 So, good morning everyone, I welcome you to lecture number seven of our course Collective Dynamics of Firms. It occurs to me that there is not a sufficient number of registered students, so we should have five registered students, at least for me to take the effort to teach here, right? So I'm here, what's your comment on this? No, no, that's a not a registered student. It's difficult for me also to handle this. Yes, please. Yes, yeah, for the first two weeks only. Okay, but this is certainly not what we, yeah, what we had in mind in the beginning. I mean, I understand that you are the wrong people to blame for, right? So that's something I certainly understand. Okay, I also understand that today is a special day because of the Easter break. I mean, I can certainly follow that other people are already on travel. I mean, that's something I certainly understand. So let's do it like this. I teach this course now, yeah, but I'm not going to continue with this in after the Easter break if there are not at least five people, yeah? So then you do it as a self-study talk. So okay, so what we did, what we did last time was to introduce a modeling part and before we introduced a modeling part and before we did, so we learned about the Starlight's fact of the dynamics of company, oh, ah, so okay, so Natalia has compiled it the wrong way. That's fine. We learned about the Starlight's fact in order to understand what should we compare to, you know, if we have our models. I do not repeat the Starlight's fact. I think that everyone knows about it and there was all of this online test, right? So everybody finished. Last week, Pavlin started introducing the modeling part and we started with very simple stochastic model. So I assume that you were here. Pavlin told me six people were here last time. So then how did he do this? Yeah, I think he's quite experienced. And we started with the most simple assumption here, namely with the random walk. Of course, we found that the random walk is unable to reproduce some of the Starlight's facts that we already know, but it was a good starting point. We were then modifying this assumption of the random walk into a multiplicative stochastic growth process and surprisingly enough this already captured some of the Starlight's facts that we have already seen. There were a few things that were not properly included in particular this issue of the growing variants, for example. We got a log-normal firm size distribution, but this was growing all the time. That was something that is not in accordance with the Starlight's facts. And therefore, today we start modifying Gbrud's assumption in order to get this right. So if you think of the course in a methodological manner, then it's like a construction kit. You start with a simple basis and then you add more tools to this to get a certain functionality. What we do here in the modeling part is not different from this. Let us first look into some empirical work that was made in the mid-50s in order to test some of the Gbrud assumption. Remember, Gbrud did his work in 1931 and this empirical study by Hart and Preiss was done in 1956. And, of course, it depends on the data of what you see. That's an important comment that you should please keep in mind. If you see deviations, it's maybe also because you have a different data set. So the sample that they used was from UK companies. Fortunately for a quite long time interval, more than 65 years, you see that the size of the firm was proxied by market valuation. That means the number of stock times the stock price. That was the proxy for the firm size. You keep in mind that depending on the proxy, you may have differences. You do not change the start out fact, but you will not see identical things. And what they looked into was the growth rate over time and the variance over time. If you think for a moment about the Gbrud dynamics, then what would you expect? This is the Gbrud dynamics and we just took the logarithm of this. That means that there was a multiplicative process and then we treated this as an iterative equation. That means there is a firm relation between x of t and x of t minus one, which can be replaced by x of t minus two and so on. At the end, we end up with something like this. In the middle, we have the product of the growth rates, which were assumed to be random numbers. Basically, the Ln of this product is nothing but the sum. What you see here is there should be a linear relationship between x0 and xt. That's what we express here. Now, let's go back to this plot here, where we plot the size in 1939 versus the size in 1950. What do you see? What's your conclusion? Comments? They are positively correlated, so we see this linear relationship, but we do not see it for all firms, so that's the comment here. In particular, here in the range of small firm sizes, this cannot be confirmed. That's an important point. If you recall our stylized facts and you know that the small sizes were quite particular. In particular, the growth rates of small firms had a large variance. Maybe if you have to explain this in the exam, maybe you point out that this can be due to the larger fluctuations of the growth rates, because this is of course only the average. There can be all the other effects. Between 1939 and 1950, there was World War II. What could have been the effect of World War II on the growth of smaller companies? This is after World War II than this data. Another argument. At least we have to talk about the small firms in particular. These two does not nicely match us. You could also say, well, maybe the data set was not very good at proxying firms of small size, because we remember that's market valuation. How many of the small firms are probably adjusted on the stock market? If we talk about the price of Nestle or something like this, that's relatively stable as compared to your startup company that just made it to the stock market and then has this heavy fluctuations. There are various reasons for this, but it's important to understand that the prediction of the G-Broad model was a straight line, and this is not met by the small firms. As I said, Natalia has compiled this with the notes, but you have the notes anyway. Please. Yes, correct. That's what I said. That's correct. Yes, that's what I said. Maybe the data set under represents the smaller firms. I mean, of course, they had a market valuation. Otherwise, they would not have been included in the data set, but it's not clear whether they are properly represented. The second thing that Hart and Prey looked into was the variance over time. You probably remember what the G-Broad model predicts. It predicts a linear growth in time. Natalia, as always, is very proper. She wrote it up for you here that you understand how to get to this variance. This is something that can be tested. It's a variance growing over time. If you go back to the paper, the paper you find in Mendeley, if you want, you can click on it and read it and see how people argued about these things in 1956 as compared to the Amaral paper of 1997 or something. It's quite interesting. These were economists, Amaral, and co-laborators with physicists. They found out when doing the statistical tests that they cannot confirm that the variance is growing over time. That means there must be something wrong with the plain G-Broad model. Then the question was, what can we do in order to include factors or terms into the G-Broad model that account for this observation and still get us the log normal distribution? That's the challenge. The assumption that they took at that time was there was an optimal firm size. The firm evolves towards an optimal size and then it stops. What an optimal size means, of course, very much depends on what industry you are talking about. Remember when we talked about entry and exit dynamics, then I discussed, lengthily, the difference of having a firm established in construction versus in insurance and other services. With the optimal size, it's a similar argument. You can understand that if you think about economies of scale, that at some point it doesn't make sense to hire more people because your revenue is not increasing proportionally to the number of people. There is a marginal revenue that goes to zero. Then why should you extend your business? That is one argument that was also used here with the term financial criteria. How was this built in to this model? Here, that's the G-Broad dynamics. They took the logarithm, so capital X means log of the small x, where the small x is the size. Then they have this one plus b as the growth rate here and then they end up with something like this. Why is there the plus b? What did they do? Yes? Which one? Correct. Yes. The ln of 1 plus x was proxied by x or b in this case. That's correct. The assumption here is still, if you test this, the assumption is that b is a small number, small compared to 1. That's important to keep in mind. Then the whole thing looks like this. If we assume that there is an optimal size, then we include in new term here, which means growth towards the mean. You look here, then you see if the x is smaller than the x minus 1. If the capital X is smaller than the x bar, which is the optimal size, then this is smaller than this. How is it? It depends on the sign of the beta whether this is positive or negative. First of all, you have to determine what the beta is. Secondly, you have to find out what the optimal size is. What they found is that if you want to calculate the variance of this process, then the beta certainly has to be less than 1. If the beta is larger than 1, then the variance grows over time. This is one mathematical condition from this. Then they went to test this against data. Is the beta less or larger than 1? That was the question. If the beta is less than 1, then the growth process stops at the optimal size. If it's larger than 1, then the distribution is extending over time. This is the data that they used. I think, as we also mentioned in the notes, you should look here into rows number 5 and 6. The B bar is basically the average over the beta. We named it beta. Here you see that the beta is indeed smaller than 1, but not for all times. Then, if you're an economist, then you go back and say, well, what is different in this area between 1907 and 1924? First of all, there was a World War I. This is your K data to what extent was UK involved in World War I, as compared to other countries, most importantly Germany and France. If probably Germany and France didn't grow so much, that was a chance for the UK companies to grow and to extend their business. This was also the time of important investment in colonialization and these kind of things. You find an argument that this extends, the distribution extends over time. But for most of the time, you see that the beta is indeed less than 1. That means the distribution grows up to a certain optimal size and then it more or less stays stable. Here you even see a stronger consolidation, which is then dubbed to World War II. This was the first attempt, the optimal firm size. This issue was also picked up by Louis Amaral and co-workers because they also saw in their lognormal distribution that the variance is stable over time. Remember that was the very first picture I showed you when there was this impressive master curve of the data about 20 years from these US manufacturing companies was quite impressive. You could already recognize that the variance was stable. They already thought about this and used the same argument but in a different model. They assumed a deeper dynamics where the growth factor was randomized in such a way here. It's a bit different. The K is a constant and the epsilon appears in the exponent. The assumption here is if the x is less than an optimal size, which has to be determined, then the dynamics follows K times e to the epsilon times t. If the x is above the optimal size, then just the constant inverts. If K is equal to 1, then you end up more or less with the G-bride model. You wrote the fluctuation in a different way but that's fine. That means if you have a K more or less similar to 1, that means 1.1, 1.2 or something like this, then you have a smooth relaxation into this optimal value. Why did he choose this model? Because the solution was already known. It was discussed in the literature as the so-called towards the mean problem. We mentioned paper by the famous Milton Friedman in the notes who already dealt with these questions. The solution of this dynamic is this. What do you see if you look into the distribution here? What do you recognize? It's something very specific. This distribution is a Laplacian distribution. Laplacian distribution, Dalat's fact number 2. That's something that we like to see from the model. They of course knew this and therefore they have tuned the whole model in such a way. This was in line with other observations. That means there is a reason to assume that there is an optimal firm size and a towards the mean problem in economy. As I have shown you in these examples, there are different ways of including this into the model. You can do this in the simple way as hard and praded. Simply beta x minus x optimal. Or you do it in this way and then you get something additional. With this, we move one step further. What was also not properly covered in the G-Broad model? If you have to name five important drawbacks of the G-Broad model, what would come to your mind immediately? What should we change? What's a very bad assumption of the G-Broad model? Do you still remember this? Please. It's a constant number of firms, for example. This was a very bad thing. I told you about enter and exodynamic of firms and then we do a model where the number of firms is always constant. That's something bad. This is true, but the question is, you can think of a number of stochastic processes, but the question is, will you end up with a log-normal distribution if you change the number of these underlying processes? The answer is no. That's important. That was one drawback. Another drawback was that there was basically no interaction between these firms. G-Broad didn't assume anything about the interaction, but we already know that, for example, if you have a certain business, then the success of your business is somehow correlated to the performance of the overall economy or the performance of the industry, these kind of things. This was all dropped. Subsequently, people started to improve the G-Broad model. On this slide, we should start with the bottom. There was an idea by Takah Yazu who said, well, we keep more or less the G-Broad model. We assume an additional term here, that's the F. I come to this model on the next slide again. Adjust all of these variables according to data. That means we do not try to improve the dynamics. We correct these assumptions by simply fitting the growth rate to the data. This was one idea. Here is another one, which was discussed in mathematics. People added more noise to this. This is called as a so-called Kasten process. You see this is the G-Broad dynamics here. Then there was an additional random force added. Any idea of what could be the economic reason of this additional random force? What could it be? Certainly random, of course. This could be a constant, not a constant, a continuous in time support, for example. Think about the government. The government subsidized your business because you are so important for the industry or for the overall performance of the economy. Every month, you get a little bit of support. How much you get depends, of course, on how the economy is doing, but you can rely on this. That's one assumption. We look into both of these cases today. Therefore, we start with some. First, what did the Japanese colleagues do? They had two data sets available. One is about U.S. companies and the other one is about non-U.S. companies. Then they had a third data set available, which contains Japanese companies, but only very large Japanese companies, as you see. They have here plotted the income of a particular Japanese company. The name is not given. In number of million yen. This is 10 to the 10 yen, probably a large number. This is income before tax, if I am correct. Let me check this. We didn't write this in the note, but I assume it was income before tax. There is a corresponding curve that gives you the rank of the company in this data set, the Japanese company. This company was ranked number three. It means it was the third largest company in Japan according to the income in 1970. Then you see that the rank heavily fluctuates. There are two reasons for the rank to fluctuate. You are performing bad, or the other performing better than you. These are the two reasons. If you don't do anything, you can still lose your rank because there are competitors doing better. You see it fluctuates quite a lot. That's a logarithmic scale here. The rank. I was wrong. What did I say? Number three. That's wrong. This is a hundred. Then this was 10. It's about number 30 here. I didn't make it right. In this data set. It's a rank with respect to these 80,000 firms. I cannot recall precisely what sector this data set was about. I assume it was where the data set was about the largest 80,000 companies in Japan. This is the kind of data that they had available in the paper. They didn't make any use of the rank, but it's interesting to see this. If you look into periods, this is not always synchronous. You can increase your income but still get a worse rank. That means you have a booming economy and the others are catching up much faster than you do. That's one of the reasons for this. All the other way around. The economy is doing badly and your competitors are hit more than you. I point to this because the rank dynamics is different from the income dynamics. It's a very interesting feature, but it's not discussed in the paper. Then the first thing that they did with the data set was exactly the same as you had to do in your exercise. They plotted the growth rates. Here you see the plot of the growth rates. They divided the data set into four different size classes. Then for each of the size classes, they plotted the growth rate. They accounted for the fact that there is a scaling relation between the variance and the size class. You already know this from your exercise. This is then the way they got everything mapped onto one master curve. What's the most irritating fact in this curve? Right. It's not symmetric. What did you expect, actually? What were you expecting? The tent, right. I was also expecting the tent. Why is this not the tent? That's something I mean. We discussed this a lot. I didn't write an email to Takayaza. I could have asked him myself. The assumption was that this only refers to larger data, to larger companies. This is probably income after tax. If you're a small company, then you do not pay the same tax rate as maybe a larger company. That means there is an indirect help for you to have larger growth rates. That's just a hypothesis. If you assume that this is income after tax, before I said income before tax, actually I don't know it precisely. I have to look up the paper or you can't also do it. We provide the paper to you. Then you understand if the tax is not proportionally applied to everyone, then the growth rate of smaller companies is, of course, larger than of larger companies because there is a correction by the government. What you can do now is you can fit your unknown functions here from the deep-roar dynamics to this curve. That's what they did. This is the dynamics that they used. Please note this additional term, the F. You see that the growth rate is split up into two parts. The alpha is just a sign plus or minus and the lambda is basically the magnitude of the growth rate, the absolute value. Here they did the following. If you divide the whole thing by this X and you assume that the X is large, which means we are just talking about large companies, then you can drop F divided by X because it's probably a very small number. Then from this you get the growth rate here by looking into the large companies. You get a proxy for this. If you divide the whole thing by X, you drop the last term and you get this. They already scaled the growth rate, as we have discussed before, in our course. This means the growth rates are then described as we usually did this. The variance of the growth rate is normalized by the sigma that they get for the largest firm sizes. That was the idea. Now, let's look a bit more in detail into this lambda times alpha. We discussed the lambda. The lambda was obtained the way I just described this. Remember the scaling R to the sigma X divided by sigma zero. What is about the alpha? The alpha was a sign. This was again fitted against the data. What they have plotted here in this plot is this is a proxy for the size. It's the income. This is the probability to grow or to shrink. For the US data set, there is a black and a white triangle. The black triangle means you have grown in one year and then you switch to shrinking in the next year. That means the sign, we are talking about the sign of the growth factor here, not the absolute amount. This means the sign is negative. This means your growth rate was negative in the previous year and became positive in the year after. Then they looked into the probability to find these transitions dependent on the size. It's more or less what you expect if you go here for example for the very large firms. Then there is a very small probability that you had a positive growth in one year and then you switched to a negative growth in the last year. The growth may be small as you recall but it may not be negative. That's an important thing. That means in this limit of large firms you have only a very small probability that this is not the x by the way, this is the growth rate. Can you send me an email that we fix this here? That's not the x, it's a growth rate. You should remind me that we fix this for one time. It's not the x, it's a dot x, it's a growth rate. I can use this slide 10 times so I probably recognize this every time. The question is then who reminds me on fixing it? This means there's a very small probability that the sign of your growth becomes negative if you're a large company. They use this limit and then of course the complementary probability is there is a very large probability that if your growth was positive before it will stay positive. This sums up to one. The same argument is used for switching from negative to positive. This can be proxied for large firms at about 25. Remember that there should be a dot. That means there is a 0.25 probability that your growth was negative in the last period and becomes positive in the next period. This means the complementary probability. That means the alpha, the sign of the growth rate is determined empirically and the lambda which was the absolute growth rate. This was also determined empirically by means of the previous slides that I've shown you with a nice master curve. They have determined everything in an empirical way. They used the G-bar dynamics and they fixed all these functions empirically from the data set. Now what can you do with this? Personally I consider this a weakest part of the whole paper but that's my personal opinion. They did Monte Carlo simulations. That's the same what you do in your exercise. You set up a number of independent stochastic processes, so 6000 here. You let them start with the delta functions. That means at t equals 0 every company has a size 100. Then you sample the coefficients of this growth dynamics from distributions that you have just determined empirically. Then you let this run. Then at the end you draw the distribution of your stochastic processes exactly as you do in the exercises and this is what you get. You start with the delta distribution and then you see that of course this delta distribution becomes this kind of log normal distribution. They do not talk about this log normal distribution. They only talk about the long-term behavior but the whole thing looks like a scaling law, like a power law. From this simulation they concluded the following. First of all the distribution becomes more or less stationary. You see at t equals 200 you have a power law behavior. Then they said well here at t equals 50 where we have this q distribution. We compare this to the actual distribution of firm sizes that we have in Japan. Then we estimate the difference between this distribution and the long-term limit and that somehow characterize our growth potential. From this simulation we see we are about here with our real distribution and the whole thing can go on up to a point where we reach this stationary distribution. Then they estimate what's the growth potential and then they map somehow their time scales to years and then they conclude that the US will have a constant growth rate for the next 100 years. I wish that the US would have this. Personally I have serious doubts about it but that's a different thing. That's at least what they did. They didn't do anything different from your exercise in this course. Okay this is a bit more interesting. Mr. Takayasa is a researcher at Sony and Sony is a company that is large enough to have its own currency as you probably know because this makes them independent of these exchange rates all over the world. They have an internal currency that they use for balancing and calculations and so on. Then one of the important question is, so what do we do with our revenue? Where should we invest our money that we earn? Shall we invest in larger companies or larger parts of our own company or should we invest in smaller parts of our own company? As we already know from our own discussion of the growth rates, there is a certain risk investing in small companies because the variance is very low. You can have high growth rates but you can also have high losses as you recall. Whereas in large companies this variance is very low. You have small growth rates, positive growth but you have also only small losses. There is a trade-off. You can have the risk reduced by investing in large companies but your audio reduced your extra profit that you make from this investment. Where is the optimum? What they did here is they used this data to calculate the cumulative income for the next five or for five years. This is the time horizon. The x of zero is the starting point and the x of n is after five years. They sum up this and that is the cumulative income. Then they plot this cumulative income over the starting value of this company, the income at time zero. Then they find the following. They find that if you are already at 10 to the 10 US dollar, which is a huge number, then of course your growth is not very high. That x of t after five years divided by the original. Let me finish just with this slide. Then you see that the growth is not very impressive of course because we are talking about big numbers. If you go for small companies then you see that the growth can be really important. You see on average you can reach really high income here. The question was then where do you have the optimal in size of the company to invest into? We discuss this right after the break. That means exactly in 10 minutes. Let us please continue. What I just said is small firms may have a larger growth in the cumulative income than larger firms relative, larger growth in the relative cumulative income, but there is also a risk associated. The question is where should we invest now? Here you see this is a plot of the distribution of these growth with the losses and the gains. You see that of course in the smallest size class we have large losses but also large gains. You see this is not really symmetric as you probably see here. These losses are a little bit less frequent than these large gains. It is a small asymmetry here. That is the plot we already discussed. What they propose then is an estimator that relates this cumulative growth. The x is the same as the i but weighted by the variance. This means sigma off. That is a function. It is not sigma times. It is sigma off and that is the argument of the sigma. Where do we know the sigma from this? That is the sigma here. This is a gain and that is a risk. This estimator weights the two. The c is of course the weight that you give to the risk in this estimator. I think they have chosen this to a particular value. That is also something we may want to print here in the notes. What is the number of c? Oh, it is one over. Oh, really. I was thinking of something like 0.25. It is plotted here. Then you see this shape. Of course, it depends on how you choose to see. You see the estimator becomes maximum for firms of this size. This is 10 to the 6 yen. According to this estimator, the best balance between still getting some revenue from the growth process but mediating the risk of losing too much. For large firms, as you see here, the risk is smaller but the relative gain is also smaller. That was the conclusion of this little exercise. Now, let me come to the more important part of this lecture. This is about an extension that was proposed by Herbert Simon. Let me go here and first use this. Herbert Simon, as I said, there are only three to four pictures in this course about famous economists. Herbert Simon is one of these. He was a very inspiring person because he did not only work in economics but also in artificial intelligence and many other fields. He did important contributions to different fields. You will probably notice that he got the Nobel Prize for pioneering research in decision-making processes in economics. That is something we are not talking about. I mentioned it that you don't think he got the Nobel Prize for the Simon model of firm growth. That is not true. He also made important contribution to artificial intelligence which was related to his research in decision-making. That is an important statement of him. A wealth of information creates a poverty of attention. At that time, though, people did not really think about this. In the times of the internet, this is a very famous quote. You have so much to look at that it is the most difficult task. How do you get people's attention for what you have to say? This course is just one example of it. That is Herbert Simon and he proposed an extension to the G. Brat model together with a mathematician, Jules, who has done some more fundamental work before and has introduced another distribution. Here, again, we have the criticisms that we have discussed before. What Simon looked at, particularly, was the constant number of firms. That is something he didn't like. He said, okay, let's assume we have an entry rate. How does this change distribution? Even if we assume that the underlying dynamics follows the G. Brat dynamics, but we have constantly entering new firms into the system, what's the impact of this? He proposed a different distribution, which is then called the Jules-Simon distribution here. I'll come to this in a moment. That's the result of his investigations. That's just an overview slide. How does the Jules-Simon distribution look like? That's a distribution I have just shown with the gamma function. If you have an original scale, not scale, it looks like a very skew distribution. If you go here, then you see in the log-log plot, you dependent on how you tune this parameter row, which is a pre-factor here of this distribution. Let me go back to this here. That's the row, but also this argument here. Dependent, you can get everything between a power law and something that's a bit more bent. It's a different kind of a skew distribution. That's not really a log-normal distribution. It's not really a power law, but dependent on how you choose the row, you can end up indifferent of these cases. That was his idea. We already discussed this. How did he get this? Natalia has prepared a few nice slides for you to follow this. I can also recommend that you read the paper, but this is really a mathematical paper. Simon was very formal in deriving this distribution. He assumed that for each of the industries, there is a minimum size for a firm to enter the market. This is different if you talk about insurance industry as compared to construction industry. We already discussed this, but there is an assumption. There is an X-min. You cannot start with one or with zero. You have to have like five or ten or something. Then Simon assumed if the firm has entered the market, the G-brot process applies. It's very important. It grows proportional to the size. He further assumed that this growth rate is independent of the size class. That means all firms have a growth rate drawn from the same distribution. Not small firms have larger or smaller growth rates, and larger firms have smaller growth rates, not at all. This is independent of the size class. The result of this process, as you already know from G-brot, is a log-normal distribution. This applies for firms that already exist, but then the important point comes. Firms are able to enter the market. Simon assumed for this model that there is a constant entrance rate. Every time interval, there is a number of firms that are born in the smallest size class, as Simon formulated. That means with a starting value of X-min, they enter the market and then they follow the G-brot dynamics. The question was what is the outcome of this process? The outcome is the dual distribution instead of the log-normal. Why? Because we have a new process here, namely the entrance rate of the market. In the following slides, we look into this a bit more into detail. For those of you who still stay with me here, I assume that you are particularly interested in the details of this model now. Therefore, we go into the details. For every size class, Simon defines this cumulative number of firms in the size class, that is Q, and N is the total market size at time G. The assumption is that there is a constant entrance rate, so N can be then used also as a substitute for the time. If you have a constant entrance rate and the total size of the market grows at a constant rate, then N is nothing but a time. This is the important variable. Then, as I said, if the firm already exists, it grows proportional to the G-brot law. X times the number of firms in this particular size class. Then the second assumption is that there is a probability, P0, that new firms enter the market. That's a small number. It shouldn't be one. It should be a small number compared to one, but it's not negligible. Then, as we already said, market growth in each time step, the total market size grows by one unit. That means the total is N. Therefore, I said N is the proxy for the time. At each time step, you enter one unit. Now, you have two chances or two choices. You can put it on this new unit on an existing firm, or you create a new firm. These are the two choices. Therefore, you introduce a probability. That means with the probability of P0, it goes into a new firm, and with the probability 1 minus P0, it goes into a firm that already exists. If you have to choose in which firm you have to put this growth unit, then of course, it's proportional to the size of the firm. That means the larger you are, the more likely it is that you, as a larger firm, get this one growth unit that's distributed at this particular time point. Is this crystal clear to everyone? That's the underlying model. That's very important. Okay. Well, Natalia made it very nice. All those she made, these very nice draws here that describe what will happen to this. So, the underlying assumption for the growth of firms is what we also call preferential attachment. This is a term that is mostly used to get that with complex networks. Who's taking this complex network? Our economic networks course? Okay, so then you know what preferential attachment is. But already Herbert Simon and of course, D. Brad used this preferential attachment idea. I remember there was part of a discussion, but then D. D. Sonnet asked Laszlo Barabasi what's really new on the preferential attachment, because already Simon and D. Brad discussed this. Then Barabasi thought for a moment and he said, well, that's true, but they didn't do it for networks. That's all the talk, right? There was, they didn't look into what's the impact on the topology of interaction, for example. That's another thing. They applied it to the size to a scalar variable. Yeah, preferential attachment. Yes, if you want to say so, yes. Okay, so let me go here. We have now to write down two equations. One is for the growth of those firms that are already existing. And the other one is for the creation of new firms, right? And as you probably recall from what I just said. Oh, where are we here? Yeah, here. So from what I just said, so P0 is the probability of having a new firm, right? So that means the equation for entering is proportional to the P0 here. And this is for the growth, which is determined by growth of existing firms, which is determined by 1 minus P. That's this one. So what are these two terms here? You see what you have on this side here, Q of x and n and t plus 1 and Q of x and n. So this is basically dQ after dt, right? So because n is the time. Every time, instance and time, you add a new growth. And that's that time n plus 1 and that's that time n. So basically you measure the difference of the Q, which is the number of firms in this given size class x over time, right? When you move to the next time step, clear? That's what's written here. And these are the two processes that are involved. What can happen? It can happen that a firm that has a size that's in size class x minus 1 is then chosen and gets this one growth unit that is distributed in this particular time interval and then moves from size class x minus 1 to size class x, right? That's a process that's described here. Or it can happen that we choose a firm of size n, sorry, of size x at time n and give this, give the one growth unit to this firm. And then this firm moves from size class x to size class x plus 1, so which means it moves away from size class x. Therefore, we have a minus here. This decreases all the number of firms that are in size class x and this increases all the number of firms that are in size class x, clear? Yeah, very clear. That's called a rate equation. It always looks like this. So this is the difference equation for creating a new firm. And then the assumption is that Simon argues that we then reach a quasi stationary state. That is usually what you do because then you can solve this numerically. Then you see how the distribution grows but Simon was not interested in how the distribution grows. He was interested in how the distribution looks like if you reach equilibrium. Therefore, he made the equilibrium assumption here. And the equilibrium assumption is that q of x at a given time n plus 1 is almost the same as q of x at a given time n. It doesn't change except for this very tiny difference n plus 1 divided by n. That's the stationary state assumption, right? And then he can solve this equation. And this is what he gets. This is the distribution for this stationary state. Remember what the stationary state, how it was defined, was written in the previous slide. And here you see the gamma function emerging. I'm not sure if everyone knows what the gamma function is. Who knows what the gamma function is? Okay, one. It's not so important, right? The gamma function... What if we print this here in the... I think we printed it in the notes somewhere. Let me just... Yes. When I introduced this the first on slide number 17, when I introduced this yield distribution the first time, then I wrote in the notes what the gamma function is. The gamma function is related to the beta function and that's a well-defined mathematical function that is listed in a table, right? Or you get it in your computer, right? You just define the argument and then you get a value, right? That means it's not so important to understand this. But you see the gamma function of x times the gamma function of rho. And what's the rho? The rho is a very important parameter. It is simply this inverse of the probability that the attachment was made to an existing firm. Remember p0 was a probability that it's given to a new firm. And 1 minus p0 is the probability that it's given to an existing firm and this is 1 over... That means this is a number that is between 1, that's the lowest value and infinity. And then we can rewrite our distribution for the stationary state like this here. So here there is a beta function already involved. And let me see what we get on the... Yeah. So this is how it looks like basically. I'm not asking the derivation or these kind of things in the exam. I'm also not asking to write you down the yield assignment distribution. That is not important. But you should understand that it looks a bit more complicated but it's well defined. The gamma function appears like what you see here. And the most important parameter, that is what you have to recall, is the rho. And the rho is related to this probability of giving something to the new firm. And here we discuss how the yield distribution depends on the rho. What Simon did and what is also the most logical application of it. He said, let me fit the rho to the data. So there is a second remark that I would like to make. So I think I had a slide before. Oh yeah. This is the slide. Okay. I don't know why it is here. So this is what I wanted to tell you. So this is the yield assignment distribution. You can proxy it by the gamma function as an argument of rho divided by x to the rho plus one. Sorry for this. So this was the yield assignment distribution before rho plus one, gamma of x, gamma of rho plus one, gamma of x plus rho plus one. And then you can separate this with reasonable mathematical arguments into something like this. If you want to remember something, then you should remember this. And it's important to recognize that for large x this converges into a power law where the rho appears as the exponent. This is a relation to all the other things that we have discussed about size distributions before, right? But you also recognize that if you have if you go for large x you cannot distinguish this finding from a power law. So you cannot confirm that Simon was right. If you want to confirm that Simon is right, then you have to go for small firms, right? Where you notice a difference between the power law and the yield distribution. But that's the most important thing on this slide. You see it? That becomes a constant here and here the rho appears in the exponent. So what Simon did now is he fitted it against data. He said, okay, the rho represents this probability of giving something to a new firm instead of to a growth. And now let me proxy this P0 by the by the net growth of small firms rather than net growth of established firms. That's basically the meaning of the P0, right? If everything gets into established firms then this goes to zero of course and the rho is one and then everything gets into established into new firms. So then this increases not too much of course because the growth of new firms is not the dominating process. That's all the important. And then you can express it as this ratio here. Then he went and checked this for data and he found that the empirical value of this ratio small g versus large capital G is about 10%. The meaning is about 10% of the growth goes into new firms. And this if you calculate this means a rho is about 1.1. And then he checked this against data from the UK and from the US. And he found out that this rho is the 1.1 for the UK and 1.2 for the US. And of course the difference is important. So this difference means that about 20% of the growth goes into the new firms in the US where us only 10% go into the growth of new firms in the UK. That is what he found out. You understood this, right? So he had data available of the growth of the economy and then he marked all these firms that are newly born doing this particular time interval and looked into their growth. And he also marked all these firms that were already there and looked into their growth in this time interval. This is how he proxied this ratio small g versus large g. And then he calculated the rho and compared the distribution. I'm not sure if we have this here. Okay, no. So let me come to the last discussion here. No. There was another data set that Simon was one of his collaborators, Bononi, then used later in 1958 to compare this theory with empirics. And he found evidence that the yield-simon distribution can be found in this data set as well. You should notice this one here. He found the evidence in particular for large firms. For small firm sizes, it looks a bit more like the log normal distribution. That's something interesting. Here it looked like a power law. If you remember the best available studies that I also presented some time ago that was by Axel who looked into the census data, what did you find? You remember this? Right. But in Axel's plot, this was always a power law, even for small firms. Remember, Amaral and Coworkers found this nice log normal distribution and Axel found a power law. Then we argued why did Axel find a power law and the other one a log normal distribution. The issue was that Axel also used the census data which had a focus on smaller firms. If you have smaller firms available, better data about smaller firms available, then you see a power law. And Simon noticed that for small sizes, he only finds a log normal distribution. I mean, he didn't say only. He said he finds a log normal distribution. And why is this the case? It's also written on the slide because he only used data of the largest 500 U.S. companies. That's the answer for this. Again, if you have the small firm sizes underrepresented in your data set, then you have a log normal distribution and you can firm this with the data. That's also what Amaral did. But if you have a very good representation of the small firm sizes, then you see the power law. But you always see the power law in the tail for large sizes, which is not a big surprise. Okay. So that means Simon concluded that there is evidence that G. Brat's assumptions hold it. Because, remember, why can he claim this? Because he was assuming that the firms in each established size class grow according to G. Brat. And then after he found this nicely meets the dual Simon distribution, which he by the way only called the dual distribution, then he indirectly concluded that G. Brat was right. Okay. Let me now come to the last point I would like to make. It's about correlations. So what we did so far is we assumed that every growth process is independent from every other growth process. That was the assumption of G. Brat. He drew this from the same distribution. We already know from the empiric that this is not correct, right? Because we know that firms, if they are smaller, face a larger variance in their growth rates than larger firms. Okay. So there is room for improvements. You can build in correlations in size, but you can also build in correlations in time. If you had a 10 percent growth rate this year, why should you have a 10 percent growth rate next year, right? So that means we can assume that growth rates are somehow correlated either to myself, to my own growth, or to my own size, or as I mentioned before, to the growth of the industry or the performance of the economy as a whole, right? So that's the assumption here. Okay. Again, Idiri and Simon were the ones who tried to build in these correlations here. And they used this kind of assumption. They said, okay, the growth of this, no, the size of the company at time t is correlated with the size of the company at t minus one. And they included these two factors. This is the growth factor, of course, because it matches x at time t minus one to x at time t. That's a multiplicative stochastic process. But they assumed that in addition to this number, they built in also an average performance. You see how it is defined here? This is the total size of the industry in the two different years. If we have a positive climate in the economy and the economy is growing, then of course, you see that this London is larger than one. So that means there is a positive drift in you growing in the next time. This can be still mediated by this random factor, but there is a bias in this. That's the assumption here. And if the economy is declining, we would see it here at this point. Then the London is less than one. That's the first assumption. And here for this random process, they assumed an additional assumption. The epsilon is again our random variable. But here, there is a memory effect. That means there is a correlation between the growth at this year and the growth at last year. And the alpha decides on how much you are affected by the growth of last year. Remember, G-brot assumed that you just can change every time. Minus 5% plus 10% just can be mixed in a random manner. So where G-brot and Simon and Jerry assumed that there is a memory effect in this. If you were well performing the last year, there is a certain inertia that affects you all this year. You may not completely drop down. That's the assumption. So by this assumption, they fitted this growth dynamics to the 96th largest American firm. You see the dataset gets worse and worse, right? Before it was 500 and now it's only 96. But nevertheless, they had this available for two different time periods. And they found that there is a memory effect. The alpha is 0.35 and the average lambda is 1.27. That means there is, in this year, there was an overall growth of the American industry, which of course affects the individual dynamics in terms of a correlation. And here is a nice interpretation what these numbers mean. So if a firm doubled its market share in the first four year period, it can expect to increase the market share by another 28% in the second year period. This refers to the meaning of the alpha because the alpha is like this. That's how they calculated it. That's an interesting assumption in extending the deep ride model. Then you cannot, of course, go and assume further correlations. You can, for example, write down your growth rate like this. Here you have a coupling to the economy as a whole. Here you have a coupling to the industry. You are in industry J. And here, this is the growth rate that we proxied by the random number. That's your performance relative to all the others. That means relative to the industry. And then you can try to fit this with macroeconomic data, for example, and so on. That means there is room for improvement. What should you remember about this extension of Ediri and Simon? It's not so bad if you remember the two names also to make a difference. You should remember in particular that there are two effects. One is the correlation with the industry or with the economy as a whole, dependent on how you define this here. Let's assume that's the industry. And there is, secondly, this memory effect. These are the two extensions. The memory effect and the coupling towards the business cycle, if you want to. That's important. And this was verified by the data here. Very last point. Now is this one. You can go and, do I still have time? Like five minutes, yes. We can now go and extend the GWAP dynamics by this additional term here, which we call A on the slide, which we discussed already before. A could be a constant or random but positive contribution towards your growth. This is your own wealth dynamics and this is probably the amount of money that your parents send you every month, independent of your grades for example. Every month you get something to sustain your living and that's the A basically. We can assume that the A is constant. We can also assume that the A is a random variable. The only assumption that is crucial is it has to be positive. Why is this important that it has to be positive? Because that is what I call on the slide an effective repulsion from zero. If I do not have the A at all, then you probably remember that I have a good probability to die out as a trajectory. You made the simulations of the log normal process. Hopefully you did this already. Okay, that's the same. Then you see that some of these trajectories really die out. The A prevents you from this. Whenever you are about to die, the A pushes you back into the system. Then you get it maybe next week. This is the meaning of the A. It always pushes you back into the system. The underlying assumption is then the number of firms is always constant because no firm is allowed to die. Whenever you are allowed to die, then the A tells you to stay in the market. That's the idea. That's what the A is doing. We can interpret this B in a different way, namely in terms of an investment process. We can say, well, every instance and time, the firm invests some proportion of its wealth or of its size into further growth, like 10%. That means the Q is 0.1. Then the market gives it a return on this investment. It's either growing or shrinking. That means the R can be minus one or plus one. If it is minus one, then it's probably quite bad because then you lose your investment. And with plus one, then it's very good because you can increase it. If we make this assumption, then what is the most probable value of you as a firm if you follow this kind of dynamics? You have a small constant investment, and then you invest some proportion of your size into the market and you get a return. We can solve this analytically. You should not be scared about the slide. The slide is very simple. It just tells you a solution, not even a derivation. So if we write down our algebra dynamics as something like this in a discrete manner, delta x is eta. Eta is a random variable time, something that depends on x plus something that depends on x. We can write down our dynamics from the previous slide as something like this. And then if we know this, then we can already predict or know the stationary probability distribution. It looks like this. That's known in mathematics. And now we go back to our dynamics of the G-brad. So what is the F? The F is simply the A, as I said, and the A is a small constant. So what is the G? If you think of the G-brad dynamics, what's written here? The G is simply x. But I wrote this to tell you that there are ways to guess the stationary distribution even if you have more complex dependencies. And this means if I insert this here, I get a stationary probability distribution that looks like this. So maybe before I plot it here, you see if I go for large x, then I have a power law. So it's plotted here on this slide. For large x, it looks like a power law. But for small x, it looks like an exponential function. And there is a most probable value here. The most probable value is defined as the one where the probability distribution has a maximum. This would not hold for the power law, for example. In the power law, you do not have a most probable value. So it can be anything. In the log-normal distribution, you do. But in the log-normal distribution, you do not have this power law here. It's a bit more bent because you don't have the A. The A pushes your growth. And now we can simply calculate for different assumptions of the R how this most probable value is affected. Remember what the R was. The R is your return on investment. It's basically the random number that the market has chosen for you after you invested the money. And here we have different assumptions about the distribution of the R. So this means it's either minus one or plus one. So this means it's a uniform distribution between minus one and plus one. And this means it's a Gaussian distribution that is centered around zero and has a small variance here. And then you can calculate the variance of the R from these assumptions. And this is what you then get here. Let me go back. So that means you can predict your most probable value from these assumptions. Now let's try to interpret this result on the last slide. Okay, just for one second here. You see the Q zero was the amount that you invest. Unfortunately, it appears in the denominator. What's the meaning of this? The more you invest, the less will be your most probable value. It means you can end up here, but the most probable value will shift into this direction. So from this model, you would deduce as an investment strategy, the best thing is to not invest anything because you know that these log-normal distributions, there is an underlying assumption that the trajectories can die out. So the best thing is you always take the A. You accumulate this nicely. You do not invest back into the system. And then your most probable value is proportional to the A and will steadily and becomes constant. But if you start to invest something, then of course you will likely lose more than what you gain. Of course, there is then this extreme probability of rare events that you can grow up to very high values. Okay, here I have described this. It's basically the investment dilemma that you basically get out from this. This is of course counter-intuitive and it tells us that this model is not correct because if you invest something and you take the risk, then it should be visible in your time. And we will discuss this when we talk about more refined models of growth and investment in lecture eight, I think. That's lecture one next time. Or maybe the week after when we talk about this. This is to conclude here an example where we use this as a construction kit. We took the G-bar dynamics and then we ask ourselves, okay, what happens if we have an additional term here? That's the A, or f of x. And then we see it's nice. We can get a power law behavior for large x, which is also confirmed by empirics. At the same time, the underlying dynamics does not fully capture what we expect from the economics. This gives us a further hint of where to improve our model, such as how you have to see it here. That means at each time we have the chance to evaluate our improvement of the model against either empirical facts or these theoretical considerations. This is only the self-study talks and these are the questions of today. With this, I close here and I wish you a very good Easter break.