there is a course economic growth and business cycle this course entails solo growth model ran model and business cyes so I've already created a playlist for solo growth model completely and for the RoR model in Solo growth model also we will see solo model without technology and with technology with convergence hypothesis and in Roman model we talk about economics of ideas and we'll do Roman model in detail you will have to read reading whichever reading is prescribed to you but model remains same whether you do it from one book or the other so you may be from whichever University or you may be preparing from whichever course this can act as a good basic reading for you so you might make notes of these recordings and also try to include points from your own read I hope this will be useful to we'll be talking about complete solo model in this recording this is a lendy recording but it will be very helpful to you if you listen to me and write notes side by side we'll be covering almost every part of uh solo model we'll start with assumptions we'll talk about the production function the production function in intensive form capital accumulation equation interpretation of this capital accumulation equation then we will derive key equation of solo model we'll talk about how do we draw solo diagram what is capital deepening and capital widening we'll talk about how the uh steady states are they going to change when uh population growth rate and uh savings rate are going to change and then between the two steady States how do we interpret the transition dynamics that is what we are interested in we'll slowly come over to then technological progress in Solo model we'll see up until now we would have seen everything without technological progress and now we'll be looking at the technological progress so we'll again derive the key equation of solo model with technological progress and solo diagram also with technological progress we'll talk about the the level effects and the growth effects and then we will end up the discussion with the convergence in Solo model that is is it true that the poor countries will catch up with the rich countries what is the distinction between the conditional and the unconditional convergence so this complete recording is uh filled up with notes my suggestion to you is pick up your pen and paper and register and start writing it out it will really help we have tried to bring all the nearly all the aspects of solo model in this and uh we hope you like this let us start today with solo model so this recording is about the different assumptions of the model and telling you about the two key equations in the solo model one is the production function and the another one is the capital accumulation equation so there are few assumptions which you making on the the outset one um please write countries consume only a single homogeneous good only a single homogeneous Goods homogeneous good right so one of the Assumption or one of the implication of this assumption is that there is no international trade in the model right because there is only a single good one of country might give you X in order to get X only what is the point we will be relaxing these assumptions later on second the technology um technology is exogenous in simple terms so it is given to you right so technology available to firms in this simple model is unaffected by the actions of the firms including R&D including R&D uh so even if research and development is happening this is not going to affect the technology so whatever technology is given that is exogenous we'll be dropping these assump later on simple thing now there are two main equations in the model so one equation we are talking about is the production function and the another one is the capital aculation equation so let me talk about the first first equation which is your production function right so we are saying the production function is of the form Y is equal to f k l and let us also assume one form of the production function which is K to the^ Alpha l^ 1 minus Alpha and Alpha is a constant which is lying between 0 to 1 which is lying between 0 to 1 so what we are assuming is that production function is given by Y is equal to F of K there are two inputs to to the out to the uh to the production process capital and labor now just think about uh it in this way that growth of the economy is not only dependent upon the amount of the inputs which you are using that is dependent upon many other things also political institutions so whether your country has a democratic setup or not right corruption there are many other things which can also affect growth but right now we not taking all of them so what we are seeing is that output is dependent upon the inputs which are being used in in this way and uh there are two inputs to the factor production uh there are two inputs to the production process and we are also assuming that technology is given to you so we will drop these assumptions later on but this is the thing a if you look at this production function which is y is equal to f k l k^ Alpha L ^ 1us Alpha and let's say I increase K andl by Lambda KL by Lambda so what happens is that uh I can take this Lambda out this is Lambda the^ Alpha Lambda ^ 1us Alpha K to ^ Alpha L ^ 1us Alpha so it is Lambda and bet so this is just Lambda to the^ 1 and what is this telling you because Lambda is completely factored out it is a homogeneous function and the degree with which it is factored out is the degree of homogenity here so the degree with which it is factored out is one and if the degree of this homogenity is equal to one you call this production function as a CRS production function you have already done this in the micro course right so there are things from the micro which will be which we will be assuming while doing macro right so because degree of homogenity is equal to 1 so this production function exhibits CRS so this production function exhibits CRS would mean what that it is the constant returns to scale production function so you were using let's say one unit of Labor one unit of capital they were producing one unit of output you use two units of Labor and two units of capital they will be producing two units of output right three units of Labor three units of capital they'll be producing three units of output so if doubling the input your output gets doubled that is the constant returns to scale if by doubling of the input your output more than doubles that is increasing returns to scale if by doubling of the input your output less than doubles that is decreasing returns to scale right so that is this is the constant returns to scale okay let us also assume so these are the assumptions also that is you're using two units of Labor and capital oh sorry you're using two inputs labor and capital okay and then the assumption is let's say the firms are paying wage rate small W per unit to labor and the rental rate small R to Capital firms in this economy P workers a wage W per unit and a rental rate R per unit of capital for one period for one period right and the the another assumption is that there is perfect competition in the economy right so the firms they are perfectly competitive producers and uh right so it is uh should I write this you can have this is the third assumption the fourth assumption would be that uh the FMS are perfectly competitive and then you have the Firs in this economy they pay the workers W per unit in the rental rate R per unit of capital for one period a there is an implication for the perfectly competitive firms that these firms they are the price takers the firms they are the price takers right and let us also assume okay so if they the price takers how will they be maximizing their profit P into y minus WL minus RK Let Me Assume P to be one right the prices of uh these price of the goods and services that is equal to one y is nothing but my production function minus WL minus r right so I need to maximize this under capital and label now when I say this that means these are my control variables I mean I can control how much of Labor and capital I want to employ D Pi by DL is DF by DL minus W = 0 and then I differentiate this Del Pi by d k which is DF by d k - R = 0 so DF by DL is what m marginal product of labor DF by Delk is what mpk marginal product of capital guys I hope you know this I mean we have done all of these things Del by Del MPL MPL is the when you use one more unit of Labor what is the addition to the total output that is what m is mpk is you use one more unit of capital what is the addition to the total output due to that so what you are seeing is that this MPL at the profit maximizing condition the first order condition MPL should be equal to W and mpk should be equal to right okay but we have written our y equals to this h y or F of k equals to this so what is d y by DL 1 - Alpha L ^ 1 - Alp - 1 K the power Alpha l^ minus Alpha capital K the^ Alpha like this we can write like this just remember this H we'll be using this later on Del by d k is going to be what Alpha K ^ Alpha minus 1 L to^ 1us huh and if I just write this as this um so if I if I just write this as this this is Alpha minus one let's say or minus of this guy so if I just take this in the denominator if I just take this in the denominator so this is if I'm writing this in the denominator so so this thing could be written as this 1 - Alpha minus 1 which is nothing but L ^ 1 - Al so I'm just writing it in this way I've just taken this in the denominator I hope you understand this right so if I want to write this bet this is equivalent to this why because this l 1 upon l^ Alp minus 1 could be written as l^ minus Alpha minus 1 which is L ^ 1 - Alpha only H okay so keep this thing in head I'll come back to this keep this thing in head I'll be coming back to this right okay so if you look at it carefully what is it that we have said that uh firms are going to employ labor up to the point where MPL is equal to W and capital up to the point where mpk is equal to R right okay now this was my ml now this is my MPL Del this was my MPL a can I write also like this just just a working what is going to be why by L what was y k ^ Alpha L ^ 1 - Alpha upon l so what is this K to^ alpha l^ 1 - Alpha - 1 so what is this K ^ Alpha L ^ minus Alpha so what is this K upon L to the power Alpha so you guys can see this so can I write this thing as 1 - alpha y by L I can write it like this similarly similarly can I write what y by K is K ^ Alpha L ^ 1 - Alpha upon K so that is k^ Alpha minus 1 l^ 1- Alpha H so I have just told you L the power 1 - Alpha is equivalent to 1 upon l^ Alpha minus 1 I've just told told you that so can I write this Al into y by K I can write this the other thing is that this is my mp and and we have just found out that MPL is equal to W right other thing is that this is what my mpk is and we have just found out that mpk is equal to R from the first order condition right what was WL plus RK then WL + RK what is this going to be what is w w is just this 1 - alpha y by L this is what w is into L plus what is r Alpha into y by K into K so what is left out by what is left out 1 - alpha y + alpha y so this is what 1 minus alpha y + alpha y cancel that out so what are you saying you are saying that the total output which is being produced which is why it is completely exhausted in giving out wages and giving out rent this is the wage Bill WL this guy is the rental bill this guy is the rental bill right so uh please write this the payments to the inputs completely exhaust the the value of output produced so that there are no economic profits to beond that there are no economic profits to be earned so what you are saying better your profits was P into Yus WL minus RK but P you have assumed to be 1 minus WL + r ab ab you have said that WL plus RK is nothing but equal to Y profits are zero your entire output which you were producing y that has gone in uh in the distribution of wage bill and the rental bill right so this is the general property of the production function with the consant returns to scale that uh that uh uh you do not get any economic profit so that entire profit is actually gone into giving wage to the labor and rent to the capital and rent to the capital you with me h okay I think uh I'll stop out here we'll talk about the capital accumulation equation and few more things about uh about uh the production function uh in the next class right thank you R let us take the discussion further for solo model which we were discussing yesterday so we said this we started with this kind of production function our production function was y = to k^ Alpha L ^ 1us Alpha and let's say 0 is Alpha is less than one 0 is less than Alpha is less than one alpha is any e constant we know that L is the labor which is employed W is the wage rate per unit of Labor so WL is the total wage bill WL upon y would be the share of wages in total output right similarly K is the capital which is employed r is the per unit Capital Total Rental Bill r k upon Y is share of rent in total output right okay guys you remember now we did this we said this that MPL is equal to 1us Alpha into y by L and this is equal to the wage rate we said this that mpk is equal to Dy by DK which is equal to alpha y by K and that is equal to the rental rate we did this yesterday so don't uh forget this huh so let me just copy this for you because I'll be using it here yeah we did this thing and let me also copy this thing here so I'm not in a hurry in doing things so you also whenever you make notes now you have to make notes with uh atmost as if you have all the time in the world then you understand because these are these lectures are not meant for any kind of entertainment it is nothing like that so go slow every time anyways so you have you said this that MPL is equal to the wage rate and MPL was equal to this you also said this so what is w into L 1- alpha y by L into l so that is 1 - Alpha into Y and what is w into L by y just divided by y so your left out with only 1- Al so 1- Alpha is nothing but the share of wages in the total output right similarly here dely by d k is alpha y by K Dy by Delk is the mpk that is equal to alpha y by K we said this that this mpk is equal to the rental rate so we can write this this way what is r k is alpha y by K into K that is equal to alpha y and what is RK upon y that is y so that is Alpha so share of rent in the total output is Alpha this was your production function right this was your production function and uh Alpha is the share of you can say r rent in the total output or share of capital in the total output and 1 minus Alpha is the share of wages or share of Labor in the total output right so what you want to say by this is that the factor Shares are constant this is the factor share factor Shares are constant this is the factor share of Labor this is the factor share of capital this is the factor share of Labor and capital okay now if I want to convert this uh production function which I have in the Intensive form of the production function then how how I can do that so intensive form means the cap output production function in intensive form intensive form or in the per capita output form or um output per labor form output per worker form these are the different names output per worker or output per capita right output per worker output per capita like this so what is output per worker capital Y by L and which we denote as small Y which we denote as small Y and beta what is capital y k to the^ Alpha L ^ 1us Alpha upon this L here so what is this this is nothing but K by L to the^ Alpha and this is nothing but small K to the^ Alpha so you have output per capita is small K to the power right small K to the power Alpha small guys this is nothing but Capital uh per labor right or Capital labor ratio right or Capital per worker this is capital per worker this is capital per worker so this is the production function in the Intensive form right and you have also assumed that 0 is less than Alpha is less than 1 how does it look like okay one if small K is zero then small Y is 0 to the^ Alpha which is zero means production function is passing through origin first thing right or you want to write this more technically intensive form production function is passing through origin one thing a second thing what is D small y by D small K Alpha small K to^ Alpha minus one bet Alpha is positive and Alpha means Alpha is greater than zero Alpha is less than 1 so Alpha minus 1 is also positive so K to the power this thing this is positive right so it means this is an increasing function this is an increasing function d2y by DK2 would be Alpha into Alpha - 1 K ^ Alpha - 1 - 1 so this would be Alpha into Alpha minus1 right into K to the power Alpha minus 2 Beta Alpha minus 1 would be negative now Alpha is less than one so Alpha minus one would be negative Alpha is positive so this guy is negative so if the second derivative is negative so you have the concave function h you have the concave function one thing so if I just want to put this on the graph small K here small y here leave the function like this this is small y = to small k^ Alpha something like this so this is a concave function right one it is passing through origin It is an increasing function and this is a concave function one thing right so yes if you add more Capital so if capital is going to increase means Capital labor ratio is going to increase means you will be able to produce more output per worker that is for short this is given by D small y by D small K so beta if K is going to increase K by L is going to increase assuming that there is no change in the labor small K is going to increase small Y is increasing for sure so that is what is meant by the first one uh second one but the increase in out output per worker within every increase in capital per worker that is going to be diminishing you've done this right in your micro course also you've done this that the marginal product of inputs is positive but it is diminishing right so diminishing marginal product so so please write with more capital per worker means now every worker has more Capital given to him so he'll be able to produce more output also every worker will be able to produce more out forms produce more output per work right but there are diminishing returns to the capital per worker so each additional unit of capital per worker is going to add less and less addition to the output per this is what you do in case of your marginal product also you remember that now you say that marginal product of labor is positive but if you keep on increasing this labor then marginal product of labor will keep on falling marginal product of labor itself is positive so you definitely need an input which has the marginal product positive because otherwise why will you use that input it's who marginal product is zero or whose marginal product is negative I mean that is not a profitable input then so and that is not profitable to use such an input in case of the marginal product is going to be negative but the increase in marginal product as labor is going to increase that is going to fall so that is what diminishing margin product is so for example you there is a fixed amount of land and you keep on increasing the amount of Labor in that land so every unit every additional unit of Labor will be adding less and less to the output that is what is meant by the diminishing margin product of labor and in terms of capital per worker what you are saying is this however there are diminishing returns to Capital per so each additional unit of capital per worker is going to add less and less amount of uh addition to the output that's an idea so you have this thing so I'll talk about the second equation of the solar model in the next class which is your capital accumulation equation right sure thank you so we started with this production function y = k ^ Alpha l^ 1- Alpha 0 is less than Alpha is less than 1 and we reached till this production function in the Intensive form right so we have done this much in the solo model up until now now we are moving on to the another equation of the solo model in which is the main equation the capital accumulation equation capital accumulation equation which is let me write the equation first and then I'll explain K dot is Sy y minus Delta K let's look at all of these terms separately K dot is the instantaneous change in K so what is instantaneous change in K let me just write first of all instantaneous change in K so instantaneous change in K could be written as like this the capital in time period t + 1 minus capital in time period T right that is the change in capital and if I want to find out the growth rate in capital growth rate in capital then don't you think I have to find this out KT + 1 minus KT upon KT so KT + 1 minus KT is just telling me the change in capital but but if I want to find out how the capital is growing over time then I will also have to divide it by KT right this instantaneous change in capital is written as K dot capital K dot or more technically this is the derivative of K with respect to time this is the derivative of capital K with respect to time D capital K by so this is just telling you I mean in time period let's say I'm just giving an example so if the capital here you have time and here you have Capital so this is just telling you at time period uh T this is the amount of the capital which you have at time period t+ one this is the amount of the capital which you so what is the change in capital this is what DK by DT is telling but for the growth rate in capital you need to find out the the uh division of this K dot with K that's what I hope you know this that is you find out the growth rate in any variable for example uh output in year 2021 output in year 2021 you want to find out the growth rate in output so what you do with respect to 2020 output you do it this way only mhm upon output 2020 this way you do it this way only yeah okay so one thing is this that uh this is the way we write the instantaneous change and the growth rate in capital then would be written as capital K Do by Capital One Thing should be clear uh the outset capital K dot is instantaneous change in K capital K dot by capital K is the growth rate in K or is or you say the proportional change in K or you say the proportional change in k okay this capital K dot when I say that capital in time period uh t+ one is KT Capital KT + 1 in time period T it was Capital KT so the change in the Capital stock is nothing but the net investment so this capital K dot is what you call as net investment net investment small s so what you are doing is what you are saying here is this in the closed economy all amount is all whatever amount is saved that is invested right so investment is equal to uh savings so whether you write gross savings or you write gross investment that means one of the same thing one thing okay what happens in the closed economy Y is equal to C + I + G this is what you get in the closed Eon so I can also write this as y - c minus okay y - cus G = to I I can write like this so bet what is what is this this is nothing but your savings or you can call as savings or gross savings what you have so from the total income the amount which you have uh which you have taken out for consumption means consumers have spent something or the amount which government has spent so whatever is left out is what your savings are and this is nothing but your gross investment this is nothing but your gross investment and this guy can I also do it this way Yus C minus t is nothing but private savings I have done nothing I have just added and subtracted T from here and T minus G is nothing but public savings public savings are government savings let think about it what is the expenditure of the government G what is the income of the government T so from the income if you spend if you deduct the expenditure is what you get is the savings Savings of the government right so adding them together private savings plus public savings you add them together what you get is the gross savings these are equal to gross Investments so whether I write gross in Gross investment or I write gross savings that means one of the same thing right so these are savings so let's say capital S are what my gross savings are they are equal to some proportion of the total income sum proportion of the total income what is that say what is that sum proportion is what savings propensity or savings ratio this is exhaust exogenously given exogenously given means given to you from outside savings propensity exogenous also this capital Y is nothing but what you remember now we have just done this in the last class total income is the total wage income plus Total Rental income that's what WL + r one thing and these gross savings are nothing but equal to invest gross savings are nothing but equal to investment why I'm writing it here so this these are gross savings and these gross savings are equal to gross investment are equal to gross investment fair enough okay then some amount of uh the capital which you have invested or some amount of capital or the machines which you have used they are subject to normal VR and tier they are subject to normal V and tier so this Delta K is nothing but the replacement cost or the or the depreciation what do you have Delta is the depreciation rate and Delta K is the uh total replacement cost I'll tell you what exactly uh this would be right so please write Delta K is the total the depreciation depreciation of Capital stock that occurs during production that occurs during production right and Delta is nothing but the depreciation rate and this is also given to you from outside so from gross when you deduct depreciation what you get is net so from gross investment when you deduct depreciation what you get is net investment right so this is what you have K dot is syy minus Delta right K dot is S5 minus Delta K so if the gross investment is more than what is needed to keep the Capital stock unchanged the uh the total capital in the economy is going to increase what what is it that I've have said if I want to keep the Capital stock unchanged it means K dot has to be equal to zero right at capital K dot equals to 0 Capital stock is intact Capital stock is intact Capital stock is intact means capital is not changing capital is not changing why because whatever you are investing that is equal to the replacement capital or replacement cost so I have uh invested let's say five machines and if I want to keep my Capital stock unchanged I have to replace this Capital by five machine so there is no addition to uh uh what do you call the Capital stock in the economy but supposedly if the replacement cost is just five machines in the economy so if you want to keep uh the Capital stock unchanged you just need five more machines so that is what my depreciation is that is what my replacement cost is but I have invested in the economy 10 machines so the investment the gross investment the number of additions of the machines is more than the replacement of the machines then the Capital stock in the economy is going to increase right if Sy Y is greater than Delta k then Delta K is greater than zero if Sy Y is less than Delta k then Delta K is less than zero again understand this looks very simple but there is there is the point here supposedly if you want to keep the Capital stock unchanged you need five machines let us say in your economy but the gross investment in the period is just for two machines so you have just invested two more machines while in order to keep the Capital stock unchanged you need five machines so the capital in the economy is going to fall so their K dot is going to fall capital K dot is going to fall I'm very particular about using the words capital and small here because very soon we're going to convert this equation in the output per worker where we will be using small y small K so be very very clear about all that so Sy is the gross investment gross savings so you're writing gross investment in terms of savings you are saying the number of machines added in the economy so how many number of machines added in the economy let's say 10 Delta K is the number of machines which need to be added to the economy to keep the capital per capital unchanged uh so let's say that is five machines so if you are adding more machines then what is needed to be added to keep the capital unchanged then definitely capital K dot is going to increase if you are adding less machine means then what is needed to keep Capital unchanged then capital is going to fall then capital is going to fall right so AB what is it that we have done we have just defined what the capital accumulation equation is and uh we have talked about what is uh this instantaneous change in K and what is the growth rate in capital so what you call as K capital K dot that is the instantaneous change in K that is nothing but DK by DT and what you call as K dot by K is nothing but the growth rate in capital or the proportional change in capital so I'll be talking about the I'll be converting this capital accumulation equation in the capital per worker form and we'll call that as the key equation of solo model and we'll be doing that in the next class right thank you up until now we have derived one we have started with this production function and from there we have derived the Intensive form of the production function small Y is equal to small K to the^ Alpha where small Y is output per capita and small K is capital labor ratio and we have also derived the capital accumulation equation capital K dot is S capital Y minus Delta capital K now we want to convert this capital accumulation equation in the per capita form we want to calculate we want to convert this capital accumulation equation in per capita form per capita form of capital accumulation equation per capita form of capital accumulation equation mhm right okay now just think about it you have like this so capital K dot is equal to Sy y minus Delta K right okay can I take a step back and can I write what is small K small K is K by L right okay can I write what is log of small k equals 2 which is log of capital K minus log of capital N yes can I take the d uh the time derivative of both the sides yes I can so I can differentiate this with respect to time I can do that so what will this come out to be 1 upon small k because the derivative of log K is what 1 upon small K derivative of log small K is what 1 upon small k d small K by DT this is 1 upon capital K D capital K by DT minus 1 upon l d capital L by DT 1 upon capital l d capital L by DT right fair enough a what is d d small K by DT we have said that this is equal to the instantaneous rate so it is small K dot by small K so D small K by DT is small K uh sorry small K dot d capital K by DT is capital K dot d l by DT is l dot this is capital K Do by capital K minus L do by L and what is small K Do by small K that is the growth rate in small K what is capital K Do by capital K growth rate in aggregate Capital what is l dot by L growth rate in labor right Fair okay now I want to tell you one thing that uh there is an assumption which says that labor force is exogenously growing at the rate n labor force is exogeneously growing at the rate n this this population growth rate of the labor force growth rate is given to you this is given to you this is exogenously given to you and this is also one of the weakness of the model if you just have a look at this why because the model is not able to distinguish between the population growth rate and the labor force growth rate so according to the model in case of the uh child is born today that is the addition to the labor force but once a child actually becomes a labor force a huge amount of time has passed but the law but the model is assuming that the population growth rate and the labor force growth rate that means one of the same thing one and the other thing is there are only two factors which are affecting the output what capital and labor out of that you you have assumed one of the factors growth rate is as given to you so everything is now dependent upon how capital is growing over time so that is also one of the weakness anyways so you are given that labor force is exogenously growing at the rate n so some of the books they write it like this that LT is equal to l e to the^ NT like this so L not is the initial amount of Labor and N is let's say uh the growth rate in labor n is the growth rate in labor and this LT is the the labor at the time at the time de L not is the initial labor so let's say I take the log of both the sides plus log of e to ^ NT bet I hope you know this rule log of e to the power some constant is a constant you take the time derivatives this will be zero because log l not is given to you that's a constant plus you differentiate this NT with respect to n uh T which is coming out to be n Only what is log of capital L by uh d by DT you take the time derivatives this will be zero because log l not is given to you that's a constant plus you differentiate this NT with respect to n T which is coming out to be n Only what is log of capital L by D by DT this is 1 upon L DL by DT what is DL by DT l dot and this is L so this is n so this is what I meant to say that uh the growth rate in labor force is equal to n growth rate in labor force is equal to n fair enough so we have done two things AB one we have written that small K dot by small K is equal to K capital K Do by capital K minus L do by L right okay and this is capital K Do by capital K and what is L do by l n what is L do by l n okay what is small K Do by small K and what is capital K dot you have written what is your capital accumulation equation which was what s y minus Delta K upon K right minus I can write like this fair enough so let me write in a proper way here so small K Do by small K is S capital Y by K minus Delta minus n right third step small K dot by small k equal to S can I divide numerator denominator by L I can capital K by L minus I can take minus common this is n plus Delta huh now what it what it becomes small K Do by small k equal to S what is capital Y by L that is small y output for labor is small y upon what is capital K by L small K minus n+ Delta right and I can write K dot as small K dot as uh just uh so this thing will become what s y minus n + Delta small K and beta what is small y you have just written small Y is the production function in the Intensive form what is that small K to the power Alpha minus n + Delta small K this is small K dot this is the key equation of the solar mod this is important for us a very important expression this guy is the key equation of solo model this guy is the key equation of solo model okay how do we interpret this so let me write the two expressions which we have derived h s small y minus n + Delta small H so let us just compare let us just compare so you have won the capital accumulation equation and this is now capital accumulation per worker equation this is the key equation of the solo model so here what is capital K dot telling you net investment here small K dot is telling you net investment per worker H here s capital Y is telling you gross investment here s small Y is telling you gross investment per worker right here Delta K is telling you replacement investment or total depreciation whatever to keep the replacement investment me replacement investment to keep the Capital stock unchanged here n plus Delta small K is telling you replacement investment per worker but there is one addition here what is Delta that is Delta so what is Delta K telling you see in I mean in every year there are n new workers which are also added so sorry the new thing is this n I'm so sorry because Delta small K is fine just say Delta capital K is your depreciation Delta small K is depreciation per worker what is NK that's an addition enk is telling you um see there are n new workers which are added every year in the economy so you have to equip these n new workers with the same number of machines so that the investment per worker it remains constant right or the net investment per worker it remains constant so for example in this class there are let's say 10 people and everyone has five units of capital and there are five new people also come into this class so if you want that everyone should have now five units per uh of capital uh each then those five new workers which have just come into the class those also should be equipped with five units of capital no so for every n worker I mean there are workers which are being added for every worker there are end new workers which are being added so you have to equip these n new workers also with the same amount of capital per worker with the same amount of capital per worker so please write this in each period and new workers would be added to the economy who were not there last period who were not there last period right so these new workers also have to be equipped with the same amount of capital to keep Capital per worker unchanged to keep Capital per worker unchanged so the idea is I'll again repeat that uh statement so in class let's say there are 10 people and five more people are now added earlier every one in the class those 10 people had five units of capital each now those five un five more people have come in case if you do not give them the same amount of capital which everyone has then the capital per worker will fall so you have to if you want that Capital per worker to remain constant then this would mean that you have to equip these five new people which have been added to the class with the same amount of capital per worker otherwise Capital per worker is going to fall so this is the key equation of solo model and this is the interpretation of this key equation of solo model right okay thank you beta solo diagram we have have two equations up until now if you have uh seen the earlier recordings one is small y = to small K to the^ alpha alpha is lying between 0 to 1 this is the production function in the Intensive form in intensive form and you have the another equation also this is the first equation and you have the another equation which is uh K do equal to S small y minus n + Delta small K small K dot equal to S small Yus n + Delta small K so this is the key equation of solo model this guy is the key equation of solo model right AA then uh before I just draw this uh what are the parameters what are the exogenous variables and what are the endogenous variables so let me write a point about them h so endogenous variables are those variables whose values are determined within the model itself so endogenous variable endogenous varibles right they are those variables whose values are determined within the whose values are determined within the mo right so for example in this casee small K small y these are endogenous variables right and uh parameters parameters parameters are um a given number right for example smallest for example Alpha for example initial value of capital per work for example population growth rate these are um the single variable and single values right they're not changing over time then you have exogenous variables exogenous variables they can vary over time but their values are determined outside the system so for parameters also the values are determined outside the system for exogenous variables also the values are determined outside the system but in case of parameters they are given exogenous variables they can also increase over time right so they may vary over time they may vary over time but whose values are determine outside the system or outside the model right so for example capital L for example capital L so you should understand what are the endogenous exogenous variables and what is it uh what are the parameters and and what you can determine and what you can't determine and what is given to you from outside so you know what is endogenous you know what are parameters and you know what are exogenous variables okay now let us say I have this expression with me this model with and on the x-axis I have uh small K which is your uh Capital per labor on the y axis I have gross investment per worker small s into small K I think I have told you last time also if you remember we have uh we have sorry this is small one we have written S capital Y is gross investment and S small Y is gross investment per worker right so this is gross investment per work fair enough and in the initial recordings of the solo model if you have been following this properly then we have derived the production function like this small yals to small K to the^ Alpha so and we have shown that this production function was concave right this production function was concave one thing let me just draw that production function again and I have the function like this Y is equal to small K to the^ Alpha right now I have y is equal small K to the^ Alpha now if I'm going to multiply this y with small s on both the sides I get small s into K to the power Alpha now this s is a constant term that is a savings propensity it is lying between 0 to one so don't you think that this Sy y or S into small K to the^ Alpha term is nothing but the dampened version of small y so this guy s small Y is s small K to the power Alpha so s small K to the power Alpha is the gross investment per worker and this is a dampened version of Y = to K small y = to K to the^ Al s small y is a dampened version of small V cve right s small Y is the dampened version of the uh small y curve right okay it is this curve is translated down by small s that's it by a factor of X so please write this curve means small s into small y curve this curve is translated down by smallness right okay then you have the another line which is your n+ Delta k line n plus Delta K this is the uh amount of investment per worker which is required to keep the capital per worker and change we have talked about this right this is your replacement level of investment per worker which is required to keep the capital per worker unchanged and uh please write n plus Delta k n+ Delta K represents the amount of new investment per worker right amount of new investment per worker required to keep required to keep the capital per worker unchanged that's what it is your K dot is what s small y minus n plus Delta okay right and uh replacement investment per worker is that so everyone has uh let's say five units of capital per worker and uh you have invested uh 10 units of capital per worker but there has been the depreciation so of the five units so you are left out with only five units of capital per month so you're left out with only five units of capital per worker so in order to keep the capital per worker unchanged the amount of machines which you need to add to keep the capital per worker unchange that is what the replacement capital is because with the normal vrn there is some amount of machines which are uh which which just get obsolete right so you need to invest again in that number so that Capital per worker remains so because if the number of machines getting obsolete are more than the number of machines which you add so the capital per worker is going to fall so if you want to keep the capital per worker unchanged you have to uh add the number of machines also and the point is this is telling you uh Sy Y is telling you the number of machines added per worker n plus Delta K is telling you number of machines need to be added per worker so if the number of machines added per worker is equal to number of machines need to be added per worker to keep the capital per worker unchanged you say small K dot is zero if the number of machines added per worker are more than what is needed to keep Capital per worker unchanged then Capital per worker will increase if the number of machines added per worker is less than the number of machines need to be added per worker to keep Capital per worker unchanged Capital per worker is going to fall that's something right so this entire curve this green curve is telling you the number of machines added per worker this red line is telling you number of machines need to be added per worker at this point the number of machines added per worker is equal to the number of machines need to be added per worker and we call this as the St steady state you call this as as the steady state where nothing is changing right Capital per worker is not growing it is it is an equilibrium at this point so when the number of machines added per worker is equal to the number of machines need to be added per worker this is the study state but what about this point kot here number of machines added per worker is more than the number of machines need to be added per worker so the capital per worker is going to increase and it is going to increase till when till the study state is reached similarly here here the number of machines added per worker is less than the number of machines need to be added per worker so the capital per worker is going to fall when Capital per worker is going to fall it is going to fall till when till the study state is reached that is at this so this is the steady state this guy is the steady state although I'm not telling you about the stability but if you look at it this is a stable equilibrium you start from anywhere you are moving towards an equilibrium point so this is a stable equilibrium this is a stable equilibrium you start from anywhere you're moving towards this point so you're starting to from the left of the stady state you're moving towards the study State you're you're starting from the right of the study State you're moving towards the study state so this is a stable equilibrium what is an unstable equilibrium you move away from an equilibrium so if you start from the left you move further away from the study State you start from the right you move further away from the steady state so that is what the unstable equilibrium is but that's not what we are doing right now I'm just telling you that this is the way you solve this is the solo diagram and I want to tell you two more points there there is something which is called the capital deepening there is something which is called Capital deepening and something which is called Capital widening Capital deepening right Capital deepening okay when the amount of capital per work amount of machines you added adding per worker is more than the amount of machines need to be added per worker then Capital per worker is is is increasing so when this change is positive you you say that the uh capital is deepening so every labor now has more amount of capital with him the when every lab has more amount of capital with him when this change when this change means um when uh when uh the number of machines you add per worker is more than the number of machines you need to add per worker then every worker has more amount of machines with them this change this difference between the number of machines added and the number of machines need to be added when this change is positive when this change is positive an economy is increasing it's Capital per worker we say that capital deepening is occurring we say that capital deepening is occurring right okay a there can also be the case that capital is increasing but labor force is also increasing capital is increasing but labor force is also increasing and when with an increase in the capital your labor force also increases then what happens is that uh that uh there is no increase in the capital per worker and that is what is called the capital widening when every labor has more machines with him that is capital deepening Capital widening is that yes capital is increasing but at the same time labor force is also increasing so the number number of machines per worker is not increasing so but the capital is increasing so Capital widening is this Capital widening right when the stock of capital is increasing at the same rate as the labor force and the depreciation rate and the depreciation rate right uh so so this is what capital deepening and capital widening is capital deepening and capital widening is so when you move from K to K Star right here in this diagram this is the capital deepening case that is every individual has more amount of capital per worker every every individual has more Capital with him that is the capital deing case so beta this is what I wanted to do in this class right thank you beta comparative Statics in Solo diagram before I start I just uh want to say sorry I mean I was not able to make video since last 3 4 days there is some problem in my eye um I hope it will be fine soon so I I'll try to complete it comparative Statics in Solo diagram so we have seen uh how do we draw solo diagram in the last class right and how do we interpret this green curve showing the number of machines added per year per worker and this red curve is showing you the number of machines need to be added per worker per year to keep Capital per worker unchanged now let us look at comparative Statics means that you are changing some of the parameters let's say you change the savings rate or you change the uh uh population growth rate right then what happens so the first case case one increase in savings rate so we'll draw the initial solo diagram first so you have Capital per worker here and you have investment per worker or replacement investment per worker gross investment per worker like this thing here so you have the green curve like this small s into small K to the^ Alpha and this curve which is your replacement investment curve n plus Delta small K so this class is assuming that you have seen the earlier recording and we know that our study state is at this point so at this uh small case star the number of machines added is equal to the number of machines uh need to be added per worker per year to keep Capital per worker unchange that is what the stady state is right okay now we are saying that savings rate is going to increase so this small s is coming only in this curve now this green curve it is not coming here or here or anywhere so what will happen is that uh this savings Curve will be because of the increase in savings the savings Curve will shift upwards right savings rate has increased so savings Curve will shift upwards from this point to this point right so the new study state is this now new study state is this now so let me call this as K star this one is small K1 star now just have a look at this let me also just point this out these are point A and B at which the number of machines added per worker is equal to the number of machines need to be added per worker okay now you just look at the initial study State at initial study state but new savings rate listen to me carefully at initial studies state but new savings rate this point steady state is initial canot start but new savings rate this is telling you what this point is telling you what this this is the point in the green curve number of machines added per worker per year this point is telling you what this red line is telling you what number of machines need to be added per worker per year so what is happening you are adding more machines per worker per year then what you need to add per worker per year to keep Capital per worker unchange so what will happen every worker will have now more Capital with it so the capital per worker is increasing and it is increasing till when it is increasing till new study state is reached so at the initial study State at the initial study State and the new savings rate the number of machines you are adding per worker per year is more than the number of machines you need to add per worker per year to keep Capital per worker unchange so what will happen every worker will now have more machines with him so Capital per worker is going to increase and it will increase till when it will increase till new stud state is reached that is the point what is the second one case two increase in population growth rate increase in population growth rate right okay a small K this is investment per worker and here you have this guy small s into small K to the^ Alpha this is telling you this curve is telling you the number of machines you're adding per worker per year this is telling you this is telling you um number of this is n+ Delta small this red line is telling you number of machines need to be added per worker per year this is the initial study State now when capital sorry this population growth rate has increased so what will happen it has increased from n to N Dash it has increased from n to N Dash right so the new study state is reached here okay now listen to me carefully at the initial study state in the new population growth rate bet these are your two study States now this one and this one this is the old study state or original study State this is the new study state so at the initial stady state capital per worker and the new population growth rate here what is happening here you are adding less number of machines per worker per year than what you need to add uh per worker per year so now every worker will have less machines available with him so Capital per worker is going to fall and it will fall till when it will fall till new study state is reached K St it will St it will it will fall till new study state is reached right okay then I would want you to just have a look at this h so please write steady state quantity of capital per worker is given by the condition K small K dot is equal to Z but small K dot key expression we have derived now s into small y minus n plus Delta small K what is small K dot zero in place of small y I can write small K to the power Alpha like this right okay now can I write it this way s upon n + Delta K upon K the^ Alpha so can I write it like this s upon n + Delta K to the power 1us Alpha so I can raise uh I can take the root of this thing for for the power of 1 upon 1us Alpha so it is s upon n + Delta to the power 1 upon 1 - Alpha is equal to K small K star so this is that level of this is the steady state level of capital perver H stady state level of capital for work mhm and what is the stady state level of output per work C so your small y star small kyar to the power alpha alpha to the power Alpha upon Alpha minus Alpha upon 1 minus so this is stady state level of output per worker mhm this is the steady state level of output per worker so you have the steady state level of capital per worker and you have the stady state level of output for worker so what is what is it that you did in this class you found out the study state level of capital per worker and output per worker and you also looked at two comparative Statics that is uh through solo diagram increase in savings rate and increase in population growth rate right thank you let us look at transition Dynamics in Solo model we have seen that at the stady state capital per worker is not grown that is what the meaning of study state is that Capital per work worker is not doing so here at this point this is what my study state level of capital per worker but here K dot is equal to zero right so K dot by K is equal to Zer so the way you have found out steady state is by putting K dot equal to Z so the moment you put K dot equal to Z so this small K dot by small K that is also equal to Z that means the capital per worker is not growing small K dot is instant years change in K small K small K dot by small K is growth rate in small K so Capital per worker is not growing in stady state so one thing is that but what is happening in off stady State position when you are not at the study state that is what we are interested in right so there can be growth when you are below stady state but then the you reach at stady State the growth is uh going to be halted right so in this model economies make grow for some time but they can't grow forever right so if there is an economy which has still not reached its steady state it will be experiencing growth so while transiting towards steady state it will experience growth right but the moment it reaches steady state then the growth is halted but that is what we are interested in what is happening when economy is not at stady state that is what we are interested right so an economy that begins with the stock of capital per worker below study State value will experience growth in small K and small y along the transition path I'll be telling you what the transition path is along the transition path right now this is along the transition path H transition part to what to the steady state to the study state right uh so the further away you are from the study State more growth you will experience closer and closer you going to reach towards the study State growth will be lower and the growth is going to completely halt it will be stopped the moment you reach steady state over time growth slows down as the economy approaches its steady state and eventually and eventually growth stops all together and eventually growth both stops all together so we'll go back to our key equation of solo model we'll go back to our key equation of solo model which is what small K dot equal to S small y minus n + Delta small so this is small K dot s but you have already derived earlier in the earlier recording that small Y is output per worker which is equal to small K to the^ Alpha so if you have not seen those recordings please do that minus n + Delta small k okay a this is instantaneous change in small K if I want to write the growth rate so can I divide both the sides by small k i can small K Dot on small K is s into small k^ Alpha minus 1 minus n plus Delta this small K small K will get canceled out this small K small K will get canceled out right so this is what we have so this is the growth rate in capital per work this so I just write it somewhere the small K dot by small K is nothing but the growth great in capital per worker a the way Capital per worker is going to grow it will have the proportional change on the output per worker also why because I hope you remember this that small Y is equal to small K to ^ Alpha right I can take log of both the sides so it is log small y = to Alpha log small K I can do it this way I can take the time log derivatives D log small y upon DT = to Alpha D log small K upon DT so this thing is what 1 upon y 1 upon small Y into D small y by DT = to Alpha 1 upon small K into D small K by DT we can write like this so what is D small y by DT small y dot upon small y Alpha what is small so D small K by DT small K dot by small K so what I'm trying to say is that whatever is happening in in growth rate of capital that is uh that is proportionally affecting the output per capita also this is also uh you need to understand this so what is the meaning of this point this point means um can I just the meaning of this point is that the growth rate of small y that is output per capita is proportional to the growth rate in capital per work right growth rate in small y is proportional to the growth rate of small CL right so the way uh I mean whatever you say about small K you can easily say almost the similar thing about small y also because the growth rate of small Y is proportional to the growth rate of small Fair a then you think about it this way I'll just rewrite this equation because I'll be needing it that is small K dot by small K is equal to S small K to the power Alpha minus 1 - n + Delta I hope you not in hurry because I am not in hurry right so these lectures are not in any way in any kind quick fixes so you will have to sit with them I'm assuming that you are writing alongside you will understand it more I can just assume and I can just advise you that it'll be helpful for you okay AA other thing is Alpha is lying between 0 to one so it is less than one so let's say you have Alpha is half right or anything so it is going to be what um um say Alpha is equal to half so Alpha - 1 is equal to half - 1 that is - half so small K to the^ minus half which will be something like 1 upon root something like this right so the idea is that as your capital is going to grow capital is going to increase capital is going to grow then uh the growth rate of capital will eventually fall right it is going to eventually fall so please write this because [Music] Alpha is less than one right s SK Rises as small K Rises so be very particular in Solo model whether you are saying small K or you saying capital K that is important for us small K as small K Rises the growth rate of small K gradually declines right growth rate of small K gradually declines so you look at this thing so this is the first term this is the second term as your small K is going to increase now this term is going to grow is is is going to fall right as your small K is increasing s is given to you from outside so you have the first term is s into K ^ Alpha minus one this is the first term right or in simp simp Le words don't write this let's say if Alpha is equal to half then it say s upon root K like this so s is given to you from outside it is exogenously given and as small K is going to increase this is eventually going to fall this s upon root K this is going to fall over time please write and uh the the the higher the level of capital per worker the lower is the average product of capital small y by small K now what do you mean by this how is this coming okay beta you remember this we started with this yes this was a production function initially now in our initial recordings yes okay then this is what K to the^ Alpha L ^ 1- Alpha upon l so this is what this is your K ^ Alpha into l^ 1 - Alpha minus into L ^ 1us Alpha minus one so this is what capital K upon L the power Alpha so that is small K to the power Alpha right this guy is small K to the power Alpha so this is what small Y is equal to small K uh sorry small Y is equal to small K to the power Alpha right fair enough okay now if I'm going to divide this and this is what this is my uh average product of the labor right but I want the average product of capital so average product of capital so that is K to the^ Alpha L the power Alp into L the power uh 1- Alpha upon this one so this is going to be what K to the power Alpha minus 1 into L ^ 1 - Al so I can write this as K ^ Alpha - 1 upon L the power L from so this is capital K by L to the power Al so this is going to be small K to the power Alpha minus one this is going to be like this right small K to the power Alpha minus one fair enough so this is my average product of capital that is y by K is small K to^ Alpha minus okay now can I divide both numerator denominator of this with L I can and it doesn't change anything so what what does this thing become capital Y by L small Y what does this thing become small K oh sorry capital K by small and and this is equal to your small K to the^ Alpha minus right this is equal to the small K to the power Alpha so this guy is nothing your AP capital K is nothing but capital Y by capital K and you're writing it in this way so what they are saying is that if you look at it this small y by small K don't be confused the small y by small k is nothing but capital Y by capital K which is nothing but average product of capital right so and this small y by small K is nothing but small K to the^ Alpha minus one which is this and what you are saying is that as you are using more Capital per worker right the average product of capital is going to fall which is true and this is true because of diminishing uh productivity of the capital diminishing returns to capital accumulation so please write this because of diminishing returns to investment because of diminishing returns to investment so this is the first term this guy is the first term and then you have the second term second term of what second term of this expression so we have you have decoded this term now let's decode this term which is n plus Delta there is no K here there's no problem at all right so it is this is the second term n plus Delta it doesn't it does not depend upon small K it does not depend upon small K now comes your diagram just have a look at this so you have small K on the horizontal axis and uh this is this line n plus Delta this is nowhere dependent upon small K and here you have investment per worker or growth rate in capital whatever and beta this is your first term this guy which which is what small s into small y by small K which is small s into small K to the^ Alpha minus one this I have told you now that as your small K is increasing this term is going down your y small y by small K is going down s is exogeneously given so s small s into small y by small K is going down this is right as your small K is increasing this is what you have done and at this particular Point here this this is at this point this thing is equal to this means small K dot by small K is equal to Zer bet this is this expression now small K dot by small K = to S into small K to^ alpha-1 minus n present so when these two are equal that is the point of small K Do by small K to be zero so that means what that means your uh your steady state that means the growth rate in capital per worker is zero here this is that point small case Stu now the point is that supposedly you are sitting here tuck tuck tuck here so what is the growth rate yeah the growth rate is small K Do by small K is this growth rate more or is this growth rate more you can say this is the growth rate at uh when Capital per work is given or is this growth rate no this is which growth rate small K dot by small K let's say two H so this is small K1 this is small K2 now you tell me one thing further away you are from the stady state more you are growing the the nearer you are going towards stady State the less you are cing right and the moment you reach towards stady State growth is Haled is that right huh so right so the and the difference between the two lines is nothing but the growth rate that's that's what it is this is you have drawn this is the first term this is the second term the difference between the two lines is nothing but the growth rate in capital per so the further please write this line the further an economy is below the study State value of small k the faster the economy grows the faster economy grows you with me so the growth rate in capital per worker at K1 is more than the growth rate in capital per worker at K2 at at small K2 right so the further you are away from the study State the faster you are going the nearer you are to the stady state the lower you are going you with me the further an economy is below its steady state value the faster it grows and the further economy is above the stady state value the faster it declines right so for example you are here so here the growth rate in decline because now you will be moving towards it so here you are moving towards the steady state from here you'll be moving towards the steady state from the right hand side so there is the fall in the growth rate here there's the fall in the growth rate here so the further you are above the stady state the faster is going to be the decline also the further an economy is above it steady state value of small k the faster small K declines the faster small K declines so this is your transitional Dynamics in Solo right I hope you would have made notes and I hope you like the video right thank you bet solo model with technological progress now in this recording we're going to do one part of uh solo model when we are bringing technological progress also into account we are trying to show that along a balanced growth path the rate of growth of output per worker and the capital per worker is equal to the rate of growth of technology so up until now we have not brought the technological progress into account while doing solo model now we are bringing technology and if you remember in Solo model in the last recordings we have we we said this that at the stady state capital per worker and output per worker they were not growing small y dot by small Y and small K Do by small K small K were both equal to zero but now what we are going to show is that they will be growing at the rate of technological progress right so technology is going to be the source of sustained economic growth it is the technology which is going to be the main source of the sustained economic growth right so to show or to generate sustained economic growth we will be uh bringing technology into account so let's have a look at that and in the process we also going to Define what is a balanced growth path so I hope you'll write alongside me please write that to generate so sustained economic growth in per capita income we must assume technological progress and we are assuming that a is the variable of the growth in technology or a is the technology variable not the growth in technology a is the technology variable so the growth in a is what the technological progress is so a is the technology variable he is the technology varable uh now we will be assuming a labor augmenting technology that is when you introduce the technology when you use technology it is going to make labor more productive it could have also happened that it can make Capital more effective but that's a different assumption so I'm writing three kinds of production Function One in which labor is augmenting other in which capital is augmenting and other where technology technological progress parameter is just multiplied with a production function and there are different names of this so in this class we are assuming labor augmented so let me write the three kinds of production function for you one is labor augmenting and which you call as harod neutral right so here when a is increasing it means technological progress is there and due to that technological progress the labor is becoming more effective the productivity of Labor is increasing when he increases technological progress occurs due to which a unit of Labor becomes more productive due to which unit of Labor becomes more productive so the kind of the production function which we are going to assume is this it is increasing since this technology parameter is now multiplied with L so it is increasing the productivity of Labor the other one is cap and we will be assuming this right labor augmenting so other one is capital augment capital agenting and this you call as solo neutral so now the technology parameter is being multiplied with K so it is increasing the productivity of capital third one is hick neutral so here the technology parameter is being just multiplied with the original production function so this was the original production function and the technology parameter is been multiplied like this so it doesn't change much anything so we are assuming that the technological progress is exogenous this is exogenously given to you right so we are not worried about what is driving this technological progress that is not what we are interested so this is exogenously given to you that technological progress is exogenously given so technology parameters is a and a do by a would be the growth in the technology so that is what we are assuming that this is exogenously given to you SG this is exogenously given to you SG suppose uh a not is the initial level of uh technology right this is the technological progress or S sorry this is the level of Technology after T periods when technology is growing at the rate of T A not is the initial level of technology technology is growing at the rate of G for T periods so the resultant level of Technology after T periods is 80 so if you take time log derivatives so first of all take log of B this log of e to the^ K is just Cas so in this case it is just GT D log of and then take differentiate this with respect to t you differentiate this thing with respect to T is just G right so what is this this is 1 Upon A D A by DT but this is just a constant so this will be zero and what is this da by DT that is a DOT by a which is equal to G so this is what has been written out here this is what has been written out here so G is the parameter which is showing the increase in technology or the growth rate in technology this is the parameter which is representing the growth rate in technology right then Y is equal to a to the power Alpha a l to the power 1 minus Alpha is given to you right so let's say I divide both sides by Al Al is the effective lab we can write like this so this is small y this is small K to the power Alpha but this is this one is output per effective labor this is capital per effective labor mhm okay there are some books which write it this way they see divide both sides by L we we've written like this so it becomes what uh a to the power 1 - Alpha K to the^ Alpha L ^ 1 - Alpha minus so it becomes a^ 1 - Alpha K to the^ Alpha l the^ 1 L the^ minus Al so this K to the^ Alpha into l^ minus Alpha is what K by L the^ Alpha like this this is my right then I take log of both the sides Alpha log of k + 1us Alpha log of a I take time log derivatives D log of small y by DT is equal to Alpha D log of small K by DT + 1 - Alpha D log of small a by DT but what what would this be 1 by small y d small y by DT alpha 1 by small k d small K by DT + 1 - alpha 1 upon capital A D capital A by DT so this thing becomes what 1 upon small Y into y dot d small y by DT is what by Dot like this so keep this thing in head uh keep this thing in head this is an important uh expression for us we'll be coming back to this we'll be coming back to this okay you tell me one thing what was our capital accumulation equation right capital K dot S capital Y minus Delta K this was our capital accumulation equation and even with the model with technological progress this particular equation is not changing I'm not saying uh about the key equation of solo model I'm talking about this guy right this capital accumulation equation doesn't change so if I divide both the sides by capital K it becomes capital K Do by capital K S capital Y by capital K minus Delta it is like this right so what is capital K Do by capital K that is the growth rate in capital K or growth in capital K right okay now if you look at this growth rate in capital K but s is exogenously given to you small s that is savings rate Delta is exogenously given to you small Delta that's given to you so if you want that the uh your growth rate of capital K to be constant then you would want y capital Y by K to grow at the constant rate that's what you want so growth rate because bet small s is exogenously given to you Delta is exogenously given to you they are not changing so if you do not want that the growth rate in capital K do not change then this would definitely mean that growth rate of capital K will not change only when capital Y by K is is not changing right that is capital Y by K is constant growth rate in capital K will be constant if and only if capital Y by K is constant if and only if capital Y byk constant one thing you tell me one thing your small uh y by K what should I write this small y by K can I write it like [Music] this okay you have capital Y by K okay I divide numerator and denominator by L so this thing will become what capital Y by L upon capital K by L but what is this this is small Y what is capital K by small K right you tell me L is growing at the constant rate this is the labor force now and if capital Y by K is constant capital Y by capital K is constant and L is given to you you have assumed it that L is growing at the const then small y by K is also constant small y by K is also constant you understood this this is important for us right okay third point beta you are saying small y by small K is constant so this would mean that when will this be constant when the growth rate in small Y and the growth rate in small K is growing at the same rate when small Y and small K small y by small K is constant right then small Y and small K are growing at the same rate they are growing at the same rate right small y by small K is constant if this needs to be constant then small Y and small K they have to grow at the same rate otherwise this will not remain constant right so what is the Balan growth part balanced growth path is the situation in which capital output consumption labor all of them they are growing at the same rate all of them they are growing at the same rate so please write this a situation in which Capital output consumption and population are growing at the same rate oh it's not same rate are growing at the constant is called is called balance growth part right it's called balance growth part so I'm not saying that the population is also growing at the same rate as capital and out nothing like that so but they're growing at the same rate and mind you when when you say that uh in the steady state steady state is also the special case of the balance growth part so although they are growing at zero but that also is a constant rate so the situation for the balance growth path is that all of them they should grow at a constant rate right okay now come back to the equation which you have derived is what you have derived youve derived this small y do by small Y is = to Alpha small K Do by small k + 1 - Alpha capital A do by capital A what is this small y do by small y growth rate in output per worker what is this small K Do by small K growth rate in capital per worker what is this capital a do by capital a growth rate in technology which you have assumed it to be G we have assumed this but that small y by K is constant so this would mean that small Y and small K should be growing at the same rate so it means what the growth rate in small Y is equal to growth rate in small k so can I write gy = to Alpha gy + 1 - Alpha G I can write like this so can I write 1 - Alpha g y = to 1 - Alpha G so this 1 - Alpha and 1- Alpha will get canceled out and you'll have gy equals to G but gy is equal to GK so therefore gy is equal to GK is equal to this thing this this is an extremely important result because what is this result telling you that the output per worker and capital per work worker they are growing at the rate of the growth in technology right uh thus and this is happening along the Balan growth part thus along the balance growth path output per worker and capital perver are growing at the constant rate that is the rate of growth of technological progress that is the rate of rate of exogenous technology uh technological change or the great growth rate of technological prose this is a very powerful result in the model without technology ology what is it that you have said in the model without technology you have said that small K Do by small K that means G small K and small y do by small y that means G small y they are equal to zero at the study State that's what you say but here what you are saying is that the growth rate in output per worker is equal to the growth rate of technology so it means that the tech technology is the source of sustained economic growth that's an idea technology is the source of sustained economy so this is what I want to do in the first part of uh your solo model with technological progress so we'll take the discussion further in the next class thank you bet solo model with technology this is part two in this series so I hope you remember that we have written this kind of production function right K to the^ Alpha a l to the power 1us Alpha right K to the^ Alpha Al L to the^ 1us Alpha where Alpha is line between 0 to 1 so if I divide both sides by a l we'll be getting this guy e to the power Alpha e ^ 1 - Alpha minus 1 mhm this is capital Y by so let us call this as y c and this guy is small K C to the power Alpha right K C to the power Alpha K C is nothing but the capital per effective Labor uh or you can also write the ratio of ratio of capital per worker to technology right ratio of capital per worker to technology okay and this is your uh output technology ratio vle is called output technology ratio so our K is capital K by L so I can write log of K as log of capital K minus log of L that is log of capital K minus log of a minus log of [Music] l d log of cap D log of small k D log of capital K minus D log of a minus D log of L so I'll take the time log derivatives and then just differentiate this with respect to time so what will this be 1 upon small k c d small K C by DT = to 1 upon capital K D capital K by DT - 1 upon da d a by DT minus 1 upon l d l by DT what is this d k d small K C by DT is what K C dot upon K upon small K this is capital K Do by capital K minus a do by A minus L do by L like this uhhuh right so you have this guy with you fair enough okay so can I write this as small K capital K Do by capital K now capital A do by capital A is g l do by L is n do you know that is the rate of of technology is G rate of growth of labor force is n this is exogeneously given to you this is what we are assuming it since throughout mhm okay now what about this guy how can I write this your this equation your capital K dot equation here there is not much change in the capital K dot equation that is Sy y minus Delta k is capital Y minus Delta K upon K minus Gus this is the capital accumulation equation beta we have written this in the last recording if you if you have uh forgotten it please have a look at that this is sorry this is small S capital Y upon capital K minus capital K capital K will get canceled out minus n + G + Delta mhm like this uhhuh okay okay now can I divide both the sides by L I can mhm I can divide this both the sides by L uh you know what I should not be dividing It by L I should rather be dividing it by Al L because to keep everything uh same no so you have what s small vcle upon small kle small K dot upon small K can I can I can I can I can I can I okay can I just multiply both the sides by small K I can so small kle dot is equal to S into to small vle minus n + G + Delta small K curve like this right so the this is almost same as the key equation of your solo model without technology progress but you have written the key equation of solo model with technological progress also equation of solo model with the technological progress right with the technological progress now the point is the diagram is also more or less same right there's there isn't much difference in case of the solo diagram also small K C and here you have your investment per worker um this thing so you remember it better your production function in the Intensive form is small y curve small K Cur to the power Alpha right and your uh this guy this is s small vcle minus n + Delta + G small K so you draw this function I have told you you in Solo model without technological progress also if you have seen that recording that this is the concave shaped production function if you haven't seen that please see to that then we have also talked about this that if you multiply both the sides of this if you multiply this production function in the Intensive form with small s on both the sides then you will get the new curve so let me just call this is y c small K C to the power Alpha then you will get the new C like this it is the dampened version of uh of vur curve because s is lying between 0 to one so this is a dampened version of that curve here you have this is what it is your n + G + Delta plus CLE n+ G Plus Delta plus CLE right uhhuh so this is your replacement level of capital per worker and where these two are equal because at the steady state K dot is equal to zero that means s k to the power Alpha like this uhhuh so you guys have want so you have um s upon n + G upon Delta this is this is K Cur to the^ 1 upon K Cur to the^ Alpha so this is K Cur to the^ 1 minus Al so at what level of K curl will these two be equal when K is equal to S small s upon n + G + Delta to the power 1 upon 1us to the power n upon uh sorry small K to the small kle star is the steady state level of capital per worker in s Sol model with technological progress and the level is given by small s upon n+ G + Delta the^ 1 upon 1us and that is coming at this pointal stop that is coming at this thing so if let's say economy start at the capital uh uh technology ratio Which is less than small kle star let's say at small CLE not right then what will happen is that eventually economy will move towards the steady state level of K that is KL Star right because this is showing you this point is showing you the amount of investment being undertaken right and this is telling you the amount of investment which needs to be undertaken to keep the capital technology ratio constant so when you are investing more than what is being needed then definitely Capital technology ratio is going to increase over time so you might write one line please right if or at kot at small kot Cur the investment being undertaken exceeds the amount needed to keep the capital technology ratio [Music] right so Capital technology ratio will increase over time till steady state is reached and where is steady state reached when small K star is given by small s upon n+ G + Delta to the^ 1 upon 1us right and at point of small CLE star the economy is growing at the balance growth path so we have talked about the balance growth path in the last recording right uh economy is at this steady state and grows along the balance growth and this is growing along the balance growth path with me right so this is uh what I wanted to do in this record thank you solo model with technological progress this is part three in this series here now we are going to see comparative Statics means we're going to assume supposedly the savings rate is going to increase how this is going to affect the steady state in technological progress one and secondly what is going to be the effect on the time paths I mean how do you draw the time paths so for the capital per effective labor let's say so comparative Statics comparative Statics right so what we are saying is that uh we are going to see how the steady state level is going to be changed H how the stady state level is going to be changed in solo model with technology when savings rate is going to change and we will be also finding out the time pass for the variables this is your s small K Alpha right and uh this guy is your n + G plus Delta K this is the initial level of stady State this is the initial level of study State K star uhhuh okay now this solo diagram you have drawn in part two already now supposedly the savings rate will increase so the savings rate will increase your uh is going to be s- K to the^ Alpha now savings rate increase from s to S Das so your Capital per effective labor this will also increase from K C uh K not star to K one star so the steady state ear was sitting here now it is sitting here so at the old steady state and the new savings rate which is given by the green line at the old stady state level of of capital per uh effective labor and the new savings rate the number of machines added is more than the number of machines need to be added so naturally the capital per worker is going to increase and this is going to increase till new study status R so there is there isn't much difference as far as uh this guy is concerned uh the case without technological progress also the case without technological progress also uh okay now can we write our expression which we derived yesterday you remember we derived this thing yesterday K small kle dot is equal to s y small K minus n + Delta + G small cable this was the key equation of solo model with technological progress this was the key equation of solo model with technological progress now can I divide both the sides by K C I can a small V upon small K minus n + G + Delta fair enough okay now you tell me one thing your vle was kle to the power Alpha this also you have derived in the first part of technological progress in which you wrote that this is the production function in the Intensive form where KL is the effective labor fair enough so if I write V by kle this is going to be k k to the power Alpha upon kle that is K to the power Alpha that is kle to the power Alpha minus right okay now can I write this expression as this guy okay so basically I have kle by K dot by K is equal to S small s k c to the^ Alpha minus 1 minus n + G+ Delta minus n + G+ Delta right minus n+ G+ Delta fair enough now what I'll do is just have a look at this this is Cle and this is your kle by kle kle dot by kle so and this is uh this is uh made up of two things one is s into K to the^ Alpha minus one and this is n+ G Plus Delta so n plus G Plus Delta is not at all dependent upon K so this is just this the straight line like this H now as K curl is going to increase and since Alpha is lying between 0 to 1 this guy is going to fall no like this this guy is going to form and when these two curves when this curve s into K to the^ Alpha minus one and this n plus G+ Delta when they are equal then beta this K dot by K is equal to zero now this is that point and this is the steady state level K star no now this is let's say the first stady state given by this now supposedly the savings rate will increase so there is a shift of this curve like this this is s-h K to the power Alpha minus one no huh so at the initial stady State and the new savings State at the initial study State and the new savings streate this difference B this difference is nothing but K dot by K when I say k i mean small kle dot by small kle uhhuh that is the growth that is growth now what you are saying is that can you see this now this growth is declining over time and again this declines till this particular point where again your these two points are equal your s- K to the^ Alpha minus one and N plus G Plus Delta are equal again at the new study state GLE star so you guys could see this at the initial steady state level of capital per effective labor and the new savings rate the growth rate temporarily increases but then again it falls it falls it falls it falls till the new study state is so at the stady state again there is no growth that is first point that is first point so keep this thing in head please write this the increase in s to S Dash raises the growth rate temporarily as the economy transits to the new study State small K small K star one right G that g was the growth rate in technology that is constant one thing since G is constant fter growth in K cve in small K curve along the transition path implies let me just write that and then I explain implies that output per worker increases more rapidly then G right that is uh small y dot by small Y is greater than G is greater than G so what you trying to say you're trying to say this what do you mean by this particular point you mean this bet this is time this is small y do by small y this you know that uh this is the growth rate in output per worker a you did this thing yesterday Now growth rate in capital per worker growth rate in output per worker in the model with technological progress they are growing at the rate of G right they're growing at the rate of G in the stady stct in the stud state in model without technological progress they were not growing in the stady state but in the model with technological progress they are growing so this is g at time period T star your uh uh uh what do you call Save savings rate has increased so suddenly the output per worker has increased suddenly output per worker has increased but then again the growth rate has started falling falling falling falling falling falling falling till it again comes back at G till it again comes back at G that's an idea uhhuh till it again comes back at G you with me so it does not have a growth effect it does not have the growth effect so the so the policy change what do you mean by policy change policy change here means increase in investment rate increase in the savings rate this increases the growth rate but it increases the growth rate only temporarily because that growth again comes back to G over the period of time right so write this line note that policy changes policy changes in the solo model increases growth rate only temp temporarily along the transition to the new stud state so it means what that the policy changes have no growth effects that's important important line for us it can come as a question that is policy changes have no long run policy changes have no long run growth effects but the second point is that the policy changes can have have the level effect policy changes can have the level effect so please write policy changes when I say policy changes it means I mean here in this example I mean uh that uh your savings rate is increasing can how level effects policy changes can have level effects so what do you mean by this so here you have time here you have log of small y right log of small y so D log of small y DT uh that is your 1 by y Dy by DT that is y do by y so it means that one thing is that output per worker was growing at the level of G till time period T till time period T and then suddenly um Suddenly at the time period t your or you can say t star at time period T star the policy has changed that now the savings rate will be S Dash so it will increase and then again it will come back to G and then again it will come back to G so you can see this better this is the level effect this guy is the level effect so at the time of the policy change output per workers started growing more rapidly and then again growth fell and it again started growing at G but now the level has increased although the growth has again so it was growing at G here it is growing at G here but in between it was growing at greater than G but the level has increased now right so so it started growing at G it again started growing at more than G but then again it again started growing at G this is what you did here also it was growing at G it was growing at greater than G and then again it started growing at G that's what you did uhhuh so here also it is growing at G greater than G and again going back at G but this is a level effect which is uh which is happening this is the level effect so so the permanent policy change can permanently raise or reduce the level of per capita output that is an idea in this huh so this is what I wanted to do in this class thank you convergence in Solo growth model what do we mean by convergence under certain circumstances backward countries would converge towards the rich countries right so in order to close the gap between the rich and the poor countries in this process of catching up when backward countries are catching up with the forward countries that process is called convergence so let me write this point and then I'll move forward please write under circum circumstances under certain circumstances backward countries would tend to grow faster then rich countries in order to close the gap between the two countries and this phenomena of catching up is called convergence this phenomena of catching up is called convergence right so what is the second point so solo growth model I mean we are making uh two assumptions about the convergence one we have two identical countries so when we say two identical countries means all the parameters are same they have same total Factor productivity growth they have same labor force growth they have same savings rate so the initial assumptions initial conditions are same right that is we have two countries so assumptions we have two identical countries same total Factor productivity growth labor force growth savings stre right and the other assumption is that although they are identical means their growth rates for total total factor of productivity are same their growth rate for labor force is same their growth rate in sorry their savings rate are same but initially one country is starting with the higher Capital per labor and hence higher output uh per labor and the another country is starting with a lower Capital per labor and hence lower output per labor right so the rich country has initially a higher level higher level of capital per worker relative to the poor country so if they're starting with the higher Capital per labor it means that they have higher output per labor also right I hope you remember we did this now small Y is equal to small K to the^ Alpha this was the production function in the Intensive form so you start with small K to the power Alpha means you have the higher Capital labor ratio it is going to transform into higher output per labor ratio and consequently higher output per labor right higher output per labor fair enough so up and until now we haven't come to the point of convergence right I'm just going to show it to you beta so you have uh I hope you remember this curve sfk curve right and this guy is your n plus Delta K and at this point you have steady state level of capital labor now rich country is starting with the higher Capital labor ratio and poor country is starting with the lower Capital laboration rich country is starting with the higher Capital labor ratio and poor country is starting with the lower Capital labor issue right now if you just have a look at it you both of them are going to converge towards skar the growth rate which the rich country requires to reach towards G star is lesser than as compared to the growth rate which the poor country would require to reach towards the kar so poor country will start converging towards the rich country both of them will converge towards Cas star rich country requires lower growth rate and the poor country requires higher growth rate to reach towards skar which you show diagrammatically as this also which you show diagrammatically as this also you have log of capital Y and T out here and this is let's say the long run growth path this is what we have is the long run growth path and uh so rich country will converge like this poor countries starting with a lower Capital labor ratio will also converge toward this this way right so what you're trying to say is that uh the countries which have the lower output per worker or income per worker they have high higher growth rates in uh income per work you get the point you want to see the negative correlation between the level of income of the country and the growth rates so what you're trying to see is I mean in the panel where you have certain countries there are some rich countries some poor countries you want to see I mean when the convergence is going to happen is when the countries which have low more income per capita they will have higher growth rates this way but empirically what they found was this that when they looked at convergence among all countries H they found it does not happen I'm making three diagrams it will make things little clear to you so here you have the percentage growth rate in income per worker and here you have income per worker right so now what I'm doing is this just see this 20 40 80 100 120 and so on here at this point you have -4 - 2 0 2 4 4 6 8 okay so one of the country whose income is less it is having a high growth rate but there are some countries whose income is less their growth rates are also low there are countries whose income is less their growth rates are low but there are some countries whose income sorry here there is one country whose income is more more but their growth rate is less but there are some countries whose income is more their growth rate is also more so the point which I want to make is that when they ran the regression or when they find out the correlation they find this that uh uh it is not necessary that the countries which have low income per worker they will also have the high percentage growth rate in income per worker you want the negative correlation but this they found that if you look at the entire data of all the countries together you do not get this point where you see the negative correlation because in case if there is negative correlation so which country will have the lower income perver poor country when will the convergence is going to happen converion is going to happen when the poor country is going to have the higher growth rates but okay this is happening here yes but this is not happening here and the higher income countries will have the lower growth rates this is happening here but this is not happening here so it is not necessary that the convergence you will be able to see convergence when you take up the data for all the countries together that's not necessary so you would like to see the negative correlation uh so please write this we would like you see a negative correlation so that poor countries are growing faster then rich countries thus catching up does catching up so here you do not see this here you do not see this that is if you look at the data for all the countries together for uh for a long time period you won't be able to see this but then what happens is that if you look at only some rich countries so you take up the subset of countries you take up only those countries which are rich I mean why are you saying this this is not happening so you look at the country like ethopia so do we see this that of course ethopia has the low uh income does it also have the high growth rate no is it able to catch up with us no but if you look at only the subset of countries supposedly you look at only um few rich countries and then you look at the data then you will find that yes there is some evidence of catching there is some evidence of convergence right so convergence please write this occurs to some extent among rich countries so among rich countries yes you see conversions happening so here just see you write it like this 50 60 90 110 and so on right minus one -.5 0.5 1 1.52 and uh 2.5 and so on so you have this so among the rich countries whose income is less you find some countries yes they have U the high growth rates among the rich countries among the rich countries whose income is more you find that yes they have the low growth rates so there is some evidence of convergence in uh what do you call Rich countries that yes you see this kind of negative correlation there in case of rich countries that yes there is among poor countries again you do not see convergence I mean this is what so convergence does not occur among poor countries right so here you started like this 2 4 10 18 you have -4 - 2 0 2 4 6 so again here you do not find any pattern no that's not necessary so you do not find convergence among the poor countries you do not find convergence among the poor countries so you do not see convergence happening in all countries you do not see conversions happening in poor countries you see conversions happening to a certain extent in rich countries here comes the difference between the unconditional and the conditional convergence so I'll be telling you about the conditional and conditional convergence in the next class but the question which we want to answer in this classes why do we actually not see that uh convergence is happening in all countries why exactly it is happening why are we not seeing convergence happening in all countries the basic reason could be that the access to technology is not same for all the countries it is not same can we say that ethopia has the same access to technology as uh us has okay can we say that India has the same access to technology what US has what UK has so the access to technology is not same so what you assumed earlier that all countries are identical they have the same savings rate they have the same uh population growth rate they have the same labor force gr rate they have the same technology no that assumption that assumption is wrong so not all countries have the same technology the technology access is different and because the access to technology is different you do not find the evidence of convergence in all countries that's an idea right so countries do not have access to same technology so our assumptions they do not and why do we think that countries do not have the same access to technology the main reason could be that the countries that do not have the same access to technology because of government intervention how government intervention is going to create the problem in technology in problem in Access of Technology the reason is government intervention may have created labor unions labor unions have got bargaining power B CA of this government intervention these labor unions will not allow firms to access the technology which is going to hamper their own employment right so here government intervention is coming because of government intervention you have labor unions because of these labor unions they will not allow firms to have an access to technology I mean which labor unions do not want one other reason could be that government intervention can also create barriers to International Trade When government intervention can create barriers to International Trade then the domestic firms if they want a better technology which the foreign firms are using they will not have an access to that so because of this reason also the convergence is not going to happen so government intervention could maybe create problems because it might create trade unions for the reasons I've told you how this will hamper the same excess of technology for all the countries secondly it can also lead to barriers to International Trade so because governments they might create barriers to international trade and why do government creates barriers to International Trade because government has the Nationalist policies government has the protectionist policies they want to protect their indigenous Industries they want to protect their industries from the foreign competition they want their Industries to grow first of all without any competition but the problem is there is no growth without competition that is always there so this is about the meaning of convergence so I'm going to connect this with solo growth model also with a diagram and I'll tell you I'll be telling you about the conditional and the unconditional convergence in the next class right thank you beta so we were discussing convergence and solo growth model we have already talked about the meaning of convergence if you remember we have IED the key equation of solo model earlier also so this is K Cur dot is syy K Cur minus n + Delta + g k i can divide both the sides by uh K so it is K cull dot by k s y k upon k kle minus n + G+ Delta what is kle dot by kle that is the growth rate in K right a what is this K curl to the power Alpha upon K curl so I'm not repeating all of this again bet but I just want to show you this is what we have already done right so it's not nothing new here we have talked about these things you remember this this guy so you started with the key equation you divided both the sides by K you got K dot by K all this so we we talking about the same thing out here as well and we'll be reaching to this diagram again but right now we just writing all of this in in the context of convergence that's what we are trying to do out here so I hope you remember that the production function in the Intensive form we are writing it this way so we are resuming the same production function capital Y is equal to capital K ^ Alpha into L ^ 1us Alpha and when you write this in the Intensive form it becomes small y k y Cur is equal to small K C to the power Alpha minus n + G+ Delta K dot by K and here you have kle dot by K C as s and when I say k i mean small k out here to the power Alpha minus 1 minus n + G+ Delta remember one thing when you say small y k Sorry small y upon small K that would mean what the average product why because this is what capital Y upon L on capital K upon L right that is nothing but the average product of capital this is nothing but the average product of Capital One thing second thing second thing as K is rising that is as capital labor ratio rise but capital is subject to the diminishing returns no so this average product is going to fall so so this small YK C upon sorry small y c upon small K curl is nothing but the average product of capital and that is nothing but small K curl to the^ Alpha minus one not this point in particular capital K to the^ Alpha minus 1 declines s keal Rises because of the diminishing returns to capital accumulation in new classical model in new classical model one thing okay how is Sol all this related to convergence but this is what you have said now as your small K is increasing this guy is going to fall now s into K C to the^ Alpha minus one this is going to fall this is falling falling falling falling falling like this for the reason I've just told you this term is independent of K C so this is just like this m this just like this okay so at this particular point there is a steady state steady state means here the growth rate in capital per worker is zero you can call this as klear you can call this as KL star fair enough now suppose that uh there are two economies one economy is starting with the lower Capital per labor call this as poor economy so what is the growth rate beta growth rate is the difference between the two terms these two terms and so this difference is the growth rate of the poor economy fair enough okay let us say there is another economy which is starting with the higher Capital labor ratio Rich economy huh again growth rate is a difference between these two terms so what is the difference here for the for the higher for the uh this economy your uh your uh Rich economy this much so this is the growth rate of poor economy this guy is the growth rate of Rich econ both of them are converging towards this point steady state both of them are converging towards the stady say so if both the economies they have the same technology right they have the same uh investment rate they have the same savings rate they have the same rates of population growth means the the parameters are same so what will happen is that the poor economy will grow fast as compared to the richon right so output per worker gap between the poor and the rich economy will will become narrower and narrower As you move on right and both the economies they will approach towards the same stady state but the assumption is that both economies have same savings rate same technology same population growth rate same investment rate all of that so the initial conditions are same the only thing which is different different is that they are starting with a different level of the capital labor rip that's it right so among the countries that have the similar initial conditions or you can say that uh that uh uh among the countries that have the same stady state the convergence hypothesis will hold poor countries will catch up with the rich countries among the countries which have similar study States the convergence hypothesis is holding that poor countries are converging towards rich countries and they're converging towards the same state state right that's an idea so this is an the important prediction of uh important prediction of solo model please write this and I want um if possible please write this in in your answer also among countries that have the same stady state the convergence hypothesis should hold that is po countries should grow faster on average then rich countries on average then rich countries right and this is a very uh reasonable assumption as far as industrialized nations are concerned they have almost the similar population growth rate similar uh savings rate similar technology level so for industrialized countries yes the convergence hypothesis is going to hold right so neoclassical model is going to predict that the convergence is going to be this right but if you're going to look for all the countries of the world together then the convergence hypothesis is not going to hold because you cannot just compare industrialized nations as compared to least developed countries right they don't have same set of Technology they don't have the same savings rate they don't have the same population growth rate so you can't say that all of them are going to converge towards the same state no right so the differen is in one more thing is one more thing is that there can be the differences in the income levels which is going to have the impact on the study State levels also so it can result in the different study States as well so please write this the differences in income levels around the world the differences in income levels around the world largely reflect the differences in steady States so because all countries they do not have the same investment rate all countries do not have the same population growth rate all countries do not have the same savings rate same technology levels so you cannot expect that all countries will converge toward the same steady state ha convergence hypothesis is not going to hold right so please write because all countries do not have the same investment rate population growth rates technology levels they are not they are not generally expected grow towards the same stady state Target the same study State Target right a this is this is one prediction of the solo model the another prediction of the solo model another prediction another prediction is what B if the economy is far below the study state level the higher is going to be the growth rate of that right uh the faster economy is going to grow while the further economy is above this the steady state level the slower economy right that is also one thing please write this the further economy is below its steady state the faster the economy should grow the further an economy is above the stady state the slower economy should grow right the slower economy should grow so these are the two main predictions of the solo model and this is what your convergence hypothesis so the unconditional uh convergence is whatever be your level of stady State whatever be the level of population growth rate uh investment rate savings rate all economies will converge towards the stud state right conditional convergence and this is not supported by data unconditional convergence is not supported by data you have seen this I've given you the examples um yesterday also in terms of a diagram conditional convergence is somewhat supported by data that is an example of the industrialized nations so you instead of instead of looking at all the countries together if you look at the subset of countries the countries which have the similar savings rate similar population growth rate similar technology levels they tend to in among those countries the poorer countries will converge towards the rich countries that's an Ting uh so that is the meaning of unconditional and the conditional convergence right okay beta so this is what I wanted to do in this class thank you so more or less I think uh in by this recording I have completed mostly what I have to do in the solo model so I'll pick up something new in the next case thank you beta hello guys in this recording on RoR model of endogenous growth we'll be talking about how this model is different from solo model what are the assumptions which are different in RoR model as compared to the solo model are the predictions different and we will also talk about RoR economy so this will consist of three sectors final goods sector intermediate Goods sector and the research sector so these are the micro found foundations in the model and then we will derive the balanced growth equation of RoR model after that we will also talk about the comparative Statics that is supposedly if there is a permanent increase in the share of Labor Force which is engaged in research then how will it affect the economy how will it affect the steady state right so these are the things which we will be doing in this model so I hope uh this will be helpful to you the RoR model so we are going to distinguish the RoR model with the solo model and the main thing about the RoR model is that it has endogenized the technology right it has endogenized the technological progress while in Solo model technological progress was exogenously given so this is the basic difference between the two right so I'll be writing out and uh you'll be writing alongside and one thing is that in the in the last three classes what we have done is that we have done about the economics of ideas Source model explanation economics of idear right so researchers who are interested in profiting from their inventions right they will search for the new ideas and RoR model has endogenized the technology uh the technological progress right so please write this economics of ideas recording then I would urge you to please see that first rumor model endogenizes technological progress by introducing the search for new ideas by introducing the search for new ideas right Advanced countries they have developed particularly because of the technological progress right and they have been able and they have been able to have a sustained economic growth for a long period of time because they were they were continuously pushing their technology forward resources labor Capital ideations Roman model is particularly designed to explain how advanced countries were able to have sustained economic growth for a very long period of time paper a model is designed explain why and how the advanced countries of the world exhibit exhibit sustained economic growth right so solo model solo model tealog exogenously given so equation production function assume y equal to K to the power Alpha a l y to the power 1us Alpha this is what the production function is K is the stock of capital ly is the stock of Labor uh and uh Y is the output and a is the stock of idea so this is the new thing so I'm just writing it this way Alpha is lying between 0 to 1 right so for a given level of a this production function exhibits CRS or a given level of Technology production function y is equal to K to the^ Alpha e l y to the power 1us Alpha exhibit CLS right just Lambda Lambda Lambda l l Lambda l y this is there right mhm this is the same thing which you are getting right so what you are getting is just this Lambda power 1- so it is Lambda to the power 1 it is Lambda to the^ 1 so what you have shown out here is and this guy is what Lambda uh y KL so by increasing capital and labor by Lambda you are getting Lambda y KL that is it is almost like you are multiplying the initial output with Lambda right so what you are getting is CRS this is what the CRS definition this is what the CRS definition but you also have to understand and this is the point where the model is differing from solo that when you are incorporating a also as an input to the production function then this production function is going to exhibit increasing returns to scale however when we recognize ideas e also input the production function then then there are increasing returns to scale so it is like labor Capital double outut more so production function is going to exhibit increasing returns with respect to all three inputs ideas labor Capital it is exhibiting constant returns to scale with respect to Capital and labor one thing and why do you think the uh you are getting increasing returns to scale particularly with respect to the ideas that is coming fundamentally from the nonrivalrous nature of the ideas which you have already talked in economics of ideas right so note that presence of increasing returns to scale results fundamentally from the nonrival RIS nature of ideas nature of IDE capital accumulation equation that is mainly similar as the solo model only that is uh have capital accumulation equation which is uh K dot is SK y minus Delta K right so I'm not repeating this thing again because I have already dis discuss this at length while I'm while I was explaining solo model expain right so Capital accumulates as uh see what is SK SK is the savings rate so capital is accumulating as people in the economy are foregoing consumption forgoing consumption means savings at some rate SK and the capital depre iates at the rate Delta small syy are the gross savings so gross savings are equal to investment s is equal to Y savings are equal to investment in the economy or is inv investment so again you might have to look back at those recordings for for solo model just may this has been explained in very clear terms all of that so this SK into Y is the gross savings but gross savings is equal to gross investment in the economy at equilibrium from gross investment when you deduct depreciation you get net investment net investment is the change in capital right your liver is growing it's some constant and exogenous rate and some exogenous rate n so it is L do by l = n l do by L = to n next equal equation equation or so this is different from solo model because solo model technology is growing exogenously at some rate but here in RoR model we will be talking about when technology is growing how technology is growing and what factors it depend so uh we will be talking about that in the next recording right CH thank you bet we are taking the discussion further about the RoR model see if you if you look at solo model one thing is true that he has taken up technological change as exogenous right so he has understood this that yes the technology is improving and but what is determining that technological Improvement that is that we don't know so long run growth is due to the technological change but what is determining that technological change that we don't know so that is outside the model that is exogenously given what RoR has tried to do he has tried to endogenize this technological change so I'll write few points first of all so if you look at solo on the one hand and you look at RoR on the other then as far as solo is concerned right he is identifying that it is the technological change or technological Improvement which is determining the long run he's identifying that he's very clear about it right but he is not providing any explanation of what is determining this technological change so as far as long run growth is concerned is determining or is determined not determining is determined by technological change but the problem is he has not explained what is bringing about this technological change he has not explained that right what is what is determining technological change he has not determined that so so we say that technological change is something which is exogenous to the model right or you can say long run growth is something which is exogenous to the model this is what solo has done but what Roman has done is that he has picked up this disadvantage of the solo model and he has tried to explain this what so he tries to say that technological change is endogenous technological changes and dogenous funing he is saying this that yes solo has done till this point he has done completely correct that long run growth is going to be determined by technological change but what RoR is saying that I'm going one step further I am trying to uh explain what is determining this technological change right so what he has said is that he accepts he accepts the solo model model result that technological progress is determining the long run growth in output per worker but the point is that he is trying to tell that what is determining this technological change so rumor attempts to explain what determines this technological change one thing right so what is this technology term so I mean he has said this okay you look at this kind of production function so Y is equal to there are labor force uh which is involved in the production of output y to the power 1us Alpha X1 to the^ Alpha plus X 2 to the power Alpha and so on to x a to the power Alpha let's say like this so what is ly ly is the number of workers reducing output y right XI is the different types of capital goods different types of capital to Main diing margin returns capital good P right that is the main difference right so it is diminishing marginal returns are applied separately to each of the capital good note that note that diminishing marginal returns lies not to Capital as a whole but separately each of the individual capital this is mainly be coming because uh you have also assumed that 0 is less than Alpha is less than this is what you have assumed so if this is there then X1 the^ Alpha is there and you just try to find out what d y by D X1 is so you will find this that because Alpha is less than 1 so there'll be diminishing marginal returns to Capital so what do you have d y by D X1 let's say so this is going to be what ly to^ 1 - Alpha into Alpha X1 the power Alpha minus one right X1 the power Alpha minus one so these are the marginal returns but diminishing marginal returns del2 2 y by X2 so it is Alpha into Alp -1 X1 the^ Alp - 2 ly to the^ Alpha minus 1us Alpha but Alpha is lying between 0 to 1 so this guy is going to be positive this is these are the marginal returns right these are the marginal returns to the marginal returns these are diminishing marginal Returns the second derivative and because Alpha is less than one so this term is going to be negative which is going to make this entire term to be negative and that is the reason we have diminishing marginal returns applied separately to each of the capital Cod right so which you can also write this thing you can write as ly^ 1- Alpha summation of I starting from 1 to a and XI like this XI the power alha I'm sorry XI the power now there are two cases one case is when a is fixed a is the technology parameter right is fixed other cases when a is not fixed so it means that is also gr it means that is also C so one let's look at the first case first is fixed if the technology is fixed uh but Ty technology is fixed and technology is not improving there are diminishing returns to the capital good so what do you think we'll happen eventually the marginal returns are going to fall fall fall fall fall fall and eventually it will taper to zeroy fix every time you use more Capital it will go on decreasing decreasing decreasing decreasing inreasing because technology itself is not growing return returns so [Music] timately and because of that the growth will taper off if a is fixed the pattern of diminishing returns the separate capital good would mean that the growth would eventually taper to zero would eventually off to zero that a is not fixed rology there'll be some workers who will be involved in this kind of research that how technology should be improved work l l is the amount of workers which are engaged in research and development which is going to improve technology right whose research and development capitals new capital goods would form so they are involved in research and development it will lead to the invention of new capital goods new capital goods of Technology a DOT is the how technology is changing over time a do d by DT that is gamma L A to the power Lambda a to the power 5 right number of workers inv cap Goods invent a do a do is nothing but this is instantaneous change in name a that is nothing but da by DT a do right capital goods and lamb what is this Lambda right so Lambda is the index of how slowly slowly diminishing marginal productivity set sin for resarch diing right prevailing value of a itself prevailing value of a itself could Newton gravitation he would have discovered giant shoulders effect giant shoulders effect giant shoulders effect giant shoulders effect something else will come right so but that will be dependent upon the the current state of Technology the same time right so RoR research sector gets rewarded with patent patents research sector research sector This research sector it gets rewarded research sector rewards things in patent research in researchers Monopoly position monop right so and wages are equated across the sectors ultimately right resources allocate economy so how many people will be there in the research sector and how many of them will be there in the output will be producing output because that's an idea right so how resources are allocated in this economy next question our resources are allocated in this economy fraction of output right so output Capal or labor me constant fraction constant fraction of output is invested in capital how many of them them will do research and how many of them will do will produce out total labor force economy labor constant fraction Sr fraction labor force constant frac output Sr into L output 1us Sr into L this is right and we'll take the discussion further of r model in the next class thank thank you Roman model of endogenous grow up until now we have already seen that uh RoR model isogen the technical growthy Reas model exogenously given Romy final goods sector intermediate Goods sector or research sector final good sector produce which is producing ideas now who is I mean what where it is going to be used these ideas will be used where these ideas will be used in order to produce capital goods capital goods these ideas will be sold to whom by the research sector these ideas will be sold to the intermediate sector by the research sector intermediate Monopoly firms research sect intermediate ex capital good intermediate sector monopolies firms research sect capitals so intermediate Goods sector capital goods final goods help finala Rec final finfly competitive firms labal in order to produce the homogeneous output final good sector output production function we write it like this r f i starting or J starting from 1 to a x g output which is being produced L Lab in order to produce y intermediate intermediate Goods right there is more than one capital good in the model summation of J starting from 1 to there is there are more than one capital good in the model there is more than more than one capital good in the model in the model summate right number a that is telling you what technology number a that is telling you the number of capital goods which are there in the economy un aggregate is what the technology is right so it measures the number of capital goods in the economy right uh right inventions ideas production function X1 to the power Alpha plus X2 to the power Alpha and so on to x a given so for aals sorry lab L X1 Lambda X1 l Lambda to^ 1us Alpha into Lambda to^ Alpha is what right that is just Lambda to the^ 1 for given it means for a given a up amount of Labor amount of each capital outut production function is is uh uh exhibiting constant returns to scale say integral 0 to a XG to^ Alpha DG you can write like this these are this is the sum of exchange so a measures the range of capital goods that are available to the final goods sector right range of capital goods that are available to the final goods sector up number ofal returns to large number of firms final good sector perf competition right okay right so Prof maximization EXC fin am ofal to produce the output Ian just for simplification the price of final output is oneus price of final output into output price of final output one output production function production function ly to the^ 1us Alpha integral 0 to a XJ to the^ Alpha DJ yeah cost or capital goods cost lab priceb there are a capital goods which you are using to 0 integral 0 to a PJ XJ X1 price P1 Capital X2 price P2 and so on and so forth say right ly to the^ 1 - Alpha X1 to the^ Alpha + X2 ^ Al P1 x1+ P2 X2 plus P2 X2 how much of Labor you will have to use right so how much of Labor you have to use it depends upon that is Del Pi by D ly so it is 1 - Alpha ly the power 1 - Alpha into -1 right and uh Alpha minus W = 0 like this right minus W = 0 so can I write this as 1 - Alpha uh L Y to the^ minus Alpha is equal to W right is equal to w okay can I write this as it is now ly to the power Alpha like this okay now uh if I can multiply this with so I can multiply both numerator denominator by L ^ 1 - Alpha right I can do that so denominator becomes what just ly now and numerator is what l^ 1 - Alpha into X1 ^ Alpha plus x^ Alpha that is nothing but y Y is what this guy ly ^ 1 - Alpha into X1 ^ Alpha plus X2 the^ Alpha that's what it is W like this so w is equal to yeah should not be putting equal to sign out here so w is equal to 1 - Alpha into y upon ly that is what it is so you will will be hiring the labor until the marginal product of labor is equal to the wage is equal to the wage what about this guy D Pi by D X1 D Pi by D X1 right so what do I have so I have X1 out here I have X1 out here okay so I can write this as l y^ 1 - Alpha into Alpha X1 the power Alpha minus 1 minus P1 = 0 minus P1 equal to 0 you with me so I can straight away write this as P1 equals to your Alpha ly the power x this thing the power Alpha minus one or for any J I can write this as PJ = to Alpha ly^ alpha-1 XG the^ Alpha minus one like this mhm so I've written this so the first condition is tell you that the firms are going to hire labor up to the point where uh the marginal product of labor is equal to W second condition says that firm is going to hire lab hire each capital good up until the point where the marginal product of La marginal product of each capital good is equal to its respective rental rate right rental rental price PJ okay so after this we'll be starting with the intermediate Goods sector which we'll be doing in the next class right thank you V so let us now take the discussion further about the intermediate sector in the RoR model so we have already seen the final good sector now we looking at the intermediate sector so intermediate sector intermediate sector monop naturally capital goods final good sector monopolist Pat research sector in capital goods final sector ultimately ultimate consum that is this particular point what is the output with with which each monopolist form is selling Capital X1 X2 X3 and so on to XJ capital good capital goods demand function given which is pjx so that is what it is so that is what the fixed cost is marginal cost isal that is what the marginal C this is what the intermediate Goods sector is looking like right profit maximization intermediate firm sorry price of the intermediate good into the quantity of intermediate good that is what the revenue minus r which is the marginal cost into the quantity produced of the intermediate good minus fixed cost they profit maximize you don't have to consider that right so I'm just writing this like this right Maxim in terms of X Maxim first function as it is into derivative of second with respect to X is 1 plus second function as it is into derivative of first with respect to X is p- x - R isal 0 - r equal 0 you be simply that is p plus x p- x is equal to R I can write like this and I can then divide both the sides by P so once I divide both the sides by P I Get x p-x upon p is equal to rpal R by P so if I want to write this in terms you going so what is p going to look like what is p going to look like it will be 1 +- x upon P into X into R here in the last class we have derived this PJ is equal to Alpha l y ^ 1- Alpha XJ the^ Alpha minus 1 so this is that firms they going to rent capital goods until the marginal product of each of the capital good is equal to the price PJ so I can write PJ as so this is from the earlier part right so if you have not seen that recording then only all of this will make sense to you XG to the power Alpha minus so I'm getting rid of these subscripts Alpha minus one so d p by D X will look like what Alpha y^ 1- alpha alpha - 1 x ^ Alp - 1 - 1 is Alpha - 2 right so I can just write this as alpha alpha minus 1 l y the^ Alpha minus 1 - alpha x the^ Alpha - 2 right this is what my DP by DX is or in in other words this is p dash this p- x can I multiply uh this with X by P Al into Alpha - 1 l y ^ 1 - Alpha x ^ Alpha - 2 into X by P and in place of P what will you write alpha x the power Alpha minus 1 to Beta to alpha alpha will get canceled out this ly^ 1us Alpha Al ly^ alha minus one will cancel out right okay so what do you have this thing will look like Alpha - 1 into x ^ Alpha - 1 into X that is your power is going to get added and this will become Alpha minus one then upon Alpha minus one again this will get like this so basically what you can write this d d p by D X is what P Das X into X by P is equal to Alpha minus one is equal to Alpha minus one so what do you have you have just written p as p as what 1 upon 1 upon 1 + p-x X by P into R so in place of p- X into X by P I can just write what I can write this as Alpha minus 1 into R so this 1 one will get canceled out this will become what 1 upon Alpha into r 1 upon Alpha into R so you have what p is equal to 1 upon Alpha into R so note that this intermediate Goods firm is going to charge a price that is simply a mark up over the marginal cost R so it is just charging 1 upon alha is two cost rge twice your cost so it is you are just charging a price that is simply a markup over the Marin cost St now this is this you have done for the particular monopolis but that is true for all monopolist right because all monopolist are same they are facing the similar kind of demand function for each of the capital good so all of these capital goods are going to be sold at the same price right demand function is also same each cap so the each capital good employed by the final goods form in the same amount by right or capital good employed by the final good firm is going to be in the same amount XJ is equal to X like this right or capital good Amal profit be same her capital good firm profit be same right that is one thing the total demand for the capital good right that has to be equal to the total supply of capital in the economy right finally please write the total demand capital from the intermediate Goods from the intermediate good firms must equal to the total total Capital stock in the economy to the total Capital stock in the economy Capital so toal Dem Capital demand that has to be equal to the total capital which is available in the economy right firm capital goods am right since capital goods are each used in the same amount X rights or her form x amount of capital a Herms x amount of capital demand total demand for Capital should be equal to the total supply of capital so what is the demand for Capital equal to then K upon a right right finction her firm capital good use that is the same amount right the final goods production function and be Rewritten using the fact at XJ equal to X Y production function a l y to the power 1us Alpha function what is the kind of the production function which you have written summation of just going back summation of J starting from one to a XJ power this is what you have written same amount of X demand same amount of X Demand right so what I can write so instead of writing summation of this thing I can just write this thing as X the^ Alpha I can just write this thing as X to the^ Alpha and what is x ^ Alpha K to ^ Alpha upon a ^ Alpha what is that a into l y^ 1 - Al e^ minus Alpha K the^ Alpha I can write like this H I can write like this so I can write like this l y to the^ 1- Alpha a^ 1- Alpha K to the power Alpha right so can't I write like this a into L1 y to the power 1 - Alpha to the power 1 - Alpha right so this is the final goods production function right and it is generating the same aggregate production function which we have been using throughout so this is the the production technology for the final good sector so this is the production technology for final goods sector right in the next class we'll be talking about the research sector so in this RoR economy you had the final goods sector you had the intermediate Goods sector and then you have the research sector the we have already talked about final good sector we have already talked about the intermediate sector in this class in the next class we'll be talking about the research sector right thank you bet so we were discussing about the RoR model of endogenous growth I'm intermediate out right so let's write alongside note that intermediate Goods supplier value of output is PJ XG is pjx am value right how much this value of output is being used P for capital goods XJ here XJ is the capital good her capital good price R rxj PJ XJ right so it is R upon PG rumor model only it will make sense last time you were able to reach till this particular Point your Alpha P equals to R Subs so it is alpal profit maximizing condition intermediate Goods suppliers small r capital r if you want you can just write at small R because otherwise this might be a confusion also right so earlier have derived the equilibrium profit maximizing condition what was that your Alpha BJ equal to R this is what we have DER Alpha P equal r p so PJ is R upon PJ is r how would you write R upon PJ that is nothing but PJ you can write R upon sorry you can write R upon Alpha that is equal to at the profit maximization so intermediate Goods FS Alpha proportion of their revenues capital goods please write so intermediate Goods form and Alpha proportion of their income if of their revenues of their revenues on capital goods capital goods that is kept as profits and then they are converting this capital goods into the final goods right so please write this so 1us Alpha proportion of their revenues are kept as profits that is spent on intermediate Goods alpha y proportion is being spent on intermediate Goods so alius Al amount that is kept as the profits right so please right if alpha y is the amount spend what spend on intermediate Goods right on intermediate Goods then alpha y is the revenue is the revenue of the intermediate Farms right uh 1 minus Alpha part is been kept as the profits right if 1- Alpha of intermediate firms profits forms Revenue are set aside profits then of that Revenue you are keeping 1 minus s pass height for profits so this is the total aggregate profit of all the intermediate forms together how many forms are there a hey this is this part is was not very clearly given so I'm just trying to break it down if there are a forms a forms a intermediate Goods forms intermediate Goods forms then what is the profit of one particular firm the profit of the all firms together is 1 minus alpha alha y profit of one particular form would be Pi upon the number of intermediate forms that is pi upon a right so if there are a forms the profit of the particular particular form is pi J which is you just divide this profit by number of forms right right so from here what you know is that the that the uh profit of the particular firm is the function of Y upon a right is proportional to y Upon A and we will be using this particular point when we'll be writing about the research sector please write this so from here we know that profit of the particular profit of the particular ter is proportional to y upon a right so what do we do out here is that from here we're going to write about the uh research sector after this particular point so that I'll take up in the next class but I'll be using this particular point right which we have just written here CH I'll be uh taking your next session next day thank you beta RoR model of endogenous growth we've been talking about this model for quite some time now and uh final good sector R economy we have talked about the intermediate Goods sector now we are talking about the research sector right so I mean you look at it anyone is free to prospect for the new ideas anybody could just find out new ideas and in our model what do we mean by new ideas new ideas are the ideas or the designs which can make the new capital goods resarch capital goods intermedi Goods or intermediate Goods final goods final goods form ultimately final goods produce is New Designs are discovered according to the following equ right so that is a doal GMA L A Lambda a a DOT is the instantaneous change in a right each technology endogenous technology here that is the instantaneous change in a or L is the proportion of Labor Force which is engaged in uh in finding out new ideas which is engaged in research right Lambda productivity Lambda is the productivity of the labor force which is engaged in research and Lambda is lying between 0 to one so it is less than one so ultimately diminishing marginal productivity resarch or dep prevailing value of a that is this is this five is showing you the externality which is attached to uh giant shoulders effect research already devel that is an idea [Music] IDE capital good inventor researcher for a new design or value present discounted value I'm getting the stream of revenues for the for future so I'm going to find out what is the present discounted value for for that investment for those profits which are discounted value of profits earned by the intermediate firm good firm say compared to the price of a patent nobody would be willing to pay right no one would be willing to pay man price of a patent is less than the present discounted value of the profits earned by thei to theice so somebody else will bid higher no someone would be willing to pay new design in that PA is the price new design right the is here is the price of a new design intermediate options right please don't write it from in your answer think from the point of view of intermediate good [Music] form so PA is invested in a bank then interest earned when investing is what RP app intermediate good options option one option two option two patent Capital right if the amount the a is invested in the patent you can earn the profits then on the profits period then sell the patent right so I mean whatever capital gain or Capital loss will occur that is going to result from the change in the price of the gain or loss that results and the change in the prise of M the pent m you'll be getting pi plus p.8 p.8 is the change in price in equilibrium idea is that it should be the case that the rate of return from both the options should be equal we equilibrium in equilibrium must be the case at the rate of return both of these Investments the same that means RPA equals to Pi + p. a so you can write the value Pi upon PA P do a upon P along the Balan growth path R is constant so what is r r is the price at which the supply of capital is equal to the demand for Capital right and that is also proportional to y by K so [Music] along R is constant right therefore Pi by PA must also be constant also be constant MH and when will Pi by PA be constant when pi and Pa both are growing at the same rate then only they will be constant which means which means that pi and PA row at the same rate the same rate right and you have already shown in the last class this is that Pi a is 1 - Alpha into alpha y by a you have shown this for a particular form right so that is proportional to y by a also we have shown that Pi is proportional so this is the point which I was telling that I'll be using that recording in this one proportional y by a and you know that the per capita output small Y and a is growing at the same rate output small Y and a go at the same rate right along the balance growth path G small Y is equal to G A is they are equal and that what comes out is that basically and and g k they are all same G small K right along the Balan growth path right so small ya a all of them are growing at the same rate so that y by a may not have to write this much in the answer but I'm just writing this will grow at a population growth rate population growth rate n population growth rate n right so Pi by a is growing at the constant rate which means pi and Pa they also have to grow at the same rate and you also know this that the per capita output small Y and a are they growing at the same rate and capital Y by a is growing at the rate of n that is the initial assumptions which we have made in the initial recordings so this would mean you can write R equals to Pi by PA + n so I can write this as r - n = Pi by PA so P = to Pi upon r- n p = to PK upon r- n right okay so this is the price of the patent along the balanced growth path D along the balance growth path right okay so we'll take the discussion further in the next class time so in this recording we're going to talk about the case that along the balance growth path the growth of the technology is given by gal to Lambda n 1 - 5 Pro this is what we going to prove so this is just using the knowledge which you have developed in the earlier classes uh of the RoR model so we are given with this production function v y = to k^ Alpha a l y to the power 1us Alpha is of course capital l y is the labor employed in the production of output L A is the labor employed in the production of research or in uh in research right so L A or l y is the labor employed in production of output right and uh a is the stock of ideas or stock of knowledge Alpha is the parameter which is lying between 0 to 1 K dot is s k minus DK so ESC the rate of savings this is D into K not derivative of K D into k d is the rate of depreciation population growth rate is also given to you L do by L is equal to n is the population growth rate a DOT is the number of new ideas which are produced that is the change in the stock of knowledge number of ideas or a a number of new ideas produced number of new ideas produced and you write a DOT as Delta Bar L A with Delta bar is the rate of discovery of new ideas right L A is the proportion of population which is engaged in research ly is the proportion of Labor which is engaged in production of output L is the proportion of population which is engaged in research or not proportion um the population engaged in research not proportion of population because it is ly plus L A is the total l so ly is the population engaged in the production of output L is the population engaged in research rate of discovery new ideas which is Delta V is Delta e the^ 5 where Delta and 5 are constants Pi greater than zero it is representing that the productivity of research increases with the stock of ideas that also that also is happening right so there are stock of ideas which have already been discovered and with those stock of ideas then new ideas are developed again represents productivity of research increas with stock of ideas already been discovered and F less than zero represent the fishing out case I less than Z would represent fishing out that is with the stock of ideas now you're not able to develop the new ideas of productivity of research is falling so I can write a DOT as Lambda this is a DOT Lambda sorry Delta l e the power Lambda e to the^ 5 H right so I'm writing it this particular way so you have a DOT you have written as Delta Bar LA Delta bar you have written as Lambda a ^ 5 and here no Delta bar you've written as Delta a the^ 5 so and here you're writing L is l a the^ Delta this lamb this La the power Lambda this Lambda can lie between 0 to one it is representing the productivity of Labor in terms of the new ideas five is representing the productivity of research in terms of new ideas this is telling you Lambda is telling you the productivity of Labor in terms of new ideas it represents the productivity of Labor in terms of new ideas H so f is the externality which is associated with standing on shoulders effect externalities externalities externality Associated by representing standing on shoulders effect standing on shoulders if so you are able to see far off because other people have developed the earlier ideas so that is what the standing on shoulders effect I've talked about the standing on shoulders uh yesterday also because he's sitting on the giant shoulders right so that is what what standing on shoulders effect is IDE that is what the standing on shoulders effect is right for Lambda externality Associated externality associated with Lambda is referred to as as stepping on two's effect two is right so it is like the decreasing returns to the physical resources in the knowledge production function that is what stepping on toes effect is the amount of labor involved in research plus the amount of Labor uh involved in uh in the output production is equal to the total labor which is available so this is the distribution of the total labor force right and uh the proportion of labor involved in research is let's say that is equal to Sr and uh this is the labor force engaged in research reduce new ideas right and this is the constant proportion we have assumed this so this is the constant proportion of the labor force which is engaged in research so 1 minus Sr is the constant proportion of the labor force which is engaged in the production of output so constant portion of Labor Force engaged in production of output that would be what 1 minus Sr 1us Sr now the point is that along the Balan growth path the rate of growth of per cap output rate of growth of Technology rate of growth of capital labor ratio that is all same that is what the balanced growth path is right so along a balanced growth path rate of growth of per cap output V small y rate of growth Capital labor ratio D small k and the growth rate of stock of ideas growth rate of stock FAS they are all growing at the same rate they're all growing at the same rate okay we have written a DOT as what Delta L A the power Lambda e the^ f right and then I can divide both sides by a l e to the power this and this thing becomes e to the power 5 1us you can you can uh write this of course this is 5 - one but I will write this as this thing Delta L A the power Lambda upon a power 1us y I can write like this and along the balance growth part G is constant so this a do by a is G along a balanc growth path a do by a is G is constant but you will have this thing as constant only if the rate of growth of numerator and the rate of growth of denominator is se but this growth rate will be constant if and only if numerator and denominator of grow at the same rate grow at the same rate so what happens is let me just write this again so a DOT by a is Delta L A the power Lambda all upon e the power 1 - 5 let me take log of both the sides log of a do by a this is log of Delta plus Lambda log of L A right minus 1 - 5 log of right so a do by a is what GA and then I take the time log derivatives so what is a do by a that is g I'm taking time log derivatives so D log of Delta by DT is z that is just a constant - 5 a by DT but G is a number that's a constant so that is this is going to be equal to zero right this is 0 plus Lambda this is what this is what 1 upon L A into d l by DT that is l a DOT by La you have I mean while doing growth Theory you should know about time log derivatives I mean this I have I think I told I've taught about this in the first class of solo model if you have not seen that recording please see that and this D log of a by DT is 1 Upon A D A by DT d by DT is a do so it is a do by a right one thing you have to understand this is not L do l a DOT upon La is not sa and is not Sr Sr is the share of the labor force engaged in the population that is not the growth rate so write one line along the Balan growth path growth rate of number of researchers what is L do a by L telling you that is the growth rate of number of researchers must be equal to the rate of growth of population that is L A do by L A is equal to n right so if the growth rate of the number of researchers is going to be more than the growth rate of the population then in that case uh the number of researchers will be more than the population and that is absurd we s bka so if it were higher the number of researchers eventually the the population which is impossible so now what can I write in place of L do a by sorry La do by La is I can write nus 1 - 5 and this uh this a do by a is what G right and from here I can write GA as just taking this thing here okay I'll take it 1 minus 5 G = to Lambda n or G = to Lambda n upon 1 - 5 right so this is what we want to prove that is the growth rate in the stop of IDE that is or the rate of technology that is equal to n upon 1us 5 in the balance along the balance that is an idea what we want to show right yesterday we derived this that growth of technology in the balanced growth path now there are few special cases which we want to talk about please write them alongside there are few special cases case one let Lambda equals to 1 I = to0 and La is constant that is the labor force engaged in the production of research or research ideas that is constant and the productivity of researchers Delta is constant right productivity of uh researchers Delta is constant and we know this that uh a do is equal to Del lamb Delta L the power Lambda e ^ 5 well I can write this as just because Lambda is 1 and this is e to the^ 0 so basically which I left out with only Delta into L MH Delta is constant La is constant so it every period of time just constant number of ideas right so the stock of ideas is constant because Delta is constant La is constant please right the economy is generating constant number of ideas this each be each period H so if you are generating the constant number of ideas each period Then a DOT by a is going to fall over time right eventually the growth rate of because what is a do by a that is the growth rate of uh the ideas right growth rate of the stock of ideas that is going to fall over time because a DOT is not growing and if that is not growing So eventually that growth rate is going to fall therefore the growth rate of stock of ideas a do e decreases over time and eventually approaching zero also the per capita growth rate of output that is also going to fall over time that is also not increasing because an economy is not operating at the sustainable growth rate without a research effort right uh and I won't say without a research effort I mean with a constant research effort because the productivity of output is subject to diminish returns if we are not not producing new ideas or we are producing ideas just at the constant rate eventually maybe in a very long term even the productivity of output is also going to fall the the the rate of growth of output that is also going to fall right is right and the growth rate of of per capita output also BS over time an economy does not operate a sustainable growth rate even with a constant resch effort even with a constant research effort so the point is that you are having a constant research effort fair enough but the problem is that over time the growth rate in the ideas that is also going to fall if you're just having the constant research effort right uh that a DOT a DOT by a will fall to zero over time because what is what is happening is that with a constant research of effort stock of ideas is increasing so a is increasing over time but a DOT is not increasing at the same rate and because of that a do by a is falling okay in my economy I will be able to produce more one of course when I'll use more inputs and two with more research If eventually the research effort growth rate is going to fall then in the very long time the per capita output growth rate that is also going to to fall and eventually that is going to be operate that is also approaching towards zero so even with a constant research effort you cannot have the sustainable growth rate of output that is one example the another example is when you have your uh case two right let us look at case two over here Lambda is 1 Y is 1 L is constant and productivity and productivity of researchers Delta it's constant mhm so how do you write e do is equal to Lambda sorry Delta L the power Lambda e^ 5 you write it like that but Lambda is 1 and 5 is one so you have this so you have a DOT by a as Delta l a a DOT by a is Delta l so a do by is the growth rate in the stock of ideas so growth rate in stock of ideas is constant Now Delta is constant La is constant so growth rate in the stock of ideas that is constant in each period right uh so please write this suggest that the growth rate stock of ideas is constant each period so if the growth rate of the research is constant then the growth rate of the output is also constant because that's of the per capita output that's y do by Y is equal to your a do by a so that is also constant therefore growth rate of of per capita output is also constant so at least it is growing maybe at the constant rate but it is still growing right per capita output is still growing so even with a constant number of researchers you still have the con la is constant the people or the labor force engaged in research that is constant so even with a constant number of researchers you still have a sustainable growth rate and even with constant number of researchers the economy operates at a sustainable growth rate right so these were the two special cases we'll take the discussion further in the next class thank you comparative Statics in RoR model so we have talked about the balanced growth equation in RoR model we have also talked about the special cases for RoR model now we are going to talk about a comparative Statics so let us say there is a permanent increase in the R&D share that is if the share of the population uh engaged in the R&D that is going to increase how is this going to affect so please write this if the share of the population engaged in R&D increases permanently right what is going to happen one we are going to assume few things just to make our life little simpler note that we have already written this point this thing here a do by a is equal to Delta l a the^ Lambda upon A1 minus 5 we have already written this in the earlier recording so you might see This Not That we have already written E Do by a = Theta l e to^ Lambda upon e^ 1 - right and when Lambda is 1 and 5 is uh zero so I can just write this as Theta L A upon l l a upon l so if Sr is the share of the population engaged in um uh research engaged in research so L is the share of the population engaged in research from the earlier recordings so and we are writing this L LA is the LA is the population engaged in result I'm sorry so Sr into the total labor force Sr is the share of population engaged in engaged in the total popul uh engaged in research let's say 25% of people are engaged in research total population is 100 so L is 25% of 100 which is 25 right so it is like this okay we have also derived this guy remember this G equal to Lambda n upon 1us 5 so remember this guy so please write in a steady state in a steady state economy grows longer balance growth part rate of technological progress = Lambda n upon 1 - 5 but we have assumed Lambda = to 1 and 5 = 0 therefore the rate of technological progress is equal to the rate of population growth g equals to n rate of technological progress is equal to the rate of population growth which is is equal to n and then you have this well we have what we have uh also note that note that a do by a is equal to Theta Sr L upon a right because when you are writing a do by equal to Theta L A by L but L by L is what Theta Sr L right I think it is it should be a I'm so sorry about it Sor a and this also you can write that as Theta L by a okay and uh what is a do by a that is equal to GA so I can write GA as Theta L by a so that would mean GA by Theta is equal to l a by a keep this thing in a head okay now suppose the share of population engaged in engaged in research is going to increase now suppose s increases from Sr Sr dash at time period zero a if the share of population is going to increase it means that that uh the of course the population engaged in research is going to increase right therefore with population of l means the the total population uh at time period Z is L KN the number of searches increases as Sr increases MH the beta if this is going to increase then La by a right jumps to a higher level what do you mean by this at time period zero the share of population engaged in research that is going to increase right so it means that people who are engaged in research that is going to increase La is going to increase so this ratio La by a is going to increase okay when you have more number of researchers of course these more number of researchers are going to produce new ideas more new ideas the additional researchers du increased number of ideas the growth of technology is also higher this point so at time period zero when the share of people who are engaged in research that is going to increase so the number of people engaged in the research in producing new ideas that is going to increase when they're going to produce when when the more people are going to be engaged in research the naturally more ideas will be generated with more ideas technology is going to improve right so the growth rate of technology is also going to be higher at this particular Point okay now now if you look at it carefully which is a do by a that is the growth of Technology this is this ratio La by a this is this ratio La by a now a DOT by a the balance growth path a do by a is your growth of technology that is equal to the population growth right okay now this thing this is also a do by a which is equal to Theta L A by A you have just written it a do by a is equal to Theta L by a where they are equal that is GA by Theta where these two are equal that is GA a by Theta H okay at time period zero what happens is that and this GA by Theta of course you can write that as Theta s r l Upon A you can write like that you can write this a do by a is nothing but GA you can write this thing as Theta Sr L by a or you can also write this as of course I mean you'll be writing this as Theta Sr L Upon A only so at time period zero at time period zero when the share of population engaged in ag engaged in research that is going to increase so I can write this as Theta uh I can write this as sr- L Upon A but here the labor force this guy and this is your uh a note right fair enough you tell me one thing this is one more thing L as Sr into total labor force so with a new share and the uh labor force at time period zero so the people engaged in in research is a your L Dash or whatever okay this particular point is labeled as X this particular point is labeled as X now at x what is happening at X you can see the growth rate of technology is more than the growth rate of population in the balanced growth path the growth rate of technology and the growth rate of population is same but at x what you can see is that growth rate of Technology a do by a is more than the growth rate of population n that is exogenously given at X technological progress E Do by E exceeds the population growth n so la by a Falls over time falls over time right La by a is falling over time technological growth rate is growing more than the growth rate of population bet po when population will increase then only the people who are engaged in research that that will increase but the rate of growth of population itself is lesser than the rate rate of growth of technology so if I look at how a is growing so a is growing at a faster rate than the growth rate of the labor force and if the labor force growth rate um is is going to grow at the slower rate than the growth of Technology then La by a is going to fall are you understanding this what I'm trying to say is this when L is increasing at the rate n then out of this L only there'll be some people who are engaged in research a is increasing at the rate whatever right that is increasing at the rate GA or whatever now this this is growing faster than this so naturally when denominator is going to grow faster than the numerator La by a is going to fall over time so when they're going to fall over time it's going to fall like this right uh so as the ratio as the ratio decreases over time the rate of Technology even this GA by this GA this a do by a that is also going to fall over time the rate of Technology gradually Falls over time until the economy returns to the balance growth path where G is equal to n where G is equal to n so if you look at it carefully I mean the time path is going to be like this time path is going to be like this here you have the growth of Technology here you have time and here you have this t or whatever right so it increases this thing till here and here you have gal to n so it increases and the same thing then gradually Falls this growth rate Falls like this this great fate Falls like this and then Jones beautifully write this therefore even if there is a permanent increase in the share of population engaged engaged in research it can increase the growth rate temporarily but eventually it falls back growth rate of Technology temporarily but or not in the long run therefore the permanent increase in the share of population engaged in in research increases the rate of technological progress but not in the long not in the long run right okay I'll take the discussion further in the next class thank you br