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I couldn't connect but

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so the Fed just hiked by 25 basis

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points.

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And

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as people expected, you know, this is

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the way that it works when there's lots

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of uncertainty essentially

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the Fed starts communicating

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what's going to do

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and the communication was still very

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clear that

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that 25 basis points was

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to be expected and and apparently I was

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reading this right now. It was released

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at

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3 minutes ago, 4 minutes ago.

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Um

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they also said that that further hikes

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are no longer guaranteed. So remember

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that we saw that expected

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hikes sort of we saw several several

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expected hikes for the next few months

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before

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the SVB mess and right after it we sort

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of saw the whole thing declining and and

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at least the minutes are consistent with

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that. Um so there we are. So not big

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uncertainty I mean the markets are

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rallying or something like that at least

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for the next 10 minutes or so but uh

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we shall see.

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Anyway, so but today we we're going to

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really start

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I'm going to I'm going to show you sort

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of the first model of economic growth.

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Uh

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And

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before I do that, who knows who that

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person is?

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No? No clue?

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He

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actually he's Robert Solow. He was an

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He's an emeritus professor at MIT.

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Together with Paul Samuelson essentially

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he's responsible for building the

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economics department at MIT. And he won

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the Nobel Prize in 1987.

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I was a student then here. Uh and

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and

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for his work primarily for his work on

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economic growth. And so what we're going

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to do in the next two three lectures are

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essentially things that Bob Solow

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developed many many years ago.

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The basic mechanism, you know, remember

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that we had this Keynesian cross before

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where we have this multiplier in the

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goods market and aggregate demand

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feeding into income and so on and so

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forth. That was sort of the start

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mechanism in in in short-run macro.

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In long-run macro growth theory

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this is sort of the the key mechanism

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and and you can think of it as the

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following. At any point in time

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an economy has, you know,

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factors of production primarily labor

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and capital.

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That capital stock, labor is more or

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less fixed or depends on population

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growth, things that are sort of

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difficult to to control or they're not

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really that endogenous to to economics.

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Not at least in the current times. Many

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centuries ago yes they were. We had this

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Malthusian theories in which you know

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population growth determined determined

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growth because

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food is scarcity and stuff like that but

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that's no longer the case fortunately.

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Uh for in most parts of the world. So

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but what can what can change over time

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and quite a bit and it depends on

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economic decisions is the capital stock.

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But at any point in time there is

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certain capital stocks which combine

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with labor give you some certain output.

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Output is income.

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Part of that income will be saved as

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we're seeing

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and that those savings will be used for

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investment. Okay?

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But investment is nothing else than

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capital accumulation.

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So

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this income will lead to saving which

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will fund investment which will change

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the stock of capital will feed into

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capital stock that will feed into income

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and so on. All this is happening very

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slowly because the capital stock

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accumulates slowly. I mean

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but but but this is what is happening

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and so all the models we're going to

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look at certainly the model we're going

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to look at in this lecture is all about

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this mechanism. Okay?

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So let's remember what we did in the

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previous lecture. We

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uh

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and I'm going to assume that population

112
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is constant. I'm going to relax that at

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the very end but assume that the

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population is constant and equal to n

115
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and remember we're not worrying about

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unemployment and stuff like that here.

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Um so output per capita or per person

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uh is y over n and we remember we had an

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production function f of k and n then

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because of constant returns to scale we

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could divide by n on both sides

122
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everything and we ended up with this

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relationship. So output per person is

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equal to an is is a is an increasing

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function of capital per person. It's an

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increasing function of capital per

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person but it's also concave

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function

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of capital per person. Why is it

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concave?

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That is why is it increasing at a

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smaller pace?

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Uh yeah.

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Decreasing

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marginal product of capital exactly.

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You know, for fixed amount of labor the

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more capital you put in in into

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production well

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output keeps expanding but by less and

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less because it has less and less labor

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to work with each each unit of capital.

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Perfect. That's very important. Uh then

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let's we're going to work in closed

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economy. I haven't opened it. I'm going

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to do that after

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uh

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quiz two.

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Um

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So and I'm going to assume also no

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public deficits so g equal to t capital

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T.

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And in that case then we know that uh

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private investment private savings equal

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to private investment. Okay? That's

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that's the way we derive the IS curve.

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Um so that's that's that's not new.

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I'm going to modify a little bit what we

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did in the short run.

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Uh um and I'm going to assume that that

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savings is proportional to income. So

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savings little s times y.

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Notice that that this is

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is different from what we did in the

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short run. In the short run remember we

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had a c0 floating around. We had a

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constant in the consumption function. So

167
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savings which was equal to income minus

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consumption also had a constant floating

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around.

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Now that that constant was important in

171
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the short-run model because you were

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approximating for a bunch of things that

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are not related to short-term income.

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Wealth you know, the price of houses,

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stuff like that. We put all that in that

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constant there.

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When you think about the long run though

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uh most of those things that we excluded

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there asset prices, stuff like that tend

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to scale with output as well. So so this

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is this are inconsistent on the surface

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but if you were to fully work out what

183
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is behind the c0 in in the consumption

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function then

185
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this is not a bad approximation. They're

186
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not that inconsistent because you

187
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endogenize things that over the long run

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scale with income. I mean, you know,

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wealth tends to rise with income and all

190
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these things tend to move together. At

191
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not at high frequency, you can have all

192
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sort of fluctuations but over the long

193
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run they tend to scale up together. So

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that's going to be our saving function.

195
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So that means that we know in

196
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equilibrium

197
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this is not investment function. We know

198
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that in equilibrium investment will be

199
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equal to uh it will be proportional to

200
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income.

201
00:07:44,800 --> 00:07:49,520
Okay? So remember what we were going

202
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through the box. We had at the top of

203
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the box we had capital that led to

204
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output. We're doing everything in terms

205
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per capita. That less led to saving

206
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and that

207
00:07:58,720 --> 00:08:05,480
funded investment. Okay? So that's

208
00:08:02,720 --> 00:08:08,280
that's what we have. So

209
00:08:05,480 --> 00:08:10,640
this growth model is really about

210
00:08:08,279 --> 00:08:12,599
uh these three functional forms and then

211
00:08:10,639 --> 00:08:13,680
a dynamic equation for the stock of

212
00:08:12,600 --> 00:08:15,520
capital.

213
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So

214
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the evolution of the stock of capital

215
00:08:18,199 --> 00:08:21,199
capital will increase because of

216
00:08:19,600 --> 00:08:22,800
investment.

217
00:08:21,199 --> 00:08:24,920
Uh that's what investment is. It's an

218
00:08:22,800 --> 00:08:28,800
increase in the stock of capital.

219
00:08:24,920 --> 00:08:30,600
Uh but but it will also decrease

220
00:08:28,800 --> 00:08:32,399
as a result of depreciation. I mean

221
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things do break up, you know.

222
00:08:32,399 --> 00:08:37,559
Uh once in a while. And so

223
00:08:36,200 --> 00:08:39,640
and different type of capital have

224
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different depreciation rates. Equipment

225
00:08:39,639 --> 00:08:44,120
depreciate much faster than structures

226
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and buildings and so on. But we're going

227
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to not going to make those distinctions

228
00:08:45,600 --> 00:08:50,159
here. But you see this tells you the

229
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capital stock at t plus one is equal to

230
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the capital stock we had before minus

231
00:08:52,720 --> 00:08:57,160
what is depreciated of that stock of

232
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capital plus any new investment we do

233
00:08:57,159 --> 00:09:00,439
today.

234
00:08:58,279 --> 00:09:00,439
Okay?

235
00:09:00,480 --> 00:09:04,039
In per worker terms and remember that

236
00:09:02,279 --> 00:09:05,439
for now I'm keeping population growth

237
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constant

238
00:09:05,440 --> 00:09:08,840
equal to zero. Not not population growth

239
00:09:07,440 --> 00:09:12,120
constant. Yeah, constant but equal to

240
00:09:08,840 --> 00:09:14,879
zero. So population is constant.

241
00:09:12,120 --> 00:09:18,279
I can divide this both sides by n, you

242
00:09:14,879 --> 00:09:21,960
know, and I get that capital per worker

243
00:09:18,279 --> 00:09:24,000
uh per worker or per person

244
00:09:21,960 --> 00:09:26,280
is equal to this expression here. I did

245
00:09:24,000 --> 00:09:30,120
two things here. I divided by n and I

246
00:09:26,279 --> 00:09:33,839
replaced replaced this I function this

247
00:09:30,120 --> 00:09:35,000
investment for savings. Okay? Because I

248
00:09:33,840 --> 00:09:37,560
know in equilibrium they have to be

249
00:09:35,000 --> 00:09:40,080
equal. So I have that.

250
00:09:37,559 --> 00:09:43,479
Um I can rewrite this, you know, just

251
00:09:40,080 --> 00:09:45,320
subtract kt over n on both sides and

252
00:09:43,480 --> 00:09:47,600
then you get at the change in capital

253
00:09:45,320 --> 00:09:50,680
per person is

254
00:09:47,600 --> 00:09:53,600
is an increasing function of savings

255
00:09:50,679 --> 00:09:55,279
and decreasing of depreciation. Okay? So

256
00:09:53,600 --> 00:09:58,639
the last step

257
00:09:55,279 --> 00:10:00,039
that is important in this model is to So

258
00:09:58,639 --> 00:10:01,759
here I have essentially a difference

259
00:10:00,039 --> 00:10:03,399
equation for capital, but we have an

260
00:10:01,759 --> 00:10:06,200
output per capita on the right-hand

261
00:10:03,399 --> 00:10:09,319
side. But it turns out that that I know

262
00:10:06,200 --> 00:10:11,560
that output per capita per person I said

263
00:10:09,320 --> 00:10:13,480
per capita per person the same thing

264
00:10:11,559 --> 00:10:14,919
per worker it's the same thing in this

265
00:10:13,480 --> 00:10:16,759
part of the course.

266
00:10:14,919 --> 00:10:19,439
Uh so this

267
00:10:16,759 --> 00:10:20,879
is equal to uh

268
00:10:19,440 --> 00:10:23,280
is a function is an increasing and

269
00:10:20,879 --> 00:10:25,519
concave function of capital

270
00:10:23,279 --> 00:10:27,439
per person. Okay?

271
00:10:25,519 --> 00:10:29,279
So this is I would say is the sort of

272
00:10:27,440 --> 00:10:30,440
fundamental equation of the Solow growth

273
00:10:29,279 --> 00:10:32,439
model.

274
00:10:30,440 --> 00:10:33,640
It says the change in the stock of

275
00:10:32,440 --> 00:10:35,080
capital

276
00:10:33,639 --> 00:10:38,199
increases

277
00:10:35,080 --> 00:10:38,200
uh with uh

278
00:10:38,879 --> 00:10:44,159
with investment of course and decreases

279
00:10:41,600 --> 00:10:46,600
with depreciation. And both of these

280
00:10:44,159 --> 00:10:48,799
expressions here are increasing

281
00:10:46,600 --> 00:10:51,840
functions of the

282
00:10:48,799 --> 00:10:53,599
stock of capital per person. Okay?

283
00:10:51,840 --> 00:10:54,759
So let's let's try to understand what is

284
00:10:53,600 --> 00:10:57,200
in here.

285
00:10:54,759 --> 00:10:57,200
So

286
00:10:57,279 --> 00:10:59,838
why

287
00:10:58,840 --> 00:11:01,839
uh

288
00:10:59,839 --> 00:11:04,120
so this is linear obviously because

289
00:11:01,839 --> 00:11:06,520
depreciation is linear. You you say say

290
00:11:04,120 --> 00:11:09,279
you you lose 5% of your stock of capital

291
00:11:06,519 --> 00:11:11,519
every year because it breaks down.

292
00:11:09,279 --> 00:11:13,799
Obviously, the more capital per person

293
00:11:11,519 --> 00:11:15,240
you have, the more units of capital

294
00:11:13,799 --> 00:11:17,399
you're going to lose. This is in units

295
00:11:15,240 --> 00:11:18,960
of capital per person. If you have a

296
00:11:17,399 --> 00:11:21,639
larger stock of capital, you're going to

297
00:11:18,960 --> 00:11:23,200
lose 5% of of a larger number is a

298
00:11:21,639 --> 00:11:25,039
larger number. So this and this is

299
00:11:23,200 --> 00:11:27,160
proportional is linear.

300
00:11:25,039 --> 00:11:30,559
Now this one remember this comes here

301
00:11:27,159 --> 00:11:33,240
from the saving function and this term

302
00:11:30,559 --> 00:11:33,919
here is equal to income per person.

303
00:11:33,240 --> 00:11:36,200
Uh

304
00:11:33,919 --> 00:11:38,039
now suppose that that you start in a

305
00:11:36,200 --> 00:11:40,680
situation where the capital stock is

306
00:11:38,039 --> 00:11:42,439
relatively low and this

307
00:11:40,679 --> 00:11:44,399
is positive.

308
00:11:42,440 --> 00:11:45,760
What does it mean that this is positive?

309
00:11:44,399 --> 00:11:47,879
I mean the implication of this being

310
00:11:45,759 --> 00:11:50,519
positive is that the stock stock of

311
00:11:47,879 --> 00:11:51,879
capital per person will be growing.

312
00:11:50,519 --> 00:11:54,960
But what does it mean that this is

313
00:11:51,879 --> 00:11:54,960
positive in words?

314
00:11:56,159 --> 00:12:00,079
I mean if you have a stock of capital,

315
00:11:58,519 --> 00:12:01,480
there are things that reduce the stock

316
00:12:00,080 --> 00:12:03,520
of capital and there are things that

317
00:12:01,480 --> 00:12:05,360
increase the stock of capital.

318
00:12:03,519 --> 00:12:06,759
This is the thing that increases the

319
00:12:05,360 --> 00:12:08,879
stock of capital that's the thing that

320
00:12:06,759 --> 00:12:10,960
reduces the stock of capital.

321
00:12:08,879 --> 00:12:13,200
So

322
00:12:10,960 --> 00:12:15,720
if this

323
00:12:13,200 --> 00:12:17,759
is greater than that what is that What

324
00:12:15,720 --> 00:12:19,680
does that mean? It's that means this is

325
00:12:17,759 --> 00:12:22,360
positive, but in words

326
00:12:19,679 --> 00:12:22,359
what is happening?

327
00:12:31,159 --> 00:12:37,000
Let me simplify it. This is remember

328
00:12:33,080 --> 00:12:37,000
this is just investment per person.

329
00:12:39,120 --> 00:12:42,159
Well, this just says

330
00:12:41,000 --> 00:12:44,279
that

331
00:12:42,159 --> 00:12:46,799
this economy in this economy there is

332
00:12:44,279 --> 00:12:49,240
more investment than destruction of

333
00:12:46,799 --> 00:12:51,439
capital due to depreciation.

334
00:12:49,240 --> 00:12:53,279
Okay? That's what this means.

335
00:12:51,440 --> 00:12:55,120
This is investment

336
00:12:53,279 --> 00:12:57,319
and and this is positive means that the

337
00:12:55,120 --> 00:12:59,039
investment that which is a function of

338
00:12:57,320 --> 00:13:01,520
saving the saving rate and stuff like

339
00:12:59,039 --> 00:13:03,439
that uh is a function of the funding

340
00:13:01,519 --> 00:13:04,960
available for investment

341
00:13:03,440 --> 00:13:07,120
is equal to the funding available for

342
00:13:04,960 --> 00:13:07,800
investment. Uh

343
00:13:07,120 --> 00:13:09,600
uh

344
00:13:07,799 --> 00:13:12,240
if this is positive, well this is

345
00:13:09,600 --> 00:13:14,120
greater than the stock of capital.

346
00:13:12,240 --> 00:13:15,879
Another way of saying it you need a

347
00:13:14,120 --> 00:13:17,440
minimum level of investment in an

348
00:13:15,879 --> 00:13:20,039
economy to maintain the stock of

349
00:13:17,440 --> 00:13:20,040
capital.

350
00:13:20,320 --> 00:13:24,040
The minimum level of investment that you

351
00:13:21,679 --> 00:13:26,679
need to maintain the stock of capital is

352
00:13:24,039 --> 00:13:27,838
equal to the depreciation. So 10 machine

353
00:13:26,679 --> 00:13:30,199
breaks

354
00:13:27,839 --> 00:13:31,520
you need to invest at least 10 machines

355
00:13:30,200 --> 00:13:34,320
in order to maintain the stock of

356
00:13:31,519 --> 00:13:36,000
capital constant. Okay?

357
00:13:34,320 --> 00:13:37,600
Now if this is positive, it means you're

358
00:13:36,000 --> 00:13:39,240
investing more than the machines that

359
00:13:37,600 --> 00:13:41,000
are breaking down.

360
00:13:39,240 --> 00:13:42,519
Now suppose you start in a situation

361
00:13:41,000 --> 00:13:44,159
where that's the case.

362
00:13:42,519 --> 00:13:45,519
So that means the stock of capital is

363
00:13:44,159 --> 00:13:48,120
growing.

364
00:13:45,519 --> 00:13:49,919
I suppose I ask you the next period do

365
00:13:48,120 --> 00:13:51,000
you think that gap will be larger or

366
00:13:49,919 --> 00:13:53,879
smaller

367
00:13:51,000 --> 00:13:53,879
than it used to be?

368
00:14:01,879 --> 00:14:04,399
Yeah, actually that's not a great

369
00:14:03,360 --> 00:14:05,560
question.

370
00:14:04,399 --> 00:14:07,759
Well

371
00:14:05,559 --> 00:14:10,519
because I'm not doing it in the right

372
00:14:07,759 --> 00:14:12,159
units for that.

373
00:14:10,519 --> 00:14:14,879
Let me ask you a

374
00:14:12,159 --> 00:14:16,838
variation of that question. Suppose we

375
00:14:14,879 --> 00:14:20,039
keep going.

376
00:14:16,839 --> 00:14:23,280
After a while, do you think that number

377
00:14:20,039 --> 00:14:25,159
will get larger or smaller? After a

378
00:14:23,279 --> 00:14:26,559
after let it run for a little for for

379
00:14:25,159 --> 00:14:28,679
quite a while.

380
00:14:26,559 --> 00:14:30,119
Do you think that number will So

381
00:14:28,679 --> 00:14:32,799
remember I said we start with some stock

382
00:14:30,120 --> 00:14:33,919
of capital. This is positive.

383
00:14:32,799 --> 00:14:35,799
If this is positive, it means that the

384
00:14:33,919 --> 00:14:37,759
capital stock is growing. That means

385
00:14:35,799 --> 00:14:41,039
this guy is growing and that guy is

386
00:14:37,759 --> 00:14:42,720
growing. And they're growing equally.

387
00:14:41,039 --> 00:14:45,719
But after a while, do you think this

388
00:14:42,720 --> 00:14:47,720
number will get smaller or bigger?

389
00:14:45,720 --> 00:14:51,639
After a long while just to make sure

390
00:14:47,720 --> 00:14:51,639
that my approximation is not bad here.

391
00:14:57,720 --> 00:15:00,360
Exactly. It's going to get smaller

392
00:14:59,320 --> 00:15:03,079
because

393
00:15:00,360 --> 00:15:04,639
this guy keeps growing linearly

394
00:15:03,078 --> 00:15:06,879
with the stock of capital and this one

395
00:15:04,639 --> 00:15:08,879
is not. It's concave, you know?

396
00:15:06,879 --> 00:15:10,439
At some point this income sort of you

397
00:15:08,879 --> 00:15:12,559
need to put a lot of capital for for

398
00:15:10,440 --> 00:15:14,400
income to keep rising and therefore for

399
00:15:12,559 --> 00:15:16,439
saving to keep rising and therefore for

400
00:15:14,399 --> 00:15:19,439
investment to keep rising. And at some

401
00:15:16,440 --> 00:15:20,160
point yes it won't be able to

402
00:15:19,440 --> 00:15:21,400
uh

403
00:15:20,159 --> 00:15:22,919
to really grow. I mean you're going to

404
00:15:21,399 --> 00:15:24,838
be using all your investment really to

405
00:15:22,919 --> 00:15:26,639
maintain the stock of capital.

406
00:15:24,839 --> 00:15:29,959
That's sort of the logic

407
00:15:26,639 --> 00:15:32,039
of the Solow model.

408
00:15:29,958 --> 00:15:33,479
And it's all in this diagram. So this is

409
00:15:32,039 --> 00:15:36,319
diagram you should really really

410
00:15:33,480 --> 00:15:38,320
understand well and control it and play

411
00:15:36,320 --> 00:15:42,120
with it and all that. It's the

412
00:15:38,320 --> 00:15:44,520
equivalent to your IS-LM model in in in

413
00:15:42,120 --> 00:15:46,759
the first part of the course.

414
00:15:44,519 --> 00:15:51,000
So look at what you have here.

415
00:15:46,759 --> 00:15:53,559
So I'm going to plot output per worker

416
00:15:51,000 --> 00:15:55,958
per worker per person against capital

417
00:15:53,559 --> 00:15:58,599
per worker here.

418
00:15:55,958 --> 00:16:01,519
And so

419
00:15:58,600 --> 00:16:02,920
this red line here

420
00:16:01,519 --> 00:16:05,679
is just

421
00:16:02,919 --> 00:16:07,240
the depreciation. Okay? This

422
00:16:05,679 --> 00:16:10,039
term here.

423
00:16:07,240 --> 00:16:12,560
And that's is a linear function of the

424
00:16:10,039 --> 00:16:14,838
capital per worker. Okay? That's what it

425
00:16:12,559 --> 00:16:14,838
is.

426
00:16:15,078 --> 00:16:18,000
Uh

427
00:16:15,919 --> 00:16:20,039
the blue line here

428
00:16:18,000 --> 00:16:22,679
is output per worker

429
00:16:20,039 --> 00:16:24,879
which as we said is a concave function

430
00:16:22,679 --> 00:16:26,759
of K over N. Remember I showed you that

431
00:16:24,879 --> 00:16:28,120
production function last in the last in

432
00:16:26,759 --> 00:16:29,319
the previous lecture.

433
00:16:28,120 --> 00:16:30,320
There you are.

434
00:16:29,320 --> 00:16:32,280
Okay?

435
00:16:30,320 --> 00:16:34,520
What is the green line?

436
00:16:32,279 --> 00:16:36,600
Is investment per worker which is equal

437
00:16:34,519 --> 00:16:40,679
to saving per worker and saving per

438
00:16:36,600 --> 00:16:43,279
worker is little s the saving rate times

439
00:16:40,679 --> 00:16:46,639
uh output. So it's little s which is a

440
00:16:43,279 --> 00:16:49,199
number like 0.1 if if if we're talking

441
00:16:46,639 --> 00:16:51,279
about the US and you know 0.4 if we're

442
00:16:49,200 --> 00:16:54,680
talking about Singapore it varies a lot

443
00:16:51,279 --> 00:16:57,480
across countries. But but uh

444
00:16:54,679 --> 00:16:59,239
but so this this green line here is

445
00:16:57,480 --> 00:17:00,839
nothing else than this blue line

446
00:16:59,240 --> 00:17:05,200
multiplied by a number that is less than

447
00:17:00,839 --> 00:17:05,200
one. That's the reason it's lower. Okay?

448
00:17:06,039 --> 00:17:10,759
Okay, good. So the point I was

449
00:17:08,119 --> 00:17:12,078
describing before is was a point like

450
00:17:10,759 --> 00:17:13,078
this.

451
00:17:12,078 --> 00:17:15,838
Remember?

452
00:17:13,078 --> 00:17:17,759
Uh the point that I was describing

453
00:17:15,838 --> 00:17:20,559
suppose the economy starts in a point

454
00:17:17,759 --> 00:17:21,920
like this one K0 over N.

455
00:17:20,559 --> 00:17:23,480
Well

456
00:17:21,920 --> 00:17:25,519
and I want to understand the dynamics of

457
00:17:23,480 --> 00:17:26,599
this economy. How will it grow over

458
00:17:25,519 --> 00:17:27,639
time?

459
00:17:26,599 --> 00:17:31,199
So

460
00:17:27,640 --> 00:17:32,360
what you have see here is that that

461
00:17:31,200 --> 00:17:36,200
uh

462
00:17:32,359 --> 00:17:38,639
at this level of capital per worker

463
00:17:36,200 --> 00:17:39,360
investment is greater than

464
00:17:38,640 --> 00:17:41,480
uh

465
00:17:39,359 --> 00:17:43,839
than depreciation.

466
00:17:41,480 --> 00:17:45,360
So that's exactly a situation where this

467
00:17:43,839 --> 00:17:46,919
is positive.

468
00:17:45,359 --> 00:17:49,959
Okay?

469
00:17:46,920 --> 00:17:52,679
That distance here

470
00:17:49,960 --> 00:17:52,679
is that.

471
00:17:52,839 --> 00:17:55,559
Okay?

472
00:17:54,000 --> 00:17:57,200
And the reason I sort of

473
00:17:55,559 --> 00:17:58,639
say I'm not going to do any local

474
00:17:57,200 --> 00:18:01,360
analysis because we could have a started

475
00:17:58,640 --> 00:18:03,560
with a K over zero over here and then

476
00:18:01,359 --> 00:18:05,039
that number is growing, but it's growing

477
00:18:03,559 --> 00:18:07,200
if you were to normalize by the stock of

478
00:18:05,039 --> 00:18:08,759
capital is is is declining. That's

479
00:18:07,200 --> 00:18:11,679
that's that but I didn't want to do that

480
00:18:08,759 --> 00:18:12,679
then. But now that's what I So let's

481
00:18:11,679 --> 00:18:14,720
look at this case. You're you're in a

482
00:18:12,679 --> 00:18:16,400
situation where this is positive. If

483
00:18:14,720 --> 00:18:18,839
this is positive

484
00:18:16,400 --> 00:18:21,600
it means the capital stock per worker is

485
00:18:18,839 --> 00:18:23,199
growing. So you're moving to the right.

486
00:18:21,599 --> 00:18:24,199
In the next period you're going to be

487
00:18:23,200 --> 00:18:25,319
here.

488
00:18:24,200 --> 00:18:26,519
That

489
00:18:25,319 --> 00:18:27,678
that means the capital stock keeps

490
00:18:26,519 --> 00:18:30,918
growing

491
00:18:27,679 --> 00:18:33,000
but by a smaller steps.

492
00:18:30,919 --> 00:18:33,679
Eventually

493
00:18:33,000 --> 00:18:36,119
uh

494
00:18:33,679 --> 00:18:39,040
the investment is entirely used

495
00:18:36,119 --> 00:18:39,039
for uh

496
00:18:39,079 --> 00:18:42,759
recovering from the depreciation of

497
00:18:40,559 --> 00:18:45,240
capital. So covering the depreciation of

498
00:18:42,759 --> 00:18:48,359
capital. And that point the capital

499
00:18:45,240 --> 00:18:52,519
stock stock stops growing. We call that

500
00:18:48,359 --> 00:18:54,639
a steady state stationary state. We stop

501
00:18:52,519 --> 00:18:55,679
Okay? So that's the steady state of this

502
00:18:54,640 --> 00:18:56,960
model.

503
00:18:55,679 --> 00:18:59,320
That means

504
00:18:56,960 --> 00:19:00,960
this economy regardless of where is I do

505
00:18:59,319 --> 00:19:02,399
analysis from the other side. Suppose

506
00:19:00,960 --> 00:19:04,480
you start from a situation like this.

507
00:19:02,400 --> 00:19:06,600
You start with a lot of capital.

508
00:19:04,480 --> 00:19:08,880
Okay? Well, if you start with a lot of

509
00:19:06,599 --> 00:19:10,399
capital in this economy

510
00:19:08,880 --> 00:19:12,159
what happens

511
00:19:10,400 --> 00:19:13,880
when here?

512
00:19:12,159 --> 00:19:15,800
Well, what happens here is that the

513
00:19:13,880 --> 00:19:17,360
investment you're putting to the ground

514
00:19:15,799 --> 00:19:19,960
in this economy is less than what you

515
00:19:17,359 --> 00:19:21,799
need to maintain the stock of capital

516
00:19:19,960 --> 00:19:23,319
which is depreciation.

517
00:19:21,799 --> 00:19:25,480
And that means the stock of capital will

518
00:19:23,319 --> 00:19:27,559
be shrinking over time.

519
00:19:25,480 --> 00:19:30,240
Okay? You're moving that way.

520
00:19:27,559 --> 00:19:33,359
So regardless of where you start in this

521
00:19:30,240 --> 00:19:35,880
economy if I I ask you the question 100

522
00:19:33,359 --> 00:19:38,119
years from now, where are you?

523
00:19:35,880 --> 00:19:39,560
You I tell you tell me I don't need to

524
00:19:38,119 --> 00:19:41,359
know where you start from. I know that

525
00:19:39,559 --> 00:19:43,319
we're going to end up around there.

526
00:19:41,359 --> 00:19:44,479
You can either you start from here, you

527
00:19:43,319 --> 00:19:46,200
go there

528
00:19:44,480 --> 00:19:47,960
from here you go there and so on. That's

529
00:19:46,200 --> 00:19:49,880
the reason we call this a steady state.

530
00:19:47,960 --> 00:19:52,759
This is where you converge in the long

531
00:19:49,880 --> 00:19:52,760
run. Okay?

532
00:19:55,000 --> 00:19:57,920
Now,

533
00:19:56,319 --> 00:19:59,319
this is already interesting because it

534
00:19:57,920 --> 00:20:01,400
tells you

535
00:19:59,319 --> 00:20:04,079
you know, at this mo- at this point

536
00:20:01,400 --> 00:20:06,000
here, the economy was growing. You know,

537
00:20:04,079 --> 00:20:08,599
the capital stock was growing and and

538
00:20:06,000 --> 00:20:10,400
and and the and output was growing. You

539
00:20:08,599 --> 00:20:11,480
see, the capital is if you start from

540
00:20:10,400 --> 00:20:13,120
here,

541
00:20:11,480 --> 00:20:15,000
the capital stock is growing, well,

542
00:20:13,119 --> 00:20:17,759
output is also growing.

543
00:20:15,000 --> 00:20:20,039
Okay? You're moving up there.

544
00:20:17,759 --> 00:20:22,359
Okay? So, you had growth.

545
00:20:20,039 --> 00:20:23,680
That kind of growth we call transitional

546
00:20:22,359 --> 00:20:26,879
growth.

547
00:20:23,680 --> 00:20:28,480
You know, it goes from one point to

548
00:20:26,880 --> 00:20:30,680
another point. It's not a permanent

549
00:20:28,480 --> 00:20:32,279
growth. It's transitional growth.

550
00:20:30,680 --> 00:20:34,560
It's the fact that you were away from

551
00:20:32,279 --> 00:20:37,399
your steady state and then you're going

552
00:20:34,559 --> 00:20:39,119
converging towards your steady state.

553
00:20:37,400 --> 00:20:40,720
A lot of the growth we observe and the

554
00:20:39,119 --> 00:20:41,919
difference of growth we observe across

555
00:20:40,720 --> 00:20:44,519
countries, remember I showed you the

556
00:20:41,920 --> 00:20:47,600
downward sloping curves and all that,

557
00:20:44,519 --> 00:20:49,559
is as a result of that. Poorer economies

558
00:20:47,599 --> 00:20:50,279
tend to have lower capital

559
00:20:49,559 --> 00:20:53,399
uh

560
00:20:50,279 --> 00:20:55,559
capital labor capital employment ratios,

561
00:20:53,400 --> 00:20:57,600
capital population ratios, and therefore

562
00:20:55,559 --> 00:20:58,799
they they tend to grow faster because

563
00:20:57,599 --> 00:21:00,159
they're catching up with their steady

564
00:20:58,799 --> 00:21:01,879
state.

565
00:21:00,160 --> 00:21:03,120
Very advanced economies that have been

566
00:21:01,880 --> 00:21:05,720
more or less in the same place for a

567
00:21:03,119 --> 00:21:08,479
long time are moving around there.

568
00:21:05,720 --> 00:21:10,160
So, there's less catching up growth.

569
00:21:08,480 --> 00:21:12,480
And that's the main responsible for the

570
00:21:10,160 --> 00:21:14,600
the downward sloping curve I showed you

571
00:21:12,480 --> 00:21:16,200
within OECD countries and even broader

572
00:21:14,599 --> 00:21:18,719
than that. Africa was a little of a

573
00:21:16,200 --> 00:21:18,720
problem there.

574
00:21:18,880 --> 00:21:23,360
Okay.

575
00:21:20,640 --> 00:21:24,360
So, that's This is an important model.

576
00:21:23,359 --> 00:21:26,000
Okay for you.

577
00:21:24,359 --> 00:21:28,399
Important diagram.

578
00:21:26,000 --> 00:21:30,319
Let's let's play a little with it. So,

579
00:21:28,400 --> 00:21:32,560
suppose that, you know, at the time,

580
00:21:30,319 --> 00:21:34,000
this is a very simple model, but

581
00:21:32,559 --> 00:21:36,599
at the time,

582
00:21:34,000 --> 00:21:37,680
the the view was that uh

583
00:21:36,599 --> 00:21:40,319
well,

584
00:21:37,680 --> 00:21:42,519
what really supports growth is saving.

585
00:21:40,319 --> 00:21:44,759
So, economies that save a lot

586
00:21:42,519 --> 00:21:46,759
grow a lot. And this sort of sort of

587
00:21:44,759 --> 00:21:48,720
makes sense here because

588
00:21:46,759 --> 00:21:50,440
investment, which is what leads to

589
00:21:48,720 --> 00:21:52,880
capital accumulation, is entirely funded

590
00:21:50,440 --> 00:21:54,200
by savings. It makes sense.

591
00:21:52,880 --> 00:21:55,920
You have more saving, you should grow

592
00:21:54,200 --> 00:21:58,240
more.

593
00:21:55,920 --> 00:22:00,640
Okay, so let's This is something we can

594
00:21:58,240 --> 00:22:03,400
do an experiment. Suppose you start at

595
00:22:00,640 --> 00:22:06,720
at at a steady state, if you will.

596
00:22:03,400 --> 00:22:09,000
And now we increase the saving rate.

597
00:22:06,720 --> 00:22:10,600
What moves?

598
00:22:09,000 --> 00:22:12,319
Which curve This is the kind of things

599
00:22:10,599 --> 00:22:13,919
you should know when you work with this

600
00:22:12,319 --> 00:22:16,200
model.

601
00:22:13,920 --> 00:22:17,720
If I change the saving rate, which curve

602
00:22:16,200 --> 00:22:19,200
moves

603
00:22:17,720 --> 00:22:20,960
in this model?

604
00:22:19,200 --> 00:22:24,200
Let me go one by one.

605
00:22:20,960 --> 00:22:24,200
Does the red line move?

606
00:22:24,880 --> 00:22:27,679
No, has nothing to do with savings.

607
00:22:26,559 --> 00:22:29,799
That's to do with depreciation. If I

608
00:22:27,679 --> 00:22:32,480
move the depreciation rate, that curve

609
00:22:29,799 --> 00:22:35,159
will move, but not

610
00:22:32,480 --> 00:22:37,799
Will the production function move?

611
00:22:35,160 --> 00:22:39,360
No. So, the blue line cannot move.

612
00:22:37,799 --> 00:22:41,200
All that will move is the green line

613
00:22:39,359 --> 00:22:43,678
because the green line is the saving

614
00:22:41,200 --> 00:22:46,000
rate times the

615
00:22:43,679 --> 00:22:47,600
the the blue line. So, if I increase the

616
00:22:46,000 --> 00:22:49,079
saving rate, I'm going to move the green

617
00:22:47,599 --> 00:22:52,439
line up.

618
00:22:49,079 --> 00:22:52,439
Okay? And that's what we have here.

619
00:22:52,480 --> 00:22:57,079
So, you see what happens is you start

620
00:22:54,839 --> 00:22:59,319
for for This was a steady state for this

621
00:22:57,079 --> 00:23:00,599
saving rate in this economy.

622
00:22:59,319 --> 00:23:02,960
Now, all of the sudden this economy

623
00:23:00,599 --> 00:23:05,159
starts saving more.

624
00:23:02,960 --> 00:23:06,840
What happens then?

625
00:23:05,160 --> 00:23:09,200
This tells you very much the story of

626
00:23:06,839 --> 00:23:11,599
Asia, the Asian miracle

627
00:23:09,200 --> 00:23:14,400
of the '60s, '70s, and so on is very

628
00:23:11,599 --> 00:23:16,240
much something like that.

629
00:23:14,400 --> 00:23:18,360
A little more complicated, but, you

630
00:23:16,240 --> 00:23:18,359
know,

631
00:23:18,480 --> 00:23:22,640
a big part of what explains sort of the

632
00:23:20,559 --> 00:23:23,879
fast growth of Asia

633
00:23:22,640 --> 00:23:25,960
uh

634
00:23:23,880 --> 00:23:28,000
during that period

635
00:23:25,960 --> 00:23:29,640
uh is that something like that happened.

636
00:23:28,000 --> 00:23:30,880
Now, why the saving rate increases and

637
00:23:29,640 --> 00:23:32,840
so on, that's all very interesting and

638
00:23:30,880 --> 00:23:34,600
so on, but but it's not what I want to

639
00:23:32,839 --> 00:23:35,879
discuss today.

640
00:23:34,599 --> 00:23:37,359
So,

641
00:23:35,880 --> 00:23:38,679
but what happens here then? So, what

642
00:23:37,359 --> 00:23:40,439
happens See, this economy was in a

643
00:23:38,679 --> 00:23:42,320
steady state, so there was no growth. It

644
00:23:40,440 --> 00:23:43,519
was growing at zero in steady state, you

645
00:23:42,319 --> 00:23:44,879
know?

646
00:23:43,519 --> 00:23:47,960
Because

647
00:23:44,880 --> 00:23:49,960
this says in a steady state output per

648
00:23:47,960 --> 00:23:51,600
per worker per remains constant and

649
00:23:49,960 --> 00:23:52,960
since we have no population work and

650
00:23:51,599 --> 00:23:54,319
growth, then that means output is not

651
00:23:52,960 --> 00:23:55,759
growing either.

652
00:23:54,319 --> 00:23:58,159
Okay? The only way you can have that

653
00:23:55,759 --> 00:23:59,960
ratio constant with the denominator not

654
00:23:58,160 --> 00:24:02,279
moving is that the numerator is not

655
00:23:59,960 --> 00:24:03,759
moving either. Okay?

656
00:24:02,279 --> 00:24:05,759
Okay, good.

657
00:24:03,759 --> 00:24:07,359
So, now

658
00:24:05,759 --> 00:24:11,000
boom, all of the sudden we get a higher

659
00:24:07,359 --> 00:24:13,759
saving rate. So, what happens now?

660
00:24:11,000 --> 00:24:13,759
What reacts?

661
00:24:14,319 --> 00:24:18,000
So, the saving rates go up. It's a

662
00:24:16,079 --> 00:24:19,439
closed economy, it means the investment

663
00:24:18,000 --> 00:24:21,079
rate will go up.

664
00:24:19,440 --> 00:24:24,120
Okay?

665
00:24:21,079 --> 00:24:24,119
What happens now?

666
00:24:29,640 --> 00:24:33,600
What does that gap tell you?

667
00:24:35,000 --> 00:24:39,319
Now, you have a positive gap there,

668
00:24:37,319 --> 00:24:41,319
which means you're investing more than

669
00:24:39,319 --> 00:24:42,919
the the what you need in order to

670
00:24:41,319 --> 00:24:45,039
maintain the stock of capital at the

671
00:24:42,920 --> 00:24:46,720
previous steady state.

672
00:24:45,039 --> 00:24:48,319
So, that means the stock of capital is

673
00:24:46,720 --> 00:24:49,799
going to start growing to the right.

674
00:24:48,319 --> 00:24:50,639
It's going to start growing.

675
00:24:49,799 --> 00:24:53,359
Okay?

676
00:24:50,640 --> 00:24:56,480
And as the stock of capital grows, then

677
00:24:53,359 --> 00:24:58,439
output per capita also grows.

678
00:24:56,480 --> 00:25:01,960
And this will keep happening until you

679
00:24:58,440 --> 00:25:01,960
reach the new steady state.

680
00:25:02,279 --> 00:25:07,079
So, a higher saving rate, so important

681
00:25:05,279 --> 00:25:08,519
conclusion there. This This as simple as

682
00:25:07,079 --> 00:25:10,919
it is

683
00:25:08,519 --> 00:25:11,720
proves something.

684
00:25:10,920 --> 00:25:13,800
Uh

685
00:25:11,720 --> 00:25:15,279
that, you know, the conventional wisdom

686
00:25:13,799 --> 00:25:18,319
that a higher saving rate would give you

687
00:25:15,279 --> 00:25:19,720
sustained growth, higher growth,

688
00:25:18,319 --> 00:25:21,559
isn't really true.

689
00:25:19,720 --> 00:25:23,079
And not certainly not in this model.

690
00:25:21,559 --> 00:25:24,480
Eventually, you'll go back to growth

691
00:25:23,079 --> 00:25:26,960
equal to zero.

692
00:25:24,480 --> 00:25:29,240
Okay? When you reach a new steady state,

693
00:25:26,960 --> 00:25:31,279
you're going to be also growing at zero.

694
00:25:29,240 --> 00:25:33,039
Okay?

695
00:25:31,279 --> 00:25:34,559
What is true, though,

696
00:25:33,039 --> 00:25:38,480
is that you get what again what is

697
00:25:34,559 --> 00:25:39,759
called transitional growth. It goes Oh,

698
00:25:38,480 --> 00:25:42,480
here you're going to start growing very

699
00:25:39,759 --> 00:25:44,240
fast, in fact. Okay? And then you're

700
00:25:42,480 --> 00:25:45,880
going to keep growing at a low slow

701
00:25:44,240 --> 00:25:47,120
lower pace until you go back to zero,

702
00:25:45,880 --> 00:25:48,720
but you're going to get lots of growth

703
00:25:47,119 --> 00:25:50,799
in the transition

704
00:25:48,720 --> 00:25:52,799
as a result of that. And it turns out in

705
00:25:50,799 --> 00:25:56,639
the data when you're looking at 20 30

706
00:25:52,799 --> 00:25:58,519
years of data, it's difficult to uh

707
00:25:56,640 --> 00:26:00,240
disentangle sort of very permanent rates

708
00:25:58,519 --> 00:26:01,440
of growth versus transitional rate of

709
00:26:00,240 --> 00:26:02,519
growth.

710
00:26:01,440 --> 00:26:04,080
This is one of the things that has

711
00:26:02,519 --> 00:26:05,759
concerned China quite a bit, you know,

712
00:26:04,079 --> 00:26:07,039
they have been they grow very very fast.

713
00:26:05,759 --> 00:26:09,079
They have been growing very very fast

714
00:26:07,039 --> 00:26:10,920
for a long time, but it's very clear

715
00:26:09,079 --> 00:26:12,639
it's becoming harder and harder for them

716
00:26:10,920 --> 00:26:14,200
to grow at the type of rate of growth

717
00:26:12,640 --> 00:26:15,600
that they had in the

718
00:26:14,200 --> 00:26:17,880
20 years ago.

719
00:26:15,599 --> 00:26:18,839
Okay? They had rates of growth of 15% or

720
00:26:17,880 --> 00:26:21,000
so.

721
00:26:18,839 --> 00:26:22,480
They had very high They had a very low

722
00:26:21,000 --> 00:26:24,240
initial capital

723
00:26:22,480 --> 00:26:26,880
population ratio,

724
00:26:24,240 --> 00:26:29,400
big population, little capital, and

725
00:26:26,880 --> 00:26:32,160
enormous saving rates.

726
00:26:29,400 --> 00:26:33,880
So, so they grew very very fast.

727
00:26:32,160 --> 00:26:36,560
They had like the green line very close

728
00:26:33,880 --> 00:26:39,679
to to the blue line, the capital stock

729
00:26:36,559 --> 00:26:40,879
very low, so they grew very very fast.

730
00:26:39,679 --> 00:26:42,320
But they have been growing very fast for

731
00:26:40,880 --> 00:26:43,720
a very long period of time, so now it's

732
00:26:42,319 --> 00:26:45,119
getting a lot harder because they're

733
00:26:43,720 --> 00:26:47,920
getting closer and closer to their

734
00:26:45,119 --> 00:26:49,439
steady state. That's the issue. Okay.

735
00:26:47,920 --> 00:26:50,440
There are other sources of growth, and

736
00:26:49,440 --> 00:26:52,000
that's what we're going to talk about in

737
00:26:50,440 --> 00:26:54,440
the next lecture,

738
00:26:52,000 --> 00:26:56,799
but but this This is something called

739
00:26:54,440 --> 00:27:00,240
the easy part of growth.

740
00:26:56,799 --> 00:27:00,240
It's sort of running out in China.

741
00:27:03,920 --> 00:27:06,160
Okay.

742
00:27:09,720 --> 00:27:13,120
And it has to run out

743
00:27:11,400 --> 00:27:15,440
in all developed economies for quite a

744
00:27:13,119 --> 00:27:15,439
while.

745
00:27:17,119 --> 00:27:21,399
Um

746
00:27:18,920 --> 00:27:21,400
good.

747
00:27:22,079 --> 00:27:25,399
Is this clear?

748
00:27:23,799 --> 00:27:28,200
It's important. I mean, a question like

749
00:27:25,400 --> 00:27:30,120
that is guaranteed in your quiz.

750
00:27:28,200 --> 00:27:31,519
It's 81.

751
00:27:30,119 --> 00:27:33,479
What happens if the saving rate does

752
00:27:31,519 --> 00:27:34,679
something?

753
00:27:33,480 --> 00:27:38,759
So,

754
00:27:34,679 --> 00:27:40,320
so so this is a plot over time or um

755
00:27:38,759 --> 00:27:42,200
so, this is a case in which you were in

756
00:27:40,319 --> 00:27:44,559
a steady state and at time T the saving

757
00:27:42,200 --> 00:27:47,840
rate goes up.

758
00:27:44,559 --> 00:27:50,720
S1 greater than S0 jump.

759
00:27:47,839 --> 00:27:52,359
Then output cannot jump.

760
00:27:50,720 --> 00:27:54,679
So, the saving rate goes up, but output

761
00:27:52,359 --> 00:27:55,919
can- cannot jump at day zero. Why?

762
00:27:54,679 --> 00:27:59,400
Why is it that output doesn't jump

763
00:27:55,920 --> 00:27:59,400
immediately to a new steady state?

764
00:28:02,079 --> 00:28:04,240
You know,

765
00:28:02,920 --> 00:28:05,880
this is the

766
00:28:04,240 --> 00:28:07,200
I'm I'm saying

767
00:28:05,880 --> 00:28:08,800
this is what will happen to output.

768
00:28:07,200 --> 00:28:10,559
You're going to start growing very fast

769
00:28:08,799 --> 00:28:12,639
early on, and then you keep growing,

770
00:28:10,559 --> 00:28:13,799
keep growing at a slower and lower pace

771
00:28:12,640 --> 00:28:15,120
because of decreasing returns to

772
00:28:13,799 --> 00:28:16,799
capital,

773
00:28:15,119 --> 00:28:19,079
uh and eventually you'll converge to a

774
00:28:16,799 --> 00:28:21,639
new steady state with with a rate of

775
00:28:19,079 --> 00:28:25,199
growth equal to zero, well, like the one

776
00:28:21,640 --> 00:28:26,840
you had before this savings shock.

777
00:28:25,200 --> 00:28:28,519
And the question I'm asking now is, why

778
00:28:26,839 --> 00:28:32,359
doesn't out- Why does output have to do

779
00:28:28,519 --> 00:28:32,359
this? Why Why doesn't it just jump?

780
00:28:35,319 --> 00:28:40,039
What would What is the only variable

781
00:28:36,839 --> 00:28:40,039
that could make it jump?

782
00:28:41,440 --> 00:28:44,000
Well, you need to look at the production

783
00:28:42,599 --> 00:28:45,158
function.

784
00:28:44,000 --> 00:28:47,200
The production function is a function of

785
00:28:45,159 --> 00:28:48,760
K over N. N is fixed. The only thing

786
00:28:47,200 --> 00:28:50,039
that can make it jump is if the capital

787
00:28:48,759 --> 00:28:52,079
stock jumps.

788
00:28:50,039 --> 00:28:53,559
But the capital stock's not jumping.

789
00:28:52,079 --> 00:28:54,918
That's a stock.

790
00:28:53,559 --> 00:28:56,720
And in order to accumulate a larger

791
00:28:54,919 --> 00:28:58,759
stock of the new steady state, you're

792
00:28:56,720 --> 00:29:00,679
going to go through a lot of flows.

793
00:28:58,759 --> 00:29:01,879
That's investment. You know, every year

794
00:29:00,679 --> 00:29:03,200
you're going to be adding a little more

795
00:29:01,880 --> 00:29:03,840
to the stock of capital on net or or or

796
00:29:03,200 --> 00:29:05,120
or

797
00:29:03,839 --> 00:29:06,639
That's the way you grow. You It's not

798
00:29:05,119 --> 00:29:10,119
that all of the sudden

799
00:29:06,640 --> 00:29:12,480
your stock of capital jumps.

800
00:29:10,119 --> 00:29:14,199
That's very much because this is a

801
00:29:12,480 --> 00:29:15,599
closed economy. If you're in an open

802
00:29:14,200 --> 00:29:18,240
economy, the capital stock can move a

803
00:29:15,599 --> 00:29:20,119
lot faster in a transition because you

804
00:29:18,240 --> 00:29:22,519
can borrow from abroad. You don't need

805
00:29:20,119 --> 00:29:24,399
to fund it all with domestic

806
00:29:22,519 --> 00:29:26,079
sources. And in fact, that's what

807
00:29:24,400 --> 00:29:28,560
typically happens

808
00:29:26,079 --> 00:29:31,079
in in in emerging markets and so on is

809
00:29:28,559 --> 00:29:32,639
they typically borrow for a long time.

810
00:29:31,079 --> 00:29:33,799
Problem is that they tend to consume it

811
00:29:32,640 --> 00:29:35,480
rather than invest it, and that's the

812
00:29:33,799 --> 00:29:37,918
reason you end up in financial crisis

813
00:29:35,480 --> 00:29:39,480
and so on. But but but in principle,

814
00:29:37,919 --> 00:29:41,560
things could go much faster if you have

815
00:29:39,480 --> 00:29:43,319
an open economy and and you have capital

816
00:29:41,559 --> 00:29:46,639
inflows into your country. But that

817
00:29:43,319 --> 00:29:48,879
you'll we'll talk about more about that

818
00:29:46,640 --> 00:29:50,240
five or six six lectures from Anyways,

819
00:29:48,880 --> 00:29:52,400
but this is what happens when I'm

820
00:29:50,240 --> 00:29:54,480
increasing the saving rate. So, yes, it

821
00:29:52,400 --> 00:29:56,440
affects the rate of growth of the

822
00:29:54,480 --> 00:29:58,319
economy during the transition,

823
00:29:56,440 --> 00:30:00,759
uh but but not in the long run. Now,

824
00:29:58,319 --> 00:30:02,519
this transition can be very long.

825
00:30:00,759 --> 00:30:06,160
Okay?

826
00:30:02,519 --> 00:30:07,119
Now, what about consumption? So, so

827
00:30:06,160 --> 00:30:09,200
uh uh

828
00:30:07,119 --> 00:30:11,119
invariably, and there's no way around

829
00:30:09,200 --> 00:30:11,680
that, if if

830
00:30:11,119 --> 00:30:14,279
uh

831
00:30:11,680 --> 00:30:16,640
given a technology and so on, if the

832
00:30:14,279 --> 00:30:18,279
saving rate goes up, then output per

833
00:30:16,640 --> 00:30:19,280
worker will go up.

834
00:30:18,279 --> 00:30:21,240
Okay?

835
00:30:19,279 --> 00:30:22,960
The question is the next question is

836
00:30:21,240 --> 00:30:25,440
what happens to consumption per worker?

837
00:30:22,960 --> 00:30:27,519
Does consumption per worker go up

838
00:30:25,440 --> 00:30:29,559
or not?

839
00:30:27,519 --> 00:30:32,000
You are inclined to say, well, I mean,

840
00:30:29,559 --> 00:30:32,879
it makes sense that it goes up because

841
00:30:32,000 --> 00:30:34,759
uh

842
00:30:32,880 --> 00:30:38,080
we have more income, no? The saving rate

843
00:30:34,759 --> 00:30:41,079
is little s times y, then consumption is

844
00:30:38,079 --> 00:30:45,359
1 minus little s times y. So, income

845
00:30:41,079 --> 00:30:45,359
goes up, consumption should go up.

846
00:30:48,799 --> 00:30:54,519
And and yes, that's a dominant source,

847
00:30:52,359 --> 00:30:57,959
but it's not all the story because

848
00:30:54,519 --> 00:30:57,960
remember I I what I told you.

849
00:31:03,079 --> 00:31:05,639
So, consumption here is going to be

850
00:31:04,519 --> 00:31:08,759
equal to

851
00:31:05,640 --> 00:31:11,880
1 minus little s

852
00:31:08,759 --> 00:31:13,960
times y, so consumption

853
00:31:11,880 --> 00:31:15,280
per person will be

854
00:31:13,960 --> 00:31:17,559
that.

855
00:31:15,279 --> 00:31:20,000
Remember that what is increasing y over

856
00:31:17,559 --> 00:31:22,799
n there, so what is making this guy go

857
00:31:20,000 --> 00:31:24,440
up, which will lead to an increase in

858
00:31:22,799 --> 00:31:25,759
consumption over n,

859
00:31:24,440 --> 00:31:28,519
is that this

860
00:31:25,759 --> 00:31:28,519
guy went up.

861
00:31:28,640 --> 00:31:32,520
And that's a force in the opposite

862
00:31:30,079 --> 00:31:33,679
direction.

863
00:31:32,519 --> 00:31:36,200
Okay?

864
00:31:33,680 --> 00:31:38,080
So, in fact, that was one of the debates

865
00:31:36,200 --> 00:31:39,799
with the

866
00:31:38,079 --> 00:31:40,879
East Asian mirror Southeast Asian

867
00:31:39,799 --> 00:31:42,680
miracle

868
00:31:40,880 --> 00:31:44,160
is that it was fueled by lots of

869
00:31:42,680 --> 00:31:45,920
savings. So, people say, okay, that's

870
00:31:44,160 --> 00:31:47,600
wonderful. Your output growth is very

871
00:31:45,920 --> 00:31:49,600
fast, but consumption growth is not so

872
00:31:47,599 --> 00:31:51,039
fast. And at some point, it may be

873
00:31:49,599 --> 00:31:53,199
hurting you. I think that they were

874
00:31:51,039 --> 00:31:55,240
right though for other reasons, but

875
00:31:53,200 --> 00:31:58,000
but

876
00:31:55,240 --> 00:32:00,319
but that's that picture makes a point.

877
00:31:58,000 --> 00:32:02,480
You know, so if if if your saving rate

878
00:32:00,319 --> 00:32:04,519
to start with, this is a general lesson.

879
00:32:02,480 --> 00:32:07,079
If the saving rate is

880
00:32:04,519 --> 00:32:08,920
you start with is very very low,

881
00:32:07,079 --> 00:32:10,519
then an increase in the saving rate will

882
00:32:08,920 --> 00:32:12,640
lead to a strong increase in consumption

883
00:32:10,519 --> 00:32:14,759
because this change is a small relative

884
00:32:12,640 --> 00:32:16,880
to the big bang you get on output.

885
00:32:14,759 --> 00:32:19,679
Because if you have low saving rate,

886
00:32:16,880 --> 00:32:21,840
that also means that the

887
00:32:19,679 --> 00:32:24,200
the capital stock is very low.

888
00:32:21,839 --> 00:32:26,439
And if the capital stock is very low, f

889
00:32:24,200 --> 00:32:27,880
prime is is very big. You know, this is

890
00:32:26,440 --> 00:32:29,880
a concave function and you're in in the

891
00:32:27,880 --> 00:32:32,440
steep part of the function.

892
00:32:29,880 --> 00:32:34,080
Later on, if saving is very high, you're

893
00:32:32,440 --> 00:32:37,080
going to tend to have capital stock very

894
00:32:34,079 --> 00:32:39,599
high, and then first of all,

895
00:32:37,079 --> 00:32:41,480
more capital won't increase output per

896
00:32:39,599 --> 00:32:43,359
worker a lot because

897
00:32:41,480 --> 00:32:45,279
because of decreasing returns,

898
00:32:43,359 --> 00:32:46,519
and and this is a big number. So, it

899
00:32:45,279 --> 00:32:47,559
starts dominating. And that's what you

900
00:32:46,519 --> 00:32:50,879
see here.

901
00:32:47,559 --> 00:32:53,000
This economy as increases saving rate,

902
00:32:50,880 --> 00:32:55,440
uh consumption per worker rises, but at

903
00:32:53,000 --> 00:32:56,839
some point, it reaches a a maximum, and

904
00:32:55,440 --> 00:32:58,880
then it starts declining.

905
00:32:56,839 --> 00:33:01,480
I mean, think of the limit. If you save

906
00:32:58,880 --> 00:33:03,120
100% of your income,

907
00:33:01,480 --> 00:33:05,160
you don't consume anything. No matter

908
00:33:03,119 --> 00:33:07,839
how much is your output, if your saving

909
00:33:05,160 --> 00:33:09,720
rate is 100%, then you're not going to

910
00:33:07,839 --> 00:33:12,480
consume anything.

911
00:33:09,720 --> 00:33:15,039
If you have no income, no saving rate,

912
00:33:12,480 --> 00:33:16,839
no savings, no income, no capital stock,

913
00:33:15,039 --> 00:33:19,639
no income, you're not going to consume

914
00:33:16,839 --> 00:33:21,599
anything either. Okay? So, you at least

915
00:33:19,640 --> 00:33:23,080
you know these two points. And since you

916
00:33:21,599 --> 00:33:24,599
know there are some positive points in

917
00:33:23,079 --> 00:33:25,599
the in the middle,

918
00:33:24,599 --> 00:33:27,359
uh you know that the curve is going to

919
00:33:25,599 --> 00:33:28,519
tend to have that that kind of change.

920
00:33:27,359 --> 00:33:30,119
It's not going to be it's going to be

921
00:33:28,519 --> 00:33:33,160
non-monotonic.

922
00:33:30,119 --> 00:33:33,159
And that's the way

923
00:33:34,119 --> 00:33:37,759
So, let me just

924
00:33:36,079 --> 00:33:40,079
play with a little a few numbers. This

925
00:33:37,759 --> 00:33:40,079
is

926
00:33:40,400 --> 00:33:43,800
Yeah, let me play with a few numbers.

927
00:33:41,679 --> 00:33:45,679
It's not that crazy.

928
00:33:43,799 --> 00:33:47,559
Uh suppose you have a a production

929
00:33:45,679 --> 00:33:49,840
function that gives equal weight to

930
00:33:47,559 --> 00:33:51,639
capital and workers. So, this production

931
00:33:49,839 --> 00:33:53,279
function.

932
00:33:51,640 --> 00:33:55,800
That's a production function of constant

933
00:33:53,279 --> 00:33:57,119
return to scale.

934
00:33:55,799 --> 00:33:58,799
It better be because that's what we're

935
00:33:57,119 --> 00:34:01,359
doing, but

936
00:33:58,799 --> 00:34:01,359
what do you think?

937
00:34:01,720 --> 00:34:06,480
Yes, no.

938
00:34:03,039 --> 00:34:09,358
The sum of the exponents is one. So,

939
00:34:06,480 --> 00:34:11,000
it's k to the 1/2 n to the 1/2. The sum

940
00:34:09,358 --> 00:34:12,519
of the exponents is one, so you know

941
00:34:11,000 --> 00:34:14,239
that

942
00:34:12,519 --> 00:34:15,960
it's proportional to the scaling factor.

943
00:34:14,239 --> 00:34:17,479
So,

944
00:34:15,960 --> 00:34:20,878
we're going to use

945
00:34:17,480 --> 00:34:21,760
as a scaling as before n, so

946
00:34:20,878 --> 00:34:23,239
um

947
00:34:21,760 --> 00:34:24,520
so we have this.

948
00:34:23,239 --> 00:34:25,918
Okay?

949
00:34:24,519 --> 00:34:29,358
This is a this is a

950
00:34:25,918 --> 00:34:31,079
f of little f of k over n is the square

951
00:34:29,358 --> 00:34:32,239
root of k over n.

952
00:34:31,079 --> 00:34:34,319
Okay?

953
00:34:32,239 --> 00:34:37,119
Minus delta k over n. So, all that I'm

954
00:34:34,320 --> 00:34:39,240
doing is I'm plugging in that function.

955
00:34:37,119 --> 00:34:39,239
Uh

956
00:34:39,440 --> 00:34:44,039
So, here only

957
00:34:41,440 --> 00:34:45,039
I'm replacing all these functions by

958
00:34:44,039 --> 00:34:46,960
by a

959
00:34:45,039 --> 00:34:49,918
a specific example, one in which this is

960
00:34:46,960 --> 00:34:52,079
a square root of k over n.

961
00:34:49,918 --> 00:34:54,480
Okay? That's a concave function, square

962
00:34:52,079 --> 00:34:54,480
root.

963
00:34:56,398 --> 00:35:00,199
Good.

964
00:34:57,960 --> 00:35:01,679
Now, do it as an exercise. If you solve

965
00:35:00,199 --> 00:35:03,000
for the steady state, how do you solve

966
00:35:01,679 --> 00:35:04,759
for the steady state? Well, set this

967
00:35:03,000 --> 00:35:06,840
equal to zero.

968
00:35:04,760 --> 00:35:07,720
That will give you the steady state.

969
00:35:06,840 --> 00:35:09,440
No?

970
00:35:07,719 --> 00:35:11,119
If the steady state is when the capital

971
00:35:09,440 --> 00:35:13,039
is not growing anymore, it's when this

972
00:35:11,119 --> 00:35:14,960
is equal to zero.

973
00:35:13,039 --> 00:35:17,559
When this is equal to zero, I can solve

974
00:35:14,960 --> 00:35:19,079
for the steady state level of k over n,

975
00:35:17,559 --> 00:35:21,199
no, from here.

976
00:35:19,079 --> 00:35:22,519
This equal to zero, I can solve for k

977
00:35:21,199 --> 00:35:25,039
over n, and I'm going to call that the

978
00:35:22,519 --> 00:35:26,320
steady state. k star.

979
00:35:25,039 --> 00:35:29,000
We typically use the stars for the

980
00:35:26,320 --> 00:35:29,960
steady states in growth theory.

981
00:35:29,000 --> 00:35:33,480
Okay?

982
00:35:29,960 --> 00:35:35,840
Well, the answer to this is is k

983
00:35:33,480 --> 00:35:39,320
uh the steady state stock of capital per

984
00:35:35,840 --> 00:35:41,840
per person is the saving rate over delta

985
00:35:39,320 --> 00:35:43,280
squared. That's what it is.

986
00:35:41,840 --> 00:35:45,600
Output

987
00:35:43,280 --> 00:35:47,880
uh per person, which is the square root

988
00:35:45,599 --> 00:35:50,880
of k over n, is therefore the square

989
00:35:47,880 --> 00:35:52,240
root of s over delta squared, so it's s

990
00:35:50,880 --> 00:35:53,640
over delta.

991
00:35:52,239 --> 00:35:55,599
Okay?

992
00:35:53,639 --> 00:35:58,440
So, in this particular model, in the

993
00:35:55,599 --> 00:36:00,480
long run, output per worker doubles when

994
00:35:58,440 --> 00:36:03,079
the saving rate doubles. Okay? If I

995
00:36:00,480 --> 00:36:06,000
double the saving rate, then output per

996
00:36:03,079 --> 00:36:06,000
worker will double.

997
00:36:07,800 --> 00:36:11,880
Notice that the stock of capital is

998
00:36:09,760 --> 00:36:12,720
is going to grow a lot more

999
00:36:11,880 --> 00:36:14,960
in the

1000
00:36:12,719 --> 00:36:15,559
when you increase the saving rate.

1001
00:36:14,960 --> 00:36:18,199
Okay?

1002
00:36:15,559 --> 00:36:18,199
It's square.

1003
00:36:19,480 --> 00:36:23,440
So, in that economy,

1004
00:36:21,159 --> 00:36:25,519
if you do increase the saving rate from

1005
00:36:23,440 --> 00:36:27,320
10 to 20%,

1006
00:36:25,519 --> 00:36:28,358
this is the way it goes.

1007
00:36:27,320 --> 00:36:31,160
Okay?

1008
00:36:28,358 --> 00:36:33,239
So, uh remember, 10 to 20% that means

1009
00:36:31,159 --> 00:36:35,079
that the the new steady state output per

1010
00:36:33,239 --> 00:36:36,759
worker will be twice what it was in the

1011
00:36:35,079 --> 00:36:39,880
previous steady state.

1012
00:36:36,760 --> 00:36:42,720
Okay? So, you go from one to two.

1013
00:36:39,880 --> 00:36:44,760
But it takes a long time.

1014
00:36:42,719 --> 00:36:48,319
And the numbers are not crazy. 50 years

1015
00:36:44,760 --> 00:36:50,960
takes you to go to the new steady state.

1016
00:36:48,320 --> 00:36:52,519
Okay? So, so that's sort of the time

1017
00:36:50,960 --> 00:36:53,880
frame we're talking about. So, it is

1018
00:36:52,519 --> 00:36:56,239
true that the saving rate will not

1019
00:36:53,880 --> 00:37:00,039
change the long run rate of growth

1020
00:36:56,239 --> 00:37:00,039
absent other mechanisms.

1021
00:37:00,358 --> 00:37:04,039
But you can grow faster than your

1022
00:37:02,400 --> 00:37:06,599
average, your steady state level for

1023
00:37:04,039 --> 00:37:09,039
quite quite some time. Okay? And and

1024
00:37:06,599 --> 00:37:13,480
again, a lot of that of the Asian

1025
00:37:09,039 --> 00:37:13,480
miracle has been of that kind.

1026
00:37:13,760 --> 00:37:17,800
This is what I was telling you of China

1027
00:37:15,400 --> 00:37:19,358
before, no? Well, yeah, you you can grow

1028
00:37:17,800 --> 00:37:22,120
very fast, especially if you have saving

1029
00:37:19,358 --> 00:37:23,519
rate much higher than 20%, I mean, 50%

1030
00:37:22,119 --> 00:37:25,719
or so.

1031
00:37:23,519 --> 00:37:28,280
But but but the rate of growth will have

1032
00:37:25,719 --> 00:37:29,719
a tendency to decline. Absent some other

1033
00:37:28,280 --> 00:37:30,880
miracle, there are a lot of the reasons

1034
00:37:29,719 --> 00:37:33,119
why we have all these fight about

1035
00:37:30,880 --> 00:37:35,039
technology and so on.

1036
00:37:33,119 --> 00:37:37,119
It has to do with cuz that's the main

1037
00:37:35,039 --> 00:37:38,358
mechanism you alternative mechanism to

1038
00:37:37,119 --> 00:37:40,400
grow.

1039
00:37:38,358 --> 00:37:42,679
It's technology. Okay? We're going to

1040
00:37:40,400 --> 00:37:44,960
talk about that in the next lecture. But

1041
00:37:42,679 --> 00:37:47,159
but this force, which is what I'm saying

1042
00:37:44,960 --> 00:37:49,960
the force, the easy part of growth, it's

1043
00:37:47,159 --> 00:37:52,039
very difficult to fight this pattern.

1044
00:37:49,960 --> 00:37:52,039
Okay?

1045
00:37:53,480 --> 00:37:57,679
So, here you have numbers

1046
00:37:55,840 --> 00:37:59,200
uh for the steady states.

1047
00:37:57,679 --> 00:38:01,199
So, if the saving rate is zero,

1048
00:37:59,199 --> 00:38:03,000
obviously, everything is zero.

1049
00:38:01,199 --> 00:38:06,679
No way around.

1050
00:38:03,000 --> 00:38:08,119
Uh if the saving rate is 0.1, 10%, then

1051
00:38:06,679 --> 00:38:10,639
in this model, capital per worker is

1052
00:38:08,119 --> 00:38:12,199
one, output per worker is one.

1053
00:38:10,639 --> 00:38:13,559
Consumption per worker didn't go from

1054
00:38:12,199 --> 00:38:15,480
zero to one. Why? Because you were

1055
00:38:13,559 --> 00:38:18,079
saving something. So, it's zero is 1

1056
00:38:15,480 --> 00:38:20,440
minus 0.1, which is the saving rate.

1057
00:38:18,079 --> 00:38:22,159
Suppose you double the saving rate.

1058
00:38:20,440 --> 00:38:23,760
Well, we know that we're going to double

1059
00:38:22,159 --> 00:38:25,079
output per worker in this economy. We

1060
00:38:23,760 --> 00:38:26,040
said that we're going to go from one to

1061
00:38:25,079 --> 00:38:27,319
two.

1062
00:38:26,039 --> 00:38:29,119
The capital stock is going to have to

1063
00:38:27,320 --> 00:38:31,280
grow a lot more to double the amount of

1064
00:38:29,119 --> 00:38:34,000
output.

1065
00:38:31,280 --> 00:38:35,600
Why is that? Decreasing returns.

1066
00:38:34,000 --> 00:38:37,239
To double output, you're going to have

1067
00:38:35,599 --> 00:38:39,000
to much more than double capital

1068
00:38:37,239 --> 00:38:42,039
because, you know, you need you're going

1069
00:38:39,000 --> 00:38:44,320
to be fighting decreasing returns.

1070
00:38:42,039 --> 00:38:46,239
What about uh consumption? Well, it

1071
00:38:44,320 --> 00:38:48,039
won't double because you're doing this

1072
00:38:46,239 --> 00:38:51,399
out of increasing the saving rate. So,

1073
00:38:48,039 --> 00:38:52,320
you get the two minus now 0.2, not 0.1.

1074
00:38:51,400 --> 00:38:56,320
Okay?

1075
00:38:52,320 --> 00:38:58,000
Minus 0.2 times two. So, you get 1.6.

1076
00:38:56,320 --> 00:38:59,039
And so on.

1077
00:38:58,000 --> 00:39:01,840
And

1078
00:38:59,039 --> 00:39:02,759
the higher you go with your saving rate,

1079
00:39:01,840 --> 00:39:04,960
uh

1080
00:39:02,760 --> 00:39:06,960
the harder it gets for capital to bring

1081
00:39:04,960 --> 00:39:07,760
along uh

1082
00:39:06,960 --> 00:39:09,199
uh

1083
00:39:07,760 --> 00:39:11,040
um

1084
00:39:09,199 --> 00:39:12,839
output per capita,

1085
00:39:11,039 --> 00:39:14,759
and the more the drag on consumption

1086
00:39:12,840 --> 00:39:17,358
because you need to be saving a lot in

1087
00:39:14,760 --> 00:39:18,800
order to maintain this high stock of

1088
00:39:17,358 --> 00:39:20,960
capital that you're having. Okay? You

1089
00:39:18,800 --> 00:39:23,080
have a very large stock of capital, that

1090
00:39:20,960 --> 00:39:25,199
means you need to save a lot just for

1091
00:39:23,079 --> 00:39:28,279
the sake of maintaining that stock of

1092
00:39:25,199 --> 00:39:30,039
capital. And so

1093
00:39:28,280 --> 00:39:31,359
little is left for

1094
00:39:30,039 --> 00:39:33,639
extra

1095
00:39:31,358 --> 00:39:35,279
output per capita. And so, you see that

1096
00:39:33,639 --> 00:39:37,400
here in this particular for this

1097
00:39:35,280 --> 00:39:39,320
particular model, when the saving rate

1098
00:39:37,400 --> 00:39:40,400
exceeds 0.5,

1099
00:39:39,320 --> 00:39:42,559
then

1100
00:39:40,400 --> 00:39:44,440
uh Uh, output obviously keeps rising

1101
00:39:42,559 --> 00:39:46,320
when you increase the saving rate, but

1102
00:39:44,440 --> 00:39:47,000
but output starts declining. So, that's

1103
00:39:46,320 --> 00:39:48,640
your

1104
00:39:47,000 --> 00:39:50,199
in the declining part.

1105
00:39:48,639 --> 00:39:52,039
And if you get to one, of course,

1106
00:39:50,199 --> 00:39:55,599
there's no consumption. So, that's a

1107
00:39:52,039 --> 00:39:55,599
that's a curve that we trace.

1108
00:39:59,639 --> 00:40:02,079
Okay.

1109
00:40:03,599 --> 00:40:07,279
Is everything clear? Now, I'm going to

1110
00:40:05,519 --> 00:40:10,039
That's a basic solo model, and that's a

1111
00:40:07,280 --> 00:40:12,800
model that again you need to control

1112
00:40:10,039 --> 00:40:14,559
completely. Okay.

1113
00:40:12,800 --> 00:40:16,680
All that I'm going to do now is very

1114
00:40:14,559 --> 00:40:19,000
simple. I'm going to just

1115
00:40:16,679 --> 00:40:22,279
modify a little bit this model

1116
00:40:19,000 --> 00:40:23,119
to uh add population growth.

1117
00:40:22,280 --> 00:40:24,480
Okay.

1118
00:40:23,119 --> 00:40:26,639
So, what happens

1119
00:40:24,480 --> 00:40:28,719
By the way,

1120
00:40:26,639 --> 00:40:32,000
for for centuries population growth has

1121
00:40:28,719 --> 00:40:35,639
been one of the main In this model,

1122
00:40:32,000 --> 00:40:37,920
we concluded that output per worker

1123
00:40:35,639 --> 00:40:40,159
was not growing.

1124
00:40:37,920 --> 00:40:41,639
What we're going to conclude in a second

1125
00:40:40,159 --> 00:40:43,639
is that

1126
00:40:41,639 --> 00:40:45,639
output per worker will not grow if

1127
00:40:43,639 --> 00:40:48,359
population is growing.

1128
00:40:45,639 --> 00:40:50,279
But that means that output is growing.

1129
00:40:48,360 --> 00:40:52,760
If population is growing and output per

1130
00:40:50,280 --> 00:40:54,880
worker is not growing, it's constant,

1131
00:40:52,760 --> 00:40:56,640
that means output is also growing. And

1132
00:40:54,880 --> 00:40:58,440
for a long time,

1133
00:40:56,639 --> 00:41:01,159
growth

1134
00:40:58,440 --> 00:41:04,119
of output, not of output per worker, was

1135
00:41:01,159 --> 00:41:06,119
driven by large population growth. And

1136
00:41:04,119 --> 00:41:07,519
sometimes you get big migration flows

1137
00:41:06,119 --> 00:41:09,159
into a country that leads sort of to

1138
00:41:07,519 --> 00:41:11,719
growth and so on.

1139
00:41:09,159 --> 00:41:13,799
Now, big parts of the world

1140
00:41:11,719 --> 00:41:15,719
have negative population growth. So, now

1141
00:41:13,800 --> 00:41:17,120
we're going through a cycle in which is

1142
00:41:15,719 --> 00:41:20,199
things are going the the other way

1143
00:41:17,119 --> 00:41:22,079
around in in in many large parts of the

1144
00:41:20,199 --> 00:41:24,399
world. I mean, this true in almost all

1145
00:41:22,079 --> 00:41:26,679
of continental Europe,

1146
00:41:24,400 --> 00:41:29,480
uh certainly in Japan, I said South

1147
00:41:26,679 --> 00:41:31,599
Korea, China,

1148
00:41:29,480 --> 00:41:34,079
and even some places Latin America.

1149
00:41:31,599 --> 00:41:36,599
Okay. So, the drug actually is is

1150
00:41:34,079 --> 00:41:38,319
against that.

1151
00:41:36,599 --> 00:41:40,639
Uh we don't have the natural force for

1152
00:41:38,320 --> 00:41:42,000
growth that we had for for many many

1153
00:41:40,639 --> 00:41:44,400
years.

1154
00:41:42,000 --> 00:41:46,320
So, let me let me introduce population

1155
00:41:44,400 --> 00:41:47,840
growth. So, assume now that that

1156
00:41:46,320 --> 00:41:50,280
population rather than being constant

1157
00:41:47,840 --> 00:41:51,480
growth growth at the rate gn, which

1158
00:41:50,280 --> 00:41:54,080
could be positive or negative. I'm going

1159
00:41:51,480 --> 00:41:56,280
to do the example for the pos a positive

1160
00:41:54,079 --> 00:41:58,880
uh population growth example.

1161
00:41:56,280 --> 00:42:00,040
So, there's no equation that changes in

1162
00:41:58,880 --> 00:42:02,240
the sense that

1163
00:42:00,039 --> 00:42:05,039
this is still true. It's still true that

1164
00:42:02,239 --> 00:42:09,119
investment equal to saving. It's still

1165
00:42:05,039 --> 00:42:12,320
true that that uh output is equal to

1166
00:42:09,119 --> 00:42:14,119
output per worker. Output is equal to f

1167
00:42:12,320 --> 00:42:16,440
of k and n,

1168
00:42:14,119 --> 00:42:17,960
and so on and so forth.

1169
00:42:16,440 --> 00:42:20,720
The the thing that

1170
00:42:17,960 --> 00:42:22,559
is a little trickier is that that, you

1171
00:42:20,719 --> 00:42:24,079
know,

1172
00:42:22,559 --> 00:42:27,679
in this model,

1173
00:42:24,079 --> 00:42:30,039
if I don't normalize things for

1174
00:42:27,679 --> 00:42:32,359
if I if I you know, in this case here

1175
00:42:30,039 --> 00:42:34,480
where population was not growing,

1176
00:42:32,360 --> 00:42:36,000
I could have just eliminated this n.

1177
00:42:34,480 --> 00:42:37,840
It's a constant, and I would have done

1178
00:42:36,000 --> 00:42:39,639
everything in in in capital in the space

1179
00:42:37,840 --> 00:42:41,000
of capital here and output here. Would

1180
00:42:39,639 --> 00:42:44,039
have been the same, just scaled by a

1181
00:42:41,000 --> 00:42:46,159
number, a constant n.

1182
00:42:44,039 --> 00:42:48,079
When I have population growth, I'm not

1183
00:42:46,159 --> 00:42:49,239
indifferent between doing one way or the

1184
00:42:48,079 --> 00:42:51,679
other.

1185
00:42:49,239 --> 00:42:54,159
Because if I don't have if I don't if I

1186
00:42:51,679 --> 00:42:56,079
do it in the space of k and y,

1187
00:42:54,159 --> 00:42:58,119
and population is growing, then all

1188
00:42:56,079 --> 00:42:59,599
these curves are moving.

1189
00:42:58,119 --> 00:43:01,480
So, it's a very unfriendly diagram

1190
00:42:59,599 --> 00:43:03,920
because my curves are all moving. As n

1191
00:43:01,480 --> 00:43:05,519
is moving, everything is moving. So, the

1192
00:43:03,920 --> 00:43:06,760
the trick in all these growth models,

1193
00:43:05,519 --> 00:43:08,639
and it's going to be even more important

1194
00:43:06,760 --> 00:43:11,120
in the next lecture, is to find the

1195
00:43:08,639 --> 00:43:13,759
right scaling of capital so there is a

1196
00:43:11,119 --> 00:43:16,759
steady state. So, you your curves are

1197
00:43:13,760 --> 00:43:18,000
not moving around as population grows.

1198
00:43:16,760 --> 00:43:20,440
It's very easy to find the scaling

1199
00:43:18,000 --> 00:43:22,960
factor. It's population.

1200
00:43:20,440 --> 00:43:25,400
Okay. So,

1201
00:43:22,960 --> 00:43:27,480
that's what I'm going to do.

1202
00:43:25,400 --> 00:43:29,519
But remember, what is different here is

1203
00:43:27,480 --> 00:43:32,119
So, I want to what I'm saying here I

1204
00:43:29,519 --> 00:43:33,880
want to get all my variables as scaled

1205
00:43:32,119 --> 00:43:35,440
by population at some point in time.

1206
00:43:33,880 --> 00:43:37,360
That's what I want to do.

1207
00:43:35,440 --> 00:43:38,440
Because I know I practice enough with

1208
00:43:37,360 --> 00:43:40,519
these things that's going to give me a

1209
00:43:38,440 --> 00:43:41,200
steady state. Okay.

1210
00:43:40,519 --> 00:43:43,519
Uh

1211
00:43:41,199 --> 00:43:45,639
um Now, what is trickier relative to

1212
00:43:43,519 --> 00:43:48,000
what I showed you before is that before

1213
00:43:45,639 --> 00:43:49,759
I just divided by n both sides and and I

1214
00:43:48,000 --> 00:43:51,960
was home.

1215
00:43:49,760 --> 00:43:55,400
Now, I can't really do that. Okay. Let

1216
00:43:51,960 --> 00:43:58,440
me divide by n t plus one both sides.

1217
00:43:55,400 --> 00:44:01,400
So, that's nice. I get my capital per

1218
00:43:58,440 --> 00:44:02,559
worker at t plus one.

1219
00:44:01,400 --> 00:44:03,920
But there's certain things that are not

1220
00:44:02,559 --> 00:44:05,079
as nice.

1221
00:44:03,920 --> 00:44:06,599
What I have on the right-hand side is

1222
00:44:05,079 --> 00:44:08,400
not what I really want. I don't want

1223
00:44:06,599 --> 00:44:11,358
capital over

1224
00:44:08,400 --> 00:44:12,960
population next period.

1225
00:44:11,358 --> 00:44:14,799
Now, my steady state's going to be in

1226
00:44:12,960 --> 00:44:17,000
the space of

1227
00:44:14,800 --> 00:44:20,560
capital over population at the same

1228
00:44:17,000 --> 00:44:23,000
time. That's my steady state.

1229
00:44:20,559 --> 00:44:24,320
So, this is not so nice.

1230
00:44:23,000 --> 00:44:26,440
So, what I have to do is I want to

1231
00:44:24,320 --> 00:44:27,960
convert this the right-hand side in

1232
00:44:26,440 --> 00:44:29,440
something that is of the kind of things

1233
00:44:27,960 --> 00:44:31,039
that I want to have.

1234
00:44:29,440 --> 00:44:33,679
So, what I'm going to do is divide and

1235
00:44:31,039 --> 00:44:35,320
multiply each of these sides by nt over

1236
00:44:33,679 --> 00:44:37,239
nt plus one.

1237
00:44:35,320 --> 00:44:40,320
So, sorry. I'm going to divide and

1238
00:44:37,239 --> 00:44:41,319
multiply each of these by nt.

1239
00:44:40,320 --> 00:44:42,480
Okay.

1240
00:44:41,320 --> 00:44:45,039
So,

1241
00:44:42,480 --> 00:44:46,559
multiply by nt, divide by nt. So, I'm

1242
00:44:45,039 --> 00:44:48,358
multiplying by one.

1243
00:44:46,559 --> 00:44:50,119
Well, and and then I can rearrange the

1244
00:44:48,358 --> 00:44:52,519
terms in this way. So, I get what I

1245
00:44:50,119 --> 00:44:55,279
want, which is capital per

1246
00:44:52,519 --> 00:44:58,199
uh person at time t, all at time t, but

1247
00:44:55,280 --> 00:44:59,960
then I get this ratio here.

1248
00:44:58,199 --> 00:45:01,879
Okay. And I can do the same for this

1249
00:44:59,960 --> 00:45:05,280
expression here.

1250
00:45:01,880 --> 00:45:05,280
Now, what is that ratio?

1251
00:45:06,480 --> 00:45:12,119
Population

1252
00:45:07,800 --> 00:45:12,120
today divided by population tomorrow.

1253
00:45:15,119 --> 00:45:17,440
Well,

1254
00:45:16,119 --> 00:45:19,480
it's one

1255
00:45:17,440 --> 00:45:21,200
over one plus the rate of growth of

1256
00:45:19,480 --> 00:45:23,559
population.

1257
00:45:21,199 --> 00:45:25,279
nt plus one is equal to nt times one

1258
00:45:23,559 --> 00:45:26,799
plus gn.

1259
00:45:25,280 --> 00:45:29,600
That's the rate of growth

1260
00:45:26,800 --> 00:45:29,600
of population.

1261
00:45:41,760 --> 00:45:46,600
Okay.

1262
00:45:43,679 --> 00:45:48,719
So, so what I have here

1263
00:45:46,599 --> 00:45:52,079
is one over one plus g

1264
00:45:48,719 --> 00:45:55,839
gn. Now, gn is not a big number.

1265
00:45:52,079 --> 00:45:57,759
So, one over one plus gn,

1266
00:45:55,840 --> 00:46:00,760
one over one plus gn is approximately

1267
00:45:57,760 --> 00:46:02,920
equal to minus gn.

1268
00:46:00,760 --> 00:46:05,120
Okay. So, one over one plus gn, gn is

1269
00:46:02,920 --> 00:46:08,358
very close to zero, is approximately

1270
00:46:05,119 --> 00:46:11,159
equal to minus gn. Okay.

1271
00:46:08,358 --> 00:46:14,279
So, that's the reason this guy became

1272
00:46:11,159 --> 00:46:16,559
that guy, approximately that guy.

1273
00:46:14,280 --> 00:46:17,519
I can do the same here, but it turns out

1274
00:46:16,559 --> 00:46:19,119
that

1275
00:46:17,519 --> 00:46:20,440
the term there's an extra term here,

1276
00:46:19,119 --> 00:46:22,920
therefore

1277
00:46:20,440 --> 00:46:25,559
uh which is equal to

1278
00:46:22,920 --> 00:46:27,920
s times gn

1279
00:46:25,559 --> 00:46:29,039
times yt over nt.

1280
00:46:27,920 --> 00:46:30,840
Well, that's second order. That's the

1281
00:46:29,039 --> 00:46:33,279
reason I'm going to drop it. Okay. It's

1282
00:46:30,840 --> 00:46:35,720
a saving rate, which is sorry it's it's

1283
00:46:33,280 --> 00:46:36,519
a it's a small number times

1284
00:46:35,719 --> 00:46:38,119
uh

1285
00:46:36,519 --> 00:46:40,000
a rate of population growth, which is a

1286
00:46:38,119 --> 00:46:41,679
number like, you know, 0.01 or something

1287
00:46:40,000 --> 00:46:43,000
like that. So, that's a small number.

1288
00:46:41,679 --> 00:46:44,639
So, I'm dropping it.

1289
00:46:43,000 --> 00:46:46,679
That's a bigger approximation than that

1290
00:46:44,639 --> 00:46:48,358
one, actually, but I'm going to do it.

1291
00:46:46,679 --> 00:46:50,199
Everything becomes a lot simpler, but

1292
00:46:48,358 --> 00:46:51,759
So, this is an approximation.

1293
00:46:50,199 --> 00:46:54,119
Okay. I'm just dropping second-order

1294
00:46:51,760 --> 00:46:54,120
terms.

1295
00:46:54,559 --> 00:46:58,199
And once I have that, I have the system

1296
00:46:56,280 --> 00:47:00,960
I want because now I have

1297
00:46:58,199 --> 00:47:02,480
a a system for the evolution of the of

1298
00:47:00,960 --> 00:47:04,440
the

1299
00:47:02,480 --> 00:47:07,480
capital per

1300
00:47:04,440 --> 00:47:10,200
po- per worker. Okay.

1301
00:47:07,480 --> 00:47:12,079
Or per person.

1302
00:47:10,199 --> 00:47:13,839
And if you see, it looks exactly as we

1303
00:47:12,079 --> 00:47:16,079
had before. Remember, this is exactly

1304
00:47:13,840 --> 00:47:18,960
what we had before.

1305
00:47:16,079 --> 00:47:22,559
s f k over We used to have n not sub-

1306
00:47:18,960 --> 00:47:24,440
subscript t. Now, it's k over nt.

1307
00:47:22,559 --> 00:47:26,000
But what is different

1308
00:47:24,440 --> 00:47:27,960
is that now, rather than having only the

1309
00:47:26,000 --> 00:47:29,400
depreciation rate here, we have the

1310
00:47:27,960 --> 00:47:32,400
depreciation rate plus the rate of

1311
00:47:29,400 --> 00:47:32,400
growth of population.

1312
00:47:32,719 --> 00:47:36,719
Why do you think we have the rate of

1313
00:47:33,840 --> 00:47:39,358
growth of population there?

1314
00:47:36,719 --> 00:47:42,319
Remember the the the economics

1315
00:47:39,358 --> 00:47:44,599
behind this expression before.

1316
00:47:42,320 --> 00:47:47,840
It was

1317
00:47:44,599 --> 00:47:49,920
This is what adds to capital.

1318
00:47:47,840 --> 00:47:52,160
To capital per worker.

1319
00:47:49,920 --> 00:47:54,240
This is what you need to maintain. What

1320
00:47:52,159 --> 00:47:55,399
takes away from capital.

1321
00:47:54,239 --> 00:47:56,639
Okay.

1322
00:47:55,400 --> 00:47:58,400
Now,

1323
00:47:56,639 --> 00:47:59,719
it's what takes away from

1324
00:47:58,400 --> 00:48:01,240
given we're doing everything in the

1325
00:47:59,719 --> 00:48:04,119
space of capital per worker, that takes

1326
00:48:01,239 --> 00:48:05,759
away from capital Oh, that's a typo.

1327
00:48:04,119 --> 00:48:07,599
There's a t there.

1328
00:48:05,760 --> 00:48:09,880
Okay.

1329
00:48:07,599 --> 00:48:09,880
t

1330
00:48:11,358 --> 00:48:14,319
Okay.

1331
00:48:12,800 --> 00:48:16,120
So, why do you think I have this gn

1332
00:48:14,320 --> 00:48:17,760
here?

1333
00:48:16,119 --> 00:48:19,400
Well, I have only one minute, so I don't

1334
00:48:17,760 --> 00:48:21,960
have time to.

1335
00:48:19,400 --> 00:48:23,519
Because if I want to maintain a stock of

1336
00:48:21,960 --> 00:48:25,119
capital

1337
00:48:23,519 --> 00:48:26,519
per worker,

1338
00:48:25,119 --> 00:48:28,039
and workers

1339
00:48:26,519 --> 00:48:29,920
are growing,

1340
00:48:28,039 --> 00:48:31,920
then I need to be growing the capital

1341
00:48:29,920 --> 00:48:33,559
stock. Even if I had no depreciation, if

1342
00:48:31,920 --> 00:48:35,840
I want to maintain the capital per

1343
00:48:33,559 --> 00:48:37,039
worker constant, and workers are

1344
00:48:35,840 --> 00:48:38,120
growing,

1345
00:48:37,039 --> 00:48:39,639
then I need to grow the stock of

1346
00:48:38,119 --> 00:48:41,599
capital.

1347
00:48:39,639 --> 00:48:43,679
So, in order to maintain the capital I

1348
00:48:41,599 --> 00:48:46,279
still need to spend what I used to spend

1349
00:48:43,679 --> 00:48:48,199
for depreciation of the capital stock.

1350
00:48:46,280 --> 00:48:50,920
But if I want to maintain the the

1351
00:48:48,199 --> 00:48:52,839
capital per worker constant, then I'm

1352
00:48:50,920 --> 00:48:53,840
going to need more investment.

1353
00:48:52,840 --> 00:48:56,559
Okay.

1354
00:48:53,840 --> 00:48:58,160
Just to make make up for that that extra

1355
00:48:56,559 --> 00:49:01,559
component.

1356
00:48:58,159 --> 00:49:03,199
So, now, set ga equal to zero. That's

1357
00:49:01,559 --> 00:49:05,599
Your diagram is exactly as before in

1358
00:49:03,199 --> 00:49:06,559
this space. Set a equal to one and

1359
00:49:05,599 --> 00:49:07,759
constant.

1360
00:49:06,559 --> 00:49:10,358
But this

1361
00:49:07,760 --> 00:49:15,200
line, the red line here, will have delta

1362
00:49:10,358 --> 00:49:15,199
plus gn. Okay. So, it rotates up.

1363
00:49:17,119 --> 00:49:20,199
So, you can play here and see what

1364
00:49:18,800 --> 00:49:21,320
happens if there's change in population

1365
00:49:20,199 --> 00:49:23,199
growth,

1366
00:49:21,320 --> 00:49:24,359
and so on and so forth.

1367
00:49:23,199 --> 00:49:26,519
It's going to be counterintuitive

1368
00:49:24,358 --> 00:49:28,358
initially because you see, if I increase

1369
00:49:26,519 --> 00:49:29,840
population growth, this curve will

1370
00:49:28,358 --> 00:49:31,840
rotate up,

1371
00:49:29,840 --> 00:49:35,000
and then it would appear as if that

1372
00:49:31,840 --> 00:49:35,000
leads to negative growth.

1373
00:49:35,760 --> 00:49:40,240
But you don't get negative growth. In

1374
00:49:37,400 --> 00:49:43,920
this diagram, you do get that

1375
00:49:40,239 --> 00:49:44,879
Y over over N will decline.

1376
00:49:43,920 --> 00:49:47,119
But that doesn't mean that you get

1377
00:49:44,880 --> 00:49:48,480
negative growth. It just means that

1378
00:49:47,119 --> 00:49:50,599
output is not growing as fast as

1379
00:49:48,480 --> 00:49:52,199
population.

1380
00:49:50,599 --> 00:49:53,279
But but both are growing. Just the

1381
00:49:52,199 --> 00:49:56,839
population is growing faster than

1382
00:49:53,280 --> 00:49:57,400
output. I'll I'll I'll start from that.

1383
00:49:56,840 --> 00:49:59,079
Uh

1384
00:49:57,400 --> 00:50:00,440
oh, I think it's after your break. So,

1385
00:49:59,079 --> 00:50:02,480
you're going to have forgotten

1386
00:50:00,440 --> 00:50:04,840
everything by then. So, I'll do a review

1387
00:50:02,480 --> 00:50:08,400
of this and then and then we

1388
00:50:04,840 --> 00:50:08,400
Okay. Have a Have a nice break.