 So good morning. I have one good news for you and That is that this is the last lecture in the course. We've made it It was easy, right? I mean probably the most difficult thing was waking up early in the morning As evident by the influx of people I Hope that a lot more of you show up After the break so that we can have the course evaluation the forms are over there. Well, let me Fix the resolution again. I understand what the problem just doesn't want to okay Finally so the forms over there will do it after the break I ask you to take them in The during the break so take them at your desks during the break and the first 15 minutes In the of the second part will be spent on on the course evaluations I am obliged to say that the course evaluations are not so much about the lecture itself, but more about the lecturer So most of the questions Will be evaluating the lecturer. So if you think That the course if you have an opinion about the course material that is the course is boring or not interesting Or it could be improved and so on you can write this down in the manual feedback section but don't expect that you get Questions evaluating the the course concept. This is evaluated in a different way. All right So Today's lecture continues kind of the what we started last time and that is the business cycles modeling business cycles Um Basically, this is the the same slide this is last lecture a quick reminder What we looked at so far was kind of a mathematical Way to get oscillations, right? We looked at basically the non-linear dynamics. We have a dynamical system We didn't specify exactly what dynamical system that is Could be everything we looked at factory supply chain For instance, and then we simply took control parameters and we saw how the Dynamics of the system it changes qualitatively in terms of bifurcations by varying the control parameters, but There was still kind of Something left out and that was more of a real application of these concepts in Terms of economics, so that's the basically the last two lectures the last three lectures actually looking at Oscillations looking at the column is a dynamical system And this is not something new Economists have been trying to do this precisely to address criticism for the equilibrium assumption That's neoclassical economics has and that is economy economy is always an equilibrium and All we can study actually is how the equilibrium changes if we change some some fundamentals right Obviously, this is not a satisfactory explanation of the real world And that's why people started modeling business cycles, but I have to say that the economists who tried to model business cycles Most of them, I would not like to say all of them because I'm not an economist. I don't know I don't have all the information but most of the economists are not neoclassical economists right, so we'll not you will not find proponents of free markets of You know efficient market hypothesis you will not find someone an economist like Eugene farmer for instance studying business cycles Right because for neoclassical economics, there is no problem if we're in a recession for instance, there is not enough supply There's not enough demand for labor Then what will happen is the wages will go down and eventually it will become profitable for employers to hire again And then the economy will bounce back so there is no problem for for neoclassical economists to explain recessions and and Business cycles, but it's still not satisfactory. So all these economists Starting actually quite long time ago through 1938 they've been trying to kind of depart from this Classical view and that's why we emphasize here. This is not an economics course, but I just like to give you some kind of economics Foundation for for doing this. It's not just because we like to Have some mathematical equation and call this variable supply and this variable demand Last lecture I introduced the Calder trade cycle. It's one of the most famous business cycle models I saved the economics Behind it. It's written nicely in Calder's paper Now I'm thinking that it may be a good idea to introduce the economics So I will probably well introduce it briefly So I will probably do it next next week at the summary lecture. I will introduce this Quickly this non-linear functions investments and savings functions and explaining kind of the economic Ration now behind the slopes and the shapes basically it would be a summary of the Calder's paper. It won't be New to you if you've read the paper Today we continue with more business cycle models And these are the models With the exception of yeah, well now Now I cannot use this this tool. So let me Let me try to open the lecture in another way Okay, now I can do this Right These are the models. We're going to be studying about we're going to be talking about With the exception of the second one It's just we don't have enough time. So all the others were going to address With the exception of the second one. So the first one How many of you have heard the name Keynes the economists the economists? Oh, wow nice so he probably you probably know his His economic idea which is not as what Popular opinion holds that he's a socialist or communist. He's actually trying to fix capitalism So his main concept is that through fiscal politics. So government spending public spending you can fight recessions Right. So there is a recession and there is not enough private investment So firms don't want to invest don't want to hire then it's the government who can step in And I always think about all the construction work going around theory all the time Although to me they sound like perfectly they they seem like perfectly good streets But they're still being reconstructed. So it could be and Probably it is some kind of Keynes concept there, but this is his main idea the government through fiscal politics Can fight recession. Obviously, this is at odds with the standard markets are most efficient all this kind of stuff so Well Keynes was targeted was criticized a lot especially by Milton Friedman. He's another famous economist Proponent of the efficient market hypothesis and so on but that's irrelevant. The point is that all these models if you want to model business cycles you kind of assume Keynes position So all these models are just mathematical representations in one way or another of his ideas Because actually he didn't leave any formal mathematical description of his ideas. So people are trying to formalize them Of course adding a little bit here and a little bit there. The first one is Samuelson. It's a Very interesting model very simple. In fact, that's why it's interesting So he simply assumes That Consumers always consume a fraction of their income It's this is called the multiplier concept. So You multiply kind of the the income by a fraction and this is what you consume at each time step Investors they invest a Fraction of the increase of consumption, right? So if consumption increases by let's say 1000 arbitrary units the increase so the differential in consumption then investors would invest a fraction of this Not I mean not a fraction but a multiple of this it could be larger than one and this is called the accelerator We'll see this in details. So the second model is well just to give you an idea the assumption here is that The economy grows trend-like. So there's a trend-like growth but Oscillation, I mean you don't see this trend In the short term you only see oscillations around the trend. So the economy grows like basically like this John Hicks It's an improvement of Samuelson model. We'll see what improvement exactly and then When you look back in in this light in retrospect, you will you will understand what this means and the good win model is different from all these other models because it Explains business cycles endogenously Probably you're all familiar with exogenous versus endogenous Only with the good win model we can get business cycles that That they're produced by the internal dynamics of the system not imposed by some External parameter. We're gonna see this So post Samuelson. I tried to oh by the way, I forgot to say this is a brand new lecture. So it was never given in the years before so these are kind of infant slides and Yesterday I was going through them again and I found a lot of typos So depending on when you downloaded the slides and printed them out You may have a slightly different version. So to be safe just download them again They would be the correct ones if you've downloaded them in the morning and printing them out just now then you're fine All right, I Tried to give you a little bit of information about each economist So for instance Paul Samuelson was already at university age of 16. You can read this You can read this at your leisure He was he's the author of the best-selling economics textbook of all time. So that's that has to come for something All right, let's get right into the model By the way, I forgot the other presenters. So I'll try to keep track of time with my with my cell phone alright, so Keynes was originally originally suggested the Idea of the multiplier and the multiplier is simply this This is the multiplier for historical reasons the parameter like alpha was I Mean came to be called the multiplier. Although it's just a fraction. I would call it a fraction, but nevertheless So the idea is that? Consumers are backward-looking. So they see what their income was in the previous time step Oh last year and they consume in currently they consume a fraction of it All right And this is the multiplier effect. This is the original suggestion by by Keynes now Samuelson came and He introduced also the idea of the accelerator So another for me, it's again just a parameter beta which could be called a fraction, but Yeah, for historical reasons, it's called accelerator. So we have the multiplier here. This is consumption and the investment now is Consists of two of two parts and you can recognize Keynes touch here. So investment consists of autonomous part Which is always there you can think of this as government spending constant government spending At each time period so it does not depend on The economic condition at the moment. It's always there This is the autonomous part and we assume it's a constant at each time period and The induced investment so the induced the induced in that induced investment. This is the it and subscript superscript ind This is what private Investment is so you induce capitalists to invest and how are they induced they're induced via the Accelerator concept this one here So they see how much consumption or demand has increased over one time period and They invest in production in order to satisfy this increase, right? So they're not Really extrapolating or doing some fancy statistics on on the demand, right? They just look at the most risen demand increase Right. So if that is positive then they will have to invest more in Production in order to satisfy this expected demand. So that's the idea accelerator and Multiplier and you can see how the two into play So if the government jump starts the process by Investing something and actually it's the next slide that I'll explain explain it now by investing something Consumption would increase right from zero suddenly we have an output an output generated Entirely by the government investing This is the autonomous part consumption positive consumption would so consumption would increase due to the multiplier Then the increase in consumption would be kind of Modulated via the accelerator and produce investment induce investment actually Induced investment would increase the output increasing the output increases consumption and This increases the differential here investment increases again Consum output increases again consumption increases again and so on and you can probably see where I'm going. It's an explosion Right and we'll see that this is in fact what can happen What in fact happens? And yes, so this bullet point basically explains this this interplay Samu son just made a simple calculation. So he did not go through computer simulations bifurcation diagrams. He just Made this simple table. So here we have the current government expenditure. Well, that is the autonomous part Right. So here is the assumption that government is providing it each time step something so in the beginning There is no consumption because there is no output. There is no investment Because there is no differential in consumption The only thing that can happen is the government jump starts the process by investing. I think this is dollars. Yes by investing one dollar It could be a bit confusing, but this is a one. It's not a capital I so investing one dollar The total the total outcome of the total output Would simply be one dollar right remember the output is equal to consumption plus investment Consumption is zero Investment is simply the government investment here. So that's one dollar. So output is one dollar Yes, so I forgot to say that alpha is 0.5. So consumers consume half of their income and Investors they simply take the accelerator to be one In the next time period Consumers would see all we have one dollar of aggregate income. So this is aggregate Income let's let's consume half of it. We consume half of it There is now a differential in consumption of point five Therefore the induced investment would be point five The government still keeps investing one dollar Therefore the total outcome of the economy or output of the economy would be two dollars And in the same way you can go on and you see what happens to the total outcome It grows grows and then it starts to decline Decline decline and then it starts growing again Starts declining again So you can probably already You can already spot that given these parameters of alpha and beta We get kind of a cyclic behavior In fact, this is not a cyclic behavior. These are damped oscillations We'll see a phase plot later on But this model can produce cyclic behavior at a very very limited parameter range, which is one of its deficiencies So you're probably wondering what what does this mean minus? Zero point one two five. So this is induced investment is negative right this simply means that The investors for capital is they do not invest at all. They just simply let things depreciate So that's why in a sense some some part of the capital good is lost because it gets depreciated and it's not covered by by induced investment it is covered though by government Investment, right? That's why the total Investment can never be less than zero Okay So this is a very simple description of the model. Let's look at how it look how it behaves In a more systematic way As I said the out the output of the economy is just equal to consumption plus investment We can rewrite the investment in the following way autonomous part or government or Yeah, kind of public spending plus induced private investment and by expressing the output In terms of alpha and beta we get that here and We can substitute We can substitute the consumption remember the consumption is equal to alpha Times the output in the previous time step So we can substitute this one this one and this one and we get the second or the difference equation For the output. It's a non homogeneous equation because of this term Anyway in the notes. I've given you The solution we solve this equation exactly in the same way as I've shown in the previous lectures We had different equations and they're solved in exactly the same way but here I've given you just in the notes the the Overall form of the solution namely the x1 if you look x1 and x2 are just given as the roots of this Polinomial because if I have to write them down, they would be very messy expressions and We this was simply confused us But if you just look at it you can immediately see that if x1 and x2 are both negative We get We get what? It's it's in the notes If x1 and x2 are negative What do we get? Do we get explosion? No We get damped oscillations to the equilibrium value Conversely if they're positive We get explosions. So only when they're kind of the interplay between x1 and x2 is is is Balances somehow the balance them out then you can get oscillations But we'll see that this is in a very limited range All right, so this is the equation the stationary state is given by y bar And you can calculate the stationary state easily by setting yt equal to yt minus 1 equal to yt minus 2 Right. This is what stationary state means Given the stationary state we just compute the consumption the steady state or stationary consumption, which is just the multiplier Alpha times the the stationary output and the investment now. Yes Yo, yeah alpha ct minus 1. Oh Yeah, so there's another type of so this should not be c, but it should be y Okay, it's simply What we're simply doing is consumption plus investment consumption Should be alpha times yt minus 1 And then this is the investment Good. Thank you. I'm impressed you were able to follow so fast Yes, so Right, this is why Good and now why So we see that this is the stationary state of the output We can calculate the stationary state of consumption Now why would the stationary state of the investment or a long-term investment? Why would it be equal only To the autonomous investment only to the government spending Where is the induced part? Where the induced part but now it's gone. So in the long term There's no private investment Why is that the case in this model let me just Uh show you The equation This is the equation for the Total investment is the sum of autonomous plus induced and suddenly the long-term Investment is autonomous. Yes Yes Yes That's true. So it's precisely this There is no differential anymore in consumption because we have a stationary consumption Makes sense And it can also be shown That this stable point or the stationary point is stable only if alpha is smaller than one over beta More interesting is the phase plane right and this is in fact So the phase plane this figure without so Without these things Is taken from the paper of samuson from samuson's paper And and these kind of phase plots I added manually. So basically When you simulate the model you can create these phase plots Unfortunately, you can see the axis This is simply phase plots on the on the x-axis here You have y of t the output of time t on the y-axis you have the output of time t minus one Right. So this is the the two Most recent outputs And what you can see is the following now on the y-axis of the large plot we have alpha This is the part that consumers consume the part of the output that consumers consume Obviously, it can never be larger than one. So if if you look carefully, there's like a bold black line at one Consumers cannot consume more than it's what's available physically Beta on the other hand can go basically in any any range. So in in a sense, this is kind of the Reaction of investors if beta is very high, they really overreact to any differential any small differential if it's small They're not sensitive at all So what we see is the following By the way I've tried to do it to put the plots Where in the regions where they belong for instance in the region b Region b is basically everything below These curves. This is region b The dynamics looks like this at region c, which is simply the The space the area between these two curves Is this one region d Is the area between the horizontal line one at one and this And now region a is just this little thing here This little area, but I couldn't put I couldn't put this plot here So have in mind that this plot refers to region a it's not that This plot refers to values of alpha which are bigger than one which obviously doesn't make sense So what we see and what you cannot see unfortunately These little arrows. I hope you can see them, but this arrow this arrow and Yes, so this arrow and this arrow they point downward so the dynamic starts from here And it finishes here Here the dynamic starts from here And ends up here this one starts from here and explodes No, sorry, this one starts from here and again goes to stable state like that And this one simply increases all the time. So the arrow points upright So what we basically what you can see is that for most of the parameter ranges If if not, yeah for most of them you either get damped oscillations like region a b and c Or exploding oscillations, which is region d That's that's the only thing you get only when alpha equals one Over beta which is that line This line here basically this one This this line. So this is when alpha is equal to one over beta only then Do you get oscillations and I've put the plot there you can see there are oscillations there So it's kind of a business cycle model, but it produces business cycles It's kind of an exception right in most cases it either explodes or dies down And that motivates a lot of the critical points on this slide So Namely the the third bullet point business cycles are only a boundary case, but moreover The cycles have a fixed duration and fixed amplitude you can show that Which is obviously again not realistic real economic cycles You can claim they more or less have an average period of I don't know four or five years But it's by no means fixed or the amplitude is by no means fixed yet in this model they are The marginal so the beta if you think about the no, sorry if you think about the alpha again, let's Let's go back just to show you Right if you think about the alpha again You can interpret alpha also is the marginal propensity to consume right if you differentiate this thing With respect to consumption then you get the marginal propensity to consume is alpha Yeah, so one unit of output induces Alpha times this unit units of consumption It's constant in the model Right alpha is constant, but of course this is not Again, really realistic. We know that Actually, it's been shown that as people become More wealthy and wealthy their consumption kind of saturates at some point There's only so much they can physically consume So alpha is by no means constant in the real economy Yes, so everything else I said And I've I've said here that as a response The non-linear Hicks model was developed now. Why is it non-linear? Or in other words, why is this model linear if you look at the equation for the yt? So yt equal as As a function of yt minus 1 and yt minus 2 you will see that it has linear coefficient. So it's a linear model therefore John Hicks developed a non-linear model Which is an extension of this one And he tried to address these critical points some of them not all you will see that some of these points still remain in the Hicks model So this is John Hicks In fact, I should have said sir John Hicks because he was knighted In his lifetime But this is the model. So his idea was very simple. He said well I mean, it's kind of a First order solution if you'd like to think about it in this way He saw that the Samuelson model produces Explosions or or complete kind of Convergence to to zero or to the autonomous investment So he said well, let's introduce upper bound and lower bound For investment and for output So one way one very easy way to prevent explosion of output Is just to limit it artificially Well, of course, he was a bit smarter than this. He just He didn't just say let's limit it artificially. He tried to give economic meaning for these bounds And you can think about it in the following way. Well, I think Yes, I will explain the bounds in the next slide So the idea was the following for Hicks now The investment was still Some of autonomous investment, but now it's not a big difference actually now the induced investment So investors do not respond to differential in consumption. They respond to differential in output It's just a kind of a generalization of this idea that investors respond to a differential of economic activity You can measure economic activity in terms of consumption or in terms of gdp, for instance So here is gdp. So it's the gross or gnp If you don't have trade so Right, so this is how the investment looks like for Hicks Consumption is still the same still accelerate the same multiplier model And then if you just do the exercise Of expressing output As a difference equation you get this This is without the bounds So there are no upper and lower bounds to the output this thing can still explode or die down All right, so this is what actually Hicks said This is the private investment or the induced investment Or rather, let's say at let's first look at the output. This is the output He puts an upper bound to the output, which is called yc. C stands for ceiling by the way Right, so the output can never be bigger Than the than the ceiling now what could be An economic reasoning for introducing a ceiling on the economic activity What do you think? Yes, so Yeah, basically that's the economic reasoning There's only so much Investment you can do at some point Making new investments Is not profitable Right, it's the so-called diseconomists of scale Right, you're limited your resources are limited employment so Employment is limited The physically the amount of people you can hire is limited Natural raw materials are limited So there is a limitation to production which you hit at some point In fact, this is one of the criticism of this model that Yes, in principle, it is possible that we hit this kind of As you said structural limit to investment But we're not sure if we can ever reach it in real life right because The examples when people have not been able to invest Because they couldn't hire more people physically or because they couldn't get more raw material Are not so many at all So but that's that's for later for now This is the ceiling now his idea for for A floor on investment Is the following so f stands for floor you see the induced investment or the private investment Can never be lower than this floor and this floor is negative So the idea was the following Um You have some kind of capital goods a certain amount of capital goods, let's say you have 10 factories for instance At each time period they depreciate as you all know so there is depreciation If you do not cover this depreciation You're kind of or let's say Yeah, so the idea was this if you do not cover this depreciation At all so you invest nothing um You're kind of Let's say this is the worst thing you can do in the sense that If You don't cover your depreciation and then in addition you sell some of your factories This would be equivalent to actively destroying capital This was Hicks idea And he didn't think that on the aggregate people or capitalists actively destroy capital So if the output Um If the output goes down if the differential in the output goes down, let's look at this This is the accelerator So if this is negative, which means that the output goes down There is a recession This tells you that you have to disinvest Disinvest meaning negative private investment So you kind of sell your capital goods on the market But if you need to disinvest more than your depreciation This would amount To actively destroying capital and and Hicks didn't think that this is What people do on the aggregate it may happen that individual people Indeed not only let their machines depreciate, but they also sell them Disinvest them But on the aggregate he didn't think this is the case. So if If you see a recession if you're a capitalist and you see a recession The worst thing that you can do is to refrain from investment So just sit let your depreciation go Let your depreciation go. So the the worst disinvestment that you can do is your depreciation Not worse on the aggregate that is That's that's how Hicks reasoned for this floor of the investment Is that clear because it it kind of At first sight it looks counterintuitive. Why shouldn't I be allowed? Not only to not cover my depreciation costs, but also to sell everything I want You are allowed to but on the aggregate Hicks argued that this doesn't happen So people just on the aggregate they just let their things depreciate Even though according to the accelerator model they should have actually sold them as well all right so now This is this is how the economy looks like for Hicks This is the induced investment and now you can immediately see what happens I will just walk you through the Business cycle creation. Let's say we have a boom right so economic economic activity rises It rises investment rises consumption rises. Everything is fine At some point we hit the ceiling of output So let's say for two periods two consecutive periods We hit the ceiling What investors would see is? No differential here at some point they would see no differential because we hit the ceiling twice This equation will tell them that their private investment has to be zero So they invest Basically nothing They let things they only cover their depreciation. That is the point So zero Induced investment means addition to the capital goods, but they still cover their depreciation So that is zero if The new investment in induced investment is zero The total investment in the economy would decrease if the total investment of the economy decreases the output In the next time period would also decrease because the output is the sum of investment plus consumption When the output decreases then this difference in the next next time period becomes negative It's not zero anymore. It's negative. So now depending on the value of the accelerator if Is negative but not smaller than the depreciation It means that capitalists they still disinvest some of their capital goods. They sell them But the amount they kind of disinvest is not larger than the depreciation if at some point This thing the amount they have to disinvest becomes larger than their depreciation They don't disinvest. They just simply don't cover their depreciation so the This is the depreciation of of capital minus if so the investment the induced investment would be negative So depreciation is not covered but Together with the autonomous investment remember government is spending all the time It makes the total investment zero exactly zero All right, and this is in fact The definition of the floor The floor when you hit the floor the total investment is zero in other words this equals this In order for for the total investment to be zero So you've hit the floor total investment becomes zero Um, and yeah, so this is the cycle You have to implement this in vensim so You you get a much better intuition how how it works with the ceiling and the floor And by the way, uh This is a new kind of a self study So I still haven't done the vensim models. There are two models that you have to develop It's the hicks model and and one more that you'll see So I still don't know if it's possible But I assume it is so Let's see All right, we'll continue after the break And let me just remind you please come and pick up the evaluation forms course evaluation forms in the break Just put them on your desks and we will Evaluate the course After the break actually so you you don't sacrifice the break All right guys, let's let's resume Um before we go into the course evaluation I have a few announcements to make one of them. Oh, no, no Yes, I was about to shut off the beamer instead of just doing this one of them is an appeal from Group v member laura. She's looking for her group members So if group members Could raise their hands who isn't group v So only you two guys for now So you can sync later. All right now the third online quiz Remember we have a third online quiz Some of you need to pass it Um The third online quiz will most likely Be go online on saturday or sunday And you have one week maximum to do it Because I need to know who's going to get the test start and who's not going to get the test start In advance and communicate this to the department So Expect expect an email in the weekend Sunday the latest yet Um With this let's go to the course evaluation. This is the information you need to put in This is the course number I hope you can read it These are zeros and not big O's And the optional thing is this And let's say we can spend about 15 minutes 10 15 minutes on this On on the evaluation I mean don't overthink it Just be honest Yes Really uh one sec. Yeah, so basically let me see It should fit Three five one No, five four one Well, this is what they gave me Okay I trust you then All right, so don't put the L Does it fit now? Let me see if I'm obliged to say something else Yeah, it's it's about me. It's not about the lecture. That's what I have to tell you But you can still put your opinion about the lecture in the Manual feedback If you're finished, please bring the forms here Or let's try to collect them in a more intelligent way Just I don't How should we do it pass them on? I just bring them here Thank you Time's up No They're anonymous as a I forgot to mention You guys are done Yeah Yeah, it does Yeah All right, so let's resume Oh, so we were this I was describing the Hicks model The column in Hicks model is given and the Hicks model is given by these four equations Thank you And the most notable thing to notice are the two Upper bound the upper the two bounds the upper bound in the lower bound The upper bound is on the output And the lower bound is on the investment Once again, let me reiterate There is only so much Output that we can produce physically And there's only so much disinvestment that we can do without actively destroying capital Which Hicks argued rational investors do not do they simply refrain from investment Now if you Calculate the steady state by the way, these are As as long as soon as you have max and min You have piecewise linear functions This is not linear in this case, but you have piecewise functions Doesn't don't concern yourselves with this The point is that mathematically this model Has not been fully analyzed yet. It's simply complicated So Papers come out all the time Finding some exotic behavior of this model Which is more of a theoretical Importance than any real practical consequences Adjusts as a side information So if we express the output in the same way as we did before in terms of The output in the previous two time steps So we get the difference equations you get the equation number one. This is the yt If you set yt equals to yt minus one To get the steady state Or fixed point you get the yfp, which is this Right, so obviously if there is no government spending at all Meaning the autonomous investment is zero Then the steady state is zero So this this this model cannot support Basically any economy unless you have government spending. That's the fixed point It can still have oscillations around the zero fixed point. That's a different issue So the two bounces I described them the first one is the So-called full employment bound on the output. So as as soon as we have full employment in the economy We cannot produce more This is as simple as that even though There is investment and there is Consumption increasing consumption and increasing investment. We cannot keep producing more because we have Basically this economies of scale as as mentioned here labor force raw materials and so on The lower bound is the so-called investment lower bound on investment. What this means is Is explained here when the output decreases too fast Which would indicate according to the accelerator that investment Or rather disinvestment have to increase more than depreciation in other words Not only will you not cover your depreciation, but you will also disinvest more Um, this would be equivalent to actively destroying capital on the aggregate Which hicks argued does not does not happen and people seem to agree with him. All right The analysis of the hicks model. We're not going to do any mathematical analysis, but just just as a Just to let you know that this has been done and people have managed to prove So this is a theorem. You don't need to understand what this says is basically there is It could be proven that there is a so-called closed set Or think about it as a space Or bounded region of space where this model the dynamics of this model resides All right, so the any fluctuations that emerge are bound Or bounded within a confined region of space and this is called k It's a closed curve anyway Right, so you don't get you can prove what this means is there are no exploding oscillations There's no there's no exploding regime, right? If there was an explosion you wouldn't be able to bound the dynamics in any finite region of space All right, so this is the proof mathematical proof That now we don't have we don't have Explosions All right, so what are yes, so what are the consequences of this so-called theorem? As I mentioned this proves that business cycles exist because the dynamics are confined um Right, so another good Outcome of this model is that fluctuations can be either periodic Or quasi periodic which is nice because we know that in real life business cycles are not perfectly periodic And in some Wilson model we always got perfect period periodicity Now I've simulated the model with r Not this is not vents. These are not venzym graphs, but r graphs you will have to produce venzym graphs like these This is for the next next week right next Tuesday we have The caldera model Anyway, so what you can see is depending on the on the parameters the multiplier alpha and the accelerator beta You can get damped oscillations So on on the y on the upper plot you have time versus the output right Okay, so If I wanted to be consistent I would have put all these time graphs on top But apparently I didn't want to be consistent. I don't know why Anyway, so this is the time development of the output. You can see it's damped oscillations the corresponding phase plot Which is just y at time t divide versus y of time t minus 1 is Eventually reaching this attractor this attracting point These are the so-called quasi periodic or a periodic cycles. So there is no clearly defined period of these cycles, but they're still bounded which is nice And by looking at by seeing this in the phase plot These fluctuations you can you can tell that they're a periodic it looks like this they look periodic But in fact if you look carefully, they're not and periodic cycles depending on the on the parameters for this alpha and this beta As you can probably guess the next logical sequence in the lecture is to criticize this model, right? We introduced the samu some model we criticized it And we saw why the hicks model needed to be developed namely to address this explosion now. We address some We we address some criticism for the hicks model Some of the criticism still remains from some also namely the alpha and beta the multiply and accelerator are constant Which is again not realistic Um More importantly, however, people have managed to show that The existence of upper bound is not necessary to produce oscillations Even if you disregard this this point Then you can criticize it further by saying existence of upper of upper bound is not even realistically provable Right, we can never prove that this upper bound exists and even if it does We can have we we can never prove that we'll reach it right, um On the aggregate that is and why is that Like a critical point of criticism. Well, people have observed that in none of the cycles in real life up to 1980 when this model was developed and even now None of the business cycles were cost by hitting an upper bound by Full employment for instance people not being able to employ capitalists not being able to employ more workers or running out of raw material So none of the business cycles have been caused by an upper bound So we may never reach it even if it exists. It's like this equilibrium concept. It may exist But we may never reach it and it's not even known whether we can reach it at all Right, um, I had a short discussion in the break about investor behavior and remember here the Reasoning was that as investment as output goes down. So there is a recession Investors just refrain from investment They just let depreciation work What happens sometimes well often in real life is exactly when there is a recession people invest Because the cost of investment is cheaper And then they can yeah, so basically they can increase their capital stock And be at an advantageous position when the boom comes This is not what happens here because there is no anticipation Included of investment behavior. So investors do not anticipate the ensuing boom after recession That that's a valid criticism obviously most importantly however in all these models even if you kind of decide to overlook all these points so far Cycly behavior is generated exogenously. So we impose these parameters alpha and beta on the system Who is to say What's what's the multiplier effect for consumers? Well, we just pick up a value. We see a business cycle and that's it, but it's an exogenous effect and we don't like this We'd like to have an endogenous effect and that's why the next model was introduced by richard goodwin It's an exo endogenously generated business cycle So a short introduction for this guy Um The most important probably is the second point He is one of the founders of endogenously modeling endogenous models of business cycles Let's get right into the model So his model is a little bit different in the sense that To generate endogenous business cycles um You need to introduce more dynamics in the model right so far in these models. We only had basically dynamics in consumption and in output But now he said the following think of two social classes capitalists and workers like the calder model Um, the workers get a some wage w. This is the real wage. That's important It's not the nominal wage is the real wage. So adjusted to inflation L is the number of people employed So I mean it may be a bit confusing when it says employment, but it's the number of people employed then obviously mu is The share of output that goes To cover costs of labor. So salaries you may think For all the employers right so w times l is simply the Total salary of all the employers of all the employees divided by the output is a total share of output that goes to to cover these costs and L over n is the share is the employment rate Okay, so that yeah, it's self-explanatory now Capitalists there's some to more assumption is that capitalists always invest everything They get so they the capitalists get some profit they save everything and by Kind of assumption of classical economics savings are used for investment You know, they're not used to just have cash in the bank, but they're used for investment. So all The profits are reinvested back into the economy This is what it means propensity to save is one Basically they always save everything And in turn everything is invested By kind of extension consumers consume everything So the consumers consume entirely their income And now if you if you write down a little bit the equations for savings and investments So as I said savings of capitalists Is equal to their investments or they invest all their savings Um This investment goes to increase the capital stock, right? So k dot increases by the amount of investment And here if you remember the solo model we have not included depreciation of capital And the original goodwin model he did not include depreciation So all the investment goes to cover to increase the capital stock and then if you express Um, where is it, right? So if you Right, so this so basically the investment would be The investment of capitalists would be their Earnings minus the the amount they pay to workers and that would be The the stuff that is left for investment So that's what they invest So basically this is an equation Linking k dot with with uh output and w and l Further assumptions These are a bit more economic assumptions now and they would require a bit more explanation now How many of you have heard of the phillips curve? Quite a lot. Yes. This is the standard standard, uh Relationship between unemployment and inflation And this is a linear phillips curve, right? Most of you know this kind of non-linear rather like this non-linear phillips curve But this is a linear phillips phillips curve. What does this mean? This Is the Increase that relative increase because we divide by w of the wage of the real wage Right. So in other words the percentage increase Of the real wage This one here and the reasoning goes as follows At times of full employment So this was the employment rate remember At times of full employment, uh workers have more bargaining power Right because they can they can argue well If you fire me Or if you don't increase my salary, I will I will leave and it will have troubles Hiring somebody else Because we have full employment already so when The employment rate is high The real the increase in real wages is also also high Right because workers have more bargaining power If we have a lot of unemployment So Basically the employment rate is very low Then workers bargaining power decreases Right. There is a lot more substitute for these workers Available Out of the unemployed population Therefore when this thing decreases The real wage Uh also decreases. So if you if you look at this, this is a curve Which basically goes like this So on the x-axis you have Um you have uh the employment rate on the y-axis You have the relative increase in real wage and it goes like that So when the employment rate increases bargaining power of workers increase and Real wage also increases. So this is the assumption for the rate of change the relative rate of change of the real wage Uh typical assumption is that capital output ratio is constant So basically what does this mean? actually capital output ratio is constant it means that out of at each time period A fixed proportion of the output is always capital Right. So if the output increases The capital goods meaning machines factories and all this kind of stuff also increase All right, and you immediately see a deficiency in the model. There is no technological progress. No innovation In real life. We know that Due to innovations the productivity of capital increase So the productivity of a machine May increase and produce more output than before but according to this the only way to increase output is to buy another machine Because the productivity of the machine is fixed There is no innovation. I just cannot refrain from uh, oh kind of restrain from uh Introducing critic criticism as I introduce the model, but we have a separate slide for the critical points. All right labor productivity now This is a more realistic assumption Gross at a constant rate Okay, so this is the amount of people employed. This is the Rate that labor productivity grows. It's alpha right so for a fixed For a fixed amount of people If that is positive the output would increase Over time because over time at each time period the productivity of each single employee increases As people become more specialized and more adept at doing their jobs Uh, this assumption is somehow justified And well, yes the population grows at a constant rate And uh, this is the rate and it's kind of an exponential growth of population I believe that's not so unrealistic Especially today probably if you want to improve the model you can introduce some kind of uh Carrying capacity restraints like in the rabbit and fox model. So you can introduce some s-curve, but For now, let's keep it simple These are the basic dynamics. So let me ask you in your opinion. What are the control parameters in this model? What are good candidates for the control parameters? Let me remind you Uh, this is the complete model. So we have Mu is the weight share V is the employment rate Um And then we have all these other things here Sigma alpha and n Because what we want to do is now to define a dynamical system Which relates our control parameters And remember control parameters are those that can be changed relatively easily. Well, yes relatively easily by policy for instance It's not actually so um Easy or let's say so intuitive this time turns out control good candidates for control parameters are The weight share of output so mu this one Which means how much of your output goes To cover the labor costs In other words, you can change w the salaries And v this is the employment rate These are things that change most most often So what we want to do now Is to have Two equations two dynamical equations coupled differential equations Relating mu and v This is what we want Now i'll show you how to get A dynamical equation a differential equation uh relating Mu dot so the rate of change of mu and and um V the employment rate In the exactly the same way as an exercise if you'd like it will not be asked on the exam You can derive the second differential equation relating v dot V dot to mu The only thing we do the only thing you need to know is the Chain rule for differentiation. That's all you need to know This is the whole exercise Well, let's see mu was defined as this So if you want an expression for mu dot you simply have to differentiate that thing You have to differentiate this thing Well, what is the with respect to time? Okay, what is the derivative of this? It's the So think of this as w times L over y well, let's look It's the derivative of w times l over y Plus w times the derivative of l over y It's the chain rule, right? So by the way, this dot refers to the derivative of this of this ratio here Not just the derivative of l divided by y So if you don't want to be confused, maybe you can put a bar Here or you can put a prime here Like I've put it here Anyway, then we continue expressing it. This is this remains the same The derivative of this whole fraction l over y Remember l over y was defined Or rather y over l Was defined as the growth of labor productivity. It was e to the power of alpha times t in the previous slide Well l over y then would be just one over e to the power of alpha t or e minus to the power of minus alpha t derivative of this Well, that's easy derivative of an exponential is simply the exponential itself times the derivative of this thing Which is minus alpha Therefore we get w alpha E remains minus This thing still remains the same This is the same now e to the this exponent is simply l over y l over y we can factor l over y Yeah, so we can factor it out in front of brackets and rewrite this expression like this Okay, so if you multiply the brackets out you would get exactly the left-hand side What is that though? Well, this was defined as the rate of change Of the relative wage, which was simply the linear Phillips curve this one Minus alpha times This is mu It's the amount You pay to all your employees divided by the output so the rage the rage the share of output that goes to cover your labor costs So it's mu Therefore we get a differential equation relating mu dot and Obviously mu but also v obviously In the same way you can go with With the v. I will not do it now To save time But it looks like like the one shown so basically our system now is these two couple differential equations And you know how to analyze it first. We tried to find the fixed points And then we tried to find uh to investigate the stability of these fixed points. We already did this in lecture 10 I believe Where we were looking at uh jacobians and stuff like that So one obvious fixed point So just set these two derivatives to zero one obvious fixed point is zero zero right if mu is zero and v is zero Everything is fine. The derivatives are zero. This is a trivial fixed point So we're not so interested in it another fixed point Would be obviously when the expression in the brackets are zero this one and this one Right. So when this is zero only when v is equal to alpha plus gamma divided by rho this one In the same way, this is zero only when mu Is equal to this So this is another fixed point And now we can calculate the Jacobian Of our system Introducing lecture 10 So the Jacobian in general The Jacobian in general is given by this What does it mean in general? So not evaluated at the fixed point, but just taking the differential. So remember this is du So We differentiate this thing with respect to mu and we get The upper left entry in the Jacobian this thing with respect to v differentiated This thing with respect to v differentiated we get The upper right entry and so on If we evaluate the Jacobian at the non-trivial fixed point So not the zero zero, but the other one that we found the Jacobian equals to this By the way, I'm sure there are no typos here because I just did it manually Without trusting that people who did it two years ago did it right So this is the Jacobian At the non-trivial fixed point. We do this. Why? To analyze the stability of the fixed point Turns out that the trace is zero. The trace is simply the sum of these two diagonal elements. It's zero The determinant is positive. It can be shown if you calculated the determinant is positive And now if you go back to lecture 10 and look at the Illustration Which relates the determinant the trace and the types of fixed points Then you can see that given these conditions the fixed point is a center Okay, what does the center mean? You'll see I believe on the next plot Yeah, they're always building something. All right This is what a center means Now this is our fixed point Our fixed point is here if we just disturb it a little bit randomly We get orbits Increasing orbits. Well, it's kind of difficult to be very precise with this But imagine kind of a spiral increasing spiral increasing spiral which eventually settles down to this center Or this circle this one Okay Now if we're at this this center and we disturb it a little bit and just by chance We disturb it In in a way that now the dynamics is here a bit up Then again an increasing spiral would go to the second circle and so on So starting from a fixed point we end up in one of these circles one of these many infinite number of circles depending on the shock And obviously depending on on initial conditions, right if the initial condition is here If we start with the initial condition here, obviously we're going to converge To that circle So that's what the center means And this is nothing else but a business cycle, right look mu on the v axis We have well on the y on the x axis we have mu on the y axis we have v So what happens is The rage the rage the The share of output which goes to employment And the employment rate fluctuate That's another way to To quantify a business cycle instead of looking at output how output oscillates you can just look at employment rate and salaries Right, so if when employment rate increases look at it here when the employment wealth Let's look at Let's look at this circle The salaries increase Here employment rate also increases That's this assumption that when employment rate increases bargaining power of of workers also increase therefore Their their uh salaries would increase as well Right, so employment rate increases bargaining power increases salaries increase too So range uh share of output that goes to salaries increase At some point, however The opposite happens The salaries decrease Whereas the employment rate increases So that's kind of a predator prey dynamics that we saw with the rabbits and foxes Well, it's not exactly predator prey. It's like predator prey So these are the business cycles And again Some criticism Um It turns out that if you introduce just small Fluctuations in the model very small But which are nevertheless economically meaningful and for instance, I've given you an example Um, if the capitalists propensity to save is not one So it's not the fact that everything they get They've they save it and invest it, but if it's a decreasing function of the wage share Meaning the more they have to pay to employer to workers The less they would save For instance, they would keep some money in the bank. Let's put it this way They would not invest everything they get Uh, then the fixed point becomes asymptotically stable And there are no business cycles anymore, right? So this is kind of a Uh, little disturbance in the assumptions of the model Which completely changes behavior That's one criticism second criticism is People have shown empirically that it is possible That this happens so labor productivity and and The sum of the labor well the increase of labor productivity and Population to be negative If that is true We get this Then the fixed point This is the fixed share of output that goes to salaries Becomes larger than one which makes no sense economically, right? You cannot pay More salaries that you have money available So that makes no sense The same thing can be shown it happens with v so, um This model is not stable in a sense To empirical data, that's the most impressive thing Oh, let's say the worst thing All right, we're almost at the end Well, I still have one minute. So let's skip this one this clock um I just want to mention a few general assumptions about the multiplier accelerated theory You know this this all the models so far except the last one had multiplier and accelerator in it And I've tried to summarize kind of the most important ones. First of all, both Accelerator accelerator multiplier assume that's Investment or supply is very elastic So if you want to ramp up production or ramp down production is response in response to supply or output differential You can do this immediately, which is obviously not realistic Furthermore the the accelerator assumes under employment Why because if you need to increase production rapidly You need to be able to hire people at ease So there must be some unemployed people just waiting for you to hire them if you want to increase production right if there are no Unemployed people available. You won't be able to increase production yet. They accelerate the concept by itself Requires this in order for it to work Another thing is there must be no unused capacity. So all capacity must be utilized at all times Why is that? Well If there is unused capacity, let's say five of your factories are not working When you see a Differential in consumption or a differential in output You will not make new investments To build new capacity. You will just reuse Or you will just use some of this unused capacity So there would be no new investments yet. You will be able to produce more And this is only possible if there's there is unused capacity. Therefore the accelerator Concept assumes all capacity is utilized at all points of time, which is again not perfectly realistic I mentioned it already Both of these concepts completely neglect expectations There were you know, we had very backward very simple backward-looking consumers. They only cared about their last salaries And we had very backward very limited Investors who only cared about the differential in the last two periods. There is no forward-looking in any way Both accelerator multiplier constants Yes Yeah, so these are the most important assumptions of the multiple accelerator theory and depending on your personal preferences You may or may not relax them So disregard them in other words Or you you might just completely disregard All these all these theories by saying they're completely economically not realistic, but it's basically a personal taste, I would say This is the last self-study You have to implement the Hicks and the Goodwin models in Vensim as I mentioned This is a new self-study. I myself haven't done it yet. I assume it's possible It doesn't look difficult So let's hope I'm right Thank you, and I'll see you next. Oh, by the way, it's something important. Sorry For next week is the q&a session. Please come with you know your questions and and try to To just ask me like Lecture 10 slides 19 this graph. I don't understand Let's keep try to keep it short and concise so that there can be as many questions as possible All right. See you next week