 So, good morning everyone. Welcome to lecture number eight of our course Collective Dynamics of Firms. I'm happy to see so many of you here, so we reached the critical size of five students. I brought a spare student here, just in case that we don't reach it, but I'm happy to see him. Okay, what would we do until now? We were talking about stylized fact that are the statistical regularities that we observe when looking into firm data. You recall all this distribution about firm size, firm growth. You recall the relation to the age of the firm and these kind of things, all about volatility and all these issues. And then we tried to understand this by providing models that are able to reproduce these stylized fact. That was a challenge for us. And we started with a very simple class of models, simply simple stochastic models. It means that because we do not know about the real growth function of a firm, we proxied it by a stochastic force, which sounds strange on the individual level, I agree. But on the aggregated or on the systemic level, we are able to reproduce some of the stylized fact. But not all of them, as you recall. Therefore, we added step by step more refined assumptions to these stochastic models in order to get to this more realistic and complete picture. And I'm not going to recall what we did last week. Particularly, we talked about correlations and growth rates and about entry and exit dynamics, this important contribution by Herbert Simon and so on. But there was one important issue missing in all of these models that we discussed so far, namely interaction of firms. We treated firms as independent entities and there was basically no correlation between the firm no interaction. And that's something we would like to do today and of course on the next week. Today we talk about competition as a basic form of interaction. And we even follow an economic theory here. That's the economic theory of free market capitalism. It was first invented in Marx's famous treatise, The Capital. I'm coming to this in a moment. If you ask me which lecture of the whole course I like most and this is certainly this one, it's a very unique lecture. It's not based on my own work, but you will not hear this again in any form somewhere else. So let me start by recalling our modeling framework. Is this okay with the light or should I dim it a bit? Remember that we are considering on the individual level and ensemble of firms. Each firm is characterized by a scalar value. That's the variable X which we dub as size and can proxy by different empirical variables. And the assumption of the dynamics is simply this and then we are left with the question what do we have to put into this function F in order to get a realistic equation? I do not discuss these issues here again because this was part of the last lectures. In particular we found already that one assumption named the law of proportionate growth was very close to the empirical observation. What does the law of proportionate growth mean? It simply means that the growth factor is proportional to the size of the firm. That is what we meant by this. But we also discussed very simple forms of indirection, namely indirection via the mean. There was a mean coupling here in the growth factors. This paper of Egeri and Simon, this was last week. And there was a little term considering a memory effect. This is all that we have discussed. If you grow last year, what is your chance that you grow this year? That was seen as a memory effect in growth or inertia if you want to use a physics term. And then you could see by comparing this assumption with empirical data what is the approximate value of the alpha. I discussed this last week. Today we go one step further by looking into a mean field coupling again, but this time related to the limited market size. The assumption is very simple. You are a firm and you grow completely independent, right? And then you grow. How do you recognize that there are other firms in the world if you do not interact with these firms? You recognize this by going to the market and realizing that no one is buying your products, right? And then you can indirectly deduce maybe there is another supplier, there is another manufacturer that produces goods of a better quality, of a better price, and therefore no one is interested in what I have to offer here, right? This is not a direct observation, it's an indirect observation. Basically, a medium which we call the market tells you indirectly about this. That's the assumption here. And the market has a finite size. It can be huge. That means for a long time you do not recognize that you have competitors, but at some point, in particular in the age of globalization, then you recognize that there are other countries and other firms which are very competitive compared to your own company. That's the idea today. How do we formalize this? For this lecture I have decided to tell you the same story three times. It means here you already have all the results, it's an important message, and then I tell you the story the second time by putting more dynamics into this, and then I tell you the story the third time by putting more economics into this. The assumption here is again the law of proportionate growth. You see proportionate to the size of the firm, and the growth factor is alpha here. In the deeper dynamics we proxied the alpha by a random variable, but here we keep the alpha as something that is associated with your firm. We later call this your fitness. And now we transform this equation into a relative equation where the relative variable is your market share. You have a size. Other firms also have a size. Then we calculate the total size of all firms and we assign you a relative variable. That's the role. Today we just talk about the role. The role is not the same as the X. You can be very huge as a firm, but you can still have a small market size, which means there are even bigger firms in the game. The second assumption here is that the growth rate, this alpha, should consist of two parts. One part which we dub here as a capital E is an expression of how well you are doing with respect to your production. It can be, as I mentioned here, a value of the quality of your goods. You are producing the best iPhone in the world. Therefore, you have a very high XEI. The second variable here, the second part of this growth rate is the K, which is negative. The negative part subtracts something from your growth rate. What is the reason for this? It could be dissipation. Decipation again is a physics term. It means you have some friction in your company and for every single growth event, you have to feed bureaucracy or administration or something like this, so some money is going away. You do not make a full profit out of your EI, of the quality of your goods, but you have to give something maybe in the form of taxes or you have to pay administrative overhead or these kind of things. Then we assume, as a third assumption, there is the conservation of the market. If we sum up all the market shares, these market shares sum up to one by definition. If we then look into the changes of the market shares for each individual firm, that's then, of course, they have to sum up zero. That means if you grow in your market share, then you probably have to lose, right? That's a very important consequence of this. By using this equation, if we plug this in with the alpha assumption, then we can find an expression for the K. This is what results formally for the K. The K is inherently related to the average value of the fitness. Now you understand what this means, so we can then rewrite this equation by going to the relative variable in such a form. This is a very known equation. It's called the Fisher-Eigen equation. Fisher suggested this equation in the 1930s, and Manfred Eigen recovered it in a different setting. In 1968 or 1970, with relation to prebiotic evolution, by the way. This is not what he got the Nobel Prize for. He discovered this after he got the Nobel Prize. Okay, what's the meaning of this? The meaning of this is also described on the next slide, but let us discuss it here. It means your market share growth as long as your fitness, the fitness of your product, the quality of your product is better than the average. Very intuitive, right? Then everyone wants to buy your product and not the product of your competitor because of the worst quality, right? But here you see that the average is constantly changing. The average fitness of all the goods depends on the market shares of the respective firms. You are a firm with a bad product, right? What will happen to your market share? Your market share decreases, right? That means the weight that you contribute to the average also decreases all the time. Therefore, the average is increasing. That means the EI is something that constantly increases over time. What does it mean? A T equals 10 years, so there are five of you who are above the average and five that are below the average. Because the influence of those who are below the average is constantly decreasing, so the mean value is constantly increasing and then it's not guaranteed that all the five of you are still above the average at 10 T equals 20, right? That's the consequence of this. That means there is a high selection pressure on you to be the best because only the best eventually will win. Therefore, this is called the winner takes all dynamics. The end is very clear. Everyone is below the average except one firm and this firm has a market share of one. This is what I have just described. The outcome is also known as the survival of the fittest. What's wrong with this? As I said, it's a very well-known dynamic that exists in all sorts of dynamics also in social dynamics and biological dynamics and so on. The problems, some of the problems are listed here already. The first problem is the problem of the constant EI. You do not need to integrate this dynamic, right? You just look into the distribution of the EI and then you can predict from the outside who is the winner, right? That means you as a firm would never enter this competition process if your EI is not the best, right? Because then you can already tell after what time you lose all your market share, right? Okay. But in fact, the EI is probably not a constant. The EI is something that has its own dynamics, but that's not considered in this particular equation here, right? If you think of the Nokia case, yeah? Nokia for a couple of years was the fittest in the market, right? It has its largest EI. And now the EI has declined and other competitors have a better fitness with the quality of the product, right? So that means the realistic assumption is to consider something for this, but we do not do this here today. The second thing is what's the economic meaning of EI, of this fitness value, right? Economists would probably criticize us at this point for biologism, right? So that's as bad as physicist, right? Physicist means we map everything to physics, right? But biologism is as bad. We map everything to biology. Economics is not fully explained by using biological insights, right? So we have to find an economic meaning for the EI. Fitness is a very abstract word that doesn't mean anything, right? The third problem which we address in the next lecture is about the outcome. Is the outcome realistic? What's your impression? Right, right. It's very unrealistic, yeah? Except with the Campbell soup for some time. And there are markets that are dominated by one firm, so this shaving company still yet, so that basically has the power to rule the whole market, right? So they could easily increase their market share to 100%. So they have 75 or something like this. So there are a few rare examples where you could assume that it goes into this direction, but looking into the general picture, it's not realistic that we have a winner takes also now. I mean, you as a firm, you would never play this game. You would say, okay, if I cannot compete with the market leader, which I cannot, then I have to create a niche, right? That's what I have to do. So that means I expand the state space, so to say, by finding my own way of producing a niche product that the market leader does not have yet. That's the idea, I mean, in a realistic scenario. Here we assume that we are all on the same surface here. That means we can all compete. There are no niches. The other question, or the last one, is the outcome desirable. In the first instance, you would say, yes, of course. If I'm the winner, yes. I have all of the market share. Great thing. That's wrong, right? Because when you kill your competitors, you also kill sources of innovation. That's an important issue. And therefore, to take the case of Juliet, Juliet supported its competitors, like Wilkinson, and there is Schick, and there are a few others. They supported them. They didn't buy them, even if they could. They supported them to stay in the market. Because they learned that a competition of, like, let's say, two or three firms is better for the whole market, for the innovation dynamics, for everything, than just having deal yet, thinking what can we do next, right? That's another thing here. Today we are addressing, number two, the economic meaning of the fitness. Next lecture, we discuss the realistic outcome where we see a skewed distribution again of the market share, and we try to understand what are the models that we produce this market share distribution. So the last question is not discussed here in this lecture, but if you go and have lectures about innovation and management and these topics, then you certainly hear more about this, what is the important relation between you and your competitors. Okay, so in order to get to the economic meaning of fitness, I'd like to refer to a particular economic theory. This is not, as in most cases, there is no consensus in economics about interpretation of these or that. Therefore, I refer to a very specific economic theory, and that's the economic theory that was laid out in Karl Marx's capital in 1867. And I think it's also useful to just discuss a few ideas behind this very famous and very big treatise. What was the idea of Marx to write this? You think a foundation of communism or something like this is completely wrong. The aim of Marx, as it was also stated in the book, is to explain the objective laws of motion of the capitalist system, to reveal the causes and the dynamics of the accumulation of capital, the growth of wage labour, the concentration of capital, competition, the tendency of the rate of profit to decline and so on. So in the first sentence, you already recognize something that sounds a bit like physics, right? The laws of motion of the capitalism system. And indeed, this is correct. So in those days, physics was seen as a paragon of all sciences. If you want to be a good social scientist, then you write up the not existing, at that time, sociology in a way like physics has set up its way of explanation. That was done by Idris Comte. He invented socio-physics because he felt sociology needs a quantitative basis. And the paragon for this quantitative basis was physics. Therefore he named it socio-physics in 1860 something, at about the same time. And Marx had the same idea. He thought, okay, if there is any rational behavior or meaning in the whole economic system, then we might be able to explain this in the same strict way as physics is doing. And in order to derive this theory, he then invented the first materialistic view of the capitalistic economy. So remember that this was quite new at that time, the capitalistic economy. But he still, of course, drew some inspiration and influence from other famous economists like Adam Smith or David Ricardo and John Stuart Mill. But he also, because he had a degree in philosophy, he also adopted the methodology of the German philosopher Hegel. Hegel wrote a very important treatise, the phenomenology of mind, where he developed dialectics. Dialectics is a specific way of arguing about the evolution of things. And Marx adopted this for his work, The Capital. In the following, we see lots of equations. So Marx never wrote any equation in these books. So these equations were proposed by other people, which I will mention afterwards. You should underline the last sentence because the conclusions that Marx drew are true for the capitalism of his time, which was the free market capitalism. So you cannot easily transfer those insights into the capitalism as we have it today. There are differences, of course. But nevertheless, the free market capitalism is, first of all, an important step in the development of capitalism. Secondly, it is the case that we are able to quantify as well in a very nice manner. So, okay. This again. So here, as I mentioned, only a few people are presented here with a picture. Marx is one of these. I'm not going to read this all for you. The important thing that you might want to recognize is in the last sentence, in order to write The Capital, which is really a book of that size. It has more than a thousand pages. He spent 12 years in the British Library, sitting there to read all the literature of the economists of his time and of previous times. He was well educated about the subject of this book. That's the first thing. And as I said, the aim was not to derive in ethical treaties or something like this. This book is not about moral and you should not exploit other people and these kind of things. Not at all. Marx said, okay, I want to be as objective as possible in order to understand the laws of motion. I'm not going to judge the laws of motion. I want to find them. That's important. And of course, the second remark here is regarding the communist manifesto. I mean, most people only know him for this. They completely ignore that he was a famous economist or he still is a famous economist. By the way, influence is growing now after a decline over the last, let's say, 10 years or so. So then it was not very fashionable to talk about Marx. But now, at least in the US, I recognize that many people get another new interest in Marx writing here. So, now let's look into what he proposes in the book. I mentioned in the note that this is a very abridged version of Marx's theory. I'm going to dance it really to the nutshell, right? Which means I simplify things, I leave out things. I'm not going to start a discussion here with you about this. It is simply to give you an idea about his arguments, right? I do not present all of his arguments. I just give you a line of reasoning. So, the main insight of Marx's wars and still is the driving force of capitalism is the exploitation of labor. As I said, this is not an ethical judgment of capitalism. This is a fact, right? That's something different. You can think whether this is good or bad. That's your own problem, right? But Marx was simply stating this as an insight, as a fact. And there's nothing wrong with this. What does he mean by this? There is an employer and there are workers. And the employer hired these workers and pays them a wage. How does he know what is a wage for the worker? This is said by the market. The employer finds out about the market value of labor and then pays these people. If he would not pay them the market value of the labor, then they would probably go somewhere else. That's the underlying assumption here. There's no idea of unemployment or these kind of things. Then these workers are in the factory and they produce some goods. And the second important issue here is that the goods or the commodities that they produce have a value that is larger than what the employer has paid them as a wage, which is absolutely satisfactory. Because you can argue yourself, why should the value of the good be higher than the market value of the labor? What's the idea behind this? No, not to make a profit. The profit is just a result. Of course, he takes the risk. He built a factory, he bought all the machines. Should he do this for nothing? Of course not. Therefore, this is a very clear and natural assumption that the market value of the good has to be larger than what he pays to the workers. But the question is how much larger does it have to be? That's the next question that we are going to answer here. There is a profit at the end, hopefully, for the producer or for the manufacturer because, of course, he made the whole thing happen. There should be at the end something for him. And as you recall, although the accumulation of capital to start this enterprise, this is an effort by itself. Most people just assume that it is very easy to put the money together. And therefore, like the German Health Minister, they discuss after they have these 5 billion euros that they give everyone 10 euro back. By arguing in this complete stupid way, you underestimate the effort that it took to get from everyone 10 euro to make it 5 billion. You don't get the same thing back. The value is in getting this together to have the equivalence of the two sums or something. So what's a commodity? That's what they produce. It's a tradeable good. Something that you can sell to others. That's very important. It can be all the service. And the commodity is called then the building unit of capitalists. That means a commodity always has a value for someone else. That is the underlying assumption for you to buy this, right? It has to have a value. And therefore, you can trade it. That's the important thing. If it is something you can only use for yourself, then it's not a commodity. So, and then Marx argues about what's called the metaphoosis of commodities. And there is a so-called commodity cycle involved money, commodity, money. And here you see you can split this into two cycles, money, commodity, and commodity, money. So, commodity, money is a sales act. You have a commodity, you sell this, and then you get some money, right? And the other money, commodity part of the cycle is a purchase act. You get some money and you go and buy some commodity, right? And then you can look into this commodity cycle in two ways. You can have CMC. That means you sell a commodity to get some money to buy something else, another commodity. That means sell in order to buy. Or you can have this kind of cycle, MCM. That means you buy some commodity in order to sell it somewhere else. So, you have to recognize the difference between these two ways of putting up the cycle. Here the money or the capital is absorbed. It's in the commodity, right? That means it's no longer available. Let's assume you bought a house, right? Then the money is put into the house. It takes the money and buys something else. And here, this is the important way, MCM, because you buy a commodity in order to sell it somewhere else. And this is the only important thing for the foundation of capitalism, because it generates capital. We will see this on the next slide again. It's important to notice that the exchange value of the commodity is the most important thing here, not the user value. It's not important what this commodity means to you. Let's assume this oil canvas of your grandmother or something like this. It's important what this commodity means to others. The exchange value. That's the first important insight. You can completely neglect the user side of this. And the other important insight is you have to circulate the money in order to make some profit. If you got stuck in this part of the cycle, then your money is absorbed in all these goods. You have to sell this in order to regain your money to buy and sell other commodities. Now comes the second important step. After this commodity cycle, Mark says, well, the simple circulation of money will not do it. That cannot be the foundation of capitalism. Of course not. You buy the commodity in order to sell it to someone else for a higher price. Why would you do it otherwise? Every trading company works like this. They buy cheap commodities somewhere and sell this to you to a higher price. That's the idea. That means the M after you have sold the commodity is not the M of before, but there is a surplus here which Mark calls a profit. And then we have to understand where this M comes from. There are differences in the value of the good. So basically what the capitalist is exploiting is a gradient in these values. For you, this commodity has a higher value than for you, for example. Why he can charge a higher price. So here I think we can skip this in order to go to the more fundamental things. So now, after we understood these two things, the commodity cycle and the generation of the surplus value, we try to formalize the whole thing. What do we mean by formalizing? We mean that we write down equations. Other people think of other ways of formalizing. They think they're formalizing if they really clearly tell you about it. We mean we write down equations. And therefore let's assume a modeling framework which is already known to you. We have a number of firms which produce something. The assumption here for the modeling is that they produce exactly the same good. And they sell this on the same market. This is important if they would produce different goods so then we would not really argue about competition. And if they sell it on different markets we can also not argue about competition. Therefore the assumption is one good and one market. So they produce a quantity per time unit. That's delta x here. That is the number of goods that they produce in a given time interval. And every firm has to spend some effort on producing this good. That's the capital omega. You can measure the effort in terms of the time spent to produce the good. That means one over omega is a measure of the efficiency. That's very important. So the omega is something you should recapture later. And then we write down a production function. So in other courses you have heard about production functions and they are named as y. So in most cases, therefore we also name it y. We understand at the same time that the y is not something constant but basically the abbreviation of a time derivative. Namely the number of goods produced per time unit. That's a very important thing. So basically what we call production is in physics terms a velocity. Quantity per time unit. Velocity of production if you want so. And then Marx, not Marx, so in we and our formal model we come to Marx later. Then, no, no, Marx suggested one important law in his book, The Capital, that's the law of value. What do we mean by this? This is a fundamental thing that I come to several times but I want you to understand. You as a capitalist you have produced the good. You spend a lot of effort into this. You paid the workers, you rented the machine and so on. This is all expressed in the omega. That's the value or the effort that you spend on this. But this doesn't tell you how much the good is worth. You can only know about the value of the good after you go to the market and then people tell you, very nice, I'll give you 10 francs for it. This is completely different from the value that you have spent into this. You understand this, right? There are always these two sides as your side and your effort and there is another side where the market tells you what the market thinks your effort is worth. That's a complete different thing. And Marx pointed out this law of value and said, okay, well, because there is one good, one commodity and there is one market, then at the end of the day all goods are sold and they are sold for the price P. That means all the produced goods times the effort or the money that you have spent on producing this equals the total amount of these goods sold at the very day times the price. That is the law of value. Everything that you brought to the market will be sold at the end of the day. So at a unique price, that's a P and then everyone gets a P, the price for the product, but of course has spent something differently to this, namely the omega i. That's a different thing. Okay, and if you rewrite this and you see what the P means, the P is nothing but a measure of the average effort you have spent on producing the good, which is a very reasonable conclusion, right? What else should the price be? The price measures the average effort spent on producing a unit of these commodities. Okay, and we know about the P only thanks to the market. The market sets the price. So now we have to come back on these two ingredients that I mentioned before. What were the two ingredients? The first one was there's a positive feedback, right? Alpha times x. And the second one there was the conservation law. So now let's plug in these two ingredients in getting this competition equation. Positive feedback. Afterwards we do conservation law. Oh no, we already did conservation law here, right? So now we come to the positive feedback. So now at the end of the day, you're getting home. So you have produced a number of goods, which was delta yi, means your goods. So the formalization is always on this differential thing. There was a deep meaning behind this, which I cannot really explain to you. It makes a lot of sense to do it like this. You'd simply accept it. And now you have produced a number of goods and for each of these you've got a price p. That's what you bring home. And then you have to pay your workers. But you also have to pay for the rent of your machinery, for the rent of the building, taxes, all these sorts of things. That means you spent your p times delta y on paying the production cost. And then hopefully something is remaining and that's in your profit. So if nothing is remaining, then you should think about your business, right? Okay. So these production costs, which we call Kappa here, have two parts, one part for the labor and another part for the machinery. Or as it is called in Marx's treatise, variable capital, that's for the labor and constant capital, that's for the machinery. So that means I can then express, so that's a p here, yeah, into these two parts. So that's what I have to pay and that's what is remaining as my profit. Or I can rewrite it like this. And now the assumption is the following. So part of my profit, I reinvest into my production. That is the positive feedback on growth. I got 100 francs left and then I spent 90 francs in expanding my production and saying 10 francs for my own consumption. That's the idea. I could choose any value. I can also spend 100% on nothing. But this is the idea. So that means you extend, let me just finish with this line. You extend your production by a share alpha of the profit. That's your investment. And if you plug this in, then you have an equation like this. So we continue with this right after the break, yeah? Okay, let us continue here. So we started with, or we ended with this important assumption here, positive feedback on growth. The capitalist gets some profit and he invests a part of this profit, namely a share alpha i into extending his production. That's the important assumption here and this leads us to this equation then. As you already see here, there is this law of proportionate growth involved, right? You see, dy after dt is proportional to y. And now let us go from this equation here to the relative market shares. I give you this hint here because you have to recognize that the y is an implicit function of time already, right? dx after dt. Therefore, it's a non-trivial way of doing the derivation and then we end up with this equation here for the market share. And now we assume that all the capitalists invest the same amount, the same ratio of their profit, let's say 50% into extending their production. Then the alpha i is equal to alpha and then this term disappears and the alpha gets into here and then we have our selection equation back with a noticeable difference. If you go back to this Eigenfischer equation, then what's the difference? The sign is not really... Yeah, the sign, good, okay, yeah. Before we named the fitness value ei and no one knew what it was, right? And here now we see what it is, namely it refers to the cost of production, right? That means by doing this exercise, we, to some extent, revealed what the cost price is at a very abstract level. So it's still a variable, right? No one knows what it is, but we have identified what plays the role of the fitness in our selection equation when we go to production and market share and that's the cost price here. But now we would like to know what the cost price is, right? That's the next step. But before let me summarize this again, it's in this slide. What we have derived by this is a competition scenario which only holds for free market capitalism and you probably remember some of the limitations of this derivation, only one good, one market and so on, right? So we have identified the cost price as the important variable that plays the role of the fitness here. We understand, although, what is the role of the cost price here? Only if the cost price of an individual capitalist is below the average effort, which is the average omega or the P as we named it, then this can increase. What is it here? Only if your cost price is less than the P that you get from selling the goods on the market, then you can survive. That's an important thing. You also understand from this equation that you have to, in order to increase your competitiveness, that you have to look into what the cost price is and how the cost price is composed in order to find measures to be more competitive. We will discuss this afterwards. But you also see that there is this increasing selection pressure. Those who have a cost price that is higher than the average cost price or the average effort, those lose in market share. Their importance is decreasing and this puts a pressure on your cost price because the average K is increasing. That means, we can put it like this in this equation, you should be thankful to those in the market who perform that bad. Because they keep the average value basically high, whereas your cost price is much lower. If all these guys disappear from the market because they are not competitive anymore, then the cost price makes a jump, the average cost price, and you are much more effective. Okay. Now, let us come to the long-term, to the long perspective, to the long version. It's always the same story. We end up with the selection equation, but in each round we understand a bit more. In the first round we understood how the selection equation should look like. In the second round we understood what is the fitness, namely the cost price, and now we try to understand what is the cost price. We go now really deep into Mark's capital. I follow a formalization that was done by Rainer Feistel. That was a very brilliant physicist, and he did this as a side work, actually, for his thesis. I mentioned it here also. This was never published. There was only a manuscript written by a typewriter. It's very difficult to find. But I got hold of this, and now translated this to use the formalization here in this course. So we should be thankful to Rainer here. Okay. The aim of this course, of this long version here, is now to get more economic insight, but also to see how a complete verbal theory is formalized. This is a lucid point here, actually. Mark's wrote everything down, but this doesn't mean that you get it into equations easily, and this was done by Rainer Feistel. Okay. And so, here, there are a few restrictions to this derivative. Let us start from the same point again. We have a number of firms. All of these firms produce the same good So the production process is basically independent on seasonal factors. We are not talking about agriculture here and other things. And the production process is continuous in time, which allows us to write down the production of a commodity. Number of goods produced per time unit. That's called production. Why is the production function? And you recall whenever you come across production functions in macroeconomics, for example, you recall that this basically is a time derivative of something. Very important. Okay. And now we have one aim. We would like to understand how this production or this output depends on the input. What is the input, labor and capital as in macroeconomics? And usually you write down why is the function of L to the power of alpha times K to the power of beta, right? That's a usual cop-duckless production function. But what you have to have in mind behind this is that we want to get an explicit expression for this and that basically we talk about the velocity of production. So labor and capital are the input variables. Now, first thing is we scale this. I'm not expecting that you recover or recall all of these steps in the derivation, not at all. I provide this for you that you see how this is formalized. This is not part of the exam, for example. I want you to understand this because you'll also learn a lot about economics in some sense. That's the aim here. So we have to consider labor and capital. And the capital that we invest is spent on two different things, as we know, to pay the workers and to rent the machine. That means we have constant capital and we have variable capital. And now we express so there is this capital C that's a constant capital that you invest in a given time period, one year, usually. And the V is the variable capital that you invest in the same time period. But we are interested in the time derivatives. So we introduce the transfer of capital into machinery. The small C after DT is basically a process. A time-dependent process at which capital is transferred to the production. And this, of course, has to be equal to the constant capital invested over the whole time period. TC is so-called the turnaround time here. This is something that you learn in production theory. What's the turnaround time for machinery to accumulate to pay off and so on. And we also introduce for the variable capital this transfer variable, the small V. And this has to match the total number of the total capital transferred over the turnaround period. These are simply two auxiliary variables to describe the dynamic process. Right? And now built on these, we use an important quantity that's called organic composition of capital. It's basically the ratio between what you spend on your machinery and what you spend on the labor. That's called organic composition of capital. And we can distinguish between the static composition and the dynamic composition. In this derivation, we are interested in the dynamic composition. The DC after DT versus the DV after DT, that is defined as an organic composition. It's in this case one over beta. Beta will play a major role afterwards. You just have to understand what it express, namely the ratio of what you spend on capital for the laborers, for the workers, and what you spend on capital for the machinery. And this is simply a definition here. We do not assume that these rates are constant. These rates can change over time. That's the definition of the small beta. Next step is we look into the production process in more detail. What is the capitalist or the manufacturer doing? He pays the workers to produce some goods and then he regains a so-called surplus value. That is what I explained on the very first slide. The workers by producing something produce a surplus value. That means the value of the goods that they produce is more than that they get what they got paid for their work. That is the important assumption here. That means I have a surplus or a value product, is it called here, which is what I had to pay for the workers plus the surplus value. Then you can define for these two variables a ratio, which is a mu. The mu is very important. In English it's called rate of exploitation. Marx never named it like this. Marx said mehrwertrate which is a very neutral term. This already sounds like something really bad. I just want to mention this. Marx didn't have this intention. He called it mehrwertrate. You understand what this is. It gives you basically the ratio of the surplus that you obtain in relation to what you had to pay for the labor of your workers. You can then express with the new and so on. It's not so important to see this. Then we can now understand what is the total individual value for the capitalist to refer. This is of course in relation what he gains from the workers and their surplus value that they produced plus what he had to spend on the machinery. It's the N here plus the C. The C is what he had to spend on the machinery. This is the individual value. That's basically if you just look on the side of the manufacturer of the capitalist this is what he basically has gotten from or invested into this production process. There is something, this W star that he has on his side. It does not mean that this is already his profit. Remember what has to happen in between. He has to take all these goods which are worth W star for him. It takes this to the market and the market tells him what the price is and there might be a real difference. You understand this. Now let us define the relative effort that the capitalist has spent. You already know that this is appearing later on. The relative effort is what we named here individual value per unit produced. Delta X is the unit produced. So X is the number of goods produced per time unit, so per individual unit. This is the effort. You can rewrite it like this and then you can plug in the equation of the W star from the previous slide. You plug it in here and then you can rewrite your production function like this. The nice thing about this is that you already got rid about this W star what should be the individual value. There is an individual value but you do not really know what it is. It means by rewriting this set of equations in this way W star disappeared. But instead you have this M star and the M star as I mentioned before is the surplus value which is not well declared either. That means the next step is to get rid of the surplus value. That's the idea. How do we express surplus value? You go back to the previous slide where I have defined the exploitation rate. Here I have the exploitation rate. That means I can rewrite this surplus value in terms of the exploitation rate and what I have to put on the labor. That means I get an equation for the surplus value like this. The V after the T is really known but I can use this additional variable that I have introduced that was called organic capital and the organic capital links the V and the C. That was the definition. How much I spent on labor versus how much I spent on the machinery. That was the organic capital. If I use these two the rate of exploitation of organic composition then I get a very nice equation which does not contain the star values anymore. Of course that's very difficult to measure for the capitalist. I have this equation, my production function dependent on my personal effort or efficiency of my production the organic capital and the exploitation rate. That's one thing missing namely the DC after the T and that's something we still have to express and in order to express this we go back to this explanation of this commodity cycle. Remember that only one part of the commodity cycle really matters. You have capital, you buy a commodity and you sell this somewhere else for hopefully a better price in order to get a profit. That was this commodity cycle. I do not repeat this again. That means the capitalist at the end gets something that covers what he has spent on the capital for the machinery that he has spent on the capital for the labor. That's his profit. This is the value that the capital expects. Now remember the idea was that some part of this is used to extend the production. These two parts are more or less fixed. You have to keep them going in order to sustain your production. You cannot cut the labor the wages for the workers and the machinery but this is something you can use to extend your production. Now let us look into this equation in a bit more detail. What I read right here, it looks a bit messy but all these equations are straightforward. You can trust me to some sense in some sense that this is correct. It is simply a reshuffling. Why? Because we want to have at the end we are reformulating this equation. Basically, that's all that we need and all the variables are already there. But we want to have one condensed variable that links the gamma the omega the beta and the mu instead of having three variables. That's done on the slide now. We try to merge them together into one scaled variable that really plays a role. We do this now by assuming that the rate of exploitation and the organic composition are the same. That's the definition how the capital spent on labor and the capital spent on machinery are linked together, namely by the organic capital. This is plugged into the value equation. This is my value equation from the previous slide. You have seen this before. Now I use this relation to get rid of the v after dt and introduce this capital. That means I can rewrite this in an equation as an equation for my profit. I simply rewrote this equation here. It looks like this. Now quite nice the whole thing. I just want to get rid of this c. I can do this by going back to this production equation here. I have an expression for the dc. I use this equation to get an expression for the dc and plug this in here and then I get this. Great. That means all these derivatives dc after dt and dv after dt they are gone. The star values are also gone. Very nice. Now I introduce an abbreviation which is called the cost price and express this like this. That's the definition of the cost price. The forward thing that covers this. Remember in the second round of our discussion we have already seen what is the meaning of the selection value, right? The meaning of the selection value was the cost price and here we have introduced the cost price and now we understand what the cost price is composed of. So for all these things efficiency rate of exploitation and organic composition they are made of the cost price. No, they make up the cost price. If you want to remember one slide for the exam then it is this slide. Here we have an expression for the cost price and here we have a plot of the cost price dependent on the cost of capital. So first of all we can interpret the profit in relation to the cost price. You get a profit if there is a difference between the value that you realize on the market that's the W without the star the W with the star that was a value for yourself, right? How much you invested and how much you like the product. And the W without the star was the value that you realize on the market which is not the same as I said, right? And if there is a positive difference between the value that you realize on the market and what you have paid for your production then you make a profit. That's the meaning of this equation very easily understood. And if you miss a DT why is the X after DT? Right? All right. So now we know what the cost price is and now we can discuss this. I think it's written on the next slide. Oh, there is a slide where we talk about this. Yeah, we talk about this later. For now we know this but there is an interpretation I give you two slides later. But this is very important here. I mean, so you are the capitalist, right? So you want to reduce your cost price. What do you do? This equation which you should remember tells you what you have to do. Before it was simply the kappa, right? So the kappa should be reduced. Fine, but how? So now we understand how the kappa could be reduced. How do you reduce the kappa? Already written here, right? So the first thing that you recognize and that every capitalist recognizes you increase the rate of exploitation. That's why it is called the rate of exploitation, right? So if you increase the muse, your kappa goes down. That's a nice thing. Then if you need to have an advice how to reduce your cost price you increase the rate of exploitation. What does this mean? Rate of exploitation was the relation with respect to what you spent on the labor, right? There was a profit, dm star divided by dv, right? That was the rate of exploitation by definition. And of course, if you decrease what you spent on the labor then you increase the rate of exploitation. Decrease what you spent on labor as compared to your profit. Then you increase the rate of exploitation, then you decrease the cost price, right? But you could also do other things. You could think about the organic composition of the capital. What was the meaning of beta? Beta was one over beta as we introduced was telling us about how much do we spend on labor we spent on machinery, right? And the beta goes down that means the one over beta goes up and then the cost price goes up. One over beta was defined let me make sure that I do not say the wrong thing I think it was so one over beta was defined as dc after dt so that means you get a one over beta increased no, you want to decrease one over beta in order to decrease the cost price, yeah? You do not want to increase the beta or decrease one over beta in order to decrease the cost price. And how do you decrease one over beta by not putting too much into the machinery? Right? Because one over beta was dc after dt. This is another hint, yeah? Because then if you have expensive machinery that is there all the time then of course your organic composition of capital is very high. And the third idea is you can think about the omega. What was the omega? We introduce one over omega as the efficiency, as you remember. Am I correct here? Let's think so. Yeah, one over omega was the efficiency or omega was the effort. Of course, if you decrease your effort or you increase your efficiency then you then you decrease the cost price. That's the next thing. That means this equation helps you to understand if you are the capitalist and you want to run your business how you become more competitive. Right? That's an important thing. Okay. Now let us just finish the derivation. So, remember again that the value of the goods for the capitalist is not the same as the value realized on the market. But the law of value as proposed by Marx tells you that the sum of both matches. That was the important step because at the end of the day that's the assumption here everything is sold. You bring your good to the market the others do the same. The market sets the price, everyone has to sell. There's a market clearing at the end. So, and this implies that then those sum of the individual prices and the sum of the values realized on the markets are the same. It does not hold for individual. It just holds for the sum of both. That is the consequence of the law of value. And the same is true for the profit. The sum over all personnel or how was it called individual values equals the sum of the profits generated on the market. This holds for the sum. That's the conservation law. Remember there are these two ingredients conservation law and positive feedback. That's the conservation law. And if we use this, then we can calculate the price. You know what the price was. The price was defined here by the value realized on the market. The value realized on the market is the number of goods sold and then we can rewrite this in such a way. That means we have an expression for the value realized on the market related to the production. And then here in this equation we have the law of value as defined by Marx. The sum over all these W stars equals the sum over all of the Ws. That means there is a match between the values on the individual side and the values on the market. Together. The sum of these. And then with this equation we can rewrite it in such a way and we get an expression for the price. The price according to this is nothing but the average effort spent. This is already known to us but we get it here back in the third round. So okay in other terms the P which is the exchange factor between the commodity and the money. That's the price. You bring your commodity to the market and you get some money back and the proportionality factor of the commodity. And this is equal to the mean effort for the production. Now let us interpret this. We have learned a few things. The first one is we know what the price represents. The price represents the mean efficiency or effort if I look into the total production of all capitalists. That is what they get from the market. So the profit equation looks like this as we have seen before. We understand that this is a P that in order to make profit you as a capitalist have to have a cost price that is lower than the price that you receive on the market or the average effort here. That's a very clear conclusion from this equation. If you are not able to have a lower cost price then you will not make any profit. That's the important thing. If you want to lower your cost price then the equation of the cost price already tells you what you have to do. We just discussed this. The effort or you can increase the labor productivity. Labor productivity means number of output per worker. That's for example a measure. Instead of having 1000 workers producing one good per hour I can have a big machinery and one worker producing 1000 goods per hour. That's a much higher quality but of course I have to pay for the machinery and then I have to find out which is a better deal. This is very clear to reduce this. You have to increase the productivity. You should get a lower organic composition. It's another way. Lower organic composition means don't put too much into the capital for renting the stuff. This is important. Think about the manufacturing industry versus the service industry. You are the internet programmer. Of course the manufacturer has all this cost for the machinery and the expensive equipment whereas you and your internet company do not have the same and we can neglect the price for the computer. This is a way to lower your organic capital composition. You spend less and less and less on the equipment and you spend more on the brain of the person. Very clear advice of what to do if you want to get more profit here. You see that the whole evolution of the economy exactly went into this direction. Or you can increase the rate of exploitation as we already discussed. That means the rate of what you pay for the labor compared to your profit. This is the meaning of it. How do you do this? You have these time-shared jobs. You employ not one person or 10 person on a 400 euro basis. This is a way to reduce the money that you have to spend on the labor because for these people you only pay flat rate for the social security and all this kind of stuff. It's much cheaper. Compared to your profit you have less money to spend on the waiters. That means we really understand from this equation what we have to do at the entrepreneur or the capitalist. That's a nice thing about this. In the first round we didn't understand what is the fitness value. It's a cost price. In the second round we didn't understand what is the cost price. What is the rate of organic capital exploitation and efficiency of production. This was actually worth the effort. Now we do the reinforcement of the accumulation of the production. Part of the production. That's the assumption that we had before. We have this messy derivation here. I did it here just to show you that you end up with this. As I said before this is an implicit derivation and this is just the full way. Then we end up with this here with this equation for the market share. And again you see that you can increase your competitiveness if you spend more of your profit into the expansion of your production. Some capitalists like to keep the profit for themselves and for consumption maybe. But the smarter way of course is to spend it on the extension of the production. This is also linked to this problem of the neoclassical economic growth model as you probably have heard in my other course on systems dynamics and complexity. There was always a decision should I reinvest this in the production or should I spend this on consumption? What's the optimal rate? Spending it on consumption or reinvesting it into the production. That's the same problem as you see here. And only if the capitalists all choose the same rate then the whole thing maps down to the known equation that we already know. Right? But even if your cost price is not really competitive you can stay in the market by reinvesting all of your profit instead of keeping it for yourself. Okay. Then if we have the assumption that all the alpha i that's the chair of what you invest into the production expansion are the same then we can simply rewrite the equation like this with an analytical solution. Very nice. So. And this is the solution here plotted over time. So all the capitalists start with the same market share 25% here and then you see that in the beginning two of these capitalists gain in market share. And of course they do this because this is on the expense of the other two capitalists who lose. But then you see exactly at that point the first capitalist who has the least income production drops out then what is not shown here the cost price makes a jump because this guy disappeared from the market. That means the competition pressure on you becomes much more higher and that's the day where for the capitalist too the things go into the wrong direction. He's no longer expanding its market share. So. The declining in this market share and only number one wins. That's a winner takes all. Okay. So. Then let me come okay. To the conclusions. So this is exactly what I have described before. Yeah. Remember that there are a number of assumptions in this model in order to get this result. So that means this is only a projection of reality. This is not the real dynamics of a real economic market. Right. One good one market all invest the same of the profit and so on. Yeah. Okay. I would like to point to the fact that they interact with each other but they do it in an indirect way. They do not really interact in a bilateral manner as we discuss in economic networks for example they interact in an indirect manner. Why are the market right? The market mediates their activity. So this is also called weak selection. So. If you go back to the original equation the original equation was dy after dt. Right. And here in the selection equation we talk about the row after dt. Row was a market share and y was a production. Right. Remember this. And that means that the capitalists still grow. All of these grow. You know. This picture does not mean that this capitalist does not grow. Capitalist grows. All of them grow. Exponentially as we know. But the market share reduces. This is a very important insight. You should not assume that the capitalist is producing less and less and less and less. That is not what I said. I said that the market share is reducing. That means the capitalist himself has the feelings that he always expands his production. Right. In fact exponentially. There is an exponential growth behind it. But there are guys in the system that grow much faster than this particular capitalist. That is the message. Therefore even that this capitalist grows he loses in market share. That is the important thing. Right. And that is the same as you go for the globalization picture. It does not mean that our Italy or France are not growing economies. Of course they grow. Right. Maybe with a small rate but they grow. But there are others that grow much, much faster. Right. Therefore the relative importance of Germany, France or Italy is declining. Despite the fact that they are all growing because they are growing not fast enough. That is the problem. And that is why we all the time talk about this growth factor. So there is also an important point to make when we talk about market share you never really disappear. Right. Your market share can be e to the minus 20. But it is not zero in this picture. This is important though. That means I was quite suggestive telling you that these capitalists disappear which is actually not true. That means the number of firms in this picture is always constant. That is important to mention. The number of firms is always constant but their market share is negligible. That is the important message here. Therefore this is called weak selection on this slide here. It is called weak selection. That is a relative dominance. But these guys never die. And there is no reason for them to die because they exponentially grow. The company is getting bigger and bigger but the others are much faster. That is the problem here. You understand the meaning of weak selection here. Weak selection means not that you are not growing weak selection means others are growing faster and therefore your relative importance is going down. That is the important message here. This is called weak selection. And there is also strong selection which I do not discuss here. Where you are really kicked off the market. That means the number of firms is fluctuating all the time. This is what Pavlin has put up for you as a self-study talk. Remember please that today we don't have an exercise. Pavlin realized this morning that he is in the US and cannot teach. Then he sent you this email. Thank you very much for your attention. I hope to see you next week.