I couldn't connect but uh so the FED just hike by 25 basis points and uh as people expected you know this is the way it works when there's lot of uncertainty essentially the f starts communicating what's going to do and the communication was very clear that that 25 basis points was uh to be expected and and apparently I was reading this right now it was released three minutes ago four minutes ago um they also said that that further hikes are no longer guarantee so remember that we saw that expected ER hikes sort of we saw several several expected hikes for the next few months before the svb mes and right after it we sort of the whole thing declining and and at least the minutes are consistent with that um so there we are so not being uncertain I imagine the markets are rallying or something like that at least for the next 10 minutes or so but we shall see anyway so but today we we're going to really start the I'm going to show you sort of the first model of economic growth uh and uh before I do that who knows who that person is no no clue he actually he's Robert solo he wasn't he an Emeritus professor at MIT together with Paul Samson essentially he's responsible for building the economics department at MIT and he won the Nobel Prize in 1987 I was a student then here H and uh and uh for his work primarily for his work on economic growth and so what we're going to do in the next two three lectures are essentially things that pop solo developed many many years ago the basic mechanism you know remember that we had this Kian cross before where we have this multiply in the Goods Market and aggregate demand feeding into income and so on so forth that was sort of the star mechanism in in in short run macro in Long Run macro growth Theory this is sort of the the key mechanism and you can think of it as the following at any point in time an economy has you know factors of production primarily labor and capital that Capital stock labor is more or less fixed so depends on population growth things that are sort of difficult to to control or they're not really that endogenous to to economics not at least in the current time times many centuries ago yes they were we had this malthusian theories in which no population growth deter determine growth because food scarcity and stuff like that but that's no longer the case fortunately for in most parts of the world so but what can what can change o over time and quite a bit and it depends on economic decisions is the capitalist stock but at any point in time there's certain capitalist stocks which combined with labor give you some certain output out put is income part of that income will be saved as we have seen and that those savings will be used for investment okay but investment is nothing else than capital accumulation so this income will lead to saving which will fund investment which will change a stock of capital will Fed into Capital stock that will fit into income and so on all this is happening very slowly because the Capital stock accumulates slowly I mean but but but this is what is happening and so all the mods we want to look at certainly the model we're going to look at in this lecture is all about this mechanism okay so let's remember what we did in the previous lecture we uh and I'm going to assume that population is constant I'm going to relax that at the very end but assume that the population is constant and equal to n and remember we're not woring about unemployment and stuff like that here um so output per capita of a person H is y/ n and we remember we had an production function f of K and N then because of Conant returns to scale we could divide by n on both sides everything and we ended up with this relationship so output per person is equal to an is a is an increasing function of capital per person it's an increasing function of capital per person but it's also concave H function of capital per person why is it concave that is why is it increasing at a smaller Pace yeah decreasing marginal product of capital exactly you know for fixed amount of Labor the more Capital you put in in into production well output keeps expanding but by less and less because it has less and less labor to work with each each unit of capital perfect that's very important uh then let's we're going to work in close economy I haven't open it I'm going to do that after uh quiz two and so and I'm going to assume also no public deficit so G equal to T capital T and in that case then we know that ER private investment private savings equal to private investment okay that that's a away with the right the is curve um so that's that's that's not new I'm going to modify a little bit what we did in the short run er um and I'm going to assume that that savings is proportional to income so savings little stimes y notice that that this is is different from what we did in the short run in the short run remember we had a czo floating around we had a constant in the consumption function so savings which was equal to income minus consumption also had a constant floating around now that that constant was important in the short run mod because it were approximating for a bunch of things that are not related to short-term income wealth you know the price of houses stuff like that we put all that in that constant there when you think about the long run though H most of those things that we excluded there asset prices stuff like that tend to scale with output as well so so this is these are inconsistent on the surface but if you were to fully work out what is behind the c0 in in the consumption function then this is not about approximation they're not that inconsistent because endogenous thinks that over the long run scale with income I mean you know wealth tends to rise with income and all these things tend to move together at not at high frequency you can have all sort of fluctuations but over the long run they tend to scale up together so that's going to be our saving function so that means that we know an equilibrium this is not investment function we know that in equilibrium investment will be equal to H it will be proportional to income okay so remember what we were through the going through the box we had at the top of the box we had Capital that led to Output we're doing everything ter per capita that Les led to savings and that funded investment okay so that's that's what we have so this growth model is really about these three functional forms and then a dynamic equation for the stock of capital so the evolution of the stock of capital capital will increase because of investment that's what investment is it's an increase in the stock of Capital H but but it will also decrease as a result of depreciation I mean things do break up no h once in a while and so uh and different type of capital have different depreciation rates equipment depreciate much faster than structures and buildings and so on but we're going to not going to make those distinctions here but you see this tells you the Capital stock at t+1 is equal to the Capital stock we had before minus what is depreciated of that stock of capital plus any new investment we do today okay in per worker terms and remember that for now I'm keeping population growth constant uh equal to zero not not population growth constant yeah constant but equal to zero so population is constant I can divide this both sides by n no and I get that Capital per worker H per worker or per person is equal to this expression here I did two things here I divided by n and I replaces replace this I function this investment for savings okay because I know in equilibrium they have to be equal so I have that um I can rewrite this know just substract KT over n on both sides and then you get a the change in capital per person is is an increasing function of savings and decreasing of depreciation okay so the last step that is important in this model is to so here I have a difference equation for Capital but we have an output per capita on the right hand side but it turns out that that I know that output per capita per person they said per capita per person the same thing per worker it's the same thing in this part of the course H so this is equal to H is a function is an increasing and concave function of capital per person okay so this is I would say is the sort of fundamental equation of the solo growth model it says the change in the stock of capital increases H with uh um with investment of course and decreases with depreciation and both of these Expressions here are increasing functions of the stock of capital per person okay so let's let's try to understand what is in here so why uh so this is linear obviously because depreciation is linear you say say you you lose 5% of your stock of capital every year because it breaks down obviously the more Capital per person you have the more units of capital you're going to lose this is in units of capital per person have a larger stock of capital you're going to lose 5% of of a larger number is a larger number so this and this is proportion is linear now this one remember this comes here from the saving function and this term here is equal to income per person H now suppose that that you start in a situation where the Capital stock is relatively low and this is positive what does it mean that this is positive I mean the implication of this being positive is that the stock stock of capital per person will be growing but what does it mean that this is positive in words I mean if you have a stock of capital there are things that that reduce the stock of capital and there are things that increase the stock of capital this is the thing that increases the stock of capital that's the thing that reduces the stock of capital so if this is greater than that what is that what does that mean that means this is positive but in words what is happening let me simplify this is remember this is just investment per person well this just says that this economy in this economy there is more investment than destruction of capital due to depreciation okay that's what this means this is investment and this is positive means that the investment that which is a function of saving the saving rate and stuff like that is a function of the funding available for investment is equal to the funding available for investment um if this is positive well this is greater than the stock of capital another way of saying it you need a minimum level of investment in an economy to maintain the stock of capital the minimum level of investment you need to maintain the stock of capital is equal to the depreciation so 10 machine breaks you need to invest at least 10 Ma machines in order to maintain the stock of capital constant okay now if this is positive it means you're investing more than the machines are breaking down now suppose you start in a situation where that's the case so that means the stock of capital is growing I suppose I ask you the next period do you think that Gap will be larger or smaller than it used to be actually that's not a great question because I'm not doing it in the right units for that let me ask you a variation of that question soose we keep going after a while do you think that number will get larger or smaller after a after let it run for a little for for quite a while do you think that number will so remember say we start for some stock of capital this is positive if this is positive means that the Capital stock is growing that means this guy is growing and that guy is growing and they're growing equally but after a while do you think this number will get smaller or bigger after a long while just to make sure that my approximation is not Paving here exactly it's going to get smaller because this guy keeps growing linearly when stock of capital and this one is not it's concave know at some point this income sort of you need to put a lot of capital for for income to keep rising and therefore for saving to keep rising and therefore for investment to keep rising and at some point yes it won't be able to H to really grow I mean you're going to be using all your investment really to maintain the stock of capital that's sort of the logic of the solo model and it's all in this diagram so this is diagram you should really really understand well and control it and play with it and all that it's the equivalent to your islm model in in in the first part of the course so look at what we have here so I'm going to plot output per worker ER per worker per person against Capital per worker here and so this this red line here is just the depreciation okay this term here and that's is a linear function of the capital per worker okay that's what it is H the blue line here is output per worker which as we said is a concave function of K Over N remember I showed you that production function last in the last in the previous lecture there you are okay what is the green line is investment per worker which is equal to saving per worker and saving per worker is little s the saving rate times a output so it's little s which is a number like 0.1 if if if we're talking about the US and you know point4 if we're talking about Singapore it varies a lot across countries but but ER but so this this green line here is nothing else than this blue line multiply by a number that is less than one that's the reason it's lower okay okay good so the point I was describing before is was a point like this remember the point I was describing is suppose that the economy starts in a point like this one k0 over n well and I want to understand the Dynamics of this economy how will it grow over time so what you have see here is that that ER at this level of capital per worker investment is greater than ER than depreciation so that's exactly a situation where this is positive okay that distance here is that okay and the reason I sort of said I'm not going to do any local analysis because we could have started with a K Over zero over here and then that number is growing but it's growing if you were to normalize by the stock of capital is it's declining that's that's that what I didn't want to do that then but now that's what so let's look at this case you're in a situation where this is positive if this is positive it means the Capital stock per worker is growing so you're moving to the right in the next period you're going to be here that that means the capital so keeps growing but by a smaller state eventually uh the investment is entirely used for a ER recovering from the depreciation of capital so covering the depreciation of capital and at that point the capital stop stock stops growing we call that a stady state stationary State we stop there okay so that's a steady state of this mode that means this economy regardless of where and do the analysis from the other side suppose that you start from a situation like this you start with a lot of capital okay well if you start with a lot of capital in this economy what happens when here well what happens here is that the investment you're putting into the ground in this economy is less than what you need to maintain the stock of capital which is the depreciation and that means the stock of capital will be shrinking over time okay you're moving that way so regardless of where you start in this economy if I you I ask you the question 100 years from now where are you you I tell you tell me I don't need to know where you start from I know that we're going to end up around there you can either you start from here you go there from here you go there and so on that's the reason we call this a stady state it's where you converge in the long run okay now this is already interesting because it tells you you know and at this Mo at this point here the economy was growing the Capital stock was growing and and and and the and output was growing you see the capitalist if you start from here the capitalist stock is growing well output is also growing okay you're moving up there okay so you had growth that kind of growth we call transitional growth know it goes from one point to another point it's not a permanent growth grow it's transitional growth it's the fact that you were away from your state state and you're going convergent towards your state state a lot of the growth we observe and the difference of growth we observe across countries remember I show you downward sloping curves and all that is as a result of that poorer economies tend to have lower capital capital labor Capital employment ratios Capital population ratios and therefore they they tend to grow faster because they're catching up with their State very advanced economies that have been more or less in the same place for a long time are moving around there so there's less catching up growth and that's the main responsible for the the downward sloping curve I show you within OCD countries and even broader than that Africa was a little of a problem there okay so that's this is an important model for you important diagram let's let's play a little with it so suppose that you know at the time this is a very simple model but at the time the The View was that well what really supports growth is saving so economies that Save A Lot grow a lot and this sort of sort of makes sense here because investment which is what leads to capital accumulation is entirely funded by savings it makes sense you have more saving you should grow more okay so let's this is something we can do experiment suppose you start at at at a say state if you will and now we increase the saving rate what moves which curve this is the kind of thing you should know when you work with this mode if I change the saving rate which curve moves in this model let me go one by one does the red line move no has nothing to do with savings has to do with depreciation if I move depreciation rate that curve will move but not will the production function move no so the Blue Line cannot move all that would move is the green line because the green line is the saving rate times the the the blue line so if I increase the saving rate I'm going to move the green line up okay that's what we have here so you see what happens is you start for for this was a St state for this saving rate in this econom now all of a sudden this economy start saving more what happen then this tells you very much the story of Asia the Asian Miracle of the 60s 70s and so on is very much something like that a little more complicated but you know a big part of what explains sort of the fast growth of Asia during that period is that something like that happened now why the saving rate increase and so on that's all very interesting and so but but it's not what I want to discuss today so but what happened here then so what happens see this economy was in a steady state so there was no growth it was growing at zero in a steady state no because this says in a stady state output per per worker remains constant and since we have no population growth then that means output is not growing either okay the only way you can have that ratio constant with the denominator not moving moving is the numerator is not moving either okay okay good so now Bo all the sudden we get a higher saving rate so what happens now what reacts so the saving rates go up it's a close economy it means the investment rate will go up okay what happens now what does that Gap tell you now you have a positive Gap there which means you're investing more than the what you need in order to maintain the stock of capital at the previous St state so that means that stock of capital is going to start growing to the right it's going to start growing okay and as the stock of capital grows then output per capita also grows and it will keep happening until you reach the new state okay so a higher saving rate so important conclusion there this this as simple as it is prove something ER that you know the conventional wisdom that the highest saving rate would give you sustained growth higher growth isn't really true and not certainly not in this model eventually you'll go back to growth equal to zero okay when you reach a new state you're going to be also growing at zero okay what is true though is that you get again what is called transitional growth because here you're going to start growing very fast in fact okay and then you're going to keep growing at a low slow lower Pace until you go back to zero but you're going to get lots of growth in the transition as a result of that and it turns out in the data when you're looking at 20 30 years of data it's difficult to H this entangle sort of very permanent rate of growth versus transitional rate of growth this is one of the things that has concerned China quite a bit no they have been they grow very very fast they have been growing very very fast for a long time but it's very clear it's becoming harder and harder for them to grow at the type of rate of growth that they had in the 20 years ago okay they had rates of growth 15% or so they had very high they had a very low initial Capital population ratio big population little capital and enormous saving rades so so they grew very very fast they had the green line very close to to the Blue Line the capital is stock very low so they grew very very fast but they have been growing very fast for a very long period of time so now it's getting a lot harder because they're getting closer and closer to the stady state that's the issue okay there are other sources of growth and that's what we're going to talk about in the next lecture but but this this is sometime calls the easy part of growth is sort of running out in China okay okay and it has run out in all developed economies for quite a while um good is this clear it's important I mean a question like that is guarantee in your p 81 what happens if the saving rate does something so so if so this is a plot over time um so this is a case in which a wear a say State and at time T the saving rate goes up S1 greater than S zero Jump Then output cannot jump so the saving rate goes up but output cannot jump at Day Zero why why is it that output doesn't jump immediately to a new state state you know this is the I'm I'm saying this is what will happen to Output you're going to start growing very fast early on and then you keep growing keep growing at a slower and lower Pace because of decrease in returns to Capital uh and eventually you'll convert to a new stud state with with the rate of growth equal to zero like like the one you had before this savings shock and the question I'm asking now is why doesn't out why does outut would have to do this why why doesn't it just jump what would what is the only variable that could make it jump well you need to look at the production function the is function of K Over N N is fixed the only thing that can make it jump is is the Capital stock jumps but the Capital stock is not jumping that's a stock and in order to accumulate a larger stock of the new stud State you're going to go to a lot of close that's investment know every year you're going to be adding a little more to the stock of capital on net that's the way you grow it's not that all of a sudden your stock of capital jumps that's very much because this is a close economy if you're in an open economy Capital stock can move a lot faster in a transition because you can borrow from abroad you don't need to fund it all with domestic sources and in fact that's what typically happens in in in emerging markets and so is they typically borrow for a long time problem is they tend to consume it rather than invest it and that's the reason you end up in financial crisis and so on but but but in principle things could go much faster if you have an open economy and and you have Capital inflows into your country but that you will talk more about that five or six six lectures from now anyways but this is what happens we're then increasing the saving rate so yes it affects the rate of growth of the economy during the transition H but but not in the long now this transition can be very long okay now what about consumption so so invariably and there's no way around that if if given a technology and so on if the saving rate goes up then output per worker will go up okay the question is the next question is what happens to consumption per worker does consumption per worker go up or not you are in cl to say well I mean it makes sense that it goes up because H we have more income no the saving rate is little stimes y then consumption is 1 minus little s time y so income goes up consumption should go up and and yes that's a dominant Source but it's not all the story because remember I what I told you so consumption here is going to be equal to 1 minus little s time y so consumption per person will be y that remember that what is increasing y Over N there so what is making this guy go up which will lead to an increase in consumption Over N is that this guy went up and that's a force in the opposite direction okay so and in fact that was one of the debates with the East Asian Mira Southeast Asian Miracle is that it was fueled by lots of savings so people say okay that's wonderful your output growth is very fast but consumption growth is not so fast and at some point it may be hurting you I think that they were right though for other reasons but but uh but that's that picture makes the point you know so if if if your saving rate to start with this is a general lesson if the saving rate is you start with is very very low then an increase in the saving rate will lead to a strong increase in consumption because this change is small relative to the big Bank you get on output because if you have low saving rate that also means that the ER the Capital stock is very low and if the Capital stock is very low F Prime is is very big you know this a concave function and you the Steep part of the function later on if saving is very high you're going to tend to have Capital stock very high and then first of all H more Capital won't increase output per worker a lot because uh because of decrease in returns and and this is a big number so it it starts dominating and that's what you see here this economy as you increas the saving rate a consumption per worker Rises but at some point it reaches a maximum and then it starts declining I mean think of the limit if you save 100% of your income you don't consume anything no matter how much is your output if your saving rate is 100% then you're not going to consume anything if you have no income no saving rate no savings no Inc no Capital stock no income you're not going to consume anything either okay so you at least you know these two points and since you know there are some positive points in the in the middle you know that the curve is going to tend to have that that kind of change it's not it's going to be non monotonic and and that's the way it is so let me just play with a little a few numbers this is yeah let me play with a few numbers not that crazy H suppose you have a a production function that gives equal weight to Capital and workers so this production function does a production function constant return to scale it better be because that's what we're doing but what do you think yes the sum of the exponents is one so it's K to the 1/2 n to the 1/2 the sum of the exponents is one so you know that it's proportional to the scaling Factor so we're going to use the as a scaling as before n so um so we have this okay this is this is a f of little f of k/ n is the square root of k/ n okay minus Delta K Over N so all that I'm doing is I'm plugging in that function ER so here only I'm replacing all these functions by by a a specific example one in which this is a square root of k/ n okay that's a concave function square root good now do it as an exercise if you solve for the steady state how do you solve for the steady state well set this equal to zero that will give you the steady state no if the steady state is when the capital is not growing anymore it's when this is equal to zero when this is equal to zero I can solve for the steady state level of K Over N no from here this equal to zero I can solve for K Over N and I'm going to call that the steady state K star we typically use stars for steady States in grow the okay well the answer to this is is K the state stock of capital per per person is the saving rate over Delta Square that's what it is output uh per person which is the square root of K Over N is therefore the square root of s over Delta Square so it's s over Delta okay so in this particular model in the long run output per worker doubles when the saving rate doubles okay if I double the saving rate then output per worker will double notice that the stock of capital is going to grow a lot more in this when you increase the saving rate okay it's Square so in that economy if you do increase a saving rate from 10 to 20% this is the way it goes okay so remember 10 to 20% that means that the new state state output per worker will be twice what it was in the previous state state okay so you go from one to two but it takes a long time and the numbers are not Cas 50 years takes you to go to the new state state okay so so that's sort of the time frame we're talking about so it is true that the saving rate will not change the longterm rate of growth absent other mechan ISM but you can grow faster than your average your state state level for quite quite some time okay and and again a lot of that of the Asian Miracle has been of that kind this is what I was telling you of China before no well yeah you you can grow very fast especially if you have saving rates much higher than 20% I mean 50% or so but but but the rate of growth will have a tendency to decline abs and some other Miracle there a lot of the reasons why we have all this fight about technology and so on it has to do with because that's the main mechanism you alternative mechanism to grow is technology okay we're going to talk about that in the next lecture but but this Force which is what I said before is the easy part of growth is very difficult to fight this pattern so here you have numbers H for the say States so if the saving rate is zero obviously everything is zero no way around if the saving rate is 0.1 10% then in this m capital per worker is one output per worker is one consumption per worker didn't go from zero to one why because you were saving something so Z is 1 minus 0.1 which is a saving rate suppose you double the saving rate well we know that we're going to double output per work in this economy we said that we're going to go from one to two the Capital stock is going to have to a lot more to double the amount of output why is that decrease in returns to double output you're going to have to much more than double Capital because you know you need you're going to be fighting decreasing in Returns what about a consumption well it won't double because you're doing this out of increasing the saving rate so you get the two minus now 02 not 0.1 okay minus point2 * 2 so you get 1.6 and so on and the higher you go with your saving rate the harder it gets for Capital to bring along um output per capita and the more the dragon consumption because you need to be saving a lot in order to maintain this High stock of capital that you're having okay you have a very large stock of capital that means you need to save a lot just for the sake of maintaining that stock of capital and so little is left for extra output per capita and so you see that here in this particular for this particular mole when the saving rate exceeds 05 then ER output obviously keeps Rising when you increase the saving rate but but output starts declining so that's your in the declining part and if you get to one of course there's no consumption so that's that's a curve that we Trace okay is everything clear now I'm going to that's a basic solo model and that's a model that again you need to control completely okay all that I'm going to do now it's very simple I'm going to just modify a little bit this model to H add population growth okay so what happens by the way for for centuries population growth has been one of the main in this model we concluded that output per worker was not growing what we're want to conclude in a second is that output per worker will not grow if population is growing but that means that output is growing if population is growing and output per worker is not growing it's constant that means output is also growing and for a long time growth of output not of output per worker was driven by large population growth and sometimes you get big migration flows into a country that lead sort of to growth and so on now big parts of the world have negative population growth so now we're going through a cycle in which things are going the the other way around in in in many large parts of the world I mean this true in almost all of Continental Europe H certainly in Japan I said South Korea China and even some places Latin America okay so the drug actually is is against the we don't have the natural force for growth that we have for for many many years so let me let me introduce population growth so assume now that that population rather been constant growth growth at the rate G which could be positive or negative I'm going to do the example for the POS a positive population growth example so there's no equation that changes in the sense that this is still true it's still true that investment equal to saving it's still true that that output is equal to Output per worker and output is equal to F of K andn and so on so forth the the thing that is a little tricker is that that you know in this model if I don't normalize things for if I if I you know in this case here where population was not growing I could have just eliminated this n it's a constant and I would have done everything in in capital in space of capital here and output here it would have been the same just scale by a number a constant n when I have population growth I'm not indifferent between doing one way or the other because if I don't have if I don't if I do it in the space of K and Y and population is growing then all these curves are moving so this see a very unfriendly diagram because my curves are all moving as n is moving everything is moved so the the trick in all these growth models and it's going to be even more important in the next lecture is to find the right scaling of capital so there a St state so you the your curves are not moving around as population growth it's very easy to find the scaling factor is population okay so that's what I'm going to do but remember what is different here is so I want what I'm saying I want to get all my variables scales by population at some point in time that's what I want to do because I know I practice enough with these things that's going to give me a steady state okay um now what is trickier relative to what I showed you before is that before I just divided by n both sides and and I was home now I can't really do that okay let me divide by nt+ one both sides so that's nice I get my Capital per worker at t+ one but there are certain things are not as nice what I have on the right hand side is not what I really want I don't want Capital over population Next Period my state state is going to be in the space of capital over population at the same time that's my state state so this is not so nice so what I have to do is I want to convert this the right hand side in something that these are the kind of things that I want to have so what I'm going to do is divide and multiply each of these sides by NT over NT + one so sorry I'm going to divide and multiply each of this by nt Okay so multiply by NT divide by NT so multiply by one well and and then I can rearrange the terms in this way so I get what I want which is capital per person at time T all at time T but then I get this ratio here okay and I can do the same for this expression here now what is that ratio population today divided by population tomorrow well is one over one plus the rate of growth of population NT + 1 is equal to NT * 1 + GN that's the rate of growth of population okay so so what I have here is 1 over 1 plus G GN now GN is not a big number so 1 / 1 + GN 1 over 1 plus GN is approximately equal to minus GN okay so 1/ 1+ GN GN is very close to zero is approximately equal to minus GN okay so that's the reason this guy became that guy approximately that guy I can do the same here but it turns out that the term there's an extra term here therefore which is equal to S * GN * yt/ NT well that's second order that's the reason I'm going to drop it okay it's a saving rate which is it's a small number times a a rate of population growth which is a number like you know 01 or something like that so that's a small number so I'm dropping it that's a bigger approximation than that one actually but I'm going to do it everything becomes a lot simpler but so this an approximation okay and just drop in second order terms and once I have that I have the system I want because now I have a system for the evolution of the of the capital per per worker okay or per person and if you see it looks exactly as we had before remember this is exactly what we had before SF K over we used to have n not sub subscri T now is K over andt but what is different is that now rather than having only the depreciation rate here we have the depreciation rate plus the rate of growth of population why do you think we have the rate of growth of population there remember the the the economics behind this expression before it was this is what adds to Capital to Capital per worker this is what you need to maintain what takes away from capital okay now it's what takes away from given we're doing everything in the space of capital per work that takes away from Capital oh that's a typo there's a t there okay T okay so why do you think I have this GN here well I have only one minute so I don't have time to because if I want to maintain a stock of capital per worker and workers are growing then I need to be growing the capital St even if I had no depreciation if I want to maintain the capital per worker constant and workers are growing then I need to grow the stock of capital so in order to maintain the capital I still need to spend what I used to spend for depreciation of the Capital stock but if I want to maintain the the capital per worker constant then I want to need more investment okay just to make make up for that that extra comp onon it so now set GA equal to zero that's your diagram is exactly as before in this space set a equal to one and constant but this line the red line here will have Delta plus GN okay so it rotates up so you can play here and see what happens if there's change in population growth and so on so forth it's going to be counterintuitive initially because you see if I increase population growth this Curve will rotate up and then it will appear as if that leads to negative growth but you don't get Negative growth in this diagram you do get that y over over n will decline but that doesn't mean that you get Negative growth it just means that output is not growing as fast as population but but both are growing just the population is growing faster than output I I'll I'll start from that ER oh I think it's after your break so you when have forgotten everything by then so I'll do a review of this and then and then we okay have a have a nice break