[00:16] I couldn't connect but
[00:18] so the Fed just hiked by 25 basis
[00:20] points.
[00:22] And
[00:23] as people expected, you know, this is
[00:25] the way that it works when there's lots
[00:27] of uncertainty essentially
[00:29] the Fed starts communicating
[00:31] what's going to do
[00:33] and the communication was still very
[00:34] clear that
[00:36] that 25 basis points was
[00:39] to be expected and and apparently I was
[00:41] reading this right now. It was released
[00:42] at
[00:43] 3 minutes ago, 4 minutes ago.
[00:45] Um
[00:47] they also said that that further hikes
[00:50] are no longer guaranteed. So remember
[00:52] that we saw that expected
[00:55] hikes sort of we saw several several
[00:57] expected hikes for the next few months
[00:59] before
[01:01] the SVB mess and right after it we sort
[01:05] of saw the whole thing declining and and
[01:07] at least the minutes are consistent with
[01:09] that. Um so there we are. So not big
[01:13] uncertainty I mean the markets are
[01:14] rallying or something like that at least
[01:16] for the next 10 minutes or so but uh
[01:18] we shall see.
[01:20] Anyway, so but today we we're going to
[01:23] really start
[01:24] I'm going to I'm going to show you sort
[01:26] of the first model of economic growth.
[01:28] Uh
[01:29] And
[01:31] before I do that, who knows who that
[01:33] person is?
[01:37] No? No clue?
[01:41] He
[01:42] actually he's Robert Solow. He was an
[01:46] He's an emeritus professor at MIT.
[01:48] Together with Paul Samuelson essentially
[01:50] he's responsible for building the
[01:52] economics department at MIT. And he won
[01:54] the Nobel Prize in 1987.
[01:57] I was a student then here. Uh and
[02:00] and
[02:02] for his work primarily for his work on
[02:05] economic growth. And so what we're going
[02:08] to do in the next two three lectures are
[02:10] essentially things that Bob Solow
[02:13] developed many many years ago.
[02:17] The basic mechanism, you know, remember
[02:20] that we had this Keynesian cross before
[02:21] where we have this multiplier in the
[02:23] goods market and aggregate demand
[02:25] feeding into income and so on and so
[02:26] forth. That was sort of the start
[02:28] mechanism in in in short-run macro.
[02:31] In long-run macro growth theory
[02:34] this is sort of the the key mechanism
[02:37] and and you can think of it as the
[02:38] following. At any point in time
[02:41] an economy has, you know,
[02:43] factors of production primarily labor
[02:45] and capital.
[02:47] That capital stock, labor is more or
[02:49] less fixed or depends on population
[02:50] growth, things that are sort of
[02:52] difficult to to control or they're not
[02:54] really that endogenous to to economics.
[02:57] Not at least in the current times. Many
[03:00] centuries ago yes they were. We had this
[03:02] Malthusian theories in which you know
[03:04] population growth determined determined
[03:06] growth because
[03:08] food is scarcity and stuff like that but
[03:11] that's no longer the case fortunately.
[03:13] Uh for in most parts of the world. So
[03:16] but what can what can change over time
[03:19] and quite a bit and it depends on
[03:21] economic decisions is the capital stock.
[03:23] But at any point in time there is
[03:25] certain capital stocks which combine
[03:27] with labor give you some certain output.
[03:29] Output is income.
[03:31] Part of that income will be saved as
[03:33] we're seeing
[03:34] and that those savings will be used for
[03:37] investment. Okay?
[03:39] But investment is nothing else than
[03:41] capital accumulation.
[03:43] So
[03:44] this income will lead to saving which
[03:47] will fund investment which will change
[03:49] the stock of capital will feed into
[03:50] capital stock that will feed into income
[03:52] and so on. All this is happening very
[03:54] slowly because the capital stock
[03:56] accumulates slowly. I mean
[03:59] but but but this is what is happening
[04:01] and so all the models we're going to
[04:03] look at certainly the model we're going
[04:04] to look at in this lecture is all about
[04:06] this mechanism. Okay?
[04:10] So let's remember what we did in the
[04:12] previous lecture. We
[04:15] uh
[04:16] and I'm going to assume that population
[04:18] is constant. I'm going to relax that at
[04:19] the very end but assume that the
[04:20] population is constant and equal to n
[04:22] and remember we're not worrying about
[04:24] unemployment and stuff like that here.
[04:26] Um so output per capita or per person
[04:31] uh is y over n and we remember we had an
[04:35] production function f of k and n then
[04:38] because of constant returns to scale we
[04:40] could divide by n on both sides
[04:43] everything and we ended up with this
[04:46] relationship. So output per person is
[04:50] equal to an is is a is an increasing
[04:52] function of capital per person. It's an
[04:54] increasing function of capital per
[04:55] person but it's also concave
[04:58] function
[04:59] of capital per person. Why is it
[05:01] concave?
[05:02] That is why is it increasing at a
[05:04] smaller pace?
[05:13] Uh yeah.
[05:17] Decreasing
[05:18] marginal product of capital exactly.
[05:21] You know, for fixed amount of labor the
[05:23] more capital you put in in into
[05:25] production well
[05:27] output keeps expanding but by less and
[05:29] less because it has less and less labor
[05:31] to work with each each unit of capital.
[05:34] Perfect. That's very important. Uh then
[05:37] let's we're going to work in closed
[05:38] economy. I haven't opened it. I'm going
[05:40] to do that after
[05:42] uh
[05:43] quiz two.
[05:44] Um
[05:45] So and I'm going to assume also no
[05:47] public deficits so g equal to t capital
[05:50] T.
[05:51] And in that case then we know that uh
[05:54] private investment private savings equal
[05:57] to private investment. Okay? That's
[05:59] that's the way we derive the IS curve.
[06:01] Um so that's that's that's not new.
[06:05] I'm going to modify a little bit what we
[06:07] did in the short run.
[06:08] Uh um and I'm going to assume that that
[06:11] savings is proportional to income. So
[06:14] savings little s times y.
[06:17] Notice that that this is
[06:20] is different from what we did in the
[06:22] short run. In the short run remember we
[06:23] had a c0 floating around. We had a
[06:25] constant in the consumption function. So
[06:28] savings which was equal to income minus
[06:30] consumption also had a constant floating
[06:32] around.
[06:33] Now that that constant was important in
[06:35] the short-run model because you were
[06:37] approximating for a bunch of things that
[06:39] are not related to short-term income.
[06:41] Wealth you know, the price of houses,
[06:44] stuff like that. We put all that in that
[06:48] constant there.
[06:50] When you think about the long run though
[06:52] uh most of those things that we excluded
[06:54] there asset prices, stuff like that tend
[06:57] to scale with output as well. So so this
[07:00] is this are inconsistent on the surface
[07:03] but if you were to fully work out what
[07:05] is behind the c0 in in the consumption
[07:07] function then
[07:09] this is not a bad approximation. They're
[07:11] not that inconsistent because you
[07:13] endogenize things that over the long run
[07:15] scale with income. I mean, you know,
[07:18] wealth tends to rise with income and all
[07:20] these things tend to move together. At
[07:22] not at high frequency, you can have all
[07:23] sort of fluctuations but over the long
[07:25] run they tend to scale up together. So
[07:29] that's going to be our saving function.
[07:31] So that means that we know in
[07:34] equilibrium
[07:35] this is not investment function. We know
[07:37] that in equilibrium investment will be
[07:40] equal to uh it will be proportional to
[07:43] income.
[07:44] Okay? So remember what we were going
[07:46] through the box. We had at the top of
[07:49] the box we had capital that led to
[07:51] output. We're doing everything in terms
[07:53] per capita. That less led to saving
[07:57] and that
[07:58] funded investment. Okay? So that's
[08:02] that's what we have. So
[08:05] this growth model is really about
[08:08] uh these three functional forms and then
[08:10] a dynamic equation for the stock of
[08:12] capital.
[08:13] So
[08:15] the evolution of the stock of capital
[08:18] capital will increase because of
[08:19] investment.
[08:21] Uh that's what investment is. It's an
[08:22] increase in the stock of capital.
[08:24] Uh but but it will also decrease
[08:28] as a result of depreciation. I mean
[08:30] things do break up, you know.
[08:32] Uh once in a while. And so
[08:36] and different type of capital have
[08:37] different depreciation rates. Equipment
[08:39] depreciate much faster than structures
[08:41] and buildings and so on. But we're going
[08:44] to not going to make those distinctions
[08:45] here. But you see this tells you the
[08:47] capital stock at t plus one is equal to
[08:50] the capital stock we had before minus
[08:52] what is depreciated of that stock of
[08:54] capital plus any new investment we do
[08:57] today.
[08:58] Okay?
[09:00] In per worker terms and remember that
[09:02] for now I'm keeping population growth
[09:04] constant
[09:05] equal to zero. Not not population growth
[09:07] constant. Yeah, constant but equal to
[09:08] zero. So population is constant.
[09:12] I can divide this both sides by n, you
[09:14] know, and I get that capital per worker
[09:18] uh per worker or per person
[09:21] is equal to this expression here. I did
[09:24] two things here. I divided by n and I
[09:26] replaced replaced this I function this
[09:30] investment for savings. Okay? Because I
[09:33] know in equilibrium they have to be
[09:35] equal. So I have that.
[09:37] Um I can rewrite this, you know, just
[09:40] subtract kt over n on both sides and
[09:43] then you get at the change in capital
[09:45] per person is
[09:47] is an increasing function of savings
[09:50] and decreasing of depreciation. Okay? So
[09:53] the last step
[09:55] that is important in this model is to So
[09:58] here I have essentially a difference
[10:00] equation for capital, but we have an
[10:01] output per capita on the right-hand
[10:03] side. But it turns out that that I know
[10:06] that output per capita per person I said
[10:09] per capita per person the same thing
[10:11] per worker it's the same thing in this
[10:13] part of the course.
[10:14] Uh so this
[10:16] is equal to uh
[10:19] is a function is an increasing and
[10:20] concave function of capital
[10:23] per person. Okay?
[10:25] So this is I would say is the sort of
[10:27] fundamental equation of the Solow growth
[10:29] model.
[10:30] It says the change in the stock of
[10:32] capital
[10:33] increases
[10:35] uh with uh
[10:38] with investment of course and decreases
[10:41] with depreciation. And both of these
[10:44] expressions here are increasing
[10:46] functions of the
[10:48] stock of capital per person. Okay?
[10:51] So let's let's try to understand what is
[10:53] in here.
[10:54] So
[10:57] why
[10:58] uh
[10:59] so this is linear obviously because
[11:01] depreciation is linear. You you say say
[11:04] you you lose 5% of your stock of capital
[11:06] every year because it breaks down.
[11:09] Obviously, the more capital per person
[11:11] you have, the more units of capital
[11:13] you're going to lose. This is in units
[11:15] of capital per person. If you have a
[11:17] larger stock of capital, you're going to
[11:18] lose 5% of of a larger number is a
[11:21] larger number. So this and this is
[11:23] proportional is linear.
[11:25] Now this one remember this comes here
[11:27] from the saving function and this term
[11:30] here is equal to income per person.
[11:33] Uh
[11:33] now suppose that that you start in a
[11:36] situation where the capital stock is
[11:38] relatively low and this
[11:40] is positive.
[11:42] What does it mean that this is positive?
[11:44] I mean the implication of this being
[11:45] positive is that the stock stock of
[11:47] capital per person will be growing.
[11:50] But what does it mean that this is
[11:51] positive in words?
[11:56] I mean if you have a stock of capital,
[11:58] there are things that reduce the stock
[12:00] of capital and there are things that
[12:01] increase the stock of capital.
[12:03] This is the thing that increases the
[12:05] stock of capital that's the thing that
[12:06] reduces the stock of capital.
[12:08] So
[12:10] if this
[12:13] is greater than that what is that What
[12:15] does that mean? It's that means this is
[12:17] positive, but in words
[12:19] what is happening?
[12:31] Let me simplify it. This is remember
[12:33] this is just investment per person.
[12:39] Well, this just says
[12:41] that
[12:42] this economy in this economy there is
[12:44] more investment than destruction of
[12:46] capital due to depreciation.
[12:49] Okay? That's what this means.
[12:51] This is investment
[12:53] and and this is positive means that the
[12:55] investment that which is a function of
[12:57] saving the saving rate and stuff like
[12:59] that uh is a function of the funding
[13:01] available for investment
[13:03] is equal to the funding available for
[13:04] investment. Uh
[13:07] uh
[13:07] if this is positive, well this is
[13:09] greater than the stock of capital.
[13:12] Another way of saying it you need a
[13:14] minimum level of investment in an
[13:15] economy to maintain the stock of
[13:17] capital.
[13:20] The minimum level of investment that you
[13:21] need to maintain the stock of capital is
[13:24] equal to the depreciation. So 10 machine
[13:26] breaks
[13:27] you need to invest at least 10 machines
[13:30] in order to maintain the stock of
[13:31] capital constant. Okay?
[13:34] Now if this is positive, it means you're
[13:36] investing more than the machines that
[13:37] are breaking down.
[13:39] Now suppose you start in a situation
[13:41] where that's the case.
[13:42] So that means the stock of capital is
[13:44] growing.
[13:45] I suppose I ask you the next period do
[13:48] you think that gap will be larger or
[13:49] smaller
[13:51] than it used to be?
[14:01] Yeah, actually that's not a great
[14:03] question.
[14:04] Well
[14:05] because I'm not doing it in the right
[14:07] units for that.
[14:10] Let me ask you a
[14:12] variation of that question. Suppose we
[14:14] keep going.
[14:16] After a while, do you think that number
[14:20] will get larger or smaller? After a
[14:23] after let it run for a little for for
[14:25] quite a while.
[14:26] Do you think that number will So
[14:28] remember I said we start with some stock
[14:30] of capital. This is positive.
[14:32] If this is positive, it means that the
[14:33] capital stock is growing. That means
[14:35] this guy is growing and that guy is
[14:37] growing. And they're growing equally.
[14:41] But after a while, do you think this
[14:42] number will get smaller or bigger?
[14:45] After a long while just to make sure
[14:47] that my approximation is not bad here.
[14:57] Exactly. It's going to get smaller
[14:59] because
[15:00] this guy keeps growing linearly
[15:03] with the stock of capital and this one
[15:04] is not. It's concave, you know?
[15:06] At some point this income sort of you
[15:08] need to put a lot of capital for for
[15:10] income to keep rising and therefore for
[15:12] saving to keep rising and therefore for
[15:14] investment to keep rising. And at some
[15:16] point yes it won't be able to
[15:19] uh
[15:20] to really grow. I mean you're going to
[15:21] be using all your investment really to
[15:22] maintain the stock of capital.
[15:24] That's sort of the logic
[15:26] of the Solow model.
[15:29] And it's all in this diagram. So this is
[15:32] diagram you should really really
[15:33] understand well and control it and play
[15:36] with it and all that. It's the
[15:38] equivalent to your IS-LM model in in in
[15:42] the first part of the course.
[15:44] So look at what you have here.
[15:46] So I'm going to plot output per worker
[15:51] per worker per person against capital
[15:53] per worker here.
[15:55] And so
[15:58] this red line here
[16:01] is just
[16:02] the depreciation. Okay? This
[16:05] term here.
[16:07] And that's is a linear function of the
[16:10] capital per worker. Okay? That's what it
[16:12] is.
[16:15] Uh
[16:15] the blue line here
[16:18] is output per worker
[16:20] which as we said is a concave function
[16:22] of K over N. Remember I showed you that
[16:24] production function last in the last in
[16:26] the previous lecture.
[16:28] There you are.
[16:29] Okay?
[16:30] What is the green line?
[16:32] Is investment per worker which is equal
[16:34] to saving per worker and saving per
[16:36] worker is little s the saving rate times
[16:40] uh output. So it's little s which is a
[16:43] number like 0.1 if if if we're talking
[16:46] about the US and you know 0.4 if we're
[16:49] talking about Singapore it varies a lot
[16:51] across countries. But but uh
[16:54] but so this this green line here is
[16:57] nothing else than this blue line
[16:59] multiplied by a number that is less than
[17:00] one. That's the reason it's lower. Okay?
[17:06] Okay, good. So the point I was
[17:08] describing before is was a point like
[17:10] this.
[17:12] Remember?
[17:13] Uh the point that I was describing
[17:15] suppose the economy starts in a point
[17:17] like this one K0 over N.
[17:20] Well
[17:21] and I want to understand the dynamics of
[17:23] this economy. How will it grow over
[17:25] time?
[17:26] So
[17:27] what you have see here is that that
[17:31] uh
[17:32] at this level of capital per worker
[17:36] investment is greater than
[17:38] uh
[17:39] than depreciation.
[17:41] So that's exactly a situation where this
[17:43] is positive.
[17:45] Okay?
[17:46] That distance here
[17:49] is that.
[17:52] Okay?
[17:54] And the reason I sort of
[17:55] say I'm not going to do any local
[17:57] analysis because we could have a started
[17:58] with a K over zero over here and then
[18:01] that number is growing, but it's growing
[18:03] if you were to normalize by the stock of
[18:05] capital is is is declining. That's
[18:07] that's that but I didn't want to do that
[18:08] then. But now that's what I So let's
[18:11] look at this case. You're you're in a
[18:12] situation where this is positive. If
[18:14] this is positive
[18:16] it means the capital stock per worker is
[18:18] growing. So you're moving to the right.
[18:21] In the next period you're going to be
[18:23] here.
[18:24] That
[18:25] that means the capital stock keeps
[18:26] growing
[18:27] but by a smaller steps.
[18:30] Eventually
[18:33] uh
[18:33] the investment is entirely used
[18:36] for uh
[18:39] recovering from the depreciation of
[18:40] capital. So covering the depreciation of
[18:42] capital. And that point the capital
[18:45] stock stock stops growing. We call that
[18:48] a steady state stationary state. We stop
[18:52] Okay? So that's the steady state of this
[18:54] model.
[18:55] That means
[18:56] this economy regardless of where is I do
[18:59] analysis from the other side. Suppose
[19:00] you start from a situation like this.
[19:02] You start with a lot of capital.
[19:04] Okay? Well, if you start with a lot of
[19:06] capital in this economy
[19:08] what happens
[19:10] when here?
[19:12] Well, what happens here is that the
[19:13] investment you're putting to the ground
[19:15] in this economy is less than what you
[19:17] need to maintain the stock of capital
[19:19] which is depreciation.
[19:21] And that means the stock of capital will
[19:23] be shrinking over time.
[19:25] Okay? You're moving that way.
[19:27] So regardless of where you start in this
[19:30] economy if I I ask you the question 100
[19:33] years from now, where are you?
[19:35] You I tell you tell me I don't need to
[19:38] know where you start from. I know that
[19:39] we're going to end up around there.
[19:41] You can either you start from here, you
[19:43] go there
[19:44] from here you go there and so on. That's
[19:46] the reason we call this a steady state.
[19:47] This is where you converge in the long
[19:49] run. Okay?
[19:55] Now,
[19:56] this is already interesting because it
[19:57] tells you
[19:59] you know, at this mo- at this point
[20:01] here, the economy was growing. You know,
[20:04] the capital stock was growing and and
[20:06] and and the and output was growing. You
[20:08] see, the capital is if you start from
[20:10] here,
[20:11] the capital stock is growing, well,
[20:13] output is also growing.
[20:15] Okay? You're moving up there.
[20:17] Okay? So, you had growth.
[20:20] That kind of growth we call transitional
[20:22] growth.
[20:23] You know, it goes from one point to
[20:26] another point. It's not a permanent
[20:28] growth. It's transitional growth.
[20:30] It's the fact that you were away from
[20:32] your steady state and then you're going
[20:34] converging towards your steady state.
[20:37] A lot of the growth we observe and the
[20:39] difference of growth we observe across
[20:40] countries, remember I showed you the
[20:41] downward sloping curves and all that,
[20:44] is as a result of that. Poorer economies
[20:47] tend to have lower capital
[20:49] uh
[20:50] capital labor capital employment ratios,
[20:53] capital population ratios, and therefore
[20:55] they they tend to grow faster because
[20:57] they're catching up with their steady
[20:58] state.
[21:00] Very advanced economies that have been
[21:01] more or less in the same place for a
[21:03] long time are moving around there.
[21:05] So, there's less catching up growth.
[21:08] And that's the main responsible for the
[21:10] the downward sloping curve I showed you
[21:12] within OECD countries and even broader
[21:14] than that. Africa was a little of a
[21:16] problem there.
[21:18] Okay.
[21:20] So, that's This is an important model.
[21:23] Okay for you.
[21:24] Important diagram.
[21:26] Let's let's play a little with it. So,
[21:28] suppose that, you know, at the time,
[21:30] this is a very simple model, but
[21:32] at the time,
[21:34] the the view was that uh
[21:36] well,
[21:37] what really supports growth is saving.
[21:40] So, economies that save a lot
[21:42] grow a lot. And this sort of sort of
[21:44] makes sense here because
[21:46] investment, which is what leads to
[21:48] capital accumulation, is entirely funded
[21:50] by savings. It makes sense.
[21:52] You have more saving, you should grow
[21:54] more.
[21:55] Okay, so let's This is something we can
[21:58] do an experiment. Suppose you start at
[22:00] at at a steady state, if you will.
[22:03] And now we increase the saving rate.
[22:06] What moves?
[22:09] Which curve This is the kind of things
[22:10] you should know when you work with this
[22:12] model.
[22:13] If I change the saving rate, which curve
[22:16] moves
[22:17] in this model?
[22:19] Let me go one by one.
[22:20] Does the red line move?
[22:24] No, has nothing to do with savings.
[22:26] That's to do with depreciation. If I
[22:27] move the depreciation rate, that curve
[22:29] will move, but not
[22:32] Will the production function move?
[22:35] No. So, the blue line cannot move.
[22:37] All that will move is the green line
[22:39] because the green line is the saving
[22:41] rate times the
[22:43] the the blue line. So, if I increase the
[22:46] saving rate, I'm going to move the green
[22:47] line up.
[22:49] Okay? And that's what we have here.
[22:52] So, you see what happens is you start
[22:54] for for This was a steady state for this
[22:57] saving rate in this economy.
[22:59] Now, all of the sudden this economy
[23:00] starts saving more.
[23:02] What happens then?
[23:05] This tells you very much the story of
[23:06] Asia, the Asian miracle
[23:09] of the '60s, '70s, and so on is very
[23:11] much something like that.
[23:14] A little more complicated, but, you
[23:16] know,
[23:18] a big part of what explains sort of the
[23:20] fast growth of Asia
[23:22] uh
[23:23] during that period
[23:25] uh is that something like that happened.
[23:28] Now, why the saving rate increases and
[23:29] so on, that's all very interesting and
[23:30] so on, but but it's not what I want to
[23:32] discuss today.
[23:34] So,
[23:35] but what happens here then? So, what
[23:37] happens See, this economy was in a
[23:38] steady state, so there was no growth. It
[23:40] was growing at zero in steady state, you
[23:42] know?
[23:43] Because
[23:44] this says in a steady state output per
[23:47] per worker per remains constant and
[23:49] since we have no population work and
[23:51] growth, then that means output is not
[23:52] growing either.
[23:54] Okay? The only way you can have that
[23:55] ratio constant with the denominator not
[23:58] moving is that the numerator is not
[23:59] moving either. Okay?
[24:02] Okay, good.
[24:03] So, now
[24:05] boom, all of the sudden we get a higher
[24:07] saving rate. So, what happens now?
[24:11] What reacts?
[24:14] So, the saving rates go up. It's a
[24:16] closed economy, it means the investment
[24:18] rate will go up.
[24:19] Okay?
[24:21] What happens now?
[24:29] What does that gap tell you?
[24:35] Now, you have a positive gap there,
[24:37] which means you're investing more than
[24:39] the the what you need in order to
[24:41] maintain the stock of capital at the
[24:42] previous steady state.
[24:45] So, that means the stock of capital is
[24:46] going to start growing to the right.
[24:48] It's going to start growing.
[24:49] Okay?
[24:50] And as the stock of capital grows, then
[24:53] output per capita also grows.
[24:56] And this will keep happening until you
[24:58] reach the new steady state.
[25:02] So, a higher saving rate, so important
[25:05] conclusion there. This This as simple as
[25:07] it is
[25:08] proves something.
[25:10] Uh
[25:11] that, you know, the conventional wisdom
[25:13] that a higher saving rate would give you
[25:15] sustained growth, higher growth,
[25:18] isn't really true.
[25:19] And not certainly not in this model.
[25:21] Eventually, you'll go back to growth
[25:23] equal to zero.
[25:24] Okay? When you reach a new steady state,
[25:26] you're going to be also growing at zero.
[25:29] Okay?
[25:31] What is true, though,
[25:33] is that you get what again what is
[25:34] called transitional growth. It goes Oh,
[25:38] here you're going to start growing very
[25:39] fast, in fact. Okay? And then you're
[25:42] going to keep growing at a low slow
[25:44] lower pace until you go back to zero,
[25:45] but you're going to get lots of growth
[25:47] in the transition
[25:48] as a result of that. And it turns out in
[25:50] the data when you're looking at 20 30
[25:52] years of data, it's difficult to uh
[25:56] disentangle sort of very permanent rates
[25:58] of growth versus transitional rate of
[26:00] growth.
[26:01] This is one of the things that has
[26:02] concerned China quite a bit, you know,
[26:04] they have been they grow very very fast.
[26:05] They have been growing very very fast
[26:07] for a long time, but it's very clear
[26:09] it's becoming harder and harder for them
[26:10] to grow at the type of rate of growth
[26:12] that they had in the
[26:14] 20 years ago.
[26:15] Okay? They had rates of growth of 15% or
[26:17] so.
[26:18] They had very high They had a very low
[26:21] initial capital
[26:22] population ratio,
[26:24] big population, little capital, and
[26:26] enormous saving rates.
[26:29] So, so they grew very very fast.
[26:32] They had like the green line very close
[26:33] to to the blue line, the capital stock
[26:36] very low, so they grew very very fast.
[26:39] But they have been growing very fast for
[26:40] a very long period of time, so now it's
[26:42] getting a lot harder because they're
[26:43] getting closer and closer to their
[26:45] steady state. That's the issue. Okay.
[26:47] There are other sources of growth, and
[26:49] that's what we're going to talk about in
[26:50] the next lecture,
[26:52] but but this This is something called
[26:54] the easy part of growth.
[26:56] It's sort of running out in China.
[27:03] Okay.
[27:09] And it has to run out
[27:11] in all developed economies for quite a
[27:13] while.
[27:17] Um
[27:18] good.
[27:22] Is this clear?
[27:23] It's important. I mean, a question like
[27:25] that is guaranteed in your quiz.
[27:28] It's 81.
[27:30] What happens if the saving rate does
[27:31] something?
[27:33] So,
[27:34] so so this is a plot over time or um
[27:38] so, this is a case in which you were in
[27:40] a steady state and at time T the saving
[27:42] rate goes up.
[27:44] S1 greater than S0 jump.
[27:47] Then output cannot jump.
[27:50] So, the saving rate goes up, but output
[27:52] can- cannot jump at day zero. Why?
[27:54] Why is it that output doesn't jump
[27:55] immediately to a new steady state?
[28:02] You know,
[28:02] this is the
[28:04] I'm I'm saying
[28:05] this is what will happen to output.
[28:07] You're going to start growing very fast
[28:08] early on, and then you keep growing,
[28:10] keep growing at a slower and lower pace
[28:12] because of decreasing returns to
[28:13] capital,
[28:15] uh and eventually you'll converge to a
[28:16] new steady state with with a rate of
[28:19] growth equal to zero, well, like the one
[28:21] you had before this savings shock.
[28:25] And the question I'm asking now is, why
[28:26] doesn't out- Why does output have to do
[28:28] this? Why Why doesn't it just jump?
[28:35] What would What is the only variable
[28:36] that could make it jump?
[28:41] Well, you need to look at the production
[28:42] function.
[28:44] The production function is a function of
[28:45] K over N. N is fixed. The only thing
[28:47] that can make it jump is if the capital
[28:48] stock jumps.
[28:50] But the capital stock's not jumping.
[28:52] That's a stock.
[28:53] And in order to accumulate a larger
[28:54] stock of the new steady state, you're
[28:56] going to go through a lot of flows.
[28:58] That's investment. You know, every year
[29:00] you're going to be adding a little more
[29:01] to the stock of capital on net or or or
[29:03] or
[29:03] That's the way you grow. You It's not
[29:05] that all of the sudden
[29:06] your stock of capital jumps.
[29:10] That's very much because this is a
[29:12] closed economy. If you're in an open
[29:14] economy, the capital stock can move a
[29:15] lot faster in a transition because you
[29:18] can borrow from abroad. You don't need
[29:20] to fund it all with domestic
[29:22] sources. And in fact, that's what
[29:24] typically happens
[29:26] in in in emerging markets and so on is
[29:28] they typically borrow for a long time.
[29:31] Problem is that they tend to consume it
[29:32] rather than invest it, and that's the
[29:33] reason you end up in financial crisis
[29:35] and so on. But but but in principle,
[29:37] things could go much faster if you have
[29:39] an open economy and and you have capital
[29:41] inflows into your country. But that
[29:43] you'll we'll talk about more about that
[29:46] five or six six lectures from Anyways,
[29:48] but this is what happens when I'm
[29:50] increasing the saving rate. So, yes, it
[29:52] affects the rate of growth of the
[29:54] economy during the transition,
[29:56] uh but but not in the long run. Now,
[29:58] this transition can be very long.
[30:00] Okay?
[30:02] Now, what about consumption? So, so
[30:06] uh uh
[30:07] invariably, and there's no way around
[30:09] that, if if
[30:11] uh
[30:11] given a technology and so on, if the
[30:14] saving rate goes up, then output per
[30:16] worker will go up.
[30:18] Okay?
[30:19] The question is the next question is
[30:21] what happens to consumption per worker?
[30:22] Does consumption per worker go up
[30:25] or not?
[30:27] You are inclined to say, well, I mean,
[30:29] it makes sense that it goes up because
[30:32] uh
[30:32] we have more income, no? The saving rate
[30:34] is little s times y, then consumption is
[30:38] 1 minus little s times y. So, income
[30:41] goes up, consumption should go up.
[30:48] And and yes, that's a dominant source,
[30:52] but it's not all the story because
[30:54] remember I I what I told you.
[31:03] So, consumption here is going to be
[31:04] equal to
[31:05] 1 minus little s
[31:08] times y, so consumption
[31:11] per person will be
[31:13] that.
[31:15] Remember that what is increasing y over
[31:17] n there, so what is making this guy go
[31:20] up, which will lead to an increase in
[31:22] consumption over n,
[31:24] is that this
[31:25] guy went up.
[31:28] And that's a force in the opposite
[31:30] direction.
[31:32] Okay?
[31:33] So, in fact, that was one of the debates
[31:36] with the
[31:38] East Asian mirror Southeast Asian
[31:39] miracle
[31:40] is that it was fueled by lots of
[31:42] savings. So, people say, okay, that's
[31:44] wonderful. Your output growth is very
[31:45] fast, but consumption growth is not so
[31:47] fast. And at some point, it may be
[31:49] hurting you. I think that they were
[31:51] right though for other reasons, but
[31:53] but
[31:55] but that's that picture makes a point.
[31:58] You know, so if if if your saving rate
[32:00] to start with, this is a general lesson.
[32:02] If the saving rate is
[32:04] you start with is very very low,
[32:07] then an increase in the saving rate will
[32:08] lead to a strong increase in consumption
[32:10] because this change is a small relative
[32:12] to the big bang you get on output.
[32:14] Because if you have low saving rate,
[32:16] that also means that the
[32:19] the capital stock is very low.
[32:21] And if the capital stock is very low, f
[32:24] prime is is very big. You know, this is
[32:26] a concave function and you're in in the
[32:27] steep part of the function.
[32:29] Later on, if saving is very high, you're
[32:32] going to tend to have capital stock very
[32:34] high, and then first of all,
[32:37] more capital won't increase output per
[32:39] worker a lot because
[32:41] because of decreasing returns,
[32:43] and and this is a big number. So, it
[32:45] starts dominating. And that's what you
[32:46] see here.
[32:47] This economy as increases saving rate,
[32:50] uh consumption per worker rises, but at
[32:53] some point, it reaches a a maximum, and
[32:55] then it starts declining.
[32:56] I mean, think of the limit. If you save
[32:58] 100% of your income,
[33:01] you don't consume anything. No matter
[33:03] how much is your output, if your saving
[33:05] rate is 100%, then you're not going to
[33:07] consume anything.
[33:09] If you have no income, no saving rate,
[33:12] no savings, no income, no capital stock,
[33:15] no income, you're not going to consume
[33:16] anything either. Okay? So, you at least
[33:19] you know these two points. And since you
[33:21] know there are some positive points in
[33:23] the in the middle,
[33:24] uh you know that the curve is going to
[33:25] tend to have that that kind of change.
[33:27] It's not going to be it's going to be
[33:28] non-monotonic.
[33:30] And that's the way
[33:34] So, let me just
[33:36] play with a little a few numbers. This
[33:37] is
[33:40] Yeah, let me play with a few numbers.
[33:41] It's not that crazy.
[33:43] Uh suppose you have a a production
[33:45] function that gives equal weight to
[33:47] capital and workers. So, this production
[33:49] function.
[33:51] That's a production function of constant
[33:53] return to scale.
[33:55] It better be because that's what we're
[33:57] doing, but
[33:58] what do you think?
[34:01] Yes, no.
[34:03] The sum of the exponents is one. So,
[34:06] it's k to the 1/2 n to the 1/2. The sum
[34:09] of the exponents is one, so you know
[34:11] that
[34:12] it's proportional to the scaling factor.
[34:14] So,
[34:15] we're going to use
[34:17] as a scaling as before n, so
[34:20] um
[34:21] so we have this.
[34:23] Okay?
[34:24] This is a this is a
[34:25] f of little f of k over n is the square
[34:29] root of k over n.
[34:31] Okay?
[34:32] Minus delta k over n. So, all that I'm
[34:34] doing is I'm plugging in that function.
[34:37] Uh
[34:39] So, here only
[34:41] I'm replacing all these functions by
[34:44] by a
[34:45] a specific example, one in which this is
[34:46] a square root of k over n.
[34:49] Okay? That's a concave function, square
[34:52] root.
[34:56] Good.
[34:57] Now, do it as an exercise. If you solve
[35:00] for the steady state, how do you solve
[35:01] for the steady state? Well, set this
[35:03] equal to zero.
[35:04] That will give you the steady state.
[35:06] No?
[35:07] If the steady state is when the capital
[35:09] is not growing anymore, it's when this
[35:11] is equal to zero.
[35:13] When this is equal to zero, I can solve
[35:14] for the steady state level of k over n,
[35:17] no, from here.
[35:19] This equal to zero, I can solve for k
[35:21] over n, and I'm going to call that the
[35:22] steady state. k star.
[35:25] We typically use the stars for the
[35:26] steady states in growth theory.
[35:29] Okay?
[35:29] Well, the answer to this is is k
[35:33] uh the steady state stock of capital per
[35:35] per person is the saving rate over delta
[35:39] squared. That's what it is.
[35:41] Output
[35:43] uh per person, which is the square root
[35:45] of k over n, is therefore the square
[35:47] root of s over delta squared, so it's s
[35:50] over delta.
[35:52] Okay?
[35:53] So, in this particular model, in the
[35:55] long run, output per worker doubles when
[35:58] the saving rate doubles. Okay? If I
[36:00] double the saving rate, then output per
[36:03] worker will double.
[36:07] Notice that the stock of capital is
[36:09] is going to grow a lot more
[36:11] in the
[36:12] when you increase the saving rate.
[36:14] Okay?
[36:15] It's square.
[36:19] So, in that economy,
[36:21] if you do increase the saving rate from
[36:23] 10 to 20%,
[36:25] this is the way it goes.
[36:27] Okay?
[36:28] So, uh remember, 10 to 20% that means
[36:31] that the the new steady state output per
[36:33] worker will be twice what it was in the
[36:35] previous steady state.
[36:36] Okay? So, you go from one to two.
[36:39] But it takes a long time.
[36:42] And the numbers are not crazy. 50 years
[36:44] takes you to go to the new steady state.
[36:48] Okay? So, so that's sort of the time
[36:50] frame we're talking about. So, it is
[36:52] true that the saving rate will not
[36:53] change the long run rate of growth
[36:56] absent other mechanisms.
[37:00] But you can grow faster than your
[37:02] average, your steady state level for
[37:04] quite quite some time. Okay? And and
[37:06] again, a lot of that of the Asian
[37:09] miracle has been of that kind.
[37:13] This is what I was telling you of China
[37:15] before, no? Well, yeah, you you can grow
[37:17] very fast, especially if you have saving
[37:19] rate much higher than 20%, I mean, 50%
[37:22] or so.
[37:23] But but but the rate of growth will have
[37:25] a tendency to decline. Absent some other
[37:28] miracle, there are a lot of the reasons
[37:29] why we have all these fight about
[37:30] technology and so on.
[37:33] It has to do with cuz that's the main
[37:35] mechanism you alternative mechanism to
[37:37] grow.
[37:38] It's technology. Okay? We're going to
[37:40] talk about that in the next lecture. But
[37:42] but this force, which is what I'm saying
[37:44] the force, the easy part of growth, it's
[37:47] very difficult to fight this pattern.
[37:49] Okay?
[37:53] So, here you have numbers
[37:55] uh for the steady states.
[37:57] So, if the saving rate is zero,
[37:59] obviously, everything is zero.
[38:01] No way around.
[38:03] Uh if the saving rate is 0.1, 10%, then
[38:06] in this model, capital per worker is
[38:08] one, output per worker is one.
[38:10] Consumption per worker didn't go from
[38:12] zero to one. Why? Because you were
[38:13] saving something. So, it's zero is 1
[38:15] minus 0.1, which is the saving rate.
[38:18] Suppose you double the saving rate.
[38:20] Well, we know that we're going to double
[38:22] output per worker in this economy. We
[38:23] said that we're going to go from one to
[38:25] two.
[38:26] The capital stock is going to have to
[38:27] grow a lot more to double the amount of
[38:29] output.
[38:31] Why is that? Decreasing returns.
[38:34] To double output, you're going to have
[38:35] to much more than double capital
[38:37] because, you know, you need you're going
[38:39] to be fighting decreasing returns.
[38:42] What about uh consumption? Well, it
[38:44] won't double because you're doing this
[38:46] out of increasing the saving rate. So,
[38:48] you get the two minus now 0.2, not 0.1.
[38:51] Okay?
[38:52] Minus 0.2 times two. So, you get 1.6.
[38:56] And so on.
[38:58] And
[38:59] the higher you go with your saving rate,
[39:01] uh
[39:02] the harder it gets for capital to bring
[39:04] along uh
[39:06] uh
[39:07] um
[39:09] output per capita,
[39:11] and the more the drag on consumption
[39:12] because you need to be saving a lot in
[39:14] order to maintain this high stock of
[39:17] capital that you're having. Okay? You
[39:18] have a very large stock of capital, that
[39:20] means you need to save a lot just for
[39:23] the sake of maintaining that stock of
[39:25] capital. And so
[39:28] little is left for
[39:30] extra
[39:31] output per capita. And so, you see that
[39:33] here in this particular for this
[39:35] particular model, when the saving rate
[39:37] exceeds 0.5,
[39:39] then
[39:40] uh Uh, output obviously keeps rising
[39:42] when you increase the saving rate, but
[39:44] but output starts declining. So, that's
[39:46] your
[39:47] in the declining part.
[39:48] And if you get to one, of course,
[39:50] there's no consumption. So, that's a
[39:52] that's a curve that we trace.
[39:59] Okay.
[40:03] Is everything clear? Now, I'm going to
[40:05] That's a basic solo model, and that's a
[40:07] model that again you need to control
[40:10] completely. Okay.
[40:12] All that I'm going to do now is very
[40:14] simple. I'm going to just
[40:16] modify a little bit this model
[40:19] to uh add population growth.
[40:22] Okay.
[40:23] So, what happens
[40:24] By the way,
[40:26] for for centuries population growth has
[40:28] been one of the main In this model,
[40:32] we concluded that output per worker
[40:35] was not growing.
[40:37] What we're going to conclude in a second
[40:40] is that
[40:41] output per worker will not grow if
[40:43] population is growing.
[40:45] But that means that output is growing.
[40:48] If population is growing and output per
[40:50] worker is not growing, it's constant,
[40:52] that means output is also growing. And
[40:54] for a long time,
[40:56] growth
[40:58] of output, not of output per worker, was
[41:01] driven by large population growth. And
[41:04] sometimes you get big migration flows
[41:06] into a country that leads sort of to
[41:07] growth and so on.
[41:09] Now, big parts of the world
[41:11] have negative population growth. So, now
[41:13] we're going through a cycle in which is
[41:15] things are going the the other way
[41:17] around in in in many large parts of the
[41:20] world. I mean, this true in almost all
[41:22] of continental Europe,
[41:24] uh certainly in Japan, I said South
[41:26] Korea, China,
[41:29] and even some places Latin America.
[41:31] Okay. So, the drug actually is is
[41:34] against that.
[41:36] Uh we don't have the natural force for
[41:38] growth that we had for for many many
[41:40] years.
[41:42] So, let me let me introduce population
[41:44] growth. So, assume now that that
[41:46] population rather than being constant
[41:47] growth growth at the rate gn, which
[41:50] could be positive or negative. I'm going
[41:51] to do the example for the pos a positive
[41:54] uh population growth example.
[41:56] So, there's no equation that changes in
[41:58] the sense that
[42:00] this is still true. It's still true that
[42:02] investment equal to saving. It's still
[42:05] true that that uh output is equal to
[42:09] output per worker. Output is equal to f
[42:12] of k and n,
[42:14] and so on and so forth.
[42:16] The the thing that
[42:17] is a little trickier is that that, you
[42:20] know,
[42:22] in this model,
[42:24] if I don't normalize things for
[42:27] if I if I you know, in this case here
[42:30] where population was not growing,
[42:32] I could have just eliminated this n.
[42:34] It's a constant, and I would have done
[42:36] everything in in in capital in the space
[42:37] of capital here and output here. Would
[42:39] have been the same, just scaled by a
[42:41] number, a constant n.
[42:44] When I have population growth, I'm not
[42:46] indifferent between doing one way or the
[42:48] other.
[42:49] Because if I don't have if I don't if I
[42:51] do it in the space of k and y,
[42:54] and population is growing, then all
[42:56] these curves are moving.
[42:58] So, it's a very unfriendly diagram
[42:59] because my curves are all moving. As n
[43:01] is moving, everything is moving. So, the
[43:03] the trick in all these growth models,
[43:05] and it's going to be even more important
[43:06] in the next lecture, is to find the
[43:08] right scaling of capital so there is a
[43:11] steady state. So, you your curves are
[43:13] not moving around as population grows.
[43:16] It's very easy to find the scaling
[43:18] factor. It's population.
[43:20] Okay. So,
[43:22] that's what I'm going to do.
[43:25] But remember, what is different here is
[43:27] So, I want to what I'm saying here I
[43:29] want to get all my variables as scaled
[43:32] by population at some point in time.
[43:33] That's what I want to do.
[43:35] Because I know I practice enough with
[43:37] these things that's going to give me a
[43:38] steady state. Okay.
[43:40] Uh
[43:41] um Now, what is trickier relative to
[43:43] what I showed you before is that before
[43:45] I just divided by n both sides and and I
[43:48] was home.
[43:49] Now, I can't really do that. Okay. Let
[43:51] me divide by n t plus one both sides.
[43:55] So, that's nice. I get my capital per
[43:58] worker at t plus one.
[44:01] But there's certain things that are not
[44:02] as nice.
[44:03] What I have on the right-hand side is
[44:05] not what I really want. I don't want
[44:06] capital over
[44:08] population next period.
[44:11] Now, my steady state's going to be in
[44:12] the space of
[44:14] capital over population at the same
[44:17] time. That's my steady state.
[44:20] So, this is not so nice.
[44:23] So, what I have to do is I want to
[44:24] convert this the right-hand side in
[44:26] something that is of the kind of things
[44:27] that I want to have.
[44:29] So, what I'm going to do is divide and
[44:31] multiply each of these sides by nt over
[44:33] nt plus one.
[44:35] So, sorry. I'm going to divide and
[44:37] multiply each of these by nt.
[44:40] Okay.
[44:41] So,
[44:42] multiply by nt, divide by nt. So, I'm
[44:45] multiplying by one.
[44:46] Well, and and then I can rearrange the
[44:48] terms in this way. So, I get what I
[44:50] want, which is capital per
[44:52] uh person at time t, all at time t, but
[44:55] then I get this ratio here.
[44:58] Okay. And I can do the same for this
[44:59] expression here.
[45:01] Now, what is that ratio?
[45:06] Population
[45:07] today divided by population tomorrow.
[45:15] Well,
[45:16] it's one
[45:17] over one plus the rate of growth of
[45:19] population.
[45:21] nt plus one is equal to nt times one
[45:23] plus gn.
[45:25] That's the rate of growth
[45:26] of population.
[45:41] Okay.
[45:43] So, so what I have here
[45:46] is one over one plus g
[45:48] gn. Now, gn is not a big number.
[45:52] So, one over one plus gn,
[45:55] one over one plus gn is approximately
[45:57] equal to minus gn.
[46:00] Okay. So, one over one plus gn, gn is
[46:02] very close to zero, is approximately
[46:05] equal to minus gn. Okay.
[46:08] So, that's the reason this guy became
[46:11] that guy, approximately that guy.
[46:14] I can do the same here, but it turns out
[46:16] that
[46:17] the term there's an extra term here,
[46:19] therefore
[46:20] uh which is equal to
[46:22] s times gn
[46:25] times yt over nt.
[46:27] Well, that's second order. That's the
[46:29] reason I'm going to drop it. Okay. It's
[46:30] a saving rate, which is sorry it's it's
[46:33] a it's a small number times
[46:35] uh
[46:36] a rate of population growth, which is a
[46:38] number like, you know, 0.01 or something
[46:40] like that. So, that's a small number.
[46:41] So, I'm dropping it.
[46:43] That's a bigger approximation than that
[46:44] one, actually, but I'm going to do it.
[46:46] Everything becomes a lot simpler, but
[46:48] So, this is an approximation.
[46:50] Okay. I'm just dropping second-order
[46:51] terms.
[46:54] And once I have that, I have the system
[46:56] I want because now I have
[46:58] a a system for the evolution of the of
[47:00] the
[47:02] capital per
[47:04] po- per worker. Okay.
[47:07] Or per person.
[47:10] And if you see, it looks exactly as we
[47:12] had before. Remember, this is exactly
[47:13] what we had before.
[47:16] s f k over We used to have n not sub-
[47:18] subscript t. Now, it's k over nt.
[47:22] But what is different
[47:24] is that now, rather than having only the
[47:26] depreciation rate here, we have the
[47:27] depreciation rate plus the rate of
[47:29] growth of population.
[47:32] Why do you think we have the rate of
[47:33] growth of population there?
[47:36] Remember the the the economics
[47:39] behind this expression before.
[47:42] It was
[47:44] This is what adds to capital.
[47:47] To capital per worker.
[47:49] This is what you need to maintain. What
[47:52] takes away from capital.
[47:54] Okay.
[47:55] Now,
[47:56] it's what takes away from
[47:58] given we're doing everything in the
[47:59] space of capital per worker, that takes
[48:01] away from capital Oh, that's a typo.
[48:04] There's a t there.
[48:05] Okay.
[48:07] t
[48:11] Okay.
[48:12] So, why do you think I have this gn
[48:14] here?
[48:16] Well, I have only one minute, so I don't
[48:17] have time to.
[48:19] Because if I want to maintain a stock of
[48:21] capital
[48:23] per worker,
[48:25] and workers
[48:26] are growing,
[48:28] then I need to be growing the capital
[48:29] stock. Even if I had no depreciation, if
[48:31] I want to maintain the capital per
[48:33] worker constant, and workers are
[48:35] growing,
[48:37] then I need to grow the stock of
[48:38] capital.
[48:39] So, in order to maintain the capital I
[48:41] still need to spend what I used to spend
[48:43] for depreciation of the capital stock.
[48:46] But if I want to maintain the the
[48:48] capital per worker constant, then I'm
[48:50] going to need more investment.
[48:52] Okay.
[48:53] Just to make make up for that that extra
[48:56] component.
[48:58] So, now, set ga equal to zero. That's
[49:01] Your diagram is exactly as before in
[49:03] this space. Set a equal to one and
[49:05] constant.
[49:06] But this
[49:07] line, the red line here, will have delta
[49:10] plus gn. Okay. So, it rotates up.
[49:17] So, you can play here and see what
[49:18] happens if there's change in population
[49:20] growth,
[49:21] and so on and so forth.
[49:23] It's going to be counterintuitive
[49:24] initially because you see, if I increase
[49:26] population growth, this curve will
[49:28] rotate up,
[49:29] and then it would appear as if that
[49:31] leads to negative growth.
[49:35] But you don't get negative growth. In
[49:37] this diagram, you do get that
[49:40] Y over over N will decline.
[49:43] But that doesn't mean that you get
[49:44] negative growth. It just means that
[49:47] output is not growing as fast as
[49:48] population.
[49:50] But but both are growing. Just the
[49:52] population is growing faster than
[49:53] output. I'll I'll I'll start from that.
[49:56] Uh
[49:57] oh, I think it's after your break. So,
[49:59] you're going to have forgotten
[50:00] everything by then. So, I'll do a review
[50:02] of this and then and then we
[50:04] Okay. Have a Have a nice break.