 Okay, good morning everyone. I apologize for the delay. I had a terrible trip. I Missed my tram and then the train was so full That I had to be like this to the wall. So yeah I Would like to spend a couple minutes on the last two slides from the last lecture Have really bad sore throat. So my voice may also be kind of weak, but if you remember we ended up last lecture With the conclusion Yes with the conclusion for our systems dynamics model for this company fast-growing electronics, yes, and now the last two slides are actually completely different The last two slides are about linking the BCG matrix If you remember the BCG matrix cash cow stars dogs and question marks to the product life cycle Okay, we have an introduced formula the product life cycle, but it's actually very simple just like with any life cycle Every product has phases and in general we can recognize four phases the I think they're mentioned here No, they're mentioned here. Okay, let's look at this graph. We'll have introduction of the product product growth Maturity So this is where the the market gets the gets mature and then saturation now personally I wouldn't call this saturation. I Would call it decline or something like that because saturation implies that Things reach kind of an assing thought right in in a technical sense, but for some reason people decided to stick with saturation However, they mean decline the maturity and saturation for me as synonymous But that's how they use it So what's shown here is for whatever product we have the profit And we have the sales and of course they more or less have the same shape Given some assumptions on on the on the costs to produce these products or the development costs. They're more or less Similar across all phases so this distance here is more or less the same But the shape is important. Please remember the shape. It's a bell shaped Curve right so we have increase In whatever product and then a saturation or yeah maturity and then saturation So we'd like to link this product life cycle. It's a life cycle. I mean the life cycle idea is very simple We'll see more live examples of life cycles today but would like to link this to the BCG matrix and the BCG matrix was You know these four different categories for products and it's basically this one Right, so if we have a hot selling product, let's let's think about what a hot-silic product is How does it start it starts? with an immature market new market and The product is is selling very fast. So there's a high growth in Still undeveloped market. So it's kind of a question mark. We started as a question mark We have this this product which may sell a lot in this much in this new market So we start the product life cycle is here, right? So the green curve is the product life cycle. So that point Would be exactly this introduction introductory phase this one the orange curve is the learning curve learning curve meaning What do you know about this product? about the technology About the the demand about the market. What do you know about this product in general? I obviously don't know very I mean you still need to know a lot. There is a lot for you to learn All right. Now if it's a hot-selling product It will move in this direction It will become a star and remember a star is a very fast growing product in a new market All right. So it's a star. We're right here. It grows very fast in the green curve We're in the growth phase. You know a little bit What you need to know is a little bit less than before And even if you look in this there's additional information here Which is the income that you get and the expenses the investments you make in this product? Obviously in the beginning you need to make a lot of investments for this question mark to become a star once it's become a star You kind of start getting a lot of money for it with relatively fewer expenses It's a hot-selling product. So the next phase. This is the growth phase. It will mature eventually What will become of it is a cash cow, right? So we're in the most. Yeah maturity phase it's a cash cow meaning that we get even more money out of it and We don't have to invest as much compared to the previous cases and when we know obviously a lot more and Then eventually even if it's a hot-selling product It will become obsolete and Goes back to dogs If it's a hot-selling product if it's not a hot-selling product So we start as a question mark. We don't know what it is. It turns out not to be a hot-selling product. Well It's a dog immediately. So the jump is here That's the most unfortunate case. All right. So this was the last few slides From last lecture let's start with a new one this is six Yes, so something is really slow with my computer today The new lecture is going to be primarily only about Modeling of the sales evolution So what happened here doesn't open. I need a new one But before that let me remind me remind you what we did last time last time we saw the Let's say the first More interesting model. It was about the inventory in the workforce dynamic and What we tried to model there was how the relatively small fluctuation in sales Transfers to relatively large fluctuations in production or in upstream and we identified two main causes for this Which together only Generate this effect separately. They don't this was the time delays basically in adjusting your workforce and adjusting your inventory and we saw during the self-study that If you Reduce the workforce adjustment to the minimum So basically you adjust as fast as possible your workforce or your production But you leave yourself some time to adjust the inventory Or the coverage the inventory coverage then you reduce the oscillations Right so that this was an important distinction if you You still have to adjust your production as fast as possible But the target production that you adjust to Take some time to determine so give yourself some time to determine that don't just take the The demand today and try to match it wait a little bit. All right Then we looked at the case study. It was very successful result They saved a lot of money the numbers are given in the last lecture and today we start with a Related topic it's about Modeling the evolution of sales Or demand sales and demand are kind of synonymous sales is more the Business term and demand is more of a economics term, but they're more or less the same So with first we'd like to see how sales look like The distribution of sales how they look like and then we'd like to model it That's the the topic for today's lecture Probably you've all heard about the S curve Have you all heard of who's heard of the S curve Okay, well less than a half so that's good. I mean we have a lot of stuff about the S curve So I don't want to be redundant Okay, so let me remind you from the last two slides the the product life cycle So we have four phases and we can map these four phases to the busy Geometrics An important thing the sales at any given point of time or the demand It's a bell-shaped curve, right? So if you take Side of the instantaneous sales they're bell-shaped, but if you take the aggregate sales So the total say sales over time There they look like s like an S curve It's an empirical fact. There's nothing we can do about it We'd like to model it All right, so this is this is what we're going to do modeling the time evolution of sales Before we start let me introduce a few basic notions. So sales I mentioned already Are equivalent to demand so when we talk about sales and demand they're the same thing One is the economics term the other one is the business term But demand is not enough. We need supply obviously or supply is equal to production They're both the same thing and we need those two things In order to have any sales in the first place and the demand and supplies you all know are matched by the market the efficient market hypothesis So at any given point of time the demand is matched by by the supply an Important factor for this matching to occur is marketing Right, so the demand side needs to be aware of what's available on the supply side And this is where marketing comes into play. You know marketing is not just fancy What happened hmm, I think my My battery is kind of down Although I didn't say anything. I don't know what happened anyway So this is the role of marketing and marketing is not just Fancy advertisements and stuff like this think about it. It's information Information about what's available on the supply side All right The life cycle that I just introduced and that I was supposed to introduce last lecture It can be applied to basically everything so we can have a life cycle in product development We can have a life cycle in sales Development life cycle in supply development, right? You can always imagine this kind of increase For example, let's take demand demand increases for a whole product It reaches some kind of a mature state and then it starts to decline This is the life cycle for demand for instance. You can apply it also for production. You ramp up your production This is the growth phase Production reaches table levels and then you run down when your product becomes obsolete So the demand the the life cycle model can be applied both to supply and to demand. It's just a I didn't show you this but that's it Okay, I know what I'm doing. I'm pressing buttons here. So let's see Let's look at this graph This is that in kind of a it's not empirical yet. It's an idealized version of how things look like but look at the red curve Disregard what the legend says the red curve could be the demand In fact, it is the demand. So demand for whatever product increases Reaches a mature state and then decreases. It's a very nice bell-shaped curve The green curve is the total sales the aggregate sales. So for instance a time hundred This point would show you all the sales all the units that were sold up to time 100 so the cumulative number of sales and they look like this. So this is the The aggregate demand Or the the cumulative demand it looks like this and the blue curve don't care about it but we'd like to model this this situation and I'm starting to introduce the model now These guys it's the famous bus innovation model. Probably you've seen it Somewhere else as well. Have you who's seen it the bus innovation model? Oh Nice Then you would you would be very interested in what I have to say all right, so the bus innovation model or Our let's say before we start with the bus innovation model our hypothesis would be obviously demand Or sales they depend on customers. So somebody needs to buy it And those that don't know yet about the product we call them adopters here The adopters are people who've already bought your product And we have the people who still haven't bought it. That's all that we have in our simple model adopters and potential adopters Which is obviously simplification, but let's not concern ourselves with that and You can immediately see the The feedback here when we increase the number of adopters The number of potential adopters decreases So this is it. This is how we model the demand life cycle Demand life cycle sales life cycle We assume It's a very popular Approach to to resort to disease spreading so people think that The adoption of a product resembles a lot how disease is spread in a population Somebody becomes infected and the infected person Interacts with a non-infected person then the non-infected person becomes infected the real disease spreading models, so they're called also the susceptible infected models S I and then there should be actually an R For recovery so Susceptible infected recovery models assume that you have susceptible people potential adopters in our case infected people actual adopters and we have those who've already recovered They will not get sick anymore. They will not Infect anyone else But the bus innovation model disregards those so for the bus innovation model. We only have adopters or Infected people and non-adopters Non-infected people and obviously the adopters infect in a sense the the potential adopters We don't have people who who have bought the product and and then they want they don't want to buy it anymore Meaning they've recovered. We don't have this So this is how it goes. We have a potential adopter and a potential adopter becomes an adopter somehow this somehow is given by that rate here the Transition rate K and it resembles a chemical reaction. So with a certain probability a potential doctor becomes an adopter We can model this a little bit more We can say well the transition rate Depends on the following things It depends on the contact rate see contact rate means How many infected people does well? Let's talk about adopters. How many adopters? Does a potential adopter meet at any given point of time per time step? So it may be for example 10 if you're very well connected. It may be zero if you're not connected at all But it's how many people Infected people you meet it in a given time step. I is the probability that meeting an infected person Would make you infected in other words meeting an adopter you may meet 10 adopters but if you're let's say you have a very Simple way of life You may not adopt their new gadgets Even though you may meet hundreds of them. So this is the probability that you will adopt Whatever these guys are offering you and that simply and is the total population and a is the number of adopters So this is the fraction of adopters So you can see the contact rate the transition probability is How many adopters you have in your population as a fraction? How many of those you meet this is the contact rate times the probability that you actually adopt This product or you become influenced in a way. So this is the Resembles the social herding effect Right. So you're influenced by others you do what they do. This is this is also called social herding Okay, so we have the number of adopters plus the number of potential adopters gives you the number of the total population and That's all we have in our population All right, and let's look at the dynamics How do the potential adopters change per time step well, it's very simple in other words how many How many adopters Come out of the potential adopters population. Well, it's very simple. It's basically this equation here So you have the number of potential adopters These potential adopters, maybe I don't know 10 they meet This fraction of the population which is infected they meet out of this population they only meet given number of people see and Multiply by the probability for infection. They they actually adopt So this is the differential equation and obviously it's always negative which means That the population of potential adopters always decreases So you can already imagine what the outcome of the model is The number of potential adopters will go to zero. There would be total adoption of this product Whether that's realistic is a different thing is a different case and obviously the how the adopters change is Simply the opposite of the potential adopters whatever comes out of the potential adopters goes to the adopters Is this clear so far? I mean, that's a very simple equation and I will repeat it if if You need me to Even if it's not intuitively clear you will have to do it in the self-study So it will become it will become clear It's one of the simplest models the bus innovation model. It has lots of flaws lots of disadvantages. We'll talk about them But to get a to get a feeling of These things you just simply need to do it and please men Note the increasing difficulty in the self-studies. So first you had to implement You had to work with a model which is in Vensim. This was the rabbit fox population. It's right there. Just open it Then you had to build a model Given the model structure in the in the slides. This was the workforce inventory model You had the all the components, but now you have to build a model just from these two differential equations It's a very simple model. You have just to stock variables But you still have to do it And there are interesting things you can do To study it Okay, so this was one way to model the demand life cycle Let me let me remind you by demand life cycle. We mean the sales evolution in time Okay, they're both the same thing Now we'd like to know how can we extend this demand? right as you as a as a As a company you would not like to reach this maturity phase and then go down But you'd like to extend the demand to generate more demand if you'd like and The next few slides are about this. How do we extend the demand life cycle or the sales evolution? and Think of it in this way Imagine that for a given Product or a given technology you have a Certain potential demand potential Okay, so imagine in the ideal case the whole world is your are your potential customers? Okay, and you want to tap into more and more customers With time so that's one way to look at it. The other way to look at it. They already said you generate new demand But you can also look at it in this way. We don't generate new demand the demand is given But we just cannot tap into all of it And and we'd like to to tap more into it. How you can do this? Well in two ways You can change You can introduce a new technology, but still address the same need for your customers Okay, so the examples in the in the next few slides are about audio technology So let's stick with audio technology The demand or let's say the need of your customers for audio technology to listen to music all kinds of stuff Is given and let's assume that potentially the whole world wants to do this But of course depending on your technology you may not be able to tap into everyone Look at this For instance This is a the vinyl records. They're a nice way to listen to music But you cannot tap into The whole population you need special care you need to take special care of those things you need special equipment to play them There are special ways how you need to store them So it's kind of a hassle not everybody is willing to do this So what you do to tap into these guys that are not willing to use that thing well is simply introduce a new technology Now we have mp3 It's a new technology, but the dress is the same need listen to music So you tap to more and more people by by introducing different technologies or new technologies And here you can see all different examples for new technologies who have the the tape the tape cassettes here CDs mp3 players Well, where are the mp3 players? There is somewhere here and then we have DVDs right so this is the basic idea You introduce new technologies to tap into more and more demand But still address the same need and things become simpler actually by introducing new technologies If we stick to this then we can talk about technology life cycle, right? You can apply the life cycle concepts to virtually anything Technology life cycle would be the same thing you introduce new technology. It's being adopted very rapidly Then there is kind of a saturation maturity in its adoption and Then nobody wants to adopt it anymore because new technologies new technology has come on the market or It's just too outdated But normally new technologies come the second way to do it is to introduce new products But still keep the same technology The idea is the same you need to address the same need for your customers, but You make it simpler for them To adopt so you introduce new products simpler products But using the same technology right and the example here is with the tape recorder in the Walkman so tape recorder Yes, so that's the tape recorder here and the Walkman. I'm not sure if it's on the slides Well anyway The tape recorder in the Walkman is the same technology, right tape Yes, okay. I agree that the Walkman needs some microelectronics as well, but the Technology which addresses the your customers needs is the same. It's tape But it's a different product. It's making it makes it easier for people to listen to music They can carry this Walkman in their pockets Instead of this big machine the tape recorder all right, and by doing all this we simply extend One way to look at it extending the demand creating new demand or tapping into unrealized demand potential and this figure illustrates what I just said here we have imagined that this is The total demand available the whole population Okay, you start with some technology and at this point of time this technology is not adopted anymore because These people here they don't want to adopt it. It's too much of a hassle for them. It's too big It takes it needs too much care stuff like that So what you do well you introduce a new technology to address to tap into these additional people the same thing you can do with products now the green curve is One technology life cycle so one of these green curves But within this technology life cycle you have many products, right? You introduce many products still with the same idea to address more and more demand all right so these were kind of a preliminary remarks into Technology adoption and product adoption. So this is technology adoption Product adoption now what we'd like to do is to look in more details Into technology adoption how new technologies get adopted or in other words, it's basically the same as How people buy this new technology or in essence it's again the sales evolution our main team for today So how do technologies get adopted First of all, you may know from experience. I'm pretty sure you know from experience that technologies can be adopted in two different dimensions It's not working. Well, that's time and space Right, so different people can adopt the same technology in different times For instance early adopters adopt the latest gadgets When they're released late adopters, they wait for the price to go down and still adopt the same technology or in space You know that different countries get the latest gadgets at different times depending on the product normally Europe gets It's one of the last in the developed country developed part of the world to get the the Apple products, I think Is that true? I think so Maybe but at least okay. The US is the first one. Let's agree on that So, yeah, that's different adoption in space and as soon as we talk about space We need to change the terminology a little bit and talk about diffusion Oh, I forgot to put my timer on Can somebody tell me how much time is left until the break? Ten minutes. All right. Now we're right on time All right So we need to change the terminology a little bit and talk about diffusion how technologies diffuse in space That's a physics concept. In fact technology diffusion is a field by itself people do research just on technology diffusion and Try to understand what factors influence the speed of this diffusion the scope of this diffusion and things like this We will not concern ourselves so much with space. We will only concern ourselves with time What makes people adopt a given technology at a given time? Or what makes people buy a given techno buy a given product at a given time? Yes Yes, yes, that's Yes, but they might these are not These two things are not Contradictory to each other so you can of course it takes some time For a diffusion process to start from here and reach that point. It takes some time, of course That's true But even if you have the same so The idea with the time dimension here is that even if you're in the same space with somebody like you and me for example We're in Switzerland You may buy the iPad before I do even though it's available at the same time for both of us and With the diffusion. Yes, there's obviously time and this is our goal now to model the technology adoption and please don't be Confused by the different terms technology adoption technology life cycle sales evolution are the same things I mean we sell this technology, right? We sell this product So it's the same thing, but it's more interesting with technology. Okay, so we'd like to to model the Technology adoption or the sales of this technology and we know what we need to get I will show you empirical evidence obviously But for now we need to get an s-shaped curve and Please remember the instantaneous rate of Sales is bell shaped right so at any given point of time if you calculate the distribution of sales you get Bell shaped curve if you integrate this distribution You get the cumulative number of sales and that's the s-shaped curve. This is what we want to model Of course if we able if we're able to model this we automatically model the bell shaped curve as well Now this is this is mentioned here the number of of sales At any given point of time or per-time interval is a normal distribution and that's also an empirical fact It has to do with the life cycle right growth maturity decline So a few a little bit of empirical evidence. This is space now and The example here is from a hybrid corn introduced in the US So apparently there is the reference to a paper. I will upload this paper in the literature section, but apparently this hybrid corn had some Beneficial ability beneficial things for the for the corn so you can get more corn using these seeds That's the basic background and what what is shown here is how different states adopted this hybrid corn Technology which was Yeah, I think it was introduced in the in the in the whole United States so you can see the state of Iowa This is the adoption of the hybrid corn right here. We have percentage of farmers Which have adopted this hybrid corn technology? Right, so we can see a nice s-shaped curve so by 1943 Everybody has adopted the hybrid corn in Iowa, but in 1943 about 40 percent of the farmers in Kentucky Have adopted the hybrid corn and obviously if you look in Texas in Alabama It's even worse. So in 1943 nobody even knew about this technology In Texas. Oh, well not 1943, but yeah well 1940 So this is how technologists the same technology can diffuse in space different states This is now diffusion in time What is shown here is? Again hybrid corn and this is number of farmers No, it's not hybrid corn. Sorry. It's some kind of Wheat to 4d wheat some special wheat wheat as in agricultural wheat and Yeah, I mean I don't know what they do in Iowa Alright, so you can see that it took 12 years Well, not 12 but 10 years For everybody to adopt this kind of new wheat So this technology took 10 years to to to get diffused to get To get adopted and the question is why we'd like to understand why it takes so much time for some technologies So much and less time for other technologies another example is Adoption of different technologies in the car industry and you have different Different car technologies like automatic transmission Power steering air conditioning and so on and so forth and you can see the different technologies get adopted in different times It takes more time for some look the automatic transmission It started like this and it took a lot of time and even it took a dip here I don't know what happened actually here Until it gets it gets adopted to 100% and I think it's It should be specific to a country Because I don't think in Europe everybody has automatic transmission But anyway, there is yes There is a paper given there Anyway, so you see that the automatic transmission took a lot of time But check this out. This is the Electronic ignition it got adopted to 100% really really fast in I think it's In about four or five years It got adopted and the automatic transmission took 20 30 35 years to get adopted different technologies also in between So this one was also quite fast. This one was slow, right? So this is how different technologies get adopted With time in fact what we can see is that the speed of adoption For newer technologies Decreases it's kind of an interesting thing to observe Even nowadays newer technologies are introduced all the time and they get less time to get adopted Right and this is a this is an example of Different technologies how much time it took to get adopted Virtually to 100% I believe so it took just one year to adopt in containers But it took 14 years to adopt centralized traffic control All right, so let's go back to the bus innovation model. It can still be used for technology adoption, right? We'd like to explain the S-curve for technology adoption. So let's go back to the bus innovation model remember we have adopters and potential adopters and We we work with the fraction of adopters. This is the number of adopters divided by the total population, right? So this is F the fraction of adopters and This so this approach to explaining the S-curve the bus innovation model and the next few models that we're going to see Basically have to deal with information Diffusion so how does information about this new technology get Spread out in the population marketing. I already mentioned this one way But the second way which you know obviously is the word of mouth effect, right? So this is personal persuasion in the context of Infections the word of mouth effect would basically be you talking to somebody and getting infected but you know distinguished between advertising or marketing which is more of a Common source of information that everybody gets and word of mouth which is based on personal interaction. So The bus innovation model is all about word of mouth Effects if you look at the equation, it's basically word of mouth. Why well, let's let's look at it So Remember we had this K the transition rate K in the previous slide now we call it beta Change of notation suddenly But we call it beta and it's equal to the contact rate again The number of people you meet at any given point of time per time period times the probability That you're influenced by these people in the case of diseases. You have no choice. You are influenced So the infection rate is beta and the bus innovation model remember is this We can transform it if we talk about relative frequencies. We can transform it to that equation now, it's a very simple transformation simply Plug-in f of t equal to this you plug it in here n of p is equal to 1 Minus and a right. So sorry not one n minus na and Divided by n It gives you 1 minus f t, right? It's a simple transformation. It's a simple transformation. So you everybody understood it Who got it? Not enough not enough. All right How much time do we have? Okay, that's enough Let's because yeah, that's that's basic thing and We will need this in the in the coming slides, right? So you have this C times I and I would just write np is n minus na Okay, okay, I will we'll continue after the break. It only takes five seconds. All right, let's start again Is there anyone from group M here Except Sylvia No one from group M Good to know alright, so This is this is basically how you substitute Let's see if I can use no I can this is how you substitute That thing you substitute this one here to get this one Remember the rate of change of the adopters is simply the opposite of that So it's simply the same expression without the minus Okay, so what I've done here. I've represented the potential adopters as n minus na right the total population minus the adopters it gives you the potential adopters we divide both sides by n Here we get the Rate of change of the fraction This is C times I this is F F of t this divided by n is One minus F again, right? So this is how you got it clear Who got it now did you get it? Oh, okay good When you solve it right so we have this equation let me remind you what this equation was in the first place It's the word of mouth right These are the potential adopters The non-infected people they meet Some fraction they meet the infected people and with a given probability beta They get convinced to adopt the new technology convinced by word of mouth Personal interactions when you solve that thing This is the expression we get is the so-called logistics curve now. You're probably asking yourselves as you should What is this? Right it was not here But now it's here It's simply a funny mathematical trick to make this more Standard in a sense, I'll show you what I mean the actual solution is In fact this Okay, this is the constant of integration. It's given by the initial condition Minus beta times t right, but indeed if you transform it in this form. So if you just make the substitution Mu is equal Is it? Yes, is it this? Yes, that's it. So if you just make this substitution You get that from here. You just get that directly and why is that interesting or let's say more useful than this well, you can say that at time t Equal to mu and Mu is a constant. So at some time t That thing would be zero the whole thing would be zero and the whole function would be one-half And if you remember how an s-curve looks like like this this equation Would give you an s-curve Which has One half here. So the maximum rate of change of this function is Here at one-half Okay, so the function starts growing growing growing Then the growth is maximum at one-half and then the growth starts to slow down So this is why an expression like that is kind of useful. You can immediately see What is that kind of? transition time Where the growth is maximum and then after that it starts to decrease and to saturate Right, so this is the solution. It's the so-called logistics curve. It reproduces the s-curve Right, so this this was the bus innovation model which assumes word of mouth effects Solution is given by that and we have an s-curve, but As I mentioned the bus innovation model is very limited. So there are some limitations Not some but they're quite substantial for instance You can see here that if in the beginning Cheers Well bless you If in the beginning we have no adopters. Nobody has adopted which is the case in real life Then the whole thing is zero. This will never change right. This is zero in the beginning The rate of change is zero. We have a constant Constant zero. So in a sense the model is unstable at zero Unstable in quotes because it's not technically unstable, but it's not realistic So the word of mouth effect Can only work if there are some people to spread the word already, right? If there are no people to spread the word It won't get spread Simple as that. So this is one thing we need We need initialization somehow. This is the so-called cold start problem Right, we need to start the process somehow Another thing The adoption of a technology depends only on this personal persuasion Between an adopter and potential adopter, but of course you can ask yourself the question Why should I adopt just because somebody else told me to? Right, so that's that's one thing and It it's not realistic in in a different sense as well in most situations what we observe is a Critical mass phenomenon or a threshold phenomenon, which means that the adoption is almost negligible But once enough people have adopted enough critical mass has accumulated the adoption is very fast But here we have constant rate of adoption this s-curve It's constant even in the beginning depending on the parameters you can model you can recreate all kinds of slopes Which is not really realistic we don't have this critical mass phenomenon It assumes homogeneous population So everybody's the same the only difference between people in this population is that some have adopted this product and some have not But eventually everybody will become an adopter Which is again not really realistic the analogy to this disease spreading is Incomplete in the sense that the real susceptible infected recovery models They assume that people get infected then they recover and they're not infected anymore. They're not contagious They will never get infected anymore But in our bus innovation model we stop at the point of getting infected Adopting it adopting the product and you stay an adopter for the rest of your life Which is again, not really not really realistic most importantly, however is this one There are no economic ingredients and this is something that you will encounter in other models in discourse, but also in In models in different courses where you have some mathematical Equations which seem to reproduce the shape you want, but there's no economic ingredient. There is no policy that you can come out with Economic policies, so there is no competition between products. We in fact we have only one product one technology There are no network effects no lock-in effects. These are important concepts When you generate a lock-in effect Then nothing else can get can get adopted but Surprisingly enough the bus innovation model fits Empirical observations very well there is plethora of literature on how the bus innovation model can fit different Different technology adoption scenarios and I'll show you one of them Which was done in fact by professor Schweitzer In the chair of systems design They use the bus innovation model as simple as that to fit or to reproduce the spread of donations After a natural disaster, right? You can think of the spread of donations again as a Spread of technology or a spread of product or sales Right, it's the conceptually. They're the same you do some you do something Based on the influence of the others You buy a product you donate money you adopt the technology. It's all the same conceptually So and they just they just use this simple model and they have a paper Which is nice? so they looked at at the data Which showed the adopter the donation behavior So it was I think it was a German database which showed some donation behavior for East Asia and the time span was July 2004 of this database. It was quite quite big database July 2004 June 2005 so it's more or less one year and the blue one is Is it mentioned somewhere? No, it's not the blue one the blue curve is the daily number of donations the number of donations The red one is the amount Right, so they're more or less matched, which means that it's not the case that one individual Donates huge amounts of money It's the fact that Many individuals donate more or less the same amount of money So they they're matched pretty closely, but so the total amount of donations Is Is it mentioned? No, this is the money. This is the total amount of money and This is the total amount of donations So what happened here? This was December 2004 between December 2004 and January 2005 in particularly it was 26th of December 2004. Do you know what happened there? Exactly, it was the tsunami in East Asia in Indonesia Right, so there was this well, it was an earthquake and then a tsunami and Look at this pike so it's not the fact that here we have no donations No, in fact, we do see this is the time span from July 2004 to December 2004 so this line There is there are still donations, but compared to this peak They're virtually Virtually zero Okay So what they found is an S-curve It's slightly strange looking S-curve not really strange, but it's prolonged at the at the high end, right? So the blue sorry the red dots are the empirical data the empirical distribution of donations or The cumulative number of donations if you'd like and this is time, right? So this is June 2005 May 2005 Everything has been donated so far Okay, so it's an S-curve and the blue one is the bus innovation model the one that I showed you with these parameters This one. Oh, sorry. Oh Yeah, it's in your notes So what we do here Please look at your notes now You have a slightly different representation of the bus innovation model and the difference is the beta So we had a beta there, but now the beta is one over tau Now don't get confused. The reason is the following. So beta shows you the probability of getting infected It's the mod it's multiplying C times I the contact rate the number of persons you meet times the probability you get infected So beta is let's say the number of infections if you'd like When you divide one over beta you get let's say the time between two success successive in Infections so the towel one over towel that you have the towel will basically be your time between infections So the longer the time the bigger the towel the smaller the beta and The more time you have to wait before there is an infection occurring So what you what will happen is your S-curve will get kind of stretched, right? The increase would be slow if the towel is very small meaning beta is very high You just have to wait a little bit between infections. So you immediately get the S-curve like this the towel The towel is simply used to Introduce the time dimension to think about time So they fit this model This data they fit it with the parameters mu Mu was over there and this is the time tau. They don't mean anything. This is the One of the critics for the best innovation model economically. They don't mean anything these parameters but They still fit the data Now do they Do we have a good fit? What do you think do you would you like this fit? Yes No, that's fine I mean if you stop yourself here if you stop yourself here and you plot it again So you would have a different aspect ratio between the X and the Y axis You would get kind of a S-curve Good-looking S-curve. Is this a good fit? It's intuitive. I mean there is no right or wrong answer. Would you accept this is a good fit? if You're a manager and I come to you and I ask you what this is a model that reproduces our sales and Based on this model. I propose that now we introduce a new product for instance. Would that be a good fit? Intuitively, what do you think it's it's not Who said yes? How many say yes and The other say no, I assume All right. Well Yes and no Aren't you a little bit suspicious what's going on here just a little bit. I mean it looks like an important Important region. I mean here. We don't care so much. This looks kind of important So it turns out that it is Important and what these guys try to do Was to understand why the bus innovation model predicts something like that the booker, but the red one is Actually deviating from it and what they found out was the beta place of role What would be the one over town? What do I mean now beta remember is the number of infected people per time period It's the multiplying the contact rate times the probability of getting infected or Let's talk about adopting in adoption terms The number of people you meet who've already adopted times the probability that they will influence you and you will adopt But as time goes the bus innovation model assumes that this bit is constant, but in reality as time goes Beta doesn't have to be constant. Maybe as time goes you're less Susceptible to worth of mouth influence Right so in the beginning when a new gadget comes out Everybody is excited about it the new adopters the early adopters are very excited about it They're very aggressive in promoting this gadget to their friends But as time goes maybe the early adopters or the adopters become less Convincing they become less enthusiastic in In spreading the word in a sense and potential adopters They're not so easily influenced anymore. So beta changes. It could be right. That's one hypothesis And they tested this hypothesis and it turns out it's true. So what these guys tried to do they tried to fit? beta This is one or one of the towel In In so yeah, ironically not ironically, but unfortunately this C is not the same as this C. Okay Differences this is the contact rate Contact rate the number of adopters you meet per time interval C is just a constant here So they tried to fit One over towel They extracted from some the extracted from the data. This is the extracted one over towel The the red one as time goes by. So you see even without fitting anything You see that one over towel goes down, which means that the time For adoption increases What could that mean well that could mean exactly this lack of interest people lose interest in this new thing in this new technology or Adopters become less enthusiastic in promoting this new new technology regardless the time increases and they've managed to fit it with with this kind of a quadratic Equation and Yes, the coefficients are not given but of course they were estimated from the data So that's how you can yes, and it's also said here in the early stage of this Disaster people are really willing to donate money. They were easily influenced by pictures in the media and by Communicating with with other people, but as time goes by they're less They become more indifferent Okay So this is a this what yes? Yeah, yeah, so the question is can we identify this time when we observe this kind of shift Um The answer is I Basically, it cannot be answered because the the question is not precise in the sense that we this is not the threshold process It's not like It's not like we have a lot of interest and then zero interest This is how the interest develops Right, so what is this point is up to you? How we how we define it right maybe for you at this point of time We can say now there is a clear Indifference in the population and now we take this time, but maybe for somebody else. It's so it's here Right, so it's kind of difficult to to pinpoint qualitatively even Where we want to be on this kind of curve was that Good enough. Oh if you if you take the The maximum of the derivative Yes, that could be one candidate That could be one candidate Yes Okay, so this was an example how the bus innovation model can be used To fit real data. It's not just corn hybrid corn, but also as they as kind of Diverses donations now Let's look at Improvements to the bus innovation model namely how to solve the cold start problem One idea so remember the bus innovation model was word of mouth effects only But one idea could be Broadcast of information For instance in the beginning when a new technology is introduced new product is introduced before anyone has adopted it We have marketing or we have advertisements. We have some kind of source of information Which Announces this new technology and this is known as the common source model Common source because the whole population has a common source of information. This could be Keynotes pitch it could be an advertisement So what what is the suggestion of the idea here is that this common source of information? Convinces at any given point of time a given fraction of the population to adopt Think of it as advertisement Advertisement convinces a given fraction of the population at any given point of time to adopt and this fraction is given by this of the non Potential adopters, of course, not the whole population This is the fraction and if you just have this Model without the word of mouth effects, but just the advertising effects Well, this is the dynamics, right? It's simply alpha times NP but NP was basically one minus FT. So this is the relative frequencies again and This is the rate of change of the adopters The resulting function is an exponential. It looks like this. This is obviously not an S curve It solves the code start problem, but it's not an S curve. It's just an exponential and I don't know why this is black, but these different curves correspond to different alphas, right? So when we increase alpha, obviously the curve increases faster okay Well, we already have the ingredients To improve the bus innovation model the bus innovation model was only word of mouth here We saw only advertisement. Well, let's combine them word of mouth and advertising, you know This is this how modeling works in general. You take little blocks and you combine them together and you see what happens And this is the code the mixed source model Mixed source because we have mixed sources of information word of mouth and advertising And it really it's simply adding the two equations together bus innovation model plus common source Yeah, plus advertising and we get the S curve Plus the benefit of no code start problem anymore Yes, so this is the S curve the adopters What is shown here is Right so in the beginning there is no there are no adopters nobody has adopted the product, but This there is certain amount of people Which we call kind of innovators Basically who are influenced by this advertisement Right Different interpretations can be given for for this common source for the advertisement you can think about one ways Advertising Relates to people who want to be at the forefront of technology right people who want to have the latest technologies The latest gadgets, so we call them innovators in a sense So these people would be influenced by the advertising and they would jump start the whole process here and Then these guys Would generate word of mouth effects in addition to that The advertising still works so in essence when we combine these two things we get The new adopters are like this and this is simply the imitators. These are only the people Who are influenced by word of mouth? They imitate what other people do All right, and they obviously develop like this and the difference between these two curves Are these people who are influenced by advertising? Okay We have an S curve now if you solve this model We have an S curve Don't concern yourself too much with this equation where we do some variable substitutions to make it look nicer but in fact, this is the S curve and Choosing the right parameters you can reproduce different Shapes of the S curve for instance look at the green one We can reproduce this kind of little dip and then increase like that and Again, this shouldn't be black Anyway, so this is the mixed horse model if you want to do it in Vensim It's not part of the cell study, but if you do it, it will be very nice. It's a very good exercise We have the two components word of mouth. This is bus And I forgot to say that bus innovation model was actually created by a person called bus Frank bus It's not random So the word of mouth effect effect, this is the bus innovation model and the advertising This is common source We combine them together and you can understand everything basically. It's simply the equations the word of mouth effect depends on The probability of getting infected I Contact rate C so C times I gives you beta or one over tau and The total population and this is the word of mouth effect. Okay, obviously the potential adopters as well And we have this reinforcing feedback here What does it mean? Well, if we increase the number of adopters this number of adopters would Promote the product more so the word of mouth effect would increase By simple numbers the word of mouth effect increases the adoption rate increases the number of adopters increase as well and We have the balancing feedback, of course the adoption rate cannot increase indefinitely The potential adopters they are influenced by advertising. They adopt the product So the more potential adopters we have the more The bigger percentage of them would be influenced by advertising the bigger percentage of them influenced by advertising The bigger the adoption rate Why is where is the where is the balancing feedback now? Oh, it's the market situation. Okay. All right So yeah, so the bigger the potential adopters The more the bigger percentage of them would be influenced by advertising The bigger the adoption rate, but then the bigger the adoption rate the less the potential adopters, you know Because we have a limited number of potential adopters Right. So these guys would get exhausted really fast Really quickly these numbers here and then this thing would go down as well Yes, so for the advertising this is alpha The advertising effectiveness if you'd like we introduced this there's a percentage of people who get influenced by advertising Which in other words is the advertising effectiveness Here and then that's the that's the mixed source model How can we improve it? We can of course improve this model a little bit. Well one way to improve it is Now remember what all this reproduces are symmetrical S curves But in real life as you saw with the automobile technology adoption They're not symmetric, right? So one way to reproduce this non-symmetric S curves is to assume heterogeneous populations Heterogeneous populations meaning every individual has a different alpha and a different beta so a different susceptibility to advertising and a different susceptibility to word of mouth effects, but this would not be a Systems dynamics model, right? Remember we only talk about representative agents representative people This would be an agent based model topic of next semester Next semester's course in collective dynamics of firms So this would not be an agent system dynamics model, but we can still do it and we will get the S curve Right, so we can define some people are rich So they will adopt more easily for instance, so their alphas would be bigger The poor people would have lower alphas and lower beta so you can imagine what will happen, right? The we let the rich people buy the products first So there would be a huge adoption really quick and then when when they've already bought the product the Relatively unriched people come into play and the S curve would be stretched out So it would be an in unsymmetric S curve So you can imagine all this kind of stuff a second way to do it is to introduce Kind of a birth rate So people are born in the population This is the same model with the birth rate. So we have a birth rate it increases the number of potential adopters and So basically Potential adopters plus adopters give you the whole population and The potential adopters are some fraction of the whole population. This is not Really needed for the model. It's just kind of an illustrative Kind of hint How the potential adopters and the adopters are linked so there are some gives you and always So if you introduce a new birth rate What what what does it mean you introduce new people? So they can be adopters Immediately they can be potential adopters, but This now opens the door to competition, right because if you have new people Coming And you have more than one products Which product would these people adopt? It's not clear There would be a self study on this So this just gives you a preliminary preliminary information What happens when you what could happen if you introduce new people? If you just keep the same model as before just introduce new people constantly refuel these guys You would not get something different you would just still get an S curve which kind of Always grows because it never gets saturated. You always get more and more people So in essence, this is not really an improvement, but it simply opens the door to thinking Okay, what can I do with these new people? Where can I allocate them to and you can only think about these things when you have more than one products more than one technology We'll see this in the coming lectures Now unfortunately this Diagram is kind of wrong and I will explain why the idea here is however To model so again improvement of the mixed-source model we'd like to model Fats or trends especially in fashion What is meant by that is so far Potential adopters become adopters and they stay adopters for the rest of their lives They never become they never give up the product. They never give up the technology or adopt a different technology They stay they stay adopters for the rest of their lives and we'd like to change that So this regard this just scratch it in your in your handouts I will update the slides by the way this regard the immigration Okay, that's not important. It's just remove it disregard this thing It's also not important What we're left with is essentially the same model potential adopters adopters, but now adopters Okay, and add a flow which goes from adopters goes back to potential adopters Like this just add a line there as a reminder. I will update the slides as I said I just saw this this morning So just add a line here going here and then this is a raid by which adopters become non-adopters become potential adopters So what can happen now is? Yes potential adopters are influenced by Okay in this model, it's just word-of-mouth effect they become adopters here Now what can happen to these adopters? Well, they can get disappointed with the technology disappointed with the product and Just give it up forever. They will never buy this technology from that company again. So they become they discard the product and They become so-called discarders or in the disease spreading model they become Recovered They've recovered from the disease. They will never get sick again. So we can forget about them And of course they have some discard rate by which they discarded Now what if you if you look at your line that you drew like that? potential adopters also May just give up the technology and wait for the next one or They might give up the technology. So don't buy the product next time Become potential adopters and adopt a different product if you have more than one product in your model Right, so this is what can happen Right and this is a way to model fats actually fashion short lift hypes We're not going to do this it just gives you an idea of the things you can do Of course, this is actually taken from from Vensim from the models there so if you'd like you can explore these models and Play with them a little bit, but it's not required. It just gives you an idea what what we can do Okay, and the last part of the lecture would be about a Different approach to the s-curve to the evolution of sales if you remember I told you that All these models so far would be concerned mainly with information spreading how information is spread word of mouth Advertising This was the the idea here as a reminder Right, this is the word of mouth effect the bass innovation model This is how it looks like you can immediately see that there is every time you see minus squared There is a saturation going on right just multiplying the brackets out. There is saturation and This is what we did so far But there is a different Way that we can approach this this whole issue and it comes from population ecology It has to do with birth rates and death rates now explain exactly what this means Let us start with the exponential growth We start from this, you know, this could be the rabbits That we saw in two lectures ago. There's some kind of growth Depending of course on this parameter gamma Right so depending on the parameter gamma if it's positive or negative this can explode or it can die down doesn't matter but now we assume and Yes, so this is gamma But now we assume that the birth rates and the death rates depend on frequency Right in the rabbit populate in the rabbit example The birth rates and the death rates were just constants But now we assume that they depend on frequency for instance. This is the frequency-dependent birth rate Frequency or density the density of people Who've already adopted the technology of who are already infected So we can assume that if there if there's no one no one in the population Is it really five minutes left? Oh my god all right frequency dependent birth rates, so If we have no population Then we have a given birth rate B, which is called the density independent birth rate logically enough, but as more people are born as More population grows the less likely they become to reproduce because you know there's crowding effects and stuff like this The same with the death rates only inverse right the more people there are The higher the death rate infections overcrowding stuff like that. So if you plug in this density dependent birth and death rates into that We get this whole thing and that's an S curve Right so but just starting from an exponential growth rate and adding The idea that the birth and death rates are frequency dependent. We get an S curve It looks like this we can change some variables. We can define the carrying capacity Which is equal to this Basically it transforms the function to the familiar Mathematical form something divided by something when it is kind of ugly. So this is much better now All right are R is this so it's the density independent net rate and This is constant doesn't matter All right, so what what is birth and birth and death in terms of technology adoption, right? It's intuitive what what it means in population terms what in what does it mean in terms of technology and adoption? I mean People don't die when they adopt a technology No, actually, that's not true either. So I mean, yeah, they they sometimes die but When we talk about birth and death what we mean from organizational theory is two things it's The so-called two stages of technology adoption legitimation phase and competition phase Legitimation phase is basically getting this technology to be adopted in the first place. You don't care about About the price you're charging in fact, you may be charging less to Excite people and get more people to adopt it So in a sense your birth rate kind of birth rate of your technology increases because you try to get more people to adopt it This is the or try to make the technology legitimate in a sense. You don't care about your money now Competition phase is after Some companies let's say have adopted your technology. They've supposedly got competitive advantage But as more and more companies adopt your technology the competitive advantage of the early adopters Would die down Right, it won't everybody will have the same technology. So you're the competitive advantage of you as an adopter Would disappear. So there is now a competition Between the people between the firms who've adopted the given technology Which means that there is less incentives for new companies to adopt that technology and this leads to Yeah lower returns for late adopters and And that leads to the saturation here, let me see if I can show you one Interesting slide. Yes So what What these so this was the birth and death This is now. I mean, it's basically the same thing about Hybrid corn again What the guy tried to do was to take Was to take this Right. So this is kind of a density independent net rate and this is some constant and try to fit the data and there is some kind of a Regrouping of that term right a Is so B is the rate of adoption The speed of adoption and a is simply positions the curve on the time scale It's not so important. This is not so important. This is the important thing the rate of adoption of your technology And the guy tried to look at this and try to fit the corn adoption pattern into this So what he did is the following taking the lock of death Leads to this right. It's also simple transformation This is the natural algorithm. So if you as the researcher take your corn data hybrid corn data You calculate that from the data and you try to do a simple regression. This is a simple regression All right, and he found this. This is the rate of acceptance. So the growth beta And you see that Some states if you remember different states adopted differently some states have a very slow adoption Some states have very like Indiana. I don't know. I Think so. Yes have very high rate of adoptions And this is the R squared value. It's how much how many percent of the Variation in your data is explained by the model. This is very good. I Mean very high values. This is the carrying capacity. Don't care about it. What carries is this So he fitted the data with using the normal linear regression. He found this So now the question is and with this I will end why Remember the original question. Why does this state? adopt a technology faster than this one for instance Okay, and what we do here is multiple regression You simply assume well, there are different factors which influence the adoption These different factors are called X right, so in his regression the X's were the size of the farm The productivity of the farm before adoption the productivity of the farm after adoption and I think Yeah, at the profitability and not just the size but those the profitability of the farm so You know, this is basically a basic a basic multiple regression. You've seen it before especially in econometrics Right. So what he did what he found this is the results, but This is the conclusion is that only large Efficient firms adopted the hybrid corn Only large and efficient firms isn't this counter-intuitive You would normally expect that Inefficient firms would adopt a new technology hoping to become efficient But in fact, it's the opposite efficient firms who've already done pretty much everything to become efficient They see this new technology as the only way to improve their efficiency. Whereas inefficient firms have a lot of things to do Besides adopting a new technology. So this was the results he found. This is the result from multiple regression and look This was his multiple regression This is a factor and this is a factor X3 is the average size of the firm It turns out that this coefficient is significantly different from zero and It's positive. Therefore the bigger the firm What this means is the bigger the firm the bigger the x3 the bigger the beta the bigger the adoption rate X8 what is x8? This is the yield of the firm Before adopting or the efficiency of the firm before adopting. Well, let's look at it. What is it? It's positive and statistically significant. This is the standard error. Therefore the bigger The x8 the bigger the yield The efficiency the bigger the adoption rate This is how you interpret the results from multiple regression and this is how he reached his conclusion And you can also this is by different states. This is different reporting districts. The conclusions are the same Only that here we have a different factor x7, which is This kind of difference, but it's again the bigger that the bigger the beta So let me recap what we did We started from the from that This is the simple bus innovation model All right, and we try to see what are the factors that influence the beat the beta or the B the the rate of acceptance And in this particular case It was an empirical study first of all, so it wasn't like a conceptual thing Like we did last time But it was an empirical study. You do a multiple regression First you do a regression you try to find out the most important thing you try to find out whether your data actually follows this if you couldn't fit this You're wrong. This is not an S curve. Your data is not an S curve But he fitted that and then try to find out what beta is and This was the multiple regression model Okay, that's That's not so important. These are just concluding remarks which you can read yourself. They're really conceptually easy One important thing I'd like to mention is that the S curve or the bus innovation model Or all the models actually that we saw today they leave out firm strategic behavior There's no strategic behavior on the on the firms side When they adopt the technology they simply either influenced by advertising or by other firms But there is no strategic behavior like I will adopt if My competitors adopt or something like this and we will work with these models in the in the coming lectures An important thing also Competition we saw competition between technologies Not between technology to competition between firms for the same technology does it promote diffusion or not If you remember this argument, I Said that it doesn't But actually the evidence is inconclusive because if you think about it If the firms like it said here If the competition between firms decreases their profits then of course they will not adopt a new technology They will have less money to do this But if they anticipate the results from competition well, they can adopt a new technology mean like at the start When they have more money, so the evidence is kind of inconclusive Yes, that's it. Thank you