The previous segment finished with a paradox, like our favorite series on TV. It finishes in the point that there is a question, and everyone is interested to see what the answer is but the answer is for the next episode. Let's see what this answer to the paradox is. The average willingness to pay in our model is indeed more than 30. We calculate it to be 37.5. However, some particular consumers, they are not willing to pay 30 or above. The average is 37.5, but there are some that are below the average. Specifically, those consumers who have valuations from 50 to 60, they have willingness to pay lower than 30 when the actual fraction of people to join the network is 50 percent, is point five. For example, if someone has a valuation of 52, then their willingness to pay is 27 is not 30. It is below 30. Same for someone that their valuation is, let's say, almost 60. This means that their willingness to pay is almost 30, still below 30. So therefore, these people will not join the service or they will quit the service, if they have already joined, if the price is 30 is above the upper bar which in our case is 100 over 4 which give us 25. So, if we have a prize that is above 25, this means that a low part of the top 50 percent consumers will decide to leave or not join the service. So, these consumers will leave the market and then once they leave the market, this means that they will be 10 percent there from 50 to 60, as we already saw. And this will drop f to zero point four instead of point five, that was before. So, now it's point four. If it's point four, however, those with valuations from 60 to 75 will abandon the market also because now, the F, the willingness to pay is multiplied by a smaller f. So, also they're not going to make the price of 30. They're not. This price is no good for them because now the network has a smaller amount of users. So, they will also drop out, dropping f further to zero point 25. If all these guys jump out. So, now you have the 75 percent of the market to drop out. So, your f has become zero point 25, 25 percent only. This phenomenon will continue since f is not satisfying our expectations f_e. This will continue till the network fails completely. Be careful, this is not going to become a smaller network if you have their own price. The network will fail completely, will be obliterated. This is very important for you to know because you will see how some real networks behave in reality. So, what is equilibrium? Let's assume my cost for setting up a network, because we will need to come up with a profit function again, in order to see what is the equilibrium decision for price. So, we cover a cost to set up a network with a maximum capacity N. Let's assume this is capital C equals to F, a fixed cost, some money that you need in order to just start the network no matter how many users it has. And this will be times c, a unit cost, times f times N, f times N is the amount of users that will actually join the network. So, it's something like we had in our previous module. It stands for Q. The amount of quantity that you will sell. So, f times N is the amount of subscriptions. So, the profit in this case would be what? We said the price is v-bar f minus v-bar f squared. We have already calculated the demand like that. Then substituting the average cost C from that will give me the average profit. And if I multiply that by the quantity of subscriptions fN, this will be my average profit, minus the F which is the fixed cost that also they have to be subtracted here. Not that they will make a difference, because will maximize that and F will just drop out in the derivative. But if we want a correct profit function, f has to be there. So, the sort of condition for this profit function is the one that I'm showing here, is again a second degree equation. It will go like in the graph that we show you in the screen. So, it will go from zero to one and there is, in the vertical axis, is the price and in the horizontal axis is the fraction. Now, this implies that for every situation in the network, there will be up to three equilibria, up to three. I know that quadratic equations have up to two solutions but here we have a third one. And the first one is a trivial equilibrium. There it is in the black dot. The trivial equilibrium is when the network is obliterated, like we said before, it goes to zero. No one, people say, "this network is okay but it's too expensive. No one is going to join." So, no one finally joins this network. So, we have a trivial equilibrium that the fraction will be zero. No one will join this network. We have a low equilibrium. There you go, at f_L, we call that because this equation indeed will have, this first equation will indeed have two roots. So, we have a low equilibrium at f and then we have a higher equilibrium, f_H, which is in our graph also. And I want you to observe that both f_L and f_H are for the same price. So, for the same price, if your price is at the point that it can intersect with these parabolic curve there, then what you have is, you have two equilibria f_L and f_H. And this gives us an interesting property of networks. So, networks can equilibrate in two different points. But it's not exactly like that. There is also a very interesting property of this equilibria. What we call a property of stability. This will be important because in 1974, Jeffrey Rohlfs noticed an interesting property of this set of equilibria especially the f_L and the f_H equilibria. Said that, okay, these two equilibria, they are different in nature from each other. He observed that the f_L equilibrium was unstable. In general, in economics, all equilibria are stable. That is, if you deviate a little bit from your equiibrium, let's say your equilibrium is here and you deviate from the equilibrium solution a little bit, then the system itself will have a natural, will develop a natural tendency for you to get back to the equilibrium situation. For example, imagine a system in perfect competition of demand and supply and accidentally you set the wrong price. Automatically, the market itself will show you that this is the wrong price. You will either observer a sharp surplus or a shortage in this market. And this shortage or surplus will push you to adjust the price so that you will get back to the equilibrium. This is the stability property of equilibria that we have in economics. However, the f_L is not one of those equilibria, the stable ones. Is an unstable. If f equals to f_e equals to f_L, the network will equilibrate. So, if you happen to find yourself exactly at f_L, then yes, you will be in an equilibrium solution. This will be in equilibrium. Once you deviate from there though, there will not be a natural tendency to return to the spot. If you deviate above this equilibrium, that is, users expect that f_e will be higher than f_L, this would cause indeed F, the actual fraction of people to join, to be higher than f_e which in turn would make a f_e to adapt upwards. And this will keep going till you reach the other equilibrium f_H. You keep growing this network till you reach f_H. And there this network will find a stable equilibrium that will be always a tendency for the network to return to f_H contrary to f_L. So, if you deviate above from f_L, you will go around the curve and you will reach f_H and you will stay there. So, deviation above f_L is very good for the network. The network will grow. On the contrary, if you expect the opposite, f_e to be smaller than the f_L, this would cause f to be smaller than f_e and therefore the network will keep becoming smaller and smaller till f will become zero. So, if you deviate below f_L, this will be catastrophic for the network because it will make it be destroyed, obliterated completely. One can say that this unstable equilibrium, f_L, is a critical mass that the network requires in order to lift-off. Meaning that, there is this important property that lots of people know first hand in business, that you need to develop a critical mass in order for your service to go well. You need this critical first mass and once you reach that, then the network will go as if it was in the autopilot. And we will see next how different networks have behaved in order to achieve this critical mass.
