when the price is zero. So, when these services given to you at a zero price, you have a valuation for the service v_i. If is positive, then you belong to those people that will join the service at the zero price to the N people. If your v_i is zero or negative, it means that you will never really join the service. So, all potential users join this means that the valuation of i would be v_i. So, if the price is zero, all N people will join. So, therefore the valuation for i will be the v_i. We assume that the valuations of different people you have several i's. So, the valuations of these people follow a uniform distribution that goes from 0 to the upper bar or as we will call it later v_bar. So, uniform distribution means that you do not have a mass of users that is around a specific valuation but all the valuations sound a uniformly distributed let's say they are in an order that starts from zero to v, and they're the same consumers to each point of valuation. Now, f_e is the fraction of N consumers who is expected to access the service at a given price p. Doesn't have to be zero given price p. So, f_e is an expected fraction that's why I have this e subscript. There it means expected. So, f_e is the expected fraction of N consumers who will join the service at the given price P. Am repeating and I am saying to them very slowly because this is kind of unusual model. I want you to understand every variable and the exact notation of the model very well because we would need those things in the next screens. So, for consumer i now, the willingness to pay is following, is the valuation that they have the product v_i multiplied by the fraction of consumers that they expect to join the service. So, v_i would be the willingness to pay if price was zero but if price is not zero, then you expect that f_e of the consumers will join. So, you have to multiply v_i by f_e in order to get their willingness to pay. This is the willingness to pay at a standard at a given price p. Notice that willingness to pay increases with expected use. If you expect the service to be hot, lots of people will join in the future you may want to jump in from early. So, your willingness to pay will increase with f_e. This is expected. This is a property of this model is not the result that we just derived, is a property of the model. What happens with our decision to join? Who is going to join and who is not going to join? Joining means that you consume a quantity of one. No joining means that you consume a quantity of zero. You will decide to go for zero not join the service in other words, if you're willing to pay f_e times v_i is smaller than the price that the service is offered. If you're willingness to pay is higher or equal to the price that the services offered you will join and you will just start your subscription to the service with one you do you're not going to have multiple subscriptions in the same service it doesn't really make a difference for you. So, those for whom the valuation, now I'm solving with respect to valuation to make a point. Those for whom the valuation is smaller than price over expected fraction will not acquire the service will not join or will not buy a subscription. And they fractioned those exact people the fraction of those people out of N would be v_i over v-bar. And this is given by the uniform distribution that we assumed before. So, v_i over v-bar people will not join the service. Those for whom their valuation is greater or equal than price over the expected fraction will join the service, will acquire the service at this prize that is offered p. Now, the fraction of those people would be one. Let's denote that by f without e. Now, this is not an expected fraction anymore. It's an actual fraction. Lets denote that by f. This would be what? One minus the people that will not join. What is a fraction of the people that they will know join? v_i over v-bar, we established that before. So, one minus v_i over v-bar would be this fraction. However, v_i now is the marginal consumer of consumers will start joining from the marginal consumer for whom v_i equals p over f_e till the consumer that has a maximum valuation v-bar. So, I can substitute disease v_i with p over f_e and this will make my life later much easier. So, the fraction now is equal to one minus those who will not join which is one over v-bar times p over f_e. Again, notice that the fraction of users increases with the expected fraction. That means that the more people who expect to join, them more likely for more people to actually join this service. Now, if I take these equation, this fraction of the people that they will buy and I solve it with respect to p, this will give me an interesting function p will be equal to v-bar times f_e times 1 minus f altogether and I can do something here. I can use these as a demand curve. But the problem that I have now is that I have an actual fraction of people that they will indeed join the service and I also have an expectation and expectation about how many people who will indeed join the service. At equilibrium, those two f_e and f should be expected to be equal. This is the rational expectations hypothesis meaning that if is different than f_e then one of the two to that job so they will become equal or these people are just not irrational. That is if the actual people that they join are less than they expected, this means that probably you don't want to do to join this product you're just you're just dropping out or you adapting your expectations to something lower. All right. So, at equilibrium, we assume that f is equal to f_e and this is an assumption that comes from rationality of consumers. Consumers at equilibrium will have their expectations verified. Therefore we can set f equal to a f_e now or f_e equal to f and we can use the above function as a demand function. P equals v-bar times f minus v-bar times f squared. What you have to be very careful. Now, look at these demand curve, this is demand for the service. This is a quadratic demand. This is kind of a strange demand. Because quadratic demands are not just decreasing they parabolic meaning that they have a shape that first goes up and then goes down or the opposite, and this is something that is a strange situation in economics. We are used in demands that they are decreasing. However, what we have here is a demand that doesn't show the quantity of products per se, shows the relation between the price and the fraction of people that they will join the service. According to the previous demand, there is no equilibrium price above v-bar over four. This is because this is the maximum of these quadratic function. Above this function, there's no equilibrium there is no function doesn't go above v-bar over four. So, if you look at these graph between price and the fraction, it will go something like that go up and then down in this case. So, the maximum is at v-bar over four. That is the network will not be sustainable for any price above v upper bar over four. You cannot charge a price higher than the 25 percent of the maximum valuation in this market. If you do these, the network will not equilibrate and you will see what the result of these is in a moment. Consider for example a network with v upper bar equal to 100 and f equal to 0.5. The maximum price for this network is 25. But what if price is 30 in this case? The average v_i for the top 50 percent of users is what? Is 50 which is the first top 50 percent user plus 100 which is that top 50 percent user, divide by two. So, you take the average of the two and then this will give you 75. So, the willingness to pay for the top 50 percent users is 37.5. This is because you have an f of 0.5 and you have to multiply 75 with 0.5 so you get 37.5. But price is 30. And now you see that the average willingness to pay is 37.5. This network is efficient but still it will not equilibrate. Wait a second. That's a paradox. Think about it and we will resolve this paradox in the next segment.