So now we'll consider two graphs to illustrate the differences between usage and flat-rate pricing. And specifically they will show the advantages of usage based pricing over flat-rate. And so, before we do that we have to define what's called the user surplus. Which is the difference between the utility that a customer gets and the cost of the provider needs to to satisfy the network. And clearly we want that to be as high as possible, right. because we want the customers to have high utility and we want the providers to incur low costs for the network. So the higher that is, the better. So under usage-based scheme let's see kind of what the user surplus is. and so, we've defined the utility, the customer, the cost of the provider, and then the surplus. So we'll consider our standard graph again, here. And just very simply we know already we, we really did this before and we'll write this as usage-based up top. That the utility is going to be A plus B in this graph and the cost is going to be just B, right? So it's the dollars per gigabyte times the number of gigabytes that are being consumed. And as for the surplus, it's just A plus B minus B, right, because this is the cost of the provider. Right because the provider, We're assuming that, again this is going to be, the cost, that the, provider's going to incur. And that is what we're looking at over here. And then if he's charging his [UNKNOWN] then the users will, you know, consume this many gigabytes. So, then the surplus is going to be A plus B minus B which is A, just as we've said before. Now let's consider under flat rate pricing what this graph looks like, so write up here flat rate. So now flat rate's interesting because even though the provider's incurring this cost, right, remember that we're doing flat rate pricing. So this isn't really what's being charged. This just at the beginning of the month this fixed amount. And everything after that is essentially free, right, so the price then is essentially dropping down to zero. Okay. So, really everyone's going to kind of consume all the way up to the top of the demand curve. Right? Because we're not here anymore. We're all the way back down at the origin. Because the price is zero for the entire month. So everybody will consume as much as possible until they can't have any more added, gain in utility. And so, If we call, we'll just draw these arrows out here. And we'll call this region in here C. Let's call this region here D. So first thing we have to do is find out what the user's utility is. So remember just like last time, the utility is just everything that's to the left of the demand curve such as this big triangle right here, right? So nothing outside the demand curve, always inside, just this area. So we have this region plus this region in here. So we have A plus B plus C, that's being the utility. Now, what's the cost incurred? Well, remember, it's how many many dollars per gigabyte times the amount that everyone's consuming. But [INAUDIBLE] [INAUDIBLE] now consuming all the way up here, right? So, the the amount that's really incurred cost wise is this, this whole box, right, right in here. So, [UNKNOWN] like this over here. And we have, cost being B plus C plus D. So now we find the surplus. We take the utility and subtract out the cost. We have A plus B plus C minus B plus C plus D. Which is going to be A minus D. 'Kay. So now, which, surplus is higher? Well, under usage-based we have a surplus of A. And under flat-rate we have a surplus of A minus D. 'Kay so, this D right here is subtracting from the surplus. And again as we said surplus is, always, we always want it to be as high as possible. Right because it's, it means that either a customer has a higher utility or the provider has lower cost. So, this area D in here, we just call Waste. It's not giving anything for us, it's just an added cost that's not counteracted by any gain in utility at all. Right, because this is just, one added cost, and utility's already I'm done here. So usage-based pricing has a higher surplus and the flat-rate pricing has a lower surplus, right? So it's just creating a waste by doing that because by using usage-based pricing, pricing, we could have a higher surplus. So now, the second graph we're going to look at we have to define different classes of users. And this is similar to something that we talked about before. Is that on our demand curve, we may have light, average and heavy users. Right, we looked at an example before with three people where one person was consuming two gigabytes per month. One was consuming six gigabytes per month, one was consuming ten. So these demos might be light average and heavy users respectively. And so different classes of users are going to have basically different demand curves. Right. so light users are going to have one where basically if it's free, they only have a certain amount of data that they would ever need anyway. And, so then they connect that and basically maybe there's one price where they all intersect at. doesn't. Again, doesn't have to be the case that they're all going to converge on the same price. But we just make that assumption just make things easier here. then there's the, there's the average user, which is a little higher, and the heavy user which is higher yet. Right, so for a given price, the light user is going to have less, least amount of demand. The average user is going to have middle, and the heavy user is going to have the most demand. So, for flat rate pricing, right, the ISP has had, needs to set a single price that it's going to charge, if you recall that. because everyone has to pay the same amount and so we can really base that on the average quantity consumed. That's really what we should do because that's the only option that we really have. We have to come up with one single price. So the best thing to do is just to basically add up everybody's contribution, write whatever we expect everyone's going to take. Average that and then base our cost off of that. And how many gigabytes per that is times the the amount that we incur per gigabyte. We give how much we would charge each person each month. So again, we have to set certain dollar per person that add up to the total cost. So, now suppose the ISP decides okay, well I'm going to charge this on the average user. And I know that I'm incurring this price, this is the dollars per gigabyte that I'm incurring. Right. So, this is the average user, right? This is what people consume on average. So everyone's going to get charged this, basically this, this rectangle right here. This is what I'm going to charge every person. Same amount, same dollars per gigabyte times the amount of gigabytes that the average user's consuming, whatever that may be. That's how much everyone's getting charged. So now let's for a second look at the light user now, okay. So this is the, we can call this R right here, which is basically the revenue of the service provider. Again, the light user is going to consume as much as really the entire demand curve. Right, because the, the light user's going to want to just take up as much as he needs. So he's going to consume up to here. And so the light user's utility is going to be this entire rectangle, in here, all the way down to the, end of this price. And so this is going to be, we'll call that U1 for the light user, utility one. But what you notice is that, this rectangle right here. This R kind of looks a lot bigger than that utility, and it should be because you're basing this price on an average user. Right? And, the light user's utility is not going to be as high as the revenue. So, the the cost of the light user is going to be really greater than his utility. So he's going to end up having a negative net utility. Or a, a negative surplus basically. So basically what we are saying is right here the utility is less than the revenue, and that's the utility of customer and the revenue of the provider. Now let's look at the heavy user and still it's the same revenue. but now the heavy user is going to have a different characteristic. Because the heavy user is going to consume all the way up to here, because this is what the heavy user needs. Because this is what the heavy user has use for. So now, the utility for the heavy user is going to be this entire rectangle. Sorry not rectangle, this entire triangle. And so we can call this red box right here, I'm just going to draw this outside, U2. Now, clearly that utility is much greater than this R region right here [INAUDIBLE] this rectangle. So we say for user two the utility is greater than the [INAUDIBLE] . So, on the one hand we have the utility being less than the revenue. On the other hand we have the utility being greater than the revenue. So then our conclusion is that light users on this side right here are subsidizing the lifestyles of the heavy user. And that's a big no-no. Remember, that was one of the things that we said we never wanted to happen. With a, with a pricing scheme, because that's not fair, right. So this does not satisfy the fairness criteria that we talked about before. So that's another reason why [INAUDIBLE] usage based pricing it's much better because we, we don't have the same issue. Because everybody's charged based upon their own consumption amount. So we would be only limited and we'd be fixed at whatever points were on the demand curve.
