 All right, good morning everyone Let me adjust the lights a little bit Is that better? Well now people in the back would tend to fall asleep. I know from personal experience All right, so Welcome to the 10th lecture three more to go And before we start as usual a few announcements now during the self study on Tuesday I Suggested that we spend half of the last lecture as a Q&A session So you can ask all kinds of questions regarding anything but then I checked the schedule of the course and it turns out that The last lecture was actually intended to be a Q&A session So we don't need to condense anything the 13th lecture there would be no new material. I would use the time to make a kind of a summary for for everything we did and I will also address the quizzes Lots of you wanted to know the right answers and why some answers were right and some were not And then we'll probably have still half the lecture for Q&A sessions for you so Please try to to have your questions In a kind of a condensed form so you can ask them directly for instance this light. I don't understand the diagram So that we can have more questions for that time So that's the last lecture before the exam. It's the exam is on the 22nd So the lecture before is on the 15th of December Yeah The quiz now, I I don't know is the quiz over now. I forgot what deadline I said It's alright. It's over so I can talk about it a little bit I Can sort one question that was the question about the policy integrated policies versus Yeah, the other types of policies because Yeah, I mean the basically None of the answers were correct Let me put it this way. I was kind of yeah, I mean, yeah, there is a very subtle semantic thing going on there and I'll talk about it, but basically everybody got one point there for free Again I Got so many emails Manual input questions will be graded by me By hand so don't worry if you get zero points. I get so many emails. Oh, I got zero points. I failed. What should I do? You should wait until I grade your manual input questions and that would probably happen in the weekend Yeah So that's it as an announcement today's lecture is going to be slightly more mathematical So those of you who wanted to have mathematics you have it and Yet the most complicated mathematical operations that we're going to do is Taking the derivative that's all All the equations that you're going to see Are simply taking the derivative setting it to zero and maximizing some function. That's all This is what? Standard economics kind of does Okay, before we start a quick reminder This is the self-study for next Tuesday. It's still from last lecture the coupled co-beb dynamics You remember the model it's in the in the last lecture today's Model is going to be the self-study for next next week All right a quick summary now a More general summary for the for the lecture what we did we looked at What kind of problems we're mainly concerned with and you remember we had this kind of ill-defined complex problems with no clear Solution space no clear algorithm to solve them and then we saw that it's actually important To have human problem solvers basically you And that's what the m-tech is actually preparing everybody for well, that's the Ideal at least to be a human problem solver to know how to tackle these problems to know how to define the problem first of all to define your goals and then to implement it properly and Implementing simply project management All right, then the next well the last part was controlling solutions and there are two ways to control solutions We saw two ways so far one was the systems dynamics approach, which is we're mainly focusing on Feedback loops, so you try to identify feedback loops in your system and And then you see how these feedback loops lead to explosion or or dying out and you know positive feedback loops always go together with negative feedback loops And so on and so forth Hopefully you now I have more experience with Vensim you can appreciate its positive and negative sites. I Certainly appreciate the negative sites but The positive sites are that it gives you a very very simple and quick way to see how these models that we're considering behave It's a very quick way If you use any other software package the learning curve in my opinion is much deeper So Vensim is a good thing as a kind of a first approach The second way to control This kind of systems is a non-linear dynamics way that is we try to identify what influences these feedback loops What parameters influence these feedback loops the so-called control parameters and These control parameters basically control the dynamics of the feedback loop and by extension the dynamics of your system And we saw bifurcations what bifurcations are by the way I was surprised to see that most people let's say the second Least successfully answered question on the quiz was what is an equilibrium point Which I thought is quite trivial But it turns out if I look at the statistics, that's the least the second least successfully answered question Which I don't understand why What is an equilibrium point? fixed point stationary point Does anybody know? Yes Exactly the dynamics of the system doesn't change. So the first whatever way you want to put it if you just want to Be very sparse you just say first of all difficult to zero If you want to be a bit more conceptual then you can say yes, the system doesn't change anymore and so on and so forth, but Apparently, I don't know it was a tough question We looked at predictability, so that means in our context chaos and Please remember that chaos is not associated with randomness with having random forces influencing your system thereby limiting your your predictability, no the Unpredictability emerges from the systems and annex and if you remember the logistics map, there was no random inputs there But something is missing Probably already saw it in your handout so far. We didn't talk too much about economics, right? We saw these models with different control parameters But there was no clear relationship to economics. We started last lecture with considering the The cobweb markets The coupled cobweb and the single cobweb and there we talked about the output market You remember this picture where you had the market of goods and services That's where firms sell their production, but we also had the input markets. That's the markets for Factors of production. That's where Basically labor is that's where labor is both in a sense capital also So today we're going to be considering or dealing with input markets Now most of you who take courses in macroeconomics microeconomics even the basic ones you would be familiar more or less with the concepts here But still for completeness Let me introduce a few few basic economic concepts. The first one is the economic theory of production So in the new classical economics and that is that means that For for the new classical economists the output of Your economy the income distribution of your economy the wages the prices are all set by the interaction Between supply and demand That's the basic idea of new classical economics. You have supply. You have demand. They try to match each other and By this process is all about market clearing mechanism. We said the price. We said the optimal level of production and so on So in this In this framework Production is nothing more than transforming the output to an input The input on output. So the input factors somehow a transform to an output and This is done basically by an abstract entity which we call production function, right and the production function here is given by By this F which takes your inputs the different X's here And gives you an output Y now it's important to see that The production function is an abstraction of the real production process We don't concern ourselves here with the managerial or an engineering problems That arise when you try to produce something we just assume That these don't exist and we're interested in how we can allocate these inputs In a way that maximizes our output. So it's a question of allocation We're not interested in Whether it's actually possible to do it from a technological point of view Basically standard economics is not concerned with this all right, so the inputs could be many but Let's say the standard is to assume only two Factors of production. Let's say the most important to capital and labor capital is all your machinery infrastructure buildings facilities and so on and so forth and labor is basically people and Yeah, so capital is denoted by K and Labor is noted by L. Therefore the production function is just this and As I mentioned here The production functions simply describes the technology behind Behind this transformation inputs to outputs it doesn't describe economic choices It doesn't describe for instance, how consumers change their preferences or how What kind of problems many managers encounter in implementing these these transformation? No, we're only the F just gives you the current type of technology if you'd like as an example you can see Production function like this simply linear combination of your inputs is a technology where all the inputs are substitutes Right, you can see that if we remove this input All we have to do So if we remove x1 all we have to do to have the same level of y is simply increase x2 That's all So inputs are completely substitutes That is the type of technology. It's not economic choice another example is This one here these are the minimum of all the inputs In this example all the inputs are essential so I have complimentary technology right for instance If x1 is zero If we set x1 to zero, so we remove x1 Your output would be zero so you cannot you cannot replace x1 by increasing anything else because you know of the functional form And then what we want to know basically with all this Exercise is what is a what production function describes real economies? That's what we want to know and this lecture is going to be about this basically who has heard The following terms cop Douglas production function Not too many Solo model All right, that's good. Not too many so it won't be boring for most of you Let me introduce first a few Basic notions probably all know them even if you don't know them. They're intuitive enough average product It's basically simply you divide your output your total output by the total input in that case we have y Total output divided by the total input Factor i x i so we get the average product marginal product which is more interesting is Basically the marginal revenue. Let's say the marginal contribution of each additional unit of input So it's simply the derivative of Your output right so if I change If I change my input x i by one unit My output changes by this much Right, so if let's say well Every additional employee produces 10 more boats Then the marginal product for the employees would be 10 Okay, now it's an empirical fact That the marginal product decreases all the time let's say it's It kind of the returns you get from each additional unit of input decrease eventually they become they could become negative Right, so what would that mean if the marginal product is negative you hire more and more people Not only are they not contributing to the output But you actually lose money for keeping them So you can think of Bureaucracy for instance, I would imagine there they have pretty negative marginal products But all right Yeah, so this is called the law of diminishing returns. It's an empirical fact Nothing complicated so far This is a picture which illustrates how the output interacts with the average product and the marginal product and It's taken from Wikipedia. In fact, if you just look for production function, you'll find this article It's pretty good. It has a lot of details. It has a lot of links to to other economic concepts and I find this this figure to be let's say the best way to summarize The slides so far production function margin protein average product So what we have here is the following. I'll try to use the other presenter Yes So what we have here is a typical quadratic or non-linear production function All right, that's Also, you can consider it an empirical fact what this means So first of all on the x-axis we have the units of input and in that case we mean units of variable input Right, we can have variable input and fixed inputs Variable input is labor, right we can relatively easily vary it in the short term at least Fixed input would be all your machinery or your facilities Which cannot be easily vary it in the short term, right? If you need 10 more people you can easily hire them But if you need a new factory You need time to to build it so The fixed inputs are more or less fixed. They don't change in the short term Here we're concerned with the units of variable input So input is variable input. What we have is The production function or the output of your economy Increases as you employ more inputs Which makes sense, but at some point it actually starts to decline, right? This is the place where the marginal product Becomes negative. So you have so many people That they're not only are they not productive anymore You basically lose money keeping them paying their wages. They're not producing anything It's not their fault by the way Imagine having a machine which can only be used by five people, but for some reason we've hired 20, right? They would be crowding So it's kind of your fault for hiring them Right, so this is a typical production function and we have so called three stages Let's let's see each stage individually Now it's difficult to see from this from this shape here, but At point a The production function has the steepest slope Right, so it's kind of an inflection. It's kind of an inflection point, right if you imagine By the way, I'm going to be using the blackboard a little bit today To make things clearer So unfortunately that won't be visible in the recording Right. So if I can make point a clear This is Something like this. So well, you can't really see it. So that inflection point So called inflection point is the point of maximum slope Right. So, I mean, yeah, you can't really see it, but It's like that So that means let's first start from zero You start employing more More inputs the slope is zero it increases the slope of the production function starts to increase increase increase At a its maximum But what is the slope of the production function? It's a question for you. What is the slope of the production function? Come again. Oh The production function is well, this is The production function This curve. So if you take I don't know five Units of input you would get whatever output that, you know, this function tells you So you see the slope of the production function starts from zero You employ more inputs more variable inputs. You hire more people the slope starts to increase increase increase increase maximum is here So again, what is the slope of the production function? midpoint I don't I don't understand Yeah, but what is it? Point a well in general what is the slope of the production function? It's the marginal product exactly. So in other words the marginal product increases increases increases it reaches a maximum here What would that mean in economic terms while you hire the first few people? They bring you increasing returns right look at the marginal product now. This is the marginal product These new people bring you increasing marginal products. So there are synergies created between these people right you create one person You create you you employ one person then you employ one more there would be synergies between these two people for instance and The total the total output would be higher Right, so there's increasing marginal products or in other words increasing returns To scale we'll see what this means at that point You know, this is just an example production function at that point the marginal product is maximum the slope Otherwise, it's maximum and then starts to decline Starts to decline decline decline at that point it becomes negative, you know, this is the marginal product now Let's look at the average product No, yeah, I have to write a few things here because by the way all the textual explanation is given in the slide But I feel that unless you see you see the equations It's kind of difficult to understand the The textual explanation not difficult of course, but it makes it easier So let's have a look. I would I mean I won't go too much into Writing these equations because I don't want to underestimate you I'm sure you can do it yourself But let's see what happens The average product is simply the output divided by the input, right? Let's try to find the maximum of the average product That's all as I mentioned the only thing we're going to do today is Derivation and finding maximums. So we differentiate this we want the first derivative of this to be zero we get the derivative of y times x minus y Divided by x squared, you know, this is just the simple rule of taking derivatives Now we want this to be zero, which means Can you see that in the back? Which means that this is zero, right? in more details Or we have this now if we integrate what we get is Plus a constant is Equal to ln of x plus a constant. Okay? Do you follow? Simply integrating Which means that y your output is now equal to To this where C is Just the sum of C1 plus C2 there are constants which can be determined from the initial conditions. We don't care about them But now you see this is an interesting result Your average product is maximized when the output is Equal to this But what is that? Let's let's focus on this Right, this is just a line This is just a line with a given slope, right? The slope of this line is e to the power of C a constant. It can be determined from the initial condition So that's just a line now. Let's go back to the slides or if you just look at your slides Let me show you right. We want the output to be equal to a line that goes through the origin That goes through the origin and has a certain slope Look at it again We want to maximize. Well, yes Let's calculate the average product now the average product is this If we substitute Our given y so these cancel out Right, so the average product it's simply the slope of this line It's a line that goes through the origin. It has a certain slope and intersects your y function We want to maximize this which means that we need a line with maximum slope But this slope this line should still intersect with your production function They should still be equal at some point and this is basically point B and the texture explanation says that It's the steepest possible line through the origin that touches the production function. Why steepest? Because of this we want to maximize the average product. This is the slope. It has to be steepest All right, so this is point B Right and the average product right the average product is this Let's calculate the marginal product What would the marginal product be at this point at this point? Well, it's a derivative of y with respect to x. What is y? Why is this? It's the derivative of this with respect to s which is e to the power of c So you see the marginal product is now equal to the average product and this is what you see on the figure Simply taking derivatives. That's all we're doing That's the point here the marginal product product is equal to the average product. What would that mean conceptually? It means that every new employee Contributes the average product. So if we stay at this point If we don't employ more inputs if we stay at this point The average product is maximum and every new employee Just contributes the average, right? So if every employee produces two boats And there are no Yeah, so if every new employee produces two boats adding just new and new employees would just increase your production by two Bolts your average product would stay two Right two boats per employee. So we're at that point If you keep adding more people though Your marginal product decreases so each new employee contributes less But the output increases right this is not a bad thing the fact that the marginal product decreases Doesn't it's not a bad thing. It's too positive marginal product Every new employee still contributes something positive. That's why the output keeps increasing At point C. It's maximum. That's when the marginal product is zero. You know if you remember from your economic courses Profit seeking firm in a competitive market always operates at marginal profit Marginal revenue equal to zero Okay, so that's the that's the thing marginal product is equal to zero. We have maximum output and so on So companies ideally want to be between actually point B and C ideally at point C So that's that's that with the function I spend I think way too much time on this So let's move quickly through the following slides. They're actually Easy returns to scale another basic concept in economics Constant returns. Well, what is returns to scale the question is if we? Increase the production factors by a certain factor Say by two with double all the other factors of production. What happens to the output? Does it double as well? does it triple or or Or less So constant returns to scale You're familiar with this I suppose constant return to scale is We in the factor by which we increase each production factor Is the same as the increase in your in your production? So if we increase all the inputs by 10% output is increased exactly by 10% increasing returns to scale Is intuitively More than 10% and decreasing is less than 10% That's That's that again a quick notion on elasticity what is In that case elasticity not of supply and demand but elasticity of the production function to the inputs Again, we define it as a percentage So the percentage change in the output divided by the percentage change of the input is this and it can also be Rewritten as the derivative of the log right derivative of natural logarithm of y is 1 over y times dy Derivative of this is 1 over x times dx and then yeah, they're equal. So that's the elasticity And this was the basic setup we need production function And and you know this kind of notions are average product marginal product now we're going to consider consider Given a specific production function and See what conclusions we can draw from it. It's a so-called cop Douglas production function The cop Douglas production function So remember I said in the beginning what we're interested in is Coming up with the production function which resembles real economy Okay, and this is one of the candidates for this is the cop Douglas production function We'll see how it was derived and we'll see why it resembles real economy But before we start I'll just give you the function. It's like this right, it's L the labor to some power Kate the capital to some power And then there is a constant which we define as let's say productivity or Technology efficiency Now you should be able to to to see that given how we defined elasticity is Alpha is the elasticity of the output to labor and Beta is the elasticity of output to capital right these are elasticity's I Mean very easy you should be able to just calculate this quantity this thing first take the derivative of y with respect to L and Calculate that you find the elasticity of labor then take the derivative of y with respect to K and You find the elasticity of capital their alpha and beta. I won't spend time deriving this for you. It's very easy now how it was developed well the Douglas Paul Douglas was interested in kind of coming up with a law or coming up with a Yeah, with basically a stylized fact about relating the input to the output of an economy So he assumed that there must be some universal law which is true for basically all economies and What he did he was not a mathematician what he did he looked at this Data from the National Bureau of Economic Research and he found That throughout these what is it nine years the share of output that was paid to labor Was pretty much constant But 75 percent so three fourth just pretty much constant. So this was one kind of evidence to him That there must be some governing law Which links input which transforms inputs to outputs and that is valid across across the board Then he contacted this mathematician Charles cop and he suggested the following form Now why did he suggest this form and? Why these numbers are one fourth and three fourth we're going to see now This is how the reasoning goes We have a certain production function y And Let's calculate the profit first of our firm the profit as this is set here Let me try to use this presenter again is This it's the price that you get for your output. So the price times your output Right. This is the in other words the revenue or sales minus what the capital cost you and Minus what the labor cost you so that's minus the cost of productions. So that's the profit. It's a simple definition revenue minus costs and The price and the cost of capital and cost of labor are constants that we need to identify So let's try to maximize this profit again the formula operation taking the derivative setting it to zero But now since the profit is actually a function of capital and labor so K and L We need to take the two partial derivatives So they're derivating the profit with respect to K and with respect to L in other words the gradient And setting this to zero. So that's what we've done here the derivative of right, so We take the derivative of this with respect to First of all the derivative of the production function just the production function with respect to Capital is this this is the partial derivative and we denote it by F subscript K and The same thing for labor right the partial derivative with respect to labor is L F L now We take the derivative of the profit with respect to L. It's this With respect to labor right this is zero. This would just be W We need it to be zero with respect to capital again. We need it to be zero So what you see here is That a profit maximizing strategy of a firm would be the following your profit Sorry your marginal Revenue, you know, this is marginal revenue of labor Right without the P is just the marginal product of labor multiplying it by the price that you get For this marginal product gives it a marginal revenue the marginal revenue that you get from one additional unit of labor should be equal To the cost of one Additional unit of labor or to the marginal cost So remember this marginal revenue equals marginal cost from profit maximizing firm And you'll see this in other courses. I'm sure the same thing for capital now We take some of the output we allocated to capital we take some of the output and we're allocated to labor in the next period if you'd like so we take some of the Some of the output and we allocate it to labor so This is your output the revenue we take alpha of the Output so some part of the output and we pay our labor with this We take beta of the output and we we pay our capital with this right So this is the total labor force that we have the the cost of the labor cost that we have We pay it by this fraction This is the total cost of the capital that we have we paid by this fraction You know, this was the assumption or the empirical fact that was observed some of the output Is taken and is used to pay salaries some of the output is taken is used to make investments in capital Now if you play a little bit by this equation, so this is the profit-maximizing strategy and this is more or less an empirical fact We go to the next slide now there's a small error. It's not dividing one by four. It's dividing two by four and In the same, please make a note in the same way. It's not dividing two by three. It's dividing one by three. I Mean if you just play with the equations you immediately see that so if we do that So let's combine two and four oops We combine two and four right what can we express from here? We can express this. I mean you can play with this We reached this To this relationship Right, so K star. I forgot to say K star and L star are the amounts of capital and labor that Maximize your profit, right? These are the optimal amounts of capital and labor just by solving these equations you can find them so we get that and In other words, so this is If you transform this a little bit then you get f of K divided by f Here is equal to beta divided by that In this so remember this this relationship Then in the same way combining one and three we get that relationship here and Now This is nothing more than the derivative of the logarithm Okay So if you if you differentiate the Logarithm of F with respect to K where you get that the partial derivative with respect to K divided by F Okay, that's that's nothing more so therefore the logarithm of F of K and L is Equal to this thing Right now you simply integrate we integrate that and we integrate that and we get this You can check that you can take the derivative of this with respect to K And we should get exactly That this relationship holds G of L is a function which depends on labor If we take the derivative partial derivative with respect to K, this is zero, right? There is no K here. So this is zero the constant is zero all we get. I mean we can take it It's just let's just do it. It's one line Yeah, I don't know where I left my chalk I'll do it here now. Let's take the derivative of that thing So the derivative of F with respect to K is this Right, this is the derivative of the logarithm. It's equal to beta times the derivative of ln of K with respect to K is one over K Everything else is zero. What is that? It's exactly that equation over there On top right, right? The same thing for the second. Okay So now we have the following relationship And look at this these two things are equal So we just equate the right-hand sides if we do that Again, let's be thorough. All right, we say beta times ln of K Plus some function plus a constant is equal to alpha ln of L plus Another function plus another constant. Okay, there is no thing here, right? So how can we make these two things equal? Well, one way to do it is if that thing is equal to this all right and This function is equal to this That's one way to make them equal and obviously I we don't care about the constants But we can make the constants equal to All right. So in that way ln of F would be equal to beta times ln of K plus the function GL which we impose To be equal to alpha times ln of L and this is what it says here plus a constant, of course So this is the constant. We don't have to set them equal, but We can if you want to but we don't have to Yeah Because look they must be equal They must be equal because the left-hand sides are the same Right, so they must be equal and one way to make them equal is to just impose what this function should be and what this function should be That's one way All right, so if G of L is equal to alpha times ln of L and H of K is equal to that Then they're equal Clear. All right. So this is exactly what I said plus a constant now we just Exponentiate both sides just to let me just finish this slide. It's really quick. We exponentiate both slides both slides both sides and We get that This is the form that we saw in the beginning and A is a constant which is e to the power of C It's just a constant. So let me recap how we did this how we came up to this conclusion We started with assuming that the firm tries to maximize it profit That was one thing second We assume that some part of some fraction of the output is taken is used to pay labor costs Some other part of the output is taken is used to pay capital costs That's all these two things we combine them We played with them and we reached to the conclusion that a production function which maximizes profit and and Allocates ratios of output to labor and capital looks like this Okay, and yeah, let me just finish this slide We can calculate the returns to scale of this function Let's just increase the inputs by a factor of lambda, right if you do that and And you use the property of the of the exponent then you reach to the conclusion that If you multiply the inputs by lambda your output is this This is the output now if you want constant returns to scale as This is also kind of an empirical fact. We'll see data later on then alpha plus beta must be one, right? If you want increasing returns to scale it would be larger than one and Obviously smaller than one for decreasing returns to scale But if you assume that it's one you don't assume it. It's an empirical fact that these guys observed then we get we get this and Yeah, that's all This function also has the property that the inputs are essential if you set any input to zero The total output is zero right zero to some power is defined to be zero, right? So we get We get the whole thing is zero so that is how The cop Douglas function was derived and in the following slides we're going to see Empirical validations for this so we'll do this after the break Let's continue Okay So what I showed you is The cop Douglas production function how it was derived why it has the form this for the following form And then we looked at conditions under which we have constant or increasing or decreasing returns to scale Just a visualization. This is how it looks like With the following parameters, so this is capital and labor what you see is if you hold labor constant And you just increase the capital the output doesn't increase I mean it the output still increases But not that much if you hold the capital constant and you increase labor. Let's just Look at that. Let's hold the labor constant here The first contour line here labor is constant capital increases The production function the output increases like that if we hold capital constant and we increase labor The output increases even less compared to before Which means? What does it mean in terms of elasticity? It means that in that particular case the output the elicited output to labor is less than the elicited output to capital right which is in fact the opposite of what you would expect and The reason is that look alpha is 0.35 So alpha was the elasticity of capital The elasticity of labor would be zero point. What is it six? 65 so yeah 65 so that has Reprocussions for your investment decisions should you invest in capital? Holding labor constant should you invest in labor only and Probably the best thing to do is to invest in both Right the largest increase is when you invest in both Like this all right, so let's see how the cob Douglas production function can be used to honor Before that before we see that this is the empirical data that the two guys worked with so these are different countries Australia and Different parts of these countries and what they saw so I actually I don't know the details what the observations are I think these are regions within the country So what they saw was that values of K? K was capital or let's say the the Ratio of output that goes to capital is this The ratio of output that goes to labor is this All right And you saw that the sum so this is alpha and beta right the sum Is more or less constant it's one so that's how I Mean that's why they posited that This function is constant returns to scale because alpha plus beta is kind of close to one and Besides what you can see also here is that The average Actually, sorry Yes, it's the other way around. Yeah, it's kind of confusing K. Yeah, please make a note K is labor And J is capital so this is labor and the average of this Where is it the average of this is about zero point six and in? Different courses you would encounter kind of slightly different Alphas and betas for the co-blogger's production function could be one fourth and three fourth could be one third and two third Depending on the on the empirical data on the consideration here. It's sixty percent which is Which is closer to two-thirds all right a Few properties of the co-blogger's production function the marginal products are positive which is nice So oh another another typo in this slide This is alpha over K Brackets the whole thing to the power of alpha All right, so just put brackets here To the power of alpha the same thing here alpha Over K brackets and now it's to the power of alpha minus one not one minus alpha It's a big mess. I know but I'll fix it In fact, I mean if you just compute this quantity you would immediately see the right result So in in in fact, yeah All right, so these are positive the marginal products are positive which is nice and they're decreasing So if you take the second derivative, it's negative. So they're decreasing Just a few properties Now let's see how this production function can be applied If we well before we see that again You would probably see the function in a different form different functional forms linear form as We just saw but also logarithmic form. You just take the logarithm. That's all and Well, yes the computation of the elastisted the elastista that I talked about is now here So you can see that the elasticity of labor is alpha and capital is beta So that's just the computation So finally we can take this production function and assume that this is the production function that governs our economy Okay, and we want to see how the economy would develop This is the so-called new classical growth model or the solo model which is Taking the production function the cobb Douglas production function applying it to an economy and see what happens to the economy is your very different Factors of production. So that's basically the rest of the lecture Yes, so these are introductory remarks not so important. Yes, the only thing to mention is that That this is basically The solo's contribution was that he assumed now that labor is a factor of production For some reason before him. This was not this was not the case. So he included labor In into the calculations Alright, let's start now First the assumptions of the model Let's quickly jump to this equation there. We have the output the total output of the country is the sum of consumption investment government expenditures and net exports in other words Take a given output And you try to divide it now What to do with this output you can consume all of it If you consume all of it, you won't have anything left For tomorrow You can invest so investment here means you invest in capital. You invest in a factor of production And in the solo growth model the investment is only in the capital So the capital factor the factor of production capital not labor So you can take some of this you invest in capital which would increase your capital stock Which hopefully increases your output in the next In the next time period Or as I said you can consume it for now Let's disregard the government the government spending and the net exports. Let's assume. It's a closed economy There is no trade. Nothing's going on in the simplest case by the way, if you take More economics courses for instance development economics It was an elective when I was when I was doing the master's degree It's the the course is a lot about the solo growth model Extensions of the solo growth model including trade here. We assume no trade close economy net exports is zero All right, but For now our output would get we take a certain output and we divide it with this we have to make a decision How much do we consume? Which means how much basically is lost? You know consumption means we just lose it. We don't We don't get any return on consumption except kind of personal satisfaction, which is kind of unquantifiable or We invest it Into capital, so if I just draw a little diagram To make things a little bit clear What we had so far are the factors of production K and L Yeah, we had K and who had L We combined these two factors somehow This is our production function and we create output All right Then we make the decision. What should we do with this output should we consume part of it? Should we invest part of it? But now you may be wondering. Well, how does C and I relate to K and L? Right, so this is the input What produces the output and that is what happens to the output once we have it? So it turns out in in this model The in the investment is used Well, it's actually So this is kind of a dashed line the investment Maybe I should have used a different color The investment is used Not labor is used to increase capital So this is the feedback, right? This is how our actions our decisions On the output side influence that is the what happens on the input side And in the solar growth model the L is assumed to be Exogenous governed exogenous, so there is kind of a Exogenous labor supply people come to the country people leave the country We we don't care about that So consumption is lost output, you know, it's Goes to nothing Investment feeds back to to capital. This is the relationship between The output side and the input side Okay Just as a conceptual overview Probably I should create some diagram and put it in the slides good Right. So as I mentioned the labor force grows exogenously right there. So this is basically the the percentage growth of Labor is constant and somebody Just got maybe Determines this this and all right. Let's analyze the model further Our production function now depends only on K and L We split the output between consumption and investment and remember the decision we need to make now is How much should we spend on consumption? How much should we spend on investment? In fact What people are mostly interested with is What is the Let me first say the following the investment is assumed to be a part of the output Right, it's a linear part of the output. So we just take some fraction from the output and we invest it back to capital So this is s the savings rate So at each time period the government has to decide how much do I save? The rest I will consume people in my country will consume and the question is what is the savings rate? Which maximizes your consumption? Because for some reason maximizing consumption is Let's say one of the most important things that people Expect from a government So that's the question we want to answer. How much would we save and Invest in capital so that our consumption is maximized now you can think if you don't save anything You maximize consumption today, but tomorrow you have nothing Right, so the actual equilibrium is zero consumption Because your capital gets destroyed If you save everything You would constantly have more and more capital because you invest everything into capital But consumption is zero all the time because you don't you don't consume anything So you need to find this kind of middle ground where consumption is maximized Yeah, and then now we're going to find this This is the whole purpose of of the following slides The feedback mechanism I already talked about It's this one you take the investment And you use it to increase your capital by the following Relationship so all the investment is simply taken and added to the capital and of course we have to account for depreciation of capital right So let's say you have a battery if you leave this battery for let's two years It will depreciate and at the end of the day it won't work Right depreciation or for non tangible assets. It's a mortization, but it's the same idea the Intrinsing value of something declines Basically physical process. So this is how your capital or the feedback mechanism how it works your capital increases or decreases By the difference of what you decide to invest minus depreciation So you can immediately see if you invest less Then what the depreciation is you cannot cover the let's say destruction of capital the intrinsic destruction of capital So your capital stock would decrease If you invest more your capital stock will increase All right, so investment is simply a fraction s of your output and minus that now what we do next is Yes, we basically try to find the level of savings the so-called golden level of savings that Maximizes your consumption. This is the whole purpose of the exercise. So let's see what happens Now first we'd like to eliminate our two factors of production we have K and L But it turns out it's more convenient to work with the ratio K and L And quickly I want to show you There is there should be three of them No, maybe not Let's use this one quickly. I want to show you what happens if we use this ratio So I will use that part Here all right So we have our production function Which is this Now let's Let's divide the whole thing by labor Okay, we divide it by labor This now y over L y over L would be Output per person or output per capita output per individual employee But this is interesting now the cop Douglas production function has constant Returns to scale or we want we basically impose the condition. It has constant returns to scale. What does that mean? It means that if we scale the factors of production by a given rate Let's scale them by one over L. All right one over L times K And one over L times L Constant returns to scale. What does it mean? What what should be the equal sign be all right? We scale the cop Douglas production function by one over L each factor of production What should it equal for constant returns to scale? Yes So what should I write F? this over L Do you agree? all of you Really, I mean K over L one so in fact we define K over L to be the capital the capital per capita or the capital per person You know, this is let's say if my country has five factories and five people living there Each person would get one factory So this is the capital per person capital per capita and this is more K in the slides so in essence What we've shown is That if we scale the production function By L one over L the output is this so it's basically One over L times F And now F This is small K So this is the production function per person Okay No, let's go back to the slides Yes, which term this one. Yeah. Yes. Oh, yeah. Yeah. Yes. Yes. Yes. Thank you. I'm happy you all follow Good So this is what's in the slides We define the capital per capita small K Okay, and then your output per person is just this F of small K Now there is another Little thing so let me see if I should derive this for you. Let's try to explain it Remember here we had the following thing Yeah, let me derive this for you and that's the last thing I will derive actually. Oh, I should Yeah Yes, I can use that part there So remember this equation. Okay, I will use different color Let's say blue the equation from the previous slide slide 18. So we had This is capital K now is equal to investment which is S times F of K and L minus This Depreciation now let's divide by L again. We divide everything by L Oh, no What we're interested in is what we're interested in is Small K how the dynamics of small K behaves right this is K over L Capital K Over L Okay, we want to find an expression which more or less looks like this. This is the capital per capita Well Again taking the derivative. So this is small K It's very small. I guess if you see it from from behind so that is Derivative of K times L. It's simply differentiating minus K times derivative of L Divided by L squared Let's continue here This is We've written. Let's yeah, this is capital K divided by L minus Capital K times L divided by L Okay, now capital K's dot divided by L is simply that thing Divided by L right. We simply divide this by L. So what we get is S times F of K L minus this Divided by L minus capital K times What is L dot divided by L? It's N. You're absolutely right It's like this Thank you actually now the calculation who said that Thanks. So basically now the calculations would would be right, right? I just take this L would cancel out. So that's an L That thing is still over L squared now L dot over L is N And then we just have a leftover factor of L K star is now equal to What is that We know what this is already It's S times F of small K. Now everything is small K now. I Just did this here. Oh the other board. All right And then we have minus K over L so capital K over L and That is N Plus plus Delta right Let me see. Yes We're almost done. What is K over L? It's small K. So this is what we wanted the dynamics of the capital per capita and This is the equation you find on Slide 19. Yes, it's like slide 19. So the next slide There you go, this is the equation So what is that the capital per capita? I mean, it makes sense. This is the investment that we do in the capital per capita and this is the cost of The capital per capita, but now the cost Would be basically the sum of the cost of capital plus the cost of labor And you can think of N is the cost of labor good So let's try to well now this is K and the consumption Per capita the consumption per person is defined in the same way as the normal consumption So let's try to find the stationary points of this of this equation So what we want to find is at what point the capital per capita does not change anymore So it's zero it means that each person gets two factories forever. It doesn't change anymore Well, we know what to do. This should be zero therefore that should be equal to that. It's right here and we call the value of the capital per capita which Satisfies this equation. We call it stationary K stationary. So this is the stationary value Right, so this is it And from here you can express s and if you substitute s right here you get that the stationary consumption Per capita, that's what we're interested in the stationary consumption per capita is equal to that It's simply very easy just replace s Express s from this equation put it here and you get that equation Why did we do all this? We finally ended up with the consumption per capita Because we want to maximize this Right, this was our original goal. Let's maximize consumption per capita again familiar procedure how to maximize a function First derivative should be zero first derivative of this is simply f prime of K Minus n plus delta and Now we call the value of K this stationary point that sets the first derivative to zero. We call it gold capital per capital per person because it maximizes consumption per capita that's golden for consumerist societies and Again familiar familiar form the marginal Product of one unit of capital per capita should be equal to the marginal cost of that unit Right, so this is again the same the same thing as we saw before Good this was for a general form of a production function with constant returns to scale Now let's plug in the Cobb Douglas production function is simply calculate the actual values of K gold and K stationary this is what we're doing now and it's simply Plugging in the right values from the Cobb Douglas production function into these equations Right, so you plug in You can express The production per person is this Yeah, you simply divide by L and so on so it's like this. This is f of K and then you simply plug in F of K into all the equations. For instance, you can plug in f of K F of K here, right you said it basically well, you said this thing to zero and Instead of f of K you put a times K to the power of alpha You make this zero and then you calculate the stationary K K stationary if you calculate K stationary, it's this So this is what is that? This is the stationary Point for the capital per Capita per person Where So basically the capital doesn't change anymore the capital per capita does not change anymore This is the stationary point the stationary value of the production is this And now if you plug in K stationary into that equation Into that equation is here. You want the first derivative of that to be equal to this and what you find at the end is that the golden ratio so the capital per capita value which maximizes your consumption is equal to this and Just comparing that in that you see that the optimal production the optimal savings rate which maximizes your consumption is alpha So what you've basically what this means is you have your empirical data you find the elasticity of capital, right? This is alpha This should be a savings rate if you want to maximize consumption This is the bottom line You will have to do the cop Douglas or the sorry the solo model is an X as a self-study in the next next cell next next Exercise and then you really understand how these different K gold and K stationary come to existence These are some computer simulations In Vensim by the way, so you'll probably have to do something like that Let's focus on this graph right, this is the production per person F of K small f of K production per person you see at some point it saturates That's when we have reached K stationary The different values represent different savings ratios Now it would be easier to understand all these graphs if we quickly jump Okay, this is this is interesting now. This is the most important graph This is the consumption per capita. This is Exactly what we're interested in consumption per person and you want to maximizing to maximize it So the green curve is saving straight of 0.3 The red curve is saving straight of 0.5 and the green sorry the blue one Yes, the blue one is 0.3 the green one is 0.7 and this is 0.5 Remember the elasticity alpha was given and It was 0.5 So this equation tells us that the optimum savings rate should be 0.5 But we've played a bit We don't choose 0.5. We choose 0.3. We get the blue curve 0.5 is the red curve and you can immediately see that this really produces the optimum the largest consumption per capital in the long term So you have to play with this Yeah, the excesses is time this is time, okay all this Also the mathematical analysis considered something evolving over time and reaching a stationary state and Then we try to find out. Okay. What is the stationary state? How can we maximize something now? What people have been using the cop Douglas or the solo model for is not just to see evolution over time It's kind of a trivial thing Because you always know that you would reach an equilibrium But what people have been using it for is to compare different countries with different savings rates with different capital per capita and see basically why some countries are poorer than other countries, what is the reason for this and This is the Domain of comparative statics in a sense. So you compare stationary Equilibriums you've already reached these equilibriums and you just compare different equilibrium solver different countries to see What can we what what can be done in order to bring one country closer to a higher equilibrium? This text here is just explanation of that graph and I'll just explain that graph without explaining the text Well, we yeah, maybe we well, let's is that confusing if I use it too much Like this spotlight I Assume no then All right. So this is the depreciation Depreciation of capital per capita Remember so D is the depreciation of your capital and is a depreciation of labor right or in other words the cost of labor You can also think of it in this way. So this is what we have to pay to support a given Capital per capita. This is the investment that curve So s a fraction times the output y and you can imagine that the output y Would be just the same it would have the same shape But it would be maybe higher. It would be like this and s times y is just Given fraction of it What we know is let well, let's start from here this point This level of capital per capita What happens is that our investment is higher than the depreciation What does that mean? It means that we can create new capital Because we cover the depreciation, but we there's also something left over to create new capital Therefore the capital stock the capital per capita would increase in the next time period it would keep increasing until this point until The investment we make are just enough to cover the depreciation of the existing capital per capita And we don't create or destroy anything. So that's the equilibrium here This is where that thing is equal to that thing and you can see that in the previous slides in the mathematics If we start from this point, however If we're here for some reason Then the investment we do is not enough To cover all the depreciation Therefore some of the capital per capita would be destroyed we couldn't invest enough and in the next time period The capital per capita would decrease eventually reaching the equilibrium Now this is the equilibrium. What happens if for some reason the country decides to save more? Well the saving curve is basically shifted up Right now S prime is bigger than S. The new equilibrium is here. It's just the intersection of these two points So now the everybody in the country Would have more capital per capita because new capital would have been created By investing more so instead of me having two factories I would probably have five But we'll see that this does not mean that I can consume more This is not Approximate for consumption Just remember this now the capital per capita Which is not always gadgets is Increases what will happen now if The cost of capital or the cost of labor increases well that curve would simply Change position like that. The new equilibrium would be here. Therefore everybody would have less capital per capita For instance if we have new people coming into the country and Would increase So the curve would shift here. So everybody with the same output would have less capital per capita. That's that's natural But now and this is the text just explains these two graphs nothing more if you understand how the shifts The shifts influence the the equilibrium points and you're fine This text now Explains that graph. I'll do the same now. I'll explain this graph and this goes back to the idea that More capital per capita does not always mean more consumption In fact, there is only one Value of the capital per capita that is the golden value that maximizes consumption and we'll see why Okay, so This is The output per person f of k small f of k the green curve Okay This is the depreciation as we saw it before now if we invest So what is first of all, what is the stationary state? We know that if we have a given investment curve Let's assume The investment curve is like that It crosses it crosses the depreciation curve here Okay Right right there Just assume that Then we invest this amount We invest this amount and The consumption is simply the output minus investment So the consumption is in fact the difference between the output curve and The depreciation curve right this difference here is the consumption and you see that The consumption is maximized here when that difference is highest right, so there's only one level of Capital per capita, which is the golden level where the difference is maximum if you have an equilibrium point here That difference will not be maximum. You would have more capital stock per capita. I would have five factories instead of two But that doesn't mean I can consume more Because to consume I need money Okay, and Mathematically it can be shown that the maximum the optimal Gold level of capital can be Graphically calculated by Taking the derivative or the slope of the production function At the point where the slope equals the slope of the Of the depreciation curve So at that point The slow the derivative here the slope of the curve is equal to that slope So it can be shown that this is how you can graphically determine it Is that clear? So more capital per capita does not mean more consumption But whether that's a good thing or bad thing depends on let's say what's important to you If you want to maximize consumption in your country today Then probably you would be happy if you have if you'd have this point here If you don't want to maximize consumption if you let's say I want to Still have fairly good consumption, but you won't have more capital per capita Then that's better So as I said this slide explains that graph Actually that slide yes, so that's like basically tells you if you have a rich country Rich country defined as more capital per capita What can the rich country do to increase consumption is to decrease investment? Right, so the country should stop creating so much capital per capita But create more consumption in the same sense the poor country here Poor country here can increase consumption by investing more Basically moving to that point So these empirical implications of the solar model should be clear by just Keeping in mind That basic logic of what happens when you move the curves around. Oh Oh, so this basic empirical value empirical results Output for workers should be higher and so on. I leave that to you. I mean it's really If you understood the graph so far, this is this is Trivial the only thing that I want to finish with so this is Yes, what I mentioned just now for rich countries and poor countries Some critics that's important first of all the saving is kind of Exogenous somebody decides what the saving rate should be it's not included in the model Second thing we completely ignore the technological progress a We completely ignored the influence of Let's say we have two countries with the same Capital per capita the same saving straight the same depreciation but different levels of technology How would what would that mean for their for their well-being we completely ignore that Of course human capital is not incorporated So education if you invest in education now You minimize consumption what you reduce consumption with the potential promise for Increased consumption in the future In fact the biggest critic of the solar model the biggest one is equilibrium as Let's say standard economics The solar model always assumes that the countries converge to whatever equilibrium and the equilibrium is determined by Depreciation rate by production function There is very weak empirical evidence that countries ever reach an equilibrium at least we haven't found maybe japan nowadays, but Except for japan we haven't found a country which has reached A stable equilibrium point without changing without any changes This has to do with the fact that the solar model ignores trade We don't have any net exports In real life, we know we have business cycles Oscillations in GDP Which is definitely not an equilibrium. So that's the biggest critic of the solar model It doesn't tell you How to reach an equilibrium it just assumes there is an equilibrium eventually would reach it if that's true or not We need empirical evidence and so far we don't have it And these are the questions, uh, wait These are the questions Yes, so that's all if you have any questions. I have time No, so thank you