 All right, let's start. This is on. This is on. Welcome to the last but one lecture. We're almost at the end of the course. Before we begin I'd like to make as usual a few announcements. Next week we're going to have the course evaluation. So please all of you come to evaluate the course, at least for the evaluation is probably going to be in the beginning of the second half so that you don't need to come so early at 8 o'clock but also those who you know looking at the podcast you know please come at least to evaluate the the course opposite to what you may think actually course evaluations matter a lot. At least they matter enough to the rectorat to put enough pressure on on professors in order to improve their courses. So this is this is really taken into consideration. It's analyzed by a whole different department in ETH which does only this to evaluate courses so to analyze this course evaluations. So this is going to be next week. Before we begin today's lecture I'd like to discuss with you the results from the online test and from the online quiz. So let's get to that. Let's go first through the online quiz. These are the aggregate results. You remember the passing barrier was 50% so it was like 10 points here. A lot of you passed. Surprisingly a lot more people failed than the first online quiz. Actually a lot of these tries or attempts are somehow empty so I don't know what happened there but people just started and didn't answer any question and then submitted. I don't know why but you will have a third chance to cover this two out of three requirement. So this is the online quiz. In fact the the results are pretty good. I can show you the question by question division. The worst question meaning the least amount of people answered this question was 56% right? So this was the question with oh yeah this was my favorite question that I came up with about the shoes and the headache. So yeah causality versus correlation and a lot of you chose it's not that you gave the wrong answer but a lot of you chose only one of the correct answers and there was a second one. But we'll discuss in details the quizzes in the last Q&A session. Alright and this is now the online survey. I'm quite happy with with my personal evaluation. A lot of you I mean if you look at this one there is a kind of a diversity in how many exercise sessions people have attended. Most of you tend to be on the high end which is good despite the fact that now we have Vensim and the exercises are not so dynamic and discussion prone as before. I'm happy that most of you attend the lectures. I'm also happy that you like my explanations which doesn't mean that they're always the best but I mean I try. Yes so I so intentionally there is no absolutely true here and this in this question. Simply because if there was absolutely true I think not not many people would select it because it's kind of an extreme opinion although it's positive. So mostly true is somehow yeah it makes me feel good. Alright now it's a joke. Now this is actually interesting. If you look at these two questions the mathematical side of systems dynamics and the business side of systems dynamics there's a huge diversity. Some of you want math some of you want less math and this is no surprise given the diversity in the audience. I'm still thinking what the best way to do this is in order to satisfy all the tastes and there was a suggestion in the in the feedback that you gave that maybe we can provide a lot more math in the notes so in this note section. We already tried to do this but only for derivation that is somehow difficult and requires further explanation but we don't do it for like simple taking derivative or maximizing a function but it may be a good idea to expand a bit more of the mathematical side in the notes for next year for instance. But yeah I'll get back to this question when I analyze the manual feedback. The cell study tasks so if you look at where was it yes this question my overall impression of the exercise was that the quality was there there's actually a significant percentage of you who are kind of on the borderline between you know they're not good and they're tolerable and I attribute this to the Vensim part of the self-studies because the Vensim part is not as exciting as in the beginning we don't get to discuss so much and if you don't actually do the self study yourself you may not understand the presentation. The presentations in most cases are simply showing you the model and quickly clicking on a few variables and then showing your graph so if you don't do it yourself you don't actually get to understand it. I don't know if there is a better way to teach Vensim or to teach modeling except doing it but maybe we can think of for the future we can think of some extended exercise sessions where we can we can discuss more but of course that has to do with resource constraints. I'm the only one in the chair who is involved in this course at the moment so that's kind of a that's kind of a yeah high workload. The pace is fine and so on okay that's fine let's go back to the manual feedback which is mostly interesting let's see alright yeah this I already mentioned more mathematical explanations in the notes this is one idea which I think is pretty good so I've tried to do this a little bit today actually yesterday at midnight so if you have downloaded the handouts sometime before midnight yesterday you have missed a few figures in the notes you may want to read download I will mention explicitly which slides so you'll know. So some of you wanted to answer or for me to or we all to answer the questions at the end of each lecture now this is this the questions at the end of each lecture were not intended to be answered in class they were intended to be answered by you and this time this year we're going to answer them in the Q&A session if you want of course if something is unclear but the main point of these questions is to be answered by you as a preparation for the exam. Dedicating the introduction to Vensim this is a good idea and we'll probably do it next year again it has to do a lot of with constraints with mainly people yeah this is the less math versus more math we can take the balanced approach and just have an average amount of math but I don't think this is a good idea a better idea would be to somehow include the math in the in the notes a lot of you wanted more real-life examples and case studies in the part with systems dynamics with the modeling you know feedback loops and stuff like that we saw couple we saw this ship building case but you wanted more so if you want to if we want to have more then we need to sacrifice some of the material I have to discuss this with with professors fights and see what we can come up with if we can incorporate a few more self studies real-life examples because just presenting you the real-life example is pointless in my opinion we have to discuss it in detail and see how the modeling works for that particular example otherwise I can just give you bullet points with real-life examples it would be useless now yeah I was in a funky mood yesterday so I coined this term lost in digitization which means that when you look at the podcast you don't see this for instance you don't see what I write on the blackboard I've been trying to use this presenter here which is like a mini keyboard right I still haven't got used to it perfectly I'm more comfortable with this one but there is always something that you miss if you don't come to the lectures so I'll try to I'll try to use this one today exclusively but you just have to accept it if you don't come to the lectures you you miss for instance the blackboard stuff in that case you can ask some people for notes or try to do it yourself right so still some of you are not really clear or don't really understand how each part of the course fits in and that's why I'm going to do a summary in the in the last lecture the Q&A session where I go through the whole course and hopefully I manage to explain what we did basically each part why we did it and so on yes also some people are not happy with the exercise presentations as I mentioned they were kind of monotonous and kind of time-pressuring for the groups we have like 15 up to 20 minutes to present and basically present the same thing this a lot has to do with resource constraints in the past we used to have two exercise sessions in the from five to six and six to seven in the afternoon but there were more people working there now it's just me so we have one exercise session and we have to somehow make do with it but it I mean obviously this has to be changed for the for the next for the next year right timing now maybe you've noticed that in the last few lectures we kind of spend a lot of time in the beginning explaining basic things and then the last few slides we don't have so much time to really go into details I realized this phenomenon and let me tell you why do it so in my opinion it's worth spending more time on the fundamentals like what is a bifurcation and why are we interested in this or that and even you know write down all the equation so that you can understand it and then if we don't have enough time to go into details in the last few slides you will have the the fundamentals to understand them by yourself as a comparison you can look at the podcasts from 2010 so last year they're also available at the at the podcast website so I just click on 2010 autumn semester and you see professor Schweitzer teaching more or less the same things and he's going in a different way so different timing progression and in my opinion as a student I could have benefited a bit more from more detailed explanations in the basic things like what is modeling and what is chaos and what is not chaos and stuff like this and then I can understand by myself let's say what is a window in the chaos for instance just by looking at the bifurcation diagram so this was my my idea to spend more time in the beginning having said that I think I spent too much time in the beginning now that I look in the podcast so I'll try to change this actually but still emphasize a bit more in the beginning again lots of questions about the exam what is going to be on the exam what is not do we need to learn every bullet point in the slides what can I say ideally yes ideally yes if you if you want a good performance just try to understand the concepts even if you don't remember the equations try to make sure that you can explain everything by words if it's not with it with equations for instance a question like when does chaos emerge in the logistics map can be answered to bite in two lines writing the equation and saying okay for values of the parameter this and that or you can answer in words just make sure that you can do that we'll talk about the exam during the Q&A session I'll try to give you some hints but don't expect a sample exam or anything like this all right I think that's all the self study for next Tuesday is actually the solo model or the newer growth model from from last week if I remember correctly there are only two groups presenting yeah you can find it actually online the self study and now we actually start today's lecture what did we do so far let me so much time I have here we looked of course in the beginning we looked at defining the problem the problem solving cycle but then the last two parts of the course were systems dynamics and non-linear non-linear dynamics system dynamics have to do all primarily with feedback cycles so we define our system we define the elements in that system and then we try to basically understand how elements are related to each other and primarily in terms of feedback processes we were not interested so much with detailed equations just kind of an overview like an architect me basically you have if I hire more people then I create a balancing feedback I would increase my production and satisfy my demand but also would create a disbalancing positive feedback in terms of stress were overwork burnout and stuff like that so these were feedback loops the non-linear dynamics part of the course is about the control of the control the influence of control parameter each feedback loop is controlled by is governed by control parameter so in the non-linear dynamics part we went deep into the structure of the system the dynamics of the system and we saw how this dynamics changes with the control parameters and most importantly we saw that the control parameters influence two things the influence the number of equilibrium states and the stability of equilibrium states this is related to the notion of bifurcations I hope this notion is already clear to you bifurcation does not just mean one stable equilibrium state now is split into two stable equilibrium states it could be anything could be destruction creation changing a stability of equilibrium set in so states and so on in the last lecture we introduced a little bit of economics to all this so we were studying this kind of basically dynamical systems with control parameters but they had no real economic interpretation there were just some kind of systems in the last lecture we looked at the production function the solo growth model and we introduced one of the basic let's say notions in economics standard economics this is the equilibrium so standard economics is only concerned with upper with the equilibrium and then how do what the new equilibrium will be if something changes it's not concerned with the transition towards an equilibrium or the transition from one equilibrium to another it just sees let's say the world as a bunch of equilibrium states and we're either in one or another but we didn't study exactly how much we we transition and in fact we we don't even know if we can even reach an equilibrium state I I don't think today's world is in any kind of equilibrium but the solo model the new classical growth model assumes that eventually all countries reach an equilibrium if we change some of the inputs we simply change the equilibrium that's all so what is missing I kind of hinted to this what is missing is the dynamics how do we reach an equilibrium do we reach an equilibrium and and so on so we're going to be studying today and the next next week the last lecture the motion towards equilibrium or in other words cycles economic cycles business cycles and you'll see that if you buy what I'm what I'm going to introduce to you then the economics or economy is not an equilibrium and will never be there will be business cycles and oscillations the whole time a quick kind of heuristic introduction to business cycles on this figure you see the now what is that yes so this is the investment in well it's I don't know it's some kind of investment investment fluctuation so yeah output let's see I believe the output is the black one like the bold line and the investment is the the normal black line so they fluctuate a lot obviously there is no notion of equilibrium here there is not even a kind of a hint that this thing may even may ever got to to an equilibrium state yes of course this is just it spans about 40 years you may say that's not enough that's a valid that's a valid point but you may also say that it's enough so now business cycles this is a definition of business cycle you can read it yourself but let's say the most distinct distinctive characteristic of a business cycle is this one you start reading from here a cycle consists of expansions followed by similar general recessions right this is this is basically a business cycle I mean it's it's very intuitive this notion and here now we have a few more realistic growth a bit more realistic curves this is on the left side we have the long-term economic growth in the US I think it's the GDP yes I know it's the GNP okay so it's the gross national product and you see it kind of grows so you may say well there are no cycles here but right but now if you take the left one and you detrend it so there is a clear trend there right this is due to inflation this is due to economic expansion in general but if you detrend it if you reduce this influence on primarily on inflation then what you see is the same gross national product now fluctuates a lot and it's a log scale so we can see more range and it fluctuates a lot actually quite significantly from it's about 50 years again no hint that this thing is ever going to reach an equilibrium state we already looked at cycles actually when we looked at product life cycles right this was like there was the introduction then growth saturation and then decline in the product in general the the business cycles can be applied to any kind of economic activity it could be salaries wages GDP GNP whatever you want and on the right you see the life cycle that we already saw with the with the products with the product introductions yeah I mean this is I think the notion of business cycle is very intuitive so we don't need to spend much time on these lights now before we continue there would be a mathematical part now about stability we already looked at when an equilibrium state is stable and I kind of gave you an intuitive explanation well stability means that small random fluctuations eventually die out this is kind of a heuristic explanation but to actually analyze mathematically whether an equilibrium state is stable and if it's not stable what is the instability is it explosion is it oscillations so business cycles basically we need to to have a more rigorous approach and this is the more rigorous approach now you remember this slide I showed you in lecture nine I believe what we have is the dynamics of a system this one and it's depicted there one example of a function f and if you look at how many times f crosses the x-line it's basically three so we have three equilibrium states x1 x2 and x3 they're also called fixed points and so on and so forth and now we're going to study how to how we can analyze the stability of these fixed points the idea is actually very simple in fact the idea is exactly what I said we perturb the equilibrium point a little bit and we see what happens to this perturbation does the perturbation die down or does it grow so what do we have here how many of you know Taylor expansion oh perfect so we take this and we expand the function f around the fixed point okay imagine the function f is any function and we want to investigate small fluctuations around the fixed point this is what we want to do well the Taylor expansion that's exactly there's a teleexpansion approximate a function in the neighborhood of a given point right so this is exactly what we need the Taylor expansion is around the fixed point is the function at the fixed point which is zero so we don't see it here but there is a term in the beginning which is the function f at the fixed point by definition it's zero plus the first derivative at the fixed point times this this is the first term of the Taylor series and then we disregard any kind of quadratic influences so this is our function it may be whatever fluctuating function at the fixed point in the neighborhood of the fixed point this is the neighborhood in a sense approximate approximated by straight line this is the slope of the line all right so I think this is clear enough now we define this difference so this is x minus the fixed point so this is kind of a small region around the fixed point we define it as delta x so delta x is one variable now it's not delta times x it's delta x so delta x is is this difference if you differentiate both sides just differentiate both sides with respect to x you get delta x the derivative of delta x plus the derivative of the fixed point is equal to derivative of x this is a constant right the fixed point is a constant so its derivative is zero right therefore delta x the derivative of the small fluctuation so this is we're interested in the dynamics of the small fluctuation or the small perturbation the derivative of the small perturbation is is well yeah so we have the x x dot is equal to delta x prime prime is again derivative but x dot is also equal to this thing right therefore we can simply say that thing is equal to that thing and we have delta x prime is equal to this the first term which we call m times this but this is by definition delta x delta x so we get a dynamic equation telling us how the small perturbation develops the solute so yeah m is just the slope of the of the line so it's the first derivative of x the fix at the fixed point the solution is very simple it's given by this and there you go if m or if the first derivative is positive at the fixed point then the small perturbation will exponentially grow the point the fixed point is unstable if it's negative here then the small fluctuation will eventually die out because this thing will eventually be zero delta x and often this first derivative is called the eigenvalue I'll show you why it's called an eigenvalue but our task when we analyze such a system is to determine m or to determine this thing if we determine this the derivative of the of the function f at the fixed point that we interested in and we just look at whether it's positive or negative then we know stability stable or not and this is just an example we take the function x squared minus u1 times x I should say that so why do we have two variables technically I should yeah technically I should say that we just call now m lambda we do this for convention because when we go to a function of two variables or coupled differential equations then we have a matrix instead of m and then we're interested in eigenvalues which by convention are denoted by lambda so yeah but you'll see that this is an example x squared minus u1 times x without even caring about let's say Taylor expansion or anything like this if you just remember we take the first derivative at the fixed point so the there are two fixed points zero and u1 I hope you can determine that right how did we get zero and u1 as fixed points anybody care to say yes so we basically set the first derivative equal to zero and we find the roots therefore the first derivative of the function is two times x minus u1 right if you take the first root two times x minus u1 at zero to x minus u1 is simply minus u1 if u1 is positive then minus u1 is negative which means that zero is a stable point conversely if at the second equilibrium point u1 to x minus u1 is u1 which is positive and that's unstable and then you can see there the the graph now we go to two-dimensional systems these are the most the most interesting ones it's basically the same idea we have some dynamics regarding x and some dynamics regarding y this could be rabbits and foxes for instance and f and well g the functions are both themselves functions of x and y so a fixed point now would be a pair of x and y right it won't be just x will be a pair of x and y let's see what's going on you know this is the fixed point x and y a fixed point means that neither x nor y changes anymore and a two-dimensional system so we have this yeah we have that so at f at the fixed point is zero and g at the fixed point is also zero in the same way as before we take a small perturbation it's exactly the same way as the one-dimensional system we kind of do it a little bit differently here we define the small perturbation to be this just different notation the small perturbation in the y direction is this we do the Taylor expansion only the first term and we get a dynamical equation regarding the small perturbation in the x direction and the small perturbation in the y direction here and now look this is y dot is equal to this v dot is equal to this now just focus on that part so this part and this part can you see how we can represent these two equations in terms of a matrix by vector matrix multiplication just for more compact form it's not we don't do any mathematical transformation just rewrite the whole thing can you can ever who can see that let's let's just be safe because then you won't understand what is Jacobian and then the whole thing kind of snowballs exactly exactly so if you write I was advised to keep a more organized writing style so this is our two equations this is df dx df dy at the fixed point at the fixed point times well that's actually big I started small and now it's a lot bigger all right so we just multiply this matrix times this vector and we get this right just matrix vector multiplication and just explaining the next slide in words this we call the Jacobian matrix of the system it's basically the gradient right so we take the partial derivatives of f the partial derivatives of g we multiply them by uv and we get the our dynamics so this is called the Jacobian it's simply writing the whole thing the one-dimensional system in two dimensions and then in matrix form for convenience all right this is exactly what I said the dynamics of u and v are some matrix times u and v and the matrix is this it's called the Jacobian the solution is again given in the same way as the one dimensional system t means transpose so uv would be kind of a column vector so uv is equal to the initial condition at time zero times exponential and now lambda here is not just a single value but it's a vector it it contains two values because we have two two coupled equations so lambda here consists of of basically two values lambda one and lambda two and these are called the eigenvalues of the matrix that's why m in the previous slide was lambda this for convention lambda one and lambda two are eigenvalues and they determine whether the fixed point is stable or not and now you can imagine we have two dimensions here x and y okay and we have two eigenvalues what happens if one of the eigenvalues one of the lambdas is positive that one is negative let's say in the x eigenvalue is positive and the y eigenvalue is negative well what will happen is you will never get no no you will never get to the x so what they say x is positive you will never get to the x direction so you're kind of repelled from the x direction but you're attracted to the y direction and okay I have a picture I'll show you this but this slide basically talks about how to find eigenvalues of a matrix how many of you know how to find eigenvalues oh that's that's good that's perfect so you know the lambda the eigenvalues so lambda is is the root of the characteristic point polynomial this is the trace of the matrix defined as the sum of the diagonal elements and this is the determinant of the matrix defined as yeah as this right so the two eigenvalues are this one and this one this is the trace and this is the determinant if both eigenvalues are real then the overall solution is given by that it's basically a linear combination of the eigenvalues right these are the two solutions this is the y in this case is the y eigenvalue and this is the v eigenvalue so let's see what happens if one of the eigenvalues is positive the other one is negative why did I go back all right I'll get back to this right this one it's it's this case if the determinant is zero is less than zero then the eigenvalues are real but have opposite signs so one is positive one is negative you can immediately see this from here right if that thing is less than zero then this this big thing is positive and that is always smaller than that so the difference would be negative here this would be negative this would be positive and this is what happens so you are always repelled from the x direction right so here you're always repelled from the x direction so you never reach no sorry you're always repelled from the y direction like this is the y but you're always attracted to the x direction so if you think about how how the dynamic the dynamics could look like well it's like this right you're attracted to the x direction eventually but always repelled from the y direction in the same way in the in the other cases so this is got a subtle note in two dimensions and this case is when you're attracted to both x and y directions meaning that both eigenvalues are negative of course it doesn't have to be straight lines it could be it could be a curve like this like that but you're always attracted could be another curve like that you're always attracted if you're always repelled from the x and y direction meaning both eigenvalues are positive you have this if so this is a nice figure actually if let's look at here if the trace is negative so where is it right if the if the trace so if that thing is negative where is it sorry yeah if this is negative we basically have complex eigenvalues right because this this then would be a complex entity and a complex eigenvalue so they're complex if the trace is negative we have attracting spirals you're somehow attracted to both directions in a very kind of oscillating way due to the complex eigenvalues the opposite case you're repelled and if both eigenvalues are zero then you get limit cycles these are called limit cycles right so what happens is this is a equilibrium point remember this is an equilibrium point if we perturb it a little bit you would have a spiral which increases increases increases and eventually settles at this limit cycle if we perturb this limit cycle now a little bit then you have an increasing spiral which increases increase and then set those to the second limit cycle so these are typical oscillations right so if this is if the trace is zero right and both eigenvalues are entirely complex so the real part is is zero and this is a classification of all possible regimes that you can have dependent on the values of the trace and determinant because the values of the trace and determinant in turn determine your eigenvalues and yeah it's not so important to write so degenerate notes degenerate notes means that one eigenvalue so both eigenvalues are equal for instance and are non-zero so that would mean that you don't have any kind of spirals but you always you always converge on on a line but it's not so important for you to know that the important things are the spirals when we get spirals limit cycles or centers and unstable and stable notes that's all good so we'll continue after the break this is another way to calculate or to illustrate the stability of fixed points but I'll talk about this after the break okay so I just want to show you another way graphical way it's entirely graphical way to calculate or not to calculate but to visualize stability or instability fixed points so far we saw rigorous mathematical analysis in terms of linear approximation Taylor series and Jacobian but now think about the following this actually Euler came up with this we have our dynamics x dot is some function of f it's yeah it's it's yeah basically a function of f now imagine this is the real x this is our x not x dot but x we want to know how this thing looks like when we plot it the solution of this thing how does it look like when we plot it well the idea is that you split your space in grid points or mesh right so the smaller the better but let's assume this is some kind of mesh or grid point and we take this one it has coordinates so it has a given coordinate x okay let's say this is x0 we want to know how how does our function look like at x0 meaning what is the slope of our function you know is it increasing decreasing is it constant what is it well it's actually given by this equation we have it right the slope is just the function f at this grid point right so it may look like this increasing slope slope of one for instance if you have this well it's kind of trivial but yeah the slope would be one then we go to the next grid point maybe the slope at that grid point given by the function f would now be this maybe then it's like this then it's like this right so we visualize how the solution actually looks like without even calculating it by just plotting by just calculating its slope at each grid point and this is the method of gradient direction fields all right the idea is that you already know how the solution behaves qualitatively because you know the slopes the slope of your solution at each point so if you have a two dimensional system two coupled differential equations x and y x and y and you take a grid point here imagine this is a grid point like this kind of square what is the slope in the x direction and what is the slope in the y direction and basically it's some of some of two vectors the x and the y and then you sum up the two vectors well the slope dy dx is simply g divided by f and you have these functions so you can calculate this is a visualization of how the system looks like qualitatively and yeah so we don't need to integrate we don't need to do anything explicitly and basically we can do pretty much the same analysis as before this is just a side remark by the way it's not a major part of what we're concerned with but in the notes you have a very nice introduction to this method it's very powerful if you're interested but we I mean don't spend too much time on it now we have all the basic ingredients to start modeling business cycles you know before we model business cycle and oscillations we need to to know how these oscillations can be produced from a dynamical system and we know now depending on some values for the eigen values we can have limit cycles or oscillations so now we can start studying actual systems and this is what this is the model for business cycle that we're going to see this lecture and the next lecture we're going to see more models for business cycles this one is the most famous the cow door model the trade cycle it's you can find the original paper by the way in the literature section I put it there last night it's very nicely written purely economic terms no major equations and you can if you're interested you can also read some of his policy requirements based on his model I'll talk about this policy requirements at the end but it's very powerful it's yeah one of the most famous ones it's based entirely on economic theory unfortunately it's somehow difficult to understand we've had a lot of experimentation with presenting this model in the last last leg last years one way or another and it's somehow difficult to understand because it's required the typical presentation requires a lot of economic theory knowledge basically what this time we've chosen to present it in a more kind of mathematical way they're not equate not not so many equations but we've left out some of the economic theory behind the model I will mention it as we go along but primarily on the slides we have the mathematical stuff you'll see what I mean so Caldor was actually he's a very famous economist who never got the Nobel Prize yeah so why that is the case I believe it has to do with his university affiliations you probably suspect that the Nobel Prize is not entirely objective especially the Nobel Prize for peace but okay right so yeah I mean it's it's very annoying he is so he was looking at the standard new classical growth model and well let's actually get back let's get to it he was looking at the standard classical new growth model and if you remember the economy in the solar model was basically output equals investment plus consumption right whatever is not consumed is invested and then we said well investment is actually equal to savings right so whatever we're not whatever is not consumed is kind of saved for investment right that's all that happened there and this is the standard the standard new classical growth model for closed economy investment plus consumption and then the investment of kind of yeah the investment is simply some kind of fraction of the output let's say fraction s we call this propensity to save and obviously the consumption is this now and in the new classical growth model the only dynamics in the model was this if you remember just the capital stock how the capital stock changes the investment that we do in the capital stock minus the depreciation of the capital stock this was the only the this was the only dynamics but then Calder was asking himself well what actually drives investment right in this model kind of consumption drives investment right we consume something and whatever is left we invested and this is for countries by the way the new classical growth model was developed for countries and in the context of firms and markets what drives investment you can say well the market clearing mechanism drives investment because if demand is higher than supply the price would increase if the price increases producers will have more incentives to produce obviously and they would invest more in production so this is the major question the Calder wanted to answer what drives investment investment is not now just determined automatically by the amount we consume but something else drives it now yeah actually this light is better Calder defined kind of two types of people not people but let's say economic actors these are the people who invest obviously and the people who consume the investors he called capitalists and the people who consume wage earners I will kind of use wage earners and workers interchangeably because wage earners is some kind of a mouthful but for Calder that's that's the two economic actors capitalists and wage earners so what happens before the the production is realized so let's say in the beginning of the time period capitalists make predictions how much consumption there will be in the economy which means which would determine how much they would need to invest right so think about what drives investment now as I said here the capitalists they decide on a given investment level x anti x anti means before the it's given in the notes right before the output has been realized in the economy so based on kind of predictions of forecasts so they decide how much to invest x anti and wage earners or workers decide how much they want to save so this is basically the subscript w is workers or wage earners capitalists right so you as a wage earner you decide with my next salary I am planning to save I don't know I'm planning to save 10% of it the other 90% I'm going to consume right so that's what the wage earners do they only consume and save the capitalists they only invest all right so but the capitalists they also they also try to estimate how much the where is it yes they also try to estimate how much the workers are going to save okay so investment actually close saving so if the capitalists decide that people are going to save that amount of money they're going to invest the same amount this is x anti everything happens x anti now the production is realized what happens well two things can happen if the capitalists underestimate the amount of say the amount of actual savings right what is actually what does this actually mean so capitalists thought that workers may save a given amount of money but then workers actually saved less which means that they consumed more than capitalists thought if they consumed more that means that capitalists produced less so they couldn't satisfy this consumption demand there would be excess demand excess demand meaning there would be unsatisfied demand for goods and services right is that notion clear this interplay between between capitalists and wage earners I'll say it again if capitalists let's say think about consumption now not just savings if capitalists underestimated the consumption level of the workers meaning now workers consume more than they should have there would be excess demand so unsatisfied demand this is the condition here right and it says it capitalists forecast that the workers will save more than they actually do so they save in fact less which means they consumed more right and this is the the definition of the so-called excess demand we have the cap the the workers decide to consume something a given amount ex ante plus the investments that capitalists want to do ex ante so this is the forecasted output of the economy right the forecasted output of the economy remember why was equal to I plus C it's here why was equal to I plus C but now we split the investment in the savings decisions we split them different people decide to invest different people decide to consume or save so the consumption that all the workers in the economy want to have plus the investment that needs to be done is the total output forecasted output of the economy obviously it may be different from the real realized output yft and now if you if you notice the production function is time dependent now in this model in the solo model it was just why there was no time dependency in the production function now have time dependency now the idea is the following the demand side or the consumption side is always satisfied so we always try to satisfy first the demand side or the consumption what does this mean well assume we have a given level of realized output let's say total amount of goods and services is 1 billion francs for instance with this we first try to satisfy the consumption okay we try to satisfy the consumption and then whatever is let whatever is left will invest it what does that mean we can if that's true then we can rewrite this whole equation like this right this is simply I minus this in brackets right so the realized output minus what wage earners wanted to consume they consume it whatever is whatever is left is the investment that capitalists should do okay but the investments that capitalists thought they should do is this all right so the well yeah okay scratch that bracket right so the difference between the investment that they should do and the investment that they thought they should do is this now y minus c is simply the savings that wage earners do right this is just y minus c so this is the savings that workers thought to do x anti why can we do this now why can we do this because why can we say that y minus c is equal to s well because the demand side or the consumption side is always realized if that weren't the case we wouldn't be able to do this we wouldn't be able to say that y minus c is equal to s but now we can say this because we know that out of our output we first always try to satisfy consumption so in other words x and t before any output is realized workers decide how much to save this amount capitalists decide how much they want to invest this amount if I is bigger than s it means that capital again I will repeat it capitalists thought that workers want to save more than they actually saved which means that capitalists thought workers want to consume less than they actually did which results in excess demand right people consume more than expected so capitalists have to invest more in the next time period in order to satisfy the success demand right and yd is called the excess demand I minus s and everything is x anti and the subsequent slides will drop these subscripts and and superscripts will drop them it would be just I and s but remember I refers to capitalists because only capitalists invest s refers to workers because they save you see now the savings decision now are done by workers the investment decisions are done by capitalists yes could you repeat again are they actually saved by workers so you're asking what you're asking what happens with this money that workers save so this so what happens to this well yes that's true so yeah the consumers think of it in this way the consumers actually decide yes true now think of it in this way the workers they don't decide how much to say they decide how much to consume okay the assumption in the model is the disconsumption is always realized oh let's not always realize but with the amount of money that we get from the economy we first try to satisfy consumption if that is negative yes of course this this could be negative that means that there is excess that this thing so if this is negative what is going on here it means that there needs to be more investment right so consumers didn't let's say wage owners could not consume everything they wanted to consume which means there must be more production in the next time period and that's why capitalists would invest if we can satisfy this or let's say if if yeah so if we can satisfy the the consumption whatever is left is basically yeah the idea is that it's safe yeah whatever is left from from consuming the actual output so in other words the savings it goes to workers it's the yeah this is this is the idea the basic idea now how the capitalists get their money is somehow yeah unclear in this model capitalists only invest that's what they do and their investment is based on on basically the excess demand how much demand is unsatisfied that's what their investment is based on it's actually given by this equation this one so if yeah I mean this is the same equation as before the amount that was saved by workers or you can think of it in this way the amount that was saved by workers need to be let's say invested back into production because yeah otherwise what would you do with it with this money so the amount that is saved by the workers need to be basically invested but not more not less and that's why the investments yeah you can think that the that the workers give back their savings to the capitalists so that they can invest you can think of it in this way but of course ex-ante capitalists didn't know how much workers wanted to save they had to make some kind of estimation right so if they overestimated the amount of savings then that thing would be positive which means that the production needs to increase to satisfy the success consumption that was required so this is the dynamics now look we have a dynamic of dynamics of production function it's not of capital only capital stock is in the solar model but also of the production function and alpha is the so-called multiplier effect which means that capitalists adjust the production so here the production in the economy depends only on the capitalists they adjust their production given this alpha now alpha is the speed of adjustment if that speed is very high you will be able to adjust almost instantaneously to the excess demand or to the consumption that you couldn't satisfy in the previous time step but it will be very unstable you will have huge fluctuations because as as this changes so the workers desired consumption hence desired savings change you would basically react or overreact similar to that model that we saw with inventory inventory utilization yeah so this bullet point basically tries to say what I just said if the amount that workers decided to save is less than the than the amount capitalists wanted to invest yeah think of it in this way workers give their savings back to capitalists so that the latter the latter can can can invest this money but of course ex-ante capitalists don't know how much money people are going to save so how much money they're going to have for an investment they have to make some kind of estimation if the estimation is bigger than what actually get from the workers it means they the consumption was higher than expected if consumption is higher than expected they have to produce more all right this is the dynamics of the model sorry this is dynamics of the output only of course we have we have the yeah we have the dynamics for K as well but let's let's look at the some qualitative analysis you see these are functions here this is the investment function and this is the savings function capitalists don't know the savings the savings functions if they knew it this amount would always be zero but they don't know it obviously the workers also don't know the investor do not know the investment function so Caldor one of his main contributions was to give sound economic justification why these two functions need to look the way he proposed the way he proposed is described in his paper basically are nonlinear functions they're kind of they have certain slopes and the reasoning why they have these slopes is purely qualitative only economic there is no equation there's nothing for instance now maybe I can show it to you at the end if we have time right but if you read his paper you would see that his main contribution is in defining these two functions and depending on what these two functions look like you get business cycles or you don't get business cycles now let's look at the simplest case these are linear functions right so this is a linear function I and S imagine we have this situation here obviously this is an equilibrium because I equals S so the investment or let's say the in the investment that capitalists wanted to do meaning the money they thought they would get from the workers is actually the same as the money they got if we perturb the system a little bit again our definition of stability we come here a little bit just a little bit here well not here what happens is that now investment is less than savings which means that yeah so investment is less than savings which means that capitalists overestimated the consumption of the workers they produce more than needs to be produced therefore in the next time period they will decrease their output so why will decrease we go back to the equilibrium similar reasoning goes if the perturbation is here capitalists underestimated the consumption level they need to produce more therefore we come back to the equilibrium this is a stable equilibrium the exact same reasoning applies if the linear curves look like this if the slopes are different this would be unstable equilibrium so if we have linear functions only we get stable equilibrium and unstable equilibrium there are no oscillations so it's not a very realistic case which doesn't mean that the model is wrong it means that the calibration if you'd like is wrong so the functions the investment in the savings functions are wrong and you can you can immediately see why it's wrong I mean if you have a huge output in the economy imagine investment that capitalists do or want to do was really linear function if you have a huge output in the economy you wouldn't expect investment to increase proportionately right because at some point you read some kind of saturation you cannot add new capital goods all the time new factories new machines and so on the return on investment actually decreases the more things you add similar to decreasing marginal returns so it doesn't make sense that this is a linear function in the same sense if you have very low output like people have almost no money they would save almost nothing right they would consume most of it on the other hand if you have very high output Caldor actually his economic reasoning is that if you have very high output so people are very very rich their savings level would kind of saturate or yeah their savings levels such rate they would always save the same percentage of output so it doesn't make sense that we have linear functions and Caldor assumed that we have nonlinear functions and here comes the difficulty in presentation in presenting this model I can either show you the nonlinear functions that Caldor proposed or I can just we can just assume that they're nonlinear and look at the mathematical kind of dynamics and we're going to do the the the latter because yeah I think it's more clear like this if you want to look at the economic justification just have a look at the paper or we can talk about this at the Q&A session all right yeah so this is what I said in fact Caldor proposed S curves for investment and savings it's it's not that he proposed some fancy dynamics just S curves all right this is how the output changes we saw this before and this is from the solo model how the capital changes so investment goes to capital goods and here are the nonlinear functions so this is a mathematical way to present the nonlinear functions but yeah in his in his paper you can see actually the pictures what we have the first one as output grows investment also grows that kind of makes sense now this is this is interesting as output grows savings also grow the justification is is kind of economically interesting and it goes like this as output grows the capital also grows so you would have more factories more machines and stuff like this so more capital goods if you have more capital goods you would be able to bring the price of consumer goods down right so if you have like a lot of factories producing semiconductors you would be able to bring the price of semiconductors down right which means that if people want to consume a given amount of a fixed amount of semiconductors per time period they can do this with less money because every the consumer goods now cost cheaper right so they can satisfy their consumption with less money which means that the savings increases the rest is saved so they have more money left to save and that's why savings increase is output right so output causes more capital goods more capital goods bring the prices of consumer goods down and that in turn leaves more money for people to save this these are the kind of economic reasonings that Caldor did in his paper now this one here as capital grows so more where is it yes these two actually are the process of what Caldor calls capital accumulation as capital grows investment actually declines this is what I mentioned before so you have more and more factories and more and more machines and capital goods your actual investment goes down because it becomes more and more expensive to build a new a new factory right your return on investment declines your return on investment decreases the same thing with savings right if you if you have this is basically the same reasoning as here right if you have more capital goods because you have more output if you have more capital goods prices go down of consumer goods people would have more money left to save and these are the non-linear functions of Caldor in his paper he analyzed the dynamics of the stability and kinds of the regimes that we get he analyzed them purely qualitatively by just thinking okay what will happen now if if output increases by this much or capital decreases by this much then this curve would shift up this curve would shift down and then we get the stable point then we get unstable point we get the limit cycle this was the kind of reasoning that he did and it takes a kind of some time to understand it that's why we're just going to analyze this system and the way that I presented it so far looking at the Jacobian calculating trace calculating determinant and see what happens I strongly suggest that you never the least read the paper it may not be so relevant for the exam but it's it's a very nice nicely written paper and you can learn some economic things as well alright so these are the system of coupled equations we calculated Jacobian as I described before the Jacobian is this so it's basically the derivative of i-s with respect to to y then with respect to k we put it here then the derivative of this with respect to y is this right so this is 0 because there's no why the derivative of this with respect to y is this the derivative of this with respect to k is i subscript k minus delta this is the Jacobian and then we simply calculate as as I showed you above the determinant and the trace the determinant is given by this you can just as an exercise you can try to do it but it's really trivial this is the trace and now if you go back to the figure with the with the classification of fixed points depending on the trace and determinant we want the determinant to be positive to exclude saddle points remember saddle points were this kind of this kind of dynamics this would mean that there's actually no stability in the model there's absolutely no stability which obviously we don't want and right so if the trace is negative so if this thing is negative and the condition for this thing being negative is exactly that then we have an asymptotically stable equilibrium obviously if that is positive we have the unstable equilibrium just look at the picture now we can ask ourselves that so this is just qualitative analysis of the equilibrium what is the slope right so actually I'll show you ideally we want to see a picture like this remember we had two couple differential equations y and k y-axis k-axis this is always how we present the solution and then we ask ourselves well how does it look like we know that on the given conditions there is like fixed point and stable and unstable but how does it look like so for instance this is a fixed point what is the slope along this ISO so isocline yeah how should I explain this all right so in this phase space so y and k we have a space so let's say we have a region of this space where capital doesn't change this is this line here and in the new note well I'll talk about the new notes but this is the this line here so capital doesn't change what does it mean if we're here exactly on this line we can never go so we can never go up the line so if we're here we can never go up this line because that would mean that the capital would have to change if we're here the only directions that we can go are to the right we can go up we can go left we can go down we can go diagonally but never along this line because that would imply that k would change the same thing for y so we're asking ourselves this is the fixed point what is that slope that slope it's exactly this what is the slope of k as y changes at the fixed point k equal to 0 if you calculate this derivative from that equation so simply divide this equation by this equation dt would cancel out and you get this that thing is always positive right yeah it's always positive then you get upward slope there you go it's a bit more funny with the y because the slope looks like this oops the slope looks like this and that could be negative positive or equal to zero what what so first of all this i k minus s k is always negative remember i k is this it's negative so as more capital goes into into the economy the investment actually decreases so that's negative but savings increase so that's positive therefore i k minus s k is always negative this is always negative the savings s y minus y i could be positive could be negative let's look at low levels of output and in caldor's paper he describes this these different regimes very low levels of output almost no money it there would be not much there would be not enough money for investment in the first place but there would be even less money for savings because people would want to consume everything so caldor's justification for this is for low levels of output this is positive okay this is positive therefore positive divided by negative is negative hence the slope is negative here the slope is negative this is negative slope for low levels of output as the output increases we have more money for investment right and the slope becomes positive so in other words we invest a little bit more than is saved and then for high levels of output the slope again becomes negative this is actually the phase diagram of the caldor model in the notes in the new notes i've presented in fact this is not a phase diagram it's kind of a almost a phase diagram in the notes in the new notes you have the actual behavior of the system depending on where you start from you can even see it from here now what happens is what does this mean this means that we go diagonally here in this direction it's just the sum of these two vectors we go diagonally here we go diagonally here and diagonally here so we kind of get this thing right now depending on the parameters this is a fixed point depending on alpha actually this is the major parameter alpha you can get exploding oscillations like this you can get dying oscillations so like this kind of a dying spiral or you can get the limit cycle and that's the interesting thing this is the the business cycle you're at the fixed point you perturb it a little bit you have an oscillation like this and eventually it stabilizes at the limit cycle and that's an it that's a business cycle right and and you can see it in the new notes the simulation of this and the self-study in fact next next self-study is the color model so you just I mean it's it's much easier to do it in Vincent because you simply have to put these two equations these two equations that's all alright so the color model is if you read the paper again I strongly suggest that it's very heavily grounded in economics all his reasonings for economic reasonings but the mechanisms which generate limit cycles or business cycles are not economic mechanisms they're actually very fundamental mechanisms and one of them is the famous van der pohe oscillator who has heard of it what is it oh okay it's basically a physical system entirely physical system let me see if the equation is given okay just a few more minutes alright so let's start with the color model this was the color model we do some kind of easy tricks we differentiate again the first derivative therefore we get the second derivative is equal to this right so now here we see that we have k dot k dot well we have k dot substitute k dot we get a second order differential equation this one this is the second order differential equation this one and if you assume this kind of assumptions don't don't be concerned too much with them and you change variables so alpha times this and alpha times i k times s it basically this is the change of variables that you do you can get to this equation which is the van der pohe equation and it's a physical system which oscillates exactly in the same way as the caldor model these are the results this is the van der pohe oscillator y k this is why this is k you see limit cycle here we start from the fixed point see this is why with time it oscillates regularly there the limit cycle can be changed it can have a different shape right here and here depending on on the alpha or mu in this case but why why why is this important why why am I showing you this the idea is that the mechanism of the caldor model which generates limit cycles is actually fundamental just it's not just economic mechanism this is not so relevant for the exam I have to tell you this but it just makes you realize that it's not an artificial thing that caldor did like yeah let's assume we have this nonlinear functions and they play in this way it's actually a real existing physical system which you can actually test you can see it and that basically gives more credibility to his model that's that's the message if of these few slides so this is all this is the new self-study and these are the questions as I said I strongly recommend reading caldor paper especially the section where he displays the nonlinear functions s and I and justifies economically why they look the way they do we can also discuss this at the Q&A session at the end I think it's very interesting thank you very much