Marketing Analytics Professor Swagato ChatterjeeVinod Gupta School of Management Indian Institute of Technology, KharagpurLecture 21 Demand Forecasting and Pricing (contd.)Hello everybody, welcome to Marketing Analytics course. This is Doctor Swagato Chatterjeefrom VGSOM, IIT Kharagpur, who is taking the course for you, we are in week four. And weare discussing demand forecasting and pricing. So, in this particular video I will talk abouta special type of forecasting technique where the product is generally durable product whichyou buy only once in a life time or probably once in a particular model at least if notin the life time. And then you so the maximum possible market share is fixed in this kindof a situation what happens? So, we call it a technology adoption modelwhere bass diffusion comes into the picture, and where I say that there are for any productthere is a product life cycle.And product life cycle the product life cycle if you have seen, there are see the productlife cycle PLC curve those in marketing 101, looks like this, the PLC curve looks likethis. And if this is the thing this is the sales, at time t and this is the time t. Firstthe sales is low and then it picks up and then it reaches maturity and then decline.So, this is the part who are basically early adaptors, sorry so these are innovators. Thenthere will be early adopters, this part is probably so these are laggards. So, innovators,early adopters, late adopters, and laggards probably something like that. So, this issomething that I try to look at. And this is same for, heavily same I would not sayexactly same, for many technology products. Now, often time we are interested to actuallyestimate this particular curve. How this particular curve will look like. And if you also seethat, if I try this as sales at time t, if I try to plot the cumulative sales. The cumulativesales looks like this.So, it goes up then gets saturated at a certain point of time. So, if it goes up, and getsaturated at a certain point of time, it might look like a S curve. So, the this is how itlooks like basically it goes up and get saturated at a certain part of time. So, this is thecumulative sales, time t and that is how it looks like. So, this is something that weare trying to model. And then there was professor Bass in, I think1960s when he has given this particular model. And he said that. So, there are early adopters,what happens in the market, there were some. So, it is also related to spreading of rumorsspreading of virus infections, actually happening of a viral message or probably creating arevolution, all of this things are related to this kind of a model world.There has to be a significant mass accumulated at the starting point. Only when that significantmass gets accumulated it will actually reach the infection point and then will go up. Ifit does not get that significant mind of the starting point, then it will not actuallyreach the highest levels. So, that is why maximum amount of technology products failbecause they fail early, and the early failure occurs because the significant mass is notgetting generated. Significant amount of demand is not gettinggenerated at the starting point of their life. So, that is something which is common andyou will see that epidemic is not very common, viral epidemic is not very common. It happensbut it if if you can actually take care of in the starting point in a in a first 100,200, 1000 patience if you can, if you can limit it within that limit then you are safe.You can survive, and if it goes out of the hand if the significant mass is getting generatedin terms of number of patients at the starting point then it will reach epidemic point ofview.So, something like that is this particular model where Bass told that, that let us saythe accumulation, the extra sales f t if the total sales possible is 1. The 100 percent.Then the percentage sales happening at time period t, will actually depend on two things.There is some innovators, p innovators and then there are q adopters.So, p is called coefficient of innovation, and q is called coefficient of adoptions.So, some guys who are innovators. So, there will be always some people who you will actuallywho will actually be there in a mass the huge mass who will be coming into coming in touchwith the particular technology and buy. Let us say p percentage( ) of the overallpopulation, are innovators. What is the innovators mean, innovators means whenever they comein contact with the particular technology, they purchases the technology or wheneverthey come in contact with a particular virus, they gets affected by the virus.Whenever they comes contact with a particular let us say viral message, they share the viralmessage. So, these are the guys who are innovators. So, at time period t, how many persons areleft at time period t, f t person’s, f t the small f t persons are actually who aregoing to adopt this particular product at time period t, fair enough.Then if the t is very small, then capital F t is basically summation 0 to t or summationor integration f t dt or it is the, in other words in normal language it is the total cumulativenumber of people, cumulative sales that up to time t. (6.25)So, up to time t total sales that has happened, or total number of people who has got adopted,is actually told by capital F t. Now, if capital F t number of people, has already bought thisparticular product, up to time period t. Then how many people are left assuming the totalis 1, 1 minus F t people are left this many people are left, and out of these many peoplep percentage, p proportion of people will actually come in contact and purchase.Each every time period. So, p into 1 minus F t. These many people purchases at this timeperiod, adopts on their own. Now, the rest of the people does not adopt on their own.They adopt based on whether other people have adopted or not. So, they actually are friendsof the already people who have adopted. So, what happens is this capital F t peoplewho have already adopted, they go and talk with others. The word of mouth, and becauseof this word of mouth there are some people gets affected. How much, which people arethere, q into F t this proportion number of people means the more the adoption happens,the more the adoption happens, the more is the chances of word of mouth. No, so if morenumber of people have actually experienced that product, then the word of mouth willhappen more. So, this is the overall word of mouth this part is the overall word ofmouth. Now, these proportion of people come fromwhere? The same the rest of the guys they will comes from the rest of the people no.They will not come from somebody else. So, this is my total rest of the people, thisis my total rest of the people. Now, out of this rest of the people, p proportion comeson their own and q into F t proportion comes by word of mouth.They do not come on their own, they come by word of mouth. So, as F t goes up, as alreadyaccumulated cells goes up, as adoption number of people who have already adopted. Let ussay WhatsApp, WhatsApp how many people will actually adopt, so let us say today is 2020and there will be how many people will adopt in let us say February 2020. How many peoplewill adopt. So, let us say if there are around I do notknow, 10 million or hundred million customers possible who have internet access, they arepossible hundred million customers. And out of them have 50 million has already adopted,and the rest is 50 million. So, as this rest of 50 million is higher, the chances of adoptionis higher as this shrinks this rest of 50 millions shrinks.The number of people who will adopt at a time period will also shrink. If instead of 50million, if it was 5 million the rest of the people. If by chance instead of 50 millionif it was 5 million then the number of people who will adopt in February 2020 will alsobe smaller in the second case. So, how many people will adopt in their... in the currenttime period, will depend on how many people is left aside till now. Who have not adoptedtill now. So, that is this part this part is that.And how many people will adopt in the current time period, will depend on some amount whichis actually the innovators, who depend on nobody which is the p and some amount whichdepends on the already adopted people. If the already adopted people is 25 million,what says the already adopted people is 5 million then the 25 million guy will havea higher word of mouth than the 5 million guy.So, that part is actually taking care of this point. What we are talking about the imitationpart. So, this is the p is called coefficient of innovation, coefficient of innovation andthis q is called coefficient of imitation. So, and these two has huge meaning. So, bysaying that what I am trying to say is that the currently the I would say just 1 minute.So, currently the what is what is the function then? The function is f t by 1 minus capitalF t. This is actually the function of Bass diffusion model is p plus q of capital F t.If you actually try to solve this is a differential equation and if you try to solve this. Andwhat is what is sales at time period t, if total number of market share is m, the totalmarket number of revenue that you create, m into small f t is the total sales that canhappen at time period t. (12.10) And the functional formula for this is, ifyou solve that you will find, m p plus q square by p. This is the part into e to the powerminus p plus qt. And this function I do not remember I am just writing it from a notethat I have. So, it is difficult to remember, 1 plus you can just google it up, e minusp plus q t square of that. So, you can just google it up.I am not saying that you have to, so but I have actually copied, I have kept a note Ihave written it there. This is the function that is there. So, this p and q endures thecompulsion formula. It has not no purpose. But there are some things why it helps. So,this is the sales function. So if you we will use this particular sales function later.So, you we have to remember this, this is the sales function. We do not need to rememberthis, but this we have note it down. So, that we can use it later that is number 1. Whatextra? So, you will see that there is a peak point, when the sales is peak. That peak salespoint is called the function is lnq log of q minus lnp log of p by p plus q. And at thistime period, the number of products that are sold, so f t is basically 1 by 4q and p plusq square. So, you will see that at the highest time period, when the if that the more isthe innovation and imitation the higher you will reach. And more is the imitation in comparisonto innovation, it will come down. So, that is something that we will get.And if I find to find out the cumulative sales, if I try to find out the cumulative salesthat was a individual sales, that was the individual sales if I try to find out thecumulative sales the formula is, so F t the formula is basically and S cumulative salesCS t is nothing but m into F t capital F t. And capital F t formula is basically 1 minuse to the power p plus qt divided by 1 plus q by p, e to the power minus p plus qt. So,something like that is the formula. And will use this formulas to estimate the effect ofthe effect of innovation and imitation in case of our technology product. (14.56)So, how do I do that? So, let us say I have a data set. Simple data set, this is the dataset that various times and we will see that 100 is the maximum possible achievable sales.And I have 12 percent of the maximum possible has been achieved in 2001, 17 percent in 2002,this cumulative 20 percent in 3, 24 percent in 4 and etcetera, etcetera. That is the valuethat I got. Now, let us say if I say that ok the cumulativevalue p is 1 and q is also 1. Then what is the cumulative value. The cumulated, so thisis my predicted. The predicted cumulative value is the formula is written here 1 minus,so is equal to just check what I am writing. 1 minus exponential of what?Exponential of something, what will I write it in this something, in the something I willwrite something into the t, the t is this. And what is this? This is p plus q. So, thisplus this and H2 and H3 I have to, I have to put a F4. So, carefully see what did Iwrite that the equations, I am just writing the equation of the this equation basicallyI am writing this equation properly. So, you have to find out that whether I amwriting correctly. And then divide it by 1 plus what is here? And exponential of somethinghere. What is here I will write q by p, so q by p means this divided by H2. H2 is andI will put a F4 sign here. F4 here and F4 here and exponential whatever I wrote insidethis one will come up to here again. So, the same p plus q into t, I will put it here andthen put a negative sign in front of it.So, if I just run this, I think I have wrote this one would be negative here. I think Ihave not run it correctly. So, this is something is the percentage. So this should be if Iam not wrong, this should be minus also just check I have forgot. So, this is somethingthat is that is the formula that we get. And if I try to, if now I try to, so this something and then this is this is in proportionalterm, this into 100 is the maximum possible thing. So, this is what I am getting say 76to 100. I will reduce it a little bit, let us say the this is 0.5 and this is also 0.5. Now,if I just try plot it, if you will see that if I try to plot this particular curve, thiscurve comes up to be as almost a S curve that I just told you.Let us say if I if I further reduce it let us say 0.25 and 0.5. Then and then try toplot it. It comes up to be a S curve. So, simple good looking S curve. But this valuesand this values are different because the actual pq value and the one that I startedwith will be different. So, how much is the error. The error is basically nothing but the 12 minus this is sqer. So,similar to previous one I am getting my error square, so I can say sqer error is sqer. Andthen what is rmse? rmse is nothing but square root of average of this particular column.And I have to somehow minimize this by changing what, by changing p and q. That is form thatI do not know. And p and q, now remember p and q has to be positive cannot be negativeI cannot have a negative proportion.So, I will solve this. And it gives me, ok so p is zero, p cannot be 0 as well. So thatis something that I have to also check, so sorry I forgot that. So I have to check thatit cannot be 0. So solve and add that. This guys has to be greater than equal to 0. Add,so now if I have this one as my constraint, now if I try to solve. It is still comingup to be 0 just 1 minute. I have done something wrong.So it might be the case that, p cannot be 0 because if p is 0 then this one comes tobe infinite. So what did I do, I wrote greater than and equal to. That is something thatI did wrong.So, this is not greater than equal to change it and make it. So, greater than equal tosome very small value .001 now solve it. So, I have to change 0.25, 0.5 and in the solverI change it to 0.001, solve it. And now I am getting some value, fair enough.So this is the predicted, this is the actual and my rmse is much lower. And these are thethese are the cases that I am getting, so initially I am not getting results but asI go ahead I getting good pretty much pretty close 89, 90, 87, 81 and so on. The primary2, 3 values are not very good. So, this is the table. So, if I just try to plot thistwo curves, see this are very close curves. Actually what I am getting. And that is how.Now, if I know that, if I have to know that at what point I will reach 99 percentage.When I will over more or less get everything. I will just simply use this one end probablydragging it up value upto 65 time period. I do not know. So 12, 13 and then I will dragit up upto 65 time period. And if I just double click on this, it will give me okay .So, you are reaching 99 at 21st time period. So, another another probably 8 time periodsand it will reach your maximum possible thing. Now, why this is important, why will I useit. If by chance I know that my pq is like this. At this point of time at 32, then Iwill know that at what time I will reach the maturity, and if I know that at what timeperiod I will reach the maturity, in how many days I will reach the maturity.I can pre plan, if you remember like iPhone 1 and iPhone 1 iPhone 2, so they keep on innovating.So, that they try to create a curve, sales curve which looks like this. So their PLCcurve is like this, it goes out which is the maturity and then another curve comes up.From here itself another curve comes up and goes. So, they do not let it come down. Theycome up with a new newer products and etc. So, this is something that is important. IfI when I am standing here, if I know that ok I am reaching maturity or not, I at allif I am reaching maturity or not then this particular thing is ok. Now, you will alsosee that this p if it by chance if the p value is very low. By chance if the p value is letus say, I am just giving an example. By chance this P value is 0.0001, then how does the curve looks like. Just try to plotthis curve, when the p value is very low. So, the p value is very low it will reach,it initially it will be absolutely 0, nobody will buy it initially. So, if nobody buysit and then even only after 30 months or 30 years somebody buys it, it is the yearly data.So, 30 years, somebody buys it then actually it fails. It will fail 2, 3… So, that issomething that is very important to understand that your p has to be higher.So, p has to be at least 0.02 to get something to get to reach upto a tenth level or somethinglike that. So, that percentage should be there in your p, which is the coefficient of innovation.And then coefficient of adoption can be something that it is also very important. But this issomething that is very important. If it is lower, then your product does notreach the maturity level, it will actually die after second time period, third time periodand max 10 time period it will die. So, that is something that will keep into account.So, that is all for sales forecasting, we called it Bass diffusion model. And laterpoint of time people have actually included much other things.So, for example now I have some error, I can also find out how these error which is notrelated to diffusion which is related to something else. So, I can find out the predictive model,but this error is predicted by the price and sales and advertisement expenditure and etc.So, how much price I have given on that particular month if or a particular year, if the priceon the particular year was higher, the demand will coming down.So, again all that rest of the part which is non-time series part can be explained byjust putting a regression equation. With this error here it is sqer error. We have to squareroot it up. So, with the error component and all other X variable that you have you tryto create a regression equation. So that is all we have about forecasting.We will go on and meet you in the next week and we will do, how to use this forecasting.So, how to use this demand function. Its various pricing problems. Will talk about that inweek 5. Thank you very much for being with me.
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