In this simple example, we'll be asking the following question. Okay, suppose you have users, all uniform, they all look the same with a certain utility function which is a weighted log utility function for certain given capacity bits per second measured, Okay, and denoted by x. I look at the log of x weighted by sigma. Okay. So, sigma parameterized is, basically, now they log utility functions. And the user, or the carriers is charging a price based on the consumption amount. This is a typical chart we've seen. There's a flat rate component g dollars,$ it doesn't matter how much you consume, plus a usage base h dollar per amount of consumption, okay. So now, if I look at the graph, it looks like this. Okay? This is g dollars, Right? And this part is h dollars per unit of consumption, say, per gigabyte. So, I'm going to look at these two functions. On the consumer side, it's the utility function sigma times log of x. And on the carrier sides, p of x, g + hx. And I'll ignore this horizontal shift in the demand because by paying g, you actually get a fixed amount of up to a certain fixed amount of data. So, I'm going to ignore this to simplify the notation. It does not change the gist of either presentation or the conclusions. So, question number one. Given the baseline usage price, is given g dollars and h dollar per gigabyte what is the demand? And second, given such demand, how to optimize the price. Now you see this feedback loop, right?. Consumers have utility functions. Carriers decide the price, Right? You give a different price to consumer, they give you a different demand. And that will shift to your price again. Now, this kind of loop we'll see a few times. We will see it in TCP, in a network context. We will see next lecture in SDP, Smart Data Pricing in a more elaborate way and the general notion of you design something and then the agents will react, was certainly even h, broader in its application, since we have seen in auction and so on. Alright. So, let's write down the equations. First of all on the consumer side, you say that, you know what I'm going to look at my net utility, Which is sigma x log of x - minus the price of ghx. Plus hx, okay? This is my net utility. I will set its derivative against x to be zero. What is the derivative? Was just sigma over x minus h equal to zero, That means, My x star is just sigma over h. Alright, that make sense? Bigger h, steeper slope, I consume less. How much less? Well, governed by sigma phi. This is the answer, by setting the derivative of that utility to be zero. Now, what about the carriers?, Price settings, that is the g and h settings, the two parameters for you to play with. The base line and the slope. Well, in this case, you are not setting the derivative to zero, instead, instead we say that this is a poor monopoly carrier. It has a price setting power. Meaning that it can squeeze your net utility itself, now the derivative itself, to zero. This assumes that this is a very powerful carrier, it got the price setting, Power, Meaning that you can set the g of h such that your utility is zero. This is clearly an extreme case of modeling monopoly power. It's not realistic in our mobile data market. But for a simple illustration, it suffices. Okay. So, this leaves no utility to the consumers. It can't be negative, cuz if it's negative, the users won't participate. Okay? And to properly use this monopoly price setting power, We'll set it to be zero. So, we can solve this second equation The second equation that says, sigma log of x equals g plus hx, So now, I got two equations. One is by the demand modeling, the others by the monopoly price setting modeling. We can stick this x into this equation and now, we have the following result. Sigma times log of sigma over h equals g plus h sigma over h equals g plus sigma. Okay. That means, We can set the flat rate g to be sigma log of sigma over h minus one. Now, this assumes that you know that, as a carrier, you know the consumers are taking the logatative function with a known sigma value. Which again, is unrealistic to assume. But, let's say, that's the case now. So, now we have the answer. Okay, so you give me a sigma. Okay. And then, I can pick out a h slope, Then that will give me a g. For example, let's say, sigma is set to be a 100 unit. Alright Then let's look at two cases. One is one h is two dollar a gigabyte. That's the slope. Then, plug in the equation, sigma and h, you get the g, which is the flat rate, $70, Alright? So, $70 a month is the baseline, then extra gigabyte, you pay $two. This implies that the revenue for ISP, gHX=$170 plus hx, a $170 a month, Okay? And the flat rate component is 41% of the total revenue. Case A. Case B. Suppose I increase h now to $five per gigabyte, okay? Then, apply it in the equations, sigma 100, h is five, G Is 30. That means an alternative is to set the baseline to be $30, and every gigabyte get $5.. This actually is quite close, reasonably close to the actual pricing points used by AT&T and Verizon Wireless. This implies, on average, the ISP monopoly power ISP under our assumption, guess a $130 as the revenue and the flat rate component is not only 23% of the total revenue. Now, you can look at many more cases. But you get the trend, the trend says, that actually, you should make h real small, okay, very shallow slope. Okay. So that you can afford to make a g, the constant part, very high. This will max out your revenue, and the flat rate part will be dominant, okay? Now, say, that's actually somewhat counterintuitive conclusion, isn't it? Indeed it is, that's because we have made up three unrealistic assumptions. Number one is that we assume there is only one bottleneck link. In lecture fourteen, we'll take this away in TCP congestion control at a much faster time scale than monthly bills. Number two is there's no capacity cost of constraints of any kind. And that was the fundamental reason why we got that somewhat counterintuitive conclusion from this simple numeric illustration. Once we put capacity either as a constraint, you can exceed a certain capacity, or as a cost. Then you see that we cannot make h arbitrarily small and g arbitrarily large, because that would induce a tremendous amount of traffic that will violate the constraints or increase the costs to the point of not making it worthwhile. The last one is that ISP was soon to be a monopoly. And therefore, price setting power to squeeze your utility as a consumer to zero. This is unrealistic. So, later in the course, we'll, perhaps we'll keep this assumption, but certainly remove these two assumptions. In fact, in the advanced material, we'll remove this assumption. In lecture fourteen, we'll remove this one. Now, having gone through this perhaps unrealistic small numerical illustration, Let's highlight that we did touch upon a very important conceptual and methodological tool called utility maximization. It generalizes the payoff function ideas we mentioned before. So far, we haven't seen network cuz there's no network constraints or coupling get, but that's coming up in lectures ahead. This utility maximization models consumer behavior. We've seen utility function, demand function, and induced elasticity of demand. We've also seen a very, very powerful notion that the pricing, whether it's on a single link of a network, it can coordinate demand versus supply. Whether that's done on a monthly bill time scale, or a 50 millisecond round trip scale of the internet. Whatever timescale, if you can design a proper pricing, then it could help just demand supply. Just like we've seen proper pricing in auction, like second price, can't correctly internalize certain type of negative externality. But improper pricing can also create a tragedy of comments. Finally, for the specific topic of why usage-based pricing, why, you know the answer. It's because of jobs inequality of capacity. Where do we go now from $ten a gigabyte? Is this the end of story or more? We'll look at smart data pricing in the next lecture.