S1: okay. first i just do a (xx) <P :04> okay so this is our, last section. you glad or are you sad? <LAUGH> okay the review session, let's write it down in this room. in your very own room. um, seven to nine P-M, Friday... the twenty-third. i emailed you guys already. okay? and today's lecture gonna be the hardest one. i can say because it hard for me this morning. for my morning section so bear with me. and if you have any questions, just raise your hand. it's okay. alright? because i have to sign a consent form too. okay? i'm gonna talk about solo growth model, and uh if i have time i'm gonna go over, question three and eight, from the problem set, and then i'll talk about discussion sections. discussion problems. okay? for, solo growth model, i listed down the components for you guys. so you can just pick it up whenever you want to use it. right? because we gonna do a lot of manipulation in the, in this model. we're not gonna have as many graphs. as i'd like. before. so we're gonna have to use all these equations to explain, the movement of the model right now. the first one, it's the labor supply. okay? we assume that everyone in the country. the whole population. are in the labor force is in the labor force. okay? L-T, and the growth rate of this L, is N. which is the population growth. right? so now i have to kind of refresh your memory a little bit about the growth rate, because what is a growth rate? suppose you want to find a growth rate for L, D-L over D-T. growth rate is how, the labor, the number of labor change over time. right? Ricardo will call this L dot T. okay? and, i'll call it, change-in-L over L-T. they're the same thing. okay? in the review notes that you gonna get you're gonna see something very complicated. that's why, i was, carefully, writing these notes last night. for you guys. if you don't understand those, fine. understand my notes and you're gonna be able to do the final. okay? because he, he assume L equals to something, and then he derive L-dot. but i already derived it for you. you have to go through the revision. so this is N. the growth rate of the labor force is N. second... the effectiveness of the labor augmenting technology, and we call that E, okay? and the growth rate of E, which shifts the change-in-E over E-T, is G. these are given, right? population growth technological growth, they are given and constant too. okay? the third one is production function we talked bout production function last time. they have constant returns to scale, the first derivative of production function, which is the slope, is gonna be positive. with both respect to labor and capital. right? increase capital, increase output. increase labor, increase output. this is Y equals to F, K over L. okay? what is the fourth one? okay? the saving function. just regular saving, these are all in the big letters. saving function is the small-S, which is the saving rates, times Y-T. how much you save out of the income. <P :06> okay?
S2: uh why don't you put, on the if it eh if it didn't pass on the (slope,) on the production function you don't have E?
<P :04> 
S1: yeah, sorry. that's good. i forgot to put E in because we just add this E, into the model. right we usually have L. but i just wanted the full model. so good question. thanks. for, number five, we have saving equals to investment in closed economy remember? in closed economy with no government. savings is gonna incr- equal to investment. so. <P :04> investment is gonna also equal to little-S times big-Y-T. okay? the last component of solo growth model. is capital accumulation. remember the bathtub last time? the change in, the total number of capital in the economy equal to, investment, which is the inflow of the water, minus, delta-K which is the outflow of the water. what's left in the tank, is the change in capital. or, is the level of capital. okay? now, if you want to study the growth model, we have to define some parameters, because it's gonna be very easy for us. if you specifying some parameters. well let's define little-K, so now distinguish my both Ks, big-K and little-K, let's define it as big-K over, the effective unit of labor. this is not only labor this is the effective unit of labor. if you have ten labor. doesn't mean that everyone works the same way, right? someone can be more effective than the others. so you you care for the effective unit of labor. when you produce, the pr- the goods. so, we have to divide it by E-T, L-T. okay? and also, we need to define, output per effective unit of labor. same thing. okay? now to study growth, we wanna find, we wanna look at the behavior of the capital accumulation. so we wanna see, how this little-K changes, over time. okay? so. we gonna do, K dot T, for Ricardo, or for me, change-in-little-K-T over K-T. mkay...? when we ha- when you wanna derive this, the ri- deri- derivative of K-T, you have to do the quotient rule again. remember? bottom multiplied by top minus top multiplied by the bottom. so let's do that again... <WRITING THROUGHOUT UTTERANCE> just enter the first, derivation. E-T, L-T, change-in-big-K, minus top, change the bottom... over, the bottom squared. okay? and you plot the change in capital, using the last equation here. put it here, and do some manipulation, you gonna get the change-in-K-T, i did it in my notes so, go through it okay? i'm just gonna skip that part, it's gonna be equal to F, F-K-T, minus delta plus N plus G, little-K-T... you have to understand the derivation of this. but you don't have to get it down like a hundred percent like you have to rewrite it. but you have to memorize this one. this is important. okay?
S3: what exactly does that mean again? th-
S1: this is, the growth rate of the capital per effective unit of labor. you wanna see, how the change in effective unit of labor contribute to growth. okay? so that's why you need this equation this is the whole solo model, this equation. tells you the whole thing. okay? so last time we put this in graph already, right? we have this graph, production function scaled down by saving rate, which gonna be the same, just like lower, because saving rate is gonna be less than one, and you know this is the shape of production function. okay? so we have this. straight lines slope of delta plus N plus G, K-T. this is K-T. in Mankiw book, in the book, they ju- they did it in three steps. they do it in three steps. they do it, first, with fixed labor, okay, so you don't have any N. N equals to zero, G equals to zero for the first step. remember? professor, the professor went over this too. so this equation, you're not gonna have N plus G. right the picture is gonna be the same. the second version, they're not including E yet. they left E out. you have only, N and the f- plus the first version. so this what you have. and the model we're doing right now is the final step so we have everything. so if you wanna do, the first two steps just take this out. and you have the same picture still. see that because this is just a slope. right? it doesn't affect anything else. so, from this, you get K-star. why? because in the steady state, okay, in the steady state, K stop growing, little-K. and little-Y too. because, Y equals to a function is a function of K, if K stops growing. the little-Y is gonna stop growing too. see that? so, in steady state, this term is zero. right? see that? so this term equals to this term, as a- the investment has to equal to the depreciation. that's when you don't get any change in capital. you have just stable capital. that's, represent by this point this K-star. okay? we have that, and if you away from K-star you going back to it remember? if you are on, on the right hand side, on the left hand side your left hand side, oh my left hand side. your left hand side. oh. this is left hand side of K-star. you're gonna have an investment which is F, S-F-K greater than, the depreciation. so you're gonna have, a positive, change-in-K so your K is going up. if you're on the right hand side, you're gonna have depreciation, over investment. right? these change gonna be negative and you're gonna have a fall in K. this is the definition of steady state. if you away from it, you coming back to it. if you at K-star you're not gonna go anywhere you're gonna stay there because that's not gonna be any change in capital. okay? now this is easy. comes to, the hardest, the harder part. which is the convergence rate. okay? the convergence rate in this case is the convergence rate, from any point in this line to K-star. we need to use the law of diminishing marginal product of capital again. if you have, small amount of capital, adding one more unit of capital, gives you a large output. but if you keep adding more and more capital, that extra output is gonna decrease. right? even though you still have an increase in output, that increase is to- is smaller and smaller. as you keep adding capital. this is the same thing. if you are here okay? if you are at this point, you have very low level of capital. if you av- if you add one extra unit of capital you gonna have high, unit of output. high extra output. this mean you have a high growth. high convergence rate. to K-star. but if you keep getting closer and closer to K-star, your increase in output is falling... right? so your convergence rate is falling and falling, when you getting toward K-star. and when you get to K-star what is your growth rate? zero. right? at this point, K stop growing. little-Y stop growing. same thing. on this side, if you further away, if you decrease your capital by one unit you decrease output by a lot. so the convergence rate, on this side, is high too even though you have a decrease your convergence rate is negative. but in absolute value. it has to be big. and it's getting smaller smaller in absolute term. until it gets to K-star. the further you goes from, K-star, the bigger convergence rate. okay? i know it gets, hard it's hard for me to kind of.... i talk about Mankiw thing, let's do the growth rate. let's see whether, if we in steady state what's gonna happen. we know that little-K and little-Y stop growing. what happen to all these big-Y big-Ks and savings and, wage, rental rate. okay? i did a lot of example here. i don't know if it's gonna be useful but, i hope so because Ricardo is writing a final on this part. and he's very mad. so we must be ready, for that. <LAUGH> <P :05> okay. first thing i wanna point out, is that, remember that the change-in-K the little-K and little-Y stop growing in steady state. so from now on, from now, the next ten minutes, everything's gonna be in steady states. okay...? what happened to, big-Y? when we wanna look at the big-Y, we have to track back to little-Y because we have to use that information. we know that the growth rate of the little-Y is zero at the steady state. right? so, little-Y, i'm gonna leave the subscript T off okay? it's gonna be equal to Y, over L-E. right, this is, how we define it. so if you wanna see what happen to big-Y, we just multiply both side by L-E, and now, we apply the growth rate. formula. <P :06> so the change-in-Y over Y equals to the change-in-little-Y over little-Y, what is this? plus, right? because when you have the multiplication. the growth rate has to be added to, added to. so change-in-L over L, plus change-in-E over E. what is the change-in-Y over Y what is the growth rate of little-Y in steady state? zero. come on you can talk.<LAUGH> you already said something. what is the change-in-L over L? 
SU-M: N
S1: N
SU-M: yes
S1: what is change-in-E over E?
SU-F: G
S1: G, right? so. the growth rate of the big-Y is, N plus G. okay?
<P :06> 
S4: on your notes it looks like you have m- you have minus 
SU-M: minus right.
S1: oh.
S4: should that be plus?
S1: oh no no no because [S4: is that ] i changed , i i used, the change the growth rate of little-Y equals to the growth rate of big-Y, minus ch- growth rate of L [S4: oh okay ] minus growth of B. so i kind of like change around. [S4: okay ] mhm. now i mug- change change it first and then take loss. but, that's the same thing... okay... and also, i wanna point out something too. at the steady state, one condition at the steady state that you need to know is from this one. if you at the steady state, this thing S-F-K equals to, the depreciation. so at the steady state S-F-K-star, is gonna equal to delta plus N plus G, K-star, right? [SU-M: yeah ] if you know the functional form. if you know the production function like, you impose Cobb-Douglas production function. right? you go- suppose you have like K, little-K-to-the-one-half, you can start from K-star. right? you can start from K-star. and the K-star will be a function of, delta plus N plus G and S. and what are they? they're constant. they're all constants, see that? constant constant constant and constant... right? so... that's why K-star is constant in the steady state. because it's a function of constant. these parameters not gonna change over time. okay? that's why K-star is constant and the growth rate of K-star is zero. the second thing. the effective unit of labor. L-T-E-T. without peeking, what is the growth rate...? come on what is the growth of L?
S8: N-plus- G
S1: yeah it's N plus G because it's equal to_ the growth rate is equal to the greater of L, plus the growth rate of E. so it's N plus G. okay you're getting better. now this is the hard one. this big-K-T. are we doing the same thing? in here. we define little-K-T, i just leave it out, big-K or L-E, okay? so big-K is gonna equal to little-K-L-E. right? what is the growth rate of big-K? what is the growth ra- the growth rate of little-K. [SU-M: zero ] in steady state? zero. what if, what is the growth rate of L?
SU-M: N
S1: N, what is the growth rate of E? 
SU-M: G
S1: G. so it's N plus G. know the trick now? okay good... number four... savings. saving is S, times big-Y... the growth rate of savings. it's gonna be equal to the growth rate of little-S, plus the growth rate of little-Y_ big-Y. sorry. so, what is the growth rate of little-S?
SU-M: zero
S1: zero. what is the growth rate of big-Y...? N plus G. right? so saving grow, at rate_ saving grows at rate N plus G. okay? so, the L-dot equal to L e- multiplied by exponential N, T is Ricardo's way. the growth rate thing is my way, and this is Professor Basu's way. so pick every one you want_ any one you want. but don't do Ricardo i think it's too much t- i mean he's too good. we kind of like this level. i'm kind of like this level. okay. well this is easy too but this is like, the sure thing. you're not gonna miss anything from here. okay? i'm in between. okay investment since investment equals to savings, the growth rate of investment has to equal to growth rate of savings too. right? so, growth rate of investment is just N plus G again. okay? but don't try to guess in multiple choice. like most of them are N plus G but you know. try to figure it out... well the next couple of, of this derivation is dealing with number eight. so let's do, number eight in the problem sets. okay? i think it's more useful to you and to me. for number one. and these are all in steady state again. right? for number one, you having, you have to prove K over Y, to be constant. what should be the starting point...? let's divide both, numerator and denominator by L-E, okay? so what is it? it's gonna be little-K over little-Y. right? because K over L is L-little-K, Y over L-E is L-little-Y. since we are in the steady state, just give them some stars. okay? now. let's, let it sit for a while, and look at our steady state condition. the condition is... S-F-K-star, remember in steady state the two thing has to be equal. what is this F of K-star...? just Y-star. right? now, you can find K-star over Y-star from this relationship. K-star, over Y-star equal to, S over, delta plus N plus G. saving rate, depreciation rate, population growth technological growth, they all constant. that's why K-star over Y-star, is constant. proof. okay? the second one. remember capital share? from the very first? Lisa? 
S5: are you gonna ask something like this like in proof in the short answer part in the, uh
S1: no i don't think so
S5: okay.
S1: i don't think so. i hope not. i'll fight for it. [SU-F: <LAUGH> ] well a little you know the easy ones are, these ones are okay right? these probably a little bit tricky. what is capital share? remember the capital this is what the capital income, plus the labor income, equal to the whole, the total output in the economy. remember this equation? right that's why we have zero economic profit. because, we have to d- divide, output, to pay for capital and labor. and we have no profit left. so, the capital share if you divide everything by Y, and the labor share equals to one. and this is our capital share that we wanna look at. so, R-K over Y, what is the growth rate of R-K over Y? i'm gonna simplify it for you a little bit. in the book, it's given that R equal to M-P-K, and then K over Y. this K, can be_ we can do the same thing over there, divide both, numerator and denominator by L-E, okay? what is the marginal product of capital...? [SU-F: (marginal) ] what_ how do you get marginal product of capital? you take production function, and you do something with it. [SU-M: integrate it ] take derivative. yes, right? so this is, F-prime little-K, and then this is little-K, this is little-Y, and since we in the steady states, everything's star. F-of-K is the function of K-star. depend on K-star only if K-star constant F-prime of K is gonna be constant, i_ we already proved, that K-star over Y-star is constant. right? so these whole thing are constant. okay? so it's proved... you wanna see this? suppose F of K-star, equal to, K-star-to-the-one-half, F-prime of K-star equal to one-half, K-star minus one-half. right? and K-star's constant? one-half's constant minus one-half's constant. so F-prime of K-star's constant, Lisa.
S5: what's the difference between little-R and big-R like how, i thought that was always 
S1: little-R? 
S5: and big-R. like the (arrangement) 
S1: big-R is the rental rate. the total payment to capital. if you subtract the depreciation rate, you gonna get the interest rates. this is big-R this is small-R. it's the definition. thing. this is the total payment. but once you subtract off depreciation this is the capital gain. okay? that's why we call interest rate, like capital gain... and i just thought of something. the big-K over Y the big-K over Y. what is the growth rate of K? we just had, [SU-F: N plus G ] N plus G, what is the growth rate of Y? [SU-F: N plus G ] N plus G since they both, have the same growth rate and they divide each other, they should be constant right? that's like, the wording, the wordy proof. this is like a mathematical proof. Tali?
S6: um, when you divided K over L by L-E, [S1: mhm ] um, you just do that. like you don't need to do anything else to the equation?
S1: no you don't have to why because it's the same thing as you multiply it by one. this is one right? L-E and L-E's one.
S6: okay
S1: okay...? that's number, that's that's question B, well since W_ this is labor share, labor share is equal to one minus capital share. so it's gonna be one minus this. one is constant, this is constant, so the labor tax also gonna be constant. right...? now, for the total payment of labor. i think you can do it better now. let's see. the uh, total payment of capital. sorry. what is the growth rate of R? the growth rate of R is, the R is M-P-K. M-P-K is F-prime K. so the growth rate of R is, zero, right? and the growth rate of K is, N plus G. so, there you have the growth rate of R-K equals to N plus G. okay? or, if y- i wanna do it my way, R-K, is gonna be equal to constant, i mean we already proved that R-K over Y is constant, on B. in problem B right? so, R-K equal to constant, times big-Y. the growth rate in R-K, do it my way, is gonna equal to the growth rate of the constant, which is zero. plus the growth rate of Y, which is, N plus G. same thing. taking like extra three minutes. okay...? what about the payment to labor? in this case it has to be W-L-E because we're paying by effective labor unit. i should put L-E here too... well but it's not gonna change any result anyways. okay? so. you'll be doing the same thing. W-L-E over Y is gonna be constant, right? so, W-L-E equals to constant times Y, the growth rate of W-L-E is gonna be the, equal to [SU-M: i appreciate it ] the, the constant, the growth rate of the constant plus the growth rate of Y. which is A-plus-B again. we looking for the growth rate of the whole thing. see that? okay. you have a question to raise?
SU-M: uh, no. no. 
S1: no? [SU-M: no. ] okay ... so any question on number eight? and now. so i've done six seven eight nine. like a little bit differently you can check, the note that, in my notes. six seven li- six seven eight nine, okay? now. oh. you ready for this? for ten. why we care about C over L. consumption over, L is the per capita consumption. you don't care how effective you are, but you you care about, how much each of us. can consume. like per person, right? you don't care how effective you are, but it just how much you can consume. so let's see what is the growth rate of C over L. this is kinda, complicated, but it's not that hard, so. you wanna find the growth rate of this, right? so consumption is equal to Y minus I, right? if you divide the whole thing by L-E, you're gonna have C-T equals little-Y, T, minus little-I-T. little-I-T, is just, saving is close to investment right? so little s- S, and Y, which is savings, you're gonna have one minus S little-Y-T. is your little consumption. little-C. see that? now. little-C-T, we got this little-C-T by dividing big-C-T, by L-E. equals to little-C-T. now. since we care only about C over L, we can have C over L equals to little-C times E. see that? now. from that equation... little-C we already figured out... C over L equals to, E times little-C. little-C is one minus S, little-Y-T. okay? now. in steady states. the growth rate of C over L, equal to the growth rate of E, plus the growth rate of one minus S plus the growth rate of Y. little-Y. what is the growth rate of E? in steady state? G. what is the growth rate of one minus S?
SU-F: zero
S1: zero
SU-M: zero
S1: what is the growth rate of little-Y?
SU-F: zero
S1: zero. in steady state. so the growth rate of C over L, is just G. okay? <P :05> okay. now let's, get to, something more economic. well, w- not much but, a little. the golden rule of steady states. as you can see, K-star... can be written as a function of th- all these parameter. like uh, well. what else? delta N G and S, right? so what, it's_ i mean, it's okay if you have different, parameters. but if you have different parameters in each state of your development, you're gonna get different K-star, right? and sometimes you can manipulate, these parameters too. like, the government can push up your saving rate or push down your saving rates, um, you know some country try to limit their population growth like China. right? so... if you have, if you can have different K-star, which one is your best K-star? we don't talk about how to converge to steady state anymore. we talking only_ okay let's see, if we are at the steady state. which steady state is the best for us? okay? so wha- how do we care? why wha- what what should be the criteria, to decide a capital the, K-star. that is the best for us?
SU-F: C-star
S1: yeah. because we care about consumption, so whichever steady state that give you the maximum consumption. you go for it. okay? so. i der- i think i derived, the condition of the steady state, Professor Basu did it once in lecture i did here again, it's just the maximum_ the maximization thing, you have, C-T equal to Y-T minus little-I-T. okay? and this little-I is, S-Y-T. what is S-Y-T? i erased it again. oh. S-Y-T at the steady state. now we talking about steady state now. s- Y is equal to this term. so we can substitute Y-star i'm gonna change it to star now, minus, N plus G plus delta, K-star. and these little-Y-stars, we can substitute in F, of K-star right, because they're equal. why? we wanna find the K-star that maximize consumption, so we have to change the parameters in this equation to be K-star so we can maximize over it. so now maximize consumption function. you take derivative of this function, set it equal to, zero. right? so. take derivative F-prime K-star, equal to, well minus N plus G mi- plus delta, equal to zero. you have F-prime K-star, equal to N plus G plus delta. this is the condition for golden rule steady states. it is the K-star where, your F-prime K-star at that point equal to the slope, of the straight line. so let's see which, point is this? it is, over there. okay suppose now you care only about K-star. this is all you care about. K-star. you in, steady state. this is still your production function, F, K-star, and you can draw the line, N plus G plus delta, K-star. okay? now. the condition is the slope of this production function. has to equal to the slope of this line. that mean the golden rule steady state has to be, where the slope, whe- when you draw the line, touches the slope of production function, it has to be parallel. to the straight line. something like, over here, right? because the slope here, equal to the slope here. satisfy the condition. so, this is your K-star golden rule. at this K-star you maximize the consumption in the economy. okay? if you away, f- if you are away from this, you're gonna have less consumption either way. if you come toward this point toward the right hand side, it's gonna be, this is the consumption. right? why? because consumption, where is it? equal to F-K-star minus, the depreciation. this is, Y this is F of K-star this is depreciation. so the difference be- between the two, is your consumption. this is the maximized consumption, maximum consumption, if you go further, to the right, it's falling. if you go further to the left it's falling. okay...? any question on, the golden rule, of steady states? okay. now. is it number three too? yeah. so i'm gonna do number three. and try to use this picture to explain number three. this is the only two i'm gonna do today sorry three and eight. okay? probably we do something later if we have time. like seven, seven's good. seven gets me. i can't do seven at the first time i'm like aw... okay. number three. if you have, a higher saving, proportion to income which is a higher saving rates basically. what's gonna happen? okay? the high saving rates. now we're back to regular case. we're not gonna be in all steady states. we're gonna be in regular case and, our goal is to go for a steady state now. okay? this is now back to, the scaled-down production function by savings. N, this is just N plus G plus delta, K. okay? now i've got my board line as my steady state... now, if you have a lower. no. here you have a, a larger savings rate. bad picture... i'll leave, that line... okay let's do this... okay. this is your original level. you at the steady state. and then, saving rate goes up. what would happen? this is just a constant, right? if saving rate goes up it pushes up this curve the production function the saving function. so this curve is going, to be like that, it's got pushed up, S-prime-F, K-star. okay now. i'm getting back to my, board line K-star. you have a higher K-star. when you cha- when you increase saving rates. you ending up, having a higher capital, a growth in capital, and a higher level of Y. since Y is a function of, little-K little-Y it's a function of little-K. okay? i'm talking about all little, now. so... what happened to consumption can you see in this picture? no you can't. right? so. let's look at this. it matters, where you start. when you change saving rates. if you are... here. okay. if you original at K-star, this is K-star in this picture, and if you have a higher K-star. suppose you move up to here... what would happen? you have a fall in consumption. right? why? why is it? it's weird right? you have a higher K. that's mean you have capital accumulation and why, you have a lower consumption, can be? because this is_ this K-star golden rule is your optimum, K-star. if you beyond that point, this point, you're gonna have too much, K-star. if you have too much capital in the economy, you know, when you have_ you gonna have very low marginal product output. uh, marginal product capital. and also. you have too much K and you know that too much K means too much depreciation. it's gonna eat up your investment. if you have to invest more you have to save more and when you have to save more you have to consume, less. right? that's why it's bad. but if you over this side, you're fine, right? if you are, y- if you have too little K-star, if you have too little capital in the economy, and you have a saving rate goes up... i mean this is gonna make your consumption goes up. what i'm doing is not exactly three. i'm trying to use s- three, to relate to this. if you don't have the same answer, don't be panic. well, for number three it's just like, if you are here, right? and savings savings rates goes down, goes up the immediate effect is just you have to consume, [SU-F: less ] less because you save more. right? but you know that K is increasing. you have capital accumulation capital growth. then you gonna have productivity growth because you can produce more. and that's the gain. the immediate gain. and at the end, i mean, the transition, at the transition, you accumulate more more and more capital, and once you get here. your capital growth rate is going back to zero again the little-K. right? so you go from zero here, positive, positive, and it's zero here again. so you enjoy, what's in between transition rate. and what's ana- what's th- the other good thing is, you end up at a higher level K. higher level Y. and your consumption's gonna increase later. because Y increases. you were gonna ask that? [SU-F: mhm ] mhm Lisa?
S5: so how can you, like if you want, oh you can't_ i'm just trying to bring them, together 
S1: don't. don't. <SS LAUGH> don't try. oh okay. see that? so if you are a policy maker. you're our policy maker what would you do? if you know, which side you're on, which side you're on. on this picture. if you're on the, if you're on your right hand side, if you're the policy maker, would you encourage the saving rate to go up or down?
SS: down.
S1: down, because you're gonna increase consumption forever. right? and that's a picture that we did yesterday in class too. your consum- your consumption's gonna jump up, and stay there forever. and if you on the left hand side, you want saving rate to go up or down?
SS: up 
S1: up because you wanna save more you have s- so little too little capital you need to, increase more capital so you need to save more. right? so that will, increase your consumption, again. so it depends on where you are. okay? on the exam, i'll make sure that it's specified which side you are on K-star if we dealing with saving rates. or you can ask. me. i'm gonna proctor your, you guys. so, i hope, that, it's better this time... and uh, what else i'm gonna_ um, i have to do this thing. this discussion Ryan?
S7: um, in the article, [S1: mhm ] you talked about uh, [S1: mhm ] the migration of people, [S1: yeah ] from the, these four 
S1: i'm gonna talk about it right now.
S7: oh you're gonna talk about that?
S1: yeah. yeah. definitely.
S7: um, only question i have is like, like, if uh, you have uh, people moving out, [S1: mhm ] like your end growth rate's gonna drop. right?
S1: no. let's let's look at it as one time moving now, see what i'm saying? like you have just... the population growth is just like, when they calculate, they calculate over years right? but, for this, uh for this part, it just like, well let's, what if they migrate for a month? like for this month, ten thousand people came in. so it's not gonna, well we not gonna, have them affect a lot of population growth. let's let's make it that simple. okay? so let's make it that L change, but N is not changing. because N is just the growth rate right? so, hm, in th- the average growth rate of every year. suppose this year you have, three percent next year you have two percent. you can have fo- four percent this year and one percent next year and you have the same average amount of, N. we assume N to be constant right now. i mean we can change N but if you change N, it's gonna tilt, if_ i think it's number seven. that you have to do. but for immigration. i don't think it's gonna affect, that much. like in case of West and East Germany. right? they're not really_ i don't know if they really migrate they can just still go back and forth. like, they probably cross the border and come_ and work and then go back. but try to keep it simple that, N is not changing because otherwise your analysis gonna go crazy. like. because you have, you know, Y goes down and you have N, shift your, straight line again. see what i'm saying? if you have a migration, an immigration, if you let N change you have to let this change too. you can't just do a simple analysis anymore. it's not that, complicated but you know it's, a little bit... not easy, okay? but you can put it in your analysis too if you already put it there. it's fine. it's fine. just make assumptions. if it's gonna change or not but i'm not gonna let it change in this case. until_ unless you are told, you know that, N is changing from zero to, ten percent something like that. if not just immigration, just let N, to be like, quite constant. because we're not gonna go this far we're just gonna look at the production function. to see what happened to Y and per capita Y. okay? well let's see if i can apply that too... hm. okay let's, let's start with number, two. i'm gonna start with number two first. and then we'll come back to the, immigration problem when we get to number, number three. you have to, i didn't, i didn't put down, i don't wanna kill more trees. <SU-M LAUGH> but if_ well you do you have, do you have the, everyone has the problems? well you must have it in front of you, right? okay. i'm gonna go for the second one first. under what circumstances would you expect countries to converge? in this case, there are two convergence now. the first convergence is the country converged to K-star, the second convergent is the two countries converged to the same point. so now we talking about, what is the condition that the two countries converge to the same point. what makes them converge to the same point? any, thoughts? Lisa?
S5: they have similar saving rates, population growth, depreciation rate and, G-D-P.
S1: yes, exactly. because you know that K-star is a function of those parameters. G N delta and S. right? if they_ if the two countries has the same lev- level of those rates, or, when do th- when you do some manipulation and those rates. give you exact same K-star. as the other countries. then definitely in the long run you know in the long run you have to get to K-star. and if you have the same K-star, you gonna converge, right? so first you must have the same K-star to be able to converge to the same point first. okay? so that's the condition. any other, comments? questions...? Jenny?
S4: would_ i don't know this just eh this is another idea that struck me is like natural resource endowments like how would that affect that like if, say one part of the country had like a lot more, um, goods. where they ever, like oil for example.
S1: like oil natural resource?
S4: would that affect [S1: well but this ] the K-star? that would
S1: this probably affect the convergence rates. like [S4: okay ] how you accumulate capital but not the K-star. [S4: okay ] because K-star we already proved that it's, only for saving [S4: okay ] and it can reflect on the saving already because if you have oil you gonna be like you know you gonna have a lot of money and you're probably, not gonna spend all of it so you're gonna save a lot. right? so it's already show in S, and definitely you know, always counting as capital, like in some [S4: mhm ] sense. so that's gonna affect your convergence rate too [S4: okay ] definitely. good. okay. any other, comments? okay let's go back to number one. in case of, Germany, East and West Germany. will it eventually converge to the same ca- per capita income? do you think? [S4: mhm ] well let's, you know look at the condition and see whether it's, converging... who think it's a yes?
S5: no
S1: who think it's a no? [SU-F: oh ] no one thinks it's a no? Lisa? you think it's a no it's a yes you said no.
S5: i said yes.
S1: okay <LAUGH> okay it's a yes. because why? i mean it can be yes or no it depends on your reasons. it's a yes because, right now they're like practically one country. they're in the s- under the same government. right? same policies, same monetary policy fiscal policies, right? so. they're probably gonna converge, the saving rate's gonna change, a little bit deprecionsh- depreciation rate might change a little bit, and ha- in the long run they're gonna converge for sure. but i don't know how long, because you know, when you read the article it's very long, to see a convergence, right?
S7: um what's the effect on total output? like i know like the, the like
S1: the total output?
S7: yeah like like the poor country's total output, like i know like, that its per capita income would like catch up like will increase [S1: mhm ] but um like, does that necessarily mean that its ac- its total output will increase?
S1: let's see. total_ what is to- what is, the little-Y? okay the per capita income. per capita not per effecti- unit of labor i lost my chalk. Y over L, is your per capita income, but if you have E here you're gonna have, little-Y here. or you can look at this, even. if this is going up, and given that labor is not falling, what happen to output?
SU-M: it'd rise 
S1: output has to go up yeah.
S7: but i thought the reason why per capita like, cuz labor actually fell.
S1: why, actually 
S7: cuz like people moved out
S1: oh no no no no no. no no no no no we not... we we look at the immigration on the West Germany side. [S7: okay ] see what i'm saying? so what happened in West Germany but this is in East Germany. you don't know whether, well the labor force it's, let's assume it's not changing. okay in this case. i don't want you to kind of, going back and forth. it depends on you can include this and say that out- total output falls but i don't, i don't wanna see that because even though labor falls, it still can be to- the total output has increased but maybe not you know, it's still gonna be able to increase right? to make this thing increase. it doesn't have to be only out fall. so we don't know. and it, it's most likely to go up why because we have more capital coming in East, East Germany. right? if you have more capital since Y equal to F-K-L maybe E, if K goes up, Y goes up too. so we not, looking at only labor right? we looking at both labor and capital. okay is that clear? is that okay?
S7: yeah
S1: okay well, i stress more when we go to three and four. [SU-M: okay ] Olivia?
S2: doesn't that undo (xx) the capital that they had, before they, when they ha- were separated countries? because it said now they are uh, they have the same problem and the same policies. but, it's now. what about before? before we knew that West Germany had less capital than [S1: yes ] East Germany 
S1: East Germany has less
S2: yes
S1: policy <LAUGH>
S2: eh East less_ East Germany has uh yeah 
S1: yeah yeah yeah yeah
S2: so it doesn't matter this?
S1: it doesn't matter where you start. right? it doesn't matter where you start. you gonna go_ you gonna converge. to each other. maybe_ why? you know why? because when you look at this, okay look at this. look at this picture the West Germany can start here. right? East Germany can start here. this is West this is East this is K, this is your K-star. what is the convergence rate here? is it higher or lower than West Germany?
SU-F: higher
S1: higher. right? once you get nearer to K-star, your convergence rate falls, right? so here has to be slower the convergence rate for West Germany has to be slower, than East, Germany anyways. so it's gonna catch up, in the long run. see what i'm saying? and, and even though it's not gonna catch up, by the time West Germany get there, but if German- West Germany, are at K-star they stop growing. they just gonna be there. and finally, East Germany's gonna be able to catch up anyways. right? because West Germany's gonna start here at K-star. and East Germany's gonna come. and catch up later anyways. okay? so finally they have to converge. seven hundred years, you can say. <SS LAUGH> something like that. well this, no offense. i was just like throwing the numbers out because you know that a convergence rate is very low. okay number three. the article notes that the regional income difference usually cause migration from poor to rich regions. okay now it comes to Ryan's uh, question. So let's let's look at just production function first. okay? what happened to total output? what's gonna happen? total output in West Germany. it will go... come on, you have more labor in West Germany [SU-F: up ] what will [SU-F: up ] happen to the total output?
SU-M: goes up
S1: goes up yes. Y equals to F-K-L, if L goes up... definitely. Y's gonna go up. because of what because we have the f- fourth derivative e- it's greater than zero. so increase in L increase Y. increase in K increase Y. but, if we divide it by L, L, L, we can do that because we have constant returns to scale function, right? it's gonna be Y equals to, F-of-K. right? now what is this small-K? we have, migration, labor force is going up in West Germany. K is, constant, pretty much, little-K, fall. so little-Y, fall. think about it, what is little-Y? little-Y is, per capita per effective unit of labor. if you have more labor, and not producing as, you know much more, you're gonna have a fall in, output per effective unit of labor for sure. even though you have an increase in Y. like Y over L-E, equal to little-Y, even though you have this increase, this can increase more and make this fall. right? so that's the case... is that, better? okay. let's see if we, let's see if we can say something. this is for you Ryan... if a an increase in West Germany, you have a smaller K-star, and you have a smaller, smaller, Y. little-Y too. so same thing. okay? and since, well but this is the steady state so, this is like, you know in a month. this is the steady state though. but you still have the same answer. see that? so K falls, little-Y falls. right? what happened to big-Y? if N increase, [SU-F: it went up ] what if the_ what is the growth rate of big-Y? 
SU-F: N plus G
S1: N plus G good. so if N goes up, the growth rate is going, up. so big-Y is going up. so same result... okay? so if you can just well you know, if this immigration is large enough to change the population growth, you still have the same result... okay. that's for labor. let's do it for capital. well, wait. now, what does it mean it means that, in West Germany... you have a fall in output. so you have a slower growth right? if... if West Germany is going to K-star, and East Germany is catching up, if West Germany is slowing down, the two countries gonna converge faster, right. so, the immigration actually speed up the convergence of the two countries, too. see that because it's slowed down the growth rate of West Germany a little bit, okay? that's one thing. that's for labor. let's look at the capital what happened? if West Germany, lend the money or capital, to East Germany. to finance their investment so they can invest more and have more capital accumulation... when they have more capital accumulation, okay...? i don't know why i erased that picture here. right? if you have higher capital, East Germany is gonna, catch up faster, right? it's gonna catch up West Germany faster because they now have extra money, to accumulate that capital. see that? so, both the immigration of labor from East Germany to West Germany, and the lending, the ability to borrow f- of East Germany to West Germany, will speed up the convergence rate between the two so East Germany can catch up faster. Jenny?
S4: what about when they have to pay 'em back. i mean is there like 
S1: well, but if you are at the K-star, if you have the same growth rate, it's expected that you gonna have money to pay back anyways. but right now you need it you need it for capital accumulation. but when you have enough capital. you know you can just do anything. right? but right now you need capital that's why, the developing country the underdeveloped country, they borrow a lot because they expect that they their capital accumulation, will help them to pay back the debt later. that's what my country is doing and now, since we devalue we can't pay back the debt anymore even though we have more capital. right? so that's the same thing. okay? so as you_ and you can tell. which way the directions, is. for labor. the labor flow from the poor coun- poor region, to the rich region. but the capital flow, from, the rich region to the poor region. and both things speed up the convergence rate of the two countries. okay? any other comments? suggestions? <P :07> so i i suggest you to do_ to read, to print out, the solution for this chapter. because it's very important that you understand everything. if_ mark the line if you don't understand. come talk to me. because you better understand every single line in there. Keith?
SU-M: are you having uh, office hours now?
S1: i don't have my office hour today. i'm n- , i'm gonna have my office hour, i'm move my office hour from, on Wednesday to Thursday because i have presentation 
SU-M: oh you just want it next week.
S1: next week Thursday
SU-M: oh i though you said on Tuesday
S1: yeah, no.
SU-M: okay
S1: and yeah i said what i wanna say in the back page of your notes... did you look?
<P :04> 
SU-F: is it just this section? 
S1: good luck on your final.
SU-F: is this just this section?
S1: no. oh yeah. you're right
SU-F: my section's this one?
S1: yeah
SU-F: oh okay
S1: this is your section. yeah... oh that's
SU-F: oh oh oh. yeah
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