 Good morning everyone Welcome to the very very last lecture in the course. I I Thought a lot about how to conduct this lecture today. So should I? Go through an extensive summary of the whole course each lecture Or should I? Just open the floor for questions should we go through the quizzes in details and things like this So here's what I come up or what I came up with I will give you a very short summary It's about it's actually four slides of the basic ideas in the course And then we we immediately start with questions I think that's the best thing for for most of you for the majority of the people Those of you who interested in more details about some of the topics Of course, we can discuss this also privately after the after the lecture But I thought that the majority of you would like to have questions answered Am I correct? more or less Yes, so one person agrees with me I thought that's the best thing for the exam because It's important. Yeah with this new Bologna system and so on and so forth. It's important not to kind of over engineer In a sense for the exam. All right over study Those of you interested in just getting a good grade would Embrace this concept those of you who want to go into more depth We can also do that, but I think the majority of you just want to You know Get a good grade All right, so that's what we're going to do About ten minutes. I'm going to spend on on the summary. Then we go through the quizzes Not every question, but at least the problematic questions And then we start with the with the Q&A session One thing to say I think the deadline for the third online quiz was yesterday midnight, right? So I will process the results right after the lecture and The people who would not get the test start You would get an email from me notifying you that you didn't get the test that so he can't come to the exam But if you I think you know best whether you've passed two out of three online tests So you don't need to wait for my email to know that you didn't get the test About the exam Ironically, I'm still not perfectly sure where it's going to take place which room This is I think too small for all people and there are regulations regarding the minimum distance between People and so on so basically I will check this today, and I'll send everyone an email With the exact location of the exam next week all right Are there any questions before we start? something unrelated to the course Yes, and you mean the the website of the chair. Oh, okay. Yeah, I didn't update it The exam is on Thursday, of course. It's the last lecture It cannot be on any other day because we just We didn't have I mean we don't have the time slot And I think for most courses the exam is in the last lecture, right Or no Okay, by the way, how much time I already forgot how much time is normally given to an m-tech exam 90 minutes or two hours Really? Hmm. All right I'll I'll get a second opinion of that Because Okay Anyway, so let's let's start I will kind of repeat myself But I will repeat what I think are the more important concepts of the course The larger part of the course was spent on This notion of controlling solutions so non-linear dynamics systems dynamics and Especially at the end when we looked at business cycles we got kind of I Feel we got too much into detail with equations with technicalities right like What kind of bifurcation is this? What is the Jacobian of that system? What is the determinant of that system? and I feel that Some of you may have lost the bigger picture as I mentioned on Tuesday actually the at the self-study All these three parts that we looked at including the last one are related to each other So the last part controlling solutions It does not exist on its own. It doesn't come first actually what comes first as we saw is First of all trying to understand what we're actually doing Okay, before putting down an equation we first tried we have to understand what we're doing what problem we're solving and we looked into The reasons why the most problems in life are not simple Or the interesting problems that we try to solve are not simple and I gave you three reasons for this I think yes three reasons first of all most interesting real-life problems are just difficult to define in Natural language you can think about defining a problem in natural language. You have a complex situation Like the Thierry Airport. This is the standard example So how do you define the problem in a natural language? If you ask the different stakeholders, they're all going to tell you a different thing Right, so this is the notion of ill-defined problems We just cannot define them let alone come up with an algorithm to solve them and One reason why we cannot define them is because they have multiple optimization criteria. They have many different viewpoints and and Kind of preferences and that's why we cannot define them This is the yes contradicting but correlated optimization criteria, right? So there Lots of times criteria are contradicting with each other. This is the worst situation For these problems As you can guess multiple perspectives generate multiple solution possibilities So I may prefer As an airline I may prefer another Runway at the Thierry Airport you as a citizen you may prefer a different airport far away from your house Right, so different solution criteria exist in exactly because we have multiple preferences all these I mean this is kind of a Heuristic explanation why problems are not simple, but all of these three things Are in a sense the source of complexity or the source of diversity that we Looked further in the course so multiple solutions multiple optimization criteria You can say difference in people Because this is all coming from difference in people it generates diversity in life and I hope I managed to communicate the notion that this is a good thing. It's on the bad thing diversity is good because Is it mentioned here? No, it's not mentioned Well diversity is a good thing because it allows Systems to adapt and to evolve if you have a system Which is static and This goes to the notion of No, it's on the next slide Well, if you have a system which doesn't change if everybody likes the same thing have the same preferences That system would be static it will not change anymore We have we won't have any progress. So diversity is good and Related to diversity is is then the notion of complexity or complex problems We looked at complexity not as as something confusing But as a systems property just like just just the systems have size color Stability instability they also have this additional property that we introduced and that is complexity We don't want to remove the complexity. We don't want to explain it away We just want to understand it and to understand how to how to control it in a sense and We we had a look at two major sources of complexity I mean complexity is not magic the fact that you get let's say in the logistics map you get This huge sensitivity to initial conditions, which eventually make the system unpredictable in the long term. This is not magic The sources of diversity we looked at were two Actually more but we just in this course we dealt with two. This is the feedback cycles You know when you have related activities and each of them feedback on on a previous one all these non-linear non-linearities introduced by the feedback cycles generate complexity I Mean above all the jet they generate unintended consequences Right and this is in essence complexity So you you do something, but you really have no idea what the consequences would be and then we looked at all these measures that management takes hiring new people has unintended consequences of diluting the quality of your workforce or Making more making people work more has the unintended consequences of stress and overwork and so on and so forth the second source of complexity Where these non-linear non-linearities which are inherent in the systems in the system dynamics in the in the In the system actually So there we looked at bifurcations and the role of critical parameters But in essence The source of complexity here was non-linearity So the system responds in a non-linear way to an input and because it's non-linear. You cannot predict it easily If everything was linear we wouldn't have any complexity any chaos nothing to have been very nice very easy So you see it's not magic. This is not wow. It's complex. It's complexity. It's it's You know, it's it's God. I don't know. It's really non-linear response to an input which makes it Hard to predict and feedbacks things feedback to previous things and you have no idea what's going on and then One notion that that I hope I manage to convey is that if you've done your best to analyze the problem to implement the problem To implement the solution before that choosing the right solution if you've done your best and still You get chaotic behavior. You get complex behavior. Let's say we look to the supply chain If you still get that it's not your fault as a manager It's a systems property, right? doesn't mean that Now you when you become managers you need to be very lax about doing your job. It just means that Sometimes there is nothing you can do and you have to kind of learn how to deal with this and how to convey Or how to communicate this to other people as well The fact that we have this chaotic behavior in most supply chain systems today Despite all the effort that goes you know from academics and from managers and from from employees simply a test to this to this notion that Complexity is could be a systems property and we just cannot reduce it So yeah, the problem-solving cycle Yeah, I'm not gonna say anything about that. I think it's clear enough What I want to spend some time on is what's in your notes? So the sources of complexity, I think there was a slight in in the lecture introducing complexity were yes feedback cycles non-linear non-linearities, but also I think we had a bullet point saying something like Complexity can arise from the interactions among individual elements from individual interactions I believe there was a bullet point. We didn't look into that Right, we didn't look at interactions between elements what we looked at Were representative agents basically the systems dynamics approach the Interaction the individual interactions or so-called agent-based Modeling is topic of these two courses, which are going to be thought Next semester their elective courses both of them, which means that the mTech department doesn't consider them to be Basic requirement in a sense to graduate When I was doing the master's degree the collective dynamics of firms was a core course and the economic networks Course didn't exist Which I'm very sad about because it's a great course the economic networks is introduces one major aspect that we completely ignored and that is Networks obviously But you see in any system in any real system economic system financial system social system people are not just Interacting with all the other people at random, but they're embedded in a network, right? So if you think let's say about the adoption model the best innovation model what we looked at were just adopters and potential adopters and an implicit assumption was that Adopters communicate with all the potential adopters Right, which is obviously not true the results can change drastically if you introduce networks And this is very hot topic today. There are a lot of practical applications to this network approach, especially in finance Where people study a lot how the different financial institutions are interlinked with each other and how this translates Then to the stability of the whole system, right? Because if you have a system, which is highly interconnected You may think this is a good thing for let's say flow of information for adaptability And it's true, but it's also true that this system is very very susceptible to just a few institutions failing and then this failure propagates basically like a plague through the whole system So the economic networks is a course which which studies a lot of practical examples Which is a good thing for those of you who like it a lot of practical examples of Of a free-life networks Not just abstract models Collective dynamics of firms is agent-based modeling is the other source of complexity. All right now let's move on as I mentioned controlling solutions the system dynamics and stuff like this is just the last part of a larger Kind of framework. The first one was defining what the problem is understanding the different Preferences of different stakeholders Coming up with solutions This is the problem service angle coming up with solution approaches and then choosing some of the better solutions and remember it's it's Your job is decision-makers to choose the right solution. There is no algorithm for this The second step after you've chosen the solution, you know what you should do But you have no idea how to do it. It's the implementing of the solution and here we looked at Kind of to straightforward or let's say standard Approaches to implement a solution. That's the project management Which is basically Structuring your project your your your hope. Yeah, your project into different sub tasks and Trying to figure out how exactly to implement each sub tasks in what sequence. Can you implement some things in parallel? Which is a good thing Can you deal with uncertainty in? In your implementation remember the critical path we use these buffers and And right now I can tell you that there is going to be For sure one question regarding the critical path method on the exam So those of you who write emails, I have no idea what's going to be on the exam. I'm totally confused You know, it's awful One question is going to be critical path method That's for sure It's not going to be enough though One question. So yeah Alright Then We're implementing our solution And then we find out That some of some parts of this solution don't work This is the quality control feedback loop that we introduced so basically The project management approach defines rigorous criteria how to evaluate your progress we have this Gateway no gateways was it It was gateways, right? Was it gateways The checkpoints Milestones milestones. Yes. Thank you Gateway I am thinking about something else. That's why I said gateway. All right Milestones. Yeah, so we check The progress at each milestone something doesn't work We go into this problem-solving cycle notion again going back and redoing things This is the quality control and the quality control The major message is that it's just another feedback loop. It's nothing else Okay finally The last part was controlling of the solution. So you've done your best to choose Some of the better solutions you've done your best to implement it on time with the desired quality within the desired time frame and Then you see that your solution basically Doesn't work as you expected So there is kind of a let's say chaotic behavior and this was the last part of the course Understanding whether it was actually your fault or whether it's a systems property And of course we took a kind of a biased perspective. We only looked at systems where it was not your fault, but there are Much many more examples in real life where it's gonna. It's going to be your fault. So yeah As I mentioned the source of complexity were the feedback cycles and the critical parameters for the bifurcations an important thing here and And and I kind of started this discussion on Tuesday the business cycle. So are they good or are they bad? right We looked at Instability or systems which are unstable Meaning that when you vary a critical parameter, you suddenly change the system behavior. We looked at this instability But we didn't actually Come to a conclusion. Is that a good thing or is that a bad thing? Some of you think that this is a bad thing because it Leads to a destruction of wealth and unemployment and stuff like that Some of you think it's a good thing because it leads to progress instability actually if you think about it it it doesn't The notion of instability means basically a system which is Able to change right? This is what instability means a system which is susceptible to change And that's a good thing if you think about instability in this way So a system which is able to change then it's a good thing because that's how How the how systems evolved if a system is static Impossible to change then It's locked kind of So the ability if you looked at instability as ability to change it's a good thing It's so it happens in real life that The change is not only upwards it also it's also downwards But the notion of instability per se is a good thing Yes We looked into Chaos I'm going to spend a few words on this one and we go through the quiz but But for now think about chaos Not as Not as random forces affecting your system Random forces that you cannot predict Chaos in particular deterministic chaos is Unpredictability that arises from the dynamics of your system which dynamics could be perfectly deterministic as we saw with the logistics map so Another hint there is going to be question about This notion of chaos on the exam Okay, so make sure you you you understand what is meant by chaos Yeah, I Want to say yes You see It's like chaos is like complex systems people cannot agree on a common definition What is complex systems? What is chaos? for us chaos is Not the result of random forces, but it's the result of completely deterministic dynamics and Non-linearities, I mean non-linearities in the dynamics obviously Some people may think that if we have random forces affecting your system, let's say random noise Some not white noise, but let's say some funny-looking noise That could generate chaos as well because it's in essence unpredictable and It's it's a it's a correct notion, but at least I I don't I Haven't come across a common definition of what chaos is So everybody understands it different. That's why I want to make it clear what we mean by chaos in this course Yes, and that yeah exactly Right Let's see what I also put here right so the last four lectures, I think we're about cycles and oscillations and What we saw there is that one if you think about all the models that we saw we generate business cycles one Commonality between all of them is time delays time delays generate oscillations Right. We looked at time delays with the inventory correction time. I think and then we looked at time delays with Consume suppliers anticipating or not anticipating but taking into consideration previous consumption So there was always time delay we generated oscillations or cycles Yeah, so If you have problems remembering What all these models were about think about time delays Because I mean why it's not magic. Why do time delays generate cycles? Well, simply because with time delay we have an associated response and that response in terms of overreaction or under reaction is What generates the mismatch between what really happens and what you think should have happened. So if you overreact You basically you think that what should have happened is a lot is a lot more than what actually happened, right? Okay, I I don't think having it. I don't have anything more to say about this. I Don't want to to waste time repeating myself so as I said what I'm going to do now is to go through the quizzes really quick You can ask me questions or I mean, I don't know how how I should do it Do you want me to ask? Do you want to ask me questions about particular quiz questions or should I go through all the quizzes through all the questions and explain? The reasoning behind every answer. What should I do? Who wants to go through who wants me to go through all the questions in all the quizzes? Well, I think it's less than the half Not every single on but it yeah, okay fine Okay, okay All right, so this is the first quiz Can you can you read the question in the back? All right? So why is it important to accept short-term deteriorations in solving a complex problem? It's a question The the the kind of the short answer is because this This is related to instability. This allows us to explore outcomes That in the long term may turn out to be better Right, we don't have unlimited unlimited Rationality or we don't have unlimited capability of exploring the future in a sense so we accept short-term deteriorations in order to escape from this kind of Suboptimal minimum suboptimal solution milestone friend diagram. Okay. Well This is just a slide What is the difference between complex and trivial problems? It's basically this It's what I said, right? We have unclear solution space and then multiple conflicting criteria and so on and so forth This is directed from the slides What are two sources of complexity? right Introducing the legs so we had nonlinear feedbacks. That's one thing This one we didn't actually Study in the course. It's next semester, but we introduced it nevertheless Okay, this is again taken from the slides the objectives Okay, first or the second order solutions. I think it's clear By the way, I I found a really nice example of a second order solution Yeah, anyway It was you know this How is this game called? Tic-tac-toe, right? You know the game. No, I have I have to show it to you. It's really ingenious We have plenty of time this lecture. So So let's say we have the following situation tac and Okay, so Let's say well We have this situation. Oh, no Oh, yeah, we have this situation So if it's my turn I can just do this, you know, it's kind of thinking outside of the box Yeah It looks much better when it's on a picture Anyway Yeah, I'm looking for a good diagram to illustrate this notion. We had the lines, but I don't like the line so well What is the right sequence of activities? That's just from the slides free flow just from the slides Ah, there was a yes, I think there was a confusion with this question Let me try to remember what it was. Um, I think I know there was something with the free float I think that if one of the wrong answers was actually Correct if you think about free float in a more generous. Yes, I forgot If you remember just ask me Uncertainty oh, yes, yes, yes the uncertainty So with my deal was the following with the critical path method you we had these buffers to incorporate Kind of Oh We didn't have the buffers, but we had we we could calculate the free float and total float So basically the times by which an activity could be delayed without affecting the whole project or without affecting the Activity immediately next in sequence so the idea here was that we can incorporate a limited notion of uncertainty by just taking this kind of free time that we get and taking the buffers and Incorporating uncertainty by just varying the size of the buffers. So if you're not sure How how much time your task is going to take you simply allocate More buffers to this task But then I Yeah, so this was the idea, but then some people and that's why the answer is This one by using the floats I think there was only one one one right answer here. Is that correct? Let's see So now you see how the quizzes are created What was it critical path? This one Was it this one? Yes critical path to Minus this Yes, there was only one correct answer And this was my idea To have uncertainty incorporated into the buffers into the floats But then some people thought well, yes Yes The slight difference which I didn't mention in the in the lecture was that with the This last thing what was it called bird? Yes There you incorporate a certainty based more on an empirical evidence So you come up with this formula how to calculate the time Based on optimistic pessimistic and whatever scenario, but it's kind of an empirical thing So based on experience you define the weights for the different for the different Cases but with the with the critical path method You can still incorporate a limited notion of uncertainty, which is basically your intuition You know how very very how variable your your Task is going to be I understand it's a bit confusing I have a memory of mentioning that floats can be used to incorporate Uncertainty in the time required for the activity, but it's definitely not explicitly mentioned as a bullet point in the slides So yeah But this was the reasoning here defined objectives Free floats uncertainty What was the main cause for small changes? Okay, I think that's that's also clear basically feedback loops What is a system? I think all of you know this already Okay, yes another Kind of a linguistic confusion was this what is the first step? so the first step I meant really the most the uppermost Bullet point in the problem-solving cycle not the first step within the first step within the first step or something like this So that also caused some confusion Okay problem-solving cycle What oh? Let me try to find the lecture slide for you which lecture is this it was it to Yes It's lecture three then No, it's lecture four. There was a slide about this. Oh Yeah, okay, so if you refer to lecture four Slight nine where control gates are explained There is a bullet point. So there there is a So what could happen at the control gate everything is fine Quality standards are adhered to move on second Something is not fine. You need more resources. You need more time Whatever Out of this Multiple things can happen first of all you can get the required resources you can kind of extend the schedule if you want but if you can't Then what happens is you kind of psychologically start thinking well Maybe that solution that we rejected in the beginning was not that bad after all it would take us less time less resources Yeah, it's it's not Quantitatively the best But we made these assumptions in evaluating and maybe our assumptions are not perfectly correct Maybe it will turn out to be okay at the end right and this is why the bullet point is Already existing alternatives become more attractive even if suboptimal And I remember spending some time on thinking on on talking about this kind of psychological trap That that you start rationalizing suboptimal solutions. So that's why I Because this is not how we introduce project management by changing your By changing your goals Exposed, I mean that's like Changing your master thesis topic in the last month. I Understand that this is what can happen in real life for sure, but this is not how we introduce project management No, the goal the goal is still the same build this factory or something what this route good, let's move on you know It's always the case when when people try to factor in previous knowledge and previous experience and Everything could be challenged basically every assumption every statement could be challenged based on on The different experiences that people have And and that's that's a good thing in education. The bad thing in in education is that this doesn't Go so well on an exam So I perfectly understand that things can be challenged and it's a good thing but Challenge them after the exam. Let me put it this way. What is the purpose of a project? I think it there was a bullet point about this in the slides Techniques bullet point, which of the following is not topping of the course Okay, so Maybe I should slightly apologize for this question although I don't find it too confusing The answer is none of the above Okay So basically everything else is a topic of the course Which means that what is not topic is none of the above Yeah, the point is that I chose that the answers should be shuffled within a question and then this got completely messed up But yes, if you're taking yeah, if you want to be technically correct None of the above refers to a so Is not addressed in the course Yes, so linear optimizations we didn't look into that for a positive. Okay, fine Characteristic of total float or maybe the total float was the confusing one. I Don't remember now online quiz to Let's go What is the role of inventory correction? There is a bullet point for this What results do you expect? Yes So I spent some time on giving him what kind of a motivation or justification not justification Well, let's say a motivation for for why we do the models we do and what we expect from them So do you want me to go through the quizzes in the break? It's not gonna require too much Participation on your side. So it's like you can still relax. All right. I Hope I made it clear that we don't want this We don't want this So basically the quote the answers are this and this Understanding the minimum ingredients required to reproduce a given behavior and I gave you this example with cooperation All right, what is a fixed point? I think that's clear Mechanism for driving comma source. This is public information. That's also clear. Oh So this I came up with this question So the idea was the following it was the relationship between Correlation the difference between correlation and causality and the notion is that the idea is that correlation does not imply causality so you You go to bed with your shoes on and then you wake up with a headache So what can we say about the situation? One thing we can certainly? Can say for sure is that oh not for sure but one thing that We cannot say for sure is this for instance. I Don't know if it's medically proven, but I guess not now Discorrelations what I presented in the statement is a correlation going to bed with shoes waking up with a headache this correlation may be due to an external factor such as going to bed drunk and forgetting to take your shoes off or You went you you were sober, but you went to bed with a headache in the first place Right, so the statement just presented you with a correlation. It did not present you with with the causal Kind of position of these two events. I did not say that first you put your shoes on and then you develop the headache Right, so you can also think well, maybe we had the headache in the first place So these are the two correct answers B and C may be correct too. They may be correct too, but they make This step the jump from correlation to causality B and C try to explain Your headache By the fact that you you have shoes Which is this jump that we want to avoid without enough evidence, of course Which is not evidence of chaos This is an evidence Sensitive to initial conditions non-linearity. I mentioned it. This is an evidence Now this is not an evidence Okay For the chaos as introduced in this lecture. It's what we discussed before Okay, structural perspective. There was a bullet point for this taking the system modeling the elements stuff like that This is again taken from the slides This is again taken from the slides What is the primary reason for developing the workforce model? Okay, that's it. Yeah, so Yeah What would it be? What would it be easier for you this size? so Any question like this would be is to remember with okay, how easy would that be to remember? Yeah, really come on Okay, so we're not gonna have these these equations No, you don't need to remember that don't worry. No need. Yeah, I mean a question could be for instance um What are two What are the two control parameters in in the Hicks model for instance, and then you say it's the multiplier And it's the accelerator Now having said that there won't be such a question, but it would be a question similar to that I mean it's it's it's logical. So Yeah, you don't you don't need to remember the equations Let me take this huge load of your chest No need to remember equations Although it would help you to reason to answer questions like these What was the primary reason? This was the primary reason What this is what we observed in the beginning? Okay In the beginning meaning we started off with this empirical fact and we tried to explain it Okay, these are all things so B C and D They came kind of they came out of the model Once we've developed it, but why we developed it because of a alright Overcrowding it's an S curve. That's that's clear. Uh-huh So another hint for the exam. I mean I might have it as well just handed you the exams now I said so much already Chaos it's related to chaos. So what what is the answer to this question? We have a system we run that system with the same initial conditions Absolutely the same initial conditions. So let's say we've we run it once I think it's the logistics map. Yes We run the logistics map We calculate all The hundredth iterations including the hundred hundredth iteration Then we run it again We start from the beginning with the same initial conditions. The question is what was the exact question Can we always predict the value? for the hundredth iteration After we've already calculated it once Given the same initial conditions and the answer is yes, we have it the logistics map is the logistics map is not Something it doesn't have dynamics which which change Given the same state variables Okay, if x is something x plus one is defined Exactly what what what it's going to be the chaos comes from sensitivity to initial conditions So if our initial conditions are slightly wrong Just by a fraction you can think of this fraction is computing kind of some kind of storage problems in your In your computer then the hundred iteration would be very far from what you computed before But given the same initial conditions and we have to be correct saying I know absolutely correct Computer precision unlimited computer precision when we calculate things then It's completely predictable once you've calculated it. You can reuse this value Hmm the the answer is this one This is a great answer So you're saying maybe predict is not the right word Okay, I agree. It may be not the right word Which is not the source of complexity Definitely linearity linearity does not produce complex behavior. Why? Well, it's simple. I mean you provide an input, you know exactly What your output is going to be it's kind of a fraction of the input By fracations, I think you already know this Okay, I think I canceled this question at the end. So I gave everybody one point. I Don't I don't know if it makes sense to go through this question, but it yeah, it's it's a bad question on my side What I meant here is So we have this Isolated policies and integrated policies isolated policies are basically kind of Do one thing let's say reduce Inventory correction time, you know, that's that's a policy another policy may be improve customer service Then an integrated policy could be do both at the same time now my thinking here was that a Policy a single policy integrated policy Isolated policy is controlled by Just one control parameter Which is not necessarily true You can have a policy Being influenced by multiple control parameters and that's actually more realistic So this whole question doesn't make sense. Don't don't worry about it rabbits and fox model You know this bass innovation model, you know this aha and I also got a question about these things The stable points especially in the last quiz so let's Let's go Let me show you what what happens So in the last lecture We have a dynamical system and basically I'm answering this question and then also The related question in the third quiz. We have the following system. Okay Maybe we have a control parameter. I don't know we don't need the control parameter for this reasoning But maybe we have a control parameter so if we plot X the solution and X is a function of t. Okay, if we plot the solution Versus its first derivative versus the rate of change then what we need to plot here is simply the function f Okay, this is the function f This is what relates X and X prime so well Let's have a look now if this is the function f these are Fixed points because the first derivative of zero is zero and then we start reasoning in the following way It's always the same way by the way What happens if we disrupt this equilibrium? We disrupt it a little bit. Let's say we go here We disrupt it and we come to that point What will be the next value of x not x dot but x well, let's have a look We increase x a little bit right we increase it and we come here We increase it a little bit and we come here What is the value of? Of x prime it's positive right this is zero Which means that our x according to that equation should continue to increase Right therefore if we just disrupt the equilibrium a little bit We go up according to our equation positive rate of change. We should keep increasing All right The same thing happens here We disrupt the x a little bit here for instance So we decrease it by a little bit Rate of change is negative. So it should keep decreasing therefore. We move away from the stable point in the same way This now would be a stable point Is it clear why? If we disrupt x a little bit so we increase it Rate of change is negative, which means that it should decrease in the next time period So we increase it by a shock the dynamics says now Go back decrease it So we go back to that point the same reasoning here the same reasoning here unstable point and This was the what was introduced in the third lecture The slope of that thing is negative the slope of fx is negative now that question We looked into this picture from a slightly different angle, but it's the same reasoning We disrupt it a little bit and and and we see what happens and that's different angle was the following We still have that system. I'll just use a different color We still have this system, but now we say We looked into the gradient systems and gradient systems are simply Yeah, it's a different way to look into things We said let's assume now that our particle or our our object of interest Follows a gradient field Now, what would that mean exactly? What was this? Yeah, let's say it's something like this. Oh That's so ugly. This is zero here. We simply rewrote the system like this Now our function f is Minus the derivative of another function V. Why do we do all this kind of mental gymnastics? Well If you look into things in this way Then you can think of V as a potential Which is somehow very relevant for physics because you have particles moving in some potentials So positive negative potential. So if you look into this and we have a ball, let's put a particle here a ball here According to that equation the rate of change of that ball Should be the negative of the slope Which means that it should follow the direction of steepest Decrease right the direction of steepest decrease of that thing is here So it it reflects physical reality if you put a ball in this kind of Shape due to gravity it will go down. It will not go up if we don't have this minus If we don't have this minus Forget about the minus and we're here if we disrupt this equilibrium a little oh, it's not really an equilibrium Yeah, it's an equilibrium if we disrupt this equilibrium a little bit Let's say here Well, so now forget about this just just keep in mind that We follow the the direction of steepest decrease because this reflects physical reality the ball has to go down due to gravity So it goes down. Therefore This is a stable point and now some people thought well Stable points are minimum all the minimum if it's a minimum Then it's a stable point But that's not true Look at our dynamics. There is a minus there and as a counter example Is a counter example look at the following system Look at into the following system. This is f of x. There is no control parameters We don't need control parameters. This is f of x now This is equal to the derivative of What is it? This is our V Yeah, yeah, okay, don't care about the constant. This is our V All right Let's plot it. Yeah Here I define it as a V We can also say that this is equal to minus minus Is that right? This is minus V of x So let's plot things V of x V of x looks something like that Let me see. This is V of x V minus V of x is Simply, let's see if I can mirror it exactly This is minus V of x And this is V of x All right, so depending on how you've defined your system you either have Your ball going down like that. Sorry you either have your ball Following D of x or following minus the other one So let's see what happens if we just follow this now. What is on the axis? That's that's also important this is V of x obviously And this is x. All right So as you all know this would be a fixed point because the derivative of x of A V of x here is zero The derivative of V of x is f of x. So when f of x is zero This is zero So a function doesn't change. This is a fixed point. Let's disrupt it a little bit We increase it by a little bit. We increase x a little bit. What happens? The derivative is negative here So we should go back. So that's a stable point Oh No, is that true? Wait a second No, no the stable points are at plus minus square root of five So that should be minus square root of five That should be Plus square root of five, okay Yes, so any crazy to no no This is a point. Oh, wait, wait No, no This is a point when v of x is zero Okay, when this is zero and that is zero for x equal to zero here and It's zero for what is the other solution? I don't know Yeah, you have to x squared. I Think it's quarter 15 or something Yeah So this is quarter 15 But that's irrelevant. We don't care about this point We only care about the point where the slope of that thing is zero the slope of that point is zero here So we disrupt it a little bit. We increase the x a little bit Okay, yeah, so we increase the x a little bit. We're here now What is the next x? What should the next x be? The next x is given by the slope of V the slope of V here is positive. This is a positive slope So the next x should increase There it goes it increases the same way here it decreases So you see now a minimum in a potential is Unstable point simply because of the way we defined our dynamics Now let's look and to the two things are equivalent. Let's look at x following that thing over there minus the blue line We increase the x a little bit What is the next x the next next x? It should be the opposite of the slope The slope here is negative. This is a negative slope The opposite of the negative slope Is positive so the point should increase so x should increase Again unstable point Yeah We put a minus in front of it to make things look like reality in s in the sense that in the sense that If we put a ball here it goes down That's that's why but the reasoning why a point is stable or not is exactly the same You perturb it a little bit and you see what the next one should be should it be Should it increase should it go back and you simply look at x dot. What is x dot? It tells you exactly what happens? Okay, I spent way too much time on this But I think it was worth it. So let's quickly go through that Rabbit Fox that are that Okay, you all know this and Finally the third online quiz The third online quiz was pretty easy. So do you think we need to go? Let's go through Wow, I Is this is very annoying? So don't don't go now doing the quiz How is the Goodwin's model different? Indogenous, it's the first endogenous model that we saw Why did Calder introduce nonlinear functions because otherwise we don't get oscillations Investment accelerator, uh-huh. So look at this. What is the investment accelerator? It is the fraction of consumption difference the difference in consumption that is That is covered by new investment. So we need to have new investments This answer the fraction of consumption Covered with existing visit. Yeah with the existing capacity is What this theory of accelerator accelerator multiplier? Assumes away Right. Remember according to the today to this theory. We never have unused capacity. We always have full capacity Exactly to avoid situations like this where we have increased in consumption But due to unused capacity we don't We do not Induce new investments. So we want to We want to remove this possibility and that's why we assume full capacity What does the market clearing mechanism do that's fine. We just set the prices Hicks model of business cycles Yes, we have the two boundaries Upper and lower bound When suppliers switch market with infinite sensitivity That means that they react to the smallest profit differentials This I talked about now. I Also think that's that's clear This is also clear Okay, this question may be a little bit confusing what is meant here is the Investment rate Right remember in the solo model In the in this standard economic models in general Savings equal investment So what we save is not kind of does not accumulate interest in you in the bank It's immediately reinvested back back. So savings equal investments And you know in the solo model when the savings rate s equals alpha s equals alpha You maximize consumption and that's the question here instead of savings rate. I Should have set investment rate Because the two things are equal Investment equal savings I just set investment and I meant investment rate Some people are confused, but fortunately it was not critical for them to pass the quiz so This is this is why the answer here is the consumption is maximized. This is basically the consumption with s when s equals alpha All right Now we can start with the questions anybody else Go Wait, wait, wait Okay, so tell me which slide I will show the the slide here. Oh come on 27 you say Yes This one Mm-hmm That's wrong. Oh, yes. So the question is If you look at this bullet point It basically tells you that steady-state consumption is not possible for the golden level of capital Which is basically wrong. It's it's a completely wrong bullet point Because I mean what is the golden Level of capital the golden level of capital is nothing else, but the equilibrium point Which maximizes your consumption? It's one of the many Stationary states that you can have in the solar model, but it's still a stationary state Which maximizes your consumption in that sense It cannot be true that steady-state consumption is not possible. It is possible because k gold is still an equilibrium point if you remember How the equilibrium point is determined it's the intersection between Investment and depreciation so that point is k star or k stationary and if we choose investment if you choose Savings to be equal to alpha Then the k that you get from the intersection is the golden k, but it's still a stationary state. So it's wrong Thank you. So scratch it other questions No, you had a question lecture seven lectures So the questions. Okay. Wait a second. Is this lecture seven? Yes, so you want to go to The questions at the end Hold on This should be the last slide if it renders Okay So so the question is In each of these different models common source model Bass innovation model mixed source model. What are the mechanisms which generate an S curve? like this So the answer is Let's let's let's go to the common source model how to explain the S curve Here we started first with Now where's the common source model? Oh My god, okay, good So in the common source model the mechanism well first of all, we don't have an S curve in the common source model That's one so if So the mechanism so the answer for the common source model is we don't have an S curve but if we go to the bus and away the mixed source model well, it's clear the mechanisms are the What was it? Was it alpha? So it's the al yes, it's the alpha from here So the mechanisms are the interplay between broadcasted public information and Interaction or this word of mouth effect here beta So this is the mechanism which generates an S curve Yeah Yeah, exactly But if you don't want to remember equations, you just remember broadcasted public information and Word of mouth effect wait a second. Where is the bus innovation model? Okay, the density. Yeah, okay. Let's let's go to the bus to the bus first. Where is the bus innovation? Did I miss it? 19 no. Oh, yes. Yes. Yes. Yes. I I Understand I understand The question is not yes, the question is not technically 100% defined properly because a More proper definition will be would have been what are the minimum mechanisms required to explain the S curve in all these models and then we wouldn't need the common source the broadcasted public information But the way it's asked Such an answer will be perfectly accepted would be would be accepted. Yes if you say the Minimal the minimally required mechanisms are these that's also fine But the question was kind of simpler than that It just asked you to go through the slide and see aha here. We have broadcasted here We have word of mouth here. We only have word of mouth to reiterate these things But if you explain it in that way, then it's also perfectly fine And in the density dependent model, it's Where was this this animals? Cold start problem It's the carrying capacity In the animals or the density dependent model other questions. Yes No, no, no, I mean I Know you have a lot of questions, but let's let's give the chance to somebody else the style of the question would be a bit more Precise I would say For instance It will ask you to describe If it you would be presented that's an example. I'm not saying it will happen You would be presented with a graph that we've seen on the slides It would be just copy paste from the slides and then you have to explain this graph So what you see how it how it was generated? And and things like this it's hard for me to say what it's theoretical It will be easy No, the thing is it it will not even ask you detailed equations You may be presented with an equation and asked to explain the parameters But definitely want you won't be presented with with let's say the Hicks Equation for the business cycles and then ask you to explain Some mathematical properties of this system No, no, you don't have to derive anything Okay lecture 7 or 12 hold on 26 26 Yes So the question is whether there is something called multiple accelerated theory There is So in the way that these concepts were introduced First the multiplier then the accelerator Yes, they are just fractions alpha beta But people refer to them as the multiplied theory because it relies on all these assumptions Okay, so when you talk about the multiplied theory, you don't just talk about alpha You just you also talk about all these assumptions So they go hand in hand in a sense No, the goodwin model doesn't have the multiplier and accelerator the Samuelson and the Hicks model they're basically kind of mathematical formulations of the Keynes ideas But the goodwin's model is completely different. It has no it has no multiplier no accelerator this is just a summary of The know of the assumptions that underlie multiplier and accelerator both models Yes, I mean they all rely on multiplying accelerator. The difference is is basically invest the two bounds. That's the difference Yes There will be no multiple choice questions All of them will be open questions. I mean, I don't know what you mean by open questions Yes, all of them would be open questions. Yes, and the exam is closed book It's closed book. You don't even need a dictionary if you could understand me then you don't need a dictionary Because lots of people ask do can I bring a dictionary? Well, why would use a dictionary for it's on the English class? You can also you cannot know you can also provide some answers in German if that would be easier for you. That's also fine Hmm, then you can ask me I would be there it won't be unsupervised exam Yes, I always can you repeat Indogenous model. That's a that's a very good question. So why do we care more about in dodging us? Oh, let's say why do we like more in dodging us models or in dodging us explanations from for phenomena than exogenous It's a good question What anybody has any opinions on that? No one has any opinions. Yes Yes, you're going into into the right direction. So the thing is that yes Yes, you both have the right intuition and then basically the answer is Exogenous explanations rely on your ability To to explain how this exogenous factor arise in the first place Right. Let's let's take a model. Let's take Samuelson model What are the exogenous factors there alpha and beta right this multiplier and accelerator they come from the outside if Somebody sets the values of these parameters to something then we get cycles So if you that that's nice, but it doesn't answer the question who sets alpha and beta Where do they come from they come from the outside? You know, it's it's this outside that we didn't explain Indogenous means that everything happens within the system so we can explain all the parameters within the system So it's it's it gives you more. I wouldn't say predictive power. I would say it gives you more explanatory power And ultimately I would say more controlling power because you can propose better policies Other questions Yes, oh definitely You have the feeling that in the quizzes how many questions required this my goal was for the quizzes to To make you know everything by heart Because the idea I mentioned this in the beginning the idea of the quizzes was simply to force you to open the slides again Not to remember them, but just to open them and find the information No, no No understanding is completely up to you the goal of the quizzes Was simply to force you to open the slides and hopefully You remembered something even if you didn't try Now for the exam though, it's a completely different thing for the exam. You won't be required to memorize anything You would be asked to understand it So if you try to understand also the answers to the quizzes then you're fine Yes Okay, so I will answer this question But first I would like to answer all the questions that are related to the slides Anything unclear? Some plot maybe draft. Yes Yeah Yes Yes So you see the I believe this is the Goodwin's model. Yes the Goodwin's model relied on The capitalist propensity to save is constant where was it Where was the introduction Yes Capitalist propensity to save is one. So everything they got it's safe and then it's reinvested Economically feasible perturbations mean that you change this assumption a little bit by economic Arguments so you can claim it shouldn't be constant. Let's make it decreasing function of the wage So the more they have to pay For for employees the less they save So in essence, it's a small It's a small change in your model From an economic point of view. You just so this is not a constant anymore. It's a decreasing function of the wage And then it leads to completely different dynamics So the model is in a sense not stable to to changes in the assumptions, you know, it's kind of stress test if you'd like for for those with quantitative background Exactly the models result are too dependent on the assumptions. So if we perturb the assumptions a little bit It all changes That's that's one criticism Other questions related to the slides anything unclear Yes, I just remembered something before you asked the question the slides where Increasing and decreasing returns to scale are defined The signs there are mixed up Actually, I can go there. You can write it down No, we have a lot of time left Do you remember? Okay, this is this is the one this is the slides Hold on hold on here these signs Are reversed so this should be less this should be greater. This is This is a lecture Which lecture was this hold on lecture 10 slide 8 Right. No, sorry slide 7 lecture 10 slide 7. Yes. No, you had a question Slide 5 lecture 12 Yes It generates and it generates an oscillatory behavior only when alpha is equal to 1 over beta So for this very particular Very particular Parameter combination you can see this from this plot Only when alpha equals to 1 over beta equals 1 over beta This happens right when we increase beta alpha decreases When did we ever talk about that lecture 8? Which slide aha You mean so your question is how do we get from this? How do we get to this okay, so okay, that's that's a that's a deeper question Continuous time model means that you can discretize your time scale infinitely So if you look at I always refer to the following thing We have our stable point here x then we perturb it a little bit What is the new value of x at the next time step? But in a continuous sense the next time step could be infinitely close to your current time step so you can discretize your interval infinitely infinitely many times and the the thing is that The smaller the time steps the more accurate you are When you discretize things look how we discretize we simply did the Taylor expansion around around Around t t plus delta t right delta t is your time step and the Taylor expansion We approximated it by just by the first term we ignored all these things So when you do the Taylor expansion and you take the time step to be one This is the discreet. This is how you compute it actually in in your computer This is the discreet dynamics and now depending on your function depending on what this is Going from continuous time to discreet time Could completely change everything One example is the exponential function if you try to approximate x prime equal to e to the power of x By this Taylor method this linear Approximation with a time step of one. It's a huge time step You would get a completely different a completely wrong result so the difference between these two models is is Is perspective if you want to be philosophical other questions? I think he was first. Oh Okay other questions about the course About the slides and I will answer this Election number 12 slide 14 Yes this one It's about At the lower bound But then which lecture is it? Isn't this 12? Okay, the lower bounds where the where was the lower bound? Hold on Yes, this one So the question is what are the upper bounds and lower bounds again? The I will be short the upper bound to in the upper bound to output Or to investment as well is the fact that as your capital stock increases You have more and more factories. You have more and more output. It becomes increasingly more expensive To to invest into new capital goods for instance if you have Let's say a small catering firm All right with ten employees And you enjoy popularity and suddenly you're employed by ETH for a huge conference Potentially getting a lot of money. It may not be beneficial for you to hire X for people to in order to satisfy this this demand So you will not be so it will not be so cheap for you to invest into let's say ten more people twenty more people Another example is if you already have let's say ten factories It will be more expensive to invest into into new factories let's say You know, you just get like ten Additional units that you need to produce but these ten additional units would require you to build a whole factory You know, it's not economically sound. So that's the upper bound to investment. You cannot Invest indefinitely. Also, you're limited obviously by resources. You cannot hire people indefinitely the lower bound to investment is the notion that when output declines investors Refrain in real life on the aggregate they refrain from disinvesting Meaning actively selling their assets away. They just don't cover their depreciation costs but they would not sell their factories Because that would be active destruction of capital and Hicks thought Reasoned that this doesn't happen on the aggregate And therefore there is a lower bound on disinvestment and that lower bound is simply equal to the depreciation cost You cannot disinvest more than your depreciation cost on the aggregate Are the questions? Yes lecture 11 slide 27 this one the question. What is the question these things? well Okay, I will answer this question If you want after the lecture, but I mean I have to write things down for you now to show you and I Don't know if if that's something that most people would like Yes Yeah, you can send me emails, but I cannot promise to answer all of them Depending on how many I get and I mean if you ask me can you repeat lecture 11? You know You can ask me short questions. No problem lecture 12 slide 1 that's like the introductory slide Okay, slide 5 This one. Oh, that's not lecture 12. What is it? Oh, come on. Oh, that's lecture 11. Sorry Slide 5 this one What is the question? It's written here and I explained it in the podcast. So just just listen to the podcast I don't want to repeat myself Yes, yes Aha, so you mean why we have this? Hi, oh why why we have this model? Why we have this cycle? Yes, yes Wait a second So this is the employment rate and this is the share of capital that goes to output. Yes Come to me after the lecture. I mean Okay So thanks. Well, wait, I I had a thank-you speech, but fine We're gonna we're gonna We're gonna skip it because we have no time. So Yeah, but no, we're not gonna skip it. It's a good speech. So I I want to thank all of you for you know Being so committed to this course. I got so many emails spotting mistakes in the slides asking me basically really identifying mistakes in the slides and Kind of trying to improve how the course is being thought by feedback and everything you not only put up with this kind of typos and and and Confusions that arise in the slides, but you also kind of you know try to Spot them actively which I think improves understanding. It also contributes to frustration, but It's a good thing to know how to deal with frustration. The point is Most of you were quite dedicated to To trying to follow this course and I hope you got something out of it No matter if you remember the math, that's not important. I hope you remember the main messages. I Wish you good luck to your exams for your exams. I wish you good luck for your studies But above all I wish you good luck for after the studies Because that's when things become interesting happy holidays and I'm pretty sure I'll see most of you around in the university So thank you. As I said, I will I will communicate the exact Room for the exam by email in the meantime you can still send me emails